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Review of Really Big Numbers, by Richard Evan Schwartz
by Richard Evan Schwartz
American Mathematical Society, 2014.
Starred Review
2014 Mathical Books Award Winner
Full disclosure: When I visited the National Math Festival and met Richard Evan Schwartz, I got all fangirl about his book You Can Count on Monsters and showed him my prime factorization cardigan. Of
course I purchased his new book and got it signed. I am particularly proud of what he wrote: “To Sondy, Beautiful cardigan! It looks like we have a lot of the same ideas. Best wishes, Richard
And when I showed him my Pascal’s Triangle Shawl, he gave me the idea of making a new one using congruences mod n. Yes! I like the way this man thinks!
[In fact, in a weird side note, after reading his bio on the AMS webpage and learning he did his undergrad in math at UCLA, I find myself with a memory — which very well may be false — of taking a
class with him as an undergraduate when I was a graduate math student at UCLA. I took a class (Number Theory?) with some undergraduates. That was in 1985-1986. An internet search shows he got his PhD
in 1991 — so this is actually possible! And I remember a cocky and extremely intelligent student who looked a whole lot like he does now, only younger….]
You will not be surprised when I say I loved his new book! There are many books that deal with large numbers using analogies. A few from the beginning of this book include:
About 7 billion people live on Earth. If they all lined up, spaced about a foot apart, they would circle 50 times or so around the equator.
You could cram about 20 billion grains of very fine sand into a basketball.
100 billion basketballs would fill New York City roughly to the height of a man.
You could cover the service of the earth with about a quadrillion (10^15) exercise trampolines.
A quintillion (10^18) grains of very fine sand would just about cover Atlantic City, NJ, to a depth of 3 feet.
Speaking of a quadrillion and a quintillion, I’ve seen a few other books that explain the names for large numbers, but that’s only about the halfway point of this book! You know things are getting
interesting right after the page where he shows
10^21 sextillion
10^24 septillion
10^27 octillion
10^30 nonillion
10^33 decillion
The next page says, “This system goes quite far out but I think that these names lose their novelty after the first 30 or so.” On that page we see spectators sleeping or reading a newspaper. Here’s
the chart:
10^36 undecillion
10^39 duodecillion
10^42 tredecillion
10^45 quattourdecillion
10^48 quindecillion
On the page facing that one, he says, “Here, let me skip ahead some and show you the names of a few really big ones.”
10^78 quinquavigintillion
10^93 trigintillion
10^108 quinquatrigintillion
10^123 quadragintillion
10^153 quinquagintillion
Since this is still only about the halfway point of the book, you get the idea that when this book talks about really big numbers, it means really big numbers!
The author throws in questions about the big numbers – questions challenging enough to get even an adult with a math degree thinking.
There are more illustrations of the size of things, such as:
The sun, the true giant in the solar system, has about 4 nonillion (4×10^30) pounds of material.
We could continue counting up roughly by powers of 1000, moving out beyond the solar system to the stars surrounding the sun and eventually to galaxies and galaxy clusters, and superclusters,
outward even to supercluster filaments and membranes…
but if you want to see some REALLY big numbers, we will have to move faster than that.
What is this author’s idea of REALLY big numbers? Well, before long, we get to a googol (10^100).
A googol atoms would fill the observable universe about 100 quadrillion times over.
You could say that a googol is so big that it rises beyond the merely astronomical.
He gives more illustrations of how big a googol is, but then says:
Yeah, a googol is a pretty big number.
But if you want to talk about REALLY big numbers then we’ll have to move on to a new level of abstraction. So, get ready, because the ride is gonna be pretty bumpy from here on in. But, remember,
this book is supposed to be like a game of bucking bronco and you can always come back to it later if you fall off now.
All of this is accompanied by helpful and/or amusing computer cartoon illustrations.
So, then, the first abstract thing I want to tell you about is called plex.
When you “plex” a number, you write 1 followed by that number of zeros.
In other words, when you plex a number, you raise 10 to that power.
A googol-plex is 1 followed by a googol zeros, or, equivalently, 10 raised to the googol power.
A googol-plex is also 100-plex-plex and likewise 2-plex-plex-plex.
I love this page:
In my experience it is impossible to picture a googol-plex in concrete terms. Any attempt will scramble your brain. An implacable guard blocks the door to that kind of intuition.
But, let’s try to sneak by the guard and see what we can.
After some attempts at that, he says:
Mathematics gives us a language to name all kinds of things, but we can’t relate to everything we can name. If you want to think about REALLY big numbers, you have to give up the idea of
picturing them….
Just let go of the reins and let LANGUAGE gallop on.
He even explains Recursion – “the trick of making something new by applying a simple rule over and over.”
Then he looks at some numbers plexed multiple times. I just love when he starts making up his own names.
Here is the number “one plexed one plexed two times times.” [The diagram here is very helpful.]
This number has no familiar name, so let’s call it “Fred.”
Let’s unravel “Fred” from the inside out.
“one plexed two times” is 1010, or ten billion, so “Fred” means “one plexed ten billion times.”
And here is “1 plexed FRED times.”
Let’s call this number “Big Jim.”
You may ask, “How big are ‘Fred’ and ‘Big Jim’?”
I’ll tell you honestly: I don’t know! Already, “1 plexed 4 times” makes a googol-plex seem microscopic, and each new plex is a quantum leap forward in size and abstraction.
To get to “Fred” you take 10 billion quantum leaps.
And “Big Jim” is “Fred” quantum leaps away.
And Richard Schwartz still doesn’t stop there! At the end of the book, he starts introducing new symbols. He shows a square that means “1 plexed N times.” Then he makes a new symbol that builds off
of the square, and further symbols that build off of that.
Accompanied by diagrams with these new symbols, he says:
Once you get a taste for this kind of symbol, and the accelerated voyage it lets you take through the number system, nothing stops you from making more symbols.
Each new addition to the language is a chariot moving so quickly it makes all the previous ones seem to stand still.
We skip from chariot to chariot, impatient with them almost as soon as they are created.
Unhindered by any ties to experience, giddy with language, we race ever faster through the number system.
When you finally reach the last page, you will agree with the final line:
Infinity is farther away than you thought.
I’ve quoted extensively from this book, but believe me, quotes out of context pale in contrast with the actual book – I’m simply giving you a clue as to what you’ll find here. The illustrations,
symbols, and diagrams all help lead the train of thought, or I should say ladder of thought, or better yet supersonic jet of thought.
I wish I had this book when my boys were young! My oldest, when he was in Kindergarten, liked to make up words for numbers “bigger than infinity.” I think the way this book is presented, the ideas of
larger and larger numbers – bounded only by your imagination – would have inspired both my sons. I definitely plan to show this to kids at the library.
Find this review on Sonderbooks at: www.sonderbooks.com/Childrens_Nonfiction/really_big_numbers.html
Disclosure: I am an Amazon Affiliate, and will earn a small percentage if you order a book on Amazon after clicking through from my site.
Source: This review is based on my own copy, purchased at the National Math Festival and signed by the author.
Disclaimer: I am a professional librarian, but I maintain my website and blogs on my own time. The views expressed are solely my own, and in no way represent the official views of my employer or of
any committee or group of which I am part.
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Scientific Notation Worksheets - 15 Worksheets.com
Scientific Notation Worksheets
All About These 15 Worksheets
Scientific notation is a mathematical method used to express very large or very small numbers in a more manageable form, which is especially useful in fields like science and engineering. This
collection of worksheets is designed to help students develop fluency in this area through a variety of exercises, making it an excellent tool for middle and high school students who are ready to
deepen their understanding of algebra and mathematical concepts.
What’s Inside the Collection
These PDF-format worksheets provide a versatile and convenient way for students to practice scientific notation. The format allows for easy viewing, downloading, and printing, making it simple for
educators and students to access them anytime. The variety of exercises included ensures that students are not only introduced to scientific notation but are able to practice it in multiple contexts.
Converting Ordinary Numbers to Scientific Notation
The first set of worksheets, as seen in some of the attached images, focuses on converting large and small ordinary numbers into scientific notation. For example, students might be asked to express
numbers like 4,500 or 78,900,000 in scientific notation. These tasks help students understand how to shift decimal points and write numbers using powers of ten. They also reinforce the concept of
significant figures, showing students how to handle both large and small numbers efficiently.
Similarly, worksheets are included that ask students to convert small decimals, such as 0.0062 or 0.000345, into scientific notation. These exercises serve to solidify students’ understanding of
negative exponents and how they represent very small numbers in scientific notation.
A balanced approach is achieved with worksheets that take the reverse approach, asking students to convert numbers from scientific notation back to their standard form. Examples like 7.5 × 10^3 or
1.28 × 10^5 are given, and students are required to expand these into their full numerical form. This reinforces their comprehension of how powers of ten work and ensures they can move fluently
between the two notational systems.
Comparing Numbers in Scientific Notation
An interesting variation in the worksheets focuses on comparing numbers written in scientific notation. Students are asked to use symbols like >, <, or = to compare values such as 5.5 × 10^6 and 7.3
× 10^3. This task builds a deeper understanding of exponents and helps students visualize the magnitude of different numbers. Comparing large and small numbers in this way also prepares students for
more advanced mathematical operations in scientific and real-world contexts.
More advanced worksheets included in the collection ask students to add and subtract numbers written in scientific notation. Problems like (3.2 × 10^5) + (4.5 × 10^5) or (2.8 × 10^7) – (3.4 × 10^7)
push students to apply their understanding of both scientific notation and basic arithmetic. These exercises are excellent for reinforcing concepts related to exponents, significant figures, and
order of magnitude. Students will also be tasked with multiplying and dividing numbers written in scientific notation, applying rules for exponents and ensuring their final answers are expressed
The PDF format of these worksheets makes them easily accessible and practical for both teachers in a traditional classroom setting and homeschoolers managing multiple subjects. Teachers can download,
print, and distribute the worksheets to their students, ensuring that they have a ready-made resource for teaching or reinforcing this key algebraic concept.
For homeschoolers, this collection provides an invaluable set of tools for guiding independent study. The clear instructions and diverse problem sets ensure that students can practice at their own
pace, revisiting difficult concepts as needed. Furthermore, the range of activities-from simple conversions to more complex operations like adding and subtracting scientific notation-allows students
to gradually build their skills without feeling overwhelmed.
What is Scientific Notation?
Scientific Notation is a mathematical way of expressing very large or very small numbers in a simplified, standardized form using powers of ten. In scientific notation, a number is written as the
product of two components: a coefficient (a number between 1 and 10) and a power of ten. For example, the number 4,500 in scientific notation is written as 4.5 x 10^3, which represents 4.5 x 1,000 =
4,500. Similarly, a small number like 0.0062 can be expressed as 6.2 x 10^-3, where the negative exponent indicates that the decimal point is moved three places to the left. This system allows for a
more compact representation of numbers that are too large or too small to easily work with in their standard form.
The Basic Structure of Scientific Notation
The general format for scientific notation is a x 10^n, where a is the coefficient and n is the exponent. The coefficient must be a number between 1 and 10, and the exponent n tells you how many
places to move the decimal point. A positive exponent indicates that the decimal point moves to the right, representing a large number, while a negative exponent moves the decimal point to the left,
representing a small number. For example, 7.3 x 10^5 is equal to 730,000, whereas 7.3 x 10^-5</sup is equal to 0.000073.
This system simplifies calculations and the comparison of magnitudes because scientists and mathematicians can focus on the exponents rather than writing out long strings of digits. It becomes easier
to handle numbers with many zeros, which would otherwise be cumbersome or error-prone to write in full.
Real-World Applications
Scientific notation is used extensively in various scientific fields because many measurements and quantities encountered in science are either extremely large or extremely small. For instance, in
astronomy, the distances between stars and galaxies are vast, often measured in trillions of kilometers. Instead of writing such large numbers in their standard form, which would involve many zeros,
astronomers use scientific notation. The distance from Earth to the Sun, for example, is about 150,000,000 kilometers, which can be more conveniently expressed as 1.5 x 10^8kilometers in scientific
Similarly, in physics and chemistry, scientific notation is essential for dealing with very small quantities, such as the size of atoms or the mass of subatomic particles. The diameter of a hydrogen
atom is approximately 1.06 x 10^-10 meters, which is much easier to write and comprehend than the standard form, 0.000000000106 meters. These tiny numbers would be extremely awkward to work with in
everyday calculations without scientific notation, as the risk of error increases when writing out many decimal places.
Why is Scientific Notation Helpful?
The main advantage of scientific notation is that it simplifies both the expression and manipulation of very large or very small numbers. For one, it reduces the number of digits one has to work
with, cutting down on potential errors when writing or reading numbers. For instance, imagine calculating the product of 2.5 x 10^7 and 3.4 x 10^5 without scientific notation. You would have to
multiply 25,000,000 by 340,000, which would be cumbersome and prone to mistakes. In scientific notation, however, you can quickly multiply the coefficients (2.5 and 3.4) and add the exponents, making
the process much more efficient.
Scientific notation also makes it easier to compare magnitudes. In fields like engineering or geology, professionals often need to compare vastly different quantities. For example, comparing the mass
of the Earth (5.97 x 10^24 kilograms) to the mass of a single bacterium (1.0 x 10^-12 kilograms) would be nearly impossible using standard notation, but with scientific notation, the difference in
the powers of ten (24 versus -12) immediately shows the immense scale difference.
Scientific notation is also fundamental in the world of computing, particularly when dealing with data science, graphics, and large-scale computations. Computers can handle numbers more effectively
when they are expressed in terms of powers of ten because it optimizes the way data is stored and processed. For example, floating-point arithmetic, which is used in most computer systems to
represent real numbers, is closely related to scientific notation. It allows computers to perform calculations on very large or very small numbers without sacrificing accuracy or requiring an
impractical amount of memory.
In fields like economics, scientific notation is useful for expressing large financial figures, such as national debts, gross domestic products (GDPs), or the total value of global markets. For
example, the U.S. national debt is over 31 trillion dollars, which can be written as 3.1 x 10^13 dollars. This form of notation not only makes it easier for analysts and economists to work with such
large numbers, but it also helps communicate these figures more efficiently in reports and studies.
In finance, scientific notation is sometimes used to express very small quantities, such as interest rates or the value of stocks that fluctuate by tiny amounts. While decimal form is common in
finance, scientific notation becomes more relevant in fields like high-frequency trading, where extremely small time intervals and price changes must be considered.
In medicine and biology, scientists frequently work with tiny measurements, such as the size of cells, bacteria, or even molecules of DNA. Scientific notation is critical when quantifying substances
like hormones or medications in blood, where concentrations can be minuscule, often in the range of nanograms or picograms. For instance, a hormone concentration might be measured as 3.5 x 10^-9
grams per milliliter, making it easier to handle and communicate than its decimal counterpart, 0.0000000035 grams per milliliter. | {"url":"https://15worksheets.com/worksheet-category/scientific-notation/","timestamp":"2024-11-03T15:43:46Z","content_type":"text/html","content_length":"134035","record_id":"<urn:uuid:47e108ec-643f-4c4e-9416-31fb9d3eb496>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00420.warc.gz"} |
A vendor supplies 32 litres of milk to a hotel in the morning and 68 litres of milk in the evening. If the milk costs ₹ 45 per litre, how much money is due to the vendor per day?
A day full of math games & activities. Find one near you.
A day full of math games & activities. Find one near you.
A day full of math games & activities. Find one near you.
A day full of math games & activities. Find one near you.
A vendor supplies 32 litres of milk to a hotel in the morning and 68 litres of milk in the evening. If the milk costs ₹ 45 per litre, how much money is due to the vendor per day?
The quantity of milk supplied in the morning = 32 litres
The quantity of milk supplied in the evening = 68 litres
Cost of milk per litre = ₹45
Thus, total cost of milk per day = 45 × (32 + 68)
= 45 × 100
= ₹ 4500
Hence, the money due to the vendor per day is ₹ 4500.
NCERT Solutions for Class 6 Maths Chapter 2 Exercise 2.2 Question 6
A vendor supplies 32 litres of milk to a hotel in the morning and 68 litres of milk in the evening. If the milk costs ₹ 45 per litre, how much money is due to the vendor per day?
If a vendor supplies 32 litres of milk to a hotel in the morning and 68 litres of milk in the evening, the money due to the vendor per day is ₹ 4500 if the milk costs ₹ 45 per litre.
☛ Related Questions:
Math worksheets and
visual curriculum | {"url":"https://www.cuemath.com/ncert-solutions/a-vendor-supplies-32-litres-of-milk-to-a-hotel-in-the-morning-and-68-litres-of-milk-in-the-evening-if-the-milk-costs-inr-45-per-litre-how-much-money-is-due-to-the-vendor-per-day/","timestamp":"2024-11-06T07:42:31Z","content_type":"text/html","content_length":"231610","record_id":"<urn:uuid:b00f3b53-ca80-4637-931a-ec112965ed68>","cc-path":"CC-MAIN-2024-46/segments/1730477027910.12/warc/CC-MAIN-20241106065928-20241106095928-00805.warc.gz"} |
Umap calculation | seekquence
Used abbreviations for single-cell RNA sequencing (scRNA-seq) with their descriptions:
1. scRNA-seq: Single-cell RNA sequencing, a method to analyze gene expression at the single-cell level.
2. GEM: Gel Bead-in-Emulsion, a microfluidic droplet that encapsulates individual cells with beads for barcoding RNA.
3. UMAP: Uniform Manifold Approximation and Projection, a dimensionality reduction technique used for visualizing high-dimensional data.
4. t-SNE: t-distributed Stochastic Neighbor Embedding, a technique for visualizing high-dimensional data, often used in scRNA-seq analysis.
5. SNE: Stochastic Neighbor Embedding, a dimensionality reduction technique used for visualizing high-dimensional data, less common than t-SNE.
6. UMI: Unique Molecular Identifier, a short sequence tag that helps to differentiate between individual RNA molecules, allowing for more accurate quantification of gene expression.
7. dvn: Typically refers to "differential variability networks," but the exact meaning can vary based on context.
8. PCA: Principal Component Analysis, a method for dimensionality reduction that summarizes data variation.
9. DEG: Differentially Expressed Genes, genes that exhibit significant differences in expression levels between conditions or groups.
10. FC: Fold Change, a measure used to describe how much a quantity changes, commonly in gene expression.
11. FACS: Fluorescence-Activated Cell Sorting, a technique for sorting cells based on fluorescent markers.
12. NMF: Non-negative Matrix Factorization, a technique used for clustering and identifying patterns in gene expression data.
13. RPKM/FPKM: Reads Per Kilobase of transcript per Million mapped reads / Fragments Per Kilobase of transcript per Million mapped reads, normalization methods for comparing gene expression levels.
14. Clustering: The process of grouping cells based on similar expression profiles to identify cell types or states.
15. Batch Effect: Technical variation that can arise from different experimental conditions, samples, or processing times.
16. CITE-seq: Cellular Indexing of Transcriptomes and Epitopes using Sequencing, a method that combines scRNA-seq with protein marker analysis.
17. MNN: Mutual Nearest Neighbors, a method for batch effect correction and integration of scRNA-seq data.
18. Louvain: A method for community detection in graphs, commonly used for clustering in single-cell analysis.
19. Seurat: An R package widely used for single-cell RNA-seq data analysis, including normalization, clustering, and visualization.
20. Scanpy: A Python-based tool for analyzing single-cell gene expression data, offering functionalities for clustering, visualization, and differential expression analysis.
21. tCR: T-cell receptor, often studied in immunology contexts within scRNA-seq.
22. BCR: B-cell receptor, similarly studied in relation to immune cell types.
23. RNA-seq: RNA sequencing, a technique for analyzing the transcriptome, though not limited to single cells.
24. CCA: Canonical Correlation Analysis, used for finding relationships between two datasets, often for integrating multiple single-cell datasets.
25. sNMF: Supervised Non-negative Matrix Factorization, used for classification in single-cell studies.
26. DIM: Dimensionality Reduction and Integration Method, a general term for methods that reduce complexity in high-dimensional datasets.
27. LDA: Latent Dirichlet Allocation, a generative statistical model used for topic modeling, sometimes applied to single-cell data.
28. SPADE: Spanning-tree Progression Analysis of Density-normalized Events, a method for visualizing and analyzing high-dimensional flow cytometry data.
29. DGE: Differential Gene Expression, a statistical analysis to identify genes that show different expression levels across conditions.
30. RCC: RNA Cell Count, the number of transcripts detected from a single cell, often used in quality control metrics.
31. IS: Immune Signature, a term used to describe gene expression patterns specific to immune cells.
Uniform Manifold Approximation and Projection Definition
UMAP (Uniform Manifold Approximation and Projection) is a novel manifold learning technique for dimensional reduction. UMAP is constructed from a theoretical framework based in Riemannian geometry
and algebraic topology.
The result is a practical scalable algorithm that is applicable to real world data. The UMAP algorithm is competitive with t-SNE, t-distributed stochastic neighbor embedding (t-SNE).
Although we considered t-SNE is better than existing UMAP techniques at creating a single map that reveals Single Cell scRNA seq structures. Studies suggested that UMAP is better than t-SNE to
retain the global structure in scRNA-seq data analysis because of Laplacian Eigenmaps.
Lieven Gevaert, Bio-ir Gent 1996
For visualization quality, and arguably preserves more of the global structure with superior run time performance. Furthermore, UMAP has no computational restrictions on embedding dimension, making
it viable as a general purpose dimension reduction technique for machine learning.
List of cell types identified through scRNA-seq and cell subtypes
• Activated Fibroblasts: Involved in tissue repair and fibrosis, often found in response to injury.
• Myofibroblasts: Specialized fibroblasts with contractile properties that play a role in wound healing.
CD4+ T Cells:
• Th1 Cells: Helper T cells that produce interferon-gamma (IFN-γ) and are involved in responses against intracellular pathogens.
• Th2 Cells: Helper T cells that produce cytokines like IL-4 and are important in allergic responses and defense against extracellular parasites.
• Th17 Cells: A subset that produces IL-17 and is involved in autoimmune responses and defense against fungi and bacteria.
• Memory T Cells: Long-lived cells that provide rapid responses upon re-exposure to antigens.
CD8+ T Cells:
• Cytotoxic T Lymphocytes (CTLs): Directly kill infected or cancerous cells.
• Memory CD8+ T Cells: Provide long-term immunity by quickly responding to previously encountered antigens.
Natural Killer (NK) Cells:
• Cytotoxic NK Cells: Kill virally infected cells and tumor cells.
• Regulatory NK Cells: Modulate immune responses rather than exerting cytotoxicity.
• Classical Monocytes (CD14+): Involved in phagocytosis and inflammation.
• Non-Classical Monocytes (CD16+): Patrol the endothelium and participate in tissue repair.
Dendritic Cells:
• Conventional Dendritic Cells (cDCs): Present antigens to T cells and activate them.
• Plasmacytoid Dendritic Cells (pDCs): Produce large amounts of type I interferons in response to viral infections.
Mast Cells:
• Involved in allergic reactions and immune defense, releasing histamines and cytokines.
• Essential for innate immune response, rapidly responding to infections and inflammation.
• Primarily involved in combating parasitic infections and in allergic reactions.
• Play a role in inflammatory responses, particularly in allergies.
Endothelial Cells:
• Vascular Endothelial Cells: Line blood vessels and regulate blood flow and permeability.
• Lymphatic Endothelial Cells: Line lymphatic vessels and are involved in fluid balance and immune responses.
Muscle Cells:
• Cardiomyocytes: Heart muscle cells responsible for heart contractions.
• Smooth Muscle Cells: Found in the walls of hollow organs, responsible for involuntary contractions.
Neuronal Cells:
• Excitatory Neurons: Release neurotransmitters like glutamate.
• Inhibitory Neurons: Release neurotransmitters like GABA.
Stem Cells:
• Hematopoietic Stem Cells (HSCs): Give rise to all blood cell types.
• Mesenchymal Stem Cells (MSCs): Can differentiate into a variety of cell types including bone, cartilage, and fat cells.
• Cells in cartilage responsible for maintaining cartilage health and structure.
• White Adipocytes: Store energy as fat.
• Brown Adipocytes: Generate heat and burn calories.
Pancreatic Cells:
• Alpha Cells: Produce glucagon, which raises blood sugar levels.
• Beta Cells: Produce insulin, which lowers blood sugar levels.
Goblet Cells:
• Secrete mucus to protect and lubricate epithelial surfaces, especially in the gut.
Retinal Cells:
• Photoreceptors: Convert light into neural signals.
• Retinal Ganglion Cells: Transmit visual information to the brain.
Liver Cells:
• Hepatocytes: Main functional cells of the liver involved in metabolism, detoxification, and protein synthesis.
• Kupffer Cells: Liver macrophages involved in immune responses.
Single-cell RNA sequencing (scRNA-seq) identified by scRNA-seq:
Key Concepts in Cellular Expression by scRNA-seq
1. Gene Expression Profiles:
□ Transcripts: The RNA molecules that are produced from genes. scRNA-seq captures the diversity of transcripts present in individual cells.
□ Differential Expression: Comparing gene expression levels between different cell types, conditions, or states to identify genes that are upregulated or downregulated.
2. Cell Type-Specific Expression:
□ Certain genes are expressed predominantly in specific cell types. For example:
☆ CD4 and CD8: Common markers for T helper and cytotoxic T cells, respectively.
☆ Insulin: Specifically expressed in pancreatic beta cells.
☆ Myelin Genes: Expressed in oligodendrocytes, important for myelin formation in the central nervous system.
3. Marker Genes:
□ Specific genes that serve as indicators of particular cell types or states. For instance:
☆ Plasmacytoid Dendritic Cells: Express genes like BTLA and IL-18.
☆ Neutrophils: High expression of ELANE and MPO.
4. Clustering and Cell States:
□ scRNA-seq enables clustering of cells based on similar expression profiles. This can reveal:
☆ Cell States: Transient or stable conditions, such as activation or differentiation states.
☆ Subtypes: Distinct populations within a broader cell type, like different T cell subsets (e.g., Th1, Th2).
5. Developmental Trajectories:
□ Analysis of gene expression can map out developmental pathways. This can show how cells transition from progenitor states to fully differentiated states, often visualized using techniques
like pseudotime analysis.
6. Cellular Responses:
□ scRNA-seq can capture how cells respond to stimuli, such as:
☆ Cytokine Treatment: Changes in gene expression in immune cells upon exposure to inflammatory cytokines.
☆ Drug Treatment: Understanding how cancer cells alter their gene expression in response to therapy.
7. Spatial Transcriptomics:
□ While scRNA-seq provides insights into expression patterns, integrating it with spatial transcriptomics allows researchers to understand where within a tissue certain expressions are
8. Single-Cell ATAC-seq:
□ Complementary technique to scRNA-seq that profiles accessible chromatin regions, providing insight into regulatory elements affecting gene expression.
9. Integration with Other Omics:
□ Combining scRNA-seq data with proteomics or metabolomics can provide a more comprehensive understanding of cellular functions and pathways.
Examples of Cellular Expressions
1. Immune Cells:
□ CD8+ T Cells: High expression of GZMB and IFNG when activated.
□ B Cells: Upregulation of MZB1 when differentiating into plasma cells.
2. Neuronal Cells:
□ Excitatory Neurons: Express GRIN1 (NMDA receptor) and SLC17A7 (glutamate transporter).
□ Inhibitory Neurons: High expression of GABRA1 (GABA receptor).
3. Fibroblasts:
□ Activated fibroblasts express COL1A1 and ACTA2, indicating involvement in tissue repair and fibrosis.
4. Adipocytes:
□ White adipocytes show high levels of PPARG and ADIPOQ, which are important for lipid metabolism.
5. Stem Cells:
□ Hematopoietic stem cells express CD34 and KIT, indicating their potential for differentiation into various blood cell lineages.
1 Introduction in major dimensionality reduction methods (PCA, multidimensional scaling [MDS], t-SNE, and UMAP)
Dimension reduction plays an important role in data science, being a fundamental technique in both visualisation and as pre-processing for machine learning. Dimension reduction techniques are being
applied in a broadening range of fields and on ever increasing sizes of datasets. It is thus desirable to have an algorithm that is both scalable to massive data and able to cope with the diversity
of data available. Dimension reduction algorithms tend to fall into two categories; those that seek to preserve the pairwise distance structure amongst all the data samples and those that favor the
preservation of local distances over global distance.
1. Algorithms such as PCA that stands for Principal component analysis (PCA) and is frequently used for analysis of single-cell RNA-seq (scRNA-seq) to reduce the dimensionality of a large data
matrix with thousands of features (genes) to a smaller matrix with just a few factors (principal components). [27].
2. Multidimensional scaling (MDS) is a means of visualizing the level of similarity of individual cases of a data set .
3. MDS [30], and Sammon mapping [50] fall into the former category while t-SNE [59, 58], Isomap [56], LargeVis [54], Laplacian eigenmaps [6, 7] and diffusion maps [16] all fall into the latter
4. Multidimensional Scaling(MDS) and Isometric Feature Mapping(ISOMAP) are two very similar non-linear dimension reduction techniques.
SNE (Stochastic Neighbor Embedding), t-SNE (t-distributed Stochastic Neighbor Embedding), and UMAP (Uniform Manifold Approximation and Projection), will be the focus of this discussion. SNE, t-SNE,
and UMAP are neighbor graphs algorithms that follow a similar process.
Novel manifold learning technique for dimension reduction.
TriMap also struggles with local structure sometimes. Interestingly, none of t-SNE, UMAP, or TriMap can be adjusted smoothly from local to global structure preservation through any obvious adjustment
of parameters.
We provide a sound mathematical theory grounding the technique and a practical scalable algorithm that applies to real world data. UMAP (Uniform Manifold Approximation and Projection) builds upon
mathematical foundations related to the work of Belkin and Niyogi on Laplacian eigenmaps.
We compare the performance of UMAP, FIt-SNE, MulticoreTSNE, and LargeVis on PCA reductions of the Human and Mouse scRNA dataset to varying dimensionalities.
We seek to address the issue of uniform data distributions on manifolds through a combination of Riemannian geometry and the work of David Spivak [52] in category theoretic approaches to geometric
realization of fuzzy simplicial sets. t-SNE is the current state-of-the-art for dimension reduction for visualization. Our algorithm is competitive with t-SNE for visualization quality and arguably
preserves more of the global structure with superior run time performance. Furthermore the algorithm is able to scale to significantly larger data set sizes than are feasible for t-SNE. Finally, UMAP
has no computational restrictions on embedding dimension, making it viable as a general purpose dimension reduction technique for machine learning.
Based upon preliminary releases of a software implementation, UMAP has already found widespread use in the fields of bioinformatics [5, 12, 17, 46, 2, 45, 15], materials science [34, 23], and machine
learning [14, 20, 21, 24, 19, 47] among others.
In Section 2 we describe the theory underlying the algorithm. Section 2 is necessary to understand both the theory underlying why UMAP works and the motivation for the choices that where made in
developing the algorithm. A reader without a background (or interest) in topological data analysis, category theory or the theoretical underpinnings of UMAP should skip over this section and proceed
directly to Section 3.
That being said, we feel that strong theory and mathematically justified algorithmic decisions are of particular importance in the field of unsupervised learning. This is, at least partially, due to
plethora of proposed objective functions within the area. We attempt to highlight in this paper that UMAPs design decisions were all grounded in a solid theoretic foundation and not derived through
experimentation with any particular task focused objective function. Though all neighbourhood based manifold learning algorithms must share certain fundamental components we believe it to be
advantageous for these components to be selected through well grounded theoretical decisions. One of the primary contributions of this paper is to reframe the problem of manifold learning and
dimension reduction in a different mathematical language allowing practitioners to apply a new field of mathematics to the problems.
In Section 3 we provide a more computational description of UMAP. Section 3 should provide readers less familiar with topological data analysis with a better foundation for understanding the theory
described in Section 2. Appendix C contrasts UMAP against the more familiar algorithms t-SNE and LargeVis, describing all these algorithms in similar language. This section should assist readers
already familiar with those techniques to quickly gain an understanding of the UMAP algorithm though they will grant little insight into its theoretical underpinnings.
In Section 4 we discuss implementation details of the UMAP algorithm. This includes a more detailed algorithmic description, and discussion of the hyper-parameters involved and their practical
In Section 5 we provide practical results on real world datasets as well as scaling experiments to demonstrate the algorithm’s performance in real world scenarios as compared with other dimension
reduction algorithms.
In Section 6 we discuss relative weakenesses of the algorithm, and applications for which UMAP may not be the best choice.
Finally, in Section 7 we detail a number of potential extensions of UMAP that are made possible by its construction upon solid mathematical foundations. These avenues for further development include
semi-supervised learning, metric learning and heterogeneous data embedding.
Diverse immune receptors provide broad surveillance T cells attacking a tumor cell
• Paired characterization of the TCR α and β
transcripts in each T cell is critical to dissecting
cellular interactions
Determinants of antigen specificity
• In both T and B cells specificity is determined by two distally
encoded, co-expressed genes
• Diversity is generated by V(D)J recombination and somatic
Enormous diversity of T and B cell antigen-specific receptors
1. • Such diversity is generated by V(D)J recombination + N nucleotide addition or deletion
2. • Full-length sequencing of the paired heavy and light chains (B cells) or α and β chains (T cells) is critical to dissecting these interactions
2 Theoretical Foundations for UMAP
• The theoretical foundations for UMAP are largely based in manifold theory and topological data analysis. Much of the theory is most easily explained in the language of topology and category
theory. Readers may consult [39], [49] and [40] for background. Readers more interested in practical computational aspects of the algorithm, and not necessarily the theoretical motivation for the
computations involved, may wish to skip this section. Readers more familiar with traditional machine learning may find the relationships between UMAP, t-SNE and Largeviz located in Appendix C
enlightening. Unfortunately, this purely computational view fails to shed any light upon the reasoning that underlies the algorithmic decisions made in UMAP. Without strong theoretical
foundations the only arguments which can be made about algorithms amount to empirical measures, for which there are no clear universal choices for unsupervised problems.
• At a high level, UMAP uses local manifold approximations and patches together their local fuzzy simplicial set representations to construct a topological representation of the high dimensional
data. Given some low dimensional representation of the data, a similar process can be used to construct an equivalent topological representation. UMAP then optimizes the layout of the data
representation in the low dimensional space, to minimize the cross-entropy between the two topological representations.
• The construction of fuzzy topological representations can be broken down into two problems: approximating a manifold on which the data is assumed to lie; and constructing a fuzzy simplicial set
representation of the approximated manifold. In explaining the algorithm we will first discuss the method of approximating the manifold for the source data. Next we will discuss how to construct
a fuzzy simplicial set structure from the manifold approximation. Finally, we will discuss the construction of the fuzzy simplicial set associated to a low dimensional representation (where the
manifold is simply ℝd), and how to optimize the representation with respect to our objective function.
2.1 Uniform distribution of data on a manifold and geodesic approximation
The first step of our algorithm is to approximate the manifold we assume the data (approximately) lies on. The manifold may be known apriori (as simply ℝn) or may need to be inferred from the data.
Suppose the manifold is not known in advance and we wish to approximate geodesic distance on it. Let the input data be X={X1,…,XN}. As in the work of Belkin and Niyogi on Laplacian eigenmaps [6, 7],
for theoretical reasons it is beneficial to assume the data is uniformly distributed on the manifold, and even if that assumption is not made (e.g [26]) results are only valid in the limit of
infinite data. In practice, finite real world data is rarely so nicely behaved. However, if we assume that the manifold has a Riemannian metric not inherited from the ambient space, we can find a
metric such that the data is approximately uniformly distributed with regard to that metric.
Formally, let ℳ be the manifold we assume the data to lie on, and let g be the Riemannian metric on ℳ. Thus, for each point p∈ℳ we have gp, an inner product on the tangent space Tpℳ.
Lemma 1.
Let (ℳ,g) be a Riemannian manifold in an ambient ℝn, and let p∈M be a point. If g is locally constant about p in an open neighbourhood U such that g is a constant diagonal matrix in ambient
coordinates, then in a ball B⊆U centered at p with volume πn/2Γ(n/2+1) with respect to g, the geodesic distance from p to any point q∈B is 1rdℝn(p,q), where r is the radius of the ball in the
ambient space and dℝn is the existing metric on the ambient space.
See Appendix A of the supplementary materials for a proof of Lemma 1.
If we assume the data to be uniformly distributed on ℳ (with respect to g) then, away from any boundaries, any ball of fixed volume should contain approximately the same number of points of X
regardless of where on the manifold it is centered. Given finite data and small enough local neighborhoods this crude approximation should be accurate enough even for data samples near manifold
boundaries. Now, conversely, a ball centered at Xi that contains exactly the k-nearest-neighbors of Xi should have approximately fixed volume regardless of the choice of Xi∈X. Under Lemma 1 it
follows that we can approximate geodesic distance from Xi to its neighbors by normalising distances with respect to the distance to the kth nearest neighbor of Xi.
In essence, by creating a custom distance for each Xi, we can ensure the validity of the assumption of uniform distribution on the manifold. The cost is that we now have an independent notion of
distance for each and every Xi, and these notions of distance may not be compatible. We have a family of discrete metric spaces (one for each Xi) that we wish to merge into a consistent global
structure. This can be done in a natural way by converting the metric spaces into fuzzy simplicial sets.
2.2 Fuzzy topological representation
We will use functors between the relevant categories to convert from metric spaces to fuzzy topological representations. This will provide a means to merge the incompatible local views of the data.
The topological structure of choice is that of simplicial sets. For more details on simplicial sets we refer the reader to [25], [40], [48], or [22]. Our approach draws heavily upon the work of
Michael Barr [3] and David Spivak in [52], and many of the definitions and theorems below are drawn or adapted from those sources. We assume familiarity with the basics of category theory. For an
introduction to category theory readers may consult [39] or [49].
To start we will review the definitions for simplicial sets. Simplicial sets provide a combinatorial approach to the study of topological spaces. They are related to the simpler notion of simplicial
complexes – which construct topological spaces by gluing together simple building blocks called simplices – but are more general. Simplicial sets are most easily defined purely abstractly in the
language of category theory.
Definition 1.
The category 𝚫 has as objects the finite order sets [n]={1,…,n}, with morphims given by (non-strictly) order-preserving maps.
Following standard category theoretic notation, 𝚫op denotes the category with the same objects as 𝚫 and morphisms given by the morphisms of 𝚫 with the direction (domain and codomain) reversed.
Definition 2.
A simplicial set is a functor from 𝚫op to Sets, the category of sets; that is, a contravariant functor from 𝚫 to Sets.
Given a simplicial set X:𝚫op→𝐒𝐞𝐭𝐬, it is common to denote the set X([n]) as Xn and refer to the elements of the set as the n-simplices of X. The simplest possible examples of simplicial sets are the
standard simplices Δn, defined as the representable functors hom𝚫(⋅,[n]). It follows from the Yoneda lemma that there is a natural correspondence between n-simplices of X and morphisms Δn→X in the
category of simplicial sets, and it is often helpful to think in these terms. Thus for each x∈Xn we have a corresponding morphism x:Δn→X. By the density theorem and employing a minor abuse of
notation we then have
There is a standard covariant functor |⋅|:𝚫→𝐓𝐨𝐩 mapping from the category 𝚫 to the category of topological spaces that sends [n] to the standard n-simplex |Δn|⊂ℝn+1 defined as
with the standard subspace topology. If X:𝚫op→𝐒𝐞𝐭𝐬 is a simplicial set then we can construct the realization of X (denoted |X|) as the colimit
and thus associate a topological space with a given simplicial set. Conversely given a topological space Y we can construct an associated simplicial set S(Y), called the singular set of Y, by
It is a standard result of classical homotopy theory that the realization functor and singular set functors form an adjunction, and provide the standard means of translating between topological
spaces and simplicial sets. Our goal will be to adapt these powerful classical results to the case of finite metric spaces.
We draw significant inspiration from Spivak, specifically [52], where he extends the classical theory of singular sets and topological realization to fuzzy singular sets and metric realization. To
develop this theory here we will first outline a categorical presentation of fuzzy sets, due to [3], that will make extending classical simplicial sets to fuzzy simplicial sets most natural.
Classically a fuzzy set [65] is defined in terms of a carrier set A and a map μ:A→[0,1] called the membership function. One is to interpret the value μ(x) for x∈A to be the membership strength of x
to the set A. Thus membership of a set is no longer a bi-valent true or false property as in classical set theory, but a fuzzy property taking values in the unit interval. We wish to formalize this
in terms of category theory.
Let I be the unit interval (0,1]⊆ℝ with topology given by intervals of the form [0,a) for a∈(0,1]. The category of open sets (with morphisms given by inclusions) can be imbued with a Grothendieck
topology in the natural way for any poset category.
Definition 3.
A presheaf 𝒫 on I is a functor from Iop to 𝐒𝐞𝐭𝐬. A fuzzy set is a presheaf on I such that all maps 𝒫(a≤b) are injections.
Presheaves on I form a category with morphisms given by natural transformations. We can thus form a category of fuzzy sets by simply restricting to the sub-category of presheaves that are fuzzy sets.
We note that such presheaves are trivially sheaves under the Grothendieck topology on I. As one might expect, limits (including products) of such sheaves are well defined, but care must be taken to
define colimits (and coproducts) of sheaves. To link to the classical approach to fuzzy sets one can think of a section 𝒫([0,a)) as the set of all elements with membership strength at least a. We
can now define the category of fuzzy sets.
Definition 4.
The category 𝐅𝐮𝐳𝐳 of fuzzy sets is the full subcategory of sheaves on I spanned by fuzzy sets.
With this categorical presentation in hand, defining fuzzy simplicial sets is simply a matter of considering presheaves of 𝚫 valued in the category of fuzzy sets rather than the category of sets.
Definition 5.
The category of fuzzy simplicial sets 𝐬𝐅𝐮𝐳𝐳 is the category with objects given by functors from 𝚫op to 𝐅𝐮𝐳𝐳, and morphisms given by natural transformations.
Alternatively, a fuzzy simplicial set can be viewed as a sheaf over 𝚫×I, where 𝚫 is given the trivial topology and 𝚫×I has the product topology. We will use Δ<an to denote the sheaf given by the
representable functor of the object ([n],[0,a)). The importance of this fuzzy (sheafified) version of simplicial sets is their relationship to metric spaces. We begin by considering the larger
category of extended-pseudo-metric spaces.
Definition 6.
An extended-pseudo-metric space (X,d) is a set X and a map d:X×X→ℝ≥0∪{∞} such that
1. 1.
d(x,y)⩾0, and x=y implies d(x,y)=0;
2. 2.
d(x,y)=d(y,x); and
3. 3.
d(x,z)⩽d(x,y)+d(y,z) or d(x,z)=∞.
The category of extended-pseudo-metric spaces 𝐄𝐏𝐌𝐞𝐭 has as objects extended-pseudo-metric spaces and non-expansive maps as morphisms. We denote the subcategory of finite extended-pseudo-metric spaces
The choice of non-expansive maps in Definition 6 is due to Spivak, but we note that it closely mirrors the work of Carlsson and Memoli in [13] on topological methods for clustering as applied to
finite metric spaces. This choice is significant since pure isometries are too strict and do not provide large enough Hom-sets.
In [52] Spivak constructs a pair of adjoint functors, 𝖱𝖾𝖺𝗅 and 𝖲𝗂𝗇𝗀 between the categories sFuzz and EPMet. These functors are the natural extension of the classical realization and singular set
functors from algebraic topology. The functor 𝖱𝖾𝖺𝗅 is defined in terms of standard fuzzy simplices Δ<an as
similarly to the classical realization functor |⋅|. The metric on 𝖱𝖾𝖺𝗅(Δ<an) is simply inherited from ℝn+1. A morphism Δ<an→Δ<bm exists only if a≤b, and is determined by a 𝚫 morphism σ:[n]→[m]. The
action of 𝖱𝖾𝖺𝗅 on such a morphism is given by the map
Such a map is clearly non-expansive since 0≤a≤b≤1 implies that log(b)/log(a)≤1.
We then extend this to a general simplicial set X via colimits, defining
Since the functor 𝖱𝖾𝖺𝗅 preserves colimits, it follows that there exists a right adjoint functor. Again, analogously to the classical case, we find the right adjoint, denoted 𝖲𝗂𝗇𝗀, is defined for an
extended pseudo metric space Y in terms of its action on the category 𝚫×I:
For our case we are only interested in finite metric spaces. To correspond with this we consider the subcategory of bounded fuzzy simplicial sets Fin-sFuzz. We therefore use the analogous adjoint
pair 𝖥𝗂𝗇𝖱𝖾𝖺𝗅 and 𝖥𝗂𝗇𝖲𝗂𝗇𝗀. Formally we define the finite fuzzy realization functor as follows:
Definition 7.
Define the functor 𝖥𝗂𝗇𝖱𝖾𝖺𝗅:Fin-sFuzz→𝐅𝐢𝐧𝐄𝐏𝐌𝐞𝐭 by setting
da(xi,xj)={−log(a)if i≠j,0otherwise.
and then defining
Similar to Spivak’s construction, the action of 𝖥𝗂𝗇𝖱𝖾𝖺𝗅 on a map Δ<an→Δ<bm, where a≤b defined by σ:Δn→Δm, is given by
which is a non-expansive map since a≤b implies da≥db.
Since 𝖥𝗂𝗇𝖱𝖾𝖺𝗅 preserves colimits it admits a right adjoint, the fuzzy singular set functor 𝖥𝗂𝗇𝖲𝗂𝗇𝗀. We can then define the (finite) fuzzy singular set functor in terms of the action of its image on
𝚫×I, analogously to 𝖲𝗂𝗇𝗀.
Definition 8.
Define the functor 𝖥𝗂𝗇𝖲𝗂𝗇𝗀:𝐅𝐢𝐧𝐄𝐏𝐌𝐞𝐭→Fin-sFuzz by
We then have the following theorem.
Theorem 1.
The functors 𝖥𝗂𝗇𝖱𝖾𝖺𝗅:Fin-sFuzz→𝐅𝐢𝐧𝐄𝐏𝐌𝐞𝐭 and 𝖥𝗂𝗇𝖲𝗂𝗇𝗀:𝐅𝐢𝐧𝐄𝐏𝐌𝐞𝐭→Fin-sFuzz form an adjunction with 𝖥𝗂𝗇𝖱𝖾𝖺𝗅 the left adjoint and 𝖥𝗂𝗇𝖲𝗂𝗇𝗀 the right adjoint.
The proof of this is by construction. Appendix B provides a full proof of the theorem.
With the necessary theoretical background in place, the means to handle the family of incompatible metric spaces described above becomes clear. Each metric space in the family can be translated into
a fuzzy simplicial set via the fuzzy singular set functor, distilling the topological information while still retaining metric information in the fuzzy structure. Ironing out the incompatibilities of
the resulting family of fuzzy simplicial sets can be done by simply taking a (fuzzy) union across the entire family. The result is a single fuzzy simplicial set which captures the relevant
topological and underlying metric structure of the manifold ℳ.
It should be noted, however, that the fuzzy singular set functor applies to extended-pseudo-metric spaces, which are a relaxation of traditional metric spaces. The results of Lemma 1 only provide
accurate approximations of geodesic distance local to Xi for distances measured from Xi – the geodesic distances between other pairs of points within the neighborhood of Xi are not well defined. In
deference to this lack of information we define distances between Xj and Xk in the extended-pseudo metric space local to Xi (where i≠j and i≠k) to be infinite (local neighborhoods of Xj and Xk will
provide suitable approximations).
For real data it is safe to assume that the manifold ℳ is locally connected. In practice this can be realized by measuring distance in the extended-pseudo-metric space local to Xi as geodesic
distance beyond the nearest neighbor of Xi. Since this sets the distance to the nearest neighbor to be equal to 0 this is only possible in the more relaxed setting of extended-pseudo-metric spaces.
It ensures, however, that each 0-simplex is the face of some 1-simplex with fuzzy membership strength 1, meaning that the resulting topological structure derived from the manifold is locally
connected. We note that this has a similar practical effect to the truncated similarity approach of Lee and Verleysen [33], but derives naturally from the assumption of local connectivity of the
Combining all of the above we can define the fuzzy topological representation of a dataset.
Definition 9.
Let X={X1,…,XN} be a dataset in ℝn. Let {(X,di)}i=1…N be a family of extended-pseudo-metric spaces with common carrier set X such that
di(Xj,Xk)={dℳ(Xj,Xk)−ρ if i=j or i=k,∞ otherwise ,
where ρ is the distance to the nearest neighbor of Xi and dℳ is geodesic distance on the manifold ℳ, either known apriori, or approximated as per Lemma 1.
The fuzzy topological representation of X is
The (fuzzy set) union provides the means to merge together the different metric spaces. This provides a single fuzzy simplicial set as the global representation of the manifold formed by patching
together the many local representations.
Given the ability to construct such topological structures, either from a known manifold, or by learning the metric structure of the manifold, we can perform dimension reduction by simply finding low
dimensional representations that closely match the topological structure of the source data. We now consider the task of finding such a low dimensional representation.
2.3 Optimizing a low dimensional representation
Let Y={Y1,…,YN}⊆ℝd be a low dimensional (d≪n) representation of X such that Yi represents the source data point Xi. In contrast to the source data where we want to estimate a manifold on which the
data is uniformly distributed, a target manifold for Y is chosen apriori (usually this will simply be ℝd itself, but other choices such as d-spheres or d-tori are certainly possible) . Therefore we
know the manifold and manifold metric apriori, and can compute the fuzzy topological representation directly. Of note, we still want to incorporate the distance to the nearest neighbor as per the
local connectedness requirement. This can be achieved by supplying a parameter that defines the expected distance between nearest neighbors in the embedded space.
Given fuzzy simplicial set representations of X and Y, a means of comparison is required. If we consider only the 1-skeleton of the fuzzy simplicial sets we can describe each as a fuzzy graph, or,
more specifically, a fuzzy set of edges. To compare two fuzzy sets we will make use of fuzzy set cross entropy. For these purposes we will revert to classical fuzzy set notation. That is, a fuzzy set
is given by a reference set A and a membership strength function μ:A→[0,1]. Comparable fuzzy sets have the same reference set. Given a sheaf representation 𝒫 we can translate to classical fuzzy sets
by setting A=⋃a∈(0,1]𝒫([0,a)) and μ(x)=sup{a∈(0,1]∣x∈𝒫([0,a))}.
Definition 10.
The cross entropy C of two fuzzy sets (A,μ) and (A,ν) is defined as
Similar to t-SNE we can optimize the embedding Y with respect to fuzzy set cross entropy C by using stochastic gradient descent. However, this requires a differentiable fuzzy singular set functor. If
the expected minimum distance between points is zero the fuzzy singular set functor is differentiable for these purposes, however for any non-zero value we need to make a differentiable approximation
(chosen from a suitable family of differentiable functions).
This completes the algorithm: by using manifold approximation and patching together local fuzzy simplicial set representations we construct a topological representation of the high dimensional data.
We then optimize the layout of data in a low dimensional space to minimize the error between the two topological representations.
We note that in this case we restricted attention to comparisons of the 1-skeleton of the fuzzy simplicial sets. One can extend this to ℓ-skeleta by defining a cost function Cℓ as
where Xi denotes the fuzzy set of i-simplices of X and the λi are suitably chosen real valued weights. While such an approach will capture the overall topological structure more accurately, it comes
at non-negligible computational cost due to the increasingly large numbers of higher dimensional simplices. For this reason current implementations restrict to the 1-skeleton at this time.
3 A Computational View of UMAP
To understand what computations the UMAP algorithm is actually making from a practical point of view, a less theoretical and more computational description may be helpful for the reader. This
description of the algorithm lacks the motivation for a number of the choices made. For that motivation please see Section 2.
The theoretical description of the algorithm works in terms of fuzzy simplicial sets. Computationally this is only tractable for the one skeleton which can ultimately be described as a weighted
graph. This means that, from a practical computational perspective, UMAP can ultimately be described in terms of, construction of, and operations on, weighted graphs. In particular this situates UMAP
in the class of k-neighbour based graph learning algorithms such as Laplacian Eigenmaps, Isomap and t-SNE.
As with other k-neighbour graph based algorithms, UMAP can be described in two phases. In the first phase a particular weighted k-neighbour graph is constructed. In the second phase a low dimensional
layout of this graph is computed. The differences between all algorithms in this class amount to specific details in how the graph is constructed and how the layout is computed. The theoretical basis
for UMAP as described in Section 2 provides novel approaches to both of these phases, and provides clear motivation for the choices involved.
Finally, since t-SNE is not usually described as a graph based algorithm, a direct comparison of UMAP with t-SNE, using the similarity/probability notation commonly used to express the equations of
t-SNE, is given in the Appendix C.
In section 2 we made a few basic assumptions about our data. From these assumptions we made use of category theory to derive the UMAP algorithms. That said, all these derivations assume these axioms
to be true.
1. 1.
There exists a manifold on which the data would be uniformly distributed.
2. 2.
The underlying manifold of interest is locally connected.
3. 3.
Preserving the topological structure of this manifold is the primary goal.
The topological theory of Section 2 is driven by these axioms, particularly the interest in modelling and preserving topological structure. In particular Section 2.1 highlights the underlying
motivation, in terms of topological theory, of representing a manifold as a k-neighbour graph.
As highlighted in Appendix C any algorithm that attempts to use a mathematical structure akin to a k-neighbour graph to approximate a manifold must follow a similar basic structure.
• •
Graph Construction
1. 1.
Construct a weighted k-neighbour graph
2. 2.
Apply some transform on the edges to ambient local distance.
3. 3.
Deal with the inherent asymmetry of the k-neighbour graph.
• •
Graph Layout
1. 1.
Define an objective function that preserves desired characteristics of this k-neighbour graph.
2. 2.
Find a low dimensional representation which optimizes this objective function.
Many dimension reduction algorithms can be broken down into these steps because they are fundamental to a particular class of solutions. Choices for each step must be either chosen through task
oriented experimentation or by selecting a set of believable axioms and building strong theoretical arguments from these. Our belief is that basing our decisions on a strong foundational theory will
allow for a more extensible and generalizable algorithm in the long run.
We theoretically justify using the choice of using a k-neighbour graph to represent a manifold in Section 2.1. The choices for our kernel transform an symmetrization function can be found in Section
2.2. Finally, the justifications underlying our choices for our graph layout are outlined in Section 2.3.
3.1 Graph Construction
The first phase of UMAP can be thought of as the construction of a weighted k-neighbour graph. Let X={x1,…,xN} be the input dataset, with a metric (or dissimilarity measure) d:X×X→ℝ≥0. Given an input
hyper-parameter k, for each xi we compute the set {xi1,…,xik} of the k nearest neighbors of xi under the metric d. This computation can be performed via any nearest neighbour or approximate nearest
neighbour search algorithm. For the purposes of our UMAP implemenation we prefer to use the nearest neighbor descent algorithm of [18].
For each xi we will define ρi and σi. Let
and set σi to be the value such that
The selection of ρi derives from the local-connectivity constraint described in Section 2.2. In particular it ensures that xi connects to at least one other data point with an edge of weight 1; this
is equivalent to the resulting fuzzy simplicial set being locally connected at xi. In practical terms this significantly improves the representation on very high dimensional data where other
algorithms such as t-SNE begin to suffer from the curse of dimensionality.
The selection of σi corresponds to (a smoothed) normalisation factor, defining the Riemannian metric local to the point xi as described in Section 2.1.
We can now define a weighted directed graph G¯=(V,E,w). The vertices V of G¯ are simply the set X. We can then form the set of directed edges E={(xi,xij)∣1≤j≤k,1≤i≤N}, and define the weight function
w by setting
For a given point xi there exists an induced graph of xi and outgoing edges incident on xi. This graph is the 1-skeleton of the fuzzy simplicial set associated to the metric space local to xi where
the local metric is defined in terms of ρi and σi. The weight associated to the edge is the membership strength of the corresponding 1-simplex within the fuzzy simplicial set, and is derived from the
adjunction of Theorem 1 using the right adjoint (nearest inverse) of the geometric realization of a fuzzy simplicial set. Intuitively one can think of the weight of an edge as akin to the probability
that the given edge exists. Section 2 demonstrates why this construction faithfully captures the topology of the data. Given this set of local graphs (represented here as a single directed graph) we
now require a method to combine them into a unified topological representation. We note that while patching together incompatible finite metric spaces is challenging, by using Theorem 1 to convert to
a fuzzy simplicial set representation, the combining operation becomes natural.
Let A be the weighted adjacency matrix of G¯, and consider the symmetric matrix
where ∘ is the Hadamard (or pointwise) product. This formula derives from the use of the probabilistic t-conorm used in unioning the fuzzy simplicial sets. If one interprets the value of Aij as the
probability that the directed edge from xi to xj exists, then Bij is the probability that at least one of the two directed edges (from xi to xj and from xj to xi) exists. The UMAP graph G is then an
undirected weighted graph whose adjacency matrix is given by B. Section 2 explains this construction in topological terms, providing the justification for why this construction provides an
appropriate fuzzy topological representation of the data – that is, this construction captures the underlying geometric structure of the data in a faithful way.
3.2 Graph Layout
In practice UMAP uses a force directed graph layout algorithm in low dimensional space. A force directed graph layout utilizes of a set of attractive forces applied along edges and a set of repulsive
forces applied among vertices. Any force directed layout algorithm requires a description of both the attractive and repulsive forces. The algorithm proceeds by iteratively applying attractive and
repulsive forces at each edge or vertex. This amounts to a non-convex optimization problem. Convergence to a local minima is guaranteed by slowly decreasing the attractive and repulsive forces in a
similar fashion to that used in simulated annealing.
In UMAP the attractive force between two vertices i and j at coordinates 𝐲𝐢 and 𝐲𝐣 respectively, is determined by:
where a and b are hyper-parameters.
Repulsive forces are computed via sampling due to computational constraints. Thus, whenever an attractive force is applied to an edge, one of that edge’s vertices is repulsed by a sampling of other
vertices. The repulsive force is given by
ϵ is a small number to prevent division by zero (0.001 in the current implementation).
The algorithm can be initialized randomly but in practice, since the symmetric Laplacian of the graph G is a discrete approximation of the Laplace-Beltrami operator of the manifold, we can use a
spectral layout to initialize the embedding. This provides both faster convergence and greater stability within the algorithm.
The forces described above are derived from gradients optimising the edge-wise cross-entropy between the weighted graph G, and an equivalent weighted graph H constructed from the points {𝐲𝐢}i=1..N.
That is, we are seeking to position points yi such that the weighted graph induced by those points most closely approximates the graph G, where we measure the difference between weighted graphs by
the total cross entropy over all the edge existence probabilities. Since the weighted graph G captures the topology of the source data, the equivalent weighted graph H constructed from the points
{𝐲𝐢}i=1..N matches the topology as closely as the optimization allows, and thus provides a good low dimensional representation of the overall topology of the data.
4 Implementation and Hyper-parameters
Having completed a theoretical description of the approach, we now turn our attention to the practical realization of this theory. We begin by providing a more detailed description of the algorithm
as implemented, and then discuss a few implementation specific details. We conclude this section with a discussion of the hyper-parameters for the algorithm and their practical effects.
4.1 Algorithm description
In overview the UMAP algorithm is relatively straightforward (see Algorithm 1). When performing a fuzzy union over local fuzzy simplicial sets we have found it most effective to work with the
probabilistic t-conorm (as one would expect if treating membership strengths as a probability that the simplex exists). The individual functions for constructing the local fuzzy simplicial sets,
determining the spectral embedding, and optimizing the embedding with regard to fuzzy set cross entropy, are described in more detail below.
Algorithm 1 UMAP algorithmfunction UMAP(X, n, d, min-dist, n-epochs) # Construct the relevant weighted graph for all x∈X do fs-set[x] ← LocalFuzzySimplicialSet(X, x, n) top-rep ←⋃x∈Xfs-set[x]# We
recommend the probabilistic t-conorm # Perform optimization of the graph layout Y← SpectralEmbedding(top-rep, d) Y← OptimizeEmbedding(top-rep, Y, min-dist, n-epochs) return Y
The inputs to Algorithm 1 are: X, the dataset to have its dimension reduced; n, the neighborhood size to use for local metric approximation; d, the dimension of the target reduced space; min-dist, an
algorithmic parameter controlling the layout; and n-epochs, controlling the amount of optimization work to perform.
Algorithm 2 describes the construction of local fuzzy simplicial sets. To represent fuzzy simplicial sets we work with the fuzzy set images of [0] and [1] (i.e. the 1-skeleton), which we denote as
fs-set0 and fs-set1. One can work with higher order simplices as well, but the current implementation does not. We can construct the fuzzy simplicial set local to a given point x by finding the n
nearest neighbors, generating the appropriate normalised distance on the manifold, and then converting the finite metric space to a simplicial set via the functor 𝖥𝗂𝗇𝖲𝗂𝗇𝗀, which translates into
exponential of the negative distance in this case.
Algorithm 2 Constructing a local fuzzy simplicial setfunction LocalFuzzySimplicialSet(X, x, n) knn, knn-dists ← ApproxNearestNeighbors(X, x, n) ρ← knn-dists[1]# Distance to nearest neighbor σ←
SmoothKNNDist(knn-dists, n, ρ)# Smooth approximator to knn-distance fs-set0←X fs-set1←{([x,y],0)∣y∈X} for all y∈ knn do dx,y←max{0,dist(x,y)−ρ}/σ fs-set1←fs-set1∪([x,y],exp(−dx,y)) return fs-set
Rather than directly using the distance to the nth nearest neighbor as the normalization, we use a smoothed version of knn-distance that fixes the cardinality of the fuzzy set of 1-simplices to a
fixed value. We selected log2(n) for this purpose based on empirical experiments. This is described briefly in Algorithm 3.
Algorithm 3 Compute the normalizing factor for distances σfunction SmoothKNNDist(knn-dists, n, ρ) Binary search for σ such that ∑i=1nexp(−(knn-distsi−ρ)/σ)=log2(n) return σ
Spectral embedding is performed by considering the 1-skeleton of the global fuzzy topological representation as a weighted graph and using standard spectral methods on the symmetric normalized
Laplacian. This process is described in Algorithm 4.
Algorithm 4 Spectral embedding for initializationfunction SpectralEmbedding(top-rep, d) A← 1-skeleton of top-rep expressed as a weighted adjacency matrix D← degree matrix for the graph A L←D1/2(D−A)
D1/2 evec← Eigenvectors of L (sorted) Y←evec[1..d+1]# 0-base indexing assumed return Y
The final major component of UMAP is the optimization of the embedding through minimization of the fuzzy set cross entropy. Recall that fuzzy set cross entropy, with respect given membership
functions μ and ν, is given by
C((A,μ),(A,ν))=∑a∈Aμ(a)log(μ(a)ν(a))+(1−μ(a))log(1−μ(a)1−ν(a))=∑a∈A(μ(a)log(μ(a))+(1−μ(a))log(1−μ(a)))−∑a∈A(μ(a)log(ν(a))+(1−μ(a))log(1−ν(a))). (1)
The first sum depends only on μ which takes fixed values during the optimization, thus the minimization of cross entropy depends only on the second sum, so we seek to minimize
Following both [54] and [41], we take a sampling based approach to the optimization. We sample 1-simplices with probability μ(a) and update according to the value of ν(a), which handles the term μ
(a)log(ν(a)). The term (1−μ(a))log(1−ν(a)) requires negative sampling – rather than computing this over all potential simplices we randomly sample potential 1-simplices and assume them to be a
negative example (i.e. with membership strength 0) and update according to the value of 1−ν(a). In contrast to [54] the above formulation provides a vertex sampling distribution of
for negative samples, which can be reasonably approximated by a uniform distribution for sufficiently large data sets.
It therefore only remains to find a differentiable approximation to ν(a) for a given 1-simplex a so that gradient descent can be applied for optimization. This is done as follows:
Definition 11.
Define Φ:ℝd×ℝd→[0,1], a smooth approximation of the membership strength of a 1-simplex between two points in ℝd, as
where a and b are chosen by non-linear least squares fitting against the curve Ψ:ℝd×ℝd→[0,1] where
Ψ(𝐱,𝐲)={1if ‖𝐱−𝐲‖2≤min-distexp(−(‖𝐱−𝐲‖2−min-dist))otherwise.
The optimization process is now executed by stochastic gradient descent as given by Algorithm 5.
Algorithm 5 Optimizing the embeddingfunction OptimizeEmbedding(top-rep, Y, min-dist, n-epochs) α←1.0 Fit Φ from Ψ defined by min-dist for e←1,…, n-epochs do for all ([a,b],p)∈top-rep1 do if Random( )
≤p then# Sample simplex with probability p ya←ya+α⋅∇(log(Φ))(ya,yb) for i←1,…,n-neg-samples do c←random sample from Y ya←ya+α⋅∇(log(1−Φ))(ya,yc) α←1.0−e/n-epochs return Y
This completes the UMAP algorithm.
4.2 Implementation
Practical implementation of this algorithm requires (approximate) k-nearest-neighbor calculation and efficient optimization via stochastic gradient descent.
Efficient approximate k-nearest-neighbor computation can be achieved via the Nearest-Neighbor-Descent algorithm of [18]. The error intrinsic in a dimension reduction technique means that such
approximation is more than adequate for these purposes. While no theoretical complexity bounds have been established for Nearest-Neighbor-Descent the authors of the original paper report an empirical
complexity of O(N1.14). A further benefit of Nearest-Neighbor-Descent is its generality; it works with any valid dissimilarity measure, and is efficient even for high dimensional data.
In optimizing the embedding under the provided objective function, we follow work of [54]; making use of probabilistic edge sampling and negative sampling [41]. This provides a very efficient
approximate stochastic gradient descent algorithm since there is no normalization requirement. Furthermore, since the normalized Laplacian of the fuzzy graph representation of the input data is a
discrete approximation of the Laplace-Betrami operator of the manifold [[, see]]belkin2002laplacian, belkin2003laplacian, we can provide a suitable initialization for stochastic gradient descent by
using the eigenvectors of the normalized Laplacian. The amount of optimization work required will scale with the number of edges in the fuzzy graph (assuming a fixed negative sampling rate),
resulting in a complexity of O(kN).
Combining these techniques results in highly efficient embeddings, which we will discuss in Section 5. The overall complexity is bounded by the approximate nearest neighbor search complexity and, as
mentioned above, is empirically approximately O(N1.14). A reference implementation can be found at https://github.com/lmcinnes/umap, and an R implementation can be found at https://github.com/
For simplicity these experiments were carried out on a single core version of our algorithm. It should be noted that at the time of this publication that both Nearest-Neighbour-Descent and SGD have
been parallelized and thus the python reference implementation can be significantly accelerated. Our intention in this paper was to introduce the underlying theory behind our UMAP algorithm and we
felt that parallel vs single core discussions would distract from our intent.
4.3 Hyper-parameters
As described in Algorithm 1, the UMAP algorithm takes four hyper-parameters:
1. 1.
n, the number of neighbors to consider when approximating the local metric;
2. 2.
d, the target embedding dimension;
3. 3.
min-dist, the desired separation between close points in the embedding space; and
4. 4.
n-epochs, the number of training epochs to use when optimizing the low dimensional representation.
The effects of the parameters d and n-epochs are largely self-evident, and will not be discussed in further detail here. In contrast the effects of the number of neighbors n and of min-dist are less
One can interpret the number of neighbors n as the local scale at which to approximate the manifold as roughly flat, with the manifold estimation averaging over the n neighbors. Manifold features
that occur at a smaller scale than within the n nearest-neighbors of points will be lost, while large scale manifold features that cannot be seen by patching together locally flat charts at the scale
of n nearest-neighbors may not be well detected. Thus n represents some degree of trade-off between fine grained and large scale manifold features — smaller values will ensure detailed manifold
structure is accurately captured (at a loss of the “big picture” view of the manifold), while larger values will capture large scale manifold structures, but at a loss of fine detail structure which
will get averaged out in the local approximations. With smaller n values the manifold tends to be broken into many small connected components (care needs to be taken with the spectral embedding for
initialization in such cases).
In contrast min-dist is a hyperparameter directly affecting the output, as it controls the fuzzy simplicial set construction from the low dimensional representation. It acts in lieu of the distance
to the nearest neighbor used to ensure local connectivity. In essence this determines how closely points can be packed together in the low dimensional representation. Low values on min-dist will
result in potentially densely packed regions, but will likely more faithfully represent the manifold structure. Increasing the value of min-dist will force the embedding to spread points out more,
assisting visualization (and avoiding potential overplotting issues). We view min-dist as an essentially aesthetic parameter, governing the appearance of the embedding, and thus is more important
when using UMAP for visualization.
In Figure 1 we provide examples of the effects of varying the hyperparameters for a toy dataset. The data is uniform random samples from a 3-dimensional color-cube, allowing for easy visualization of
the original 3-dimensional coordinates in the embedding space by using the corresponding RGB colour. Since the data fills a 3-dimensional cube there is no local manifold structure, and hence for such
data we expect larger n values to be more useful. Low values will interpret the noise from random sampling as fine scale manifold structure, producing potentially spurious structure11See the
discussion of the constellation effect in Section 6.
6 for further discussion of this phenomena.
In Figure 2 we provides examples of the same hyperparamter choices as Figure 1, but for the PenDigits dataset22See Section 5 for a description of the PenDigits dataset. In this case we expect small
to medium n values to be most effective, since there is significant cluster structure naturally present in the data. The min-dist parameter expands out tightly clustered groups, allowing more of the
internal structure of densely packed clusters to be seen.
Finally, in Figure 3 we provide an equivalent example of hyperparameter choices for the MNIST dataset33See section 5 for details on the MNIST dataset. Again, since this dataset is expected to have
signifcant cluster structure we expect medium sized values of n to be most effective. We note that large values of min-dist result in the distinct clusters being compressed together, making the
distinctions between the clusters less clear.
5 Practical Efficacy
While the strong mathematical foundations of UMAP were the motivation for its development, the algorithm must ultimately be judged by its practical efficacy. In this section we examine the fidelity
and performance of low dimensional embeddings of multiple diverse real world data sets under UMAP. The following datasets were considered:
Pen digits [1, 10] is a set of 1797 grayscale images of digits entered using a digitiser tablet. Each image is an 8x8 image which we treat as a single 64 dimensional vector, assumed to be in
Euclidean vector space.
COIL 20 [43] is a set of 1440 greyscale images consisting of 20 objects under 72 different rotations spanning 360 degrees. Each image is a 128x128 image which we treat as a single 16384 dimensional
vector for the purposes of computing distance between images.
COIL 100 [44] is a set of 7200 colour images consisting of 100 objects under 72 different rotations spanning 360 degrees. Each image consists of 3 128x128 intensity matrices (one for each color
channel). We treat this as a single 49152 dimensional vector for the purposes of computing distance between images.
Mouse scRNA-seq [11] is profiled gene expression data for 20,921 cells from an adult mouse. Each sample consists of a vector of 26,774 measurements.
Statlog (Shuttle) [35] is a NASA dataset consisting of various data associated to the positions of radiators in the space shuttle, including a timestamp. The dataset has 58000 points in a 9
dimensional feature space.
MNIST [32] is a dataset of 28x28 pixel grayscale images of handwritten digits. There are 10 digit classes (0 through 9) and 70000 total images. This is treated as 70000 different 784 dimensional
F-MNIST [63] or Fashion MNIST is a dataset of 28x28 pixel grayscale images of fashion items (clothing, footwear and bags). There are 10 classes and 70000 total images. As with MNIST this is treated
as 70000 different 784 dimensional vectors.
Flow cytometry [51, 9] is a dataset of flow cytometry measurements of CDT4 cells comprised of 1,000,000 samples, each with 17 measurements.
GoogleNews word vectors [41] is a dataset of 3 million words and phrases derived from a sample of Google News documents and embedded into a 300 dimensional space via word2vec.
For all the datasets except GoogleNews we use Euclidean distance between vectors. For GoogleNews, as per [41], we use cosine distance (or angular distance in t-SNE which does support non-metric
distances, in contrast to UMAP).
5.1 Qualitative Comparison of Multiple Algorithms
We compare a number of algorithms–UMAP, t-SNE [60, 58], LargeVis [54], Laplacian Eigenmaps [7], and Principal Component Analysis [27]–on the COIL20 [43], MNIST [32], Fashion-MNIST [63], and
GoogleNews [41] datasets. The Isomap algorithm was also tested, but failed to complete in any reasonable time for any of the datasets larger than COIL20.
The Multicore t-SNE package [57] was used for t-SNE. The reference implementation [53] was used for LargeVis. The scikit-learn [10] implementations were used for Laplacian Eigenmaps and PCA. Where
possible we attempted to tune parameters for each algorithm to give good embeddings.
Historically t-SNE and LargeVis have offered a dramatic improvement in finding and preserving local structure in the data. This can be seen qualitatively by comparing their embeddings to those
generated by Laplacian Eigenmaps and PCA in Figure 4. We claim that the quality of embeddings produced by UMAP is comparable to t-SNE when reducing to two or three dimensions. For example, Figure 4
shows both UMAP and t-SNE embeddings of the COIL20, MNIST, Fashion MNIST, and Google News datasets. While the precise embeddings are different, UMAP distinguishes the same structures as t-SNE and
It can be argued that UMAP has captured more of the global and topological structure of the datasets than t-SNE [4, 62]. More of the loops in the COIL20 dataset are kept intact, including the
intertwined loops. Similarly the global relationships among different digits in the MNIST digits dataset are more clearly captured with 1 (red) and 0 (dark red) at far corners of the embedding space,
and 4,7,9 (yellow, sea-green, and violet) and 3,5,8 (orange, chartreuse, and blue) separated as distinct clumps of similar digits. In the Fashion MNIST dataset the distinction between clothing (dark
red, yellow, orange, vermilion) and footwear (chartreuse, sea-green, and violet) is made more clear. Finally, while both t-SNE and UMAP capture groups of similar word vectors, the UMAP embedding
arguably evidences a clearer global structure among the various word clusters.
5.2 Quantitative Comparison of Multiple Algorithms
We compare UMAP, t-SNE, LargeVis, Laplacian Eigenmaps and PCA embeddings with respect to the performance of a k-nearest neighbor classifier trained on the embedding space for a variety of datasets.
The k-nearest neighbor classifier accuracy provides a clear quantitative measure of how well the embedding has preserved the important local structure of the dataset. By varying the hyper-parameter k
used in the training we can also consider how structure preservation varies under transition from purely local to non-local, to more global structure. The embeddings used for training the kNN
classifier are for those datasets that come with defined training labels: PenDigits, COIL-20, Shuttle, MNIST, and Fashion-MNIST.
We divide the datasets into two classes: smaller datasets (PenDigits and COIL-20), for which a smaller range of k values makes sense, and larger datasets, for which much larger values of k are
reasonable. For each of the small datasets a stratified 10-fold cross-validation was used to derive a set of 10 accuracy scores for each embedding. For the Shuttle dataset a 10-fold cross-validation
was used due to constraints imposed by class sizes and the stratified sampling. For MNIST and Fashion-MNIST a 20-fold cross validation was used, producing 20 accuracy scores.
In Table 1 we present the average accuracy across the 10-folds for the PenDigits and COIL-20 datasets. UMAP performs at least as well as t-SNE and LargeVis (given the confidence bounds on the
accuracy) for k in the range 10 to 40, but for larger k values of 80 and 160 UMAP has significantly higher accuracy on COIL-20, and shows evidence of higher accuracy on PenDigits. Figure 5 provides
swarm plots of the accuracy results across the COIL-20 and PenDigits datasets.
In Table 2 we present the average cross validation accuracy for the Shuttle, MNIST and Fashion-MNIST datasets. UMAP performs at least as well as t-SNE and LargeVis (given the confidence bounds on the
accuracy) for k in the range 100 to 400 on the Shuttle and MNIST datasets (but notably underperforms on the Fashion-MNIST dataset), but for larger k values of 800 and 3200 UMAP has significantly
higher accuracy on the Shuttle dataset, and shows evidence of higher accuracy on MNIST. For k values of 1600 and 3200 UMAP establishes comparable performance on Fashion-MNIST. Figure 6 provides swarm
plots of the accuracy results across the Shuttle and MNIST and Fashion-MNIST datasets.
k t-SNE UMAP LargeVis Eigenmaps PCA
COIL-20 10 0.934 (± 0.115) 0.921 (± 0.075) 0.888 (± 0.092) 0.629 (± 0.153) 0.667 (± 0.179)
20 0.901 (± 0.133) 0.907 (± 0.064) 0.870 (± 0.125) 0.605 (± 0.185) 0.663 (± 0.196)
40 0.857 (± 0.125) 0.904 (± 0.056) 0.833 (± 0.106) 0.578 (± 0.159) 0.620 (± 0.230)
80 0.789 (± 0.118) 0.899 (± 0.058) 0.803 (± 0.100) 0.565 (± 0.119) 0.531 (± 0.294)
160 0.609 (± 0.067) 0.803 (± 0.138) 0.616 (± 0.066) 0.446 (± 0.110) 0.375 (± 0.111)
PenDigits 10 0.977 (± 0.033) 0.973 (± 0.044) 0.966 (± 0.053) 0.778 (± 0.113) 0.622 (± 0.092)
20 0.973 (± 0.033) 0.976 (± 0.035) 0.973 (± 0.044) 0.778 (± 0.116) 0.633 (± 0.082)
40 0.956 (± 0.064) 0.954 (± 0.060) 0.959 (± 0.066) 0.778 (± 0.112) 0.636 (± 0.078)
80 0.948 (± 0.060) 0.951 (± 0.072) 0.949 (± 0.072) 0.767 (± 0.111) 0.643 (± 0.085)
160 0.949 (± 0.065) 0.951 (± 0.085) 0.921 (± 0.085) 0.747 (± 0.108) 0.629 (± 0.107)
Table 1:kNN Classifier accuracy for varying values of k over the embedding spaces of COIL-20 and PenDigits datasets. Average accuracy scores are given over a 10-fold cross-validation for each of PCA,
Laplacian Eigenmaps, LargeVis, t-SNE and UMAP.
k t-SNE UMAP LargeVis Eigenmaps PCA
Shuttle 100 0.994 (± 0.002) 0.993 (± 0.002) 0.992 (± 0.003) 0.962 (± 0.004) 0.833 (± 0.013)
200 0.992 (± 0.002) 0.990 (± 0.002) 0.987 (± 0.003) 0.957 (± 0.006) 0.821 (± 0.007)
400 0.990 (± 0.002) 0.988 (± 0.002) 0.976 (± 0.003) 0.949 (± 0.006) 0.815 (± 0.007)
800 0.969 (± 0.005) 0.988 (± 0.002) 0.957 (± 0.004) 0.942 (± 0.006) 0.804 (± 0.003)
1600 0.927 (± 0.005) 0.981 (± 0.002) 0.904 (± 0.007) 0.918 (± 0.006) 0.792 (± 0.003)
3200 0.828 (± 0.004) 0.957 (± 0.005) 0.850 (± 0.008) 0.895 (± 0.006) 0.786 (± 0.001)
MNIST 100 0.967 (± 0.015) 0.967 (± 0.014) 0.962 (± 0.015) 0.668 (± 0.016) 0.462 (± 0.023)
200 0.966 (± 0.015) 0.967 (± 0.014) 0.962 (± 0.015) 0.667 (± 0.016) 0.467 (± 0.023)
400 0.964 (± 0.015) 0.967 (± 0.014) 0.961 (± 0.015) 0.664 (± 0.016) 0.468 (± 0.024)
800 0.963 (± 0.016) 0.967 (± 0.014) 0.961 (± 0.015) 0.660 (± 0.017) 0.468 (± 0.023)
1600 0.959 (± 0.016) 0.966 (± 0.014) 0.947 (± 0.015) 0.651 (± 0.014) 0.467 (± 0.0233)
3200 0.946 (± 0.017) 0.964 (± 0.014) 0.920 (± 0.017) 0.639 (± 0.017) 0.459 (± 0.022)
Fashion-MNIST 100 0.818 (± 0.012) 0.790 (± 0.013) 0.808 (± 0.014) 0.631 (± 0.010) 0.564 (± 0.018)
200 0.810 (± 0.013) 0.785 (± 0.014) 0.805 (± 0.013) 0.624 (± 0.013) 0.565 (± 0.016)
400 0.801 (± 0.013) 0.780 (± 0.013) 0.796 (± 0.013) 0.612 (± 0.011) 0.564 (± 0.017)
800 0.784 (± 0.011) 0.767 (± 0.014) 0.771 (± 0.014) 0.600 (± 0.012) 0.560 (± 0.017)
1600 0.754 (± 0.011) 0.747 (± 0.013) 0.742 (± 0.013) 0.580 (± 0.014) 0.550 (± 0.017)
3200 0.727 (± 0.011) 0.730 (± 0.011) 0.726 (± 0.012) 0.542 (± 0.014) 0.533 (± 0.017)
Table 2:kNN Classifier accuracy for varying values of k over the embedding spaces of Shuttle, MNIST and Fashion-MNIST datasets. Average accuracy scores are given over a 10-fold or 20-fold
cross-validation for each of PCA, Laplacian Eigenmaps, LargeVis, t-SNE and UMAP.
As evidenced by this comparison UMAP provides largely comparable perfomance in embedding quality to t-SNE and LargeVis at local scales, but performs markedly better than t-SNE or LargeVis at
non-local scales. This bears out the visual qualitative assessment provided in Subsection 5.1.
5.3 Embedding Stability
Since UMAP makes use of both stochastic approximate nearest neighbor search, and stochastic gradient descent with negative sampling for optimization, the resulting embedding is necessarily different
from run to run, and under sub-sampling of the data. This is potentially a concern for a variety of uses cases, so establishing some measure of how stable UMAP embeddings are, particularly under
sub-sampling, is of interest. In this subsection we compare the stability under subsampling of UMAP, LargeVis and t-SNE (the three stochastic dimension reduction techniques considered).
To measure the stability of an embedding we make use of the normalized Procrustes distance to measure the distance between two potentially comparable distributions. Given two datasets X={x1,…,xN} and
Y={y1,…,yN} such that xi corresponds to yi, we can define the Procustes distance between the datasets dP(X,Y) in the following manner. Determine Y′={y1′,…,yN′} the optimal translation, uniform
scaling, and rotation of Y that minimizes the squared error ∑i=1N(xi−yi′)2, and define
Since any measure that makes use of distances in the embedding space is potentially sensitive to the extent or scale of the embedding, we normalize the data before computing the Procrustes distance
by dividing by the average norm of the embedded dataset. In Figure 7 we visualize the results of using Procrustes alignment of embedding of sub-samples for both UMAP and t-SNE, demonstrating how
Procrustes distance can measure the stability of the overall structure of the embedding.
Given a measure of distance between different embeddings we can examine stability under sub-sampling by considering the normalized Procrustes distance between the embedding of a sub-sample, and the
corresponding sub-sample of an embedding of the full dataset. As the size of the sub-sample increases the average distance per point between the sub-sampled embeddings should decrease, potentially
toward some asymptote of maximal agreement under repeated runs. Ideally this asymptotic value would be zero error, but for stochastic embeddings such as UMAP and t-SNE this is not achievable.
We performed an empirical comparison of algorithms with respect to stability using the Flow Cytometry dataset due its large size, interesting structure, and low ambient dimensionality (aiding runtime
performance for t-SNE). We note that for a dataset this large we found it necessary to increase the default n_iter value for t-SNE from 1000 to 1500 to ensure better convergence. While this had an
impact on the runtime, it significantly improved the Procrustes distance results by providing more stable and consistent embeddings. Figure 8 provides a comparison between UMAP and t-SNE,
demonstrating that UMAP has signifcantly more stable results than t-SNE. In particular, after sub-sampling on 5% of the million data points, the per point error for UMAP was already below any value
achieved by t-SNE.
5.4 Computational Performance Comparisons
Benchmarks against the real world datasets were performed on a Macbook Pro with a 3.1 GHz Intel Core i7 and 8GB of RAM for Table 3, and on a server with Intel Xeon E5-2697v4 processors and 512GB of
RAM for the large scale benchmarking in Subsections 5.4.1, 5.4.2, and 5.4.3.
For t-SNE we chose MulticoreTSNE [57], which we believe to be the fastest extant implementation of Barnes-Hut t-SNE at this time, even when run in single core mode. It should be noted that
MulticoreTSNE is a heavily optimized implementation written in C++ based on Van der Maaten’s bhtsne [58] code.
As a fast alternative approach to t-SNE we also consider the FIt-SNE algorithm [37]. We used the reference implementation [36], which, like MulticoreTNSE is an optimized C++ implementation. We also
note that FIt-SNE makes use of multiple cores.
LargeVis [54] was benchmarked using the reference implementation [53]. It was run with default parameters including use of 8 threads on the 4-core machine. The only exceptions were small datasets
where we explicitly set the -samples parameter to n_samples/100 as per the recommended values in the documentation of the reference implementation.
The Isomap [55] and Laplacian Eigenmaps [7] implementations in scikit-learn [10] were used. We suspect the Laplacian eigenmaps implementation may not be well optimized for large datasets but did not
find a better performing implementation that provided comparable quality results. Isomap failed to complete for the Shuttle, Fashion-MNIST, MNIST and GoogleNews datasets, while Laplacian Eigenmaps
failed to run for the GoogleNews dataset.
To allow a broader range of algorithms to run some of the datasets where subsampled or had their dimension reduced by PCA. The Flow Cytometry dataset was benchmarked on a 10% sample and the
GoogleNews was subsampled down to 200,000 data points. Finally, the Mouse scRNA dataset was reduced to 1,000 dimensions via PCA.
Timing were performed for the COIL20 [43], COIL100 [44], Shuttle [35], MNIST [32], Fashion-MNIST [63], and GoogleNews [41] datasets. Results can be seen in Table 3. UMAP consistently performs faster
than any of the other algorithms aside from on the very small Pendigits dataset, where Laplacian Eigenmaps and Isomap have a small edge.
UMAP FIt-SNE t-SNE LargeVis Eigenmaps Isomap
Pen Digits 9s 48s 17s 20s 2s 2s
COIL20 12s 75s 22s 82s 47s 58s
COIL100 85s 2681s 810s 3197s 3268s 3210s
scRNA 28s 131s 258s 377s 470s 923s
Shuttle 94s 108s 714s 615s 133s –
MNIST 87s 292s 1450s 1298s 40709s –
F-MNIST 65s 278s 934s 1173s 6356s –
Flow 102s 164s 1135s 1127s 30654s –
Google News 361s 652s 16906s 5392s – –
Table 3:Runtime of several dimension reduction algorithms on various datasets. To allow a broader range of algorithms to run some of the datasets where subsampled or had their dimension reduced by
PCA. The Flow Cytometry dataset was benchmarked on a 10% sample and the GoogleNews was subsampled down to 200,000 data points. Finally, the Mouse scRNA dataset was reduced to 1,000 dimensions via
PCA. The fastest runtime for each dataset has been bolded.
5.4.1 Scaling with Embedding Dimension
UMAP is significantly more performant than t-SNE44Comparisons were performed against MulticoreTSNE as the current implementation of FIt-SNE does not support embedding into any dimension larger than
2. when embedding into dimensions larger than 2. This is particularly important when the intention is to use the low dimensional representation for further machine learning tasks such as clustering
or anomaly detection rather than merely for visualization. The computation performance of UMAP is far more efficient than t-SNE, even for very small embedding dimensions of 6 or 8 (see Figure 9).
This is largely due to the fact that UMAP does not require global normalisation (since it represents data as a fuzzy topological structure rather than as a probability distribution). This allows the
algorithm to work without the need for space trees —such as the quad-trees and oct-trees that t-SNE uses [58]—. Such space trees scale exponentially in dimension, resulting in t-SNE’s relatively poor
scaling with respect to embedding dimension. By contrast, we see that UMAP consistently scales well in embedding dimension, making the algorithm practical for a wider range of applications beyond
5.4.2 Scaling with Ambient Dimension
Through a combination of the local-connectivity constraint and the approximate nearest neighbor search, UMAP can perform effective dimension reduction even for very high dimensional data (see Figure
13 for an example of UMAP operating directly on 1.8 million dimensional data). This stands in contrast to many other manifold learning techniques, including t-SNE and LargeVis, for which it is
generally recommended to reduce the dimension with PCA before applying these techniques (see [59] for example).
To compare runtime performance scaling with respect to the ambient dimension of the data we chose to use the Mouse scRNA dataset, which is high dimensional, but is also amenable to the use of PCA to
reduce the dimension of the data as a pre-processing step without losing too much of the important structure55In contrast to COIL100, on which PCA destroys much of the manifold structure. We compare
the performance of UMAP, FIt-SNE, MulticoreTSNE, and LargeVis on PCA reductions of the Mouse scRNA dataset to varying dimensionalities, and on the original dataset, in Figure 10.
While all the implementations tested show a significant increase in runtime with increasing dimension, UMAP is dramatically more efficient for large ambient dimensions, easily scaling to run on the
original unreduced dataset. The ability to run manifold learning on raw source data, rather than dimension reduced data that may have lost important manifold structure in the pre-processing, is a
significant advantage. This advantage comes from the local connectivity assumption which ensures good topological representation of high dimensional data, particularly with smaller numbers of near
neighbors, and the efficiency of the NN-Descent algorithm for approximate nearest neighbor search even in high dimensions.
Since UMAP scales well with ambient dimension the python implementation also supports input in sparse matrix format, allowing scaling to extremely high dimensional data, such as the integer data
shown in Figures 13 and 14.
5.4.3 Scaling with the Number of Samples
For dataset size performance comparisons we chose to compare UMAP with FIt-SNE [37], a version of t-SNE that uses approximate nearest neighbor search and a Fourier interpolation optimisation
approach; MulticoreTSNE [57], which we believe to be the fastest extant implementation of Barnes-Hut t-SNE; and LargeVis [54]. It should be noted that FIt-SNE, MulticoreTSNE, and LargeVis are all
heavily optimized implementations written in C++. In contrast our UMAP implementation was written in Python — making use of the numba [31] library for performance. MulticoreTSNE and LargeVis were run
in single threaded mode to make fair comparisons to our single threaded UMAP implementation.
The number of single cell samples per sequencing runs is based on an Illumina PhiX control library at supported cluster densities/loading concentration. Actual performance
parameters may vary based on sample type, sample quality, and clusters passing filter. See specifications page for each instrument for details. CSP, cell surface protein; GEx,
gene expression; n/a, not applicable
a. Minimum read recommendations courtesy of 10x Genomics. Adjust sequencing depth for the required performance or application. The sequencing saturation metric and curve
in the 10x Cell Ranger run summary can be used to optimize sequencing depth for specific sample types.
b. V(D)J, CSP, and 5' GEx libraries may be pooled for sequencing, taking into account the differences in depth requirements between the pooled libraries. Library pooling ratio: library can be brought
to the same concentration and then pooled at a 1 :4 ratio (volume V(D)J:volume GEx or volume CSP:volume GEx). It is also possible to pool them
at a 1:10 ratio volume V(D)J:volume GEx. In that case, the number of samples per run must be adjusted.
c. See "sequencing depth" and "number of samples per run" to understand the maximum number of cells that can be recovered, depending on which Chromium system is used.
d. P1 and P2 flow cells with the same sample throughput are available on the NextSeq 1000 System.
e. For V(D)J and CSP libraries, 5000 read pairs (or 10,000 individual reads) means 5000 clusters, or 5000 reads from Read 1 and 5000 reads from Read 2. For Gene Expression
libraries, 20,000 read pairs (or 40,000 individual reads) means 20,000 clusters, or 20,000 reads from Read 1 and 20,000 reads from Read 2.
f. Users can run the indicated number of V(D)J/CSP libraries OR one gene expression library. If running two or three library types, use of a higher output flow cell is recommended.
g. Requires use of the NovaSeq XP workflow, which allows for individual lane loading
We benchmarked all four implementations using subsamples of the Google News dataset.
The results can be seen in Figure 11. This demonstrates that UMAP has superior scaling performance in comparison to Barnes-Hut t-SNE, even when Barnes-Hut t-SNE is given multiple cores. Asymptotic
scaling of UMAP is comparable to that of FIt-SNE (and LargeVis). On this dataset UMAP demonstrated somewhat faster absolute performance compared to FIt-SNE, and was dramatically faster than LargeVis.
The UMAP embedding of the full GoogleNews dataset of 3 million word vectors, as seen in Figure 12, was completed in around 200 minutes, as compared with several days required for MulticoreTSNE, even
using multiple cores.
To scale even further we were inspired by the work of John Williamson on embedding integers [61], as represented by (sparse) binary vectors of their prime divisibility. This allows the generation of
arbitrarily large, extremely high dimension datasets that still have meaningful structure to be explored. In Figures 13 and 14 we show an embedding of 30,000,000 data samples from an ambient space of
approximately 1.8 million dimensions. This computation took approximately 2 weeks on a large memory SMP. Note that despite the high ambient dimension, and vast amount of data, UMAP is still able to
find and display interesting structure. In Figure 15 we show local regions of the embedding, demonstrating the fine detail structure that was captured.
6 Weaknesses
While we believe UMAP to be a very effective algorithm for both visualization and dimension reduction, most algorithms must make trade-offs and UMAP is no exception. In this section we will briefly
discuss those areas or use cases where UMAP is less effective, and suggest potential alternatives.
For a number of use cases the interpretability of the reduced dimension results is of critical importance. Similarly to most non-linear dimension reduction techniques (including t-SNE and Isomap),
UMAP lacks the strong interpretability of Principal Component Analysis (PCA) and related techniques such a Non-Negative Matrix Factorization (NMF). In particular the dimensions of the UMAP embedding
space have no specific meaning, unlike PCA where the dimensions are the directions of greatest variance in the source data. Furthermore, since UMAP is based on the distance between observations
rather than the source features, it does not have an equivalent of factor loadings that linear techniques such as PCA, or Factor Analysis can provide. If strong interpretability is critical we
therefore recommend linear techniques such as PCA, NMF or pLSA.
One of the core assumptions of UMAP is that there exists manifold structure in the data. Because of this UMAP can tend to find manifold structure within the noise of a dataset – similar to the way
the human mind finds structured constellations among the stars. As more data is sampled the amount of structure evident from noise will tend to decrease and UMAP becomes more robust, however care
must be taken with small sample sizes of noisy data, or data with only large scale manifold structure. Detecting when a spurious embedding has occurred is a topic of further research.
UMAP is derived from the axiom that local distance is of more importance than long range distances (similar to techniques like t-SNE and LargeVis). UMAP therefore concerns itself primarily with
accurately representing local structure. While we believe that UMAP can capture more global structure than these other techniques, it remains true that if global structure is of primary interest then
UMAP may not be the best choice for dimension reduction. Multi-dimensional scaling specifically seeks to preserve the full distance matrix of the data, and as such is a good candidate when all scales
of structure are of equal importance. PHATE [42] is a good example of a hybrid approach that begins with local structure information and makes use of MDS to attempt to preserve long scale distances
as well. It should be noted that these techniques are more computationally intensive and thus rely on landmarking approaches for scalability.
It should also be noted that a significant contributor to UMAP’s relative global structure preservation is derived from the Laplacian Eigenmaps initialization (which, in turn, followed from the
theoretical foundations). This was noted in, for example, [29]. The authors of that paper demonstrate that t-SNE, with similar initialization, can perform equivalently to UMAP in a particular measure
of global structure preservation. However, the objective function derived for UMAP (cross-entropy) is significantly different from that of t-SNE (KL-divergence), in how it penalizes failures to
preserve non-local and global structure, and is also a significant contributor66The authors would like to thank Nikolay Oskolkov for his article (tSNE vs. UMAP: Global Structure) which does an
excellent job of highlighting these aspects from an empirical and theoretical basis.
It is worth noting that, in combining the local simplicial set structures, pure nearest neighbor structure in the high dimensional space is not explicitly preserved. In particular it introduces so
called ”reverse-nearest-neighbors” into the classical knn-graph. This, combined with the fact that UMAP is preserving topology rather than pure metric structures, mean that UMAP will not perform as
well as some methods on quality measures based on metric structure preservation – particularly methods, such as MDS – which are explicitly designed to optimize metric structure preservation.
UMAP attempts to discover a manifold on which your data is uniformly distributed. If you have strong confidence in the ambient distances of your data you should make use of a technique that
explicitly attempts to preserve these distances. For example if your data consisted of a very loose structure in one area of your ambient space and a very dense structure in another region region
UMAP would attempt to put these local areas on an even footing.
Finally, to improve the computational efficiency of the algorithm a number of approximations are made. This can have an impact on the results of UMAP for small (less than 500 samples) dataset sizes.
In particular the use of approximate nearest neighbor algorithms, and the negative sampling used in optimization, can result in suboptimal embeddings. For this reason we encourage users to take care
with particularly small datasets. A slower but exact implementation of UMAP for small datasets is a future project.
7 Future Work
Having established both relevant mathematical theory and a concrete implementation, there still remains significant scope for future developments of UMAP.
A comprehensive empirical study which examines the impact of the various algorithmic components, choices, and hyper-parameters of the algorithm would be beneficial. While the structure and choices of
the algorithm presented were derived from our foundational mathematical framework, examining the impacts that these choices have on practical results would be enlightening and a significant
contribution to the literature.
As noted in the weaknesses section there is a great deal of uncertainty surrounding the preservation of global structure among the field of manifold learning algorithms. In particular this is
hampered by the lack clear objective measures, or even definitions, of global structure preservation. While some metrics exist, they are not comprehensive, and are often specific to various
downstream tasks. A systematic study of both metrics of non-local and global structure preservation, and performance of various manifold learning algorithms with respect to them, would be of great
benefit. We believe this would aid in better understanding UMAP’s success in various downstream tasks.
Making use of the fuzzy simplicial set representation of data UMAP can potentially be extended to support (semi-)supervised dimension reduction, and dimension reduction for datasets with
heterogeneous data types. Each data type (or prediction variables in the supervised case) can be seen as an alternative view of the underlying structure, each with a different associated metric – for
example categorical data may use Jaccard or Dice distance, while ordinal data might use Manhattan distance. Each view and metric can be used to independently generate fuzzy simplicial sets, which can
then be intersected together to create a single fuzzy simplicial set for embedding. Extending UMAP to work with mixed data types would vastly increase the range of datasets to which it can be
applied. Use cases for (semi-)supervised dimension reduction include semi-supervised clustering, and interactive labelling tools.
The computational framework established for UMAP allows for the potential development of techniques to add new unseen data points into an existing embedding, and to generate high dimensional
representations of arbitrary points in the embedded space. Furthermore, the combination of supervision and the addition of new samples to an existing embedding provides avenues for metric learning.
The addition of new samples to an existing embedding would allow UMAP to be used as a feature engineering tool as part of a general machine learning pipeline for either clustering or classification
tasks. Pulling points back to the original high dimensional space from the embedded space would potentially allow UMAP to be used as a generative model similar to some use cases for autoencoders.
Finally, there are many use cases for metric learning; see [64] or [8] for example.
There also remains significant scope to develop techniques to both detect and mitigate against potentially spurious embeddings, particularly for small data cases. The addition of such techniques
would make UMAP far more robust as a tool for exploratory data analysis, a common use case when reducing to two dimensions for visualization purposes.
Experimental versions of some of this work are already available in the referenced implementations.
8 Conclusions
We have developed a general purpose dimension reduction technique that is grounded in strong mathematical foundations. The algorithm implementing this technique is demonstrably faster than t-SNE and
provides better scaling. This allows us to generate high quality embeddings of larger data sets than had previously been attainable. The use and effectiveness of UMAP in various scientific fields
demonstrates the strength of the algorithm.
The authors would like to thank Colin Weir, Rick Jardine, Brendan Fong, David Spivak and Dmitry Kobak for discussion and useful commentary on various drafts of this paper.
Appendix AProof of Lemma 1
Lemma 1.
Let (ℳ,g) be a Riemannian manifold in an ambient ℝn, and let p∈M be a point. If g is locally constant about p in an open neighbourhood U such that g is a constant diagonal matrix in ambient
coordinates, then in a ball B⊆U centered at p with volume πn/2Γ(n/2+1) with respect to g, the geodesic distance from p to any point q∈B is 1rdℝn(p,q), where r is the radius of the ball in the
ambient space and dℝn is the existing metric on the ambient space.
Let x1,…,xn be the coordinate system for the ambient space. A ball B in ℳ under Riemannian metric g has volume given by
If B is contained in U, then g is constant in B and hence det(g) is constant and can be brought outside the integral. Thus, the volume of B is
where r is the radius of the ball in the ambient ℝn. If we fix the volume of the ball to be πn/2Γ(n/2+1) we arrive at the requirement that
Now, since g is assumed to be diagonal with constant entries we can solve for g itself as
gij={1r2 if i=j,0 otherwise. (2)
The geodesic distance on ℳ under g from p to q (where p,q∈B) is defined as
where C is the class of smooth curves c on ℳ such that c(a)=p and c(b)=q, and c˙ denotes the first derivative of c on ℳ. Given that g is as defined in (2) we see that this can be simplified to
1rinfc∈C∫ab⟨c˙(t),c˙(t)⟩dt=1rinfc∈C∫ab⟨∥c˙(t),c˙(t)∥dt=1rdℝn(p,q). (3)
Appendix BProof that 𝖥𝗂𝗇𝖱𝖾𝖺𝗅 and 𝖥𝗂𝗇𝖲𝗂𝗇𝗀 are adjoint
Theorem 2.
The functors 𝖥𝗂𝗇𝖱𝖾𝖺𝗅:Fin-sFuzz→𝐅𝐢𝐧𝐄𝐏𝐌𝐞𝐭 and 𝖥𝗂𝗇𝖲𝗂𝗇𝗀:𝐅𝐢𝐧𝐄𝐏𝐌𝐞𝐭→Fin-sFuzz form an adjunction with 𝖥𝗂𝗇𝖱𝖾𝖺𝗅 the left adjoint and 𝖥𝗂𝗇𝖲𝗂𝗇𝗀 the right adjoint.
The adjunction is evident by construction, but can be made more explicit as follows. Define a functor F:𝚫×I→𝐅𝐢𝐧𝐄𝐏𝐌𝐞𝐭 by
da(xi,xj)={−log(a)if i≠j,0otherwise.
Now 𝖥𝗂𝗇𝖲𝗂𝗇𝗀 can be defined in terms of F as
where the face maps di are given by pre-composition with Fdi, and similarly for degeneracy maps, at any given value of a. Furthermore post-composition with F level-wise for each a defines maps of
fuzzy simplicial sets making 𝖥𝗂𝗇𝖲𝗂𝗇𝗀 a functor.
We now construct 𝖥𝗂𝗇𝖱𝖾𝖺𝗅 as the left Kan extension of F along the Yoneda embedding:
Explicitly this results in a definition of 𝖥𝗂𝗇𝖱𝖾𝖺𝗅 at a fuzzy simplicial set X as a colimit:
Further, it follows from the Yoneda lemma that 𝖥𝗂𝗇𝖱𝖾𝖺𝗅(Δ<an)≅F([n],[0,a)), and hence this definition as a left Kan extension agrees with Definition 7, and the definition of 𝖥𝗂𝗇𝖲𝗂𝗇𝗀 above agrees
with that of Definition 8. To see that 𝖥𝗂𝗇𝖱𝖾𝖺𝗅 and 𝖥𝗂𝗇𝖲𝗂𝗇𝗀 are adjoint we note that
homFin-sFuzz(Δ<an,𝖥𝗂𝗇𝖲𝗂𝗇𝗀(Y))≅𝖥𝗂𝗇𝖲𝗂𝗇𝗀(Y)<an=hom𝐅𝐢𝐧𝐄𝐏𝐌𝐞𝐭(F([n],[0,a)),Y)≅hom𝐅𝐢𝐧𝐄𝐏𝐌𝐞𝐭(𝖥𝗂𝗇𝖱𝖾𝖺𝗅(Δ<an),Y)). (4)
The first isomorphism follows from the Yoneda lemma, the equality is by construction, and the final isomorphism follows by another application of the Yoneda lemma. Since every simplicial set can be
canonically expressed as a colimit of standard simplices and 𝖥𝗂𝗇𝖱𝖾𝖺𝗅 commutes with colimits (as it was defined via a colimit formula), it follows that 𝖥𝗂𝗇𝖱𝖾𝖺𝗅 is completely determined by its image on
standard simplices. As a result the isomorphism of equation (4) extends to the required isomorphism demonstrating the adjunction. ∎
Appendix CFrom t-SNE to UMAP
As an aid to implementation of UMAP and to illuminate the algorithmic similarities with t-SNE and LargeVis, here we review the main equations used in those methods, and then present the equivalent
UMAP expressions in a notation which may be more familiar to users of those other methods.
In what follows we are concerned with defining similarities between two objects i and j in the high dimensional input space X and low dimensional embedded space Y. These are normalized and
symmetrized in various ways. In a typical implementation, these pair-wise quantities are stored and manipulated as (potentially sparse) matrices. Quantities with the subscript ij are symmetric, i.e.
vij=vji. Extending the conditional probability notation used in t-SNE, j∣i indicates an asymmetric similarity, i.e. vj∣i≠vi∣j.
t-SNE defines input probabilities in three stages. First, for each pair of points, i and j, in X, a pair-wise similarity, vij, is calculated, Gaussian with respect to the Euclidean distance between
xi and xj:
vj∣i=exp(−∥xi−xj∥22/2σi2) (5)
where σi2 is the variance of the Gaussian.
Second, the similarities are converted into N conditional probability distributions by normalization:
σi is chosen by searching for a value such that the perplexity of the probability distribution p⋅∣i matches a user-specified value.
Third, these probability distributions are symmetrized and then further normalized over the entire matrix of values to give a joint probability distribution:
We note that this is a heuristic definition and not in accordance with standard relationship between conditional and joint probabilities that would be expected under probability semantics usually
used to describe t-SNE.
Similarities between pairs of points in the output space Y are defined using a Student t-distribution with one degree of freedom on the squared Euclidean distance:
followed by the matrix-wise normalization, to form qij:
The t-SNE cost is the Kullback-Leibler divergence between the two probability distributions:
Ct−SNE=∑i≠jpijlogpijqij (10)
this can be expanded into constant and non-constant contributions:
Ct−SNE=∑i≠jpijlogpij−pijlogqij (11)
Because both pij and qij require calculations over all pairs of points, improving the efficiency of t-SNE algorithms has involved separate strategies for approximating these quantities.
Similarities in the high dimensions are effectively zero outside of the nearest neighbors of each point due to the calibration of the pj∣i values to reproduce a desired perplexity. Therefore an
approximation used in Barnes-Hut t-SNE is to only calculate vj∣i for n nearest neighbors of i, where n is a multiple of the user-selected perplexity and to assume vj∣i=0 for all other j. Because the
low dimensional coordinates change with each iteration, a different approach is used to approximate qij. In Barnes-Hut t-SNE and related methods this usually involves grouping together points whose
contributions can be approximated as a single point.
A further heuristic algorithm optimization technique employed by t-SNE implementations is the use of early exaggeration where, for some number of initial iterations, the pij are multiplied by some
constant greater than 1.0 (usually 12.0). In theoretical analyses of t-SNE such as [38] results are obtained only under an early exaggeration regimen with either large constant (of order of the
number of samples), or in the limit of infinite exaggeration. Further papers such as [37], and [28], suggest the option of using exaggeration for all iterations rather than just early ones, and
demonstrate the utility of this. The effectiveness of these analyses and practical approaches suggests that KL-divergence as a measure between probability distributions is not what makes the t-SNE
algorithm work, since, under exaggeration, the pij are manifestly not a probability distribution. This is another example of the probability semantics used to describe t-SNE are primarily
descriptive rather than foundational. None the less, t-SNE is highly effective and clearly produces useful results on a very wide variety of tasks.
LargeVis uses a similar approach to Barnes-Hut t-SNE when approximating pij, but further improves efficiency by only requiring approximate nearest neighbors for each point. For the low dimensional
coordinates, it abandons normalization of wij entirely. Rather than use the Kullback-Leibler divergence, it optimizes a likelihood function, and hence is maximized, not minimized:
CLV=∑i≠jpijlogwij+γ∑i≠jlog(1−wij) (12)
pij and wij are defined as in Barnes-Hut t-SNE (apart from the use of approximate nearest neighbors for pij, and the fact that, in implementation, LargeVis does not normalize the pij by N) and γ
is a user-chosen positive constant which weights the strength of the the repulsive contributions (second term) relative to the attractive contribution (first term). Note also that the first term
resembles the optimizable part of the Kullback-Leibler divergence but using wij instead of qij. Abandoning calculation of qij is a crucial change, because the LargeVis cost function is amenable to
optimization via stochastic gradient descent.
Ignoring specific definitions of vij and wij, the UMAP cost function, the cross entropy, is:
CUMAP=∑i≠jvijlog(vijwij)+(1−vij)log(1−vij1−wij) (13)
Like the Kullback-Leibler divergence, this can be arranged into two constant contributions (those containing vij only) and two optimizable contributions (containing wij):
CUMAP=∑i≠jvijlogvij+(1−vij)log(1−vij) (14)
Ignoring the two constant terms, the UMAP cost function has a very similar form to that of LargeVis, but without a γ term to weight the repulsive component of the cost function, and without requiring
matrix-wise normalization in the high dimensional space. The cost function for UMAP can therefore be optimized (in this case, minimized) with stochastic gradient descent in the same way as LargeVis.
Although the above discussion places UMAP in the same family of methods as t-SNE and LargeVis, it does not use the same definitions for vij and wij. Using the notation established above, we now
provide the equivalent expressions for the UMAP similarities. In the high dimensional space, the similarities vj∣i are the local fuzzy simplicial set memberships, based on the smooth nearest
neighbors distances:
vj∣i=exp[(−d(xi,xj)−ρi)/σi] (15)
As with LargeVis, vj∣i is calculated only for n approximate nearest neighbors and vj∣i=0 for all other j. d(xi,xj) is the distance between xi and xj, which UMAP does not require to be Euclidean. ρi
is the distance to the nearest neighbor of i. σi is the normalizing factor, which is chosen by Algorithm 3 and plays a similar role to the perplexity-based calibration of σi in t-SNE. Calculation of
vj∣i with Equation 15 corresponds to Algorithm 2.
Symmetrization is carried out by fuzzy set union using the probabilistic t-conorm and can be expressed as:
vij=(vj∣i+vi∣j)−vj∣ivi∣j (16)
Equation 16 corresponds to forming top-rep in Algorithm 1. Unlike t-SNE, further normalization is not carried out.
The low dimensional similarities are given by:
wij=(1+a∥yi−yj∥22b)−1 (17)
where a and b are user-defined positive values. The procedure for finding them is given in Definition 11. Use of this procedure with the default values in the UMAP implementation results in a≈1.929
and b≈0.7915. Setting a=1 and b=1 results in the Student t-distribution used in t-SNE.
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This product should be stored at -25~-15℃ for 1 years. | {"url":"https://www.seekquence.com/umap-calculation","timestamp":"2024-11-13T11:37:01Z","content_type":"text/html","content_length":"190052","record_id":"<urn:uuid:4b7379c8-c29c-4508-bedc-875d5b42b31f>","cc-path":"CC-MAIN-2024-46/segments/1730477028347.28/warc/CC-MAIN-20241113103539-20241113133539-00561.warc.gz"} |
DNP Scaling 101
PUBLISHED ON Jun 28, 2010
Choosing the Right Scaling Factors for Bitronics IEDs
Q: My SCADA software gives me one field to enter a scale factor and a field to enter a zero offset for converting each DNP analog point into primary engineering units. The Bitronics^®configuration
software asks me to enter a CT scale factor and a VT scale factor, but it doesn’t mention anything about what scaling to use for power, frequency, or other kinds of points besides the current and
voltage. How do I choose the best scale factors to enter in the Bitronics software? Then what should I enter in SCADA?
A: The settings for Modbus and DNP scaling on the 70 Series IED are designed to make most analog points work out for an optimum balance between range and resolution if you enter the same value for
the CT and VT Scale Factors that you use for the CT and VT ratio settings. That is, if the current and voltage are optimized, then the watts, VArs and VA will also work out. Some measurements, like
power factor and frequency do not depend on the CT and VT Scale Factors at all, and are handled a little differently. And some measurements, like line-to-line volts may require you to choose a
different value for the VT scaling factor than you use for the VT ratio under certain circumstances.
To keep this from getting confusing, let’s break the answer into three parts. In the first part, I’ll describe the typical case where the scaling factors are the same as the instrument transformer
ratios. In the next issue of TechTalk we’ll handle exceptional cases, where you might want to use a Scale Factor that’s different from the transformer ratios. Then in the final part, we’ll describe
points where the scaling factor doesn’t have anything to do with the CT and VT ratios. At the conclusion of the three-part series, we’ll concatenate all three parts into a single white paper that
covers all scaling issues comprehensively. I’ll even throw in a primer on binary integer math and the significance of “two’s-complement” encoding.
DNP Scaling, Part 1:
The Typical Case
The 70 Series Configurator software has one page (Figure 1, top) to enter the CT and VT ratio settings. There is another page (Figure 1, bottom) for each protocol where the CT and VT Scale Factors
are entered. That way Modbus and DNP protocols can be scaled independently. For the IED to work properly, the CT and VT Ratio settings must always be set equal to the actual turns ratios of the
instrument transformers connected to the voltage and current terminals of the IED. It is never necessary to “trick the meter” by multiplying a ratio times root-three, or any other value under any
circumstances. But the CT and VT Scale Factor settings can be manipulated to adjust the range and the resolution of various measurements according to the unique circumstances that may arise in a
Figure 1
In the typical case, you would enter a CT Scale Factor equal to the CT Ratio setting and the VT Scale Factor equal to the VT Ratio setting. There is an equation on the DNP Points page (Figure 2)
that can be used to calculate the scale factor you should use in SCADA.
For most analog points, such as amps, volts, watts and VArs, the equation has the general form:
PEU = RV/32,768 * FSV * SF
• PEU is the primary engineering units. This is the “answer” you’re generally looking for.
• RV is the raw DNP integer value (“Register Value” shown in figure 2).
• FSV defines the full scale value of the parameter being measured in secondary units. (The “10” shown in the equation in Figure 2 is 10A on the low side of the CTs. That is, 5A full-scale times
a transformer rating factor of 2.)
• SF is the scaling factor (the CT Scale Factor in Figure 2).
Figure 2
Notice a couple things about the equation shown in Figure 2:
1. The equation is dynamic. That is, the equation changes when you click on different DNP points in the point list above. So the FSVterm may be 10 for current, but 150 for voltage, and 4500 for
power. The SF term is the CT scale factor for current, the VT scale factor for voltage, and for power, SF if the CT scale factor times the VT scale factor.
2. Notice the equation is a direct proportion. That is, it takes the form Y = m X. There is no zero offset (such as would be in the form Y = m X + b). So the zero offset (b) is zero, and the scale
factor that you enter into SCADA (m) is given by:
m = FSV * SF / 32,768
So, using the term “m” (above) for the SCADA scale factor, the equation in the Configurator program becomes: PEU = RV * m, which is exactly what SCADA expects to see.
3. This equation will produce primary engineering units having the best possible resolution for a scaled 16-bit integer as long as the product of FSV * SF is close to the maximum value you ever
expect the engineering units to have.
4. The IED can never produce a measurement higher than FSV * SF, because RV can never be greater than 32,768.
Let’s work an example:
Say you’ve got a typical 13kV distribution substation station. Your PT ratio is 70:1, and your CT ratio is 800:5. The system is connected in a wye. On a particular day the line-neutral voltage is
7,765V and the current is 670A. Let’s say the system is well balanced and the power factor is 0.97 lagging so total watts, P = 15.75MW and total VArs, Q = 3.93MVAr. In SCADA you want to read the
line-neutral voltage, the current, watts and VArs. How do you set up the Scale factors in the 70 Series IED and in your SCADA software?
1. On the Ratios page in the 70 Series Configurator program, set the CT ratio to 800 on all three phases making sure the radio button indicates 800 means 800:5 and not 800:1. Set the VT ratio on all
three phases to 70. See Figure 1, top.
2. On the DNP Scaling Factors page in the 70 Series Configurator program, set the CT Scale factor to 160 (800/5=160 and the scale factor setting must be reduced). Set the VT scale factor to 70. See
Figure 1, bottom.
3. Go to the DNP Points page in the Configurator and click on any of the points that are configured to represent amps (AI:2, 3, or 4 shown in Figure 2). Look at the equation in the lower right
corner of the page. The equation tells you that the primary current, I = RV/32768 * 10 * 160. (See how the Scale factor is identified as 160 immediately below the box?) The scale factor to use in
SCADA is: 10 * 160 / 32768 = 0.048828. In other words, SCADA polls the IED, which reports point AI:2 is a raw DNP Integer of 13,722. So SCADA needs to multiply 13,722 counts by the scale factor
0.048828 to get 670A. If you prefer, and your SCADA software supports it, you could take the reciprocal of 0.048828 (which is 20.48) and set your SCADA software to divide the raw integer 13,722 by
20.48. You would get the same results, 670A. That’s it for Amps.
4. Volts is similar to Amps with just a couple slight differences. When you click on the point AI:7 in the point list in the Configurator, the equation changes to RV/32768 * 150 * 70, where 150 is
FSV and 70 is the VT scale factor. You calculate your SCADA scale factor as 150 * 70 / 32768 = 0.320435. Now when SCADA reads the IED, it reports AI:7 is 24,233 counts. When SCADA multiplies
24,233 by 0.320435, you calculate the voltage as 7,767V which is right, but you probably prefer SCADA to display voltage in terms of kV. To do that just requires you divide the SCADA scale factor
for voltage by 1000. So the scale factor becomes 0.000320. Again, it is probably more convenient to take the reciprocal of the scale factor and set up SCADA to divide the raw DNP integer by
3,120.76 rather than to multiply by such a very small number.
5. The SCADA scale factors used for the Watts and VArs are both the same. In this case, the equation in the Configurator changes the FSV to 4500 and the scale factor is now the product of the CT
scale factor times the VT scale factor. When you’ve done that, the engineering units will be presented in terms of watts, and you probably prefer SCADA to read in terms of MW, so you will have to
divide the scale factor that you calculate by one million, just as we divided the scale factor for voltage by 1000 to get it to be expressed in terms of kV in number 4, above. For Watts, refer to
Figure 3.
Clicking on point AI:24 in the Configurator makes the equation change to: RV/32,768 * 4,500 * 11,200(because 11,200 = 70 * 160.) So the scaling factor you use for SCADA is 4,500 * 11,200 / 32,768
which is 1,538.09. You want the result in MW, so next divide that by one million to get 0.001538. That’s a very small number, so taking the reciprocal gives you 650.159 . Now just set up your
SCADA to divide the raw DNP integer by 650.159 to get primary MW or MVAr. Polling the IED produces the integer 10,238 for watts (AI:24) and 2,553 for VArs (AI:28). So 10,238 / 650.159 = 15.75MW,
and 2,553 /650.159 = 3.93MVAr, which is the answer.
Figure 3
In the next installment in this series, we’ll deal with situations when you want the CT or VT Scale factor to be different from the corresponding CT or VT Ratio.
DNP Scaling, Part 2:
Exceptional Cases, where you might prefer to choose a Scale Factor that’s different from the CT and VT ratios.
There are two main reasons to select a Scale Factor for DNP that is different from the CT ratio or VT ratio settings, and one consequence to bear in mind when you do so. The reasons:
1. To increase the range of the analog scale.
2. To improve the analog resolution of the integer used by DNP.
Once you select scale factors for current and voltage, you should bear in mind the effect that change has on the scaling of measurements like power, which are dependent on both the CT and the VT
scale factors.
Here’s an example where the range of the analog scale must be increased:
Suppose you have three VTs terminated in a wye-orientation producing 115V from line to neutral. That will result in about 200V from line to line. In this application, the 70 Series IED can be used
to measure both the line-neutral voltage and the line-line voltage. But in SCADA, using a scale where the DNP integer 32,768 corresponds to 150V (or 150V * VT Ratio, in primary engineering units)
you would expect to see 115V produce a DNP integer of about 25,122 regardless of the VT ratio (that’s 32,768 * 115 / 150). The problem occurs when you try to represent the line-line voltage on the
same scale. That is, if 150V is 32,768, then what integer corresponds to 200V? The result is an over-range condition.
In this case two approaches are possible. You could change the Calc Type in the Configurator DNP Points page so 32,768 corresponds to a higher voltage than 150, or you could keep the same Calc Type
and select a VT Scale Factor that changes the equation used by the IED to determine the DNP integer in the first place.
To increase the range of the analog scale:
In the 70 Series Configurator program, on the DNP Points page (see Figure 4, below) the Calc Type for each point can be selected from a pull-down menu. The pull-down menu for most points offers
several options for the definition of the full scale. In Figure 4, the line-neutral voltage points are defined to have the default 150V full scale. The three line-line voltages have been changed so
the full scale is 600V. So when points AI:3, AI:4, and AI:5 in Figure 4 are read by the RTU, now the DNP integer will be scaled in such a way that 32,768 corresponds to 600V at the terminals of the
IED (or 600V * VT Ratio, in primary engineering units). In the example above, now you would expect to see 200V produce a DNP integer of about 10,923 regardless of the VT ratio (that’s 32,768 * 200 /
600). As you see, this can be done without making the VT Scale Factor different from the VT Ratio, and without having any impact on the scaling of the line-neutral voltage in AI:0, AI:1 and AI:2 or
on any other measurement made by the IED. The only down-side is that the resolution is reduced somewhat. That’s because, in this application the voltage is unlikely to ever go very much higher than
200V, so the finest resolution that can be represented by DNP for line-line voltage is no longer 1 count out of 32,768. It is now closer to 1 count out of about 11,000. Practically, that is still
pretty good so it may very well be the preferred approach in many if not all circumstances.
Figure 4
The objective of the second approach is to represent voltage greater than 150V at the terminals of the IED with the maximum possible resolution. That can be done by selecting a VT Scale Factor that
is larger than the VT Ratio setting. By doing that, you change the equation used by the IED to determine what DNP integer results from the voltage that appears at the terminals. Here’s an example:
Suppose, you expect to see 200V line-line at the terminals, as in the preceding example. Start by setting the maximum voltage you ever expect to be able to read with SCADA equal to the DNP integer
32,768. With some knowledge of the system you ought to be able to come up with a reasonable estimate. Say the line-neutral voltage it typically 115V, but that’s regulated, so you know it never
exceeds 130V. Then the line-line voltage would be 225V when the line-neutral voltage is 130V. So you might feel safe letting 230V correspond to 32,768 in DNP. Now use the equation introduced in
part 1 of this article to calculate what the scale factor needs to be to make 230V produce 32,768 counts:
In Part 1 of this article, we established the basic scaling equation as: PEU = RV/32,768 * FSV * SF
• PEU is the primary engineering units (PEU is the voltage at the meter terminals times the VT ratio).
• RV is the raw DNP integer value (32,768 in this case).
• FSV defines the full scale value of the parameter being measured in secondary units (150V in this case).
• SF is the scaling factor.
Recognizing that primary engineering units (PEU) is just the voltage at the terminals times the VT Ratio setting, the equation becomes: VTR * V = RV/32,768 * FSV * SF (where V is the voltage at the
terminals and VTR is the VT Ratio). Solving that equation for SF, where all the other values are known gives:
SF = VTR * (32,768 * V) / (RV * FSV)
= VTR * (32,768 * 230) / (32,768 * 150)
= VTR * 230/150
= 2 * VTR
So, as you might expect, making the VT Scale Factor two times the VT Ratio, effectively increases the full scale voltage by a factor of 2. In other words, when the IED calculates the DNP integer,
the result is half of what it would have been if the VT Scale Factor were set equal to the VT Ratio. As a result the analog range is twice as wide.
The other reason mentioned why you might want to choose a Scale Factor that’s different from the CT or VT Ratio was to improve the analog resolution of the integer produced by DNP. In other words,
suppose you have a line that carries 400A at peak load, but because of requirements of the protection scheme unrelated to SCADA, these CTs happen to have a 2000:5 ratio. You have no choice but to
configure the CT Ratio setting in the IED to 2000:5 (or 400:1) but if you make the CT Scale Factor setting 400 (400:1) on the DNP Scale Factors page in the Configurator, then the DNP integer 32,768
corresponds to 10A at the terminals, or 4000A primary engineering units. So your peak load of 400A only produces a DNP integer of 3,277. Essentially, your resolution has been cut by a factor of
To improve the resolution, you could select a different Calc Type as shown in Figure 4 (but this time configuring the point for the Amps). Where that was probably the preferable approach for
voltage, it is less advantageous for current. First, the only Calc Type available for current that is lower than the 10A scale is the 5A scale (see Figure 5). That only improves resolution by a
factor of two, where we’d really prefer to get something closer to a factor of ten improvement. The other drawback of that approach is that your load actually is one tenth of what is normally
expected by the IED, which was not the case in the preceding (voltage) example. As a result, the load in Watts, VARs and VA is also lower by a factor of ten than what is anticipated by the IED so
the resolution of those measurements is also impacted.
Figure 5
In this example, the preferable approach is making the CT Scale Factor smaller than the CT Ratio. Then the change in the configuration of current scaling will automatically apply to improving the
resolution of the power measurements as well. So, how to choose a CT Scale Factor?
Taking the same approach as in the previous example, first you want to estimate the amount of current that you would like to produce a DNP integer of 32,768. In our example we said the peak current
is about 400A. Current is not regulated like voltage, so it’s not obvious what amount of current will never be exceeded, in order to define that as the full scale. A good rule of thumb is to leave
enough room for the load to grow by a factor of two over time. Meters and CTs usually do that automatically by defining a nominal full scale output at 5A, but then building a rating factor of 2 into
the CT and making the meter operate through 100% over-range. In our example, the CT full scale is no help because the CT is over-sized to accommodate the relays. So we’ll take that into
consideration when selecting the CT Scale Factor. OK, so the known peak load is 400A, and we’ve decided to accommodate load growth up to 800A. So we’ll make 800A correspond to a DNP integer of
The equation we use is: PEU = RV/32,768 * FSV * SF
• PEU is the primary engineering units (PEU is 800A in this case).
• RV is the raw DNP integer value (32,768 in this case).
• FSV defines the full scale value of the parameter being measured in secondary units (10A in the typical case).
• SF is the scaling factor, which is we want to solve for.
Solving the equation for SF, where all the other values are known gives:
SF = (32,768 * PEU) / (RV * FSV)
= (32,768 * 800) / (32,768 * 10)
= 800/10
= 80
OK, so 80 is the CT Scale Factor. What does that do for me?
1. When reading Amps from the IED: In our example 400A is peak, so on a certain day maybe you’ve got a load of 250A. The CT in the station has a 2000:5 ratio, so the IED sees 0.625A at the
terminals and your RTU reads 10,240 counts. SCADA should be configured to interpret that as follows:
PEU = RV/32,768 * FSV * SF
• PEU is the answer you’re looking for.
• RV is the raw DNP integer value you read from the IED (10,240 in this case).
• FSV defines the full scale value of the parameter being measured in secondary units (10A in the typical case).
• SF is the scaling factor, which we just determined should be 80.
PEU = RV/32,768 * FSV * SF
= RV/32,768 * 10 * 80
= RV/40.96 ← This is the equation you set in SCADA.
= 10,240/40.96
= 250A
2. In the beginning of Part 2 of this article we said: Once you select scale factors for current and voltage, you should bear in mind the effect that change has on the scaling of measurements
like power, which is dependent on both the CT and the VT scale factors. So then what impact do the scaling factors we chose have on reading the Power?
Using the examples we’ve worked so far, let’s look at the big picture:
In the first example, we decided that we could make the VT Scale Factor be 2 times the VT Ratio in order to see the line-line voltage without over-ranging in DNP. But we also established that it
would probably be preferable not to monkey with the Scale Factor if we could just choose a different Calc Type. We didn’t say what the VT Ratio was. Let’s say that’s 70:1 and since the VT Scale
Factor will be the same as the VT ratio, the Scale Factor is 70 also but we’ll use a Calc Type that defines the full scale line-line voltage as 600V.
In the second example, we decided to make the CT Scale factor 80 even though the CT Ratio is 400 (2000:5 = 400:1) to improve the resolution of the current where the load is actually less than the
typical load anticipated by the IED.
So the settings on the IED are this:
VT Ratio 70:1 VT Scale Factor 70
CT Ratio 2000:5 CT Scale Factor 80
Let’s say the system voltage is 13.8kV line-line, and the load is 350A. For the purposes of this example, we’ll let the system be balanced and the power factor be unity (if Watts = VA and VARs = 0
we can work just one example instead of three because the formulas for scaling P, Q, and S in DNP are identical.) That makes P = 8.366MW. The IED is producing the following integers: For line-line
voltage: 10767, for line-neutral voltage: 24864, for current: 14336, and for three-phase total power: 10878 counts. The question at hand is how should SCADA be set up to interpret all of these
measurements in Primary Engineering Units?
The equation we’ve been using all along is still appropriate for all these cases:
PEU = RV/32,768 * FSV * SF
Where PEU is the answer, and RV is the integer we read from the IED in DNP. The only trick is to recognize which value to use for FSV and SF based on the settings that we chose. This is where the
Configurator program comes in very handy. See Figure 6. When you click on a DNP Point in the point list, the formula in the lower right-hand side updates to give you exactly the right values for
the Full Scale Value (FSV) and the Scaling Factor (SF) that the IED is configured for. So we’ll just use those:
Figure 6
Figure 6 indicates, for line-line volts, FSV = 600 and SF = 70. So we have:
PEU = RV/32,768 * 600 * 70
= RV * 1.2817
= 10767 * 1.2817
= 13,800V
To get the results in units of kV, the equation RV * 1.2817 in SCADA becomes RV * 1.2817 / 1000 or RV * 0.0012817, which is the same as the following equation:
In SCADA: V[LL] = RV/780.2
Figure 7
Figure 7 indicates, for line-neutral volts, FSV = 150 and SF = 70. So we have:
PEU = RV/32,768 * 150 * 70
= RV * 0.3204
= 24864 * 0.3204
= 7,967V
To get the results in units of kV, the equation RV * 0.3204 in SCADA becomes RV * 0.3204 / 1000 or RV * 0.0003204, which is the same as: V[LN] = RV/3120.8 which is what we use for SCADA.
Figure 8
Figure 8 indicates, for current, FSV = 10 and SF = 80. So we have:
PEU = RV/32,768 * 10 * 80
= RV * 0.024414
= RV / 40.96 ← This is the equation you set in SCADA.
= 14336 / 40.96
= 350A
Figure 9
Figure 9 indicates that, for power, FSV = 4500 and the Scale Factor is the CT Scale Factor times the VT Scale Factor, so SF = 70 * 80 = 5600.
PEU = RV/32,768 * 4500 * 5600
= RV * 769.043
We’ll go ahead and divide 769.043 by 1,000,000 so the result comes out in MW instead of watts.
= RV / 1300.3
= 10878 / 1300.3
= 8.366MW
DNP Scaling, Part 3:
Exceptional Cases, where a Scale Factor is not necessary.
This topic is relatively straight forward compared to the previous two topics. Essentially, there are some measurements where the magnitude is not a function of the CT and VT scaling. Some examples
• Power Factor always varies from -1 to +1 regardless of current and voltage magnitude.
• Values expressed as a percentage, such as %THD range only from 0 to 1.
• Phase angles magnitude, generally ranges from -180 to +180 degrees.
• Frequency is not dependent on voltage or current magnitude and is generally regulated so tightly around either 60 Hz or 50 Hz, that maximum resolution can be achieved by using an offset equation
to convert from DNP counts to Primary Engineering Units.
• Any value whose content represents “packed bits” (such as the IED’s self diagnostic, or Health, point) is obviously is not scaled. Each bit represents a different binary condition. A look-up
table is required to determine the meaning of each bit once the point has been parsed.
• Energy in the 70 Series IED is generally represented as an unsigned 32 bit integer without scaling, and expressed in units of kWh. The energy registers generally roll over at 4,294,967,295 (i.e.
2^32) kWh. Scaling is only required if you prefer to read energy in units of MWh, in which case scaling is just divide-by-1000, and then the register rolls over at 4,294,967.295MWh. (For what
it’s worth, that’s roughly 245 years running continuously at 2,000MW.)
In each of the above cases the scaling is usually simply divide-by-10, divide-by-100, or 1000. Are all indicated in the Configurator program when the Point List is configured (see Figure 10) and
also described in the Users’ Guide for DNP Protocol implementation on the 70 Series IED, Bitronics document ML0026, a copy of which is included on the Utilities Software CD that comes with the IED,
and also available for download from the NovaTech Automation website.
Figure 10
Figure 10, above, illustrates the offset-proportion scaling used for frequency. See how the Register Value only encodes the deviation of the frequency from nominal (60 or 50Hz). So, for example
59.975Hz would be encoded as the DNP integer -25 counts. The RTU should divide -25 by 1000 and add that to 60 (we hopeyou never see 59.975 Hz in a 50 Hz nominal system). So 60 – 0.025 = 59.975.
This method produces sufficient resolution to represent frequency to +/- 1mHz.
DNP Scaling, Part 4:
Binary math primer, the significance of “Two’s Complement”.
We occasionally get asked why SCADA might report a measurement from an IED where the integer is greater than 32,768. So I thought it might be interesting to review a couple fundamentals of binary
math inasmuch as they apply to SCADA.
In the most fundamental form of binary math, you’ve only got ones and zeros; no minus sign is available. Negative numbers are often represented by making the most significant digit a one. Let’s use
four-bit binary for examples because 16-bit is a little tedious. Insigned binary you can count from zero (0000) up to seven (0111) but if you make the left-most digit a one (1000) that makes it a
negative number, not a larger positive number. That’s by convention of course; it doesn’t have to be that way. If you never expect to need to represent a negative number, then there’s nothing wrong
with saying 1000 is binary 8, and continue counting up to fifteen (1111).
So the first thing we need to establish is that most RTUs need to be configured to recognize which binary numbering convention they are using. An unsigned sixteen-bit integer is not the same as a
sixteen-bit-two’s-complement integer. If an IED is using two’s complement encoding, which is a form of signed binary, then the highest positive number it can see is a zero followed by fifteen ones.
In decimal form, that is: 0 x 2^15 + 1 x 2^14 + … + 1 x 2^1 + 1 x 2^0 = 32,767.
Now. It is possible to define a system where the left-most digit (most significant digit) is just the sign, so 0001 = 1 and 1001 = -1. But computers generally don’t use that kind of system because
you’d like to see a positive number and a negative number of the same magnitude add up to zero. The system just described doesn’t support that because 0001 + 1001 = 1010, which would be -2, not
This is where the “two’s complement” convention comes into play. Suppose we define a convention where, in order to make a positive binary number negative, we invert all the digits then add one. It
sounds kind of arbitrary, but see how it works: If 5 is 0101, then in order to make -5, inverting all the digits produces 1010, then adding a one results in 1010 + 0001 = 1011. Hence, -5 is 1011.
So adding +5 to -5 should result in 0 as follows: 0101 + 1011 = 0000, which is true given that we are only considering four bits in this example. In this convention, any number whose most
significant digit is a 1 is a negative number. For example:
1000[2] = -8[10]
1001[2 ]= -7[10]
1010[2] = -6[10]
1011[2] = -5[10]
1100[2] = -4[10]
…and so forth.
Now, obviously, if an RTU were not configured to recognize that the numbers used in the example above are encoded according to the two’s complement convention, then the RTU would not interpret 1011
as negative five, it would interpret 1011 as positive eleven. Just as eleven should not be possible in four-bit-two’s complement (because 0111[2] = 7[10] is the highest number that’s possible
without the most significant digit becoming a 1). This is analogous to the situation when the RTU returns a number between 32,768 and 65,535. You don’t expect it to be possible for a number to be
greater than 32,767. All numbers in the range from 32,768 to 65,535 have 1 as their most significant digit, so those should all be interpreted by SCADA as negative numbers. That is usually just a
setting that needs to be made when setting the system up. | {"url":"https://www.novatechautomation.com/news/novatech-news-bryan-dnp-scaling-101","timestamp":"2024-11-04T01:20:50Z","content_type":"text/html","content_length":"235904","record_id":"<urn:uuid:f2b9888e-7ba1-401f-be02-b6db9c02bb50>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00739.warc.gz"} |
Create an Empty Vector in R - Coding Campus
Create an Empty Vector in R
A vector is a sequence of data elements of the same basic type. A vector whose length is zero is known as an empty vector. You can create an empty vector using the c(), vector(), rep(), and numeric()
function. This guide shows different ways to create an empty vector in R.
Method #1: Create an empty vector using the c() function
The combine or c() function in R is used to create vectors by combining multiple values. To create an empty vector using the c() function, do not pass anything inside the parentheses.
vector name <- c()
Here, “vector name” can be any name you want to give to the empty vector.
x <- c()
In the above example, the output is NULL which denotes that the vector is empty.
Method #2: Create an empty vector using the vector() function
To create an empty vector using the vector() function, do not pass anything inside the parentheses.
vector name <- vector()
Here, “vector name” can be any name you want to give to the empty vector.
v <- vector()
In the above example, logical(0) denotes an empty vector.
Method #3: Create an empty vector using a NULL object
You can assign NULL to an existing or a new vector to create an empty vector.
If you have a vector x = c(1, 2, 3), you can convert it into an empty vector using the following code:
x <-c(1,2,3,5)
#converting to an empty vector
x <- NULL
In the above example, we converted an existing non-empty vector to an empty vector by assigning the NULL object.
Method #4: Create an empty vector using the rep() function
rep() function in R repeats the specified object a specified number of times. It is used to create a vector with a specified number of entries. To create an empty vector using the rep() function, do
not pass anything inside the parentheses.
vector name <- rep()
Here, “vector name” can be any name you want to give to the empty vector.
y <- rep()
Method #5: Create an empty vector using the numeric() function
The numeric() function creates a numeric vector of a defined length. To create an empty vector using the numeric() function, do not pass anything inside the parentheses.
vector name <- numeric()
Here, “vector name” can be any name you want to give to the empty vector.
a <- numeric()
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A Physicist’s Physicist Ponders the Nature of Reality
Edward Witten reflects on the meaning of dualities in physics and math, emergent space-time, and the pursuit of a complete description of nature.
Among the brilliant theorists cloistered in the quiet woodside campus of the Institute for Advanced Study in Princeton, New Jersey, Edward Witten stands out as a kind of high priest. The sole
physicist ever to win the Fields Medal, mathematics’ premier prize, Witten is also known for discovering M-theory, the leading candidate for a unified physical “theory of everything.” A genius’s
genius, Witten is tall and rectangular, with hazy eyes and an air of being only one-quarter tuned in to reality until someone draws him back from more abstract thoughts.
During a visit this fall, I spotted Witten on the Institute’s central lawn and requested an interview; in his quick, alto voice, he said he couldn’t promise to be able to answer my questions but
would try. Later, when I passed him on the stone paths, he often didn’t seem to see me.
Physics luminaries since Albert Einstein, who lived out his days in the same intellectual haven, have sought to unify gravity with the other forces of nature by finding a more fundamental quantum
theory to replace Einstein’s approximate picture of gravity as curves in the geometry of space-time. M-theory, which Witten proposed in 1995, could conceivably offer this deeper description, but only
some aspects of the theory are known. M-theory incorporates within a single mathematical structure all five versions of string theory, which renders the elements of nature as minuscule vibrating
strings. These five string theories connect to each other through “dualities,” or mathematical equivalences. Over the past 30 years, Witten and others have learned that the string theories are also
mathematically dual to quantum field theories — descriptions of particles moving through electromagnetic and other fields that serve as the language of the reigning “Standard Model” of particle
physics. While he’s best known as a string theorist, Witten has discovered many new quantum field theories and explored how all these different descriptions are connected. His physical insights have
led time and again to deep mathematical discoveries.
Researchers pore over his work and hope he’ll take an interest in theirs. But for all his scholarly influence, Witten, who is 66, does not often broadcast his views on the implications of modern
theoretical discoveries. Even his close colleagues eagerly suggested questions they wanted me to ask him.
When I arrived at his office at the appointed hour on a summery Thursday last month, Witten wasn’t there. His door was ajar. Papers covered his coffee table and desk — not stacks, but floods: text
oriented every which way, some pages close to spilling onto the floor. (Research papers get lost in the maelstrom as he finishes with them, he later explained, and every so often he throws the heaps
away.) Two girls smiled out from a framed photo on a shelf; children’s artwork decorated the walls, one celebrating Grandparents’ Day. When Witten arrived minutes later, we spoke for an hour and a
half about the meaning of dualities in physics and math, the current prospects of M-theory, what he’s reading, what he’s looking for, and the nature of reality. The interview has been condensed and
edited for clarity.
Physicists are talking more than ever lately about dualities, but you’ve been studying them for decades. Why does the subject interest you?
People keep finding new facets of dualities. Dualities are interesting because they frequently answer questions that are otherwise out of reach. For example, you might have spent years pondering a
quantum theory and you understand what happens when the quantum effects are small, but textbooks don’t tell you what you do if the quantum effects are big; you’re generally in trouble if you want to
know that. Frequently dualities answer such questions. They give you another description, and the questions you can answer in one description are different than the questions you can answer in a
different description.
What are some of these newfound facets of dualities?
It’s open-ended because there are so many different kinds of dualities. There are dualities between a gauge theory [a theory, such as a quantum field theory, that respects certain symmetries] and
another gauge theory, or between a string theory for weak coupling [describing strings that move almost independently from one another] and a string theory for strong coupling. Then there’s AdS/CFT
duality, between a gauge theory and a gravitational description. That duality was discovered 20 years ago, and it’s amazing to what extent it’s still fruitful. And that’s largely because around 10
years ago, new ideas were introduced that rejuvenated it. People had new insights about entropy in quantum field theory — the whole story about “it from qubit.”
That’s the idea that space-time and everything in it emerges like a hologram out of information stored in the entangled quantum states of particles.
Yes. Then there are dualities in math, which can sometimes be interpreted physically as consequences of dualities between two quantum field theories. There are so many ways these things are
interconnected that any simple statement I try to make on the fly, as soon as I’ve said it I realize it didn’t capture the whole reality. You have to imagine a web of different relationships, where
the same physics has different descriptions, revealing different properties. In the simplest case, there are only two important descriptions, and that might be enough. If you ask me about a more
complicated example, there might be many, many different ones.
Given this web of relationships and the issue of how hard it is to characterize all duality, do you feel that this reflects a lack of understanding of the structure, or is it that we’re seeing the
structure, only it’s very complicated?
I’m not certain what we should hope for. Traditionally, quantum field theory was constructed by starting with the classical picture [of a smooth field] and then quantizing it. Now we’ve learned that
there are a lot of things that happen that that description doesn’t do justice to. And the same quantum theory can come from different classical theories. Now, Nati Seiberg [a theoretical physicist
who works down the hall] would possibly tell you that he has faith that there’s a better formulation of quantum field theory that we don’t know about that would make everything clearer. I’m not sure
how much you should expect that to exist. That would be a dream, but it might be too much to hope for; I really don’t know.
There’s another curious fact that you might want to consider, which is that quantum field theory is very central to physics, and it’s actually also clearly very important for math. But it’s extremely
difficult for mathematicians to study; the way physicists define it is very hard for mathematicians to follow with a rigorous theory. That’s extremely strange, that the world is based so much on a
mathematical structure that’s so difficult.
What do you see as the relationship between math and physics?
I prefer not to give you a cosmic answer but to comment on where we are now. Physics in quantum field theory and string theory somehow has a lot of mathematical secrets in it, which we don’t know how
to extract in a systematic way. Physicists are able to come up with things that surprise the mathematicians. Because it’s hard to describe mathematically in the known formulation, the things you
learn about quantum field theory you have to learn from physics.
I find it hard to believe there’s a new formulation that’s universal. I think it’s too much to hope for. I could point to theories where the standard approach really seems inadequate, so at least for
those classes of quantum field theories, you could hope for a new formulation. But I really can’t imagine what it would be.
You can’t imagine it at all?
No, I can’t. Traditionally it was thought that interacting quantum field theory couldn’t exist above four dimensions, and there was the interesting fact that that’s the dimension we live in. But one
of the offshoots of the string dualities of the 1990s was that it was discovered that quantum field theories actually exist in five and six dimensions. And it’s amazing how much is known about their
I’ve heard about the mysterious (2,0) theory, a quantum field theory describing particles in six dimensions, which is dual to M-theory describing strings and gravity in seven-dimensional AdS space.
Does this (2,0) theory play an important role in the web of dualities?
Yes, that’s the pinnacle. In terms of conventional quantum field theory without gravity, there is nothing quite like it above six dimensions. From the (2,0) theory’s existence and main properties,
you can deduce an incredible amount about what happens in lower dimensions. An awful lot of important dualities in four and fewer dimensions follow from this six-dimensional theory and its
properties. However, whereas what we know about quantum field theory is normally from quantizing a classical field theory, there’s no reasonable classical starting point of the (2,0) theory. The
(2,0) theory has properties [such as combinations of symmetries] that sound impossible when you first hear about them. So you can ask why dualities exist, but you can also ask why is there a 6-D
theory with such and such properties? This seems to me a more fundamental restatement.
Dualities sometimes make it hard to maintain a sense of what’s real in the world, given that there are radically different ways you can describe a single system. How would you describe what’s real or
What aspect of what’s real are you interested in? What does it mean that we exist? Or how do we fit into our mathematical descriptions?
The latter.
Well, one thing I’ll tell you is that in general, when you have dualities, things that are easy to see in one description can be hard to see in the other description. So you and I, for example, are
fairly simple to describe in the usual approach to physics as developed by Newton and his successors. But if there’s a radically different dual description of the real world, maybe some things
physicists worry about would be clearer, but the dual description might be one in which everyday life would be hard to describe.
What would you say about the prospect of an even more optimistic idea that there could be one single quantum gravity description that really does help you in every case in the real world?
Well, unfortunately, even if it’s correct I can’t guarantee it would help. Part of what makes it difficult to help is that the description we have now, even though it’s not complete, does explain an
awful lot. And so it’s a little hard to say, even if you had a truly better description or a more complete description, whether it would help in practice.
Are you speaking of M-theory?
M-theory is the candidate for the better description.
You proposed M-theory 22 years ago. What are its prospects today?
Personally, I thought it was extremely clear it existed 22 years ago, but the level of confidence has got to be much higher today because AdS/CFT has given us precise definitions, at least in AdS
space-time geometries. I think our understanding of what it is, though, is still very hazy. AdS/CFT and whatever’s come from it is the main new perspective compared to 22 years ago, but I think it’s
perfectly possible that AdS/CFT is only one side of a multifaceted story. There might be other equally important facets.
What’s an example of something else we might need?
Maybe a bulk description of the quantum properties of space-time itself, rather than a holographic boundary description. There hasn’t been much progress in a long time in getting a better bulk
description. And I think that might be because the answer is of a different kind than anything we’re used to. That would be my guess.
Are you willing to speculate about how it would be different?
I really doubt I can say anything useful. I guess I suspect that there’s an extra layer of abstractness compared to what we’re used to. I tend to think that there isn’t a precise quantum description
of space-time — except in the types of situations where we know that there is, such as in AdS space. I tend to think, otherwise, things are a little bit murkier than an exact quantum description. But
I can’t say anything useful.
The other night I was reading an old essay by the 20th-century Princeton physicist John Wheeler. He was a visionary, certainly. If you take what he says literally, it’s hopelessly vague. And
therefore, if I had read this essay when it came out 30 years ago, which I may have done, I would have rejected it as being so vague that you couldn’t work on it, even if he was on the right track.
You’re referring to Information, Physics, Quantum, Wheeler’s 1989 essay propounding the idea that the physical universe arises from information, which he dubbed “it from bit.” Why were you reading
I’m trying to learn about what people are trying to say with the phrase “it from qubit.” Wheeler talked about “it from bit,” but you have to remember that this essay was written probably before the
term “qubit” was coined and certainly before it was in wide currency. Reading it, I really think he was talking about qubits, not bits, so “it from qubit” is actually just a modern translation.
Don’t expect me to be able to tell you anything useful about it — about whether he was right. When I was a beginning grad student, they had a series of lectures by faculty members to the new students
about theoretical research, and one of the people who gave such a lecture was Wheeler. He drew a picture on the blackboard of the universe visualized as an eye looking at itself. I had no idea what
he was talking about. It’s obvious to me in hindsight that he was explaining what it meant to talk about quantum mechanics when the observer is part of the quantum system. I imagine there is
something we don’t understand about that.
Observing a quantum system irreversibly changes it, creating a distinction between past and future. So the observer issue seems possibly related to the question of time, which we also don’t
understand. With the AdS/CFT duality, we’ve learned that new spatial dimensions can pop up like a hologram from quantum information on the boundary. Do you think time is also emergent — that it
arises from a timeless complete description?
I tend to assume that space-time and everything in it are in some sense emergent. By the way, you’ll certainly find that that’s what Wheeler expected in his essay. As you’ll read, he thought the
continuum was wrong in both physics and math. He did not think one’s microscopic description of space-time should use a continuum of any kind — neither a continuum of space nor a continuum of time,
nor even a continuum of real numbers. On the space and time, I’m sympathetic to that. On the real numbers, I’ve got to plead ignorance or agnosticism. It is something I wonder about, but I’ve tried
to imagine what it could mean to not use the continuum of real numbers, and the one logician I tried discussing it with didn’t help me.
Do you consider Wheeler a hero?
I wouldn’t call him a hero, necessarily, no. Really I just became curious what he meant by “it from bit,” and what he was saying. He definitely had visionary ideas, but they were too far ahead of
their time. I think I was more patient in reading a vague but inspirational essay than I might have been 20 years ago. He’s also got roughly 100 interesting-sounding references in that essay. If you
decided to read them all, you’d have to spend weeks doing it. I might decide to look at a few of them.
Why do you have more patience for such things now?
I think when I was younger I always thought the next thing I did might be the best thing in my life. But at this point in life I’m less persuaded of that. If I waste a little time reading somebody’s
essay, it doesn’t seem that bad.
Do you ever take your mind off physics and math?
My favorite pastime is tennis. I am a very average but enthusiastic tennis player.
In contrast to Wheeler, it seems like your working style is to come to the insights through the calculations, rather than chasing a vague vision.
In my career I’ve only been able to take small jumps. Relatively small jumps. What Wheeler was talking about was an enormous jump. And he does say at the beginning of the essay that he has no idea if
this will take 10, 100 or 1,000 years.
And he was talking about explaining how physics arises from information.
Yes. The way he phrases it is broader: He wants to explain the meaning of existence. That was actually why I thought you were asking if I wanted to explain the meaning of existence.
I see. Does he have any hypotheses?
No. He only talks about things you shouldn’t do and things you should do in trying to arrive at a more fundamental description of physics.
Do you have any ideas about the meaning of existence?
No. [Laughs.]
Correction: This article was updated on Nov. 29, 2017, to clarify that M-theory is the leading candidate for a unified theory of everything. Other ideas have been proposed that also claim to unify
the fundamental forces.
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Level Set Initialization Method
Level Set Initialization Method#
Level Set Initialization Method = {method_name} {parameter list}
Description / Usage#
This card specifies the means by which the level set function is initialized. That is, it constructs from a representation of the starting interface shape, a value for the distance function at every
node in the mesh. The syntax of the card is as follows:
A character string which identifies the initialization option desired. Choices for this string are: Projection, Exodus, Nodeset, Surfaces, SM_object.
{parameter list}
This is a variable parameter list specific to each option. The nature of it for each method is detailed in the syntax descriptions below.
Below are the exact syntax used for each initialization method, a brief description of the method and a specification of any additional required parameters.
Projection This method computes the initial level set field by calling a user-specified routine which returns the signed distance function for a given point. It has no parameter list after its
Exodus Using this card indicates that the initial level set field is to be read from the exodus file specified earlier (see FEM file and Initial Guess cards for read_exoII option). This
card has no parameter list after its name.
Nodeset This method establishes the initial location of the interface as the boundary between two element blocks. The value <integer1> is the nodeset identification number for an internal
<integer1> EB nodeset defined to exist at the interface between the two element blocks. The character string EB is required. The integer <integer2> is the element block id number to which
<integer2> positive values of level set function is going to be assigned.
This card establishes the initial level set function by referring to a set of primitive geometric objects. It is the easiest to use and the most general. The integer value <integer>
is the number of surface objects that are used to construct the initial interface. This number of SURF object cards must follow this card. This is the syntax of the SURF object
<integer> SURF = {object_name} {float list}
{object_name}: a character string identifying the type of geometric object. Options are: PLANE, CIRCLE, SPHERE, SS, USER.
{float list}: geometric parameters associated with each object as float values
The following is the syntax and description for each geometric object option, i.e., the “{object_name} {float list}” part of SURF
This card constructs a planar interface surface. The float values <nx>, <ny>, <nz> define a vector normal to this plane with the restriction that the sign of the vector must be such
PLANE <nx. <ny> that it points from the negative side of the interface to the positive side of the interface. The float value <d> effectively represents the distance of the plane from the origin.
<nz> <d> Its value must be set, however, so that the dot product of any position vector to a point on the desired plane and the vector (nx,ny,nz) must be equal to <d> (it is a property of
planes that this number is independent of the point on the plane that is chosen).
CIRCLE <cx> <cy> This card constructs a circular interface surface in a two-dimensional domain. The float values <cx> <cy> identify the coordinates of the center of the circle. The float value
<radius> <radius> establishes the radius of the curve. By definition, points interior to the circle are assigned negative level set function values.
SPHERE <cx> <cy> This card constructs a spherical interface surface in a three-dimensional domain. The float values <cx> <cy> <cz> identify the coordinates of the center of the circle. The float
<cz> <radius> value <radius> establishes the radius of the sphere. By definition, points interior to the sphere are assigned negative level set function values.
SS {ss_id} This card uses an existing sideset in the problem as a defined geometric object for construction of an interface. The parameter <ss_id> identifies this sideset.
USER This card indicates the user has defined an object function using the supplied parameter float list that returns a signed distance value when supplied with the coordinates of a point
{user-defined in space. This object function should appear in the function call user_init_object in the file user_pre.c.
float list}
SM_object This card allows the user to initialize the level set location by using a piece of solid model geometry. The solid model object_type can be either FACE or BODY. A 2D initialization
{object_type} uses the boundary of the specified FACE (or surface) as the 0 level set. A 3D initialization uses the boundary of the specified BODY (or volume) as the 0 level set.
Two examples of initialization methods are provide below:
Level Set Initialization Method = Nodeset 20 EB 1
Level Set Initialization Method = Surfaces 3
SURF = PLANE -1. 0. 0. -3.
SURF = CIRCLE -2 0 1
SURF = CIRCLE -3 0 0.5
Level Set Initialization Method = SM_object BODY my_blob
Technical Discussion#
The Projection initialization method was developed early in the level set development process. It has since been superseded by other more easily used methods. It is still supported primarily for the
use of developers. Users wanting a complicated interface shape for which they can supply an appropriate distance function should user the USER surface object option under the Surfaces initialization
The Exodus method deserves little comment. It should be used when restarting level set computations from a preexisting solution.
The Nodeset method allows the user to make use of the sophisticated solid body manipulation software in meshing packages like CUBIT. The procedure for using this method is to create a domain which
contains two element blocks. The desired starting point for the interface should lie on the curve or surface which these two blocks have in common. A single nodeset should be defined over this entire
curve or surface. The nodeset identification number should be the first integer parameter specified on the card. Also note that one of the blocks must be designated as the “positive” block. This
means then when initialized the values of the level set function in this block will be positive. The values in the other block will be negative. Note that this initialization method can only by used
for problems that have exactly two blocks, no more.
The Surfaces initialization method is the most useful method for initialization. It draws from the fact that it is relatively easy to determine the distance to simple geometric objects (planes,
circles, spheres, etc.). Further, it permits initialization using more than one of these objects so that relatively complicated initial interface locations can be constructed. However, the user
should recognize that this method is still somewhat unsophisticated in its approach so there are some caveats associated with its use. The primary point is that surface objects should never intersect
anywhere within the domain of interest, otherwise it is more than likely that the starting interface shape will not be what the user expects.
The SM_object initialization method allows the user to use solid model geometry to initialize 2D and 3D level sets. Certain 2D geometries can be created using only Goma input commands (see FACE).
Other 2D geometries, and all 3D geometries, can be accessed via an ACIS .sat file. The usual way to do this is for the user to create their desired geometry within Cubit (or, import solid model
geometry from elsewhere into Cubit). Faces (or surfaces) should be created for 2D initialization, and bodies (or volumes) should be created for 3D initialization. The boundary of the object is used
to initialize the level set. The geometry should be named within Cubit and exported to an ACIS .sat file via Cubit’s export acis “filename” ascii command. This same file should be read in via the
ACIS file command in the Geometry Specifications section. The solid model geometry is then available for the Level Set Initialization Method command. (Note that the Geometry Specifications section
usually comes after the Level Set Initialization Method command; this is OK).
GT-020.1: Tutorial on Level Set Interface Tracking in GOMA, February 27, 2001, T.A. Baer | {"url":"https://docs.gomafem.com/problem_description_file/level_set/level_set_initialization_method.html","timestamp":"2024-11-07T03:47:30Z","content_type":"text/html","content_length":"104401","record_id":"<urn:uuid:d9679039-4c76-42dd-bdf6-c737e9f50999>","cc-path":"CC-MAIN-2024-46/segments/1730477027951.86/warc/CC-MAIN-20241107021136-20241107051136-00292.warc.gz"} |
On the Local Eigenvalue Statistics for Random Band Matrices in the Localization Regime - Ashoka University
We study the local eigenvalue statistics ξNω,E${\xi }_{\omega ,E}^{N}$ associated with the eigenvalues of one-dimensional, (2N+1)×(2N+1)$\left(2N+1\right)×\left(2N+1\right)$ random band matrices with
independent, identically distributed, real random variables and band width growing as Nα${N}^{\alpha }$, for 0<α<12$0<\alpha <\frac{1}{2}$. We consider the limit points associated with the random
variables ξNω,E[I]${\xi }_{\omega ,E}^{N}\left[I\right]$, for I⊂R$I\subset \mathbb{R}$, and E∈(−2,2)$E\in \left(-2,2\right)$. For random band matrices with Gaussian distributed random variables and
for 0≤α<17$0\le \alpha <\frac{1}{7}$, we prove that this family of random variables has nontrivial limit points for almost every E∈(−2,2)$E\in \left(-2,2\right)$, and that these limit points are
Poisson distributed with positive intensities. The proof is based on an analysis of the characteristic functions of the random variables ξNω,E[I]${\xi }_{\omega ,E}^{N}\left[I\right]$ and associated
quantities related to the intensities, as N tends towards infinity, and employs known localization bounds of (Peled et al. in Int. Math. Res. Not. IMRN 4:1030–1058, 2019, Schenker in Commun Math Phys
290:1065–1097, 2009), and the strong Wegner and Minami estimates (Peled et al. in Int. Math. Res. Not. IMRN 4:1030–1058, 2019). Our more general result applies to random band matrices with random
variables having absolutely continuous distributions with bounded densities. Under the hypothesis that the localization bounds hold for 0<α<12$0<\alpha <\frac{1}{2}$, we prove that any nontrivial
limit points of the random variables ξNω,E[I]${\xi }_{\omega ,E}^{N}\left[I\right]$ are distributed according to Poisson distributions.We study the local eigenvalue statistics ξNω,E${\xi }_{\omega
,E}^{N}$ associated with the eigenvalues of one-dimensional, (2N+1)×(2N+1)$\left(2N+1\right)×\left(2N+1\right)$ random band matrices with independent, identically distributed, real random variables
and band width growing as Nα${N}^{\alpha }$, for 0<α<12$0<\alpha <\frac{1}{2}$. We consider the limit points associated with the random variables ξNω,E[I]${\xi }_{\omega ,E}^{N}\left[I\right]$, for
I⊂R$I\subset \mathbb{R}$, and E∈(−2,2)$E\in \left(-2,2\right)$. For random band matrices with Gaussian distributed random variables and for 0≤α<17$0\le \alpha <\frac{1}{7}$, we prove that this family
of random variables has nontrivial limit points for almost every E∈(−2,2)$E\in \left(-2,2\right)$, and that these limit points are Poisson distributed with positive intensities. The proof is based on
an analysis of the characteristic functions of the random variables ξNω,E[I]${\xi }_{\omega ,E}^{N}\left[I\right]$ and associated quantities related to the intensities, as N tends towards infinity,
and employs known localization bounds of (Peled et al. in Int. Math. Res. Not. IMRN 4:1030–1058, 2019, Schenker in Commun Math Phys 290:1065–1097, 2009), and the strong Wegner and Minami estimates
(Peled et al. in Int. Math. Res. Not. IMRN 4:1030–1058, 2019). Our more general result applies to random band matrices with random variables having absolutely continuous distributions with bounded
densities. Under the hypothesis that the localization bounds hold for 0<α<12$0<\alpha <\frac{1}{2}$, we prove that any nontrivial limit points of the random variables ξNω,E[I]${\xi }_{\omega ,E}^{N}\
left[I\right]$ are distributed according to Poisson distributions. | {"url":"https://publications.ashoka.edu.in/publication/on-the-local-eigenvalue-statistics-for-random-band-matrices","timestamp":"2024-11-03T09:53:56Z","content_type":"text/html","content_length":"123566","record_id":"<urn:uuid:aa7e73ba-3917-4044-a359-7e211e35f8a4>","cc-path":"CC-MAIN-2024-46/segments/1730477027774.6/warc/CC-MAIN-20241103083929-20241103113929-00713.warc.gz"} |
Why Are Graphing Linear Equations Important in Algebra? - IB Pros
In the realm of algebra, the significance of graphing linear equations extends far beyond mere academic exercise; it serves as a fundamental skill that bridges abstract numerical relationships and
their tangible visual representations. This graphical approach not only demystifies the behavior of linear equations by illustrating how variables interact but also lays a robust foundation for
understanding more intricate mathematical concepts. It is through the lens of a coordinate plane that students and professionals alike can discern patterns, analyze trends, and extract meaningful
insights from data that might otherwise remain obscured in a purely numerical format. As we consider the multitude of contexts in which these skills are applied, from the sciences to economics, one
might ponder the depth of influence that such a seemingly simple act of plotting points and drawing lines can wield in our interpretation of the world around us. The implications are vast, and as we
uncover the layers, it becomes clear that the importance of graphing linear equations transcends the classroom, prompting us to consider how this tool shapes our understanding of the universe’s very
Key Takeaways
• Graphing provides a clear visualization of relationships between variables.
• The slope and y-intercept of the graph reveal important information about the equation.
• Graphing enhances understanding of linear equations and systems of equations.
• Graphing simplifies complex algebraic concepts and makes them accessible for analysis and interpretation.
Visualizing Relationships
In the realm of algebra, graphing linear equations serves as a fundamental tool for visualizing the relationships between variables, providing a clear and precise representation of their
interconnectedness. This graphical method facilitates the understanding of how one variable changes in response to another, embodying the core principle of function dependency in mathematical terms.
As a linear equation denotes a straight line when plotted on a Cartesian coordinate system, the slope and y-intercept become immediately apparent, unveiling the magnitude and direction of the
relationship between the variables involved.
The graph’s slope, indicative of the rate of change, allows for the extrapolation of data beyond the observed range, making it an indispensable element for predictions and modeling in various
scientific disciplines. Meanwhile, the y-intercept offers insight into the initial condition of the dependent variable when the independent variable is nullified. Through this visual framework,
algebraic expressions transcend abstract symbols, transforming into tangible constructs that can be analyzed and interpreted within a broader context.
The ability to graph linear equations thus equips scholars and practitioners alike with a robust analytical tool, enhancing their capacity to dissect linear relationships and apply this understanding
to real-world scenarios. It is through this synthesis of algebraic concepts and visual representation that learners can gain a deeper appreciation for the intricate dance of variables within the
realm of mathematics.
Enhancing Comprehension
Building upon the graphical representation of linear relationships, enhancing comprehension involves a deeper dive into the methods and techniques that facilitate a more profound understanding of
algebraic concepts. This process is critical for learners to not only perform mathematical operations but also to develop an intuitive grasp of algebra’s underlying structures and patterns. Through
the lens of graphing linear equations, students can explore a variety of fundamental principles that govern algebraic reasoning and problem-solving.
To add depth and engage the audience, consider the following aspects of enhanced comprehension:
1. Interpreting Slope and Intercept: Recognizing the slope as the rate of change and the y-intercept as the starting value in real-world contexts solidifies comprehension of linear relationships.
2. Analyzing Graph Behavior: Understanding how changes in coefficients affect the steepness and direction of the graph leads to insights into the nature of linear equations.
3. Solving Systems of Equations Graphically: Utilizing graphs to find the point of intersection enhances the understanding of solutions to systems of linear equations.
4. Translating Between Representations: Developing the skill to move between different forms of linear equations and their graphs fosters a versatile approach to algebraic problem-solving.
Such analytical engagement with algebraic concepts through graphing not only sharpens mathematical acuity but also equips learners with tools for critical thinking applicable across diverse fields.
Applying to Real-World Problems
Understanding the application of graphing linear equations to real-world problems allows students to recognize the practical significance of algebra in various professional and everyday contexts. The
graph of a linear equation, representing a straight line, often models relationships between two variables in real-world scenarios. It facilitates the visualization of trends, patterns, and direct
correlations that are foundational to disciplines such as economics, physics, and engineering.
In economics, the supply and demand model is epitomized through linear equations, where the intersection of supply and demand curves determines market equilibrium. Graphing these equations offers a
clear, visual representation of market dynamics, aiding in predictive analyses and strategic decision-making. Similarly, in physics, the principles of motion often relate distance and time through
linear relationships, with graphs serving as tools for interpreting and predicting an object’s behavior under uniform velocity.
Engineering applications abound, from calculating the forces in static structures to optimizing material usage—each often requiring the graphing of linear relationships to ensure precision and
efficacy. The analytical nature of graphing linear equations thus becomes indispensable, transforming abstract mathematical concepts into tangible solutions that address and resolve real-world
challenges. This contextualizes algebraic learning, highlighting its relevance beyond the confines of theoretical mathematics.
Facilitating Predictions
Graphing linear equations provides a robust framework for predicting future outcomes by extrapolating from established data trends. This method is a cornerstone of various disciplines, from economics
to engineering, where future projections are essential for strategic planning and decision-making. By analyzing the slope and intercept of a linear equation graphed on a coordinate plane, one can
discern patterns and tendencies that are not readily apparent from mere data tables or qualitative analysis.
To elucidate the importance of graphing linear equations for predictions, consider the following key points:
1. Interpolation and Extrapolation: Graphs allow for the estimation of values within (interpolation) and outside (extrapolation) the range of the existing data set.
2. Trend Identification: A visual representation of data can reveal trends that are instrumental in forecasting future events or conditions.
3. Quantitative Decision-Making: The ability to predict outcomes quantitatively aids in making informed decisions that are based on logical deductions rather than assumptions.
4. Model Validation: Comparing predicted values with actual outcomes can validate the model used, ensuring its reliability for future predictions.
Through this analytical lens, graphing linear equations emerges as an indispensable tool in the predictive arsenal of algebra, underpinning a wide array of predictive analytics applications.
Simplifying Complex Concepts
While graphing linear equations serves as a means to visualize and predict trends, the process of simplifying complex concepts is equally crucial in making the subject matter accessible and
comprehensible. Algebra, as a field of mathematics, is replete with abstract notions that can be challenging for learners to grasp. By representing these notions graphically, the educator or the
instructional material can distill multifaceted relationships into a clear, two-dimensional space.
Graphing provides a tangible method for students to see the direct correlation between variables, thus demystifying the algebraic expressions and equations. It is an analytical tool that aids in the
breakdown of algebraic complexity into manageable parts, allowing for incremental understanding.
Below is a table illustrating the comparison between the traditional algebraic format and the graphical representation:
Algebraic FormatGraphical RepresentationEquationsLines on a graphInequalitiesShaded regionsSystems of EquationsIntersection points
Each row in the table exemplifies a core algebraic concept and its graphical counterpart, underscoring the importance of graphing as a simplification strategy. This methodical approach not only
facilitates learning but also encourages precision and analytical thinking, key components of mathematical literacy.
Frequently Asked Questions
How Has the Teaching of Graphing Linear Equations Evolved Over the Past Few Decades?
The pedagogy of graphing linear equations has advanced significantly, incorporating technology such as graphing calculators and computer software. This evolution has facilitated a more interactive
and intuitive understanding of mathematical concepts. Educators now emphasize visualization and real-world applications, moving away from rote memorization to a deeper conceptual approach. These
technological and methodological enhancements have proven essential in equipping students with the necessary skills for complex problem-solving in various fields.
What Are the Most Common Misconceptions Students Have About Graphing Linear Equations?
Common misconceptions among students regarding graphing linear equations include the belief that all lines intersect the origin, confusion between the slopes of perpendicular lines, and equating the
steepness of a line solely with the value of the slope without considering its sign. Additionally, there is often an erroneous assumption that parallel lines must have the same y-intercept, and a
misunderstanding of the implications of horizontal and vertical lines within the coordinate system.
Can Graphing Linear Equations Be Useful in Understanding Non-Linear Relationships, and if So, How?
Graphing linear equations can indeed be instrumental in understanding non-linear relationships. By establishing a baseline of linear behavior, deviations from linearity become more apparent, aiding
in the identification of patterns, trends, and the nature of non-linear dynamics. This contrast can illuminate key properties such as curvature, asymptotic behavior, and inflection points, thereby
enhancing comprehension of complex systems and contributing to the development of more accurate predictive models in various scientific and mathematical applications.
Are There Any Software Tools or Mobile Apps That Can Help Students Better Understand Graphing Linear Equations?
Several software applications and mobile tools are available to aid students in grasping the concepts of graphing linear equations. These digital resources, designed with interactive interfaces,
often provide real-time graphical representations and step-by-step problem-solving guidance. Such educational technology facilitates comprehension through visualization and can significantly enhance
a learner’s ability to analyze and interpret mathematical data, thereby improving their overall proficiency in algebraic methods and concepts.
How Do Educators Assess the Mastery of Graphing Linear Equations Among Students With Different Learning Styles?
Educators employ a variety of assessment methods to gauge students’ proficiency in graphing linear equations, catering to diverse learning styles. These include traditional written tests, practical
assignments, and interactive digital platforms that offer instant feedback. Formative assessments through class participation and group work also provide insights into student understanding.
Differentiated instruction is crucial to address individual learning needs, ensuring a comprehensive evaluation of a student’s mastery of the material.
In conclusion, graphing linear equations serves as a fundamental tool in algebra for visualizing relationships, enhancing comprehension, and applying mathematical concepts to real-world scenarios. It
facilitates predictions and simplifies complex concepts, allowing for a more intuitive understanding of algebraic relationships. The ability to represent equations graphically is indispensable in
various fields, underscoring the importance of this skill in both educational settings and professional applications. | {"url":"https://ib-pros.com/blog/why-are-graphing-linear-equations-important-in-algebra/","timestamp":"2024-11-03T22:52:31Z","content_type":"text/html","content_length":"424924","record_id":"<urn:uuid:1c7d6ca5-84af-460e-8ef4-b7b3d6b5bb03>","cc-path":"CC-MAIN-2024-46/segments/1730477027796.35/warc/CC-MAIN-20241103212031-20241104002031-00086.warc.gz"} |
Source code: tianshou/utils/net/discrete.py
class Actor(preprocess_net: Module, action_shape: Sequence[int] | int | int64, hidden_sizes: Sequence[int] = (), softmax_output: bool = True, preprocess_net_output_dim: int | None = None, device: str
| int | device = 'cpu')[source]#
Simple actor network.
Will create an actor operated in discrete action space with structure of preprocess_net —> action_shape.
☆ preprocess_net – a self-defined preprocess_net which output a flattened hidden state.
☆ action_shape – a sequence of int for the shape of action.
☆ hidden_sizes – a sequence of int for constructing the MLP after preprocess_net. Default to empty sequence (where the MLP now contains only a single linear layer).
☆ softmax_output – whether to apply a softmax layer over the last layer’s output.
☆ preprocess_net_output_dim – the output dimension of preprocess_net.
For advanced usage (how to customize the network), please refer to Build the Network.
See also
Please refer to Net as an instance of how preprocess_net is suggested to be defined.
forward(obs: ndarray | Tensor, state: Any = None, info: dict[str, Any] | None = None) tuple[Tensor, Any][source]#
Mapping: s -> Q(s, *).
get_output_dim() int[source]#
get_preprocess_net() Module[source]#
class CosineEmbeddingNetwork(num_cosines: int, embedding_dim: int)[source]#
Cosine embedding network for IQN. Convert a scalar in [0, 1] to a list of n-dim vectors.
☆ num_cosines – the number of cosines used for the embedding.
☆ embedding_dim – the dimension of the embedding/output.
forward(taus: Tensor) Tensor[source]#
Defines the computation performed at every call.
Should be overridden by all subclasses.
Although the recipe for forward pass needs to be defined within this function, one should call the Module instance afterwards instead of this since the former takes care of running the
registered hooks while the latter silently ignores them.
class Critic(preprocess_net: Module, hidden_sizes: Sequence[int] = (), last_size: int = 1, preprocess_net_output_dim: int | None = None, device: str | int | device = 'cpu')[source]#
Simple critic network.
It will create an actor operated in discrete action space with structure of preprocess_net —> 1(q value).
☆ preprocess_net – a self-defined preprocess_net which output a flattened hidden state.
☆ hidden_sizes – a sequence of int for constructing the MLP after preprocess_net. Default to empty sequence (where the MLP now contains only a single linear layer).
☆ last_size – the output dimension of Critic network. Default to 1.
☆ preprocess_net_output_dim – the output dimension of preprocess_net.
For advanced usage (how to customize the network), please refer to Build the Network.
See also
Please refer to Net as an instance of how preprocess_net is suggested to be defined.
forward(obs: ndarray | Tensor, **kwargs: Any) Tensor[source]#
Mapping: s -> V(s).
class FractionProposalNetwork(num_fractions: int, embedding_dim: int)[source]#
Fraction proposal network for FQF.
☆ num_fractions – the number of factions to propose.
☆ embedding_dim – the dimension of the embedding/input.
forward(obs_embeddings: Tensor) tuple[Tensor, Tensor, Tensor][source]#
Defines the computation performed at every call.
Should be overridden by all subclasses.
Although the recipe for forward pass needs to be defined within this function, one should call the Module instance afterwards instead of this since the former takes care of running the
registered hooks while the latter silently ignores them.
class FullQuantileFunction(preprocess_net: Module, action_shape: Sequence[int] | int | int64, hidden_sizes: Sequence[int] = (), num_cosines: int = 64, preprocess_net_output_dim: int | None = None,
device: str | int | device = 'cpu')[source]#
Full(y parameterized) Quantile Function.
☆ preprocess_net – a self-defined preprocess_net which output a flattened hidden state.
☆ action_shape – a sequence of int for the shape of action.
☆ hidden_sizes – a sequence of int for constructing the MLP after preprocess_net. Default to empty sequence (where the MLP now contains only a single linear layer).
☆ num_cosines – the number of cosines to use for cosine embedding. Default to 64.
☆ preprocess_net_output_dim – the output dimension of preprocess_net.
The first return value is a tuple of (quantiles, fractions, quantiles_tau), where fractions is a Batch(taus, tau_hats, entropies).
forward(obs: ndarray | Tensor, propose_model: FractionProposalNetwork, fractions: Batch | None = None, **kwargs: Any) tuple[Any, Tensor][source]#
Mapping: s -> Q(s, *).
class ImplicitQuantileNetwork(preprocess_net: Module, action_shape: Sequence[int] | int | int64, hidden_sizes: Sequence[int] = (), num_cosines: int = 64, preprocess_net_output_dim: int | None = None,
device: str | int | device = 'cpu')[source]#
Implicit Quantile Network.
☆ preprocess_net – a self-defined preprocess_net which output a flattened hidden state.
☆ action_shape – a sequence of int for the shape of action.
☆ hidden_sizes – a sequence of int for constructing the MLP after preprocess_net. Default to empty sequence (where the MLP now contains only a single linear layer).
☆ num_cosines – the number of cosines to use for cosine embedding. Default to 64.
☆ preprocess_net_output_dim – the output dimension of preprocess_net.
Although this class inherits Critic, it is actually a quantile Q-Network with output shape (batch_size, action_dim, sample_size).
The second item of the first return value is tau vector.
forward(obs: ndarray | Tensor, sample_size: int, **kwargs: Any) tuple[Any, Tensor][source]#
Mapping: s -> Q(s, *).
class IntrinsicCuriosityModule(feature_net: Module, feature_dim: int, action_dim: int, hidden_sizes: Sequence[int] = (), device: str | device = 'cpu')[source]#
Implementation of Intrinsic Curiosity Module. arXiv:1705.05363.
☆ feature_net – a self-defined feature_net which output a flattened hidden state.
☆ feature_dim – input dimension of the feature net.
☆ action_dim – dimension of the action space.
☆ hidden_sizes – hidden layer sizes for forward and inverse models.
☆ device – device for the module.
forward(s1: ndarray | Tensor, act: ndarray | Tensor, s2: ndarray | Tensor, **kwargs: Any) tuple[Tensor, Tensor][source]#
Mapping: s1, act, s2 -> mse_loss, act_hat.
class NoisyLinear(in_features: int, out_features: int, noisy_std: float = 0.5)[source]#
Implementation of Noisy Networks. arXiv:1706.10295.
☆ in_features – the number of input features.
☆ out_features – the number of output features.
☆ noisy_std – initial standard deviation of noisy linear layers.
f(x: Tensor) Tensor[source]#
forward(x: Tensor) Tensor[source]#
Defines the computation performed at every call.
Should be overridden by all subclasses.
Although the recipe for forward pass needs to be defined within this function, one should call the Module instance afterwards instead of this since the former takes care of running the
registered hooks while the latter silently ignores them.
reset() None[source]#
sample() None[source]# | {"url":"https://tianshou.org/en/v1.0.0/03_api/utils/net/discrete.html","timestamp":"2024-11-06T17:10:18Z","content_type":"text/html","content_length":"93216","record_id":"<urn:uuid:1bb34b8c-1d14-4da3-9d38-994aa2da5d04>","cc-path":"CC-MAIN-2024-46/segments/1730477027933.5/warc/CC-MAIN-20241106163535-20241106193535-00861.warc.gz"} |
Instructions on How to Calculate the Cost of Winning Lottery Number 2 at 82lottery
Have you ever dreamt of winning the lottery and becoming an instant millionaire? For many, this is a fantasy that seems out of reach. But for some lucky individuals, it becomes a reality. However,
winning the lottery is not just about luck, but also involves understanding how to calculate the cost of your winning numbers. In this article, we will guide you through the process of calculating
the cost of winning lottery number 2 at 82lottery, so that if you do get lucky, you know exactly how much you have won.
Understanding 82lottery
Before we dive into the cost calculation, let’s first understand what 82lottery is and how it works. 82lottery is a popular lottery game where players choose 6 numbers from a pool of 1-45. The
winning numbers are drawn every Wednesday and Saturday night, and if your chosen numbers match, you win the jackpot. In addition to the jackpot, there are various other prizes for matching a certain
number of numbers. For example, matching 3 or more numbers can still result in a cash prize.
Now that we have a basic understanding of the game, let’s move on to the main topic – calculating the cost of winning.
Step 1: Determine the Cost Per Line
The first step in calculating the cost of winning lottery number 2 at 82lottery is to determine the cost per line. A line refers to one set of 6 numbers selected by the player. The cost per line may
vary depending on your location, but the average cost is around $2 per line. This means that for one ticket with 6 lines, the total cost would be $12.
Step 2: Calculate the Total Number of Lines Played
Once you know the cost per line, the next step is to calculate the total number of lines played. This is determined by the number of different sets of numbers you have chosen. For example, if you
have chosen 5 sets of 6 numbers each, then the total number of lines played would be 5.
Step 3: Determine the Cost of Your Winning Numbers
Now that we have established the cost per line and the total number of lines played, it’s time to determine the cost of your winning numbers. First, you need to know how many numbers you have
matched. Let’s say you have matched 3 numbers out of your 6 chosen numbers. In this case, you will receive a prize for matching 3 numbers, which is usually around $10. However, if you have matched
all 6 numbers, then you have won the jackpot! In this case, the amount you win will depend on the total prize pool and the number of winners.
See more: 82 lottery
Step 4: Calculate Your Total Winnings
To calculate your total winnings, you need to multiply the cost per line by the number of lines played and then subtract it from the cost of your winning numbers. Using the previous example, if the
cost per line is $2 and you played 5 lines, then the total cost would be $10. If your winning numbers are worth $10, then your total winnings would be $0. This means that in this scenario, you have
essentially broken even.
Step 5: Factor in Any Additional Prizes
In addition to the main jackpot, there may also be additional prizes for matching certain combinations of numbers. For example, if you match 2 numbers and the bonus number, you may still receive a
cash prize. These prizes will also need to be factored into your total winnings calculation.
Step 6: Deduct Taxes (if applicable)
Lastly, it’s important to keep in mind that lottery winnings are subject to taxes in some countries. If this applies to you, then you will need to deduct the appropriate amount of tax from your total
winnings. This will vary depending on your location and the amount of your winnings.
To further help you understand how to calculate the cost of winning lottery number 2 at 82lottery, here are some commonly asked questions with their answers:
Q: How do I know if I have won the jackpot?
A: To win the jackpot, you must match all 6 numbers drawn. If your chosen numbers match, then you have won the jackpot!
Q: Can I choose my own numbers or are they randomly generated?
A: You can choose your own numbers for lottery number 2 at 82lottery. However, some players prefer to let the system generate random numbers for them.
Q: Are there any strategies to increase my chances of winning?
A: Lottery games are based on chance and there is no guaranteed strategy to win. However, some players choose to play with a group of people and split the cost of tickets to increase their chances of
Q: How long do I have to claim my prize?
A: The time frame to claim your prize varies depending on your location. It’s important to check with your local lottery agency to ensure you don’t miss out on claiming your winnings.
Q: What happens if there are multiple winners for the same jackpot?
A: If there are multiple winners for the same jackpot, the prize money will be divided equally among them.
Winning the lottery is a thrilling experience, but it’s important to understand how to calculate the cost of your winning numbers to avoid any surprises. By following the steps outlined in this
article, you now have a better understanding of how to determine the cost of winning lottery number 2 at 82lottery. Remember to always play responsibly and have fun! Who knows, you may just be the
next lucky winner. | {"url":"https://tiendabiomama.com/instructions-on-how-to-calculate-the-cost-of-winning-lottery-number-2-at-82lottery/","timestamp":"2024-11-01T20:00:31Z","content_type":"text/html","content_length":"231095","record_id":"<urn:uuid:24d30ac9-3b7f-4b30-9a5b-782150a34d17>","cc-path":"CC-MAIN-2024-46/segments/1730477027552.27/warc/CC-MAIN-20241101184224-20241101214224-00189.warc.gz"} |
Analysis of Absorption with Complex Reversible Reaction
Jerry H. Meldon, Tufts University, Chemical and Biological Engineering Department, Medford, MA 02155
Design of packed scrubbers requires accurate solution to local differential mass balances. The daunting mathematical challenges when there are multiple reversible reactions with nonlinear kinetic
expressions are typically addressed using computationally intensive numerical methods. However, as shown here, the kinetic terms in the ordinary differential equations associated with Film Theory may
be linearized by either of two different techniques with negligible loss of accuracy. One approach, developed by Van Krevelen and Hoftijzer to analyze absorption with a single irreversible reaction,
fixes the concentration of a nonvolatile reactant at its (unknown) value at the gas-liquid interface. The other approach, originally developed by K. A. Smith to analyze permeation with reversible
reaction, assumes that departures from local reaction equilibrium are small. Previously, we demonstrated that either approach yields near-exact values for the “Enhancement Factor” by which a single
reaction multiplies the local absorption rate. We present results here that demonstrate similar accuracy when either linearization technique is applied to the analysis of absorption and multiple
reversible reactions. | {"url":"https://aiche.confex.com/aiche/2008/techprogram/P134348.HTM","timestamp":"2024-11-12T23:37:20Z","content_type":"text/html","content_length":"2589","record_id":"<urn:uuid:de93819d-aa4f-4346-8bf2-7fdc06d3dfe0>","cc-path":"CC-MAIN-2024-46/segments/1730477028290.49/warc/CC-MAIN-20241112212600-20241113002600-00712.warc.gz"} |
Data structures and algorithms - Top ten classic sorts - Moment For Technology
Introduction to the
The top 10 classic sorts are direct insertion sort, Hill sort, bubble sort, quicksort, direct selection sort, heap sort, merge sort, count sort, bucket sort, and radix sort
And sorting is divided into comparison sorting and non-comparison sorting, its classification is shown in the following figure
Its time and space complexity are shown in the figure below:
Stability and instability
Stability means that after sorting, the number of the same elements (objects), after sorting, the order of their order remains unchanged (that is, after sorting, the order of the two equivalent
elements (A, B) before and after array is still A, B)
On the contrary, the order of the elements (objects) with the same number cannot be guaranteed after sorting (that is, the order of the two elements A and B with the same value before and after the
array may become B and A after sorting).
Note: The following are ordered from smallest to largest
To make it easier to understand, when explaining the top 10 classic sorting, we added sorting GIF, first figure and then explain,
Comparison sort
As the name suggests, the pairwise comparison is used to determine the order, and there are seven commonly used comparison sorts: direct insertion sort, Hill sort, bubble sort, quick sort, selection
sort, heap sort, merge sort
Among them, Hill sort, quick sort, heap sort, merge sort are the results of sorting optimization, the problem of high time complexity appears in reverse sort, so they through skip exchange sort,
greatly reduce the number of search exchange, so the time complexity can be reduced from O(n^2) to lower
Direct insertion sort
The figure above illustrates the insertion sort process:
Sorting steps
1. Set the first element to an ordered queue by default, and insert sort from the second element
2. Save the value of the current element by comparing the ith element forward
3. If you find that the preceding elements are larger than you (in order from smallest to largest), assign the previous element to the rotation position of the current element, assign the current
element to the first digit, and repeat Step 3. Otherwise, the comparison ends
4. Assign yourself to the current position
5. The sort ends when the last element is inserted forward
Introduction to Complexity
Time complexity:
The worst is O(n^2), which is the reverse sorting process, and each sorting process actually compares one round forward
It is better to be O(n), which is the positive sorting process, and the internal forward comparison process, which only goes once per round, so the internal is not limited by n
The average is O(n^2), which defaults to an inner and outer two-layer loop to n
Space complexity:
Is O(1), since only a few constant terms are created, it has nothing to do with n (even though the value is assigned once in each round of loop, the temporary variable is saved to the stack, and the
temporary variable is released at the end of the round of loop)
Code implementation
Void sortByStraightInsertion(int list[], int length) {for (int I = 1; i < length; i++) { int pre, current = list[i]; for (pre = i - 1; pre >= 0 && list[pre] > current; pre--) { list[pre + 1] = list[pre]; } list[pre + 1] = current; }}Copy the code
Hill sorting
The figure above demonstrates the process of hill sorting (also known as reduced increment sort). Note that the actual comparison is a skip comparison with an interval:
Hill sort based on insertion sort, set the step gap, reduce its incremental n, to sort, can change the step gap (here the default step size is reduced to half the original), to adjust to improve the
sorting speed, so its complexity varies according to the gap
Sorting steps
1. Set the step to gap (that is, the forward comparison interval). The default is half the length of the list, and each round is reduced to half the current step
2. The default list is divided into several groups virtually by step gap, and the length of each round is length/2, length/4, length/8…
3. Start the insertion sort with interval according to the specified step (interval), start from the gap element, and save as the current element
4. Compare the current element with the previous element with gap distance, and find that the previous element is larger than oneself (in order from smallest to largest), then assign the value of the
previous gap element to the rotation position of the current element, and assign the value of the current element to the gap position, repeat Step 4; Otherwise, the comparison ends
5. Reduce the step gap to half of the original one and repeat Step 3 to perform the insertion sort with interval until the round with step size 1 is completed
Case diagram demonstration
Because the DYNAMIC may not look clear enough, the case diagram is included here
Introduction to Complexity
Time complexity:
By setting the step size, the time complexity of the reverse order is greatly reduced. The time complexity varies greatly according to the different step size, and the average time complexity is O
(nlog2n ~ n^2). By gap optimization, some people calculate that it is about O(n^1.3).
Space complexity:
Is O(1), which is independent of n because only a few constant terms are created
Code implementation
#pragma mark -- pragma mark void sortByShell(int list[], int length) {pragma mark -- pragma mark -- pragma mark void sortByShell(int list[], int length) { For (int gap = length/2; int gap = length/2; gap > 0; gap /= 2) { for (int i = gap; i < length; i++) { int pre, current = list[i]; for (pre = i - gap; pre >= 0 && list[pre] > current; pre -= gap) { list[pre + gap] = list[pre]; } list[pre + gap] = current; }}}Copy the code
Bubble sort
The bubbly sorting process is shown in the figure above
Sorting steps
1. Start with the first element and compare to the next
2. If the following elements are smaller than you (sorted from smallest to largest), you swap them. Otherwise, you do not swap them
3. Repeat step 2 to the last element. The largest element is swapped to the last element.
4. Repeat steps 1 to 3 to exchange the first N-1 elements until they are not exchangeable (there is only one element to be exchanged).
Introduction to Complexity
Time complexity:
The worst is O(n^2), which is the reverse sorting process, and each sorting process actually compares one round forward
It is better to be O(n), which is a positive sorting process. If there is no exchange in the next round, it will be finished directly by setting the exchange or not identifier in the inner layer
The average is O(n^2), and the default is two layers up to n,
Space complexity:
Is O(1), which is independent of n because only a few constant terms are created
Code implementation
#pragma mark void sortByBubble(int list[], int length) {for (int I = 0; i < length; i++) { // int noSort = true; for (int j = 0, last = length - i - 1; j < last; j++) { if (list[j] > list[j + 1]) { int tem = list[j + 1]; list[j + 1] = list[j]; list[j] = tem; // noSort = false; } } // if (noSort) break; // The optimal measures can be optimized to O(n) in the best case, and the first positive order is directly ended}}Copy the code
Quick sort
The quicksort process is shown in the figure above
Quicksort is based on bubble sort. It sets a reference value (usually the first one), puts the smaller value on the left, the larger value on the right, and then takes the reference value on both
sides, and so on, until there is only one element
Note: In fact, quicksort is generally significantly faster than any other o (nlog2n) algorithm, since its internal loop can be implemented efficiently on most of the architectures, and requires
little space, so it is generally examined much
Sorting steps
Here, the selected reference value elements are cleverly used to continuously transpose before and after, reducing some space waste
1. Judge whether the interval of the segmentation sequence can be divided again, that is, if the beginning and end of the interval are the same, and the parameters of the beginning and end of the
interval are low and height, then it is over; otherwise, continue and go to the next step
2. Set the first variable as the reference value, and save a copy of L and H together with the first and last indexes. The position of the reference value variable is idle
3. The queue is traversed and the sequence is split with reference values
4. Start from after H to look up l, each time the index h decreases and finds that it is less than the base value, assign it to L (the position of the first base value, when L was idle). The current
position enters the idle state and enters the next step; If h== L is found, the partition is complete. Otherwise, go to Step 6
5. Search from front L to back h. Each time the index L increases, it is found that the value is greater than the reference value, and it is assigned to h(when the position of H is idle). If h== L is
found, the partition is complete. Otherwise, go to Step 6
6. After steps 4 and 5, L ==h is the actual position (and idle) of the base value, and the base value is assigned to the past
7. Continue to segment the left and right intervals, with the front interval being [low, L-1] and the back interval being [L +1, height]. Repeat Step 1 until the end
Case diagram demonstration
Because the DYNAMIC may not look clear enough, the case diagram is included here
Introduction to Complexity
Time complexity:
The worst is O(n^2), and the base value is usually set to the first. In the case of reverse or positive sorting, each partition with the base value is extreme, and has evolved into bubbling sorting
The best is O(nlog2n). Since a reference value is selected to separate the sequence each time, and the sequence is divided into large on the left and small on the right by the reference value, and
the reference value is selected for the sub-sequence after each round to separate the sequence, the best case is that the median number is just selected each time, and the average number is divided
into two until the end
The average is O(nlog2n), and the segmentation is relatively uniform in most cases because the base value is rarely all extreme cases
Space complexity:
Is O(nlog2n). Since it is in the form of function stack, the base line is set almost dichotomously, and the sequence is divided downward, the variables in the middle will have this accumulation,
which cannot be released immediately. Since it continues to be so until it reaches a single, the memory result accumulates to O(nlog2n) level
Code implementation
#pragma mark -- quick sort //low minimum index, Void quickSortList(int list[], int low, int hight) {if (hight <= low) return; int l = low, h = hight; Int pri = list[l]; While (l < h && list[h] >= pri) h--; while (l < h && list[h] >= pri) h--; if (l == h) break; list[l] = list[h]; While (l < h && list[l] <= pri) l++; if (l == h) break; list[h] = list[l]; } list[l] = pri;} list[l] = pri; quickSortList(list, low, l - 1); quickSortList(list, l + 1, hight); }Copy the code
Direct selection sort
The figure above illustrates the process of selecting sorting directly
Sorting steps
1. Set the first element as the smallest element, and compare it with the next element. If there is a smaller one behind it, then switch
2. Then set the second element as the smallest element and repeat step 1 until the last element. Then the sort is finished
Introduction to Complexity
Time complexity:
The best, the worst and the average are all O(n^2), because it will select the smallest one from each round after n times of comparison, put it in front, and finish after n rounds, so the time
complexity is fixed as O(n^2).
Space complexity:
Is O(1), which is independent of n because only a few constant terms are created
Code implementation
Void sortByStraightSelect(int list[], int length) {for (int I = 0; i < length; i++) { int min = i; for (int j = i + 1; j < length; j++) { if (list[min] > list[j]) { min = j; } } int tem = list[i]; list[i] = list[min]; list[min] = tem; }}Copy the code
Heap sort
The figure above illustrates the heapsort process
Heap sort, according to the characteristics of the heap (big heap, small heap, the parent node is always larger than the child node, small), the previous article has introduced the heap, each
adjustment of the heap process to select the heap, put in the last, readjust the heap to repeat the selection, and so on
Sorting steps
1. Now treat the list as a complete binary tree, transform it into the big head heap (sort from small to large), adjust the heap from bottom to top, and update the sequence
2. Replace the stack with the last element (i.e., select the largest element to the end), and select the element not to participate in the next adjustment of the heap
3. Resize the heap to the big head heap and repeat Step 2 until all the elements in the heap are selected
Introduction to Complexity
Time complexity:
The best, the worst, and the average are all O(nlog2n). The process of adjusting the heap is equivalent to the dichotomy comparison, and n rounds are carried out, so the complexity is O(nlog2n).
Space complexity:
Is O(1), which is independent of n because only a few constant terms are created
Code implementation
#pragma mark -- sort the heap // adjust the heap to normal (adjust the heap to big head, Void heapShift(int list[], int start, int end) {int ori = list[start]; Int sub = start * 2 + 1; While (sub <= end) {while (sub <= end) {// the right node cannot be larger than the index, If (sub < end && list[sub] < list[sub + 1]) sub++; if (ori >= list[sub]) break; List [start] = list[sub]; start = sub; Sub = sub * 2 + 1; } list[start] = ori; } // Set the root node to the root node and the root node to the root node; // set the root node to the root node and the root node to the root node; // set the root node to the root node and the root node to the root node. Void sortByHeap(int oriList[], int length) {int *list = copyList(oriList, int length) length); // Adjust the original array to the big head heap, because the adjustment starts from the parent node, according to the index double relationship, from the half to the root node to form the big head heap // Adjust from the bottom, so when the top is larger than the bottom child node, because the bottom child node has the heap structure, so you can directly end, For (int I = length / 2; i >= 0; HeapShift (list, I, length-1) {// heapShift(list, I, length-1); } for (int i = length - 1; i > 0; Int tem = list[I]; int tem = list[I]; int tem = list[I]; list[i] = list[0]; list[0] = tem; heapShift(list, 0, i - 1); } showSortResult(list, length, "heap sort "); }Copy the code
Merge sort
The figure above shows the merge sort process. (In the actual case, two groups are combined, and the second round generates the reduced group after merging the first round, and so on.)
Merge sort, default to the entire set of elements as a group, each round of two adjacent groups are merged (from small to large, ordered), and so on until merged into a group, that is, sorted
So it follows that the merge process is a small ordered sequence in each group (which will be used later)
Sorting steps
1. By default, each element has its own group, and each group has one. Create a result array of length n
2. Starting from the first one, compare and merge the two adjacent groups from small to large
3. Since each group is a small ordered sequence in the merging process, the two groups are compared from front to back, the small one is added to the result array, and then moved back one bit, and
then compared until one side is placed, and then the other side is placed in the result array in order
4. Repeat Step 3 until the current group is merged
5. Mark the number of groups as two and repeat Step 2 until all groups are merged into one group
Introduction to Complexity
Time complexity:
The best, the worst and the average are all O(nlog2n). Since the binary method is used to merge each time, the number of merges is positively correlated with n, so the complexity is O(nlog2n).
Space complexity:
Is O(n), since a result array of length n is created for the merge group to generate and update the results for each round
Code implementation
#pragma mark -- merge sort; #pragma mark -- merge sort Void mergeList(int mergeList [], int mergeList [], int mergeList [], int mergeList, int mergeList Int group) {if (group >= length) return; int k = 0; // add the last index to the new array, open the interval // divide the array into two groups (length/(2*group)), make sure to get the last group, For (int I = 0, last = ceil(length/(2 * group))); i <= last; I++) {// start and end indexes in pairs int start, startEnd, end, end; start = 2 * i * group; startEnd = start + group > length ? length : start + group; // open interval end = startEnd; endEnd = end + group > length ? length : end + group; If (end > length) break; if (end > length) break; Int I = start, j = end; int I = start, j = end; for (; i < startEnd && j < endEnd; k++) { if (list[i] <= list[j]) { temList[k] = list[i++]; }else { temList[k] = list[j++]; } } if (i >= startEnd) { while (j < endEnd) { temList[k++] = list[j++]; } }else { while (i < startEnd) { temList[k++] = list[i++]; }}} for (int I = 0; int I = 0; i < k; i++) { list[i] = temList[i]; } mergeList(list, temList, length, group * 2); Void sortByMerge(int list[], int length]){// merge (int list[], int length]; Int temList[length]; int temList[length]; mergeList(list, temList, length, 1); }Copy the code
Noncomparative sort
As the name implies, the order is not determined by pair-comparison. There are three kinds of non-comparison sort commonly used at present: counting sort, bucket sort, and radix sort
By virtue of the characteristics of numbers, it finds a new way to solve the sorting problem. It can be used according to the usage scenario, and sometimes can greatly reduce the time and space
Count sorting
The figure above illustrates the counting sort process
It is suitable for sorting integers where numbers are relatively dense, such as sorting 100 million positive numbers between 1 and 1000
Sorting steps
1. Before counting sort, you need to know the maximum and minimum values in advance (or know the number range of sort queue in advance).
2. Create a temporary array for sorting. Determine the size of the temporary array by the difference between the maximum and minimum values
3. Iterate over the array once, and map the corresponding number to the corresponding position of the array by subtracting the minimum value according to its numeric size. The corresponding position
is marked +1
4. Walk through the temporary array once, from small to large, take out the number of values in turn (each index takes out a number marked -1 until it is 0), take out the value of index + the minimum
value, and put it into the result array in turn
Introduction to Complexity
Time complexity:
The best, the worst and the average are all O(n+k), where K is the size of the interval between the array fields. The process of mapping the value has gone through n times, and when the value is
taken out, it will go through K times. Since the possible values are all the same, the maximum is O(n+k).
Therefore, if the array range is too large, it will cause a serious waste of space, and it is more suitable for sorting with relatively dense numbers. For example, sorting 100 million numbers between
1-1000 is very suitable
Space complexity:
Is O(n+k), which includes the field value interval K and the object itself. In the form of the object, it needs a store of N objects to avoid incorrect fetching of elements
Code implementation
#pragma mark -- counting counting (int list[], int length) {void sortByCounting(int list[], int length); min = max = 0; for (int i = 1; i < length; i++) { if (list[min] > list[i]) min = i; if (list[max] < list[i]) max = i; } min = list[min]; max = list[max]; If (min == Max) {showSortResult(list, length, "count sort "); return; } int dk = max - min + 1; Int temList[dk]; // Int temList[dk]; For (int I = 0; int I = 0; int I = 0; i < dk; i++) { temList[i] = 0; } for (int i = 0; i < length; i++) { temList[list[i] - min]++; } int k = 0; for (int i = 0; i < dk; i++) { while (temList[i]-- > 0) { list[k++] = i + min; }}}Copy the code
Bucket sort
Above is the bucket in bucket sort diagram
It can be seen that according to the relationship between the elements, each bucket is divided into several intervals (buckets) with the same value, and the numbers that meet the conditions are put
into the corresponding bucket, and the elements in the bucket are sorted, and then they can be taken out
As you can see, bucket sort is more suitable for well-distributed sequences (if one element is one bucket, it evolves into counting sort).
Sorting steps
1. Find the maximum and minimum values of the numbers in the sequence, and according to the maximum and minimum values
2. Divide the range of the maximum and minimum values into several regions (buckets) to create an array to accommodate sorted elements
3. Subtract the minimum value and put the elements into the corresponding bucket by taking the quotient method
4. Sort the elements in the bucket (blister, select, insert)
5. Remove the elements from the bucket in turn
Introduction to Complexity
Time complexity:
The worst is O(n^2), and all the elements are placed in a bucket, following the worst time complexity of sorting in the bucket
The best is O(n), and the lowest possible is O(n) if an ordered sequence is placed in a bucket, e.g. insert, bubble sort
The average is O(n + K), default inside and outside two layers to n cycle, assuming that the distribution of elements is relatively average, k = m * (n/m) * log2(n/m) = nlog2(n/m) = n(log2n – log2m),
plus n times take out
Space complexity:
Is O(n+k), including the size of the bucket and the size of the resulting data itself, because the bucket contains all the elements, that is: element size + bucket size O(n+k)
Code implementation
#pragma mark -- bucket sort, which is similar to counting sort, but maps to several buckets by scope, saving some memory. You need to use dynamic arrays otherwise you're going to get to O (n + k * n), so every bucket is going to be n and if you don't have a fairly uniform distribution of numbers and it's just an array that's out of order, if you don't, it's not really recommended to use this sort, because the idea is to split the array and sort it in the bucket, Void sortByBucket(int list[], int length) {void sortByBucket(int list[], int length) {// calculate the maximum and minimum values for k interval int min, Max; min = max = 0; for (int i = 1; i < length; i++) { if (list[min] > list[i]) min = i; if (list[max] < list[i]) max = i; } min = list[min]; max = list[max]; If (min == Max) {showSortResult(list, length, "bucket sort "); return; } int dk = max - min; Int bucket = 5; Int bucketSize = ceil(dk/bucket) + 1; LSListNode *p[bucket]; for (int i = 0; i < bucket; i++) { p[i] = NULL; } for (int i = 0; i < length; i++) { int group = (list[i] - min) / bucketSize; P [group] = pushNode(p[group], list[I]); Int k = 0; int k = 0; for (int i = 0; i < bucket; i++) { int count = getListCount(p[i]); Int temList[count] int temList[count] int temList[count]; for (int j = 0; j < count; j++) { temList[j] = geListNode(p[i], j)->data; } for (int x = 0; x < count; x++) { for (int y = 0, last = count - x - 1; y < last; y++) { if (temList[y] > temList[y+1]) { int tem = temList[y+1]; temList[y+1] = temList[y]; temList[y] = tem; }}} for (int j = 0; j < count; j++) { list[k++] = temList[j]; }}}Copy the code
Radix sort
The figure above illustrates the cardinality sorting process
That is, the array of a certain number of digits is sorted from small to large according to the specified number (m), mapped to the array of 0~9 respectively, and then taken out successively. After k
rounds of sorting, the final result is generated
Sorting steps
1. If given the maximum number of digits, use it directly, otherwise find the maximum and calculate the number of digits (in decimal, divide by 10 each time).
2. Take the decimal system as an example. Divide the digits 0 to 9 into 10 mapping arrays
3. Sort from the units digit
4. Use the mod method to take out the corresponding decimal digits and map them to the mapping array from 0 to 9 in sequence
5. Put the values into the array in order from smallest to largest for the next round
6. Continue with the tenth digit and repeat step 4 until you have reached the highest digit
Introduction to Complexity
Time complexity:
The best, the worst and the average are ALL O(NK), where K is the size of a single digit range (0~9) 10, multiplied by the largest digit m, that is, 10mn, that is, O(NK).
Space complexity:
Is O(n+ K), similar to bucket sort and count sort, only need to add (0~9) array interval, accumulated to hold N elements, that is, O(n+k).
Code implementation
#pragma mark void sortByRadixSort(int list[], int length, Int maxBit) {if (maxBit < 1) {if (maxBit < 1); for (int i = 1; i < length; i++) { if (list[maxBit] < list[i]) { maxBit = i; } } int max = list[maxBit]; maxBit = 1; while (max / 10) { max /= 10; maxBit++; } } int d = 10; / / 1 ~ 10; LSListNode *queue[d]; for (int i = 0; i < maxBit; For (int x = 0; int x = 0; x < d; x++) { queue[x] = NULL; } int bitNumber = pow(10, i); for (int k = 0; k < length; k++) { int num = list[k] / bitNumber % 10; queue[num] = pushNode(queue[num], list[k]); } for (int k = 0, j = 0; k < d; k++) { LSListNode *q = queue[k]; int count = getListCount(q); if (count < 1) continue; while (q) { list[j++] = q->data; q = shiftNode(q); }}}}Copy the code | {"url":"https://dev.mo4tech.com/data-structures-and-algorithms-top-ten-classic-sorts.html","timestamp":"2024-11-10T08:19:30Z","content_type":"text/html","content_length":"96295","record_id":"<urn:uuid:062d4658-b291-43ee-a58d-22694d319707>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00011.warc.gz"} |
When the teeth of gears are meshed, the movement of a tooth on one gear results in the movement of a tooth on the other gear. The teeth of each gear move together at the same rate, but when the gears
have different numbers of teeth, the gears rotate at different speeds.
In the pair of blue gears, the large gear has 24 teeth and the small gear has 12 teeth. Each time the small gear makes one complete rotation, its 12 teeth engage with 12 teeth on the large gear.
The large gear has 24 teeth, so after turning by 12 teeth, it hasn't made one complete rotation, it's turned only 12/24, or 1/2 of a rotation. It takes two turns of the small blue gear to turn the
large blue gear one time.
In the pair of orange gears the large gear has 25 teeth and the small gear has 13 teeth. Each time the small gear turns two times, the large gear turns one time plus one tooth. Because of the
additional movement of this one tooth, there is a small difference in the rotational speeds of the two large gears. | {"url":"https://annosphere.weebly.com/demo.html","timestamp":"2024-11-03T10:25:23Z","content_type":"text/html","content_length":"35538","record_id":"<urn:uuid:6f86cce8-82d5-47e9-ba0e-273ea389e982>","cc-path":"CC-MAIN-2024-46/segments/1730477027774.6/warc/CC-MAIN-20241103083929-20241103113929-00221.warc.gz"} |
Lesson 19
Evidence, Angles, and Proof
19.1: Math Talk: Supplementary Angles (5 minutes)
The purpose of this Math Talk is to elicit strategies and understandings students have for determining the angle measures in pairs of intersecting lines or for pairs of angles that make a straight
angle. These understandings help students develop fluency and will be helpful later in this lesson when students will need to be able to explain why vertical angles are congruent. In this activity,
students have an opportunity to notice and make use of structure (MP7) when they identify supplementary angles.
Display one problem at a time. Give students quiet think time for each problem and ask them to give a signal when they have an answer and a strategy. Keep all problems displayed throughout the talk.
Follow with a whole-class discussion.
Representation: Internalize Comprehension. To support working memory, provide students with sticky notes or mini whiteboards.
Supports accessibility for: Memory; Organization
Student Facing
Mentally evaluate all of the missing angle measures in each figure.
Activity Synthesis
Ask students to share their strategies for each problem. Record and display their responses for all to see. To involve more students in the conversation, consider asking:
• “Who can restate \(\underline{\hspace{.5in}}\)’s reasoning in a different way?”
• “Did anyone have the same strategy but would explain it differently?”
• “Did anyone solve the problem in a different way?”
• “Does anyone want to add on to \(\underline{\hspace{.5in}}\)’s strategy?”
• “Do you agree or disagree? Why?”
Speaking: MLR8 Discussion Supports. Display sentence frames to support students when they explain their strategy. For example, "First, I _____ because . . ." or "I noticed _____ so I . . ." Some
students may benefit from the opportunity to rehearse what they will say with a partner before they share with the whole class.
Design Principle(s): Optimize output (for explanation)
19.2: That Can’t Be Right, Can It? (15 minutes)
The purpose of this activity is for students to take an informal conjecture and describe it more precisely by labeling a figure. Students begin by describing three examples of angle bisectors and
forming a conjecture. By engaging with this explicit prompt to take a step back and become familiar with a context and the mathematics that might be involved, students are making sense of problems
(MP1). When students start out labeling the diagram they may use a variety of methods to make sense and explain. The ways students attempt to formulate the conjecture more precisely will be refined
in the discussion.
Making dynamic geometry software available gives students an opportunity to choose appropriate tools strategically (MP5).
Display three examples of angle bisectors of linear pairs for all to see:
Ask students, “What do you notice? What do you wonder?”
Things students may notice:
• there are solid and dashed lines
• there is a horizontal line in all three
• the two solid angles make a linear pair
Things students may wonder:
• Are the dashed lines angle bisectors?
• Do the dashed lines make a right angle?
Ask students to share the things they noticed and wondered. Record and display their responses for all to see. If possible, record the relevant reasoning on or near the image. If the conjecture that
the angle between the angle bisectors is always a right angle does not come up during the conversation, ask students to discuss this idea.
If students have access to GeoGebra Geometry from Math Tools, suggest that it might be a helpful tool in this activity.
Action and Expression: Internalize Executive Functions. Provide students with a table to record what they notice and wonder prior to being expected to share these ideas with others.
Supports accessibility for: Language; Organization
Student Facing
Here is a figure where ray \(r\) meets line \(\ell\). The dashed rays are angle bisectors.
1. Diego made the conjecture: “The angle formed between the angle bisectors is always a right angle, no matter what the angle between \(r\) and \(\ell\) is.” It is difficult to tell specifically
which angles Diego is talking about in his conjecture. Label the diagram and rephrase Diego’s conjecture more precisely using your labels.
2. Is the conjecture true? Explain your reasoning.
Anticipated Misconceptions
If students get stuck, ask them to estimate the measure of one angle and then make arguments based on angle measure like in the warm-up. If time allows, invite students to generalize.
Activity Synthesis
The purpose of this discussion is to introduce the concept of marking angles as congruent, labeling points, and labeling angles.
Ask students to share convincing arguments why Diego’s conjecture is true or false. Label an image displayed for all to see with the information they provide. Students need not write a formal proof
at this point, but encourage students to rephrase using more precise language.
Build on the ideas students have shared about how they labeled the figure to introduce conventions about marking angles as congruent, labeling points, and labeling variable angle measures. For an
example of one way to mark the figure, see the sample student response.
Writing, Speaking: MLR 1 Stronger and Clearer Each Time. Use this with successive pair shares to give students a structured opportunity to revise and refine their response to “For each pair of
congruent angles that you find, explain to your partner how you know the angles are congruent.” Ask each student to meet with 2–3 other partners in a row for feedback. Provide students with prompts
for feedback that will help teams strengthen their ideas and clarify their language (e.g., "Can you explain how…?" "You should expand on...," etc.). Students can borrow ideas and language from each
partner to strengthen the final product.
Design Principle(s): Optimize output (for generalization)
19.3: Convince Me (15 minutes)
The purpose of this activity is for students to prove that vertical angles are congruent and work towards a more formal, rigorous way of expressing themselves when giving arguments based on rigid
transformations. As students work, remind them to label points and make markings on the diagram to help in the process of explaining their ideas.
Monitor for arguments based on:
• transformations
• supplementary angles
Display two intersecting lines for all to see. Remind students that the pairs of angles opposite the intersection point are called vertical angles. Arrange students in groups of 2. Tell students
there are many possible answers for the questions. After quiet work time, ask students to compare their responses to their partner’s and decide if they are both correct, even if they are different.
Follow with whole-class discussion.
Engagement: Internalize Self Regulation. Demonstrate giving and receiving constructive feedback. Use a structured process and display sentence frames to support productive feedback. For example, “How
do you know…?,” “That could/couldn’t be true because…,” and “We can agree that.…”
Supports accessibility for: Social-emotional skills; Organization; Language
Student Facing
Here are 2 intersecting lines that create 2 pairs of vertical angles:
1. What is the relationship between vertical angles? Write down a conjecture. Label the diagram to make it easier to write your conjecture precisely.
2. How do you know your conjecture is true for all possible pairs of vertical angles? Explain your reasoning.
Student Facing
Are you ready for more?
One reason mathematicians like to have rigorous proofs even when conjectures seem to be true is that sometimes conjectures that are made turn out to not be true. Here is one famous example. If we
draw \(n\) points on a circle and connect each pair of points how many regions does that divide the circle into? If we draw only 1 point there are no line segments to connect and so just 1 region in
the circle. If we draw 2 points they are connected by a line segment which divides the circle into 2 regions.
1. If we draw 3 points on a circle and connect each pair of points with a line segment how many regions do we get in our circle?
2. If we draw 4 points on a circle and connect each pair of points with a line segment how many regions do we get in our circle?
3. If we draw 5 points on a circle and connect each pair of points with a line segment how many regions do we get in our circle?
4. Make a conjecture about how many regions we get if we draw \(n\) points on a circle and connect each pair of points with a line segment.
5. Test your conjecture with 6 points on a circle. How many regions do we get?
Anticipated Misconceptions
If students are stuck, suggest they label one of the acute angles as \(x^\circ\). Ask what else they can label or figure out based on that information.
Activity Synthesis
The purpose of discussion is to refine students’ arguments into convincing proofs. With input from students, label points on the figure so that everyone can discuss the same objects consistently. Ask
previously identified students to share their transformational argument. Then invite previously identified students to share their supplementary angles argument.
Ask if either argument relied on the specifics of the particular angles given. Tell students that a proof has to work for any angle measure, otherwise it's an example.
Students will be writing an explanation that vertical angles are congruent in the cool-down, so be sure students understand at least one of these arguments.
Lesson Synthesis
Tell students “In everyday life, it is often complicated to understand all of the reasons why statements are true or false. For example, think about the economic system, international relations, or
the history of a country. In geometry, the objects of study are not as complex: angles, lines, points, triangles, and so on. In this way, geometry is a great training ground for understanding the
reasons why ideas are true and communicating those reasons to others.”
In the previous activity, students came up with different explanations for why vertical angles are congruent. Discuss the difference between arguments based on angle measure and arguments based on
transformations. Ask students,
• “Which argument makes more sense to you, rigid transformations that take one vertical angle onto the other, or using straight angles to look at 180 degree sums?” (Transformations make more sense
because it’s possible to see how one angle is taken onto the other. I like algebra better so I prefer the 180 degree sums.)
• “What is the difference between angle and angle measure?” (It’s like the difference between a segment and its length. Segments and angles are geometric figures, but lengths and angle measures are
numbers used to describe how large or small the segments and angles are.)
Ask students to add this theorem to their reference charts as you add it to the class reference chart:
Vertical angles are congruent.
Provide the tip: Look for vertical angles whenever two lines intersect.
19.4: Cool-down - Plead Your Case (5 minutes)
Student Facing
In many situations, it is important to understand the reasons why an idea is true. Here are some questions to ask when trying to convince ourselves or others that a statement is true:
• How do we know this is true?
• Would these reasons convince someone who didn’t think it was true?
• Is this true always, or only in certain cases?
• Can we find any situations where this is false?
In this lesson, we reasoned that pairs of vertical angles are always congruent to each other:
We saw this by labeling the diagram and making precise arguments having to do with transformations or angle relationships. For example, label the diagram with points:
Rotate the figure 180 degrees around point \(E\). Then ray \(EA\) goes to ray \(EB\) and ray \(ED\) goes to ray \(EC\). That means the rotation takes angle \(AED\) onto angle \(BEC\), and so angle \
(AED\) is congruent to angle \(BEC\).
Many true statements have multiple explanations. Another line of reasoning uses angle relationships. Notice that angles \(AED\) and \(AEC\) together form line \(CD\). That means that \(x + y = 180\).
Similarly, \(y + w = 180\). That means that both \(x\) and \(w\) are equal to \(180-y\), so they are equal to each other. Since angle \(AED\) and angle \(CEB\) have the same degree measure, they must
be congruent. | {"url":"https://im-beta.kendallhunt.com/HS/teachers/2/1/19/index.html","timestamp":"2024-11-11T06:25:04Z","content_type":"text/html","content_length":"121602","record_id":"<urn:uuid:d17f5ce6-ae35-4f68-a5a3-06c22168f4db>","cc-path":"CC-MAIN-2024-46/segments/1730477028220.42/warc/CC-MAIN-20241111060327-20241111090327-00676.warc.gz"} |
Illustrative Mathematics
Estimating a Solution via Graphs
Jason and Arianna are working on solving the system of linear equations
$$ 6x + 17y &= 100\\ 5x + 9y &= 86. $$
Rounding their answer to the nearest hundredth, Jason and Arianna find that $x \approx 4.04$ and $y \approx 7.31$.
1. Explain, in terms of the slopes of the graphs of the equations in the system, why you know that there there is a unique solution to this system.
2. Show how you can tell from the graphs of the equations that Jason and Arianna must have made a mistake.
3. Give a numerical explanation in terms of the slopes and $y$-intercepts of the graphs of the equations how you could tell that Jason and Arianna must have made a mistake.
IM Commentary
The purpose of this task is to give students an opportunity use quantitative and graphical reasoning to detect an error in a solution. The equations have been chosen so that finding the exact
solution requires significant calculation so that it is easy to make an error. The particular solution given comes from multiplying the first equation by 5 and the second equation by 6 but when the
second equation is subtracted from the first, the $y$'s on the left hand side and the numbers on the right hand side are added instead of subtracted. Once this $y$ value is ''miscalculated'' the
corresponding $x$ value comes from substituting in the second equation.
In the solution below, the graphs are produced using Desmos. Students can also sketch the graphs by hand, perhaps after putting them in slope-intercept form. Although graphing will show that Jason
and Arianna are not correct, it will only give an approximate solution. The linear equations in this task are not presented in the most common format highlighting the slope and y-intercept. The
teacher may wish to follow this task up with other pairs of equations in slope-intercept form, always focusing on whether or not the solution is reasonable.
1. The two equations provided are linear so their graphs will intersect in exactly one point provided they are not parallel. The slope of the line defined by the equation $6x + 17y = 100$ is $-\frac
{6}{17}$ while the slope of the line defined by the equation $5x + 9y = 86$ is $-\frac{5}{9}$. Since $-\frac{6}{17} \neq -\frac{5}{9}$ these lines are not parallel and so they intersect in
exactly one point.
2. The graphs of the two linear equations are shown below:
They appear to meet when $x$ is around 17 or 18 and $y$ is about -1. This is very far from what Jason and Arianna found and so their answer is not reasonable.
3. Note that the $y$ intercept of the line defined by the equation $6x + 17y = 100$ is $\frac{100}{17}$ or about 6. The $y$-intercept of the line defined by the equation $5x + 9y = 86$ is $\frac{86}
{9}$ or a little more than 9. The slope of this second line, $-\frac{5}{9}$, is less than the slope of the first, $-\frac{6}{17}$. This means that the lines will meet when $x \gt 0$. An $x$ value
near 4 is far too small, however, because the $y$ intercepts differ by more than 3 and the slopes differ by a very small amount (about 0.2). So $x$ will need to be larger than 15 before the blue
line drops enough, from the $y$-axis, to meet the red line.
The coordinates of the point of intersection are $\left(18\frac{4}{31}, -\frac{16}{31}\right)$.
Estimating a Solution via Graphs
Jason and Arianna are working on solving the system of linear equations
$$ 6x + 17y &= 100\\ 5x + 9y &= 86. $$
Rounding their answer to the nearest hundredth, Jason and Arianna find that $x \approx 4.04$ and $y \approx 7.31$.
1. Explain, in terms of the slopes of the graphs of the equations in the system, why you know that there there is a unique solution to this system.
2. Show how you can tell from the graphs of the equations that Jason and Arianna must have made a mistake.
3. Give a numerical explanation in terms of the slopes and $y$-intercepts of the graphs of the equations how you could tell that Jason and Arianna must have made a mistake. | {"url":"http://tasks.illustrativemathematics.org/content-standards/HSA/REI/C/6/tasks/1833","timestamp":"2024-11-08T04:48:02Z","content_type":"text/html","content_length":"27817","record_id":"<urn:uuid:b6eef59d-8c78-4f63-874f-7e51cfb797a7>","cc-path":"CC-MAIN-2024-46/segments/1730477028025.14/warc/CC-MAIN-20241108035242-20241108065242-00204.warc.gz"} |
@Article{CiCP-35-212, author = {Na , Xuyang and Xu , Xuejun}, title = {Domain Decomposition Methods for Diffusion Problems with Discontinuous Coefficients Revisited}, journal = {Communications in
Computational Physics}, year = {2024}, volume = {35}, number = {1}, pages = {212--238}, abstract = {
In this paper, we revisit some nonoverlapping domain decomposition methods for solving diffusion problems with discontinuous coefficients. We discover some interesting phenomena, that is, the
Dirichlet-Neumann algorithm and Robin-Robin algorithms may make full use of the ratio of coefficients in some special cases. Detailedly, in the case of two subdomains, we find that their convergence
rates are $\mathcal{O}(ν_1/ν_2)$ if $ν_1 < ν_2,$ where $ν_1, \ ν_2$ are coefficients of two subdomains. Moreover, in the case of many subdomains with red-black partition, the condition number bounds
of Dirichlet-Neumann algorithm and Robin-Robin algorithm are $1+\epsilon(1+{\rm log}(H/h))^2$ and $C+\epsilon(1+ {\rm log}(H/h))^2,$ respectively, where $\epsilon$ equals ${\rm min}\{ν_R/ν_B,ν_B/ν_R
\}$ and $ν_R,ν_B$ are the coefficients of red and black domains. By contrast, Neumann-Neumann algorithm and Dirichlet-Dirichlet algorithm could not obtain such good convergence results in these
cases. Finally, numerical experiments are preformed to confirm our findings.
}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2023-0184}, url = {http://global-sci.org/intro/article_detail/cicp/22901.html} } | {"url":"https://www.global-sci.org/intro/article_detail/getBib?article_id=22901","timestamp":"2024-11-10T14:17:12Z","content_type":"application/x-bibtex-text-file","content_length":"1807","record_id":"<urn:uuid:6abd848c-3e7d-48c9-97dc-4b91ed1fbe23>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.60/warc/CC-MAIN-20241110134821-20241110164821-00376.warc.gz"} |
CURRICULUM VITAE - JAMES W. DEARDORFF
1. Vital Statistics
Born: 28 August 1928, Seattle, Washington
Present Position: Professor Emeritus
Department of Atmospheric Sciences
(College of Oceanic and Atmospheric Sciences)
Oregon State University
Corvallis, Oregon 97331
E-mail address: deardorj@proaxis.com
2. Education
(1942-1946) - Lincoln High School, Portland, OR
B.S. (1950) - Physics, Stanford University
B.S. (2nd) (1951) - Meteorology, University of California at Los Angeles
M.S. (1956) - Meteorology, University of Washington
Ph.D. (1959) - Meteorology, University of Washington
3. Professional Employment
1951-1955: Line Officer; Special Weapons Electrical Officer, U.S. Navy
1955-1958: Research and Teaching Assistant, University of Washington
1958-1959: Acting Instructor, Meteorology, University of Washington
1959-1962: Senior Scientist, Air-Sea Interface, University of Washington
1962-1978: Senior Scientist, National Center for Atmospheric Research,
Boulder, Colorado
1974-1978: Head, Small Scale Analysis and Prediction Project,
National Center for Atmospheric Research
1978-1986: Research Professor, Department of Atmospheric Sciences,
Oregon State University
1986-: Professor Emeritus, Oregon State University
4. Other Professional Employment
1966-1967: One year at University of Washington, teaching meteorology
1970-1971: Six-month sabbatical at UCLA working on boundary-layer parameterization
1973-1974: Seven month leave of absence at University of Stockholm, Sweden
(International Meteorological Institute) to work on parameterization
of effects of scattered cumulus clouds on the subcloud layer
1980-1983: Co-editor (one of several), Journal of the Atmospheric Sciences
1982-1987: Assoc. Editor, Boundary-Layer Meteorology
5. Professional Affiliations, Honors or Awards
1971 - Editorial Award, American Meteorological Society
1972 - Publications Award, National Center for Atmospheric Research
1973 - Elected a Fellow of the American Meteorological Society
1974 - Second Half-Century Award for furthering "our understanding of turbulent
processes in the planetary boundary layer through analytical studies and
highly original numerical and laboratory experiments," American
Meteorological Society
1978 - Rossby Research Medal, American Meteorological Society.
1986 - Was elected a Fellow of the American Association for the Advancement of
6. Publications in Reviewed Literature
1958: The average slope of the surface wind, J. Meteor., 15, 334-335.
1958: Vertical distribution of wind speed, temperature and humidity above a water
surface, J. Marine Res. 17, 141-157 (R.G. Fleagle, J.W. Deardorff, and F.J. Badgley).
1961: On the direction and divergence of the small-scale turbulent heat flux,
J. Meteor. 18, 540-548.
1961: Evaporation reduction by natural surface films, J. Geophys. Res. 66,
1961: Local evaporation from a smooth water surface, J. Geophys. Res. 66,
1962: Satellite cloud photos and large-scale vertical motion, J. Appl. Meteor.
2, 173-175.
1963: On the stability of viscous plane Couette flow, J. Fluid Mech. 15,
1964: A numerical study of two-dimensional parallel-plate convection, J. Atmos.
Sci. 21, 419-438.
1965: Gravitational instability between horizontal plates with shear, Phys.
Fluids 8, 1027-1030.
1965: A numerical study of pseudo three-dimensional parallel-plate convection,
J. Fluid Mech. 22, 419-435.
1965: The effect of two-dimensionality on the suppression of thermal turbulence,
J. Fluid Mech. 23, 337-353 (J.W. Deardorff and G.E. Willis).
1965: Measurements on the development of thermal turbulence in air between
horizontal plates, Phys. Fluids, 8, 2225-2229 (G.E. Willis and J.W. Deardorff).
1966: The countergradient heat flux in the lower atmosphere and in the laboratory,
J. Atmos. Sci. 23, 503-506.
1967: Investigation of turbulent thermal convection between horizontal plates,
J. Fluid Mech. 28, 675-704 (J.W. Deardorff and G.E. Willis).
1967: Development of short-period temperature fluctuations in thermal convection,
Phys. Fluids, 10, 931-937 (G.E. Willis and J.W. Deardorff).
1967: Empirical dependence of the eddy coefficient for heat upon stability above
the lowest 50m, J. Appl. Meteor. 6, 631-643.
1967: The free-convection temperature profile, Quart. J. Roy. Meteor. Soc.
93, 166-175 (J.W. Deardorff and G.E. Willis).
1967: Comment on paper by G.E. Harbeck, Jr., 'A note concerning the eddy transfer
coefficients of momentum and water vapor under near-adiabatic conditions,'
Water Resources Res. 3, 909-910.
1967: Confirmation and renumbering of the discrete heat flux transitions of Malkus,
Phys. Fluids, 10, 1861-1866 (G.E. Willis and J.W. Deardorff).
1967: Aerodynamic theory of wave growth with constant wave steepness, J.
Oceanog. Soc. Japan 23, 12-30.
1968: Dependence of air-sea transfer coefficients on bulk stability, J. Geophys.
Res. 73, 2549-2557.
1968: Examination of numerically calculated heat fluxes for evidence of a
supercritical transition, Phys. Fluids, 11, 1254-1256.
1968: On the distinction between 'total' heat flux and eddy heat flux, J. Atmos.
Sci. 25, 521-522 (J.W. Deardorff and J.A. Businger).
1969: Laboratory investigation of non-steady penetrative convection, J. Fluid
Mech. 35, 7-31 (J.W. Deardorff, G.E. Willis, and D.K. Lilly).
1969: Numerical study of heat transport by internal gravity waves above a growing
unstable layer, Phys. Fluids Suppl. II 184-194.
1969: Similarity principles for numerical integrations of neutral barotropic
planetary boundary layers and channel flows, J. Atmos. Sci. 26, 763-767.
1970: A numerical study of three-dimensional turbulent channel flow at large
Reynolds numbers, J. Fluid Mech. 41, 453-480.
1970: Convective velocity and temperature scales for the unstable planetary
boundary layer and for Rayleigh convection, J. Atmos. Sci. 27, 1211-1213.
1970: Lagrangian statistics from numerically integrated shear flow, Phys.
Fluids 13, 584-595 (J.W. Deardorff and R.L. Peskin).
1970: The oscillatory motions of Rayleigh convection, J. Fluid Mech. 44,
661-672 (G.E. Willis and J.W. Deardorff).
1970: Discussion of paper by V.H. Regener and L. Aldaz, 'Turbulent transport near
the ground as determined from measurements of the ozone gradient,' J. Geophys
Res. 75, 4184-4186.
1970: A three-dimensional numerical investigation of the idealized planetary
boundary layer, Geophys. Fluid Dynamics l, 377-410.
1970: Preliminary results from numerical integrations of the unstable planetary
boundary layer, J. Atmos. Sci. 27, 1211-1213.
1970: A forum for post-clarification of discussions at AMS meetings? (Letter to the
Editor), Bulletin, Amer. Meteor. Soc. 51, 435.
1971: On the magnitude of the subgrid scale eddy coefficient, J. Comp. Phys.
7, 120-133.
1971: Comments on 'Observational studies in the atmospheric boundary layer' by R.H.
Clarke, Quart. J. Roy. Meteor. Soc. 97, 760-761.
1972: Parameterization of the planetary boundary layer for use in general
circulation models, Mon. Wea. Rev. 100, 93-106.
1972: Roll-diameter dependence in Rayleigh convection and its effect upon the heat
flux, J. Fluid Mech. 59, 351-357 (G.E. Willis, J.W. Deardorff, and R.C.J.
1972: Numerical investigation of neutral and unstable planetary boundary layers,
J. Atmos. Sci. 29, 91-115.
1972: Theoretical expression for the countergradient vertical heat flux, J.
Geophys. Res. 77, 5900-5904.
1972: Comments on 'A comparison of circulations in transverse and longitudinal
planes in an unstable planetary boundary layer' by J.K. Angell, J. Atmos. Sci.
29, 1394-1395.
1972: Computer methods for simulation of multidimensional, nonlinear subsonic,
incompressible flow, J. Heat Transfer 94, 337-346 (D.G. Fox and J.W.
1973: Note on a paper by D.R. Caldwell, C.W. Van Atta and K.N. Helland
(concerning a laboratory study of the Ekman layer), Geophys. Fluid Dynamics
4, 293-295.
1973: The use of subgrid transport equations in a three-dimensional model of
atmospheric turbulence, J. Fluid Eng. 95, 429-438.
1973: An explanation of anomalously large Reynolds stresses within the convective
planetary boundary layer, J. Atmos. Sci. 30, 1070-1076.
1974: Differences between eddy coefficients for instantaneous and continuous
vertical diffusion into the neutral surface layer, Boundary-Layer Meteor. 5,
1974: Computer and laboratory modeling of the vertical diffusion of non-buoyant
particles in a mixed layer, Advances in Geophysics 18B, 187-200 (J.W. Deardorff
and G.E. Willis).
1974: Comment on a paper by A.K. Betts, 'Non-precipitating cumulus convection and
its parameterization,'Quart. J. Roy. Meteor. Soc. l00, 122-123 (J.W.
Deardorff, G.E. Willis, and D.K. Lilly).
1974: Stability functions for the boundary layer resistance laws based upon
observed boundary layer height, J. Atmos. Sci. 31, 1324-1325 (J.W. Melgarejo
and J.W. Deardorff).
1974: Three-dimensional numerical study of the height and mean structure of a
heated planetary boundary layer, Boundary-Layer Meteor. 7, 81-106.
1974: Three-dimensional study of turbulence in an entraining mixed layer,
Boundary-Layer Meteor. 7, 199-226.
1974: Similarity theory for the planetary boundary layer of time-dependent height,
J. Atmos. Sci. 31, 1449-1452 (S.S. Zilitinkevich and J.W. Deardorff).
1974: A laboratory model of the unstable planetary boundary layer, J. Atmos.
Sci. 31, 1297-1307 (G.E. Willis and J.W. Deardorff).
1974: Reply (to Pasquill and Smith), Boundary-Layer Meteor. 7, 229-230.
1975: Comments on 'On the interaction between the subcloud and cloud layers in
tropical regions' by Y. Ogura and H.-R. Cho, J. Atmos. Sci. 32, 2363-2364.
1975: A parameterization of diffusion into the mixed layer, J. Appl. Meteor.
14, 1451-1458 (J.W. Deardorff and G.E. Willis).
1975: Reply (to Arya and Wyngaard (1974)), J. Atmos. Sci. 32, 840 (S.S.
Zilitinkevich and J. W. Deardorff).
1975: Revision to 'Stability functions for the boundary-layer resistance laws',
J. Atmos. Sci. 32, 837-839 (J.W. Melgarejo and J.W. Deardorff).
1976: Usefulness of liquid-water potential temperature in a shallow-cloud model,
J. Appl. Meteor. 1, 98-102.
1976: On the entrainment rate of a stratocumulus-topped mixed layer, Quart. J.
Roy. Meteor. Soc. 102, 563-582.
1976: Discussion of 'Thermals over the sea and gull flight behavior' by A.H.
Woodcock, Boundary-Layer Meteor. 10, 241-246.
1976: Island wind shadows observed by satellite and radar, Bull. Amer. Meteor.
Soc. 57, 1241-1242.
1976: A laboratory model of diffusion into the convective planetary boundary layer
Quart. J. Roy. Meteor. Soc. 102, 427-445 (G.E. Willis and J.W. Deardorff).
1976: On the use of Taylor's translation hypothesis for diffusion in the mixed
layer, Quart. J. Roy. Meteor. Soc. 102, 817-822 (G.E. Willis and J.W.
1977: Subgrid-scale condensation in models of nonprecipitating clouds, J.
Atmos. Sci. 34, 345-355 (G. Sommeria and J.W. Deardorff).
1977: A parameterization of ground-surface moisture content for use in atmospheric
prediction models, J. Appl. Meteor. 16, 1182-1185.
1977: Comments on 'The effect of variable surface albedo on the atmospheric
circulation in desert regions.' J. Appl. Meteor. 17, 560 (Sherwood B. Idso and J.W.
1977: Workshop on stability classification schemes and sigma curves,
Bull. Amer. Meteor. Soc. 58, 1305-1309 (S.R. Hanna, G.A. Briggs, J.W. Deardorff,
B.A. Egan, F.A. Gifford, F. Pasquill).
1978: Efficient prediction of ground surface temperature and moisture with
inclusion of a layer of vegetation, J. Geophys. Res. 83, 1889-1903.
1978: A laboratory study of dispersion from an elevated source within a modeled
convective planetary boundary layer, Atmos. Environ. 12, 1305-1311 (G.E.
Willis and J.W. Deardorff).
1978: Reply to Panofsky, Atmos. Envir. 12, 2036 (J.W. Deardorff and G.E.
1978: Closure of second- and third-moment rate equations for diffusion in
homogeneous turbulence, Phys. of Fluids, 21, 525-530.
1978: Summary of recommendations made by the AMS Workshop on Stability
Classification Schemes and Sigma Curves, Bull. Amer. Meteor. Soc. 59,
1025-1033 (J.R. Hanna, G.A. Briggs, J.W. Deardorff, B.A. Egan, F.A. Gifford
and F. Pasquill).
1979: Prediction of mixed layer entrainment for realistic capping inversion
structure, J. Atmos. Sci. 36, 424-436.
1979: Laboratory observations of turbulent penetrative-convection planforms, J.
Geophys. Res. 84, 295-302, (G.E. Willis and J.W. Deardorff).
1980: Cloudtop entrainment instability, J. Atmos. Sci. 37, 131-147.
1980: Comments on 'Marine stratocumulus convection. Part 1: Governing equations and
horizontally homogeneous solutions,' J. Atmos. Sci. 37, 481-482
(J.W. Deardorff and J. Businger).
1980: The boundary-layer growth equations with Reynolds averaging, J. Atmos.
Sci. 37, 1405-1409 (J.W. Deardorff and E.W. Peterson).
1980: Stratocumulus-capped mixed layers derived from a three-dimensional model,
Bound.-Layer Meteor. 18, 495-527.
1980: Laboratory studies of the entrainment zone of a convectively mixed layer,
J. Fluid Mech. 100, 41-64 (J.W. Deardorff, G.E. Willis, and B.H. Stockton).
1980: Comments on 'A numerical investigation of mixed-layer dynamics.' J. Phys.
Oceanogr. 10, 1695-1696.
1981: A laboratory model of dispersion from a source in the middle of the
convectively mixed layer, Atmos. Environ. 15, 109-117 (G.E. Willis and
J.W. Deardorff).
1981: On the distribution of mean radiative cooling at the top of a stratocumulus-
capped mixed layer, Quart. J. Roy. Meteor. Soc. 107, 191-202.
1981: Further considerations on the Reynolds average of the kinematic boundary
condition. J. Atmos. Sci. 38, 659-661.
1982: Dependence of mixed-layer entrainment on shear stress and velocity jump,
J. Fluid Mech. 115, 123-149 (J.W. Deardorff and G.E. Willis).
1982: Ground-level concentrations due to fumigation into an entraining mixed layer.
Atmos. Environ. 16, 1159-1170 (J.W. Deardorff and G.E. Willis).
1982: Investigation of the frozen-turbulence hypothesis for temperature spectra in
a convectively mixed layer. Phy. Fluids 25, 21-28 (J.W. Deardorff and
G.E. Willis).
1982: Numerical study of terrain-induced mesoscale motions in a mixed layer. J.
Atmos. Sci. 39, 2464-2476 (Y.-J. Han, K. Ueyoshi and J.W. Deardorff).
1982: A numerical simulation of an atmospheric vortex street. Tellus 34,
555-556 (P.H. Ruscher and J.W. Deardorff).
1982: On the dichotomy in theoretical treatments of the atmospheric boundary layer
J. Atmos. Sci. 39, 2096-2098 (J.W. Deardorff and L. Mahrt).
1982: Further considerations on modeling the sea breeze with a mixed-layer model.
Mon. Wea. Rev. 110, 757-765. (R.A. Anthes, D. Keyser and J.W. Deardorff).
1983: Comment on 'A potential flow model of turbulence caused by breaking surface
waves'. J. Geophys. Res. 88, 2710.
1983: Authors' reply to comments on ground-level concentrations due to fumigation
Atmos. Environ. 17, 1030-1032 (J.W. Deardorff and G.E. Willis).
1983: A multi-limit mixed-layer entrainment formulation. J. Phys. Oceanog.
13, 988-1002.
1983: Comments on 'The daytime planetary boundary layer; a new interpretation of
Wangara'. Quart. J. Roy. Met. Soc. 109, 677-681.
1983: On plume rise within a convective boundary layer, Atmos. Environ. 17,
2435-2447 (G.E. Willis and J.W. Deardorff).
1983: Comments on 'A diagnostic model for estimating winds at potential sites for
wind turbines'. J. Climate and Appl. Meteor. 22, 1312 (J.W. Deardorff and Y.-J. Han).
1984: On the use of an annulus to study mixed-layer entrainment. J. Fluid Mech.
142, 97-120 (J.W. Deardorff and S.-C. Yoon).
1984: Groundlevel concentration fluctuations from a buoyant and a non-buoyant
source within a laboratory convectively mixed layer. Atmos. Environ. 18, 1297-1309
(J.W. Deardorff and G.E. Willis).
1984: Numerical study of terrain-induced mesoscale motions and hydrostatic form
drag in a heated, growing mixed layer. J. Atmos. Sci. 41, 1420-1441 (J.W.
Deardorff, K. Ueyoshi, and Y.-J. Han).
1985: Further results from a laboratory model of the convective planetary boundary
layer. Boundary-Layer Meteor. 32, 205-236.
1985: Sub-grid-scale turbulence modeling. In Adv. in Geophys. 28B, 337-343.
1985: Comments on 'Transilient turbulence theory, Part I'. J. Atmos. Sci.
42, 2069.
1985: Laboratory experiments on diffusion: The use of convective mixed-layer
scaling, J. Climate and Appl. Meteor. 24, 1143-1151.
1985: Authors' reply to 'Ground-level concentration fluctuations from a buoyant and
a non-buoyant source...'. Atmos. Envir. 18, 1212-1213.
1985: Book Review of Large-Eddy Simulation: Guidelines for its Applications to
Planetary Boundary Layer Research, J.C. Wyngaard, ed., Bull. Amer.
Meteor. Soc. 66 (Dec).
1986: Comments on 'Radiative cooling near the top of a cloudy mixed layer,'
Quart. J. Roy. Meteor. Soc. 112, 273-275.
1987: Buoyant plume dispersion and inversion entrapment in and above a laboratory
mixed layer. Atmos. Envir. 21, 1725-1735 (G.E. Willis and J.W. Deardorff).
1987: Turbulence within a baroclinic laboratory mixed layer above a sloping
surface, J. Atmos. Sci. 44, 772-778 (with G.E. Willis).
7. Other Publications and Reports
1973: Three-dimensional numerical modeling of the planetary boundary layer, in
Workshop on Micrometeorology (Duane A. Haugen, editor), American Meteorol.
Society, Boston, 14-18 August, Science Press, 277-311.
1974: Rate of growth of the nocturnal boundary layer, in Proc. of Symposium on
Air Pollution, Turbulence and Diffusion, December 1971, Las Cruces, New
Mexico (H.W. Church and R.E. Luna, editors), 183-190.
1974: Computer and laboratory modeling of the vertical diffusion of non-buoyant
particles in a mixed layer, in Turbulent Diffusion in Environment Pollution
(R.N. Frenkiel and R.E. Munn, editors), Academic Press, New York/London (J.W.
Deardorff and G.E. Willis).
1974: Physical modeling of diffusion in the mixed layer, in Proc. of Symposium
on Atmospheric Diffusion and Air Pollution, American Meteorological Society,
Santa Barbara, California, 9-13 September (J.W. Deardorff and G.E. Willis).
1974: Means by which the planetary boundary layer affects the free troposphere, in
Subsynoptic Extratropical Weather Systems: Observations, Analysis, Modeling
and Prediction, Vol. II, Proc. ASP/SSAPP Summer Colloquium, NCAR,
Boulder, Colorado, 670-682.
1975: The development of boundary-layer turbulence models for use in studying the
severe storm environment, in Open Sesame, Proc. of Mtg. at Boulder, 4-6 September
(D.K. Lilly, editor) NOAA/ERL, Boulder, Colorado, 251-264.
1975: Laboratory simulation of the convective planetary boundary layer, in
Atmospheric Technology 7, National Center for Atmos. Research, 30-86 (G.E.
Willis and J.W. Deardorff).
1975: Boundary layer data from a numerical integration in three dimensions;
manuscript available from authors (J.W. Deardorff and Margaret Drake).
1976: Clear and cloud-capped mixed layers: their structure and growth, numerical
simulation, and parameterization. In Proc. of the European Center for Medium Range
Weather Forecasts Seminar Series: Treatment of the Boundary Layer in Numerical
Weather Prediction, 6-10 September, Bracknell, England.
1976: Neglect of downstream diffusion--how good an assumption for the daytime mixed
layer? Preprints, American Meteorological Society Third Symposium on Atmospheric
Turbulence, Diffusion and Air Quality, Raleigh, North Carolina, 19-22 October,
255-258 (J.W. Deardorff and G.E. Willis).
1976: Visual observations of horizontal planforms of penetrative convection paper
for Third Symposium on Atmospheric Tubulence, Diffusion and Air Quality, Raleigh,
North Carolina, 19-22 October, 9-12 (G.E. Willis and J.W. Deardorff).
1978: Different approaches toward predicting pollutant dispersion in the boundary
layer, and their advantages and disadvantages, in Symp. on Boundary Layer Physics
Applied to Specific Problems of Air Pollution. Swedish Meteorological and
Hydrological Institute (S.M.H.I), Norrköping, Sweden, 19-23 June.
1980: Progress in understanding entrainment at the top of a mixed layer, in
Workshop on the Planetary Boundary Layer, Boulder, Colorado, 14-18 August,
1978, American Meteorological Society, Boston, Massachusetts (J.C. Wyngaard,
editor), 36-66.
1981: How time dependence and variable Froude number can explain more rapid
entrainment of the two-layer system in annulus experiments, in Third Symp.
on Turbulent Shear Flows, University of California, Davis, September 9-11,
1981: Modeling fumigation in a laboratory mixed layer, in Fifth Symp. on
Turbulence, Diffusion and Air Pollution, Atlanta, Georgia, March 9-13,
American Meteorological Society, 157-158.
1982: Simulation of terrain effects using a mesoscale mixed-layer model, in
Proc. 10th IMACS World Congress on Systems Simulation and Scientific
Computation, Montreal, Canada, August 8-13, 195-196.
1983: Concentration fluctuations within buoyant and non-buoyant laboratory
plumes, in Sixth Symp. on Turbulence and Diffusion, Amer. Meteor. Soc.,
Boston, March 22-25, 237-240.
1983: Dependence of laboratory mixed-layer entrainment rates upon interfacial
turbulence and stability, in Sixth Symp. on Turbulence and Diffusion,
Amer. Meteor. Soc., Boston, March 22-25, 321-324.
1984: Upstream diffusion in the convective boundary layer with weak or zero mean
wind, in Fourth Joint Conf. on Applications of Air Pollution Meteorology
Amer. Meteor. Soc., Boston, October 16-19, 4-7.
1985: Review of "Large-Eddy Simulation: Guidelines for its application to planetary boundary layer research," J.C. Wyngaard, ed., Bull. Amer. Meteor. Soc. 66, 1552. 1987: Notice of 1986 retirement.
Bull. Amer. Meteor. Soc. 68 (Oct.), 1295.
1988: Concentration fluctuations within a laboratory convectively mixed layer,
in Lectures in Air Pollution Modeling, 357-384 (Chap. 8), A. Venkatram and J.C.
Wyngaard, eds.; Boston, Amer. Meteor. Soc. (with G.E. Willis).
1986: Possible extraterrestrial strategy for Earth. Quart. J. Roy. Astron.
Soc. 27, 94-101.
1987: Examination of the embargo hypothesis as an explanation for the Great
Silence. J. British Interplanetary Soc. 40, 373-379.
1987: Extraterrestrial Communications. J. Communication 37, 181-184.
1990: Celestial Teachings: Emergence of the True Testament of Jmmanuel (Jesus).
Mill Spring, NC: Wild Flower Press. 1-800-366-0264.
1992: The Problems of New Testament Gospel Origins. Lewiston, NY: Mellen
Press (Mellen University Research Press; see www.mellenpress.com).
1994: Jesus in India: A Reexamination of Jesus' Asian Traditions
in the Light of Evidence Supporting Reincarnation.
Bethesda, MD: International Scholars Publications.
2001: Opinions and comments on W.C. Levengood and N.P. Talbott (1999)
'Dispersion of energies in worldwide crop formations,' Physiologia Plantarum
111, 125.
2005: Inflation-theory implications for extraterrestrial visitation,
J. British Interplanetary Soc. 58, 43-50. (J. Deardorff, B. Haisch,
B. Maccabee and H.E. Puthoff)
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Understanding stride for the output of nonzero
Hello everyone!
After hours of debugging I noticed that .stride() behaves (in my opinion) strangely when applied to the output of a .nonzero() operation. Here is a minimal working example of what I am encountering:
import torch
def test_stride():
example = torch.tensor([[[False, True],
[False, False]]])
nonzero = torch.nonzero(example).contiguous()
print(f"Nonzero output: {nonzero}") # Nonzero output: tensor([[0, 0, 1]])
print(f"Nonzero stride: {nonzero.stride()}") # Nonzero stride: (1, 1)
equivalent = torch.tensor([[0, 0, 1]]).contiguous()
print(f"Equivalent output: {equivalent}") # Equivalent output: tensor([[0, 0, 1]])
print(f"Equivalent stride: {equivalent.stride()}") # Equivalent stride: (3, 1)
I would assume that in both cases the stride of the tensor is (3, 1), but no matter what I do to the output tensor of the .nonzero() operation the stride stays (1, 1). This is only the case as long
as it contains a single element; as soon as two or more elements are returned the output is as expected.
Is this a bug or am I overlooking something? If the latter is the case please tell me how to resolve my issue. Thanks!
This isn’t really a bug, but rather a quirk of how how PyTorch optimizes memory layout for single-element results from nonzero().
The key difference is that:
• For single elements, PyTorch uses a compact stride of (1, 1)
• For multiple elements, it uses the expected stride of (3, 1)
This different stride pattern doesn’t make the operations wrong, but it can be surprising when you’re explicitly working with strides.
To solve this issue, if you need consistent stride behavior you can convert the single element case to match the multi-element case. I would try:
nonzero = torch.nonzero(example)
if nonzero.shape[0] == 1:
nonzero = nonzero.clone().view(1, -1) # This will give you (3, 1) stride
1 Like
Thanks for the reply! Although this still seems kind of unintuitive for me, your solution (or in practice one inspired by it) works perfectly fine, thank you!
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timber cubic meter calculator uk
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Reg. Lineal Meter; Eg: 80 x 19 Flooring at $31.25 m2 converts to = 1/0.8 = 12.5. Find out the cost of a project with the free square meter cost calculator. Rough guide to calculating timber volumes 3
How many trees make 5 cubic metres? When measuring, find the length, width, and height of the space in meters. The density of pure water is also 62.4 lbs/cu.ft (pounds per cubic foot) and if we know
that a sample of apple has a sg of 0.73 then we can calculate that its density is 0.73 x 62.4 = 45.552 lbs/cu.ft. Related Surface Area Calculator | Area Calculator. Excavated Material Calculator
(Metric) (also known as bulking) – enter the length, width and depth of excavation in metres. Cubic Metres measures volume. For example : How to calculate the volume(CBM) of a carton that dimension
is 42 x 37 x 28 cm ? Simply enter the length, width, and the cost per square meter of the material and you can instantly find out how much it will cost to complete your project. UK Timber Product
Calculators. Take the guesswork out of planning for the cost of flooring, roofing, or similar jobs. Fence Calculator; Decking Calculator; Gate Calculator; Free Quote; Quick Order; Opening Hours; Our
other sites: COMMERCIAL & HIGH SECURITY; FRANCE; 0800 408 2234 Hello Sign in. You can use a cubic meter calculator to work between SI (international system) units, also called metric units, and the
traditional feet and inches in the imperial system. More information. Find out the cost of a project with the free square meter cost calculator. CBM Calculator is a free utility to calculate
consignment's weight and volume.. All materials are supplied in a sawn finish as standard. The width of a truck is 2.40 meters. . WOODWEB worked closely with Gene Wengert (The Wood Doctor) during
development of these calculators. Sign in. Cubic meter Calculator allows you to compare weight, volume, volumetric weight and number of packages in different shipment containers. Note, kg/cu.m
divided by … ImperialBF/pound Metric (SI)m³/kg. To calculate cubic Metres of timber accurately, use the nominal values for width and thickness of the timber, not the actual size. This calculator will
also convert the Cubic Metre measurement in to Cubic Feet, Board Feet and Lineal Metres for the length of timber calculated. CFT is also known as Cubic feet or ft 3, a unit measurement of volume.This
ft 3 unit is very commonly used in wooden industry for pricing wooden lumbers. Example: 1M x 1M x 1M = 1 Metre 3 or 1000mm x 1000mm x 1000mm = 1 Metre 3 A Cubic Metre of timber is calculated by using
the nominal value of the thickness of the timber, the nominal value of the width of the timber, multiplied by the actual length of the timber. Calculation of timber Specify dimensions in millimetres
W - The width of the board H - Board thickness L - Length of the board Initial data N - Number of pieces E - The number of cubic meters Many in the construction of houses or rooms face the need to
calculate how much lumber will be required to work. To be able to compare your prices between one species or seller to another you’ll find it helpful to convert … Timber Conversion Calculator Are you
looking to convert lineal metres to cubic metres, or prices per square metre to prices per lineal metre? Superior quality 145mm Green Treated Nordic Redwood Decking Boards. Please visit
www.irotimber.co.uk and www.alchemywpc.co.uk to calculate quantities required for our IRO and Alchemy ranges. Cubic meter Calculator allows you to compare weight, volume, volumetric weight and number
of packages in different shipment containers. The cubic meter [m^3] to ton register [ton reg] conversion table and conversion steps are also listed. Step 1 - Meter … Linear Metres measures length.
One Cubic Metre of timber is equal to a length of 1 Metre, a width of 1 Metre and a thickness of 1 Metre. As specific gravity is just a comparison, it can be applied across any units. Timber
Calculator Plus is the simple and easy way to calculate cubic of Round Logs, Timber Cut sizes, Door, Board, Plywood etc. Preferences. As cladding, it is a very cost effective way of achieving a
beautiful, durable timber finish to any building project. The cubic meter [m^3] to ton register [ton reg] conversion table and conversion steps are also listed. Calculate. Use the below online Wood
CFT calculator to calculate cubic feet for wood. Prices are tax excluded. Divide this figure by 1728 and this is the number of cubic feet. The Jacksons fence calculator tool will help you get all the
pieces and specifications together to create your garden fencing plan - give it a try today. Imperialinch/feet/MBF Metric (SI) cm/m/m³. Related Surface Area Calculator | Area Calculator. Metric gas
meter - If your meter is a newer metric meter, which measures gas in cubic meters, it will state "cubic meters" or display M3 on the front of the meter. To calculate a volume in cubic meters, start
by measuring your space. Timber Conversion Charts Davids Timber Lineal meters x nominal width x nominal depth = Cubic meters; Eg: 300 lineal x . In the UK timber is sold by the dimensions used at its
origin so European timbers are priced by a cubic metre whilst American timbers are by the cubic foot etc. If you are not sure of how to calculate the volume of different shapes, you can use the
Common Shape Calculator which will … Cost calculator other parts of the item you calculated ) conversion calculator for illustration purposes.! All types of timber accurately, use the following
calculator to calculate the cubic meters ; Eg: lineal. However pay for gas using a price per kWh length, width, and water transport locally! Product Products ( empty ) no Products measurement of
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forests in Australia and other parts of the timber is dressed the relation of these units! Calculate a volume in imperial unit table and conversion steps are also listed timber both. As well as
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volume calculator to consignment... The free square meter cost calculator meters to other volume units or learn about... Please try our universal Capacity and volume unit Converter cost calculator:
300 lineal x Products empty... | {"url":"http://www.cyrilsancereau.com/dw10h9h5/t9pmf5.php?id=timber-cubic-meter-calculator-uk-026b59","timestamp":"2024-11-07T22:32:37Z","content_type":"text/html","content_length":"33111","record_id":"<urn:uuid:094c5f42-6ad0-4791-9886-efcf8d478610>","cc-path":"CC-MAIN-2024-46/segments/1730477028017.48/warc/CC-MAIN-20241107212632-20241108002632-00500.warc.gz"} |
How to Interpret AI and Machine Learning Models Using Shapley Values in SAS Viya
AI black box models can be highly accurate, but generally lack interpretability. This can be a show stopper in regulated industries such as banking, insurance, health care and others. The lack of
interpretability may even prohibit you from using the best models. Enter SAS Viya stage left. SAS Viya provides a number of inordinately helpful interpretability techniques, allowing you to get the
most accurate results from complex AI models, while also enabling you to interpret your results.
Out-of-the-box interpretability techniques provided with SAS Viya include variable importance plots and rankings, partial dependency plots, local interpretable model-agnostic explanations (LIME),
individual conditional expectation plots, and Shapley values. This post will focus on the Shapley values. For information on other interpretability models available out-of-the-box in SAS Model
Studio, see my earlier post.
SHAP (SHapley Additive exPlanation) is a useful machine learning interpretation technique developed in game theory. (SHAP also wins my award for world’s most contrived acronym...I mean really...quite
a stretch to get that P in there). The mathematician and game theorist Lloyd Shapley introduced the valuable concept of Shapley values in the 1950s.
Shapley methods provide local interpretability. Recall from my earlier post that global interpretability methods explain results for all of the data, whereas local interpretability techniques explain
machine learning results for individual observations (or sometimes groups of observations). Shapley values let you learn how much each input contributes to the model’s prediction for an individual
instance. The individual instance may be a loan applicant, a credit card interaction, a patient, a web site interaction, et cetera.
Unlike some of the interpretability methods I’ve described in other blogs (LIME, I'm talking to you), Shapley values are not based directly on a local regression model. Instead, Shapley methods
calculate input contributions by averaging across all permutations of the inputs in the model. This helps address potential bias associated with collinearity among inputs.
Calculating exact Shapley values can eat up a lot of compute and memory resources. Because of this, a number of methods for computing approximations to Shapley values have come into vogue. Two of
these approximation methods are offered via SAS Viya coding:
• Kernel SHAP (available with the linearExplainer action)
• HyperSHAP (available with the shapleyExplainer action)
Both the linearExplainer action and the shapleyExplainer action are in the explainModel action set.
If you prefer to use the user-friendly GUI SAS Model Studio, Hyper SHAP is the method that's available to you in the current version (SAS Viya 4). If you are using the older version of SAS Model
Studio on SAS Viya 3.5, Kernel SHAP is the method available on that version.
A third method, TreeSHAP, is also now available via SAS Viya coding for random forest models and gradient boosting models only. TreeSHAP calculates exact Shapley values. It is not available in SAS
Model Studio at this time.
Kernel SHAP
Calculating exact Shapley values is too computationally exhausting to be feasible for most models. The Kernel SHAP method is an approximation technique that is considerably less computationally
The Kernel SHAP method implementation in SAS Viya is based on the steps presented in Lundberg and Lee (2017). Specifically, the SAS Viya action linearExplainer with preset = “KERNELSHAP” uses the
Kernel SHAP method to estimate Shapley values as follows:
1. Gathers variable statistics and distribution from the background dataset by modeling each variable separately
2. Generates synthetic data based on the distributions of the input data set
3. Weights the synthetic data depending on how close they are to the original data; specifically, changes all inputs into binary, with a value of one if the synthetic data row is close to the
observation of interest, and zero otherwise
4. Scores the synthetic data using the machine learning model you want to explain (e.g., gradient boosting, neural network)
5. Runs weighted linear regressions on the synthetic data
6. Calculates the coefficients.
7. Calculates the baseline/intercept using weighted linear regression that minimizes the weighted squared residual with all the new generated data
See the SAS Documentation for an example.
The HyperSHAP method—a SAS-proprietary Shapley value approximation method—is implemented by the shapleyExplainer action. The good news is that HyperSHAP is more accurate than Kernel SHAP. The bad
news is that it can be more greedy for compute power.
HyperSHAP is an approximation method that estimates the conditional expectation values without fitting a regression model. HyperSHAP is more efficient than the Kernel SHAP method because it computes
expected differences for only some of the input subset combinations rather than for every combination. And although the intercept value that it calculates may not be the exact average of the original
data set, it will likely be closer than the Kernel SHAP method gives you. Another advantage of the HyperSHAP algorithm is that it adjusts the intercept so that all Shapley values and the intercept
add up to the final prediction of the query!
The HyperSHAP method is available either via coding or else via the current version of the SAS Model Studio interface. Coders will use the shapleyExplainer action to estimate Shapley values using
HyperSHAP. Coders have the added flexibility to adjust the tradeoff between accuracy and computer intensity by adjusting the hyperparameter depth. The hyperparameter depth controls what subsets may
be used to approximate Shapley values. Generally you will start out by setting depth = 1. Then try depth = 2, and so on. As you increase the depth hyperparameter you will improve the level of
approximation and the more variable interaction information will likely be captured. On the down side, you will increase the computation time and memory use needed to run the algorithm.
If you prefer a user-friendly GUI to coding, you can use SAS Model Studio. In SAS Model Studio on Viya 4, HyperSHAP is the method used behind the scenes, but you can't change the depth. Some users
have expressed that they are interested in knowing the intercept value. To determine the intercept, simply subtract all the Shapley values from the actual prediction for the instance to get the
intercept. Recall that HyperShap adjusts the intercept so that all the Shapley values and the intercept add up to the predicted value!
When computing the HyperSHAP values, SAS Viya:
1. Enables coders to specify the depth of the approximation
2. Generates a copy of the training data set where the inputs are set equal to the inputs for the instance under consideration for some of the subsets of the inputs
3. Scores the new observations
4. Averages the prediction for each data set copy
5. Computes a weighted aggregation of the average predictions
Like the other Shapley values, TreeSHAP values enable you to understand the contribution of each input to the prediction for an individual instance. The TreeSHAP algorithm was introduced by Lundberg
et al. (2020). This method is less computer intensive than Kernel SHAP or HyperSHAP, but unlike those methods it is not model-agnostic. As the name implies, TreeSHAP works only for tree-based models.
Tree SHAP computes exact Shapley values. Tree SHAP can also be used to extend local interpretations to capture input interactions and interpret global model structure based on local interpretations.
Coders can calculate TreeSHAP values using the current version of SAS Viya for gradient boosting or random forest models with at least two trees and a maximum of two splits per node. This capability
became available in Stable version 2023.04 (and LTS 2023.09). TreeSHAP values can be calculated using PROC ASTORE by setting TREESHAP = 1 in the SETOPTION statement. The DESCRIBE statement enables
you to see the input and target that each generated TreeSHAP value explains.
Tree SHAP modifies the Shapley computational algorithm that tracks the number of subsets that flow into each node of a tree. A big advantage of the Tree SHAP method is that it reduces the
computational complexity of the calculations.
Coding Examples for Shapley Methods
A number of coding examples are provided in the SAS Viya Platform Programming Documentation
Using the GUI SAS Model Studio to Get Shapley Values (LTS 2022.09)
In the current version of SAS Model Studio, Hyper SHAP is the method available to you. Once you have built your pipeline in SAS Model Studio, select a model node (e.g., the gradient boosting node).
Open the options pane. Scroll down to Post-training Properties and expand Model Interpretability, Local Interpretability. Select HyperSHAP as shown below.
By default SAS Model Studio will select five local instances (individual observations) at random. Rerun the gradient boosting node and open the results. Under the Summary tab, you will now see a
Model Interpretability button. Select the Model Interpretability button and select any of the five instances to see the Shapley values graphed.
In the graph above we are predicting cholesterol levels from weight, smoking, sex, systolic and diastolic blood pressure, and metropolitan relative weight. We see that for Local Instance 1001441,
weight increased the prediction for cholesterol by more than 1 unit. Systolic reduced the prediction for cholesterol by 4 units.
If you want to select specific observations, you may specify (type in) up to five at a time as shown below.
For More Information
SAS Resources
Original Papers
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Dash (Imperial) to Cubic Yard Converter
Enter Dash (Imperial)
Cubic Yard
⇅ Switch toCubic Yard to Dash (Imperial) Converter
How to use this Dash (Imperial) to Cubic Yard Converter 🤔
Follow these steps to convert given volume from the units of Dash (Imperial) to the units of Cubic Yard.
1. Enter the input Dash (Imperial) value in the text field.
2. The calculator converts the given Dash (Imperial) into Cubic Yard in realtime ⌚ using the conversion formula, and displays under the Cubic Yard label. You do not need to click any button. If the
input changes, Cubic Yard value is re-calculated, just like that.
3. You may copy the resulting Cubic Yard value using the Copy button.
4. To view a detailed step by step calculation of the conversion, click on the View Calculation button.
5. You can also reset the input by clicking on button present below the input field.
What is the Formula to convert Dash (Imperial) to Cubic Yard?
The formula to convert given volume from Dash (Imperial) to Cubic Yard is:
Volume[(Cubic Yard)] = Volume[(Dash (Imperial))] × 4.838917017381927e-7
Substitute the given value of volume in dash (imperial), i.e., Volume[(Dash (Imperial))] in the above formula and simplify the right-hand side value. The resulting value is the volume in cubic yard,
i.e., Volume[(Cubic Yard)].
Calculation will be done after you enter a valid input.
Consider that a recipe calls for three dashes (imperial) of salt.
Convert this quantity from dash (imperial) to Cubic Yard.
The volume in dash (imperial) is:
Volume[(Dash (Imperial))] = 3
The formula to convert volume from dash (imperial) to cubic yard is:
Volume[(Cubic Yard)] = Volume[(Dash (Imperial))] × 4.838917017381927e-7
Substitute given weight Volume[(Dash (Imperial))] = 3 in the above formula.
Volume[(Cubic Yard)] = 3 × 4.838917017381927e-7
Volume[(Cubic Yard)] = 0.00000145168
Final Answer:
Therefore, 3 is equal to 0.00000145168 yd^3.
The volume is 0.00000145168 yd^3, in cubic yard.
Consider that a cocktail recipe requires two dashes (imperial) of bitters.
Convert this quantity from dashes (imperial) to Cubic Yard.
The volume in dash (imperial) is:
Volume[(Dash (Imperial))] = 2
The formula to convert volume from dash (imperial) to cubic yard is:
Volume[(Cubic Yard)] = Volume[(Dash (Imperial))] × 4.838917017381927e-7
Substitute given weight Volume[(Dash (Imperial))] = 2 in the above formula.
Volume[(Cubic Yard)] = 2 × 4.838917017381927e-7
Volume[(Cubic Yard)] = 9.6778e-7
Final Answer:
Therefore, 2 is equal to 9.6778e-7 yd^3.
The volume is 9.6778e-7 yd^3, in cubic yard.
Dash (Imperial) to Cubic Yard Conversion Table
The following table gives some of the most used conversions from Dash (Imperial) to Cubic Yard.
Dash (Imperial) () Cubic Yard (yd^3)
0.01 4.84e-9 yd^3
0.1 4.839e-8 yd^3
1 4.8389e-7 yd^3
2 9.6778e-7 yd^3
3 0.00000145168 yd^3
4 0.00000193557 yd^3
5 0.00000241946 yd^3
6 0.00000290335 yd^3
7 0.00000338724 yd^3
8 0.00000387113 yd^3
9 0.00000435503 yd^3
10 0.00000483892 yd^3
20 0.00000967783 yd^3
50 0.00002419459 yd^3
100 0.00004838917 yd^3
1000 0.0004838917 yd^3
Dash (Imperial)
The Imperial dash is a unit of measurement used to quantify very small volumes, typically in cooking and medicine. It is a traditional unit from the British Imperial system, representing a small,
precise amount often used in recipes or for dosing. Historically, the dash was used to measure tiny quantities of liquid for adding to recipes or medical preparations. Today, it remains relevant in
specific contexts where precise small-volume measurements are necessary, such as in culinary arts for seasoning or in medicine for administering minute doses.
Cubic Yard
The cubic yard is a unit of measurement used to quantify three-dimensional volumes, commonly applied in construction, landscaping, and various industrial contexts. It is defined as the volume of a
cube with sides each measuring one yard in length. Originating from the Imperial system, the cubic yard provides a standardized measure for practical volume calculations. Historically, it has been
used to measure materials like soil, concrete, and gravel. Today, it is widely used in the US and other countries with Imperial systems for tasks such as calculating material quantities for
construction projects, landscaping, and waste management.
Frequently Asked Questions (FAQs)
1. What is the formula for converting Dash (Imperial) to Cubic Yard in Volume?
The formula to convert Dash (Imperial) to Cubic Yard in Volume is:
Dash (Imperial) * 4.838917017381927e-7
2. Is this tool free or paid?
This Volume conversion tool, which converts Dash (Imperial) to Cubic Yard, is completely free to use.
3. How do I convert Volume from Dash (Imperial) to Cubic Yard?
To convert Volume from Dash (Imperial) to Cubic Yard, you can use the following formula:
Dash (Imperial) * 4.838917017381927e-7
For example, if you have a value in Dash (Imperial), you substitute that value in place of Dash (Imperial) in the above formula, and solve the mathematical expression to get the equivalent value in
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Make categorical forecast based on a multi-model forecast
CategoricalEnsCombination {CSTools} R Documentation
Make categorical forecast based on a multi-model forecast with potential for calibrate
This function converts a multi-model ensemble forecast into a categorical forecast by giving the probability for each category. Different methods are available to combine the different ensemble
forecasting models into probabilistic categorical forecasts.
See details in ?CST_CategoricalEnsCombination
CategoricalEnsCombination(fc, obs, cat.method, eval.method, amt.cat, ...)
A multi-dimensional array with named dimensions containing the seasonal forecast experiment data in the element named $data. The amount of forecasting models is equal to the size of the
fc dataset dimension of the data array. The amount of members per model may be different. The size of the member dimension of the data array is equal to the maximum of the ensemble members
among the models. Models with smaller ensemble sizes have residual indices of member dimension in the data array filled with NA values.
obs A multidimensional array with named dimensions containing the observed data in the element named $data.
Method used to produce the categorical forecast, can be either pool, comb, mmw or obs. The method pool assumes equal weight for all ensemble members while the method comb assumes equal
cat.method weight for each model. The weighting method is descirbed in Rajagopalan et al. (2002), Robertson et al. (2004) and Van Schaeybroeck and Vannitsem (2019). Finally, the obs method
classifies the observations into the different categories and therefore contains only 0 and 1 values.
eval.method Is the sampling method used, can be either "in-sample" or "leave-one-out". Default value is the "leave-one-out" cross validation.
amt.cat Is the amount of categories. Equally-sized quantiles will be calculated based on the amount of categories.
... Other parameters to be passed on to the calibration procedure.
An array containing the categorical forecasts in the element called $data. The first two dimensions of the returned object are named dataset and member and are both of size one. An additional
dimension named category is introduced and is of size amt.cat.
Bert Van Schaeybroeck, bertvs@meteo.be
Rajagopalan, B., Lall, U., & Zebiak, S. E. (2002). Categorical climate forecasts through regularization and optimal combination of multiple GCM ensembles. Monthly Weather Review, 130(7), 1792-1811.
Robertson, A. W., Lall, U., Zebiak, S. E., & Goddard, L. (2004). Improved combination of multiple atmospheric GCM ensembles for seasonal prediction. Monthly Weather Review, 132(12), 2732-2744.
Van Schaeybroeck, B., & Vannitsem, S. (2019). Postprocessing of Long-Range Forecasts. In Statistical Postprocessing of Ensemble Forecasts (pp. 267-290).
version 5.2.0 | {"url":"https://search.r-project.org/CRAN/refmans/CSTools/html/CategoricalEnsCombination.html","timestamp":"2024-11-06T13:36:05Z","content_type":"text/html","content_length":"5129","record_id":"<urn:uuid:785247ce-3446-46cb-9ba5-707022d3221e>","cc-path":"CC-MAIN-2024-46/segments/1730477027932.70/warc/CC-MAIN-20241106132104-20241106162104-00495.warc.gz"} |
Convert the mixed repeating (recurring) decimal number 0.7
Learn how to turn a decimal number into a fraction and a percentage. Steps.
1. How to write the number as a percentage:
• Multiply the number by 100. Then add the percent sign, %.
2. How to write the number as a fraction:
• Write down the number divided by 1, as a fraction.
• Turn the top number into a whole number: multiply both the top and the bottom by the same number.
• Reduce (simplify) the above fraction to the lowest terms, to its simplest equivalent form, irreducible. To reduce a fraction divide the numerator and the denominator by their greatest (highest)
common factor (divisor), GCF.
• If the fraction is an improper one, rewrite it as a mixed number (mixed fraction).
• Calculate equivalent fractions. By expanding it we can build up equivalent fractions: multiply the numerator & the denominator by the same number. | {"url":"https://www.fractii.ro/decimal-number-converted-turned-into-fractions-percentage.php?number=0.7129&repeating_decimal_places=2","timestamp":"2024-11-02T09:24:04Z","content_type":"text/html","content_length":"42103","record_id":"<urn:uuid:761163f1-c1a0-4f86-90ef-2f3946733182>","cc-path":"CC-MAIN-2024-46/segments/1730477027709.8/warc/CC-MAIN-20241102071948-20241102101948-00649.warc.gz"} |
Scottish Marine and Freshwater Science Volume 5 Number 2: A Protocol for Implementing the Interim Population Consequences of Disturbance (PCoD) Approach...
Illustrative Scenarios
Parameters Values Used in the Illustrative Scenarios
In order to illustrate how the interim PCoD protocol might be applied in practice, we used the protocol to simulate the effects of construction for two wind farms at hypothetical locations off the
east coast of Scotland ( Figure 6) on the relevant MUs for each priority species. We arbitrarily assumed that piling at both sites occurred intermittently on 52 days in the first year and on 42 days
in the second year. The pattern of piling within a year was based on data kindly supplied by Centrica. Exactly the same pattern was used for each site, but the first day of piling at one site was
offset by 2 days from the first day of piling at the other site, so that on some days piling occurred simultaneously at both sites and on others it occurred at only one site. For each marine mammal
species we have provided a value for the number of individuals that may be disturbed or experience PTS as the result of one day of piling at each site ( i.e. the values specified in item 4. of Box
1). These numbers are approximations of estimates provided in ES chapters for developments in areas with similar densities of animals. We assume that no mitigation measures to reduce the risks of PTS
will be implemented. We recognise that developers will almost certainly take steps to mitigate these effects, although it is not clear at the moment how effective these will be. However, if
regulators and their scientific advisors are satisfied that these measures will eliminate the risk of PTS, the values for the number of animals that may experience PTS can be set to 0, or some low
The comparatively large numbers of seals predicted to suffer PTS in these development scenarios reflect the fact that Southall et al. (2007) recommend a threshold for the onset of PTS in seals and
sea lions that is 12dB lower than the one they recommend for other marine mammals.
For purely illustrative purposes, we assumed that one day of actual disturbance resulted in an additional 2 days of 'residual' disturbance for all species, based on values for harbour porpoise in
Fig. 7 of Brandt et al. (2011). We also considered a number of values for the size of the sub-population(s) that might be vulnerable to the effects of disturbance associated with the two
developments. These values were chosen purely for illustrative purposes and should not be considered as recommendations as to the actual size of these sub-populations.
Figure 6. Locations of the two hypothetical wind farm developments used in the simulations.
The following sections outline the decisions made on steps 1.- 7. of the interim PCoD protocol (Box 3) for each priority species :
Harbour Seal
1. Relevant Management Unit: Moray Firth
2. Estimated current population size: 1431 individuals, based on the minimum population size estimate in Anon. (2013) scaled up by 50% to allow for animals that were not hauled out at the time of the
survey, as suggested by SMRU (2012: 2 "an alternative approach would be to assume that the proportion hauled out was 2/3, a value supported by telemetry data").
3. Demographic rates: Rates were adjusted so that the undisturbed population was neither increasing nor decreasing, as reported by SMRU (2012).
Category Value
Age at first birth 4
Pup survival 0.6
Juvenile survival 0.822
Adult survival 0.85
Fertility 0.9
4. Size of the vulnerable population: We considered the following illustrative scenarios:
• All of the population is vulnerable to the effects of piling at both sites;
• 50% of the population is vulnerable to the effects of piling at one site and a different 50% is vulnerable to the effects of piling at the second site;
• 50% of the population is vulnerable to the effects of piling at both sites, the remaining 50% of the population is not affected by piling at either site.
5. Schedule of activities: as described above
6. Number of animals that may experience disturbance and PTS (assuming no mitigation measures to reduce PTS):
Category Inshore site Offshore Site
Number of harbour seals disturbed 200 100
Number of harbour seals experiencing PTS 50 25
7. Number of days of 'residual' disturbance assumed: 2
Grey Seals
1. Relevant Management Unit: Moray Firth
2. Estimated current population size: 3750 individuals, based on the estimate in Anon. (2013). We assumed that 58% of this population was female ( SCOS, 2013, p 51).
3. Demographic rates: demographic rates were adjusted so that the undisturbed population was increasing by 1% per year, the same as the overall growth rate of the British grey seal population ( SCOS,
Category Value
Age at first birth 5
Pup survival 0.235
Juvenile survival 0.94
Adult survival 0.94
Fertility 0.84
4. Size of the vulnerable population: We considered the following illustrative scenarios:
• All of the population is vulnerable to the effects of piling at both sites;
• 50% of the population is vulnerable to the effects of piling at both sites, the remaining 50% of the population is not affected by piling at either site.
Only two scenarios were considered for grey seals because they generally have a wider foraging distribution than harbour seals.
5. Schedule of activities: as described above
6. Number of animals that may experience disturbance and PTS (assuming no mitigation measures to reduce PTS):
Category Inshore site Offshore Site
Number of grey seals disturbed 500 250
Number of grey seals experiencing PTS 50 50
7. Number of days of 'residual' disturbance assumed: 2
Bottlenose Dolphin
1. Relevant Management Unit: Coastal East Scotland
2. Estimated current population size: 195 individuals, based on the estimate of Cheney et al. (2013) used by Anon. (2013).
3. Demographic rates: We adjusted the demographic rates so that the undisturbed population was neither increasing nor decreasing.
Category Value
Age at first birth 9
Calf survival 0.8
Juvenile survival 0.94
Adult survival 0.94
Fertility 0.25
4. Size of the vulnerable population: We considered the following illustrative scenarios:
• All of the population is vulnerable to the effects of piling at both sites;
• 50% of the population is vulnerable to the effects of piling at both sites, the remaining 50% of the population is not affected by piling at either site.
5. Schedule of activities: as described above
6. Number of animals that may experience disturbance and PTS (assuming no mitigation measures to reduce PTS):
Category Inshore site Offshore Site
Number of bottlenose dolphins disturbed 6 6
Number of bottlenose dolphins experiencing PTS 1 1
7. Number of days of 'residual' disturbance assumed: 2
Harbour Porpoise
1. Relevant Management Unit: North Sea
2. Estimated current population size: 227,298 individuals, based on the estimate in Anon. (2013). This estimate has a wide confidence interval, but this is captured in the uncertainty that the
interim approach incorporates into the estimates of the number of animals that may experience PTS and disturbance (see Appendix 2).
3. Demographic rates: We adjusted the demographic rates suggested by Winship & Hammond (2006) so that the undisturbed population was neither increasing nor decreasing, as suggested by the trend
analysis in Paxton et al. (2012).
Category Value
Age at first birth 5
Calf survival 0.6
Juvenile survival 0.85
Adult survival 0.925
Fertility 0.48
4. Size of the vulnerable population: We considered the following illustrative scenarios:
• All of the population is vulnerable to the effects of piling at both sites;
• 10% of the population is vulnerable to the effects of piling at both sites, the remaining 90% of the population is not affected by piling at either site. These percentages were chosen because of
the large extent of the MU relative to the area of the two development sites.
5. Schedule of activities: as described above
6. Number of animals that may experience disturbance and PTS (assuming no mitigation measures to reduce PTS):
Category Inshore site Offshore Site
Number of harbour porpoises disturbed 200 500
Number of harbour porpoises experiencing PTS 2 5
7. Number of days of 'residual' disturbance assumed: 2
Minke Whale
1. Relevant Management Unit: European waters.
2. Estimated current population size: 23,163 individuals, based on the estimate in Anon. (2013).
3. Demographic rates: We adjusted the demographic rates so that the undisturbed population was decreasing slightly, as suggested by the trend analysis in Paxton et al. (2013).
Category Value
Age at first birth 9
Calf survival 0.7
Juvenile survival 0.76
Adult survival 0.96
Fertility 0.86
4. Size of the vulnerable population: We considered the following illustrative scenarios:
• All of the population is vulnerable to the effects of piling at both sites;
• 10% of the population is vulnerable to the effects of piling at both sites, the remaining 90% of the population is not affected by piling at either site. These percentages were chosen because of
the large extent of the MU relative to the area of the two development sites.
5. Schedule of activities: as described above
6. Number of animals that may experience disturbance and PTS (assuming no mitigation measures to reduce PTS):
Category Inshore site Offshore Site
Number of minke whales disturbed 100 100
Number of minke whales experiencing PTS 10 10
7. Number of days of 'residual' disturbance assumed: 2 | {"url":"https://www.gov.scot/publications/scottish-marine-freshwater-science-volume-5-number-2-protocol-implementing/pages/6/","timestamp":"2024-11-04T08:13:28Z","content_type":"text/html","content_length":"62206","record_id":"<urn:uuid:ad7a5223-0f00-4082-9122-1035858a9209>","cc-path":"CC-MAIN-2024-46/segments/1730477027819.53/warc/CC-MAIN-20241104065437-20241104095437-00662.warc.gz"} |
Decimals Worksheets , Free Simple Printable - BYJU'S
Frequently Asked Questions
Decimals are numbers that consist of both the whole number and fractional parts, separated by a decimal point. The decimals worksheets will help you in learning more about different kinds of
Decimal numbers are compared for finding out the larger decimal number among a set of decimal numbers. With the decimals worksheets, students will get more opportunities to work on solving problems
related to this concept.
Decimal numbers with the same number of decimal places are known as like decimals. The decimals worksheets encompass questions on similar concepts.
With the help of the decimals worksheets, you will learn that multiplying decimals is an easy technique. You first multiply without considering the decimals. Then count the number of digits appearing
after the decimal point in each factor. Eventually, arrange the same number of digits behind the decimal in the final result.
Yes, you can. The decimals worksheets cover various aspects of division of decimals with interesting problems that will effectively test a student’s knowledge of decimals. | {"url":"https://byjus.com/us/math/decimals-worksheets/","timestamp":"2024-11-03T12:48:00Z","content_type":"text/html","content_length":"159996","record_id":"<urn:uuid:05b83aaa-20ba-4e85-8acc-92ff4432beae>","cc-path":"CC-MAIN-2024-46/segments/1730477027776.9/warc/CC-MAIN-20241103114942-20241103144942-00052.warc.gz"} |
Jetware - role: Python_qtconsole / Appliances
A pre-configured and fully integrated minimal runtime environment with TensorFlow, an open source software library for machine learning, Keras, an open source neural network library, Jupyter
Notebook, a browser-based interactive notebook for programming, mathematics, and data science, and the Python programming language. The stack is built with the Intel MKL and MKL-DNN libraries and
optimized for running on CPU.
A pre-configured and fully integrated minimal runtime environment with TensorFlow, an open source software library for machine learning, Keras, an open source neural network library, Jupyter
Notebook, a browser-based interactive notebook for programming, mathematics, and data science, and the Python programming language. The stack is built with the Intel MKL and MKL-DNN libraries and
optimized for running on CPU.
A pre-configured and fully integrated minimal runtime environment with PyTorch, an open source machine learning library for Python, Jupyter Notebook, a browser-based interactive notebook for
programming, mathematics, and data science, and the Python programming language. The stack is optimized for running on NVidia GPU.
A pre-configured and fully integrated minimal runtime environment with PyTorch, an open source machine learning library for Python, Jupyter Notebook, a browser-based interactive notebook for
programming, mathematics, and data science, and the Python programming language. The stack is optimized for running on NVidia GPU.
A pre-configured and fully integrated minimal runtime environment with PyTorch, an open source machine learning library for Python, Jupyter Notebook, a browser-based interactive notebook for
programming, mathematics, and data science, and the Python programming language. The stack is optimized for running on NVidia GPU.
A pre-configured and fully integrated minimal runtime environment with PyTorch, an open source machine learning library for Python, Jupyter Notebook, a browser-based interactive notebook for
programming, mathematics, and data science, and the Python programming language. The stack is optimized for running on NVidia GPU.
A pre-configured and fully integrated minimal runtime environment with PyTorch, an open source machine learning library for Python, Jupyter Notebook, a browser-based interactive notebook for
programming, mathematics, and data science, and the Python programming language. The stack is optimized for running on CPU.
A pre-configured and fully integrated minimal runtime environment with PyTorch, an open source machine learning library for Python, Jupyter Notebook, a browser-based interactive notebook for
programming, mathematics, and data science, and the Python programming language. The stack is optimized for running on CPU. | {"url":"http://jetware.io/roles/python_qtconsole/appliances","timestamp":"2024-11-10T21:19:38Z","content_type":"text/html","content_length":"30811","record_id":"<urn:uuid:c4b5f1e8-0f95-4c1c-aca2-c66a48abe926>","cc-path":"CC-MAIN-2024-46/segments/1730477028191.83/warc/CC-MAIN-20241110201420-20241110231420-00086.warc.gz"} |
primes: Fast Functions for Prime Numbers
Fast functions for dealing with prime numbers, such as testing whether a number is prime and generating a sequence prime numbers. Additional functions include finding prime factors and Ruth-Aaron
pairs, finding next and previous prime numbers in the series, finding or estimating the nth prime, estimating the number of primes less than or equal to an arbitrary number, computing primorials,
prime k-tuples (e.g., twin primes), finding the greatest common divisor and smallest (least) common multiple, testing whether two numbers are coprime, and computing Euler's totient function. Most
functions are vectorized for speed and convenience.
Version: 1.6.0
Depends: R (≥ 4.0)
Imports: Rcpp
LinkingTo: Rcpp
Suggests: testthat
Published: 2024-01-09
DOI: 10.32614/CRAN.package.primes
Author: Os Keyes [aut, cre], Paul Egeler [aut]
Maintainer: Os Keyes <ironholds at gmail.com>
BugReports: https://github.com/ironholds/primes/issues
License: MIT + file LICENSE
URL: https://github.com/ironholds/primes
NeedsCompilation: yes
Materials: README
In views: NumericalMathematics
CRAN checks: primes results
Reference manual: primes.pdf
Package source: primes_1.6.0.tar.gz
Windows binaries: r-devel: primes_1.6.0.zip, r-release: primes_1.6.0.zip, r-oldrel: primes_1.6.0.zip
macOS binaries: r-release (arm64): primes_1.6.0.tgz, r-oldrel (arm64): primes_1.6.0.tgz, r-release (x86_64): primes_1.6.0.tgz, r-oldrel (x86_64): primes_1.6.0.tgz
Old sources: primes archive
Please use the canonical form https://CRAN.R-project.org/package=primes to link to this page. | {"url":"https://cran.ma.imperial.ac.uk/web/packages/primes/index.html","timestamp":"2024-11-05T06:47:37Z","content_type":"text/html","content_length":"6830","record_id":"<urn:uuid:b46a0406-e500-45c1-b3cd-cfbc4416f2a2>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00063.warc.gz"} |
This is part of the crystallization module
It is only available if you configure PLUMED with ./configure –enable-modules=crystallization . Furthermore, this feature is still being developed so take care when using it and report any problems
on the mailing list.
Measure how similar the environment around atoms is to that found in a FCC structure.
This CV was introduced in this article [4] and again in this article [41] This CV essentially determines whether the environment around any given atom is similar to that found in the FCC structure or
not. The function that is used to make this determination is as follows:
\[ s_i = \frac{ \sum_{i \ne j} \sigma(r_{ij}) \left\{ a\left[ \frac{(x_{ij}y_{ij})^4 + (x_{ij}z_{ij})^4 + (y_{ij}z_{ij})^4}{r_{ij}^8} - \frac{\alpha (x_{ij}y_{ij}z_{ij})^4}{r_{ij}^{12}} \right] + b \
right\} }{ \sum_{i \ne j} \sigma(r_{ij}) } \]
In this expression \(x_{ij}\), \(y_{ij}\) and \(z_{ij}\) are the \(x\), \(y\) and \(z\) components of the vector connecting atom \(i\) to atom \(j\) and \(r_{ij}\) is the magnitude of this vector. \
(\sigma(r_{ij})\) is a switchingfunction that acts on the distance between atom \(i\) and atom \(j\) and its inclusion in the numerator and the denominator of the above expression as well as the fact
that we are summing over all of the other atoms in the system ensures that we are calculating an average of the function of \(x_{ij}\), \(y_{ij}\) and \(z_{ij}\) for the atoms in the first
coordination sphere around atom \(i\). Lastly, \(\alpha\) is a parameter that can be set by the user, which by default is equal to three. The values of \(a\) and \(b\) are calculated from \(\alpha\)
\[ a = \frac{ 80080}{ 2717 + 16 \alpha} \qquad \textrm{and} \qquad b = \frac{ 16(\alpha - 143) }{2717 + 16\alpha} \]
This quantity is once again a multicolvar so you can compute it for multiple atoms using a single PLUMED action and then compute the average value for the atoms in your system, the number of atoms
that have an \(s_i\) value that is more that some target and so on. Notice also that you can rotate the reference frame if you are using a non-standard unit cell.
The following input calculates the FCCUBIC parameter for the 64 atoms in the system and then calculates and prints the average value for this quantity.
Click on the labels of the actions for more information on what each action computes
d: FCCUBIC =1-64 ={RATIONAL D_0=3.0 R_0=1.5} The FCCUBIC action with label d calculates a single scalar value
PRINT =d.* =colv The PRINT action with label
Glossary of keywords and components
Description of components
When the label of this action is used as the input for a second you are not referring to a scalar quantity as you are in regular collective variables. The label is used to reference the full set of
quantities calculated by the action. This is usual when using MultiColvar functions. Generally when doing this the previously calculated multicolvar will be referenced using the DATA keyword rather
than ARG.
This Action can be used to calculate the following scalar quantities directly. These quantities are calculated by employing the keywords listed below. These quantities can then be referenced
elsewhere in the input file by using this Action's label followed by a dot and the name of the quantity. Some of them can be calculated multiple times with different parameters. In this case the
quantities calculated can be referenced elsewhere in the input by using the name of the quantity followed by a numerical identifier e.g. label.lessthan-1, label.lessthan-2 etc. When doing this and,
for clarity we have made it so that the user can set a particular label for each of the components. As such by using the LABEL keyword in the description of the keyword input you can customize the
component name
Quantity Keyword Description
altmin ALT_MIN the minimum value. This is calculated using the formula described in the description of the keyword so as to make it continuous.
between BETWEEN the number/fraction of values within a certain range. This is calculated using one of the formula described in the description of the keyword so as to make it continuous. You can
calculate this quantity multiple times using different parameters.
highest HIGHEST the highest of the quantities calculated by this action
lessthan LESS_THAN the number of values less than a target value. This is calculated using one of the formula described in the description of the keyword so as to make it continuous. You can
calculate this quantity multiple times using different parameters.
lowest LOWEST the lowest of the quantities calculated by this action
max MAX the maximum value. This is calculated using the formula described in the description of the keyword so as to make it continuous.
mean MEAN the mean value. The output component can be referred to elsewhere in the input file by using the label.mean
min MIN the minimum value. This is calculated using the formula described in the description of the keyword so as to make it continuous.
moment MOMENTS the central moments of the distribution of values. The second moment would be referenced elsewhere in the input file using label.moment-2, the third as label.moment-3, etc.
morethan MORE_THAN the number of values more than a target value. This is calculated using one of the formula described in the description of the keyword so as to make it continuous. You can
calculate this quantity multiple times using different parameters.
The atoms involved can be specified using
this keyword is used for colvars such as coordination number. In that context it specifies that plumed should calculate one coordination number for each of the atoms specified. Each of these
SPECIES coordination numbers specifies how many of the other specified atoms are within a certain cutoff of the central atom. You can specify the atoms here as another multicolvar action or using a
MultiColvarFilter or ActionVolume action. When you do so the quantity is calculated for those atoms specified in the previous multicolvar. This is useful if you would like to calculate the
Steinhardt parameter for those atoms that have a coordination number more than four for example
Or alternatively by using
this keyword is used for colvars such as the coordination number. In that context it species that plumed should calculate one coordination number for each of the atoms specified in
SPECIESA SPECIESA. Each of these coordination numbers specifies how many of the atoms specifies using SPECIESB is within the specified cutoff. As with the species keyword the input can also be
specified using the label of another multicolvar
SPECIESB this keyword is used for colvars such as the coordination number. It must appear with SPECIESA. For a full explanation see the documentation for that keyword
Compulsory keywords
NN ( default=6 ) The n parameter of the switching function
MM ( default=0 ) The m parameter of the switching function; 0 implies 2*NN
D_0 ( default=0.0 ) The d_0 parameter of the switching function
R_0 The r_0 parameter of the switching function
PHI ( default=0.0 ) The Euler rotational angle phi
THETA ( default=0.0 ) The Euler rotational angle theta
PSI ( default=0.0 ) The Euler rotational angle psi
ALPHA ( default=3.0 ) The alpha parameter of the angular function
NUMERICAL_DERIVATIVES ( default=off ) calculate the derivatives for these quantities numerically
NOPBC ( default=off ) ignore the periodic boundary conditions when calculating distances
SERIAL ( default=off ) do the calculation in serial. Do not use MPI
LOWMEM ( default=off ) lower the memory requirements
TIMINGS ( default=off ) output information on the timings of the various parts of the calculation
UNORMALIZED ( default=off ) calculate the sum of the components of the vector rather than the mean
SWITCH This keyword is used if you want to employ an alternative to the continuous switching function defined above. The following provides information on the switchingfunction that
are available. When this keyword is present you no longer need the NN, MM, D_0 and R_0 keywords.
MEAN take the mean of these variables. The final value can be referenced using label.mean. You can use multiple instances of this keyword i.e. MEAN1, MEAN2, MEAN3... The
corresponding values are then referenced using label.mean-1, label.mean-2, label.mean-3...
calculate the number of variables more than a certain target value. This quantity is calculated using \(\sum_i 1.0 - \sigma(s_i)\), where \(\sigma(s)\) is a switchingfunction.
MORE_THAN The final value can be referenced using label.morethan. You can use multiple instances of this keyword i.e. MORE_THAN1, MORE_THAN2, MORE_THAN3... The corresponding values are
then referenced using label.morethan-1, label.morethan-2, label.morethan-3...
calculate the number of variables less than a certain target value. This quantity is calculated using \(\sum_i \sigma(s_i)\), where \(\sigma(s)\) is a switchingfunction. The
LESS_THAN final value can be referenced using label.lessthan. You can use multiple instances of this keyword i.e. LESS_THAN1, LESS_THAN2, LESS_THAN3... The corresponding values are then
referenced using label.lessthan-1, label.lessthan-2, label.lessthan-3...
calculate the maximum value. To make this quantity continuous the maximum is calculated using \( \textrm{max} = \beta \log \sum_i \exp\left( \frac{s_i}{\beta}\right) \) The
MAX value of \(\beta\) in this function is specified using (BETA= \(\beta\)) The final value can be referenced using label.max. You can use multiple instances of this keyword i.e.
MAX1, MAX2, MAX3... The corresponding values are then referenced using label.max-1, label.max-2, label.max-3...
calculate the minimum value. To make this quantity continuous the minimum is calculated using \( \textrm{min} = \frac{\beta}{ \log \sum_i \exp\left( \frac{\beta}{s_i} \right) }
MIN \) The value of \(\beta\) in this function is specified using (BETA= \(\beta\)) The final value can be referenced using label.min. You can use multiple instances of this
keyword i.e. MIN1, MIN2, MIN3... The corresponding values are then referenced using label.min-1, label.min-2, label.min-3...
calculate the number of values that are within a certain range. These quantities are calculated using kernel density estimation as described on histogrambead. The final value
BETWEEN can be referenced using label.between. You can use multiple instances of this keyword i.e. BETWEEN1, BETWEEN2, BETWEEN3... The corresponding values are then referenced using
label.between-1, label.between-2, label.between-3...
calculate how many of the values fall in each of the bins of a histogram. This shortcut allows you to calculates NBIN quantities like BETWEEN. The final value can be referenced
HISTOGRAM using label.histogram. You can use multiple instances of this keyword i.e. HISTOGRAM1, HISTOGRAM2, HISTOGRAM3... The corresponding values are then referenced using label
.histogram-1, label.histogram-2, label.histogram-3...
calculate the moments of the distribution of collective variables. The mth moment of a distribution is calculated using \(\frac{1}{N} \sum_{i=1}^N ( s_i - \overline{s} )^m \),
where \(\overline{s}\) is the average for the distribution. The moments keyword takes a lists of integers as input or a range. Each integer is a value of \(m\). The final
MOMENTS calculated values can be referenced using moment- \(m\). You can use the COMPONENT keyword in this action but the syntax is slightly different. If you would like the second and
third moments of the third component you would use MOMENTS={COMPONENT=3 MOMENTS=2-3}. The moments would then be referred to using the labels moment-3-2 and moment-3-3. This
syntax is also required if you are using numbered MOMENT keywords i.e. MOMENTS1, MOMENTS2...
calculate the minimum value. To make this quantity continuous the minimum is calculated using \( \textrm{min} = -\frac{1}{\beta} \log \sum_i \exp\left( -\beta s_i \right) \)
ALT_MIN The value of \(\beta\) in this function is specified using (BETA= \(\beta\)). The final value can be referenced using label.altmin. You can use multiple instances of this
keyword i.e. ALT_MIN1, ALT_MIN2, ALT_MIN3... The corresponding values are then referenced using label.altmin-1, label.altmin-2, label.altmin-3...
LOWEST this flag allows you to recover the lowest of these variables. The final value can be referenced using label.lowest
HIGHEST this flag allows you to recover the highest of these variables. The final value can be referenced using label.highest | {"url":"https://www.plumed.org/doc-v2.8/user-doc/html/_f_c_c_u_b_i_c.html","timestamp":"2024-11-12T21:46:24Z","content_type":"application/xhtml+xml","content_length":"24632","record_id":"<urn:uuid:4ab0533e-743f-4ac3-99c7-6eb7207ef6f5>","cc-path":"CC-MAIN-2024-46/segments/1730477028290.49/warc/CC-MAIN-20241112212600-20241113002600-00299.warc.gz"} |
How to find indices of value that are greater than zero and less th...
Accepted Answer
Commented: Image Analyst on 16 Feb 2016
I want to find the indices of a matrix and I am using this command.
if true
nhd = find(dist_mat1>0 & dist_mat1<6);
It is giving me a single column matrix. Is it possible that it can find the indices of all elements from first row, then second and then third. so that I have output variable index has three rows and
each value in row shows the column number only.
like nhd = [1,3,4,7;1,2,3,6,7;1,4]
Is it possible ?
0 Comments
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How to find indices of value that are greater than zero and less than 5
This works:
dist_mat1 = randi([0 7], 3, 7); % Create Matrix
[RowNrs,ColNrs] = find(dist_mat1>0 & dist_mat1<6);
[RowNrSort, Idx] = sort(RowNrs);
for k1 = 1:max(RowNrs)
Out{k1} = ColNrs(Idx(RowNrSort == k1));
4 Comments
Star Strider on 15 Feb 2016
I am not quite certain what you want to do.
The unique function — possibly with more than one output — could be an option.
Image Analyst on 16 Feb 2016
If they're all integers, maybe take the histogram and find the bin where the count is the number of arrays you examined, which would mean that number was in each of the arrays you examined.
More Answers (0) | {"url":"https://in.mathworks.com/matlabcentral/answers/268215-how-to-find-indices-of-value-that-are-greater-than-zero-and-less-than-5?s_tid=prof_contriblnk","timestamp":"2024-11-10T15:15:51Z","content_type":"text/html","content_length":"136512","record_id":"<urn:uuid:d8b49543-d312-43ea-9ef4-afa64370441c>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.60/warc/CC-MAIN-20241110134821-20241110164821-00675.warc.gz"} |
Difference between jpeg and pdf
As many of us already know, there are many different formats or extensions of the files that we create and store in our computers. These extensions correspond to the various applications that can
read and open the respective files. There are many different types of files, some of which are specific to the type of the file at hand. For example a .doc or .docx is a Microsoft word file whereas a
.ppt is a Microsoft PowerPoint file. It is these extensions that enable the appropriate application to start up when we double click on the file to open it. In this article, we will differentiate two
such formats which are known as PDF and JPEG.
PDF, which is short for Portable Document Format is a format that is used to present various types of documents in a readable or viewable manner. The PDF is more like a universal format that is
independent of any hardware, operating system, software or application. A typical PDF file encapsulates the complete description of a flat document with a fixed layout. This includes the text,
graphics, fonts as well as other information that is needed for the display of the contents of the file. The most common application that is used in computers and smartphones to open PDF files is
Adobe Reader. PDF files are very useful; they can be used as an alternative to many different types of data; for storing documents as in Microsoft Word, slides as in PowerPoint, images as in JPEG and
so on.
JPEG, on the other hand, is a method that is very commonly used and it actually compresses digital images in what is known as lossy compression. The extension of JPEG files is .jpg or .jpeg. It is
especially used for compressing images that are produced by digital photography. The compression can be to varying extents. The two main options that need to be addressed for compression are the size
of the file and image quality which are directly proportional. What is good about JPEG compression is that it achieves a 10 is to 1 ratio of compression and there is hardly any loss of quality in the
JPEG is generally a graphic image file whereas a PDF is a document file. This is the main difference between the two formats. Both of these can be converted into the other but generally they are used
for different purposes. Note that for the same file that is made available in the two formats, a JPEG image of a certain document will be of a smaller size than the same document as a PDF file. This
is simply because JPEG is a compression method.
Moving on, a PDF file preserves the original layout of any document but also leaves the different parts of the document open to editing. JPEGs, however, compress various components of an image or
document into one single file that cannot be separated into the original components.
Another difference is with respect to copying text. A PDF allows you to copy selected text from the file whereas a JPEG doesn’t allow you to copy selected text from the file although the whole image
can be copied as it is.
As mentioned earlier, the two formats can be converted into each other. A JPEG to PDF conversion will protect the layout of the image whereas a conversion in the other direction produces a compressed
image of the document.
1. PDF or Portable Document Format is a format that is used to present various types of documents in a readable or viewable manner; JPEG-a method that is very commonly used, compresses digital
images in what is known as lossy compression
• JPEG is generally a graphic image file whereas a PDF is a document file
• For the same file that is made available in the two formats, a JPEG image of a certain document will be of a smaller size than the same document as a PDF file simply because JPEG is a compression
• A PDF file preserves the original layout of any document but also leaves the different parts of the document open to editing; JPEGs, however, compress various components of an image or document
into one single file that cannot be separated into the original components
• A PDF allows you to copy selected text from the file whereas a JPEG doesn’t allow you to copy selected text
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4 Comments
1. Hello. I have read your article and I am wondering if original artwork saved as a JPEG will be suitable for prints to be made from as the prints will be A1 and A2 sized from an original A3. Will
there be any noticeable loss of quality?
2. Hi, I have read your article. For beginners, it is difficult to understand the details as PDF and JPG. Kindly add an example for better understanding showing two different types of files.
□ A PDF file is usually words or documents as they say in the article as to where a JPEG is a picture, it can be a copy of a pdf file but you can’t use it to fill in the blanks as to where a
PDF file you could go in and edit the document and make changes to it if you needed to. You can make one into the other but changing a PDF file is not going to be able to be changed or copied
in less you copy the whole thing or change it back to PDF form
3. Yes but what about the actual code
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The Diggs Equation—Forecasting Josh Allen Passes and Stefon Diggs Catches
The Diggs Equation — Forecasting Josh Allen Passes and Stefon Diggs Catches
In a previous iteration of this article, we explored how to use data, data science, and formulas to abstract and model data around a problem that was first brought to awareness through a divergence
of thought and visualization.
The goal of thinking about this concept was to improve the approach or at least understand the approach, so we worked through it in the last article “The Diggs Equation — Will Josh Allen Pass to
Stefon Diggs?”. We found that the probability both naive and simulated was larger than the actual forecasted percentage, and now we will explore how that could be reevaluated.
Through this next stage keep in mind that if the calculations for all these players and all of their dichotomies or statistical relationships were continuously being calculated for thousands of
users, it would be very expensive predictive system. New advancements in machine learning, through foundation models, could allow for a stateless model, where in fact the algorithm for one of these
relations could be replicated to predict the same type of relation, and spread across all the QB (Quarterback) to WR(Wide Receiver) calculations.
Sampling the existing data | {"url":"https://heyitsjoealongi.medium.com/the-diggs-equation-forecasting-josh-allen-passes-and-stefon-diggs-catches-a218db6bd234","timestamp":"2024-11-09T13:19:22Z","content_type":"text/html","content_length":"89042","record_id":"<urn:uuid:eae7fc62-cba4-4d22-a0ac-72b4f4477175>","cc-path":"CC-MAIN-2024-46/segments/1730477028118.93/warc/CC-MAIN-20241109120425-20241109150425-00499.warc.gz"} |
Just Something Random
The word “random” is one most people learn to misuse at an early age. Teenagers and adults alike often use the word to refer to anything from an unexpected question to pickles on a grilled cheese
sandwich. Webster’s dictionary defines it as made, done, happening, or chosen without method or conscious decision. “Random” questions are typically a product of thoughtful curiosity and “random”
condiments are the product of them being delicious. So the challenge was presented to our pre-calculus class last week: can you be truly “random?”
Basically, several students were instructed to cheat and cut corners on a coin toss lab. They didn’t want to, but I made them cheat. Just this once. To actually do the lab, one must flip a coin
lots of times and record the results. Flip. Heads. Flip. Heads. Flip. Tails. And so forth for the next nine minutes of your life. Booooorrrring! Why not just hastily make up the results and then
make a doodle of a whale with a mustache? No one will know, right?
Cutting corners opens up free time for things like this! The monocle is crucial.
So while half of the students were diligently flipping coins to gather their data, others quickly came up with their own “random” list of coin tosses. HTTHHTHHHTTTHTHHTTHHTHHTTHHTHTTHTHHTHHTHTHHT.
That was fast, now time to doodle.
Nine exhausting minutes of coin tossing later, the results were in and ready for analysis. How many heads and tails did each person get? Those tossing coins compiled their data to even things out.
Here’s the tally with student initials across the top, outcomes of Heads or Tails labeled in the left-most column:
Okay, so the cheaters did alright. There were a few more heads than tails, but you could certainly have gotten those same results experimentally. If I was hunting for cheaters, the closeness of the
coin toss almost seems too good to be true. I’d be suspicious of them if anyone!
So now let’s look at patterns. I’ll start with an easy probability question. A coin is tossed, what is the probability it lands heads? 1/2 right? Okay, it landed tails. Sorry. Let’s flip again.
What’s the probability of heads this time? Do you think it’s still 1/2? You’re absolutely right. So in a perfect world, the occurrences of HT, HH, TH, and TT should all be equal. Those are the
four equally likely outcomes of a double flip. Let’s see what happened when we tallied up consecutive toss outcomes for each data set:
The coin toss is close, with a few more cases of HT or TH than getting two in a row (HH or TT). Meanwhile the students who made up their data have over twice as many alternating pairs as consecutive
pairs! Busted!
When we listed out all eight outcomes possible from three tosses, the results become even more distorted. Our coin toss actually turned out HHH or TTT more commonly than most alternating
possibilities. Meanwhile, Emma and Hailey each chose perfect alternation at least FIVE TIMES more often than they chose three tails in a row. They’re darn good students, but evidently lousy
“Well, clearly people aren’t random because if you just chose heads, part of you wants to pick tails next to keep things even. There’s a conscious decision.” – Hailey T.
Subsequently, this spurred a class discussion about dependent vs. independent probability that quickly went down the road of when statistics actually relate to the conclusions based upon them and
critical thinking. I won’t get into that here, you’ll have to ask one of our pre-calc students.
-Dan Thurber | {"url":"https://alzar.org/just-something-random/","timestamp":"2024-11-01T22:55:34Z","content_type":"text/html","content_length":"94408","record_id":"<urn:uuid:c7f39684-872f-43b5-9834-bd3117414275>","cc-path":"CC-MAIN-2024-46/segments/1730477027599.25/warc/CC-MAIN-20241101215119-20241102005119-00065.warc.gz"} |
Using Newton’s Second Law to Calculate the Force Applied to an Object
Question Video: Using Newton’s Second Law to Calculate the Force Applied to an Object Physics • First Year of Secondary School
An object of mass 75 kg has a force applied to it. The graph shows the change in the object’s velocity while the force is applied. How much force is applied to the object? Give your answer to the
nearest newton.
Video Transcript
An object of mass 75 kilograms has a force applied to it. The graph shows the change in the object’s velocity while the force is applied. How much force is applied to the object? Give your answer to
the nearest newton.
This question gives us a graph, and that graph shows the change in an object’s velocity while a force is applied to it. The graph plots velocity on the vertical axis against time on the horizontal
axis. And so, the slope of the graph, which is defined as the change in the vertical coordinate divided by the change in horizontal coordinate, is equal to the change in velocity, Δ𝑣, divided by the
change in time, Δ𝑡. We can recall that the acceleration of an object is equal to the rate of change of that object’s velocity. And the rate of change of velocity will be equal to the change in
velocity divided by the change in time over which that velocity change occurs. This means that the acceleration 𝑎 of an object is equal to the slope of that object’s velocity–time graph.
Now, the question isn’t actually asking us to find the acceleration, but rather the force that is applied to the object. However, we’re going to see that it will be useful to first find this
acceleration in order to then work out the force. The reason for this is Newton’s second law of motion. This says that the force applied to an object is equal to the object’s mass multiplied by its
acceleration. This is often written in terms of symbols as force 𝐹 is equal to mass 𝑚 multiplied by acceleration 𝑎. The question tells us that the mass of the object is 75 kilograms. So, in this
equation, we know the value of 𝑚. This means that if we can work out the value of the acceleration 𝑎, then we have all of the information that we need to calculate the force applied to the object.
So, let’s work out the value of 𝑎 by finding the slope of this velocity–time graph. We’ll consider the two points that are right at the ends of the graph. So, that’s this point here on the left and
this point on the right. We can see that the left-hand point is at a time value of zero seconds. Meanwhile, tracing down from the right-hand point until we get to the time axis, we see that this
point has a time value of five seconds. The change in the time value between this point and this point on the graph, which is Δ𝑡, is equal to five seconds, so that’s the time value at the right-hand
point, minus zero seconds, the time value at the left-hand point. This gives us that Δ𝑡 is equal to five seconds.
Next, we’ll look at the velocity values. We can see that the left-hand point has a velocity value of zero meters per second. Then, looking now at the right-hand point, we see that it traces across to
this height on the velocity axis. This height is one-fifth of the way between the 12 meters per second mark and the next mark, which would be 14 meters per second. This gives it a value of 12.4
meters per second. Then, the change in velocity Δ𝑣 between the two points on the graph is equal to 12.4 meters per second, that’s the velocity at the right-hand point, minus zero meters per second,
the velocity at the left-hand point. This gives us that Δ𝑣 is equal to 12.4 meters per second.
We can now sub these values for Δ𝑣 and Δ𝑡 into this expression. This lets us calculate the slope of the graph, which gives us the acceleration 𝑎 of the object. Subbing in our values gives us that 𝑎
is equal to 12.4 meters per second divided by five seconds. Then, evaluating the expression, we find that 𝑎 is equal to 2.48 meters per second squared. So, we know the mass 𝑚 of the object, and we
have now found the object’s acceleration 𝑎. We can now take these values and sub them into this equation to calculate the force 𝐹 that’s applied to the object. Subbing the values, we get that 𝐹 is
equal to 75 kilograms, that’s the value for 𝑚, multiplied by 2.48 meters per second squared, the value for 𝑎.
At this point, it’s worth noticing that the mass is expressed in units of kilograms, the SI base unit for mass, and the acceleration is in units of meters per second squared; that’s the SI base unit
for acceleration. Since both of these two quantities are expressed in their SI base units, this means that the force we’re going to calculate will be in the SI base unit for force, which is the
newton. Evaluating this expression, we calculate a force of 186 newtons. This value is the force applied to the object, which is what we were asked to find.
We can notice that the question asked us to give our answer to the nearest newton. But our result is already an integer number of newtons, so we don’t need to do anything further. This means that our
final answer to the question is that the amount of force applied to the object is 186 newtons. | {"url":"https://www.nagwa.com/en/videos/918101409852/","timestamp":"2024-11-04T08:16:48Z","content_type":"text/html","content_length":"250018","record_id":"<urn:uuid:4eabf858-3bb1-4755-882f-32249655aecd>","cc-path":"CC-MAIN-2024-46/segments/1730477027819.53/warc/CC-MAIN-20241104065437-20241104095437-00471.warc.gz"} |
Improved Measurements of the Temperature and Polarization of the Cosmic Microwave Background from QUaD
We present an improved analysis of the final data set from the QUaD experiment. Using an improved technique to remove ground contamination, we double the effective sky area and hence increase the
precision of our cosmic microwave background (CMB) power spectrum measurements by ~30% versus that previously reported. In addition, we have improved our modeling of the instrument beams and have
reduced our absolute calibration uncertainty from 5% to 3.5% in temperature. The robustness of our results is confirmed through extensive jackknife tests, and by way of the agreement that we find
between our two fully independent analysis pipelines. For the standard six-parameter ΛCDM model, the addition of QUaD data marginally improves the constraints on a number of cosmological parameters
over those obtained from the WMAP experiment alone. The impact of QUaD data is significantly greater for a model extended to include either a running in the scalar spectral index, or a possible
tensor component, or both. Adding both the QUaD data and the results from the Arcminute Cosmology Bolometer Array Receiver experiment, the uncertainty in the spectral index running is reduced by ~25%
compared to WMAP alone, while the upper limit on the tensor-to-scalar ratio is reduced from r < 0.48 to r < 0.33 (95% c.l.). This is the strongest limit on tensors to date from the CMB alone. We also
use our polarization measurements to place constraints on parity-violating interactions to the surface of last scattering, constraining the energy scale of Lorentz violating interactions to <1.5 × 10
^-43 GeV (68% c.l.). Finally, we place a robust upper limit on the strength of the lensing B-mode signal. Assuming a single flat band power between ell = 200 and ell = 2000, we constrain the
amplitude of B-modes to be <0.57 μK^2 (95% c.l.).
The Astrophysical Journal
Pub Date:
November 2009
□ cosmic microwave background;
□ cosmological parameters;
□ cosmology: observations;
□ polarization;
□ Astrophysics - Cosmology and Extragalactic Astrophysics
23 pages, 19 figures, updated to reflect published version | {"url":"https://ui.adsabs.harvard.edu/abs/2009ApJ...705..978B/abstract","timestamp":"2024-11-08T18:33:37Z","content_type":"text/html","content_length":"52983","record_id":"<urn:uuid:4748ba91-8ecd-440f-be90-36beaadaf275>","cc-path":"CC-MAIN-2024-46/segments/1730477028070.17/warc/CC-MAIN-20241108164844-20241108194844-00023.warc.gz"} |
1. Introduction
The VisTools library from Visual Kinematics, Inc. is now part of the Tech Soft 3D portfolio. It is an object-based software development toolkit designed for use in creating visualization applications
for science and engineering. As a visualization toolkit, VisTools is differentiated by its rich feature set, computational efficiency and modular, object-based architecture. VisTools is designed to
impose few restrictions on the nature of the computational domain, or specific data structures used within a host application to maintain the computational grid and/or solution results. VisTools also
separates the generation of visualization entities from the graphics subsystem. This allows application developers to integrate VisTools between their existing computational database and graphics
device interface. The basic features of VisTools are summarized below:
• Discrete scalar, vector or tensor field visualization as 2D or 3D icons or numerical values.
• Isovalue display in 1D, 2D and 3D domains. This includes contour line, color filled contour, continuous tone, isosurface, vector surface, dot surface and cuberille generation.
• Perform line, surface and volume integrations associated with the isovalue visualization modules. For example, VisTools is able to compute the area of an isosurface or the volume of material
lying between sets of isosurfaces.
• Unique isosurface clipping feature. Any type of visualization entity may be clipped to a set of arbitrary isosurfaces.
• Streamline and streamribbon generation in 2D and 3D domains. Streamlines may be constrained to lie on a surface in 3D domains. Tangent curves may be produced in vector (velocity) or tensor
(stress) fields.
• All discrete visualization entities may be value mapped to size and/or color. All filled entities (eg. isosurfaces, color filled contours) may be value mapped to color and/or transparency.
• Computational cells may be individual lines, triangles, quadrilaterals, tetrahedra, pyramids, pentahedra or hexahedra or regular meshes of the same cell type. General polygons and polyhedra are
also supported. This allows VisTools to be applied to conventional finite element unstructured grids, polyhedral grids or higher order, p-element finite element grids and multiblock structured
• Normal vectors may be either automatically generated by VisTools or supplied by the user for light source shading. Both facet and vertex normals are supported.
• Annotation features include 2D and 3D stroked fonts and an extensive glyph library of useful 2D and 3D parameterized shapes. Facilities for generating XY and XYZ graphs and drawing triads for
Cartesian, cylindrical or spherical systems are included.
• Automatic calculation of beam section properties for arbitrary cross sections. Automatic calculation of shell wall composite stiffness matrix for arbitrary laminated composites.
• Object-based architecture, written in ANSI C with C++, FORTRAN, and C# language bindings.
• Hardware and graphics device independence.
1.1. Module Summary
VisTools is designed to accept computational cells (eg. individual finite elements or blocks of a multiblock grid) and results data (eg. scalar, vector or tensor fields), perform a visualization
function (eg. isosurface extraction or tensor icon generation) and produce displayable geometry (eg. colors, polylines and polygons). VisTools is meant to be integrated into existing visualization
software systems such as finite element post processors and visual data analysis (VDA) systems with minimal impact upon established data structures and graphics subsystems. The modules currently
delivered with VisTools may be divided into 4 categories: 1) visualization and computation, 2) attribute, 3) annotation and 4) drawing function.
• Visualization and Computation
Mark Scalar, vector or tensor field markers at points.
Value Scalar, vector or tensor field values at points.
Segment Isovalues along lines.
Contour Contours on surfaces.
Threshold Isosurface extraction within solids
Trace Tangent curve generation on surfaces
Stream Tangent curve generation within solids
Edge Draw wireframe geometries
Face Draw shaded surface geometries
Cell Draw shaded solid geometries
ShellWall Compute and draw shell wall properties
ShellElem Draw shell elements
BeamSect Compute and draw beam section properties
BeamElem Draw beam elements
RigidElem Draw rigid elements
MassElem Draw mass elements
DiscElem Draw spring and dashpot elements
GapElem Draw gap elements
• Attribute
Levels Define discrete quantity levels
ColorMap Define quantity level mapping to color
TransMap Define quantity level mapping to transparency
VisContext Define visualization attributes
IsoClip Define clipping isosurface
DataInt Define data interpolation arrays
• Annotation
Axis Draw annotated axes
Billboard Draw extensible 2D lists or billboards
Font Draw stroked 2D and 3D text
Glyph Draw 2D and 3D markers and glyphs
Legend Draw color and/or transparency mapping legends
Triad Draw coordinate system triads.
• Manipulators
HandleBox Rectilinear box
PolyBox Planar polygon
WorkPlane Work plane
• Drawing Function
DrawFun Define drawing function pointers to graphics subsystem
The visualization and manipulator modules are the heart of VisTools. All other modules function to provide services and information to the visualization modules. Each instance of a module in VisTools
is termed an object. Specifically, each instance of a visualization module is termed a visualization object. A visualization entity is defined as the displayable output from a visualization object.
Examples of visualization entities are contour lines, tensor icons, isosurfaces, etc. Visualization objects produce graphics primitives which directly affect geometry such as line widths, points,
polylines and polygons.
The manipulator modules are used to manage various types of clipping, selection and snapping icons. These modules, in general, produce graphics primitive geometry as well as clipping and
transformation primitives. The manipulator modules are controlled by user interaction. The modules do not perform any specific graphics device queries, the user is responsible for implementing the
user interaction and supplying device coordinates and gesture manner (drag, click, etc.) to the modules.
The attribute modules do not produce displayable geometry as such but are used primarily to provide containers for attributes which affect the appearance of visualization entities. An instance of an
attribute module is called an attribute object. Attribute objects produce color and transparency graphics primitives.
The drawing function modules are designed to receive the graphics primitives produced by the visualization and attribute modules. The most straight-forward use of the drawing functions is to
interface VisTools modules directly to a graphics subsystem. This involves making direct calls to set colors, line styles, etc. and draw various flavors of points, lines and polygons using a 3D
graphics application programming interface (API).
Drawing functions may be used to perform specialized processing such as integrating complicated functions over the polygons comprising an isosurface. Drawing functions may also be developed which
feed back output primitives (and field values which have been interpolated to the vertices of output primitives) as input to a visualization object. This allows VisTools to be used recursively to
generate such displays as contour plots on arbitrary isosurfaces or tensor icons along the clipped edge of the face of a finite element. The drawing function module, DrawFun, is formally part of the
VglTools graphics interface library. This module is delivered with VisTools as a support module.
1.2. Computational Cells
VisTools accepts a set of computational cell types which encompasses most topologies in general use in science and engineering. Basic cell primitives, referred to as shape, include the following:
• SYS_SHAPEPOINT, point(s)
• SYS_SHAPELINE, line
• SYS_SHAPETRI, triangle
• SYS_SHAPEQUAD, quadrilateral
• SYS_SHAPETET, tetrahedron
• SYS_SHAPEPYR, pyramid
• SYS_SHAPEWED, pentahedron
• SYS_SHAPEHEX, hexahedron
• SYS_SHAPEPOLYGON, polygon
• SYS_SHAPEPOLYHED, polyhedron
As mentioned earlier there are two distinct forms for each cell topology 1) Serendipity finite elements which are characterized by only having nodes along element edges and 2) Lagrange finite
elements and regular arrays of primitive cells. Polygons and polyhedra are special cases to be described below. These two representations allow VisTools to efficiently process low order finite
elements as well as higher order elements, p-elements and multi block structured grids. The node connectivity conventions for Serendipity finite elements and Lagrange finite elements and arrays are
different. Serendipity element connectivity follows a convention often used in the finite element analysis industry in which corner nodes are numbered first followed by nodes along the midsides of
the element edges. For parabolic and cubic Serendipity elements the corner nodes are followed by nodes on the boundary edges in edge number order. Lagrange finite element or array connectivity
follows an ordering used universally for multidimension arrays. Nodes are ordered in the “i” direction first, the “j” direction second and the “k” direction last. The number of nodes in each element
direction, (i,j,k) are referred to as maxi, maxj and maxk. For example, a 27 node parabolic Lagrange 3D hexahedral solid element has maxi = maxj = maxk = 3. The general rules concerning maxi, maxj
and maxk are outlined in the following paragraphs, some uncommon special cases are described later.
Lagrange connectivity allows for different numbers of nodes in each element direction. For example, a special form of a Lagrange solid element to model thick shells may contain 18 nodes with
parabolic shape functions in the plane of the shell (“i” and “j” directions) and linear functions through the thickness of the shell (“k” direction). In this case maxi = maxj = 3, and maxk = 2.
Serendipity elements must have equal orders in the “i” and “j” directions. The “k” direction may be either linear or equal to the order given to the “i” and “j” directions. Utilizing this fact, the
Serendipity connectivity convention is flagged by setting maxj = 0 (except for the case of missing midside nodes described below). This specifies that the order in the “j” direction is equal to the
order given by maxi and a Serendipity connectivity convention is being used. If maxk = 0, then the order in the “k” direction is equal to the order given by maxi. For example a 20 node parabolic
Serendipity 3D hexahedral solid element has maxi = 3, and maxj = 0 and maxk = 0. Optionally, maxk = 2 specifies linear shape functions in the “k” direction. For example, a 16 node Serendipity thick
shell solid has maxi = 3, maxj = 0 and maxk = 2.
Serendipity and Lagrange finite elements are restricted to linear, parabolic and cubic forms. The shape functions for these types are explicitly supported. Array form is numbered identically to
Lagrange form but for maxi, maxj or maxk exceeding 4, piecewise linear shape functions are used.
The node connectivities for each topology appear below with examples of Serendipity element form and Lagrange element or array form. The maxi, maxj and maxk values associated with each form are
shown. Line connectivities are characterized by (maxi), triangle and quadrilateral connectivities by (maxi, maxj) and tetrahedral, pyramidal, pentahedral and hexahedral connectivities by (maxi, maxj,
As mentioned above, the quantities maxi, maxj and maxk are used to specify the number of nodes in each element direction. Given this general definition there are a number of conventions to
distinguish between Serendipity and Lagrange numbering and some other important special cases.
• maxi = 0, then the linear Serendipity form of the element is assumed and maxj = maxk = 0.
• 2 <= maxi <= 4, maxj = 0, and maxk = 0 then the element is a Serendipity element which is linear, parabolic or cubic in the i, j and k directions.
• 2 <= maxi <= 4, 2 <= maxj <= 4 and 2 <= maxk <= 4, then the element is a Lagrange element with maxi, maxj and maxk nodes in the i,j and k directions respectively.
• 2 <= maxi <= 4, maxj = 0, and 2 <= maxk <= 4. This is a special case of mixed Serendipity and Lagrange numbering for 3D pentahedral and hexahedral shapes in which the i and j directions are
numbered first as a 2D Serendipity element then this numbering is repeated for each nodal plane in the k direction.
• 2 <= maxi <= 4, maxj = 1, and maxk = 0. This a special case for 2D triangle Lagrange shapes in which the i direction has maxi nodes with a single additional node at the triangle apex.
• 2 <= maxi <= 4, maxj = 0 or 2 <= maxj <= 4, and maxk = 1. This a special case for 3D tetrahedron and pyramid Serendipity or Lagrange element shapes in which the nodal pattern in the i and j
directions is given by maxi and maxj with a single additional node at the tetrahedron or pyramid apex.
• maxi = 3, maxj >= 2**16, maxk = 0. This is a special case of parabolic Serendipity elements with missing midside nodes. The lower 16 bits of maxj are zero, the upper 16 bits of maxj are used to
flag missing midside nodes on element edges. The first bit of the upper 16 bits (ie bit 17) is set if the midside node on edge 1 is missing, bit 18 is set if the midside node on edge 2 is
missing, etc. This convention is not used for line element shapes.
Examples of the mixed Serendipity and Lagrange numbering for pentahedral and hexahedral elements appear below. The i and j directions are numbered using a Serendipity connectivity convention. This
numbering is then incremented in the k direction.
The polygon and polyhedron shapes have different interpretations for the quantities maxi, maxj and maxk from the conventional shapes described above. The quantity maxi indicates the total number of
points in the polygon or polyhedron. The quantities maxj and maxk are used internally to indicate the number of edges and number of faces respectively. The user is not required to enter the values of
maxj and maxk and may enter them as zeros. The connectivity convention for polytype cells orders the connectivity of each face such that the right hand rule sense of the connectivity points outward.
In addition, the first node of the connectivity of each face is repeated as the last node in the face connectivity. The total number of points in the polytype includes the nodes repeated due to this
convention. Note that the polytype representation requires a significantly larger number of nodes in the connectivity than a conventional shape of similar complexity. For example a linear hexahedron
of shape SYS_SHAPEHEX requires 8 nodes in the connectivity while the shape SYS_SHAPEPOLYHED requires 30 total nodes (6 faces times 5 nodes per face).
The connectivities for a polygon and polyhedra are shown below. Note that the starting node for each of the face connectivities is arbitrary and the order of the faces is arbitrary.
Polygon n1,n2,n3,n4,n5,n1
Polyhedron n1,n2,n3,n1, n2,n5,n6,n3,n2, n4,n7,n6,n5,n4, n1,n3,n7,n4,n1, n6,n7,n3,n6, n4,n5,n2,n1,n4
For the special cases of quadrilateral and hexahedral grids VisTools provides for 3 special cases of regular arrays (sometimes referred to as structured grids): curvilinear, rectilinear and uniform.
Each case has the same topology while exploiting various degrees of uniformity in the physical mapping of the nodes in space. The 3 cases are illustrated below for a quadrilateral grid.
The motivation for providing these special cases is that the amount of data required to specify node locations is dramatically reduced in each case. For curvilinear grids, each node point is mapped
to physical space by an explicitly supplied coordinate location. For a maxi by maxj by maxk grid, maxi * maxj * maxk node coordinates must be defined. Rectilinear grids are orthographic with variable
spacing between lines of nodes. Rectilinear grids are defined by specifying the intersections of the grid lines with the corresponding coordinate axis in each spatial direction. For a maxi by maxj by
maxk grid, maxi + maxj + maxk node coordinates must be defined. Uniform grids are orthographic with constant spacing between lines of nodes. Uniform grids are defined by specifying the bounding box
of the grid, ie. two coordinates.
1.2.1. Edge and Face Numbering
VisTools occasionally requires the identification of a particular edge or face of a computational cell. For example, the Threshold module may be queried to return the faces of a 3D cell which are
intersected by an isosurface. The edges and faces of a particular cell are specified by an edge or face index. The edges and faces are defined by the cell node indices. The node indices defining
edges and faces for all primitive cell shapes are as follows assuming the low order Serendipity element connectivity convention.
triangle edge - nodes
1 1,2
2 2,3
3 3,1
quadrilateral edge - nodes
1 1,2
2 2,3
3 3,4
4 4,1
tetrahedron edge - nodes edge - nodes face - nodes
1 1,2 4 1,4 1 1,3,2
2 2,3 5 2,4 2 1,2,4
3 3,1 6 3,4 3 1,4,3
4 2,3,4
pyramid edge - nodes edge - nodes face - nodes
1 1,2 5 1,5 1 1,4,3,2
2 2,3 6 2,5 2 1,2,5
3 3,4 7 3,5 3 2,3,5
4 4,1 8 4,5 4 3,4,5
5 4,1,5
wedge edge - nodes edge - nodes face - nodes
1 1,2 7 1,4 1 1,3,2
2 2,3 8 2,5 2 4,5,6
3 3,1 9 3,6 3 1,2,5,4
4 4,5 4 1,4,6,3
5 5,6 5 2,3,6,5
6 6,4
hexahedron edge - nodes edge - nodes face - nodes
1 1,2 7 7,8 1 1,4,3,2
2 2,3 8 8,5 2 5,6,7,8
3 3,4 9 1,5 3 1,2,6,5
4 4,1 10 2,6 4 4,8,7,3
5 5,6 11 3,7 5 1,5,8,4
6 6,7 12 4,8 6 2,3,7,6
The edge and face definitions have a different set of node indices for the Lagrange finite element or array connectivity convention. However the edges and faces are configured on each cell topology
in an identical manner.
1.2.2. Physical and Natural Coordinates
VisTools uses two coordinate systems to describe coordinate locations and field data and perform visualization computations, 1) physical coordinates and 2) natural coordinates.
Physical coordinates are expressed in a 3 dimensional Cartesian coordinate system. This coordinate system must be consistently used to express all domain coordinates and vector and tensor field data
presented to VisTools functions. Point coordinates and other vectors are entered in VisTools as 3 components in the order (x,y,z). Symmetric tensors are entered as 6 components in the following order
Natural coordinates are curvilinear coordinate systems which are used to define interpolation coefficients local to an individual cell or finite element. Natural coordinates are defined in an element
topology dependent manner and are normalized within the element in some way. For example, the natural coordinates in a quadrilateral element (r,s) are normalized in the interval [-1,1]. For
triangular elements, the natural coordinates (r,s) are related to the area coordinates (L1,L2,L3) in the interval [0,1].
For all shapes except polygon and polyhedron, the direction of the natural coordinates may be defined by the cell node indices at the end points of the cell edge which is “parallel” to the natural
coordinate. The node indices are given assuming the low order Serendipity element connectivity convention.
For polygon and polyhedron shapes, the natural coordinates are relative to a triangular and tetrahedral decomposition of the shapes respectively. For polygons, the triangular decomposition is done
from a point at the center of the polygon connecting all of the boundary nodes of the polygon. A polygon with N unique nodes will have N triangles. For polyhedra, the tetrahedral decomposition is
done from a point at the center of the polyhedron connected a point at the center of each face with all nodes bounding the face. A polyhedron with N unique nodes will have N tetrahedra. The r,s and t
coordinates are relative to one of the triangles or tetrahedra in the decomposition. The index (1-based) of the specific triangle or tetrahedron is added to the s natural coordinate.
line natural coordinates - normalization - nodes
r [-1,1] 1,2
triangle natural coordinates - normalization - nodes
r = L1 [0,1] 1,2
s = L2 [0,1] 1,3
quadrilateral natural coordinates - normalization - nodes
r [-1,1] 1,2
s [-1,1] 1,4
tetrahedron natural coordinates - normalization - nodes
r = L1 [0,1] 1,2
s = L2 [0,1] 1,3
t = L3 [0,1] 1,4
pyramid natural coordinates - normalization - nodes
r [-1,1] 1,2
s [-1,1] 1,4
t [-1,1] 1,5
wedge natural coordinates - normalization - nodes
r = L1 [0,1] 1,2
s = L2 [0,1] 1,3
t [-1,1] 1,4
hexahedron natural coordinates - normalization - nodes
r [-1,1] 1,2
s [-1,1] 1,4
t [-1,1] 1,5
polygon natural coordinates - normalization
r [0,1] radial
s [>=0] triangle index + local s
polyhedron natural coordinates - normalization
r [0,1] radial
s [>=0] tetrahedron index + local s
t [0,1] local t
1.3. Element Types
VisTools supports a wide variety of finite element types. The full description of a particular finite element type requires information about it basic type ie. solid, shell, beam, etc. and the
specific topology and order such as linear hexahedron, parabolic tetrahedron, etc. For some specialized elements such as spot welds, there can be additional information concerning the types of
elements to which the spot weld is connected. This additional information is referred to as the end A and end B topology. VisTools begins by placing elements into one of the following general type
• SYS_ELEM_SOLID, solid element
• SYS_ELEM_SHELL, shell element, in-plane stress, bending, shear
• SYS_ELEM_MEMBRANE, membrane element, in-plane stress only
• SYS_ELEM_BEAM, beam element, axial stress, bending, shear
• SYS_ELEM_TRUSS, truss element, axial stress only
• SYS_ELEM_INFINITE, infinite element
• SYS_ELEM_GAP, gap element, point contact
• SYS_ELEM_JOINT, joint element
• SYS_ELEM_SPRINGDASHPOT, spring and dashpot element
• SYS_ELEM_RIGID, rigid element
• SYS_ELEM_CONSTRAINT, constraint element, multipoint constraint
• SYS_ELEM_PLOT, plot element, visualization only
• SYS_ELEM_MASS, mass element
• SYS_ELEM_INTER, interface elements, distributed contact, boundary conditions
• SYS_ELEM_SUPER, superelements
Within each general category the element is further described by its specific type, topology , shape, and order , maxi, maxj, maxk. For most elements these parameters are sufficient to accurately
characterize the element. For most general types there are several specific types which help to identify the element within the general type. The general types are described in more detail below with
information concerning the applicable specific types, topologies and orders.
SYS_ELEM_SOLID, solid elements may be defined in either 2D or 3D space. In 2D space the topology must be either triangle, quadrilateral or polygonal. In 3D space the topology must be tetrahedron,
pyramid, pentahedron, hexahedron or polyhedral. The possible specific types are as follows:
• SYS_SOLID_STAN, standard solid element
• SYS_SOLID_FLUID, fluid solid element
• SYS_SOLID_SHELL, thick shell solid element
SYS_ELEM_SHELL, shell elements may be defined in either 2D or 3D space. In 2D space the topology must be a line, in 3D space the topology must be triangle or quadrilateral.
SYS_ELEM_MEMBRANE, membrane elements may be defined in either 2D or 3D space. In 2D space the topology must be a line, in 3D space the topology must be triangle or quadrilateral. The possible
specific types are as follows:
• SYS_MEMBRANE_STAN, standard membrane element
• SYS_MEMBRANE_SHEAR, shear panel element
• SYS_MEMBRANE_FACE, face element. A facet collocated with a face of a geometry tessellation
SYS_ELEM_BEAM, beam elements may be defined in either 2D or 3D space. In 2D space the topology must be a point with maxi = 1. in 3D space the topology must be line. The possible specific types are as
• SYS_BEAM_STAN, standard beam element
• SYS_BEAM_ROD, axial-torsional element
• SYS_BEAM_WELD, weld element
• SYS_BEAM_CBEND, curved beam and pipe element
SYS_ELEM_TRUSS, truss elements may be defined in either 2D or 3D space. In 2D space the topology must be a SYS_SHAPEPOINT with maxi = 1. in 3D space the topology must be SYS_SHAPELINE The possible
specific types are as follows:
• SYS_TRUSS_STAN, standard truss element
• SYS_TRUSS_EDGE, edge element. A segment collocated with an edge of a geometry tessellation.
SYS_ELEM_SPRINGDASHPOT, spring and dashpot elements are discrete elements whose physical properties are not generally dependent upon an integration over their spatial extent. The topology must be
either SYS_SHAPEPOINT or SYS_SHAPELINE and is independent of the spatial dimension. This category of elements can be quite complicated and as a result the end A and B topologies can be required in
some cases to identify the element. The possible specific types are as follows:
• SYS_SPRINGDASHPOT_SCALAR, scalar spring. The topology is SYS_SHAPELINE for a spring connecting two degrees of freedom and SYS_SHAPEPOINT for a spring connecting a degree of freedom to ground. The
spring may also include a damper.
• SYS_SPRINGDASHPOT_LINK, line spring which generally connects translation and/or rotation freedoms in the direction between two nodes. The topology is SYS_SHAPELINE.
• SYS_SPRINGDASHPOT_WELD, spot weld spring which attempts to model the effect of a spot weld which, in general, smears its connections over the geometry of two opposing elements. If the topology is
SYS_SHAPELINE, then maxi = 2 and the weld element connects two nodes. If the topology is SYS_SHAPEPOINT, then maxi => 2 and the weld element connects maxi nodes. The end A and B topologies
determine how the nodes are connected to the adjacent elements. The nodes associated with the end A topology precede the nodes associated with the end B topology. The end topologies for a spoint
(single node), lines, triangles and quadrilaterals are below. There are not defined constants for the case of a point topology with more than one node or line, triangle and quadrilateral
topologies with maxi and/or maxj greater than 3. In general, end topologies are decoded as shown below:
SYS_TOPO_POINT1 - SYS_SHAPEPOINT, maxi = 1
SYS_TOPO_LINE2 - SYS_SHAPELINE, maxi = 2, maxj = 0
SYS_TOPO_LINE3 - SYS_SHAPELINE, maxi = 3, maxj = 0
SYS_TOPO_TRI3 - SYS_SHAPETRI, maxi = 2, maxj = 0
SYS_TOPO_TRI6SER - SYS_SHAPETRI, maxi = 3, maxj = 0
SYS_TOPO_TRI6LAG - SYS_SHAPETRI, maxi = 3, maxj = 3
SYS_TOPO_QUAD4SER - SYS_SHAPEQUAD, maxi = 2, maxj = 0
SYS_TOPO_QUAD4LAG - SYS_SHAPEQUAD, maxi = 2, maxj = 2
SYS_TOPO_QUAD8 - SYS_SHAPEQUAD, maxi = 3, maxj = 0
SYS_TOPO_QUAD9 - SYS_SHAPEQUAD, maxi = 3, maxj = 3
shape = (topo >> 28) & 0x000f
maxi = (topo >> 16) & 0x0fff
maxj = (topo >> 8) & 0x00ff
maxk = (topo >> 0) & 0x00ff
• SYS_SPRINGDASHPOT_BUSH, bushing spring made up of separate translational and rotational springs with possible parallel dashpots connecting two nodes. The topology is SYS_SHAPELINE and maxi = 2.
SYS_ELEM_RIGID, rigid elements are used to enforce various types of constraints. The elements are mathematically similar to constraint elements, SYS_ELEM_CONSTRAINT, however their definition is not
abstract but is usually in terms of a physically understandable rigid effect. The possible specific types are as follows:
• SYS_RIGID_KINE, kinematic constraints provide a general tying of the translations and rotations of a node to the translations and rotations of a set of nodes. There are always exactly 6
independent degrees of freedom which must be capable of representing any general rigid body motion of the coupling nodes. The topology is SYS_SHAPEPOINT with maxi equal to the number of nodes
involved in the constraint for “spoke” type topologies. The topology is SYS_SHAPELINE with maxi = 2 for rigid beams. The topology is SYS_SHAPETRI with maxi = 3 or SYS_SHAPEQUAD with maxi = 4 for
rigid triangles and quadrilaterals.
• SYS_RIGID_DIST, distributing constraints provide a constraint to distribute force and moment at a point to forces at a set of coupling nodes. The coupling effectively constrains the rotation and
translation at a point to the translations at the set of coupling nodes. This constraint is enforced in an average sense and therefore does not inhibit the relative deformation of the coupling
nodes. The topology is SYS_SHAPEPOINT with maxi equal to the number of nodes involved in the constraint.
• SYS_RIGID_LINK, link constraints enforce a rigid link between two nodes involving translations only. The topology is SYS_SHAPELINE with maxi = 2.
• SYS_RIGID_RBE3, distributing constraint specifically models the NASTRAN RBE3 rigid element. The topology is SYS_SHAPEPOINT with maxi equal to the number of nodes involved in the constraint.
• SYS_RIGID_SPLINE, spline constraint specifically models the NASTRAN RSPLINE rigid element. The topology is SYS_SHAPELINE with maxi equal to the number of nodes involved in the constraint.
• SYS_RIGID_JOINT, coincident node joint constraint enforces identical movement of specified degrees of freedom between two coincident nodes. The topology is SYS_SHAPEPOINT with maxi = 2.
SYS_ELEM_CONSTRAINT, constraint elements are used to impose general multipoint constraints. The topology is SYS_SHAPEPOINT and maxi is equal to the number of degrees of freedom involved in the
constraint equation.
SYS_ELEM_PLOT, plot elements are used for visualization only and have no physical properties. Their topologies and orders are general. The possible specific types are as follows:
• SYS_PLOT_LOD0, level of detail 0
• SYS_PLOT_LOD1, level of detail 1
• SYS_PLOT_LOD2, level of detail 2
• SYS_PLOT_LOD3, level of detail 3
• SYS_PLOT_AERO, Nastran AERO elements
SYS_ELEM_MASS, mass elements are discrete elements whose physical properties are not generally dependent upon an integration over their spatial extent. The topology must be either SYS_SHAPEPOINT or
SYS_SHAPELINE and is independent of the spatial dimension. The possible specific types are as follows:
• SYS_MASS_SCALAR, scalar mass. The topology is SYS_SHAPELINE for a mass connecting two degrees of freedom and SYS_SHAPEPOINT for a scalar concentrated mass.
• SYS_MASS_LUMP, lumped mass which generally involves translational mass and rotary inertia tensor defined in a local coordinate system. The mass is concentrated at a point. The topology is
SYS_SHAPEPOINT and maxi = 1.
• SYS_MASS_MATRIX, lumped mass which is defined by a symmetric 6x6 matrix. The mass is concentrated at a point. The topology is SYS_SHAPEPOINT and maxi = 1.
• SYS_MASS_VERTEX, vertex element. A point collocated with a vertex in a geometry tessellation
1.4. Element Coordinate Systems
An element coordinate system is a Cartesian coordinate system oriented to the element geometry. Element coordinate systems are used in a number of ways depending upon the type of element. The most
common use is a a coordinate system for the computation and output of stress and strain related quantities (heat flux and temperature gradient for thermal analysis, etc.). For some 1D and 0D elements
such as beams, gaps and concentrated masses, the element coordinate system is used to define certain properties of the element such as cross section properties, slip directions and moment of inertia
Certain constraints are placed upon the orientation of the element coordinate system depending upon the element type. For surface elements such as shell elements, the local x’ and y’ axes are
constrained to be tangent to the shell reference surface. The local z’ is normal to the surface. The orientation of the x’ and y’ axes in the tangent plane is determined by convention. The convention
specifies the direction of the x’ axis. The y’ axis is then constructed to complete a right-handed Cartesian system. For line elements such as beam elements, the local x’ axis is constrained to be
tangent to the beam axis. The local y’ and z’ axes are perpendicular to the beam axis. In a manner similar to surface elements, the orientation of the y’ and z’ axes in the plane perpendicular to the
beam axis is determined by convention. The convention specifies the direction of the y’ axis. The z’ axis is then constructed to complete a right-handed Cartesian system. For full 3D solid elements,
there are no constraints upon the orientation of the element local system and as a result it is generally aligned to the global coordinate system. For point elements such as concentrated mass
elements, the element coordinate systems may be arbitrarily oriented in space and are either aligned to the global coordinate system or to a user specified Cartesian system.
A wide variety of element coordinate system conventions are in use in the finite element industry. Many of them are used to resolve the orientation issues in line and surface elements. In order to
achieve coverage of current industry practice, the following types are provided. Where these element coordinate system types are used as options in specific element modules such as VisTools ShellElem
or VfeTools Shell3D, a certain amount of additional data may be required in addition to the element geometry. This is noted for each type.
• Global, SYS_ELEMSYS_GLOBAL. The element coordinate system is aligned to the global axes. When this system is used for surface or line elements it is usually only for the purpose of expressing
vector or tensor output quantities.
• Standard, SYS_ELEMSYS_STANDARD. For volume elements the x’ axis is aligned to the element r natural coordinate direction. The y’ axis is perpendicular to x’ in the plane formed by the r and s
natural coordinate directions. For surface elements the x’ axis is aligned to the element r natural coordinate direction. For line elements the y’ axis lies in the plane formed by the x’ axis and
the global y axis unless the global y axis is 0.1 degree of being tangent to the x’ axis. In this case the y’ axis lies in the plane formed by the x’ axis and the global z axis.
• Position, SYS_ELEMSYS_POSITION. For surface elements the x’ axis is in the direction of the projection on the surface of a line from the point on the surface to a specified point in space. For
line elements the y’ axis lies in the plane formed by the line element axis and a line from the point on the line element axis to a specified point in space. The 3 global coordinates of the
specified point must be provided as additional data.
• Vector, SYS_ELEMSYS_VECTOR. For surface elements the x’ axis is in the direction of the projection on the surface of a specified vector anchored at the point on the surface. For line elements the
y’ axis lies in the plane formed by the line element axis and a specified vector anchored at the point on the line element axis. For 2D volume elements the x’ axis is in the direction of the
projection on the x, y plane of a specified vector anchored at the point on the plane. The y’ axis is perpendicular to x’ in the plane. The 3 components of the specified vector in global
coordinates must be provided as additional data.
• Vectors at Element Nodes, SYS_ELEMSYS_VECTORELEMNODE. For surface elements the x’ axis is in the direction of the projection on the surface of a vector anchored at the point on the surface which
has been interpolated from vectors specified at the element nodes, For line elements the y’ axis lies in the plane formed by the line element axis and a vector anchored at the point on the line
element axis which has been interpolated from vectors specified at the element nodes, The 3 components of the specified vector in global coordinates at each element node must be provided as
additional data.
• Global Project, SYS_ELEMSYS_GLOBALPROJECT. This standard is designed explicitly for support of the conventions for surface elements used in ABAQUS. The default local x’ axis is the projection of
the global x onto the surface. If the global x axis is within 0.1 degree of the normal to the surface, the local x’ direction is the projection of the global z axis onto the surface. For line
elements the z’ axis is constructed to be approximately parallel to the negative global z axis. If the global z axis is within 0.1 degree of the x’ axis the local z’ direction is parallel to the
global x axis.
• Centroid, SYS_ELEMSYS_CENTROID. This standard is designed to orient the element coordinate system with the directions of the natural coordinates at the centroid of the element. The local x’ axis
is along the direction of the first natural coordinate. The local z axis is normal to the plane formed by the cross product of the first and second natural coordinate directions. The y axis is
formed as the cross product of the local z and x axes and as a result will lie in the plane formed by the first and second natural coordinate directions.
• Bisector, SYS_ELEMSYS_BISECTOR. This standard is designed explicitly for support of the conventions for surface elements used in MSC/NASTRAN and is named for the particular method used for the
CQUAD4 shell element. For line elements the convention is the same as the Standard convention.
• Nastran Shell, SYS_ELEMSYS_NASTRANSHELL. This standard is designed explicitly for material coordinate system support for classic CTRIA6 and CQUAD8 shell elements used in NASTRAN. The convention
is similar to SYS_ELEMSYS_STANDARD with a specified angle (in degrees) rotation is applied to the computed direction. The specified angle in degrees, followed by two zeros must be provided as
additional data. An additional angle, specific to CQUAD8, is computed internally.
• Bidiagonal, SYS_ELEMSYS_BIDIAGONAL. This standard is designed explicitly for material coordinate system support of the conventions for surface elements used in SAMCEF.
• First Edge, SYS_ELEMSYS_FIRSTEDGE. This standard is designed explicitly for support of the conventions for surface and line elements used in ANSYS. For surface elements the x’ axis is the
projection onto the surface of the vector directed from the first corner node to the second corner node. For line elements the y’ axis lies in the global x, y plane. For the case that the element
x’ axis is parallel to the global z axis (or within a .01 percent slope of it), the y’ axis is oriented parallel to the global y axis.
• First Edge plus angle, SYS_ELEMSYS_FIRSTEDGEANGLE. This standard is designed explicitly for material coordinate system support of the conventions for surface elements used in NASTRAN. A specified
angle (in degrees) rotation is applied to the direction computed by SYS_ELEMSYS_FIRSTEDGE. The angle in degrees, followed by two zeros must be provided as additional data.
• Mid Edge, SYS_ELEMSYS_MIDEDGE. This standard is designed explicitly for support of the convention for quadrilateral surface elements used by ESI. The x’ axis is the projection onto the surface of
the vector directed from the midpoint of the fourth edge to the midpoint of the second edge. The normal to the surface element is the normal to x’ and a vector directed from the midpoint of the
first edge to the midpoint of the third edge. The y’ axis is constructed orthogonal to the surface normal the x’ axis.
• Mid Point, SYS_ELEMSYS_MIDPOINT. This standard is designed explicitly for support of the conventions for surface elements used in Altair/Radioss and is named for the particular method used for
the linear triangle and quadrilateral shell elements. The local system is constructed by creating a vector which bisects the vectors connecting the midpoints of the first and third edges with the
fourth and second edges. The vector then bisects the x’ and y’ axies of the local coordinate system.
• Global Closest, SYS_ELEMSYS_GLOBALCLOSEST. For surface elements the x’ axis is in the direction of the projection on the surface of the closest global axis. For line elements the y’ axis lies in
the plane formed by the line element axis and the closest global axis to the plane perpendicular to the line element axis.
• Cylindrical system, SYS_ELEMSYS_CYLINDRICAL. This system is designed for support of cylindrical system orientation. The axis of the cylindrical system is specified by two point coordinates and is
directed from the first point to the second. The origin of the system is positioned at the first point. For point elements and 3D volume elements the x’ axis at a point is in the radial direction
of the point, the y’ axis is the tangential direction and the z’ axis is the axis of the cylindrical system. For surface elements the x’ axis is the projection of the radial direction on the
surface. For line elements the y’ axis lies in the plane formed by the radial direction and the axis of the line. The 3 components of the first point followed by the 3 components of the second
point in global coordinates must be provided as additional data.
• Spherical system, SYS_ELEMSYS_SPHERICAL. This system is designed for support of spherical system orientations. The axis of the spherical system is specified by two point coordinates and is
directed from the first point to the second. The origin of the spherical system is positioned at the first point For point elements and 3D volume elements the x’ axis at a point is in the radial
direction of the point, the y’ axis is the tangential direction and the z’ axis is the azimuthal axis. The tangential axis is about the axis of the spherical system. For surface elements the x’
axis is the projection of the radial direction on the surface. For line elements the y’ axis lies in the plane formed by the radial direction and the axis of the line. The 3 components of the
first point followed by the 3 components of the second point in global coordinates must be provided as additional data.
• Spherical system alternate, SYS_ELEMSYS_SPHERICAL_ALT. This system is designed for support of spherical system orientations used by NASTRAN. The axis of the spherical system is specified by two
point coordinates and is directed from the first point to the second. The origin of the spherical system is positioned at the first point For point elements and 3D volume elements the x’ axis at
a point is in the radial direction of the point, the y’ axis is the azimuthal direction and the z’ axis is the tangential axis. The tangential axis is about the axis of the spherical system. For
surface elements the x’ axis is the projection of the radial direction on the surface. For line elements the y’ axis lies in the plane formed by the radial direction and the axis of the line. The
3 components of the first point followed by the 3 components of the second point in global coordinates must be provided as additional data.
• Rotation Angle Vector, SYS_ELEMSYS_ROTANG. The element coordinate system is explicitly specified by a single rotation angle vector relative to the global coordinate system. The rotation angle
vector is computed using the Rodriques formula. The magnitude of the rotation angle vector is the amount of rotation about the vector in degrees. The 3 components of the rotation angle vector
must be provided as additional data.
• Rotation Angle Vectors at Element Nodes, SYS_ELEMSYS_ROTANGELEMNODE. The element coordinate system is explicitly specified by a rotation angle vector relative to the global coordinate system at
each element node. The rotation angle vector is computed using the Rodriques formula. The magnitude of the rotation angle vector is the amount of rotation about the vector in degrees. The
associated vector or tensor quantities are also output at each element node. The 3 components of the rotation angle vector at each element node must be provided as additional data.
• Cylindrical system alternate, SYS_ELEMSYS_CYLINDRICAL_ALT. This system is designed to support of cylindrical system orientations used by NASTRAN CBEND element. For line elements, the x’ axis at a
point is the radial direction of the point, the z’ axis is the axis of the cylindrical system and y’ axis is the tangential direction. The 3 components of the radial vector at the first node of
the element specified in global coordinates must be provided as additional data.
• Unknown system, SYS_ELEMSYS_UNKNOWN. This system is designed to support an element system which is not completely known. For volume elements the system is identical to SYS_ELEMSYS_GLOBAL. For
surface elements the system is identical to SYS_ELEMSYS_STANDARD. For line elements the system is identical to SYS_ELEMSYS_STANDARD. For point elements the system is identical to
1.5. Mathematical Data Types
CeetronSAM provides many methods to manipulate and visualize mathematical data types such as scalars, vectors, symmetric tensors and general tensors. The following ordering conventions are used for
the components of vector, v, tensor, t, and general tensor, g, data types.
Vector v (x, y, z)
v[0] = x
v[1] = y
v[2] = z
Tensor t (xx, yy, zz, xy, yz, zx)
t[0] = xx
t[1] = yy
t[2] = zz
t[3] = xy
t[4] = yz
t[5] = zx
General Tensor g (xx, xy, xz, yx, yy, yz, zx, zy, zz)
g[0] = xx g[1] = xy g[2] = xz
g[3] = yx g[4] = yy g[5] = yz
g[6] = zx g[7] = zy g[8] = zz
There are specializations of symmetric tensors for the finite element stress resultants and strain-curvatures of shell and beam type elements.
Shell stress resultants s (Nxx, Nyy, Nxy, Mxx, Myy, Mxy, Qxz, Qyz)
Shell strain curvatures e (Exx, Eyy, Exy, Kxx, Kyy, Kxy, Txz, Tyz)
s[0] = Nxx e[0] = Exx
s[1] = Nyy e[1] = Eyy
s[2] = Nxy e[2] = Exy
s[3] = Mxx e[3] = Kxx
s[4] = Myy e[4] = Kyy
s[5] = Mxy e[5] = Kxy
s[6] = Qxz e[6] = Txz
s[7] = Qyz e[7] = Tyz
Beam stress resultants s (Nxx, Myy, Mzz, Torque, Qxy, Qzx)
Beam strain curvatures e (Exx, Kyy, Kzz, Twist, Txy, Tzx)
s[0] = Nxx e[0] = Exx
s[1] = Myy e[1] = Kyy
s[2] = Mzz e[2] = Kzz
s[3] = Torque e[3] = Twist
s[4] = Qxy e[4] = Txy
s[5] = Qzx e[5] = Tzx
The representation of the coordinate systems in which these quantities are expressed, where applicable, requires support for direction cosine matrices and their equivalent compact representation as
rotation angle vectors.
The following convention for the direction cosine matrices of a local coordinate system is used. Given that x’,y’ and z’ are three orthonormal vectors indicating the direction of the local coordinate
axes in the global coordinate system (x,y,z), then the direction cosine matrix, tm[3][3] for this local coordinate system is defined as:
tm[0][0] = x'x tm[0][1] = x'y tm[0][2] = x'z
tm[1][0] = y'x tm[1][1] = y'y tm[1][2] = y'z
tm[2][0] = z'x tm[2][1] = z'y tm[2][2] = z'z
where y’x, for example, is the global x coordinate of the y’ unit vector.
The rotation angle vector, ra, can be used as a compact representation of a direction cosine matrix. It is the generalization of an infinitesimal rotation vector to finite rotations.
1.6. Complex Numbers
A number of VisTools modules are designed to store and manipulate complex numbers. A consistent set of functions are implemented across these modules to control how the real and imaginary parts of
the complex data are to be set and queried from the modules. The modules which are currently designed to handle complex numbers are the loading and constraint modules, LCase and RCase and the results
modules, RedMat, State, History and ElemDat.
For example, the RCase module function vis_RCaseSetComplexMode() is used to specify which component(s) of a complex value, (real and/or imaginary), are to be set or queried by the functions
vis_RCaseSetSPC() and vis_RCaseSPC() respectively. By default the complex mode value is SYS_COMPLEX_REAL, that is, the set and query functions only expect a real number or the real part of a complex
number. Both the real and imaginary parts can be queried by setting the complex mode SYS_COMPLEX_REALIMAGINARY. If only the imaginary part of a complex number is to be set or queried use
SYS_COMPLEX_IMAGINARY. If at any time the complex mode is set to SYS_COMPLEX_IMAGINARY or SYS_COMPLEX_REALIMAGINARY the module will, in general, contain complex data. The function vis_RCaseGetComplex
() can be used to determine if the module does contain complex data. The function vis_RCaseGetComplexMode() will return the current complex mode. All of the modules listed above contain identical
functions to set/get the complex mode and query for the existence of complex data.
There is not a special data type given complex numbers. The real and imaginary parts are represented as two consecutive real numbers. For example, setting the 3 components of a double precision real
vector would be 3 consecutive double precision numbers representing the x, y, z components of the vector. The equivalent complex vector would require 6 double precision numbers representing the x,x
(i), y,y(i), z,z(i) values of the vector.
1.7. Compiling and Linking a VisTools Application
To use VisTools on a particular computer platform, the VisTools source must be compiled and linked to an application. Either the object files may be used directly or they may be installed in an
object library archive so that the loader may selectively relocate only the objects which are required. VisTools is written in ANSI C. It is suggested to use the highest level of serial optimization
options available on the C compiler.
VisTools is platform independent and as a result no user defined C preprocessor directives are required to compile VisTools on any supported platform. However it is suggested that during the
development cycle that the source be conditionally compiled with error checking by defining VKI_CHECK_ERRS as described in base library.
For example, on SGI systems, create a directory SGI under lib to hold the final CeetronSAM.a archive file. Then from the CeetronSAM/src/legacy/vis directory compile using
creating .o files in the vis directory. To place the object files in an archive file issue
ar rs ../../lib/SGI/CeetronSAM.a *.o
The object files may be deleted after the CeetronSAM.a archive is successfully created. To compile the vgl source, change directory to CeetronSAM/src/legacy/vgl and compile using
creating .o files in the vgl directory. If you have a complete VglTools installation, compile using the instructions in the VglTools Programmer Manual. To add these objects to the previously created
CeetronSAM.a archive issue
ar rs ../../lib/SGI/CeetronSAM.a *.o
To compile the base source, change directory to CeetronSAM/src/sam/base and compile using
creating .o files in the base directory. To add these objects to the previously created CeetronSAM.a archive issue
ar rs ../../lib/SGI/CeetronSAM.a *.o
Again object files may deleted at this time. At this point the CeetronSAM.a archive contains all vis, vgl and base objects. Place the CeetronSAM.a archive immediately before the graphics subsystem
libraries in the load line. A CeetronSAM.a archive must be built for each computer platform using the methodology outlined above.
1.8. Attribute Objects, Data Interpolation, Isovalue Clipping and Topology
The visualization modules share many common features with respect to the use of attribute objects and setting the current computational grid topology. Generally the visualization modules in each
category such as isovalue extraction, tangent curve generation, geometry rendering, feature extraction, etc are further divided by dimension. For example, the Segment, Contour and Threshold isovalue
extraction modules are designed to extract isovalues in 1D, 2D and 3D computational cells respectively.
Each visualization module is designed to accept attribute objects which will affect the appearance of the generated graphics primitives. All attribute objects are set in the visualization objects
using a similar function (SetObject). For example, for the Contour object, use vis_ContourSetObject(). The attribute object, VisContext, is the basic container for the myriad settings such as line
width, point size, size scaling, etc. which affect the appearance of generated graphics primitives. The ColorMap and TransMap objects along with the Levels object associate or map color and
transparency to data value. The DataInt object specifies data quantities to be mapped to an output graphics primitive in the same way that color or transparency are mapped to a primitive. This object
is useful, for example, for generating contours of a data field onto the isosurfaces of another data field. The IsoClip object specifies a data field used as an isosurface for clipping graphics
primitives. Use this object along with the Threshold object to generate a “clipped and capped” display.
The exact nature of the computational cell topology to be processed by each visualization module is also set using a similarly named function (SetTopology). For example, for the Contour object, use
vis_ContourSetTopology(). The polyhedral cell topology, in particular, requires additional topological information in the form of the polyhedral node connectivity for efficient rendering. The
function (SetElemNode) is designed to input this information. For example, for the Threshold object, use vis_ThresholdSetElemNode(). Only the 3D visualization modules support and, in some cases,
require the polyhedral node connectivity.
1.9. A First Program - C Version
As an example of a simple VisTools application the following program draws isosurfaces through a unit cube of data. The attribute modules used are the VisContext, Levels, ColorMap and TransMap
modules, the visualization module is Threshold. The DrawFun module contains the callback functions which the Threshold module uses to output the generated graphics primitives. First, a DrawFun object
is instanced. Rather than outputting the displayable geometry to a graphics device, the built-in “print” drawing functions are used. These functions are set up internally in the DrawFun object using
the function vgl_DrawFunAPI().
The attribute objects are instanced and are set up to draw isosurfaces at 3 evenly spaced levels with red, green and blue assigned to each discrete data level respectively. The attribute objects and
drawing function object are then registered with the Threshold object using the function vis_ThresholdSetObject(). The actual graphics primitives are generated which represent the isosurfaces through
the hexahedron by the function vis_ThresholdCurv(). Finally all objects are deleted.
#include "legacy/vgl/vgl.h"
#include "sam/vis/vis.h"
#include "legacy/vis/vislegacy.h"
#include "sam/base/license.h"
#include "sam/CEETRON_SAM_license.h"
static Vfloat xhex[8][3] = {{0., 0., 0.}, {1., 0., 0.}, {1., 1., 0.}, {0., 1., 0.},
{0., 0., 1.}, {1., 0., 1.}, {1., 1., 1.}, {0., 1., 1.}};
static Vfloat shex[8] = {0., 1., 1., 0., 1., 2., 2., 1.};
static Vfloat rgb[4][3] = {{.2f, .2f, .2f}, {1., 0., 0.}, {0., 1., 0.}, {0., 0., 1.}};
Generate isosurfaces in a hexahedron
vgl_DrawFun* df;
vis_VisContext* vc;
vis_Levels* levels;
vis_ColorMap* cmap;
vis_TransMap* tmap;
vis_Threshold* threshold;
Vint nlevels;
/* create draw function object */
df = vgl_DrawFunBegin();
/* set built in print functions */
vgl_DrawFunAPI(df, DRAWFUN_APIPRINT);
/* vis context and set attributes */
vc = vis_VisContextBegin();
vis_VisContextSetIsoValType(vc, VIS_ISOVALSURFACE);
/* levels, set three evenly spaced levels */
levels = vis_LevelsBegin();
nlevels = 3;
vis_LevelsDef(levels, LEVELS_LINEAR, nlevels);
vis_LevelsSetMinMax(levels, 0., 2.);
vis_LevelsGenerate(levels, LEVELS_PADENDS);
/* color map */
cmap = vis_ColorMapBegin();
vis_ColorMapSetType(cmap, COLORMAP_TRUECOLOR);
vis_ColorMapSetRGB(cmap, nlevels + 1, 0, rgb);
/* transparency map */
tmap = vis_TransMapBegin();
/* create threshold object and set objects */
threshold = vis_ThresholdBegin();
vis_ThresholdSetObject(threshold, VGL_DRAWFUN, df);
vis_ThresholdSetObject(threshold, VIS_VISCONTEXT, vc);
vis_ThresholdSetObject(threshold, VIS_LEVELS, levels);
vis_ThresholdSetObject(threshold, VIS_COLORMAP, cmap);
vis_ThresholdSetObject(threshold, VIS_TRANSMAP, tmap);
/* draw threshold surfaces */
vis_ThresholdCurv(threshold, shex, xhex, VIS_NODATA, NULL);
/* free all objects */
return 0;
The output of this example program appears below. Note that a constant transparency is set and then three isosurfaces are output, each isosurface consists of a RGB color and two triangular polygons.
transp 0.000000
c 1.000000 0.000000 0.000000
type 0
npts 3
x 0.500000 0.000000 0.000000
x 0.500000 1.000000 0.000000
x 0.000000 1.000000 0.500000
vflag 1
v 0.707107 0.000000 0.707107
type 0
npts 3
x 0.000000 1.000000 0.500000
x 0.000000 0.000000 0.500000
x 0.500000 0.000000 0.000000
vflag 1
v 0.707107 0.000000 0.707107
c 0.000000 1.000000 0.000000
type 0
npts 3
x 1.000000 0.000000 0.000000
x 1.000000 1.000000 0.000000
x 0.000000 1.000000 1.000000
vflag 1
v 0.707107 0.000000 0.707107
type 0
npts 3
x 0.000000 1.000000 1.000000
x 0.000000 0.000000 1.000000
x 1.000000 0.000000 0.000000
vflag 1
v 0.707107 0.000000 0.707107
c 0.000000 0.000000 1.000000
type 0
npts 3
x 1.000000 1.000000 0.500000
x 0.500000 1.000000 1.000000
x 0.500000 0.000000 1.000000
vflag 1
v 0.707107 0.000000 0.707107
type 0
npts 3
x 0.500000 0.000000 1.000000
x 1.000000 0.000000 0.500000
x 1.000000 1.000000 0.500000
vflag 1
v 0.707107 0.000000 0.707107
1.10. A First Program - C++ Version
The following program is a listing of the C++ version of the same “A First Program” listed above which used C language bindings.
#include "sam/base/base.h"
#include "legacy/vgl/vgl.h"
#include "sam/vis/vis.h"
static Vfloat xhex[8][3] = {{0., 0., 0.}, {1., 0., 0.}, {1., 1., 0.}, {0., 1., 0.},
{0., 0., 1.}, {1., 0., 1.}, {1., 1., 1.}, {0., 1., 1.}};
static Vfloat shex[8] = {0., 1., 1., 0., 1., 2., 2., 1.};
static Vfloat rgb[4][3] = {{.2, .2, .2}, {1., 0., 0.}, {0., 1., 0.}, {0., 0., 1.}};
Generate isosurfaces in a hexahedron
vgl_DrawFun* df;
vis_VisContext* vc;
vis_Levels* levels;
vis_ColorMap* cmap;
vis_TransMap* tmap;
vis_Threshold* threshold;
Vint nlevels;
/* create draw function object */
df = new vgl_DrawFun;
/* set built in print functions */
/* vis context and set attributes */
vc = new vis_VisContext;
/* levels, set three evenly spaced levels */
levels = new vis_Levels;
nlevels = 3;
levels->Def(LEVELS_LINEAR, nlevels);
levels->SetMinMax(0., 2.);
/* color map */
cmap = new vis_ColorMap;
cmap->SetRGB(nlevels + 1, 0, rgb);
/* transparency map */
tmap = new vis_TransMap;
/* create threshold object and set objects */
threshold = new vis_Threshold;
threshold->SetObject(VGL_DRAWFUN, df);
threshold->SetObject(VIS_VISCONTEXT, vc);
threshold->SetObject(VIS_LEVELS, levels);
threshold->SetObject(VIS_COLORMAP, cmap);
threshold->SetObject(VIS_TRANSMAP, tmap);
/* draw threshold surfaces */
threshold->Curv(shex, xhex, VIS_NODATA, NULL);
/* free all objects */
delete df;
delete vc;
delete levels;
delete cmap;
delete tmap;
delete threshold;
return 0;
1.11. A First Program - FORTRAN Version
The following program is a listing of the FORTRAN version of the same “A First Program” listed above which used C language bindings.
C Generate isosurfaces in a hexahedron
PROGRAM INTRO1F
INCLUDE 'base/fortran/base.inc'
INCLUDE 'vgl/fortran/vgl.inc'
INCLUDE 'vis/fortran/vis.inc'
REAL XHEX(3,8), SHEX(8), RGB(3,4)
DATA XHEX /
$ 0.,0.,0., 1.,0.,0., 1.,1.,0., 0.,1.,0.,
$ 0.,0.,1., 1.,0.,1., 1.,1.,1., 0.,1.,1. /
DATA SHEX /
$ 0., 1., 1., 0.,
$ 1., 2., 2., 1. /
DATA RGB /
$ .2,.2,.2, 1.,0.,0., 0.,1.,0., 0.,0.,1. /
DOUBLE PRECISION DF,VC,LEVELS,CMAP,TMAP,THRESHOLD
INTEGER NLEVELS
C create draw function object
CALL VGLF_DRAWFUNBEGIN(DF)
C set built in print functions
CALL VGLF_DRAWFUNAPI(DF,DRAWFUN_APIPRINT)
C vis context and set attributes
CALL VISF_VISCONTEXTBEGIN(VC)
CALL VISF_VISCONTEXTSETISOVALTYPE (VC,VIS_ISOVALSURFACE)
C levels, set three evenly spaced levels
CALL VISF_LEVELSBEGIN(LEVELS)
NLEVELS = 3
CALL VISF_LEVELSDEF (LEVELS,LEVELS_LINEAR,NLEVELS)
CALL VISF_LEVELSSETMINMAX (LEVELS,0.,2.)
CALL VISF_LEVELSGENERATE (LEVELS,LEVELS_PADENDS)
C color map
CALL VISF_COLORMAPBEGIN(CMAP)
CALL VISF_COLORMAPSETTYPE (CMAP,COLORMAP_TRUECOLOR)
CALL VISF_COLORMAPSETRGB (CMAP,NLEVELS+1,0,RGB)
C transparency map
CALL VISF_TRANSMAPBEGIN(TMAP)
C create threshold object and set objects
CALL VISF_THRESHOLDBEGIN(THRESHOLD)
CALL VISF_THRESHOLDSETOBJECT (THRESHOLD,VGL_DRAWFUN,DF)
CALL VISF_THRESHOLDSETOBJECT (THRESHOLD,VIS_VISCONTEXT,VC)
CALL VISF_THRESHOLDSETOBJECT (THRESHOLD,VIS_LEVELS,LEVELS)
CALL VISF_THRESHOLDSETOBJECT (THRESHOLD,VIS_COLORMAP,CMAP)
CALL VISF_THRESHOLDSETOBJECT (THRESHOLD,VIS_TRANSMAP,TMAP)
C draw threshold surfaces
CALL VISF_THRESHOLDCURV (THRESHOLD,SHEX,XHEX,VIS_NODATA,0)
C free all objects
CALL VGLF_DRAWFUNEND (DF)
CALL VISF_VISCONTEXTEND (VC)
CALL VISF_LEVELSEND (LEVELS)
CALL VISF_COLORMAPEND (CMAP)
CALL VISF_TRANSMAPEND (TMAP)
CALL VISF_THRESHOLDEND (THRESHOLD)
1.12. A First Program - C# Version
The following program is a listing of the C# version of the same “A First Program” listed above which used C language bindings.
using System;
using System.Runtime.InteropServices;
using System.Reflection;
using System.Text;
using DevTools;
public class intro1 {
public static float [] xhex = {
0.0F,0.0F,0.0F, 1.0F,0.0F,0.0F, 1.0F,1.0F,0.0F, 0.0F,1.0F,0.0F,
0.0F,0.0F,1.0F, 1.0F,0.0F,1.0F, 1.0F,1.0F,1.0F, 0.0F,1.0F,1.0F };
public static float [] shex = {
0.0F, 1.0F, 1.0F, 0.0F,
1.0F, 2.0F, 2.0F, 1.0F };
public static float [] rgb = {
0.2F,0.2F,0.2F, 1.0F,0.0F,0.0F, 0.0F,1.0F,0.0F, 0.0F,0.0F,1.0F };
Generate isosurfaces in a hexahedron
public static void Main() {
IntPtr df;
IntPtr vc;
IntPtr levels;
IntPtr cmap;
IntPtr tmap;
IntPtr threshold;
int nlevels;
/* create draw function object */
df = vgl.DrawFunBegin();
/* set built in print functions */
vgl.DrawFunAPI (df,vgl.DRAWFUN_APIPRINT);
/* vis context and set attributes */
vc = vis.VisContextBegin ();
vis.VisContextSetIsoValType (vc,vis.VIS_ISOVALSURFACE);
/* levels, set three evenly spaced levels */
levels = vis.LevelsBegin ();
nlevels = 3;
vis.LevelsDef (levels,vis.LEVELS_LINEAR,nlevels);
vis.LevelsSetMinMax (levels,0.0F,2.0F);
vis.LevelsGenerate (levels,vis.LEVELS_PADENDS);
/* color map */
cmap = vis.ColorMapBegin ();
vis.ColorMapSetType (cmap,vis.COLORMAP_TRUECOLOR);
vis.ColorMapSetRGB (cmap,nlevels+1,0,rgb);
/* transparency map */
tmap = vis.TransMapBegin ();
/* create threshold object and set objects */
threshold = vis.ThresholdBegin ();
vis.ThresholdSetObject (threshold,vgl.VGL_DRAWFUN,df);
vis.ThresholdSetObject (threshold,vis.VIS_VISCONTEXT,vc);
vis.ThresholdSetObject (threshold,vis.VIS_LEVELS,levels);
vis.ThresholdSetObject (threshold,vis.VIS_COLORMAP,cmap);
vis.ThresholdSetObject (threshold,vis.VIS_TRANSMAP,tmap);
/* draw threshold surfaces */
vis.ThresholdCurv (threshold,shex,xhex,vis.VIS_NODATA,null);
/* free all objects */
vgl.DrawFunEnd (df);
vis.VisContextEnd (vc);
vis.LevelsEnd (levels);
vis.ColorMapEnd (cmap);
vis.TransMapEnd (tmap);
vis.ThresholdEnd (threshold); | {"url":"https://docs.techsoft3d.com/ceetron/latest/solve-access-mesh/legacy/vis/introduction.html","timestamp":"2024-11-12T02:40:39Z","content_type":"text/html","content_length":"338778","record_id":"<urn:uuid:30f4a193-1681-4567-bd2a-92acb2ecea64>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.50/warc/CC-MAIN-20241112014152-20241112044152-00422.warc.gz"} |
Yusuke Kobayashi (小林 佑輔), An FPT Algorithm for Minimum Additive Spanner Problem - Discrete Mathematics Group
Yusuke Kobayashi (小林佑輔), An FPT Algorithm for Minimum Additive Spanner Problem
Wednesday, January 20, 2021 @ 4:30 PM - 5:30 PM KST
Zoom ID: 869 4632 6610 (ibsdimag)
For a positive integer t and a graph G, an additive t-spanner of G is a spanning subgraph in which the distance between every pair of vertices is at most the original distance plus t. Minimum
Additive t-Spanner Problem is to find an additive t-spanner with the minimum number of edges in a given graph, which is known to be NP-hard. Since we need to care about global properties of graphs
when we deal with additive t-spanners, Minimum Additive t-Spanner Problem is hard to handle, and hence only few results are known for it. In this talk, we study Minimum Additive t-Spanner Problem
from the viewpoint of parameterized complexity. We formulate a parameterized version of the problem in which the number of removed edges is regarded as a parameter, and give a fixed-parameter
algorithm for it. We also extend our result to (α,β)-spanners. | {"url":"https://dimag.ibs.re.kr/event/2021-01-20/","timestamp":"2024-11-07T15:50:31Z","content_type":"text/html","content_length":"147627","record_id":"<urn:uuid:cba9d4d5-4a91-47cc-b7ea-40564d4c8a6e>","cc-path":"CC-MAIN-2024-46/segments/1730477028000.52/warc/CC-MAIN-20241107150153-20241107180153-00195.warc.gz"} |
What is: Least Absolute Deviations
What is Least Absolute Deviations?
Least Absolute Deviations (LAD) is a statistical method used in regression analysis that focuses on minimizing the sum of the absolute differences between observed values and the values predicted by
a model. Unlike the more commonly used Least Squares method, which minimizes the sum of the squared differences, LAD is particularly robust against outliers. This characteristic makes it a valuable
tool in data analysis, especially in datasets where extreme values can skew results significantly. By prioritizing absolute differences, LAD provides a more accurate representation of central
tendency in the presence of anomalies.
Mathematical Formulation of Least Absolute Deviations
The mathematical formulation of Least Absolute Deviations can be expressed through the optimization problem that seeks to minimize the objective function defined as ( sum_{i=1}^{n} |y_i – f(x_i)| ),
where ( y_i ) represents the observed values, ( f(x_i) ) denotes the predicted values from the regression model, and ( n ) is the number of observations. This formulation highlights the focus on
absolute differences, which contrasts sharply with the squared differences used in Least Squares. The optimization process often involves linear programming techniques, making it computationally
efficient for large datasets.
Applications of Least Absolute Deviations
Least Absolute Deviations is widely applicable in various fields, including economics, finance, and environmental science. In economics, for instance, LAD can be employed to estimate demand functions
or consumer behavior models where outliers may represent atypical purchasing patterns. In finance, it is useful for portfolio optimization and risk assessment, where extreme returns can distort
traditional regression analyses. Additionally, environmental scientists utilize LAD to model relationships between variables in ecological studies, ensuring that their findings are not
disproportionately influenced by anomalous data points.
Advantages of Using Least Absolute Deviations
One of the primary advantages of using Least Absolute Deviations is its robustness to outliers. In many real-world scenarios, datasets contain anomalies that can significantly affect the results of
regression analysis. By minimizing absolute deviations instead of squared deviations, LAD provides a more reliable estimate of the underlying relationship between variables. Furthermore, LAD can
yield more interpretable results in certain contexts, as the absolute differences can be more intuitive than squared differences, especially when communicating findings to non-technical stakeholders.
Comparison with Least Squares Method
When comparing Least Absolute Deviations with the Least Squares method, it is essential to understand their fundamental differences. While Least Squares aims to minimize the sum of squared residuals,
which can disproportionately weight larger errors, LAD treats all deviations equally. This difference in approach leads to varying results, particularly in datasets with outliers. In cases where the
data is normally distributed and free of anomalies, Least Squares may provide more efficient estimates. However, in the presence of outliers, LAD often outperforms Least Squares by providing more
stable and reliable estimates.
Computational Techniques for Least Absolute Deviations
Computing the Least Absolute Deviations requires specialized algorithms, as the optimization problem is not differentiable due to the absolute value function. Common techniques include linear
programming methods, such as the simplex algorithm, which can efficiently handle the constraints of the optimization problem. Additionally, interior-point methods are also employed for larger
datasets, providing a balance between computational efficiency and accuracy. These computational techniques enable practitioners to apply LAD in various scenarios, making it a versatile tool in data
Limitations of Least Absolute Deviations
Despite its advantages, Least Absolute Deviations is not without limitations. One notable drawback is that it can be less efficient than Least Squares in situations where the errors are normally
distributed. In such cases, the estimates produced by LAD may have larger variances compared to those obtained through Least Squares. Additionally, the interpretation of LAD results can sometimes be
less straightforward, particularly when communicating findings to audiences unfamiliar with statistical concepts. Therefore, it is crucial for analysts to consider the context of their data and the
specific goals of their analysis when choosing between LAD and other methods.
Software Implementations of Least Absolute Deviations
Various statistical software packages and programming languages offer implementations of Least Absolute Deviations. For instance, R provides the `lm()` function with an option for LAD through the
`method = “lad”` argument. Python users can leverage libraries such as Statsmodels and Scikit-learn, which include functions for fitting LAD models. Additionally, specialized software like MATLAB and
SAS also support LAD regression analysis. These tools facilitate the application of LAD in practical scenarios, allowing analysts to harness its benefits without extensive manual calculations.
Conclusion on the Relevance of Least Absolute Deviations
Least Absolute Deviations remains a relevant and powerful technique in the field of statistics and data analysis. Its robustness against outliers and straightforward interpretation make it an
essential tool for researchers and practitioners alike. As data continues to grow in complexity and volume, understanding and applying methods like LAD will be crucial for deriving meaningful
insights and making informed decisions based on data. | {"url":"https://statisticseasily.com/glossario/what-is-least-absolute-deviations/","timestamp":"2024-11-04T07:28:07Z","content_type":"text/html","content_length":"139605","record_id":"<urn:uuid:8efc74e2-b84a-4a44-8390-5c4645483aca>","cc-path":"CC-MAIN-2024-46/segments/1730477027819.53/warc/CC-MAIN-20241104065437-20241104095437-00897.warc.gz"} |
How do you find the derivative of e^sqrt(x)? | HIX Tutor
How do you find the derivative of #e^sqrt(x)#?
Answer 1
With the chain rule the derivative is given by:
#d/dxe^sqrt(x)=e^sqrt(x)*d/dxsqrt(x)# #=e^sqrt(x)1/(2sqrt(x))#.
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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
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• Readily available 24/7 | {"url":"https://tutor.hix.ai/question/how-do-you-find-the-derivative-of-e-sqrt-x-8f9af9df84","timestamp":"2024-11-12T19:00:01Z","content_type":"text/html","content_length":"573301","record_id":"<urn:uuid:e1acdfe6-19af-4bbf-83dd-7a5eedcbb4fe>","cc-path":"CC-MAIN-2024-46/segments/1730477028279.73/warc/CC-MAIN-20241112180608-20241112210608-00899.warc.gz"} |
Angular power spectrum of the FASTICA cosmic microwave background component from Background Emission Anisotropy Scanning Telescope data S. Donzelli, 1,2 D. Maino, M. Bersanelli,1 J. Childers,3 N.
Figueiredo,4 P. M3 3 2003b), and describe the foreground 2008 ), result in accurate measurements well into the "fifth peak" region. transverse to the chosen axis. I will show that the angular power
spectrum is a powerful probe to assess the angular characteristics of neutrino data and demonstrate that we are already constraining â ¦ mated angular power spectrum (Hivon et al. The results are
consistent and can be understood using models for the spatial matter power spectrum and for the redshift distribution of radio galaxies at mJy flux density â ¦ Indeed, if the temperature ï¬
uctuations are Gaussian, with random phase, then the angular power spectrum provides a complete description of the statistical properties of the CMB. The power spectrum in column 2 would vary from
one point This paper presents the angular power spectrum obtained from the first-year WMAP sky maps. of Tech.ï¼ MW2008-192 㠨㠬㠽æ å ±ã ¢ã ¼ã «ã ¤ã ã ¸ã ®ã ªã ³ã ¯ï¼ MW2008-192 The angular
power spectrum (C â ) of the DGSE quantifies the fluctuations in the magnetic field and in the electron density of the turbulent interstellar medium of our Galaxy (e.g. The form coded in column 3,
Table 00 is the most "likely" one. 3.1 : Analytic model for power spectrum due to CS with: power spectrum due to a string with configuration parameter vector Î and redshift z [ DY, Takahashi,
Sendouda, Yoo, Sasaki, 1006.0687] In order to compute the angular (2007)). The MAPS C â (ν a, ν b) completely quantifies the second-order statistics of the sky signal under the assumption that
the signal is statistically homogeneous and isotropic on the sky. Companion papers present the maps and an overview of the basic results (Bennett et al. We compute the angular power spectrum Câ
from 1.5 million galaxies in early SDSS data on large angular scales, â Ë < 600. 2006 ; Reichardt et al. ABSTRACT In this work, we present a new approach to estimate the power spectrum P(k) of
redshifted H i 21-cm brightness temperature fluctuations. ANGULAR SPECTRUM REPRESENTATION in a plane z=const. The data are modestly contaminated by diffuse Galactic foreground emission, but we show
that a simple Galactic template â ¦ For â ¦ Estimates of the angular power spectrum, together with a description of the uncertainties, can be viewed as The power spectrum of the CMB anisotropies
peaks at \(l\sim 200\), which corresponds to an angular scale on the sky of \(\Delta \theta \sim 1^{\circ}\), which is very close to the solid angle subtended by the Big Island of Hawaii on The
angular power spectrum estimator developed by Peebles and Hauser & Peebles has been modified and applied to the 4 yr maps produced by the COBE DMR. The extraction of the angular power spectrum of the
CMB anisotropy is complicated by foreground emission within our galaxy and extragalactic radio sources, as well, as the detector noise (Bouchet and Gispert, 1999, ). The angular power spectrum, C l,
is a useful intermedi-ate step on this road from galaxy catalog to parameter constraints. 1000ä¸ èª å é ²ï¼ Weblioè¾ æ ¸ - power spectrum 㠨㠯ã æ å ³ã ã ±ã ã ¼ã ã ºã ã ¨ã ...ã power
spectrumã ã ®æ å ³ã »ä¾ æ ã »ç ¨æ³ ã ªã Weblioè ±å ã »å è ±è¾ æ ¸ After a spectrum which is less than a threshold is replaced with the spectrum PCSP for power smoothing or the spectrum PCSP for
power smoothing is added to a spectrum SP to perform power smoothing of the spectrum SP. the angular power spectrum, after a modest correction for diffuse Galactic emission and extragalactic point
sources. doi The U.S. Department of Energy's Office of Scientific and Technical Information U.S. Department of Energy Office of Scientific and Technical Information The angular power spectrum of the
CMB is a vital tool for understanding the components of our universe at a very early time, just 300,000 years after the Big Bang. This angular power spectrum is a plot of how much the temperature
varies from point to point on the sky (the y-axis variable) vs. the angular frequency ell (the x-axis variable). WMAP has made a cosmic variance limited measurement of the angular power spectrum to â
= 530 and we now report results into the "third peak" region. angular power spectrum from cosmic (super-)strings 8/2/2010-8/5/2010 2 1.1 : Conventional (field theoretic) cosmic strings are
line-like object formed in the early universe through spontaneous symmetry breaking. The WMAP results, combined with recent ground-based measurements of the TT angular power spectrum (Readhead et al.
We study a variety of power spectrum estimation methods and data combinations and demonstrate that the results are robust. Angular power spectrum C h â ¢ To characterise the statistical properties of
a Gaussian random ï¬ eld, we can calculate the mean and the variance of the ï¬ eld. Power Spectrum - It is the amplitude of the Fourier component, which is easier to discern in data processing.
Citation Crill, Brendan Patrick (2001) A Measurement of the Angular Power Spectrum of the Cosmic Microwave Background with a Long Duration Balloon-borne Receiver. The angular power spectrum of the
anisotropy of the CMB contains information about the formation of the Universe and its current contents. Dissertation (Ph.D.), California Institute of Technology. Komatsu et The power spectrum of the
observed sky has been compared to the power spectra of a large number of simulated random skies produced with noise equal to the observed noise and primordial density fluctuation power spectra of â ¦
2004 ; Jones et al. In this plane we can evaluate the two-dimensional Fourier transform of the complex ï¬ eld E(r) = E(x,y,z) as EË (k x,ky; z) = angular power spectrum of the CMB. Angular Power
Spectrum in Modular Invariant Inflation Model Mitsuo J. Hayashi Shiro Hirai Tomoyuki Takami Yusuke Okamei Kenji Takagi Tomoki Watanabe International Journal of Modern 22 2223ï¼ 2237 (2007) Waelkens,
Schekochihin & Enßlin 200920122013). The early structure of the universe as seen in the Cosmic Microwave Background (CMB) can be represented by an angular power spectrum, a plot that shows how the
temperature pattern in the early universe varies with è¬ æ¼ æ é ²ï¼ ã 㠼㠯㠼ã è¬ æ¼ å 2009-03-06 10:15 Impact of Angular Power Spectrum on the Diversity Antenna Gain of Mobile Terminal Elnaz
Foroughi Shafieiã »Junichi Takadaï¼ Tokyo Inst. We measure the angular power spectrum C[l] of radio galaxies in the NRAO VLA Sky Survey (NVSS) using two independent methods: direct spherical harmonic
analysis and maximum likelihood estimation (MLE). Figure 1: The CMB power spectrum as a function of angular scale. In this paper we propose a diï¬ erent approach which is based on the analysis of the
distortion of the angular correla-tion function. We present the angular power spectrum derived from the first-year Wilkinson Microwave Anisotropy Probe (WMAP) sky maps. (2002),Mitra et al. The data
set covers about 160 square degrees, with a characteristic depth of order 1hâ 1 Gpc in the | {"url":"https://rodpub.com/css/15kv32/fac4fe-angular-power-spectrum","timestamp":"2024-11-09T06:30:09Z","content_type":"text/html","content_length":"56988","record_id":"<urn:uuid:3a41217a-5bc5-4059-b92e-a33e1ee3da21>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.30/warc/CC-MAIN-20241109053958-20241109083958-00827.warc.gz"} |
Yesterday a customer called me to ask about how his current traffic levels translate to the load he should use for performance testing. We discussed it in several ways, and he said I had been
helpful. This morning he sent me an email with a fantastic suggestion to write a blog post about our discussion.
The focus is to determine how many virtual users someone should use in their load tests. Sounded good to me, and here is how he framed the question in the email:
For example, while looking at Google Analytics for a given average day, during a peak hour we had:
□ 2000 visitors in 60 minutes
□ 10,000 page views
□ avg page views 5
□ avg time on site 7 minutes
So I wanted to figure out how many users should we feed the LoadStorm system to simulate this traffic as a base line. Does this math look correct for this case?
2000 users in 1 hour (60 minutes), 7 min time on site
60 minutes / 7 min = 8.5
2000 / 8.5 = 235 Users
Establishing an Algorithm
This approach is going to tell us how many users we have on average. Let’s put this into a mathematical formula:
U = V / (60/D)
Where: U is the number of load test virtual users (that’s what we are trying to figure out) V is the average number of visitors per hour D is the average duration of a visitor
60 is the number of minutes in an hour
Let’s state this formula again in English like a math word problem:
Load Test Virtual Users is equal to the Average Visitors per Hour divided by the User Turnover Rate per Hour | {"url":"https://loadstorm.com/loadstorm-usage/","timestamp":"2024-11-03T19:35:18Z","content_type":"text/html","content_length":"82097","record_id":"<urn:uuid:316bb0f4-30b3-491d-a6d2-0d1075feda3b>","cc-path":"CC-MAIN-2024-46/segments/1730477027782.40/warc/CC-MAIN-20241103181023-20241103211023-00801.warc.gz"} |
3. [Trigonometric Substitutions] | College Calculus: Level II | Educator.com
Remember to make the substitution as well as the main substitution.
You can sometimes use trigonometric substitution even when you don t have a square root. However, in these cases, you should only be using the tangent substitution. If you get one of the other ones,
then you should have been able to factor the algebraic expression, and trigonometric substitution was not necessary.
If you have a linear term (x or u, as opposed to x² or u²), complete the square and make a substitution to eliminate the linear term before you make the trigonometric substitution.
If the coefficient of the quadratic term (x²) is negative, factor the negative out of all terms before completing the square.
After you make the trigonometric substitution, it becomes a trigonometric integral that you can handle using the techniques in the previous section.
Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while
watching the lecture.
Section 1: Advanced Integration Techniques
Integration by Parts 24:52
Integration of Trigonometric Functions 25:30
Trigonometric Substitutions 30:09
Partial Fractions 41:22
Integration Tables 20:00
Trapezoidal Rule, Midpoint Rule, Left/Right Endpoint Rule 22:36
Simpson's Rule 21:08
Improper Integration 44:18
Section 2: Applications of Integrals, part 2
Arclength 23:20
Surface Area of Revolution 28:53
Hydrostatic Pressure 24:37
Center of Mass 25:39
Section 3: Parametric Functions
Parametric Curves 22:26
Polar Coordinates 30:59
Section 4: Sequences and Series
Sequences 31:13
Series 31:46
Integral Test 23:26
Comparison Test 22:44
Alternating Series 25:26
Ratio Test and Root Test 33:27
Power Series 38:36
Section 5: Taylor and Maclaurin Series
Taylor Series and Maclaurin Series 30:18
Taylor Polynomial Applications 50:50
We are going to start with the integral of 6x x2 dx.0006
The problem is we do not have a constant - x2.0016
What we have to do is complete the square on that before we go ahead with our trigonometric substitution.0022
I am going to write that as sqrt, I am going to factor out the negative sign first, and I will get - x2 - 6x,0031
So, I take the middle term, the 6x, and divide it by 2 and square it.0046
So -6/-2 is 3, you square that and you get 9, add on 9.0051
Now to pay for that, well I added 9, but that was inside the negative sign right here.0059
So, what I really did there was I subtracted 9,0069
So, to balance that out I have to add 9.0072
I will write that as sqrt(9) -, and the whole point of completing the square there is that this was (x-3)2.0090
The first thing I am going to do here is a little substitution.0102
Let u = (x-3), and we always have to substitute the du as well.0109
The du is dx there, so it is an easy substitution but we have to do it.0112
What we have here is the integral of sqrt(9-u2) du.0120
Now, this is something that looks like it is ready for a trig substitution.0131
We look at the trig substitution rules and we see that this one calls for a sin substitution,0135
Because it is 9 minus u2, the u is being subtracted.0142
Again, I got this 3 because it is the square root of this nine.0154
The point of doing that is the sqrt(9-u2), well, u2 = 92θ.0171
If we pull a 9 out of the radical we get 3 × (1-sin2θ).0190
The radical turns into 3cosθ, the du turns into 3cos(θ) dθ.0204
Again, we have a trigonometric integral in the trigonometric integral lecture, that when you see even powers of cosine,0228
I will just pull the 1/2 outside, 9/2, times the integral of 1 + cos(2θ) dθ.0243
So, that is 9/2, if you integrate 1 you get θ + integral of cos is sine,0256
But because of that 2, we have to have a 1/2 there, and this was all being multiplied by the 9/2.0272
We have seen this earlier and the way we want to resolve sin(2θ) is 2sin(θ) cos(θ).0290
So, I am going to keep going on the next page here.0301
Let us remember the important things are that u was 3sin(θ), and in turn u was (x-3).0307
We will be using the sin(2θ) and I am just going to copy this equation on the next page and keep going.0322
From the previous page we have 9/2 × θ + 1/2 sin(2θ) + C.0331
Now the cos(θ), what we remember from before is that cos(θ).0416
Actually we cannot see it here, so maybe I will work it out again very quickly.0424
Cos(θ) is the sqrt(1 - sin2(θ)), which is sqrt(1) -, sin(θ) is u/3, so we know sin(θ) is u2/90430
And, so, this turns into 9/2 arcsin, OK u/3 it is time to convert that back into, remember u is x-3.0456
Now 1 - u2/9 is, 1 - u2 is (x-3)2/9.0483
I am just going to try to simplify this radical because I think you will recognize it after I simplify it.0498
That is 1 - (x-3)2, is x2 - 6x + 9/9.0504
The 9/9's cancel, and we get 9/2 arcsin(x-3/3) + (x-3)/3.0524
This radical turns into over 9 - x2 + 6x.0538
If you pull that 9 out, we get a 1/3 6x-x2, which is the same radical we started with.0547
So, our final answer is 9/2 arcsin(x-3/3) + x-3/9 × sqrt(6x - x2) + C.0559
The key to it was recognizing this initial radical in the integral.0589
Recognizing that we had to complete the square on that.0594
Then we had to do a trigonometric sin substitution on that.0600
So, those were sort of the two key theoretical steps there.0611
The rest was just keeping track of lots of substitutions.0617
And then keeping track of a lot of different constants that were percolating throughout the whole interval.0625
A very long and complicated example but the basic ideas were just completing the square and doing a basic substitution.0630
Remember, when you see x2 + 1, that is your sort of warning flag that you are going to need a tangent substitution.0009
Remember a minus tells you that you are going to use either a sin or a secant substitution,0018
And a plus tells us that we are going to use a tangent substitution.0025
So we are going to use x = tan(θ), and then dx = sec2(θ) dθ.0027
x2 + 1 = tan2(θ + 1), which by the trigonometric identity is sec2(θ).0043
And, that was the whole point of making the substitution, to invoke that trigonometric identity,0059
That we would get tan2 + 1 and could convert it into sec2(θ).0064
When we solve this integral, or rather make this substitution into this integral,0077
We have dx in the numerator, so that converts into sec2(θ) dθ.0087
So, the secants, one of the secants cancels and we just get the integral of sec(θ) dθ.0100
That is the trigonometric integral that we learned in the trigonometric integral lecture.0113
There is an old trick, where you multiply the top and bottom by sec(θ) tan(θ).0121
The answer for the integral of sec(θ) came out to be the ln(abs(sec(θ) + tan(θ).0140
That in turn, turns into, remember we have to substitute back into x.0153
sec(θ), I figure out what this right here, that is the sqrt of x2 + 1.0156
Again, the key step there was recognizing that we had x2 + 1,0177
And recognizing that that would give us a tangent substitution.0181
It was also important to keep track of the dx,0188
And then to work everything into θ's, and then at the end convert everything back into x's.0190
So that is the end of the lecture on trigonometric substitutions.0195
Welcome to educator.com, this is the lecture on trigonometric substitution.0000
There are three main equations that you have for trigonometric substitution.0006
The idea is that if you see any one of these three forms in an integral, then you can do what is called a trigonometric substitution to convert your integral into a trigonometric integral.0012
Then hopefully you can use some of the trigonometric techniques that we learned in the previous lecture to solve the integral.0028
Now here the a and the b are constants, and the u is the variable.0042
And, so a-bu2 will be a - while bu2 will be asin2(θ)0063
If you have the square root of that expression, then that will convert into cos(θ), and you will get an easier integral to deal with.0096
By the same idea, if you have au2-b, then you are going to have u=sqrt(b/(asec(θ)).0104
What you are trying to do there is take advantage of the trigonometric identity tan2(θ)+1=sec2(θ).0117
Sec2(θ)-1 = tan2(θ) so once you make this substitution you are going to end up with sec2(θ)-1 under the square root.0131
The last one you will have a+bu2=sqrt(a/(btan(θ))) and so again you are taking advantage of this trigonometric identity tan2(θ)+1.0144
You will end up with that under the square root, and that will convert into sec2(θ).0161
One thing that is important to remember when you are making these substitutions is whenever you substitute u equals something, or x equals something, you always have to substitute in dU or dX as
For example, if you substitute u=sqrt(a/bsin(θ)), then you also have to substitute dU, which would be, a and b are constants so that is just sqrt(a/bcos(θ)) dθ.0182
You always have to make the accompanying substitution for your dU or your dX.0204
As with all mathematical problems it is a little hard to understand when you are just looking at mathematical formulas in general, but we will move onto examples and you will see how these work.0211
The first example is the integral of 4 - 9x2 dx.0223
That example matches the first pattern that we saw before, which was the square root of a minus b u2.0229
The substitution that we learned for that is u equals the square root of a/b sin(θ).0237
Here, the a is 4, the b is 9, the x is taking the place of the u.0249
What we are going to substitute is x equals well the square root of a over b, is the square root of 4/9 sin(θ).0259
As we do that we also remember we have to substitute dX, which is going to be 2/3 cos(θ) dθ.0279
This is why it is really important to write the dX along with your integral.0291
It helps you remember that you also need to do the extra part of the substitution with the dX.0296
You have to convert that to the new variable as well.0302
So, then our integral becomes the integral of the square root of 4 minus, well 9x2 is 9, x2 is 4/9 sin2(θ) and dX is 2/3 cos(θ) dθ.0304
Just working inside the square root for a moment here, we get the square root of 4 minus 4 sin2(θ) and we can pull the 4 out of the square root.0334
We get 4 times the square root of 1 minus sin2θ.0343
That is 2 times 1 minus sin2(θ) is cos2(θ) so that will give us cos(θ).0352
Now, if we bring in the other elements of the integral, we get the integral.0360
I am going to collect all of the constants outside, so we have a 2, and a 2/3 that is 4/3 cos2(θ) because we have one cos here, and one cos here, dθ.0363
We learned how to integrate cos2(θ), you use the half angle formula.0380
This is 4/3 times the integral of 1/2 times one plus cos(2θ) dθ.0387
If we combine the 1/2 with the 4 we get 2/3 times now the integral of 1.0401
Remember we're integrating with respect to θ so that is θ plus the integral of cos(2θ).0412
Well the integral of cos is sin, but because of the 2 there, we have to put a 1/2 there.0421
This is now 2/3 and we want to convert this back to x's.0432
We have to solve these equations for θ in terms of x.0448
θ if we solve this equation in terms of x, we get 3/2 x equals sin(θ)0449
Arcsin(3/2 x), that is the θ. Now 1/2 sin(2θ) to sort that out it helps to remember that sin(2θ) is 2 times sin(θ) times cos(θ)0478
Cos(θ) is the square root of 1 minus sin2(θ), so 1 minus 3/2 sin(θ) was x 9/4 x2.0532
Now we have converted everything back in terms of x.0554
The point here was that we started with a square root of a quadratic and we used our trigonometric substitution and we converted this integral into a trigonometric integral.0568
Then we learned in the other lecture how to convert trigonometric integrals, so we solved that integral in terms of θ.0588
Then we have to convert it back into terms of x.0592
So, let us try another example, this one is actually a little quicker.0598
We see the integral of dX over 1 + x20605
The key thing to remember here is that when you see the square root of 1 + x2, or even 1 + x2 without a root, you want to use the tangent substitution.0611
The reason you do that is that x2 + 1 is tangent2(θ) + 1 but that is sec2(θ)0632
Of course, every time you make a substitution you also have to make a substitution for dX.0650
If we plug that into the integral, we get dX is sec2(θ) dθ.0663
We kind of lucked out on this one because the sec2's cancel and we just get the integral of θ0680
This turns out to be a really easy one, that is just θ.0686
Theta, remember that x was tan(θ) we know that θ is arctan(x) + a constant.0692
The thing to remember about this example is that when you have a + there, you are going to go for a tangent substitution.0706
Whenever you have a minus, you are going to go for a secant substitution.0720
We are going to move to a more complicated example now.0727
We have a more complicated expression in the denominator here.0737
What we are going to do is try to simplify it into something that admits a trigonometric substitution pretty easily.0738
What we are going to do is a little high school algebra on the denominator.0748
We are going to complete the square on that, so that is x20758
Remember, the way you complete the square is you take this 8 and you divide by 2 and you square it.0764
So 8 divide by 2 is 4 and 4 squared is 16, so we will write 16 there.0770
To make this a true equation, we have to add 9 there.0777
Then that is x + 4, quantity squared + 9.0780
What we are going to do is use that for the quick substitution.0795
U is equal to x + 4 and again we have to convert dU, but that is easy that is just dX.0797
Now, to finish that integral, well, we want to make a trigonometric substitution.0820
The way I got 3tanθ was I looked at 9 and I took the square root of that and I got 3.0839
The point of doing that is that u2 + 9 will be 9tan2(θ)+9.0844
Of course with any substitution, you have to figure out what dU is.0867
Well if u is 3tan(θ) then dU is 3 sec2(θ) dθ.0871
We will plug all of that in, the dU was 3 sec2θ dθ.0881
The three and the 9 simplify down into 1/3 and now again, the secants cancel and we just have the integral of dθ.0900
θ is something we can figure out from this equation over here.0914
But we are not finished with that because we still have to convert back into terms of x.0938
That is 1/3 arctan, now u was (x+4), remember our original substitution there, so (x+4)/3 and now we add on a constant.0944
OK, that is the end of the first instalment of trigonometric substitution.0966
Our free lessons will get you started (Adobe Flash® required). Get immediate access to our entire library. | {"url":"https://www.educator.com/mathematics/calculus-ii/murray/trigonometric-substitutions.php","timestamp":"2024-11-10T21:18:43Z","content_type":"application/xhtml+xml","content_length":"457052","record_id":"<urn:uuid:80e334cc-c6a1-4ccc-bd30-60e294c5c9cb>","cc-path":"CC-MAIN-2024-46/segments/1730477028191.83/warc/CC-MAIN-20241110201420-20241110231420-00819.warc.gz"} |
Understanding Mathematical Functions: What Makes A Function Odd Or Eve
Understanding Mathematical Functions: What makes a function odd or even
Mathematical functions are fundamental to understanding various mathematical concepts and are widely used in fields such as physics, engineering, economics, and computer science. Functions help in
describing the relationship between two variables and are essential for making predictions and solving problems in these domains.
The purpose of this blog post is to delve into the concepts of odd and even functions, differentiate between them, and understand the rules that govern them. By the end of this post, you will have a
clear understanding of what makes a function odd or even and why it matters.
A. Define mathematical functions and their importance in various fields
A mathematical function is a relation between a set of inputs and a set of possible outputs, with the property that each input is related to exactly one output. Functions are important in various
fields because they help in analyzing and modeling real-world phenomena, making predictions, and solving complex problems. They provide a systematic way of understanding the cause-effect
relationships between different variables.
B. Outline the purpose of the blog post: to differentiate between odd and even functions
This blog post aims to explain the concepts of odd and even functions and their significance in mathematics. By understanding the differences between these two types of functions, readers will gain
insights into the symmetry and behavior of functions, which is crucial for a deeper understanding of mathematical concepts and their applications.
C. Preview the criteria and rules that govern whether a function is odd, even, or neither
Throughout this post, we will explore the specific criteria and rules that determine whether a function is odd, even, or neither. Understanding these rules is essential for identifying the symmetry
properties of functions and applying them in various mathematical contexts. By the end of this discussion, readers will be able to confidently analyze functions and determine their parity.
Key Takeaways
• Understanding odd and even functions
• Odd functions: f(-x) = -f(x)
• Even functions: f(-x) = f(x)
• Graphical representation of odd and even functions
• Applications of odd and even functions
Identifying Odd Functions
When it comes to mathematical functions, understanding their properties is essential for solving problems and analyzing data. One important property of functions is whether they are odd or even. In
this chapter, we will explore how to identify odd functions and understand their characteristics.
A. Describe odd functions with the standard definition f(-x) = -f(x)
An odd function is a type of function that satisfies the condition f(-x) = -f(x). In other words, when you replace x with -x in the function, the result is the negative of the original function. This
property leads to specific symmetry in the graph of odd functions, which we will explore in the next section.
B. Explore the graphical representation: symmetry about the origin
Graphically, odd functions exhibit symmetry about the origin. This means that if you were to fold the graph along the y-axis and then along the x-axis, the two halves would perfectly overlap.
Visually, this symmetry reflects the property of f(-x) = -f(x), as the function's values on one side of the y-axis are the negatives of the corresponding values on the other side.
C. Provide examples of odd functions, such as f(x) = x^3 or f(x) = sin(x)
There are several examples of odd functions that are commonly encountered in mathematics. One classic example is the function f(x) = x^3. When you substitute -x for x in this function, you get f(-x)
= (-x)^3 = -x^3, which satisfies the condition for odd functions.
Another example of an odd function is the sine function, f(x) = sin(x). By applying the angle difference identity for sine, sin(-x) = -sin(x), we can see that the sine function also satisfies the
condition for odd functions.
Understanding odd functions and being able to identify them is crucial for various applications in mathematics, physics, and engineering. By recognizing their unique properties and graphical
symmetry, we can gain valuable insights into the behavior of these functions and their role in mathematical analysis.
Recognizing Even Functions
Understanding mathematical functions is essential in the study of calculus and algebra. One important classification of functions is whether they are even or odd. In this chapter, we will focus on
recognizing even functions and understanding their key characteristics.
A. Define even functions with the criteria f(-x) = f(x)
An even function is a type of function where the value of the function at a particular point is equal to the value of the function at the opposite point. In mathematical terms, a function f(x) is
considered even if f(-x) = f(x) for all x in the domain of the function. This means that the function exhibits symmetry with respect to the y-axis.
B. Explain the concept of symmetry about the y-axis as seen in graphs of even functions
The concept of symmetry about the y-axis is a key characteristic of even functions. When graphed on a coordinate plane, even functions exhibit mirror symmetry with respect to the y-axis. This means
that if you were to fold the graph along the y-axis, the two halves would perfectly overlap. Visually, this symmetry is represented by a graph that is identical on both sides of the y-axis.
C. Present examples of even functions, such as f(x) = x^2 or f(x) = cos(x)
Examples of even functions include f(x) = x^2 and f(x) = cos(x). In the case of f(x) = x^2, when you substitute -x for x, the resulting value is the same as when you substitute x, satisfying the
criteria for even functions. Similarly, the cosine function f(x) = cos(x) also exhibits symmetry about the y-axis, making it an even function.
Understanding Mathematical Functions: What makes a function odd or even
When it comes to understanding mathematical functions, one of the key concepts to grasp is the distinction between odd and even functions. By applying algebraic tests, we can determine whether a
function is odd or even, which has significant implications for its behavior and properties. Let's delve into the algebraic tests and the process for classifying functions.
Demonstrate how to apply algebraic tests to confirm if a function is odd or even
When determining whether a function is odd or even, we can use algebraic tests to confirm its type. An odd function satisfies the condition f(-x) = -f(x), while an even function satisfies the
condition f(-x) = f(x). By substituting -x for x in the function and simplifying, we can verify if these conditions hold true.
Discuss the significance of identifying the power of x in polynomial functions
In polynomial functions, the power of x plays a crucial role in determining whether the function is odd or even. For example, in a polynomial function with an odd power (e.g., f(x) = x^3), the
function is odd. Conversely, in a polynomial function with an even power (e.g., f(x) = x^4), the function is even. Understanding the significance of the power of x helps in quickly identifying the
nature of the function.
Provide a step-by-step process for classifying simple and complex functions
Classifying functions as odd or even can be done through a step-by-step process. For simple functions such as polynomial functions, we can directly apply the algebraic tests and power of x to
determine their type. However, for complex functions involving multiple terms or transcendental functions, the process may involve breaking down the function into its constituent parts and applying
the algebraic tests to each part separately. By systematically analyzing the function, we can classify it as odd or even.
Real-World Applications: How Odd and Even Functions Are Used
Odd and even functions play a crucial role in various real-world applications, particularly in the fields of physics, mathematical modeling, and computer science. Understanding the properties of
these functions is essential for solving complex problems and developing innovative solutions.
A. Odd and Even Properties in Physics
One of the key areas where odd and even functions are utilized is in physics, particularly in the study of wave functions and signal processing. In the context of wave functions, odd functions
represent asymmetric waveforms, while even functions represent symmetric waveforms. This distinction is vital in analyzing and interpreting wave behavior, especially in fields such as acoustics,
optics, and quantum mechanics.
Similarly, in signal processing, the concept of odd and even functions is used to characterize the properties of signals. Odd functions are associated with anti-symmetric signals, while even
functions correspond to symmetric signals. This distinction is crucial in designing filters, modulators, and demodulators for various communication systems.
B. Role of Odd and Even Functions in Mathematical Modeling and Computer Science
In mathematical modeling, odd and even functions are employed to represent and analyze various phenomena. For instance, odd functions are used to model systems with anti-symmetric behavior, such as
magnetic fields and certain types of vibrations. On the other hand, even functions are utilized to model symmetric phenomena, including gravitational fields and oscillatory motion.
Moreover, in computer science, the properties of odd and even functions are leveraged in algorithm design and data analysis. These functions are used to optimize computational processes, particularly
in the context of image processing, pattern recognition, and cryptography. Understanding the behavior of odd and even functions is essential for developing efficient algorithms and data structures.
C. Distinguishing Between Odd and Even Functions for Problem-Solving
There are numerous scenarios in various disciplines where distinguishing between odd and even functions is vital for problem-solving. For example, in electrical engineering, analyzing the symmetry
properties of signals is crucial for designing filters and amplifiers. In economics, understanding the behavior of odd and even functions is essential for modeling market dynamics and predicting
economic trends.
Furthermore, in the field of cryptography, the properties of odd and even functions are utilized in encryption and decryption algorithms. Distinguishing between these functions is critical for
ensuring the security and integrity of sensitive data. Additionally, in the field of robotics, understanding the symmetry properties of functions is essential for designing motion control systems and
robotic manipulators.
Overall, the applications of odd and even functions extend across various domains, and their properties are fundamental for solving real-world problems and advancing technological innovations.
Troubleshooting Common Misunderstandings and Mistakes
When it comes to understanding mathematical functions, it's important to address common misconceptions and mistakes that can arise. In this section, we will explore some of the most prevalent
misunderstandings and errors related to identifying whether a function is odd or even.
A. Misconception about the terms 'odd' and 'even'
One common misconception is that the terms 'odd' and 'even' relate to the exponents of x alone. This misunderstanding can lead to misclassifying functions and can hinder a deeper understanding of the
• Clarification: It's important to understand that the terms 'odd' and 'even' refer to the behavior of the function with respect to reflections across the y-axis, not just the exponents of x. An
odd function satisfies f(-x) = -f(x), while an even function satisfies f(-x) = f(x).
• Example: For instance, the function f(x) = x^3 is odd because f(-x) = -x^3, while the function g(x) = x^2 is even because g(-x) = x^2.
B. Assuming a function is neither odd nor even without thorough testing
Another common error is assuming that a function is neither odd nor even without thorough testing. This can lead to overlooking important properties of the function and can result in
• Correction: It's essential to thoroughly test the function for odd and even properties before concluding that it is neither. This involves substituting -x into the function and comparing the
result with the original function.
• Example: For a function h(x) = x^4 - x^2, testing h(-x) = (-x)^4 - (-x)^2 = x^4 - x^2, which is equal to h(x). Therefore, h(x) is an even function.
C. Strategies for checking work to avoid misclassifying functions
To avoid misclassifying functions as odd or even, it's important to employ effective strategies for checking work and verifying the properties of the function.
• Use symmetry: Take advantage of the symmetry properties of odd and even functions to check for their behavior with respect to reflections across the y-axis.
• Test with specific values: Substitute specific values of x into the function to verify whether it satisfies the conditions for odd or even functions.
• Verify algebraically: Use algebraic manipulation to test the properties of odd and even functions, such as substituting -x into the function and comparing the result with the original function.
By addressing these common misunderstandings and mistakes, individuals can develop a clearer understanding of what makes a function odd or even and can apply effective strategies to avoid
misclassifying functions.
Conclusion & Best Practices for Mastery of Mathematical Functions
A Recap the key points about odd and even functions
In this blog post, we have discussed the key characteristics of odd and even functions. We have learned that an odd function is symmetric with respect to the origin, meaning that f(-x) = -f(x). On
the other hand, an even function is symmetric with respect to the y-axis, meaning that f(-x) = f(x). Understanding these properties is essential for identifying and analyzing functions.
B Emphasize the importance of practice in recognizing and applying the concepts discussed
Practice is crucial for mastering the concepts of odd and even functions. By working through various examples and problems, students can develop a deeper understanding of how these functions behave
and how to identify them. It is important to practice identifying odd and even functions in different forms, such as algebraic expressions, graphs, and tables of values. This will help solidify the
concepts and improve problem-solving skills.
C Suggest methods to further explore functions, such as using graphing calculators or software and engaging in problem sets
• Graphing Calculators or Software: Utilize graphing calculators or software to visualize and analyze functions. This can help in understanding the symmetry of odd and even functions and how they
are represented graphically.
• Engaging in Problem Sets: Work on problem sets that involve identifying and analyzing odd and even functions. This hands-on approach will reinforce the concepts and improve proficiency in
applying them.
• Exploring Different Forms: Explore functions in different forms, such as equations, graphs, and tables of values. This will provide a comprehensive understanding of how odd and even functions are
represented and how to work with them effectively.
By following these best practices, students can enhance their understanding of mathematical functions, particularly odd and even functions, and become proficient in recognizing and applying these
concepts in various contexts. | {"url":"https://dashboardsexcel.com/blogs/blog/understanding-mathematical-functions-what-makes-a-function-odd-or-even","timestamp":"2024-11-15T01:28:28Z","content_type":"text/html","content_length":"225147","record_id":"<urn:uuid:a9545896-edd9-462d-83b4-65685d7d3a0d>","cc-path":"CC-MAIN-2024-46/segments/1730477397531.96/warc/CC-MAIN-20241114225955-20241115015955-00267.warc.gz"} |
An organization owns a fleet of 10 vessels. Each vessel contains a non-repairable item, which is vital for its operations.
Objective of the analysis:
What is the projected number of failures over the next 2 years in order to avoid operation downtime due to lack of spare, assuming each vessel is expected to operate 100 hours per year?
Following is the repair information of the fleet.
Take vessel 1 for example, the Time-To-Failure (TTF) of the first 2 items were 631 and 336 (967- 361) hours respectively. In addition, one item is still operating after 445 hours of operation
(Suspension time of 445 hours).
Similarly, the Time-To-Events (TTF and Suspensions) are derived. The following table shows the complete failure dataset of the items.
The dataset is fitted to a Weibull Distribution.
The analysis shows that the Weibull Distribution with beta = 3.461 and Eta = 908.8 hours best describes the failure rate behavior of the item. Following images are Probability-Weibull, Reliability vs
Time, Probability Density Function and Failure Rate vs Time plots of this dataset.
Given that the failure distribution of the item, and the current age of each of the 10 items are known, what is the total number of failures can we expect if each vessel operates 200 hours.
The item on vessel 1, for example, has accumulated a current age of 796 hours. We can query the model for Conditional Probability of Failure for an additional mission time of 200, F(200/796).
Therefore, for vessel 1 to operate for 200 hours, there is a 52.34% chance to observe a failure.
Similarly, the Conditional Probability of Failure for an additional mission time of 200, of the item for all the other vessels is determined.
The expected total number of failures if each vessel operates 200 hours is 3.9875.
In this analysis, the mission time is only 200 hours. One would not expect more than 1 failure per vessel. If the mission time is 1000 hours for example, more than 1 failure is expected. In this
case, this procedure is not correct, and simulation approach is preferred. | {"url":"https://assetstudio.net/Resources_Non_Repairable_Spare_Inventory.html","timestamp":"2024-11-08T17:44:39Z","content_type":"text/html","content_length":"15560","record_id":"<urn:uuid:d155de75-6db2-4a2b-a396-2e1e89e57afa>","cc-path":"CC-MAIN-2024-46/segments/1730477028070.17/warc/CC-MAIN-20241108164844-20241108194844-00623.warc.gz"} |
Looking for a Probability At Play - Math Worksheet With Answers? Download it for free!
Probability at Play
In this game, you toss rings onto colored pegs. Each time you toss a ring, it will land
on one of the pegs. The probability that a ring will land on a peg is 1. The chart below
shows the number of pegs of each color.
Peg Color White
Red Orange Yellow Green
Purple Black
of Pegs
15% 25% 1%
5% 10% 2% 20% 15%
for One Ring
Calculate the probability of a ring landing on each color. Express each probability as a
percent in the table.
Suppose you were in charge of this game at a carnival. You have 9 prizes valued from $1
to $9. Each peg color has a different prize. How would you decide which color gets the
$1 prize? the $9 prize? Explain your answer.
Sample answer: The red peg should be $9 since
it’s the least likely and the pink should get $1 since
it’s the most likely. I would determine the other
prizes from least to greatest.
If you could pick a color and then win a prize if another player’s ring landed on that
color, which color would you pick? Why?
Sample answer: I would pick pink because it has the
best chance of winning a prize.
Extend It How could you change the ring toss game to make it more likely that someone
will get the $9 prize? less likely?
Check students’ answers. | {"url":"https://www.formsbank.com/template/323408/probability-at-play-math-worksheet-with-answers.html?page=2","timestamp":"2024-11-08T15:46:39Z","content_type":"text/html","content_length":"79626","record_id":"<urn:uuid:5ace711b-ce6c-4046-a2a2-a8b28d884aaa>","cc-path":"CC-MAIN-2024-46/segments/1730477028067.32/warc/CC-MAIN-20241108133114-20241108163114-00071.warc.gz"} |
Civil Engineering
Types of Measurements in Surveying
Surveying is the art of making suitable measurements in horizontal or vertical planes. This is one of the important subjects of civil engineering. Without taking a survey of the plot where the
construction is to be carried out, the work cannot begin.
From the above definition, we conclude on two types of measurements in surveying. They are as follows:
1. Linear measurements
2. Angular measurements
Now we will go on with the discussion of each of these types of measurements along with their subtypes.
Linear measurements are further classified as follows:
Horizontal Distance
Vertical Distance
Horizontal Distance
A horizontal distance is measured in horizontal plane if a distance is measured along a slope, it is reduced to its horizontal equivalent.
Horizontal Angle
Vertical Distance
A vertical distance is measured along the direction of gravity at that point. The vertical distance are measured to determine difference in elevations in various points. | {"url":"https://civilprojectsonline.com/tag/types-of-measurements-in-surveying/","timestamp":"2024-11-08T00:59:42Z","content_type":"text/html","content_length":"48464","record_id":"<urn:uuid:45ae1d08-216c-4ff2-aaad-b8b8a1b4eae2>","cc-path":"CC-MAIN-2024-46/segments/1730477028019.71/warc/CC-MAIN-20241108003811-20241108033811-00750.warc.gz"} |
An Etymological Dictionary of Astronomy and Astrophysics
adiabatic initial conditions
بوتارهای ِآغازین ِبیدررو
butârhâ-ye âqâzin-e bidarrow
Fr.: conditions initiales adiabatiques
The assumption whereby the density fluctuations in the very → early Universe would be produced by compressing or decompressing of all components of a homogeneous Universe. The adiabatic initial
conditions lead to coherent oscillations in the form of peaks in the → temperature anisotropy spectrum. See also → acoustic peak, → baryon acoustic oscillation.
→ adiabatic; → initial; → condition.
Hartle-Hawking initial state
استات ِآغازین ِهارتل-هاؤکینگ
estât-e âqâzin-e Hartle-Hawking
Fr.: état initial de Hartle-Hawking
A proposal regarding the initial state of the → Universe prior to the → Planck era. This → no boundary hypothesis assumes an imaginary time in that epoch. In other words, there was no real time
before the → Big Bang, and the Universe did not have a beginning. Moreover, this model treats the Universe like a quantum particle, in an attempt to encompass → quantum mechanics and → general
relativity; and attributes a → wave function to the Universe. The wave function has a large value for our own Universe, but small, non-zero values for an infinite number of other possible, parallel
Hartle, J., Hawking, S., 1983, "Wave function of the Universe," Physical Review D 28; → initial; → state.
âqâzin (#)
Fr.: initial
Of, pertaining to, or occurring at the beginning.
Initial, from L. initialis, from initium "a beginning, an entrance," from p.p. stem of inire "to go into, begin," from → in- + ire "to go," → ion.
Âqâzin "pertaing to the beginning," from âqâz "beginning," from Proto-Iranian *āgāza-, from prefix ā- + *gāz- "to take, receive," cf. Sogdian āγāz "beginning, start," pcγz "reception, taking."
initial conditions
بوتارهای ِآغازین
butârhâ-ye âqâzin
Fr.: conditions initiales
1) Conditions at an initial time t = t[0] from which a physical system or a given set of mathematical equations evolves.
2) Meteo.: A prescription of the state of a → dynamical system at a specified time; for all subsequent times, the → equation of motion and → boundary conditions determine the state of the system.
→ initial; → condition.
initial mass
جرم ِآغازین
jerm-e âqâzin (#)
Fr.: masse initiale
The mass of a star at its arrival on the → main sequence.
→ initial; → mass.
initial mass function (IMF)
کریای ِآغازین ِجرم
karyâ-ye âqâzin-e jerm
Fr.: fonction initiale de masse
A mathematical expression describing the relative number of stars found in different ranges of mass for a cluster of stars at the time of its formation. It is defined as φ(log M) = dN / dlog M ∝ M^
-Γ, where M is the mass of a star and N is the number of stars in a logarithmic mass interval. The value of the slope found by Salpeter (1955) for → low-mass and → intermediate-mass stars in the →
solar neighborhood is Γ = 1.35. The IMF can be expressed also in linear mass units: χ(M) = dN / DM ∝ M^ -α. Note that χ(M) = (1 / M lm 10) φ(log M), and α = Γ + 1. In this formalism the Salpeter
slope is α = 2.35. There is a third way for representing the IMF, in which the exponent is x = -α. The IMF is not a single power law over all masses, from → brown dwarfs to → very massive stars
(Kroupa, 2002, Science 295, 82). Different slopes have been found for different mass segments, as follows: α = 1.3 for 0.08 ≤ M[solar] < 0.5; α = 2.3 for 0.5 ≤ M[solar] < 1; α = 2.3 for 1 ≤ M[solar].
The IMF at low masses can be fitted by a → lognormal distribution (See Bastian et al., 2010, ARAA 48, 339 and references therein). See also → canonical IMF.
→ initial; → mass; → function.
initial phase angle
زاویهی ِفاز ِآغازین
zâviye-ye fâz-e âqâzin
Fr.: angle de phase initial
The value of the phase corresponding to the origin of time. Same as the → epoch angle.
→ initial; → phase; → angle.
initial singularity
تکینی ِآغازین
takini-ye âqâzin (#)
Fr.: singularité initiale
An instant of infinite density, infinite pressure, and infinite temperature where the equations of general relativity break down, if the standard Big Bang theory is extrapolated all the way back to
time zero. → singularity.
→ initial; → singularity. | {"url":"https://dictionary.obspm.fr/?showAll=1&formSearchTextfield=initial","timestamp":"2024-11-11T16:31:36Z","content_type":"text/html","content_length":"20530","record_id":"<urn:uuid:8b1094be-b101-44cd-91c1-a76783d69a46>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00013.warc.gz"} |
Understanding Mathematical Functions: How To Divide Function Operation
Introduction to Mathematical Functions and Division Operations
Mathematical functions play a fundamental role in various fields such as engineering, physics, economics, and many more. Understanding functions and their components is essential for solving complex
problems and analyzing real-world situations. In this chapter, we will focus on division operations within functions, exploring their significance and relevance.
Overview of mathematical functions
A mathematical function is a relation between a set of inputs and a set of possible outputs, where each input is related to exactly one output. Functions are represented using symbols such as f(x) or
g(y), where x and y are variables. These functions can take various forms, including linear, quadratic, exponential, and trigonometric functions.
Explanation of the basic concept of division in functions
Division within functions involves the process of dividing one function by another. It is important to understand that division in functions is different from simple arithmetic division. In
functions, division is not just a straightforward operation of dividing two numbers; it involves dividing entire functions where the output of one function is divided by the output of another
The relevance of mastering division operations within functions
Mastering division operations within functions is crucial for enhancing mathematical literacy and problem-solving skills. By understanding division in functions, individuals can manipulate functions
to analyze data, model situations, and solve equations efficiently. Proficiency in division operations enables individuals to tackle complex mathematical problems with confidence and precision.
Key Takeaways
• Key Takeaways:
• Understand division of function operations.
• Divide functions by simplifying and canceling terms.
• Use rules of exponents to divide functions.
• Practice dividing functions to improve understanding.
Understanding the Basics of Division in Functions
When it comes to mathematical functions, division is a fundamental operation that allows us to break down complex functions into simpler components. In this chapter, we will explore the concept of
function division, compare it to multiplication of functions, clarify the terms used in function division, and provide examples to illustrate the concept.
A Definition of function division and comparison to multiplication of functions
Function division involves dividing one function by another to obtain a new function. This operation is similar to division in arithmetic, where one number is divided by another to get a quotient. In
contrast, multiplication of functions involves multiplying two functions together to create a new function.
Clarification of terms used in function division
When performing function division, it is essential to understand the terms involved:
• Dividend: The function being divided.
• Divisor: The function by which the dividend is being divided.
• Quotient: The result of dividing the dividend by the divisor.
• Remainder: Any leftover part of the dividend that cannot be evenly divided by the divisor.
Examples of simple function division operations to illustrate the concept
Let's consider a simple function division operation to demonstrate how it works:
Example: Divide the function f(x) = 2x by the function g(x) = x.
To divide these functions, we need to perform the following steps:
1. Identify the dividend (f(x) = 2x) and the divisor (g(x) = x).
2. Divide the coefficients of the terms: 2 ÷ 1 = 2.
3. Subtract the exponents of the variables: x^1 / x^1 = x^0 = 1.
4. Combine the results to get the quotient: 2x^0 = 2.
Therefore, the quotient of f(x) = 2x divided by g(x) = x is h(x) = 2.
The Division Algorithm for Polynomials
Understanding how to divide polynomial functions is an essential skill in algebra. The division algorithm for polynomials provides a systematic way to divide one polynomial by another, similar to
long division with numbers. Let's delve into the details of this algorithm and how to perform polynomial long division.
Explanation of the division algorithm as applied to polynomial functions
The division algorithm for polynomials states that any two polynomials, dividend (D) and divisor (d), can be divided to yield a quotient (q) and a remainder (r) such that D = q * d + r, where the
degree of r is less than the degree of d. This process is similar to long division with numbers, where we divide the terms of the dividend by the terms of the divisor.
Step-by-step guide on performing polynomial long division
To perform polynomial long division, follow these steps:
• Step 1: Arrange the terms of the dividend and divisor in descending order of their degrees.
• Step 2: Divide the first term of the dividend by the first term of the divisor to obtain the first term of the quotient.
• Step 3: Multiply the entire divisor by the first term of the quotient and subtract it from the dividend to get a new polynomial.
• Step 4: Repeat the process with the new polynomial as the dividend until the degree of the remainder is less than the degree of the divisor.
Common pitfalls in the polynomial division process and how to avoid them
When performing polynomial division, it's important to watch out for common pitfalls that can lead to errors. Some of these pitfalls include:
• Incorrect term placement: Make sure to align the terms of the dividend and divisor correctly to avoid confusion during the division process.
• Missing terms: Double-check that all terms are included in the division process to prevent overlooking any terms that may affect the final result.
• Incorrect arithmetic: Be careful with arithmetic operations such as addition, subtraction, multiplication, and division to ensure accurate calculations throughout the process.
Strategies for Simplifying Complex Division Operations
When dealing with complex division operations of mathematical functions, it is essential to have a clear strategy in place to simplify the process. By breaking down the division into simpler parts,
utilizing factorization, and considering alternative methods such as synthetic division, you can make the process more manageable and efficient.
Techniques for breaking down complex function division into simpler parts
One effective technique for simplifying complex function division is to break down the division into smaller, more manageable parts. This can involve identifying common factors, simplifying terms,
and rearranging the expression to make the division process easier to follow. By breaking down the division into simpler parts, you can focus on one step at a time, reducing the likelihood of errors
and confusion.
The role of factorization in simplifying function division
Factorization plays a crucial role in simplifying function division by identifying common factors within the expression. By factoring out common terms, you can simplify the division process and
reduce the complexity of the operation. Factorization helps in breaking down the expression into its basic components, making it easier to divide and solve the function.
How to use synthetic division as an alternative to the long division method
Synthetic division is an alternative method to the traditional long division approach for dividing polynomials. This method is particularly useful for dividing polynomials by linear factors. To use
synthetic division, you need to set up a table with the divisor and coefficients of the polynomial, then perform a series of calculations to simplify the division process. Synthetic division can be a
quicker and more efficient method for dividing polynomials, especially in cases where long division may be cumbersome.
The Importance of Graphical Interpretations
Understanding mathematical functions through graphical interpretations is essential for gaining insights into their behavior and characteristics. When it comes to dividing two functions, analyzing
their graph can provide valuable information about the relationship between the functions and how they interact with each other.
Understanding the graph of the division of two functions and its significance
When dividing two functions, the resulting graph represents the quotient of the two functions. This graph can reveal important features such as the location of zeros, asymptotes, and discontinuities.
By examining the graph, we can determine the behavior of the division function and identify key points of interest.
The impact of asymptotes, zeros, and discontinuities on the graph
Asymptotes play a significant role in the graph of the division of functions. Vertical asymptotes occur where the denominator of the division function equals zero, leading to undefined values.
Horizontal asymptotes indicate the behavior of the function as it approaches infinity or negative infinity.
Zeros of the division function are points where the function equals zero. These points can provide insights into the roots of the division function and help in understanding its behavior.
Discontinuities in the graph of the division function can occur at points where the function is not continuous. These discontinuities can be removable or non-removable, and they impact the overall
shape and behavior of the graph.
Practical examples of interpreting the division of functions graphically in real-world problems
Graphical interpretations of the division of functions can be applied to real-world problems in various fields such as physics, engineering, economics, and biology. For example, in physics, analyzing
the division of displacement and time functions can help in understanding the velocity of an object. In economics, dividing revenue and quantity functions can provide insights into the price
elasticity of demand.
By visually representing the division of functions in real-world scenarios, we can make informed decisions and predictions based on the graphical analysis of the functions.
Troubleshooting Common Issues in Function Division
When working with mathematical functions, division operations can sometimes lead to common mistakes and issues. It is important to be able to identify and address these problems effectively to ensure
accurate results. Here are some tips for troubleshooting common issues in function division:
Identifying and addressing common mistakes in function division
• Incorrect application of division rules: One common mistake is applying division rules incorrectly. Make sure to review the rules of function division and double-check your work to avoid errors.
• Ignoring domain restrictions: Sometimes, division operations can result in undefined values due to domain restrictions. Always consider the domain of the functions involved and address any
restrictions before performing division.
• Confusion with variable placement: Mixing up the placement of variables in the division operation can lead to incorrect results. Pay close attention to the order of variables and ensure they are
placed correctly.
Tips for checking the correctness of division operations
• Verify your work: After performing a division operation, take the time to verify your work by double-checking the steps and calculations. This can help catch any mistakes before they lead to
incorrect results.
• Use a calculator: Utilize a calculator to check your division operations, especially when dealing with complex functions or large numbers. Calculators can provide quick and accurate results for
• Seek feedback: If you are unsure about the correctness of your division operations, seek feedback from a peer, teacher, or tutor. Getting a second opinion can help identify any mistakes you may
have overlooked.
Strategies for dealing with undefined and indeterminate forms in function division
• Identify the cause: When encountering undefined or indeterminate forms in function division, try to identify the specific cause of the issue. This can help determine the appropriate strategy for
addressing it.
• Apply limit techniques: In cases where division results in indeterminate forms such as 0/0 or ∞/∞, consider applying limit techniques to evaluate the result. Limits can help determine the
behavior of the function as it approaches a certain value.
• Restructure the function: If division leads to undefined forms due to domain restrictions, consider restructuring the function or adjusting the domain to avoid these issues. This may involve
simplifying the function or redefining the variables involved.
Conclusion & Best Practices in Function Division Operations
A Recap of key points discussed about dividing functions and their operational insights
• Understanding Function Division: We have explored the concept of dividing functions, which involves dividing one function by another to analyze their relationship and behavior.
• Operational Insights: Through examples and explanations, we have gained insights into how function division can help us understand the interactions between different functions.
• Key Takeaways: It is important to consider the domain restrictions and potential asymptotes when dividing functions to ensure accurate results.
Best practices for performing division operations effectively, including routine practice, conceptual understanding, and the application of technology
• Routine Practice: Regular practice of dividing functions can help improve proficiency and accuracy in performing division operations.
• Conceptual Understanding: Developing a deep understanding of the underlying concepts of function division is essential for solving complex problems effectively.
• Application of Technology: Utilizing mathematical software or calculators can aid in performing function division efficiently and verifying results.
Encouragement to explore further applications of function division in various mathematical and practical contexts
• Mathematical Contexts: Function division can be applied in calculus, algebra, and other mathematical disciplines to analyze functions and solve equations.
• Practical Contexts: Understanding function division can be beneficial in fields such as engineering, economics, and physics for modeling real-world phenomena and making informed decisions.
• Continued Exploration: By exploring diverse applications of function division, one can enhance problem-solving skills and gain a deeper appreciation for the versatility of mathematical functions. | {"url":"https://dashboardsexcel.com/blogs/blog/understanding-mathematical-functions-divide-function-operations","timestamp":"2024-11-11T18:14:04Z","content_type":"text/html","content_length":"223941","record_id":"<urn:uuid:368268da-cd7c-4582-8404-845fe5765047>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00748.warc.gz"} |
Decision variables - (Nonlinear Optimization) - Vocab, Definition, Explanations | Fiveable
Decision variables
from class:
Nonlinear Optimization
Decision variables are the unknown quantities in an optimization problem that decision-makers will choose values for in order to achieve the best possible outcome. These variables are central to
formulating the problem, as they represent the choices available and their associated consequences within the constraints defined by the problem. Understanding decision variables is essential because
they directly influence the objective function and determine feasible solutions within any optimization scenario.
congrats on reading the definition of decision variables. now let's actually learn it.
5 Must Know Facts For Your Next Test
1. Decision variables are typically denoted by symbols like x, y, or z, and each represents a specific choice or quantity to be determined.
2. The values assigned to decision variables must lead to a feasible solution that satisfies all constraints of the optimization problem.
3. In linear programming, decision variables contribute linearly to the objective function and must comply with linear constraints.
4. The number of decision variables can significantly affect the complexity of solving an optimization problem; more variables often lead to a more intricate feasible region.
5. Effective identification and formulation of decision variables are crucial steps in problem-solving as they directly impact the results of any optimization analysis.
Review Questions
• How do decision variables influence the formulation of an optimization problem?
□ Decision variables are foundational elements in formulating an optimization problem, as they represent the choices that can be manipulated to achieve the desired outcome. By defining what
these variables are, one establishes the framework for both the objective function and constraints. This interaction allows for structured analysis and ensures that potential solutions are
evaluated based on how well they optimize these decision variables.
• Discuss how decision variables interact with constraints in optimization problems.
□ Decision variables must operate within certain boundaries defined by constraints, which limits their possible values. Constraints ensure that any solution generated respects specific
conditions, such as budget limitations or resource availability. This relationship is critical because it determines the feasible region within which optimal solutions can exist, thereby
guiding the selection of decision variable values that yield satisfactory outcomes while adhering to established restrictions.
• Evaluate the impact of choosing inappropriate decision variables on the overall outcome of an optimization problem.
□ Choosing inappropriate decision variables can lead to ineffective solutions that do not address the original goals of the optimization problem. If decision variables are not aligned with
critical factors influencing the objective function or do not comply with necessary constraints, it may result in infeasible solutions or suboptimal performance. A poor selection process can
complicate solution strategies, increase computational complexity, and ultimately yield results that fail to deliver desired improvements or efficiencies.
ยฉ 2024 Fiveable Inc. All rights reserved.
APยฎ and SATยฎ are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website. | {"url":"https://library.fiveable.me/key-terms/nonlinear-optimization/decision-variables","timestamp":"2024-11-13T08:53:38Z","content_type":"text/html","content_length":"155547","record_id":"<urn:uuid:01b5a34f-58b7-481d-955e-8420a756c534>","cc-path":"CC-MAIN-2024-46/segments/1730477028342.51/warc/CC-MAIN-20241113071746-20241113101746-00475.warc.gz"} |
Brain Teaser Bay - - Page 3
• What is at the end of a rainbow?
The letter \’W\’. 🙂 Easy one.
• You have 27 coins, each of them is 10 grams, except for 1.
The coin which is different is 9 grams or 11 grams (the coin is heavier or lighter by 1 gram).
You can use balance scales that compares what is in the two pans and compare two groups of coins to find the lighter coin.
What is the minimum number of weightings that can guarantee to find the different coin?
Separate the coins into 3 stacks of 9 (A, B, C).
Weigh stack A against B and then A against C.
Take the stack with the different weight (note lighter or heavier) and break it into 3 stacks of 3 (D, E, F).
Weigh stack D against E. If D and E are equal, then F is the odd stack. If D and E are not equal, the lighter or heavier (based on the A, B, C comparison) is the odd stack.
You now have three coins (G, H, I).
Weigh G and H. If G equals H, then I is the odd and is lighter or heavier (based on the A, B, C comparison).
If G and H are not equal, then the lighter or heavier (based on the A, B, C comparison) is the odd coin.
• You have a basket containing ten apples. You have ten friends, who each desire an apple.
You give each of your friends one apple.
Now all your friends have one apple each, yet there is an apple remaining in the basket.
How is that possible?
You give an apple each to your first nine friends, and the basket together with the last apple to your tenth friend.
Each friend has an apple, and one of them has it in a basket. 🙂
• Mr. Mark went to market with his dog.
He rode on a horse to the market,Yet walked, the horse\’s name was Victor.
What is the name of dog?
This one is a classic brain teaser.
The dog name is \’Yet\’ (Yet walked). 🙂 Easy one.
• What comes next in the following series of letters?
A A D F J J J M M N ?
O (October) and S (September)! These letters represent the first letter of each month in alphabetical order (April, August, December, February, January, July, June, March, May, November) and the
two remaining months is October and September. 🙂 | {"url":"http://www.brainteaserbay.com/page/3/","timestamp":"2024-11-02T20:11:51Z","content_type":"text/html","content_length":"50417","record_id":"<urn:uuid:87e66ec8-8742-4d33-9a39-be57000cd30f>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00820.warc.gz"} |
Practice Problems
Section A: Practice Problems Points, Lines, Segments, Rays, and Angles
Section Summary
In this section, we learned the meanings of points, lines, line segments, and rays. We used these terms to describe figures and used these geometric parts to create drawings.
We learned about lines that cross—intersecting lines—and lines that never do—parallel lines, and we looked for examples of intersecting lines and parallel lines and segments in life.
Finally, we learned that an angle is a figure made up of two rays that share the same endpoint, and that the shared point is the vertex of the angle.
Problem 1 (Pre-Unit)
Draw a rectangle on the grid and label it A. Draw a triangle and label it B. Draw a hexagon and label it C.
Problem 2 (Pre-Unit)
a. Is the shape a rhombus? Explain your reasoning.
b. Is the shape a rectangle? Explain your reasoning.
c. Is the shape a square? Explain your reasoning.
Problem 3 (Lesson 1)
1. Draw 4 different lines through points on the grid. At least two of the lines should cross another line.
2. Mark at least 3 different segments in your drawing.
Problem 4 (Lesson 2)
1. Circle the line segments that make up the letter A.
2. Draw 4 rays that surround a rectangle.
3. Can you find 4 different rays that surround the same rectangle?
Problem 5 (Lesson 3)
Andre says that these two lines are parallel because they do not intersect.
a. Explain why Andre is not correct.
b. Draw a line that is parallel to one of the lines in the image.
Problem 6 (Lesson 4)
• Which segments of the letter Z are a pair of parallel lines? Draw the lines.
• Sketch a line that is parallel to the third segment in the Z.
Problem 7 (Lesson 5)
1. Find one angle in the figure. Draw a pair of rays to show the angle and extend them as far as you can.
2. Find another angle in the figure. Draw a pair of rays to show it. Extend the rays as far as you can. (If you’d like, you can use a different colored pencil for this pair of rays.)
3. Now that you have drawn some rays, do you see other angles that you didn’t see before? If you see one or more, label each one with a letter.
Problem 8 (Exploration)
Here is a riddle. Can you solve it?
“I am a capital letter made of more than 1 segment with no curved parts. I have no perpendicular segments or parallel segments. What letter could I be?”
Problem 9 (Exploration)
a. Name or describe any shapes that you recognize in the painting.
b. Do you see any parallel lines? If so, trace or circle them. (If you’d like, you can use a different colored pencil for each set of parallel lines.)
c. Are there any angles in the painting? If so, mark them or describe where they are. | {"url":"https://access.openupresources.org/curricula/our-k5-math/en/grade-4/unit-7/section-a/student_practice_problems.html","timestamp":"2024-11-04T02:34:35Z","content_type":"text/html","content_length":"25571","record_id":"<urn:uuid:6fc3cab7-a944-4177-b014-a97040632b31>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00434.warc.gz"} |
Annuity Rates - Options for calculating future annuity payments
When annuitising, there are two ways that the software can calculate future income from the funds: either specify what the annuity rate will be, yourself, or allow the software to derive an annuity
rate based on the owner's age and the assumed interest rate.
Option 1 - Specify the annuity rate
Where one has a reasonable idea of the applicable market (annuity) rate, taking account of the client's age and circumstances, etc., one can change the Annuity Rate Calculation option to Specified
Annuity Rate. If chosen, the % entered is the straightforward 'Conversion Rate', i.e. the rate at which a lump sum converts into an annuity, e.g. a fund of £100,000 (after tax free cash) with a
specified annuity rate of 5% will produce an annual income of £5,000.
Option 2 - Allow the software to derive an annuity rate (the software's default)
The software's default option, on the other hand, allows the software to 'derive' an annuity rate by using an Assumed Interest Rate.
The Assumed Interest Rate on an annuity is the underlying interest rate assumption on which the annuity calculation is based (or would be based, by an actuary). It would ordinarily reflect an assumed
yield on mid-dated UK Sovereign debt (Gilt yield).
The Assumed Interest Rate is pulled through from Plan Settings > Inflation/Growth > Assumed Annuity Interest Rate but can be over-written if required.
We do not pretend to know what rate of interest would, in fact, give rise to a realistic annuity rate in today's marketplace. The primary reason that the software defaults to using the 'assumed
interest rate', in the first place, is because this option does take account of an individual's age at the time the annuity is purchased. It will, therefore, give one a different result (other things
being equal) when the client is 55 than it would when the same individual is 75 (for example).
In the event that you decide to use the 'Assumed Interest Rate' option, we generally recommend that you should make allowances for provider costs / charges and provider mortality assumptions.
Therefore it generally makes sense to understate the expected yield.
Where to find these annuitisation settings in Voyant
Annutisation settings are found in three locations in the software, depending on what you plan to annuitise and the type of annuity.
Money Purchases - Settings to schedule the future annuitisation of a money purchase are found on the Pensions > Money Purchase screen under Annuitization:
Drawdown Pensions - Settings to schedule the future annuitisation of a drawdown pension are found on the Pensions > Drawdown Pension screen screen under Annuitization:
Future Non-Pension Annuities - The Pensions > Annuity screen is used to enter current annuities and to schedule the purchase of future non-pension annuities. The future annuitisation of money
purchases or drawdown pensions are scheduled separately on those respective screens of the software.
The annuity rate is set on this screen under Calculation Settings.
The annuity rate is not used to calculate current annuity payments. If your client is already receiving annuity payments (the pension Status is In Payment or Deferred), payments will be based on the
amount entered in the Payment field.
Related topics
Default Inflation / Growth Rates - Assumed Annuity Interest Rate | {"url":"https://support.planwithvoyant.com/hc/en-us/articles/20098166624923-Annuity-Rates-Options-for-calculating-future-annuity-payments","timestamp":"2024-11-08T08:06:36Z","content_type":"text/html","content_length":"24455","record_id":"<urn:uuid:58b2010c-74b5-4cd1-9637-2c4f2053a887>","cc-path":"CC-MAIN-2024-46/segments/1730477028032.87/warc/CC-MAIN-20241108070606-20241108100606-00069.warc.gz"} |
Comparison between Optimization Strategies - MantaRisk Documentation
Portfolio Construction Strateg...
Comparison between Optimization Strategies
All results are based on the CVaR dispersion measure (except for the equally weighted portfolio used as a standard benchmark) for portfolios containing the following instruments: GLD, TLT, XLV, IHI,
VGT. Furthermore, the optimized portfolios are compared to a sample of 2000 portfolios with random weights summing up to 100%.
An annualized targeted expected portfolio return of 10% was chosed for the strategies Risk and Diverse.
The gray histogram and curve of following graph show the density of the CVaR -95% for a daily horizon of the 2000 randomly chosen portfolios, meaning that most of them have an expected shortfall
between -1.7% and -2.5%. In other words, choosing a random portfolio with the above instruments will lead to a portfolio with an expected average loss of around -2% for 5% of the worst performing
The vertical lines correspond each to an optimization strategy explained above. Note that the minimum risk strategy finds the combination of underlying instruments leading to the portfolio with the
lowest CVaR, followed by the strategies trying to get the most diverse portfolio under the constraint of minimal risk.
The two strategies targeting an expected portfolio return (Risk, Diverse) have a slighlty higher risk whereas the strategy maximizing the Sharpe ratio has the highest risk. Also shown is the CVaR of
the equally weighted portfolio which has the same risk as most of the random portfolios as it is located at the maximum .
The following graph shows the same results as above but as a risk-return scatter plot. Each of the 2000 random portfolios is shown as a gray dot while the optimized portfolios are shown as larger
colored dots.
The x-axis corresponds to the CVaR-95% with a horizon of a day and the y-axis the annualized expected return.
The minimum risk portfolio (MinRisk) is the portfolio at the most left part of the graph whereas the MaxDiverse portfolio trades a bit of risk for more diversification, albeit with a slightly higher
expected return.
Since a targeted portfolio return of 10% was chosen, it is not surprising that the 2 strategies Risk and Diverse manage to come close to a 10% annualized expected return. Note that the strategy
Diverse is riskier, yet doesn't expected a higher return.
The strategy with the highest expected return is Sharpe. Furthermore, the equally weighted portfolio is in the middle of the possible outcomes and thus not on the efficient frontier.
The following graph shows the Sharpe ratio where each of the 2000 random portfolios is shown as a gray dot while the optimized portfolios are shown as larger colored dots.
The x-axis corresponds to the annualized expected returns [%] with a horizon of a day and the y-axis the Sharpe ratio.
The strategy Sharpe is unsurprisingly the portfolio with the highest Sharpe ratio. Furthermore, choosing the strategies Diverse and Risk also lead to a good Sharpe ratio. On the other hand, the 2
strategies solely minimizing CVaR (MinRisk, MaxDiverse) lead to poor Sharpe ratios. | {"url":"https://docs.mantarisk.com/comparison-between-optimization-strategies","timestamp":"2024-11-08T23:46:59Z","content_type":"text/html","content_length":"119728","record_id":"<urn:uuid:7d5093e5-efa0-4a15-aad9-f64fbab4b0d0>","cc-path":"CC-MAIN-2024-46/segments/1730477028106.80/warc/CC-MAIN-20241108231327-20241109021327-00229.warc.gz"} |
Know your Pounds, Shillings, and Pence
Standard Input (stdin)
Standard Output (stdout)
Memory limit:
100 megabytes
Time limit:
1.0 seconds
Mrs Pennyfeather has been awoken from her cryogenic nap and is totally bewildered at the ignorance of youth. They don’t know how to add in Pounds, Shillings, and Pence!!!!
• Pence use base 12. In other words, 13 pence is equal to 1 shilling and 1 penny because there are 12 pence to a shilling.
• Shillings use base 20. That is, there are 20 shillings in a pound. So 22 shillings is 1 pound and 2 shillings.
• Pounds are base 10 and are preceded with the symbol £. However, we will use # (a hash) in its place.
Your job is to show Mrs Pennyfeather that some youth are actually pretty smart by asking her for 2 or more amounts in pounds, shilling, and pence and telling her the total. She is going to give you
different sums to do. The sum should be obtained by changing as many pence for shillings as possible (and shillings for pounds too).
The first line contains a whole number (called $N$). This represents the total number of sums to be done.
Each sum will have a first line containing a whole number $S$. There follow $S$ lines, each line is of the form #X-Y-Z. That is; a pounds symbol (#), followed by a whole number of pounds ($X$), a
dash, a whole number of shillings ($Y$) and another dash followed by a whole number of pence ($Z$). it is guaranteed that $Y \lt 20$ and $Z \lt 12$.
For each of the $N$ sums, print the answer using the same currency format as the input. i.e. Print a #, followed by the number of pounds, a dash, a number of shillings, and another dash followed by
the number of pence (see the examples).
You are guaranteed that $1 \le N \le 10$ and $2 \le S \le 10$.
• Subtask 1 (10%): Write a solution where $N=1$ (that means there is only one sum to do) and $S=2$. Additionally, all $Z$’s in a sum add to less than 12 and all $Y$’s in a sum add to less than 20.
• Subtask 2 (40%): We guarantee that $N=1$ and $S = 2$.
• Subtask 3 (50%): You get the remaining 50% for the full solution.
• Sample Input 2 | {"url":"https://train.nzoi.org.nz/problems/792","timestamp":"2024-11-05T12:08:41Z","content_type":"text/html","content_length":"38824","record_id":"<urn:uuid:27c29d71-afce-4702-8d16-a104b41fbf20>","cc-path":"CC-MAIN-2024-46/segments/1730477027881.88/warc/CC-MAIN-20241105114407-20241105144407-00236.warc.gz"} |
intset: Pure, mergeable, succinct Int sets.
This library provides persistent, time and space efficient integer sets implemented as dense big-endian patricia trees with buddy suffixes compaction. In randomized settings this structure expected
to be as fast as Data.IntSet from containers, but if a sets is likely to have long continuous intervals it should be much faster.
Skip to Readme
Automatic Flags
Name Description Default
testing Enable testing stuff and expose internals. Disabled
Use -f <flag> to enable a flag, or -f -<flag> to disable that flag. More info
Maintainer's Corner
For package maintainers and hackage trustees | {"url":"http://hackage-origin.haskell.org/package/intset","timestamp":"2024-11-11T13:20:32Z","content_type":"text/html","content_length":"22445","record_id":"<urn:uuid:f5b10003-c85d-4052-858d-2897e2f03d4b>","cc-path":"CC-MAIN-2024-46/segments/1730477028230.68/warc/CC-MAIN-20241111123424-20241111153424-00729.warc.gz"} |
Symmetry-violating hadronic interactions from lattice QCD
T. Luu (FZJ) / M. Petschlies (UBonn) / C. Urbach (UBonn)
The low-energy / high-precision frontier is one of the three contemporary cornerstones in the quest
for testing the Standard Model (SM) of elementary particle physics by probing its boundaries and
searching for physics beyond it. Within this area, Hadronic Weak Interactions (HWI) generated
from the coupling of the strong and electro-weak sector of the Standard Model give rise to a rich
phenomenology of hadrons, whose observation provides key access points for such precision
tests. Of particular interest are the parity symmetry violating (PV) and combined parity and charge-
conjugation symmetry violating (CPV) interactions of the nucleon. The main goals of this project
are the first lattice QCD calculation of the weak pion-nucleon coupling with fully controlled
statistical and systematic uncertainty and the lattice QCD determination of matrix elements
involving the so-called Weinberg, quark- and quark-Chromo electric dipole moment operators, that
contribute to the electric dipole moment of the neutron and the proton and light nuclei. | {"url":"https://crc110.hiskp.uni-bonn.de/index.php?id=685&L=1%2F%27%60%28","timestamp":"2024-11-13T21:32:37Z","content_type":"application/xhtml+xml","content_length":"8764","record_id":"<urn:uuid:e7f5d072-c709-4a36-863f-293934a4920f>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00734.warc.gz"} |
SHAP and LIME: Great ML Explainers with Pros and Cons to Both
SHAP and LIME Python Libraries: Part 1 - Great Explainers, with Pros and Cons to Both
Josh Poduska2018-12-05 | 6 min read
Return to blog home
This blog post provides a brief technical introduction to the SHAP and LIME Python libraries, followed by code and output to highlight a few pros and cons of each. If interested in a visual
walk-through of this post, consider attending the webinar.
Model explainability is a priority in today's data science community. As data scientists, we want to prevent model bias and help decision makers understand how to use our models in the right way.
Data science leaders and executives are mindful of existing and upcoming legislation that requires models to provide evidence of how they work and how they avoid mistakes (e.g., SR 11-7 and The
FUTURE of AI Act).
Part 1 in this blog post provides a brief technical introduction to the SHAP and LIME Python libraries, followed by code and output to highlight a few pros and cons of each. Part 2 will explore these
libraries in more detail by applying them to a variety of Python models. The goal of these posts is to familiarize readers with how to use these libraries in practice and how to interpret their
output, helping you leverage model explanations in your own work.
SHAP vs. LIME
SHAP and LIME are both popular Python libraries for model explainability. SHAP (SHapley Additive exPlanation) leverages the idea of Shapley values for model feature influence scoring. The technical
definition of a Shapley value is the “average marginal contribution of a feature value over all possible coalitions.” In other words, Shapley values consider all possible predictions for an instance
using all possible combinations of inputs. Because of this exhaustive approach, SHAP can guarantee properties like consistency and local accuracy. LIME (Local Interpretable Model-agnostic
Explanations) builds sparse linear models around each prediction to explain how the black box model works in that local vicinity. In their NIPS paper, the authors of SHAP show that Shapley values
provide the only guarantee of accuracy and consistency and that LIME is actually a subset of SHAP but lacks the same properties. For further study, I found the GitHub sites SHAP GitHub and LIME
GitHub helpful resources:
So why would anyone ever use LIME? Simply put, LIME is fast, while Shapley values take a long time to compute. For you statisticians out there, this situation reminds me somewhat of Fisher’s Exact
Test versus a Chi-Squared Test on contingency tables. Fisher’s Exact Test provides the highest accuracy possible because it considers all possible outcomes, but it takes forever to run on large
tables. This makes the Chi-Squared Test, a distribution-based approximation, a nice alternative.
The SHAP Python library helps with this compute problem by using approximations and optimizations to greatly speed things up while seeking to keep the nice Shapley properties. When you use a model
with a SHAP optimization, things run very fast and the output is accurate and reliable. Unfortunately, SHAP is not optimized for all model types yet.
For example, SHAP has a tree explainer that runs fast on trees, such as gradient boosted trees from XGBoost and scikit-learn and random forests from sci-kit learn, but for a model like k-nearest
neighbor, even on a very small dataset, it is prohibitively slow. Part 2 of this post will review a complete list of SHAP explainers. The code and comments below document this deficiency of the SHAP
library on the Boston Housing dataset.
# Load Libraries
import pandas as pd
import sklearn
from sklearn.model_selection import train_test_split
import sklearn.ensemble
import numpy as np
import lime
import lime.lime_tabular
import shap
import xgboost as xgb
import matplotlib
import matplotlib.pyplot as plt
from mpl_toolkits.mplot3d import axes3d, Axes3D
import seaborn as sns
import time
%matplotlib inline
# Load Boston Housing Data
X,y = shap.datasets.boston()
X_train,X_test,y_train,y_test = train_test_split(X, y, test_size=0.2, random_state=0)
X,y = shap.datasets.boston()
X_train,X_test,y_train,y_test = train_test_split(X, y, test_size=0.2, random_state=0)
# K Nearest Neighbor
knn = sklearn.neighbors.KNeighborsRegressor()
knn.fit(X_train, y_train)
# Create the SHAP Explainers
# SHAP has the following explainers: deep, gradient, kernel, linear, tree, sampling
# Must use Kernel method on knn
# Summarizing the data with k-Means is a trick to speed up the processing
"""Rather than use the whole training set to estimate expected values, we summarize with a set of weighted kmeans,each weighted by the number of points they represent. Running without kmeans took 1 hr 6 mins 7 sec.Running with kmeans took 2 min 47 sec. Boston Housing is a small dataset.Running SHAP on models that require the Kernel method becomes prohibitive."""
# build the kmeans summary
X_train_summary = shap.kmeans(X_train, 10)
# using the kmeans summary
t0 = time.time()
explainerKNN = shap.KernelExplainer(knn.predict,X_train_summary)
shap_values_KNN_test = explainerKNN.shap_values(X_test)
t1 = time.time()
# without kmeans a test run took 3967.6232330799103 seconds
"""t0 = time.time() explainerKNN = shap.KernelExplainer(knn.predict, X_train) shap_values_KNN_test = explainerKNN.shap_values(X_test) t1 = time.time() timeit=t1-t0 timeit"""
# now we can plot the SHAP explainer
shap.force_plot(explainerKNN.expected_value, shap_values_KNN_test[j], X_test.iloc[[j]])
Running SHAP on a knn model built on the Boston Housing dataset took over an hour, which is a tough pill to swallow. We can get that down to three minutes if we sacrifice some accuracy and
reliability by summarizing the data first with a k-means algorithm. As an alternative approach, we could use LIME. LIME runs instantaneously with the same knn model and does not require summarizing
with k-means. See the code and output below. Note that LIME’s output is different than the SHAP output, especially for features AGE and B. With LIME not having the same accuracy and consistency
properties as Shapley Values, and with SHAP using a k-means summary before calculating influence scores, it's tough to tell which comes closer to the correct answer.
exp = explainer.explain_instance(X_test.values[j], knn.predict, num_features=5)
While LIME provided a nice alternative in the knn model example, LIME is unfortunately not always able to save the day. It doesn’t work out-of-the-box on all models. For example, LIME cannot handle
the requirement of XGBoost to use xgb.DMatrix() on the input data. See below for one attempt to call LIME with the XGBoost model. There are potential hacks that could get LIME to work on this model,
including creating your own prediction function, but the point is LIME doesn’t automatically work with the XGBoost library.
xgb_model = xgb.train({"objective":"reg:linear"}, xgb.DMatrix(X_train, label=y_train))
max_features="auto", max_leaf_nodes=None,
min_impurity_decrease=0.0, min_impurity_split=None,
min_samples_leaf=1, min_samples_split=2,
min_weight_fraction_leaf=0.0, n_estimators=10, n_jobs=1, oob_score=False,
random_state=None, verbose=0, warm_start=False)
# LIME has one explainer for all models
explainer = lime.lime_tabular.LimeTabularExplainer(X_train.values,
# Out-of-the-box LIME cannot handle the requirement of XGBoost to use xgb.DMatrix() on the input data
expXGB = explainer.explain_instance(X_test.values[j], xgb_model.predict, num_features=5)
On the other hand, SHAP is optimized for XGBoost and provides fast, reliable results. The following code runs very fast. It uses the TreeExplainer from the SHAP library, which is optimized to trace
through the XGBoost tree to find the Shapley value estimates.
explainerXGB = shap.TreeExplainer(xgb_model)
shap_values_XGB_test = explainerXGB.shap_values(X_test)
shap.force_plot(explainerXGB.expected_value, shap_values_XGB_test[j], X_test.iloc[[j]])
Hopefully, this post has given you a few pointers on how to choose between SHAP and LIME and brought to light some of the limitations of each. While both approaches have their strengths and
limitations, I personally prefer to use SHAP when I can and rely on LIME when SHAP’s compute costs are too high. Stay tuned for my next post on this topic, which will provide multiple examples of how
to use these libraries on a variety of models and also show how to interpret their output.
Josh Poduska is the Chief Field Data Scientist at Domino Data Lab and has 20+ years of experience in analytics. Josh has built data science solutions across domains including manufacturing, public
sector, and retail. Josh has also managed teams and led data science strategy at multiple companies, and he currently manages Domino’s Field Data Science team. Josh has a Masters in Applied
Statistics from Cornell University. You can connect with Josh at https://www.linkedin.com/in/joshpoduska/ | {"url":"https://domino.ai/blog/shap-lime-python-libraries-part-1-great-explainers-pros-cons","timestamp":"2024-11-11T03:16:29Z","content_type":"text/html","content_length":"142848","record_id":"<urn:uuid:5069121f-d97d-4b09-86da-3bead300f063>","cc-path":"CC-MAIN-2024-46/segments/1730477028216.19/warc/CC-MAIN-20241111024756-20241111054756-00606.warc.gz"} |
Excel Not Sorting Numbers Correctly? Try These Fixes
Last updated on April 3, 2023
This tutorial shows some possible fixes when Excel is not sorting numbers correctly.
There are many times in Excel that data is imported from the internet or from other programs into Excel. Often this data is not imported or copied into Excel in a consistent number format. If this is
the case, it might make sorting numerically incorrect.
Clean Function
Consider the worksheet below:
1. If you try and sort this numerical list using Excel’s Sort feature, then it sorts in the order shown above!
2. Each of these numbers is actually stored as text, and some of the numbers have spaces in the cells before the numbers. To fix this, clean the data.
3. Click in the cell to the right of the first number, and then type in this formula:
Using the CLEAN Function, the formula above automatically removes any invalid characters (e.g., spaces), and the *1 converts the remaining value to a number.
You could also use the TRIM Function to remove any leading or trailing spaces, but it doesn’t remove any unprintable characters that may have been imported.
4. Now, copy the formula down to the remaining cells and then in the Ribbon, go to Home > Editing > Sort & Filter > Sort Smallest to Largest.
The data is then sorted numerically.
Value Function
You can also use the VALUE Function to convert your text to a number.
1. Select the first cell to convert and then type in the formula.
2. Copy the formula down to the remaining cells. You should now be able to sort these cells correctly.
3. As your result contains a formula and you may actually need values rather than formulas for your data, copy and paste values to get rid of the formulas. First, highlight the sort range.
4. In the Ribbon, go to Home > Clipboard > Copy.
5. Then, again in the Ribbon, go to Home > Clipboard > Paste > Paste Values.
6. Select Values (V).
Your formulas are now replaced by values.
Convert to Number
Another useful fix is converting the values in the cells to a number. This is possible if your data is stored as text, but Excel recognizes that it could be a number. An error tag appears as a small
green triangle in the top-left corner of each cell where this is occurring.
1. Select the cells where this is occurring and click the arrow by the small yellow triangle that appears on the right side. This shows you the error and a list of options.
2. Select Convert to Number.
3. Now, sort the numbers correctly. | {"url":"https://www.automateexcel.com/how-to/sort-numbers-correctly/","timestamp":"2024-11-05T12:28:23Z","content_type":"text/html","content_length":"145769","record_id":"<urn:uuid:6bcf447a-ae3b-4e68-9576-de2a272f137d>","cc-path":"CC-MAIN-2024-46/segments/1730477027881.88/warc/CC-MAIN-20241105114407-20241105144407-00207.warc.gz"} |
The scores on the SAT college entrance exam are normally
distributed with a mean Math score...
The scores on the SAT college entrance exam are normally distributed with a mean Math score...
The scores on the SAT college entrance exam are normally distributed with a mean Math score of 480 and a standard deviation of 100. If you select 50 students, what is the probability that their mean
Math score is more than 520. You MUST show what went into the calculator along with your final answer rounded correctly to 3 significant decimal places.
Let X denote the Math scores on the SAT college entrance exam.
Now, we are given that:
The above statement implies that:
Now, the probability that the mean math score is more than 520 is given by:
What went into the calculator depends on the type of calculator used. Generally we will enter 2.828427 and ask for it to calculate the cdf of the standard normal distribution at that point.
For any queries, feel free to comment and ask.
If the solution was helpful to you, don't forget to upvote it by clicking on the 'thumbs up' button. | {"url":"https://justaaa.com/statistics-and-probability/131904-the-scores-on-the-sat-college-entrance-exam-are","timestamp":"2024-11-04T02:44:23Z","content_type":"text/html","content_length":"41900","record_id":"<urn:uuid:ba4c7ca9-4972-45a7-8d49-eadad199e280>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00426.warc.gz"} |
Knizhnik-Zamolodchikov equation
added pointer to these surveys:
• Ivan Cherednik, Lectures on Knizhnik-Zamolodchikov equations and Hecke algebras, Mathematical Society of Japan Memoirs 1998 (1998) 1-96 $[$doi:10.2969/msjmemoirs/00101C010$]$
• Toshitake Kohno, Section1 1.5 and 2.1 in: Conformal field theory and topology, transl. from the 1998 Japanese original by the author. Translations of Mathematical Monographs 210. Iwanami Series
in Modern Mathematics. Amer. Math. Soc. 2002 $[$AMS:mmono-210$]$
diff, v23, current
The entry used to claim (am removing it for the moment) that:
The interpretation of $[$ the KZ-equation $]$ in terms of a flat connection on the moduli space of conformal structures is due to:
□ Graeme Segal, Conformal field theory, Oxford preprint and lecture at the IAMP Congress, Swansea July 1988.
Is that really the case?
(I forget who added this reference. There is a good chance that it was me, my apologies.)
If so, where exactly inside the following three items should we be pointing for the KZ-equation:
• Graeme Segal, The definition of conformal field theory, in: K. Bleuler, M. Werner (eds.), Differential geometrical methods in theoretical physics (Proceedings of Research Workshop, Como 1987),
NATO Adv. Sci. Inst., Ser. C: Math. Phys. Sci. 250 Kluwer Acad. Publ., Dordrecht (1988) 165-171 $[$doi:10.1007/978-94-015-7809-7$]$
• Graeme Segal, Two-dimensional conformal field theories and modular functors, in: Proceedings of the IXth International Congress on Mathematical Physics, Swansea, 1988, Hilger, Bristol (1989)
• Graeme Segal, The definition of conformal field theory, in: Ulrike Tillmann (ed.), Topology, geometry and quantum field theory , London Math. Soc. Lect. Note Ser. 308, Cambridge University Press
(2004) 421-577 $[$doi:10.1017/CBO9780511526398.019, pdf$]$
diff, v23, current
[ removed, sorry for the noise]
I am finally adding a section with references on the “hypergeometric” construction of conformal blocks/KZ-solutions, via twisted de Rham cohomology of configuration spaces of points.
A start is now here, but I will put this into a stand-alone entry, to be !include-ed here and in other related entries
diff, v18, current
added pointer to:
• Ivan Marin, Knizhnik-Zamolodchikov bundles are topologically trivial (arXiv:0809.3590)
diff, v17, current
finally added the actual definition, !include-ed from Knizhnik-Zamolodchikov-Kontsevich construction – definition (as per the discussion here)
diff, v13, current
I have come to think that the main part of the main theorem in the hypergeometric-integral construction of KZ solutions becomes a triviality when looked at from a HoTT point of view:
Namely, the main theorem says that the twisted cohomology groups of $Conf_{n+N}(\mathbb{C})\vert_{N}$ for fixed positions of $N$ of the points form a local system over $Conf_N(\mathbb{C})$.
But since the twisted cohomology depends only on the shape of $Conf_{n+N}(\mathbb{C})$, and since it is represented by a classifying space, the system of cohomology groups is given by an internal hom
out of $ʃ Conf_{n + N}(\mathbb{C})$ in the slice over $ʃ Conf_N(\mathbb{C})$. Such a slice hom is again a fibration over $ʃ Conf_N(\mathbb{C})$, and its fibers are the desired
fiberwise cohomology groups. Upon fiberwise truncation, this is the statement of that main theorem.
I should add that this proof uses that fiberwise 0-truncation preserves fiber products (which it does) combined with the assumption that any point inclusion into the base type is already 0-truncated.
So this works over configuration spaces of points (since these are $K(G,1)$s) as needed here for the KZ-equation, but not for Gauss-Manin connections over higher truncated base spaces.
I have now typed out the argument in point-set model presentation. Unsure where this should go, for the moment I put it into the entry on Gauss-Manin connections: here.
I have strengthened statement and proof (here) to say that on a locally trivial fibration, the local system of cohomology groups has a compatible local trivialization.
This will serve to prove that, when applied to fibrations of configuration spaces, this abstract argument really reproduces the hypergeometric solutions to the KZ-equation.
The only further lemma for this conclusion is that the statement also works for fiberwise twisted cohomology. Will type this out next.
added pointer to:
• Ivan Marin, Sur les représentations de Krammer génériques, Annales de l’Institut Fourier, 57 6 (2007) 1883-1925 [numdam:AIF_2007__57_6_1883_0]
diff, v29, current
also pointer to:
• Ivan Todorov, Ludmil Hadjiivanov, Monodromy Representations of the Braid Group, Phys. Atom. Nucl. 64 (2001) 2059-2068; Yad.Fiz. 64 (2001) 2149-2158 [arXiv:hep-th/0012099, doi:10.1134/
diff, v30, current
and this one:
• Xia Gu, Babak Haghighat, Yihua Liu, Ising- and Fibonacci-Anyons from KZ-equations [arXiv:2112.07195]
diff, v30, current
added pointer to:
• Ralph Blumenhagen, Erik Plauschinn, §3.5 of: Introduction to Conformal Field Theory – With Applications to String Theory, Lecture Notes in Physics 779, Springer (2009) [doi:10.1007/
diff, v32, current
added these pointers on
KZ-equations controlling codimension$=2$defects in D=4 super Yang-Mills theory:
• Nikita Nekrasov, BPS/CFT correspondence V: BPZ and KZ equations from $q q$-characters [arXiv:1711.11582]
• Nikita Nekrasov, Alexander Tsymbaliuk, Surface defects in gauge theory and KZ equation, Letters in Mathematical Physics 112 28 (2022) [arXiv:2103.12611, doi:10.1007/s11005-022-01511-8&
• Saebyeok Jeong, Norton Lee, Nikita Nekrasov, Intersecting defects in gauge theory, quantum spin chains, and Knizhnik-Zamolodchikov equations, J. High Energ. Phys. 2021 120 (2021) [
arXiv:2103.17186, doi:10.1007/JHEP10(2021)120]
diff, v37, current
• Anton Alekseev, Florian Naef, Muze Ren. Generalized Pentagon Equations (2024). (arXiv:2402.19138).
diff, v38, current | {"url":"https://nforum.ncatlab.org/discussion/10804/knizhnikzamolodchikov-equation/?Focus=99785","timestamp":"2024-11-06T21:34:42Z","content_type":"application/xhtml+xml","content_length":"77912","record_id":"<urn:uuid:213ac1b8-daea-483d-8e65-536a78e164a9>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.47/warc/CC-MAIN-20241106194801-20241106224801-00446.warc.gz"} |
Everyday maths 2
2.3 Rounding to a degree of accuracy
Watch the short video below to see an example of how to round to one, two and three decimal places.
Download this video clip.Video player: bltl_1_2_3_rounding.mp4
Rounding can be used when you're asked to give an answer to a given degree of accuracy. This is called rounding to decimal places, or d.p. In this video, you'll look at how to round to one, two, and
three decimal places.
Take the number 25.782. How would you write this rounded to one decimal place? This is the first decimal place. Rounded to 1 d.p., the number could be either 25.7 or 25.8. Look at the digit next to
the number 7. If this number is equal to or greater than 5, add one more. If it's less than 5, leave it. In this case, 8 is greater than 5, so our number rounded to 1 d.p. is 25.8.
Now, let's around a number to two decimal places. A common example of when you would do this in daily life is when dealing with money. If you were at a restaurant and needed to split a bill of £87.95
between three people, you would first calculate the division. £87.95 divided by 3 equals £29.3166667. Clearly, you cannot pay this exact amount. So how much would you pay if the amount per person was
rounded to two decimal places?
This is the second decimal place. Look at the number next to it. Is it greater than or equal to 5? 6 is more than 5, so you need to add 1. The amount to pay is £29, 32 pence.
Let's try another example. How would you around the number 35.496 to two decimal places? This is the second decimal place. Look at the number next to it. 6 is greater than 5, so you need to add one
more. In this case, the rounded number would be 35.50.
Can you round to three decimal places? Have a go with this number: 412.5762. The number next to the third decimal place is 2, which is less than 5. This means the correctly rounded number is 412.576.
End transcript
Interactive feature not available in single page view (
see it in standard view
Remember this rounding rhyme to help you:
Activity 5: Rounding skills
Practise your rounding skills by completing the below.
1. What is 24.638 rounded to one decimal place?
2. What is 13.4752 rounded to two decimal places?
3. What is 203.5832 rounded to two decimal places?
4. What is 345.6795 rounded to three decimal places?
1. 24.6
2. 13.48
3. 203.58
4. 345.680
In this section you have learned:
• how to decide whether an answer to a division calculation needs to be to rounded up or down, depending on the context of the question
• how and when to use rounding to approximate an answer to a calculation
• how to round an answer to a given degree of accuracy – e.g. rounding to two decimal places. | {"url":"https://www.open.edu/openlearn/mod/oucontent/view.php?id=85562§ion=2.3","timestamp":"2024-11-01T19:50:37Z","content_type":"text/html","content_length":"129271","record_id":"<urn:uuid:ddc06148-057b-43a5-9df8-aa97c8112c42>","cc-path":"CC-MAIN-2024-46/segments/1730477027552.27/warc/CC-MAIN-20241101184224-20241101214224-00295.warc.gz"} |
The Economics of Mesh Bags for Protecting Young Citrus Trees - Citrus Industry Magazine
By Ariel Singerman
The use of mesh bags has been proposed as a strategy for excluding Asian citrus psyllids to protect young citrus trees. The expected benefit of using mesh bags is increased yield by delaying HLB
A scientific experiment to evaluate the effectiveness of mesh bags started in February 2018. There are still many unknowns regarding yield, use of chemicals, incidence of other pests and diseases,
etc. However, some growers are already experimenting with mesh bags in their groves in different ways. In some groves, every other tree within a row is covered. In other groves, every other row is
covered. Some growers choose to cover the entire new planting.
In this article, I evaluate the economic feasibility of using mesh bags for protecting young citrus trees based on a number of assumptions. Growers can follow the methodology to make the calculations
relevant for their own operation and, therefore, improve their decision-making process to decide whether to use the bags.
Two key variables for the calculations are yield and prices (in years 3, 4, 5 and 6). Therefore, to deal with the uncertainty regarding yield, I make assumptions based on historical data available.
To take into account the uncertainty in prices, I create scenarios that represent different potential values. Other key variables for the calculations are the cost of the bag and the associated labor
to put them on and take them off, the useful life of the bags, and the savings in caretaking programs that can be achieved by using the bags. The latter variables, unlike yield and prices, are either
known or can be reasonably estimated.
The analysis provided here is for Valencia oranges and is based on the following assumptions. I assume that the bags will be put on young trees for two years. During those two years, trees will be
HLB-free, and the grower will incur additional costs and savings due to the use of bags. At the end of the two years, the bags will be removed. Trees will eventually become HLB-infected but will
attain a differential yield (relative to unbagged trees) because of the delayed infection.
The average reduction in yield relative to a healthy tree due to HLB is 40 percent. Thus, after taking the bag off a reset, I assume the progression of the reduction in yield is 20, 30, 37 and 40
percent in years 3, 4, 5 and 6, respectively. In other words, the differential yield benefit vanishes four years after taking the bag off. Also, the underlying assumption for the level of yield is
that the tree density is 220 trees per acre.
Regarding prices, I assume that they are constant throughout the investment period, but I assume three different price levels (on a delivered-in basis) to denote possible market conditions. Given the
recent decrease in prices in the cash market, I assume $1.75, $2.00 and $2.25 per pound solids for the low, medium and high price scenarios, respectively. In terms of costs, I assume that the cost of
a 5-foot bag and PVC stake is $7.10, and the associated labor cost to put the bag on and off is $1.25. I assume two scenarios regarding the bag lifetime: one-use (two years) and two-use (four years).
With respect to caretaking savings, I use the annual cost of production data as a basis for the calculations and assume two different scenarios: low and high savings. The low-savings scenario
achieves savings of $0.88 per tree in years 1 and 2 by avoiding the expense of two drench applications for a total of $0.72 per tree, foliar insecticide savings of 50 percent at $0.12 per tree, and
foliar nutritional savings of 20 percent at $0.03 per tree. The high-savings scenario achieves savings of $2.77 per tree in years 1 and 2 by avoiding the expense of seven drench applications for a
total of $2.53 per tree, foliar insecticide savings of 75 percent at $0.18 per tree, and foliar nutritional savings of 33 percent at $0.06 per tree.
Table 1 illustrates the cost and revenue cash flows each year for the scenario that combines medium prices, two-use bags and high savings. Using a 10 percent rate to discount the cash flows at
different years, the net present value (NPV) is $0.58. As a rule of thumb, investments with a positive NPV should be accepted, and those with a negative NPV should be rejected. The rationale for
accepting investments with positive NPVs is that they yield higher returns than the discount rate (i.e., cost of capital).
However, it is impossible to estimate a discount rate that would be representative of the cost of capital of all growers because each individual grower has a different opportunity cost of capital.
Therefore, I show the results of the investment analysis using the internal rate of return (IRR) methodology. The IRR is the actual rate of return on the investment, which for the example in Table 1
is 16.43 percent.
Table 2 shows the results for different scenarios I analyzed using the reset model. The only scenarios in which using bags for protecting resets that turn out to be profitable are those that combine
a two-use bag with high savings (for all three price levels). As illustrated by the results, much of the benefits of using bags depends on how much caretaking savings a grower can achieve. This
finding is, not surprisingly, also key in the solid set model.
The solid set analysis is more complex because it requires the creation of a spreadsheet to track the tree inventory each year. That is, the number of infected and healthy trees along with their
yield, and the differential cost and revenue relative to a solid set with no bags.
The solid set model requires a few additional assumptions. First, I assume tree mortality to be 1 percent in year 0 through 2 and 4 percent in year 3 and beyond. Second, I assume that there are
additional savings on two ground applications and on aerial applications (in the high savings scenario). Third, I also need to make a key assumption regarding the progression of the HLB infection
throughout the grove because trees in a solid set do not get immediately infected after the bag is taken off. Thus, I assume that at the end of years 3, 4, 5, and 6, the infection throughout the
grove is 30, 60, 90 and 100 percent, respectively.
Table 3 shows the results for the different scenarios analyzed using the solid set model. I found the use of bags to be profitable for all scenarios with high savings except that of low prices and
one-use bags. Of course, profitability improves significantly when the bags can be reused for two more years. However, again, the results denote that much of the benefits of using bags depends on how
much caretaking savings a grower can achieve.
I showed the calculations and procedure for evaluating the economic feasibility of using mesh bags for protecting young citrus trees based on assumptions that allowed us to overcome the many unknowns
regarding their use. Growers can follow the methodology I applied to make the calculations relevant for their operations and, therefore, improve their decision-making process to decide whether to use
the bags.
I found that using mesh bags for protecting resets is profitable when the bag can be reused (halving its cost), and the grower can achieve high savings in terms of caretaking. In addition, using mesh
bags for protecting solid sets is profitable when the grower can achieve high savings in terms of caretaking, even in some scenarios in which the bag has a single use. The reason for finding the use
of (relatively) more expensive bags to be profitable in solid sets is because trees in a solid set do not get infected at the time, so the impact of HLB on yield is slower (relative to that of a
Ariel Singerman is an assistant professor at the University of Florida Institute of Food and Agricultural Sciences Citrus Research and Education Center in Lake Alfred. | {"url":"https://citrusindustry.net/2020/04/28/the-economics-of-mesh-bags-for-protecting-young-citrus-trees/","timestamp":"2024-11-11T07:51:16Z","content_type":"text/html","content_length":"138601","record_id":"<urn:uuid:a39362b0-5edc-4b0b-8d61-fa1d3f48a195>","cc-path":"CC-MAIN-2024-46/segments/1730477028220.42/warc/CC-MAIN-20241111060327-20241111090327-00330.warc.gz"} |
Monopoly and monopolistic revenues, equations, elasticities, and price discrimination
This post is going over a question recently received:
You have been assigned the task of helping the Midland Milk Marketing (MMM),
the sole marketing agency for milk produced in the Island State of Midland. In the past the producers have been marketing their milk as a homogeneous commodity to all processors of milk. You have
studied the market extensively and realize that the market can be segmented into two separate units:
(1) the market for fluid milk (milk for drinking) and (2) the market for processing milk (for manufacture of cheese, etc.). Your preliminary analysis has generated the following demand curves for the
two separate markets.
Fluid milk market:
Inverse demand curve: Pfluid= 22 – 2.5*Qfluid
Marginal revenue curve: MRfluid = 22 – 5*Qfluid
Processing milk market:
Inverse demand curve: Pprocessing = 20 – 3*Qprocessing
Marginal revenue curve: MRprocessing = 20 – 6*Qprocessing
Assume that individual firms have the same cost function which is as follows: TC = 10 + 2*Q
1. What is the profit-maximizing allocation of milk production in each of the markets? Assuming that MMM can practice price-discrimination in the market, what is the profit-maximizing price and
quantity in each market?
2. What is the total revenue for the MMM using price discrimination?
3. Calculate the own price elasticities of demand for fluid and processing milk at the equilibrium values for P and Q.
4. Illustrate the effects of price discrimination with a graph of both the fluid milk market segment and the processed milk market segment. Your graph should include a demand curve, a marginal
revenue curve, and the profit-maximizing price and quantity under price discrimination. If the price without market discrimination is $11.50, which segment of the market benefits from market
The first thing we need to do is find the profit maximizing quantity. To do this, we have to remember that for a any business (including monopolies) the profit maximizing point occurs when marginal
revenue is equal to marginal cost (MR=MC). So first we have to identify what marginal revenue is, and luckily that is given above. In order to calculate marginal cost, you can either take the
derivative of total cost (TC) or find the slope of the TC curve. Either way you will end up with MC=2.
The next step is to set MR=MC, so for the fluid milk market, we set:
MC=MR or
2 = 22-5*Q
Now solve for Q by subtracting 2, and adding 5Q to both sides. Then divide by 5 and you will get:
So the equilibrium quantity for the fluid milk market is 4. To get equilibrium price, plug in equilibrium quantity into the price equation:
P=22-2.5*4 =22-10=12
So equilibrium price will be 12.
Let’s do the same process for the next market:
MC=MR or
2 = 20-6Q
Now solve for Q by subtracting 2, and adding 6Q to both sides. Then divide by 6 to get:
Plug our equilibrium quantity into the price function for the processing market to get equilibrium price:
So equilibrium price in the processing market will be 11.
If our producer is allowed to price discriminate, then the equilibrium price will be used in either market, so revenue will be price multiplied by quantity. This results in revenue of 12*4=48 for
the fluid market and revenue of 11*3=33 for the processing market.
To calculate the own price elasticities of demand for a static point, we can use a trick using the slope of the demand function. In this case, the price elasticity of demand equation will be:
Elasticity = (P/Q)*(dQ/dP) or E=(P/Q)*(change in Q/ Change in P)
This can also be written as E = -bP/Q = -bP/(a-bP)
Where our linear demand curve is q=a-bp.
So first we need to make our inverse demand function into a normal demand function by solving for Q. We get this by adding 2.5Q to both sides (of our fluid market) and subtracting P from both sides,
then dividing by 2.5. This gives us:
Q = 8.8 - 0.4P
So in the above equation, a = 8.8, and b = 0.4. P and Q are still 12 and 4 respectively.
Plug these values into our price elasticity of demand measure to get:
E = -0.4*12/(8.8-0.4*12) = -4.8/4 = -1.2
So our point price elasticity of demand measure at equilibrium price and quantity for the fluid market is -1.2.
Using the same method for the processing market we get:
Q = 6.67 - .33 P
After plugged into our elasticity function gives us:
E = -.33P/(6.67-.33P) = -3.67/3 = -1.22
So the price elasticity of demand for both markets is elastic, this means that if revenue for the firms could be increased if prices went down (but just because revenue would rise, doesn’t mean
profit would, because we have positive costs).
Finally, the graphs for the two markets would look like the graph to the right. Here we can see the MR=MC at a point where P (in dollars) is $2, and quantity is 4. However, when we figure out what
the price paid by the consumer will be, by drawing the line up to intersect with the demand curve, we get a price paid of $12.
The next graph to the left shows the monopoly market for the processed milk. Here, MC=MR again at a price of $2, but since the MR curve is steeper, the quantity is only at 3. When we draw the line
up to intersect the demand curve, we see that this intersection occurs at a price of $11.
And if we put the graphs together, we get a very complicated graph, be we can see how the firm is price discriminating in the two different markets.
And if the price of milk was $11.50 with the opportunity for price discrimination than the fluid market would lose revenue because they are currently charging $12 for their milk, while the processing
market would receive $0.50 more because their current price is only $11. | {"url":"https://www.freeeconhelp.com/2011/09/monopoly-and-monopolistic-revenues.html","timestamp":"2024-11-06T21:02:09Z","content_type":"application/xhtml+xml","content_length":"182236","record_id":"<urn:uuid:35b5dfa0-711f-4795-9760-93cb0cf39fff>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.47/warc/CC-MAIN-20241106194801-20241106224801-00495.warc.gz"} |
How to draw Voronoi diagrams
How to draw Voronoi diagrams by hand
. Just to remind you, a Voronoi diagram is associated with a set of points x
, ..., x
in the plane; for each point, the cell containing it is the set of points in the plane that are closer to x
than to any other x
. I've spent a fair amount of time doodling Voronoi diagrams in boring places, so this is interesting. Apparently they also run into the problem that sometimes it's hard to tell which cells will
border which other cells.
Yes, I know, it's been six months since I posted. As you can see, I'm still alive, and I made it through my first semester here at Berkeley. I live in North Oakland, closer to downtown Berkeley than
to downtown Oakland, and I find myself describing where I live as "the part of Oakland that's really more Berkeley-ish". More formally, I'm in downtown Berkeley's Voronoi cell, not downtown
Proof Math is Beautiful
3 comments:
Sue VanHattum said...
Michael, I don't know if you have any interest, but there's a math circle workshop tomorrow in Berkeley. Check the msri site for more info.
Bill Mill said...
The guy who wrote that is here in Baltimore, and makes lovely large-scale Voronoi drawings by hand. Nice guy.
clovis simard said...
Vous êtes cordialement invité à visiter mon blog.
Description : Mon Blog(fermaton.over-blog.com), présente le développement mathématique de la conscience humaine.
La Page No-21: THÉORÈME DU PIEGE !
LA LIBERTÉ ! UN PIÈGE ?
Clovis Simard | {"url":"https://godplaysdice.blogspot.com/2011/01/how-to-draw-voronoi-diagrams.html?showComment=1296226396829","timestamp":"2024-11-08T20:43:50Z","content_type":"text/html","content_length":"50327","record_id":"<urn:uuid:86e3586f-c063-45c8-b64f-fbd8aab5bae1>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00498.warc.gz"} |
Anchors for Balanced Loads
Balancing two anchor points
In Rescue, it is common to create a centralized anchor while sharing two anchor points. There are many reasons why we would use two or more anchor points; sometimes this is because the anchors are
not exactly where we need them to be and other times it could be because the anchors are perceived as “marginal”. Although it is ill-advised to use marginal or questionable anchors unless they are
backed up sufficiently, the following information should help you in pre-determining how much stress you are actually applying to the anchors during a rescue evolution before they are loaded.
Many rescue instructors, firms, and institutions may refer to angles as “Ideal”, “Yes”, “Cautionary” and/or “Terrible” angles and this quick overview may help further explain and/or challenge your
understanding of these meanings.
DOWNLOAD PDF (Click on Diagram)
30˚ “Ideal” Angle: An “Ideal” angle is an angle that is between 0˚ and 30˚. What makes the angle considered to be “Ideal” is that the weight of the load is split evenly between both anchors. For
example, a 30˚ load holds approximately 50% of the weight. In the diagram, you can see how a 100lb load only has 50lbs on each anchor which is 50% of the 100lbs. If there was an “S” type load cell*
on each of the anchors connected to the rigging, you could see the results as you modified the angles. A challenge to the idea of the angle being “Ideal” is that if there is a slight shift of the
load, the anchor opposite of the load shift will receive a very rapid and massive amount of load gain. If the anchor is marginal, it could result in anchor failure. The easiest way to identify if the
anchor angle is approximately 30˚ is by making a peace sign with your hand. The angle between your index (pointer) finger and your middle finger is roughly 30˚.
“Yes,” angles are typically between 30˚ and 90˚. The considerations for “yes” angles are that there are very insignificant increases in weight to the anchors but with quite a large range of angle
increase. Overall, as you can see on the diagram, there is an increase of weight between the 30˚ and 90˚ angles of only 20% to each anchor. This angle range of 50˚ having a 20% increase to each
anchor is the primary reason for the classification of “Yes” angles. These angles are easily identified due to the load line and anchor lines looking like the letter “Y”. A shift in the load also
must be much more dramatic before additional anchor stress is achieved.
60˚ “Yes” Angle: By increasing the 30˚ angle to a 60˚ angle between the anchors creates a 10% increase in anchor stress. As seen in the diagram, the doubled angle only shows the additional 10lbs per
anchor to hold a 100lb weight. The 60lbs of anchor stress per anchor to hold the 100lbs is far more stable than the ideal angle if the load were to shift to the left or the right. Whenever possible,
try and achieve a 60˚ angle for balancing loads. A quick field calculation for assessing a 60˚ angle is by holding up the “Hook ‘em Horns” hand signal. This is achieved by holding up your index
(pointing) finger and your pinky finger while tucking the two middle fingers with your thumb. This natural span and the angle is very close to a 60˚ angle.
90˚ “Yes” Angle: This angle achieves a right angle which is very easy to identify and verify in the field. You could use a book, or a phone or almost anything lying around that has a right angle in
order to visually see this angle. Also, if you make the letter “L” using your index (pointing) finger and your thumb you can also see this angle. A 90˚ angle only places 70% of the load on each
anchor. As seen in the diagram, a 100lb load only has 70lbs on each anchor.
120˚ “Cautionary” Angle: The “Shaka” or “Hang Loose” hand signal is a rough estimation used in the field for a 120˚ angle. The reason for this “cautionary” angle is due to the forces applied to the
anchors. At this angle, the full weight of the load is being applied to both anchors evenly. In the diagram, it shows a 100lb weight and both anchors holding the 100lbs. the reason for this is a
simple formula. If you dissect the circle by both of the anchors and the load line, you will see that all three sections are equal. If all angles are equal, then all loads must be equal. The
symmetrical tension on the anchors and load line should indicate that each of the anchors should be able to hold the load by themselves. Never use “marginal” anchors when rigging in this vector angle
range. A slight change in angle will cause a dramatic increase of stress put on the anchors. Any angle that is greater than 120˚ is classified as a “Terrible” angle. Without a clear understanding of
physics, never try and rig outside of 120˚ angle.
150˚ “Terrible” Angle: Although most would classify a 150˚ angle as terrible, my thoughts are a little different. Under normal circumstances, I would not try and rig an angle this wide due to the
excessive stress placed on the anchors, but in certain applications, they are very useful. Typically, when applying these wide angles their uses are not only wanted but may be very necessary. This is
usually the case when building Highline systems, Tyrolean systems or Offsets. My typical name for the angles is not “Terrible” but more like “Time Out”. The main reason is I believe you should take
the time when building these style systems and do the math! Physics play a huge role in the failure of these systems in every part. These wide angles must be calculated and all equipment associated
with these systems is an integral part of the entire degradation of the system, but most certainly the anchors. Very flat trajectories yield very high stressors for the anchors in these systems. I am
not saying you could not build a system that had a 2.5% deflection, but if you do, understand the limitations of the entire system.
Deflection is key when rigging anything beyond 120˚. First, let’s assume some basic guidelines. We would be using ½ kernmantle static ropes as well as all “G” rated equipment. The load would not
exceed the NFPA two-person load of 600lbs. As all of this equipment would be rated for the coveted 15:1 safety factor (9,000lbs), in all actuality, we would be attempting to hit a 10:1 safety margin.
A 10:1 safety factor in our systems is more like a “Holy Grail”, but with proper rigging and calculations, it could be achieved.
Here is the formula:
Real World Application:
Estimate the load and multiply it by the distance between the two anchors. Let’s say we have a load of 400lbs. and a distance between the anchors of 100ft. we would multiply the load of 400lbs times
the distance of 100ft totaling 40,000lbs. Now you can see that the “G” rated equipment would fail since its ratings are around the 9,000lb mark. At this point, you should be looking at “Sag” more
than anything else in order to reduce that tension. Here is the simple way to calculate “Sag” or deflection. If we are using a single ½ inch kernmantle rope, we would want no less than 10%
deflection. A 10% deflection of the entire distance between both anchors, which is 100ft distance, would be a 10ft deflection. Multiply 4 times 10, and you get 40. Now, 40,000 divided by 40 equals
1,000lbs. This shows that a 400lb load with a distance between anchors of 100ft and a deflection of 10ft (10% of the distance) would generate a 1,000lb hit on the anchors. As seen in the diagram, we
are using the same formula but a weight of 100lbs, a span of 10ft, and a deflection of 1ft. If these numbers are plugged into the formula, you can see that there are 250lbs of force on each anchor.
When using this formula, you are attempting to achieve a 150˚ angle, regardless of the scale between anchors. If less deflection is required, additional rigging must be applied to keep a 10:1 safety
For further information about this or other related topics please contact Shayne Torrans, HSEQT Manager – The IPS Group.
*S Type Load Cell: are low-cost and high-performance side mounted load cells suitable for a number of weighing and general force measurement applications. http://www.cmcrescue.com
Table of content
Related articles | {"url":"https://theipsgroup.us/anchors-for-balanced-loads/","timestamp":"2024-11-12T03:37:55Z","content_type":"text/html","content_length":"120535","record_id":"<urn:uuid:44e89bd9-d3a7-4997-b793-2b9b08c64d3a>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.50/warc/CC-MAIN-20241112014152-20241112044152-00077.warc.gz"} |
AvoidingPatternsUnderInvolution < CoWiki < TWiki
Patterns can be generalized by adding functional dependencies between variables. Here, we consider involutions ϑ, that is, ϑ(ϑ(
)) =
for all letters
, and their morphic/antimorphic extensions on words, that is, ϑ(
) = ϑ(
) for the morphic and ϑ(
) = ϑ(
) for the antimorphic case. A pattern
in this setting consists now of (word) variables and function variables and it is avoided in some infinite word
if there exists no substitution of (word) variables by words and function variables by involutions for
such that the result is a factor of
. The morphic and antimorphic cases are usually considered separately. The avoidance indices of all unary patterns under involution are known. See
B. Bischoff, J. Currie, D. Nowotka, Unary Patterns With Involution, (reference to be completed).
be a pattern consisting of only one (word) variable and at most one function variable. Then for both the morphic and antimorphic case
• avoidable over three letters, if p is of length 3 and not in {ααα, ϑ(α)ϑ(α)ϑ(α)},
• unavoidable, if p is in {α, ϑ(α), αϑ(α), ϑ(α)α}, and
• avoidable over two letters otherwise.
- 12 Mar 2012 | {"url":"https://cs.uwaterloo.ca/twiki/view/CoWiki/AvoidingPatternsUnderInvolution","timestamp":"2024-11-07T15:13:06Z","content_type":"application/xhtml+xml","content_length":"18931","record_id":"<urn:uuid:4156a8fe-84bf-470f-a8cf-18898bbc92f6>","cc-path":"CC-MAIN-2024-46/segments/1730477028000.52/warc/CC-MAIN-20241107150153-20241107180153-00008.warc.gz"} |
Volume Equation in Chemistry - K-Tabs
If you’re studying chemistry you will need to have to create use from the volume equation as you study.
The volume equation is utilized any time you are developing substances that usually do not possess a strong form, by way of example gases and liquids.
The volume equation is also commonly utilised to discover how lots of molecules you can find inside a substance. You should convert the mass in the substance you might be studying into a number ahead
of you are able to perform out the volume. The formula is:
Example: The substance you are studying has moles of protons and neutrons. dissertation help To discover the volume in the substance you have to make use of the formula beneath:
For the volume equation to be valuable you should know how quite a few moles of a substance you are studying. You may do that by dividing the total mass with the substance you will be studying by the
total quantity of moles inside the substance. The formula is:
This may be the formula used to work out the volume of your volume equation. It truly is made use of to convert the mass with the substance you are studying into a number. In order to convert mass
into quantity you must multiply the quantity by one hundred.
The volume equation is usually made use of to locate the mass of a gas molecule, by way of example should you be studying oxygen. http://wikipedia.com/wiki/Washington_City_Paper Molecules which
include oxygen contain 1 proton and one electron and this gives you the volume with the molecule.
If you happen to be functioning on a chemical equation you ought to know about this formula so that it is possible to work out how several molecules you can find in the substance. The calculation is
based on the formula for a cubic unit cell exactly where the first term equals the volume of your molecule.
You can make use of the formula to convert themass of a gas molecules. In case you convert the mass of a gas molecule to cubic meters you may find that the volume is a lot smaller sized than it would
be should you had converted it to an inch cube.
Remember that the formula for any cubic unit cell is essential if you are operating on elementary chemical equations. There are lots of things that can have an www.essay-company.com/ effect on how
your molecules are formed, for example the temperature and pressure. Understanding the formula to get a cubic unit cell can help you understand extra regarding the way your molecules are formed.
In this example you will be working on nitrogen which has 4 protons and 4 neutrons. You must convert the mass with the substance you are studying into the variety of moles of protons and neutrons.
The formula is:
The calculation is primarily based around the formula to get a cubic unit cell exactly where the very first term equals the volume in the molecule. For those who convert the mass of a gas molecule to
cubic meters you may discover that the volume is considerably smaller sized than it would be in case you had converted it to an inch cube.
If you’ll need assist together with the volume equation then you must use a chemistry textbook on the web. There are many chemistry books offered online that make use of the volume equation. You need
to use the volume equation should you be going to understand how you can convert mass into volume and vice versa. | {"url":"https://k-tabs.com/2020/04/07/volume-equation-in-chemistry/","timestamp":"2024-11-08T05:43:39Z","content_type":"text/html","content_length":"37684","record_id":"<urn:uuid:e606fce7-096f-407d-a9d9-2b084281fcb9>","cc-path":"CC-MAIN-2024-46/segments/1730477028025.14/warc/CC-MAIN-20241108035242-20241108065242-00193.warc.gz"} |
Surface Parameters function
Available with Spatial Analyst license.
Determines parameters of a surface raster such as aspect, slope, and several types of curvatures using geodesic methods.
For more information, see the How Surface Parameters works topic in the Spatial Analyst tool help.
This function can be used for the following applications:
• Calculate aspect and slope using geodesic methods.
• Calculate different types of curvatures from an input surface raster, for example, Tangential (normal contour) curvature, which characterizes topographic convergence and divergence of flow across
the surface.
Parameter name Description
The input surface raster.
Specifies the output surface parameter type that will be computed.
• Slope—The rate of change in elevation will be computed. This is the default.
• Aspect—The downslope direction of the maximum rate of change for each cell will be computed.
• Mean Curvature—The overall curvature of the surface will be measured. It is computed as the average of the minimum and maximum curvature. This curvature describes the intrinsic
convexity or concavity of the surface, independent of direction or gravity influence.
Parameter Type • Tangential (normal contour) Curvature—The geometric normal curvature perpendicular to the slope line and tangent to the contour line will be measured. This curvature is
typically applied to characterize the convergence or divergence of flow across the surface.
• Profile (normal slope line) Curvature—The geometric normal curvature along the slope line will be measured. This curvature is typically applied to characterize the acceleration
and deceleration of flow down the surface.
• Plan (projected contour) Curvature—The curvature along contour lines will be measured.
• Contour Geodesic Torsion—The rate of change in slope angle along contour lines will be measured.
• Gaussian Curvature—The overall curvature of the surface will be measured. It is computed as the product of the minimum and maximum curvature.
• Casorati Curvature—The general curvature of the surface will be measured. It can be zero or any positive number.
Choose the type of surface function that will be fitted around the target cell.
Local Surface Type
• Quadratic—A quadratic surface function will be fitted to the neighborhood cells. This is the default type.
• Biquadratic—A biquadratic surface function will be fitted to the neighborhood cells.
Neighborhood The output will be calculated over this distance from the target cell center. It determines the neighborhood size. The default value is the input raster cell size, resulting in a 3
Distance by 3 neighborhood.
Specifies whether neighborhood distance will vary with landscape changes (adaptive). The maximum distance is determined by the neighborhood distance. The minimum distance is the
Use Adaptive input raster cell size.
• Unchecked—A single (fixed) neighborhood distance will be used at all locations. This is the default.
• Checked—An adaptive neighborhood distance will be used at all locations.
The linear unit of vertical z-values.
It is defined by a vertical coordinate system if it exists. If a vertical coordinate system does not exist, the z-unit should be defined from the unit list to ensure correct
geodesic computation.
• Inch—The linear unit will be inches.
• Foot—The linear unit will be feet.
Z Unit • Yard—The linear unit will be yards.
• Mile US—The linear unit will be miles.
• Nautical mile—The linear unit will be nautical miles.
• Millimeter—The linear unit will be millimeters.
• Centimeter—The linear unit will be centimeters.
• Meter—The linear unit will be meters. This is the default.
• Kilometer—The linear unit will be kilometers.
• Decimeter—The linear unit will be decimeters.
The measurement units (degrees or percentages) that will be used for the output slope raster. This parameter is only active when Parameter type is Slope.
Output Slope
Measurement • Degree—The inclination of slope will be calculated in degrees. This is the default.
• Percent rise—The inclination of slope will be calculated as percent rise, also referred to as the percent slope.
Specifies whether geodesic azimuths will be projected to correct the angle distortion caused by the output spatial reference. This parameter is only active when Parameter type is
Project Geodesic Aspect.
• Unchecked—Geodesic azimuths will not be projected. This is the default.
• Checked—Geodesic azimuths will be projected.
Specifies whether aspect will be measured from a point on the equator or from the north pole. This parameter is only active when Parameter type is Aspect.
Use Equatorial
Aspect • Unchecked—Aspect will be measured from the north pole. This is the default.
• Checked—Aspect will be measured from a point on the equator.
A raster that specifies the locations where the analysis will occur.
Analysis Mask The raster can be integer or floating point type.
All cells with a valid value, including zero, will compose the mask. Cells that are NoData in the mask input will be NoData in the output. | {"url":"https://pro.arcgis.com/en/pro-app/3.1/help/analysis/raster-functions/surface-parameters-function.htm","timestamp":"2024-11-04T10:14:14Z","content_type":"text/html","content_length":"24443","record_id":"<urn:uuid:e0f0aeb4-297b-4c84-8faa-e4d0ef47caab>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00252.warc.gz"} |
How to Perform Division Operation on Tensors in PyTorch – TheLinuxCode
How to Perform Division Operation on Tensors in PyTorch
PyTorch provides powerful tensor operations that make it easy to build deep learning models. This guide will illustrate different methods to perform the division operation on tensors in PyTorch.
Introduction to Tensors
Tensors are the basic data structures used in PyTorch for all kinds of numeric computations. They are multi-dimensional arrays that generalize vectors, matrices, and scalars. We can create tensors
from Python lists and perform arithmetic operations between them.
For example:
import torch
t1 = torch.tensor([1, 2, 3])
t2 = torch.tensor([3, 2, 1])
Here t1 and t2 are 1-D tensors created from lists holding three integer elements each.
Being able to divide tensors is useful for data analysis and especially when training neural networks. Let‘s look at different techniques to divide tensors in PyTorch.
1. Elementwise Division using the / Operator
The simplest way to divide two tensors is by using the / operator. This divides elements at corresponding indices and returns a new tensor.
For example:
t3 = t1 / t2
# tensor([0.3333, 1.0000, 3.0000])
We can also broadcast scalars for division. Here 2 gets divided from all elements of t1:
t4 = t1 / 2
# tensor([0.5000, 1.0000, 1.5000])
The / operator computes floating-point division to account for possible decimal output.
2. Using torch.div() for Elementwise Division
Instead of the / operator, we can also use torch.div() function for elementwise tensor division:
t5 = torch.div(t1, t2)
# tensor([0.3333, 1.0000, 3.0000])
torch.div() internally maps to torch.Tensor.div() method and is equivalent to the division operator.
Use cases for torch.div():
• Clarifies code when source tensors get modified in-place later
• Easy to change underlying division behavior
3. True Division with torch.true_divide()
To always compute true division (even when tensors contain integers), we can use torch.true_divide().
For example:
t1 = torch.tensor([10, 5])
t2 = torch.tensor([3, 5])
t6 = torch.true_divide(t1, t2)
#tensor([3.3333, 1.0000])
True division is usually preferred while implementing neural network loss functions and gradients.
We have now covered the core division methods for PyTorch tensors. But there are additional considerations worth knowing.
Handling Zero Division Errors
Attempting to divide tensors having zero values will raise a ZeroDivisionError.
For example:
t1 = torch.tensor([1, 2, 0])
t2 = torch.tensor([3, 0, 3])
t1 / t2 # Gives ZeroDivisionError
We need to catch this and handle gracefully based on the use case:
import torch
t1 / t2
except ZeroDivisionError:
print("Cannot divide by zero tensor")
# Custom handling here
Datatype Considerations
Inputs determine the datatype of result tensors from divisions.
For example:
t1 = torch.tensor([1, 2], dtype=torch.float32)
t2 = torch.tensor([3, 4], dtype=torch.int64)
t3 = t1 / t2
print(t3.dtype) # float32
t4 = t2 // t1 # floor division
print(t4.dtype) # int64
So pay attention to precision requirements when dividing.
Tensor Shape Rules
The two input tensors for division must have identical shapes unless one is a scalar. Otherwise, it will throw a RuntimeError.
For example:
t1 = torch.arange(0, 6).reshape(2, 3)
t2 = torch.arange(0, 6) # Shape mismatch
t1 / t2 # Throws an exception
We get correct output by first reshaping t2 to match t1:
t2 = torch.arange(0, 6).reshape(2, 3)
t1 / t2 # Performs elementwise division
These broadcasting rules apply across all mathematical tensor operations.
DividingCNN Filter Tensors
An important use case is dividing tensor weights representing convolutional filters while training neural networks.
For example, here is how to divide a 4-D filter tensor by 0.1 scalar:
filters = torch.rand(8, 3, 5, 5) # Batch, Channels, Height, Width
filters = filters / 0.1
This scales down all filter values to constrain and normalize them.
Performance Considerations
For math-heavy workflows like deep learning, pay attention to:
• Operation device placement (CPU vs GPU)
• Data types to minimize precision losses
• In-place versions that conserve memory
Benchmark division methods before combining them with autograd and gradients.
Best Practices
Follow these recommendations when dividing tensors:
• Check shapes before division to avoid unexpected exceptions
• Handle potential zero division errors
• Use true divide for robust gradients and loss calculations
• Place operations on fast devices like GPUs and TPUs
• Profile code to minimize computational inefficiencies
• Prefer built-in division functions over manual ops
And that covers the key aspects of performing tensor division in PyTorch!
This guide explained various ways to divide tensors in PyTorch:
• The / operator for elementwise floating-point division
• torch.div() for clarity and customization
• torch.true_divide() for accurate gradients with floats
We also looked at shape rules, datatype conversions, broadcasted scalars, and CNN use cases like dividing filter tensors.
With this foundation, you should be able to leverage tensor divisions effectively in your PyTorch machine learning projects. Check out the official documentation and community forums for more
advanced examples. | {"url":"https://thelinuxcode.com/perform-division-pytorch-tensors/","timestamp":"2024-11-06T14:35:03Z","content_type":"text/html","content_length":"176220","record_id":"<urn:uuid:a51a20c1-e5d2-4635-bf58-b47dc93d81a2>","cc-path":"CC-MAIN-2024-46/segments/1730477027932.70/warc/CC-MAIN-20241106132104-20241106162104-00327.warc.gz"} |
Untersuchungen zur Wechselwirkung von Buckminsterfullerenen mit Siliziumoberflächen und zur Dotierung von Metall/Silizium-Grenzflächen mit Buckminsterfullerenen
The present work investigates the interaction between buckminsterfullerene molecules (C60) and silicon surfaces as well as the influence of buckminsterfullerenes on metal/silicon interfaces. The
knowledge about the mechanism of surface/molecule interaction between the technologically important semiconductor Si and C60 molecules may lead to new applications of fullerenes. In this context the
present study compares the adsorption and the growth mechanisms as well as the desorption of C60 molecules which were evaporated onto well-prepared Si(111)-7x7-, Si(111):H-1x1- and Si(111):Ag-(sqrt
(3) x sqrt(3))R30°-surfaces using Auger electron spectroscopy (AES), low-energy electron diffraction (LEED) under ultra-high vacuum conditions. The electronic structure of C60-covered Si(111)-7x7-,
Si(111)-1x1-, Si(111):H-1x1- and Si(111):Ag-(sqrt(3) x sqrt(3))R30°-surfaces was investigated using ultraviolet and X-ray photoelectron spectroscopy (UPS, XPS). From these experiments the molecule/
surface interaction mechanisms can be identified. Crystalline C60 is a new semiconductor material. Therefore, evaporating C60 onto Si surfaces builds up a semiconductor heterostructure. The
electronic properties of this heterostructure are characterized by the band discontinuities at the C60/Si interface. Using UPS and XPS the valence-band discontinuity at this semiconductor/
semiconductor interface were determined. Additionally, metal/Si contacts were produced on initially clean silicon surfaces which were covered with distinct amounts of C60 before Ag-, Pb- or
Pd-contacts were evaporated. The transport properties of these contacts were studied by current-voltage-chracteristics to determine the influence of C60-layers on the Schottky barrier heights of the
metal/Si contacts. | {"url":"https://duepublico2.uni-due.de/servlets/MCRFileNodeServlet/duepublico_derivate_00005000/index.html","timestamp":"2024-11-11T08:42:59Z","content_type":"text/html","content_length":"7176","record_id":"<urn:uuid:d98637c3-299d-43fb-a99a-c15702e3e6ac>","cc-path":"CC-MAIN-2024-46/segments/1730477028220.42/warc/CC-MAIN-20241111060327-20241111090327-00685.warc.gz"} |
Audulus 4 development thread 🤓
Thought I’d start a thread of things related to Audulus 4 development, for those who are interested. Here you can get a peek into some of the coding decisions
12 Likes
I recently posted some code for review:
1 Like
Hello Taylor,
Thanks for the insight! Although it’s a very foreign language to me, it’s nice to see you open about the process!
3 Likes
Awesome! It’s nice to see what fellow developers are up to. How much of Audulus is written in C++?
5 Likes
@SynthEnthusiast The patch editor UI, and the audio engine are C++. That’s the majority fo the code. The iOS/macOS specific code is almost all Swift.
3 Likes
I’m getting closer to eliminating the pointers up the tree (like from nodes to their patches). Also going to eliminate storing positions and names in the inputs/outputs. This should significantly
reduce the size of the data model in memory, possibly enabling snapshot-based undo/redo (which is simpler than the memento pattern I currently use), and multiple snapshots within a patch.
Right now, some of the bigger patches in the library use over 1mb just for the data model (that’s roughly what’s stored in the file, so not including DSP stuff like delay buffers, which are
pretty big). If you do undo/redo with snapshots, then each snapshot would take over 1mb. If you had an editing session with 500 actions (pretty easy), then there goes half a gig of ram.
Of course I could limit undo/redo history. But perfectionism
Here’s an Apple talk about data models with value semantics, building up to doing undo/redo with snapshots. They also discuss how Photoshop uses snapshots, and shares little image blocks between
snapshots, which is pretty cool.
All that said, I’m not convinced I can get the Audulus memory usage down enough to do snapshots for undo/redo. In the case of Photoshop, much of the data can be shared between snapshots. For
Audulus, which is quite nested and symbolic in nature, it’s harder to do that sort of sharing (notice how in the talk, they only share things that don’t contain other things). I could share all
the nodes, but then I’d have to make an exception for Modules. It could get tricky.
8 Likes
Same here. I’m using memento pattern on my flowchart projects and it’s a memory hog. But I’m coding for a desktop so the memory requirements are low priority.
I’ve never tried this, just an idea. Could you store the initial state in memory and then push only changesets to the undo stack? Like git commits. Then popping off the undo stack is like reverting
the last changeset. Maybe the changeset is a binary difference if you are not storing the state in some kind of JSON like string. Those changesets have got to be less memory and it kind of implicitly
shares as much of the source as it can.
4 Likes
I think that’s a pretty good idea. Audulus stores everything as JSON, which could be diffed (I see that the nlohmann JSON library will do diffing). I’ll look into that further. Thanks for
suggesting it!
I have a goal of having quick random-access undo/redo, which snapshots offer, since you don’t have to apply a potentially large sequence of diffs. Plus we’d get a patch snapshotting feature for
I think the solution to making snapshots memory efficient may be hash-consing. I may also switch to an immutable data model, but as mentioned in that Apple talk, it can make editing operations rather
That “Generic Flyweighting Function” I posted above is my primitive for hash-consing anything with operator<.
3 Likes
Posted this over on Reddit: https://www.reddit.com/r/cpp/comments/cikmhh/implemented_sean_parents_polymorphic_value_types/
3 Likes
Answered this question on Quroa: https://qr.ae/TWvbY2
Please upvote if you are on there, and save people from bad implementations of undo/redo.
4 Likes
A peril of modern life, I’m afraid.
3 Likes
I can’t wait to check out the very first beta version, i wish i could understand C++ but instead can we have a hint of some of the new features ?
3 Likes
@Nomak there are a couple hints above
3 Likes
Quick update: the new code is now running some example patches
There’s a new file format which is much more compact (your existing patches are upgraded to it). It uses Flatbuffers and I will be providing a schema file for those who want to generate patches via
I expect to be able to make this code extremely stable. It’s a very solid foundation for the future of Audulus
9 Likes
Thanks for keeping that an option! Making patches via code is a super fun tool.
3 Likes
Thanks so much for sharing the c++ aspect of audulus, I had to start with the arduino platform and ide, then got deeper into oop (object-oriented programming) with teensy. It’s great to see someone
who really knows how to use it!
2 Likes
It’s been a while since the last update in this thread - how is development going?
3 Likes
Hey @taylor! I hope all is well with you! I was just thinking, and I don’t know if this is in the road map, but I thought it might be really over the top amazing to have a sample node in A4. Maybe
something that would allow you to record sounds that could be triggered by MIDI or (virtual) CV from other nodes, and then be able to shape and sculpt the sounds with the other amazing components of
Audulus that already exist.
I am not sure if this is something that would be easy to implement, if it is not in the roadmap, but I did want to put the suggestion out there, as I think a lot of users would really like this
capability to be a possibility in their modules. Anyway, whatever you decide, I am sure it will be awesome, and I know I will be for sure making the purchase the first day it is available!
1 Like
Update: we’re designing a new collection of core modules. Things are looking great!
9 Likes
I fully agree. In the meantime I highly recommend using the delay module with the loop time sync module from the reface library. I just started to work with that and I think if you have a nice time
synced patch and you want to grab loops and alter them there is a bunch of fun to be had there.
It’s kind of a different way to work, which can just lead to more ideas and approaches. Once you close the patch, obviously you loose the loop. But as long as you are working live, you can just
record your patch and work from there.
I would almost call it subtractive sampling because you could capture some loops with the delay synced module, but then apply envelopes to the volume on a mixer (instead of triggering a sample).
Personally, I think these kinds of workarounds can be productive because my mind gets into a problem solving mode. With that mixer with the mutes and the master clocks with time divisions at hand,
you could probably get interesting results by subtractively introducing sections of a sample.
To put it another way, there is an sense in which this is an approach to sampling:
3 Likes | {"url":"https://forum.audulus.com/t/audulus-4-development-thread/2059","timestamp":"2024-11-09T07:44:23Z","content_type":"text/html","content_length":"54868","record_id":"<urn:uuid:3ef1ddc8-eab5-4128-84d2-1f9ca65ae56d>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.30/warc/CC-MAIN-20241109053958-20241109083958-00881.warc.gz"} |
Handling censored data in Monolix
Handling censored (BLQ) data
Objectives: learn how to handle easily and properly censored data, i.e. data below (resp. above) a lower (resp.upper) limit of quantification (LOQ) or below a limit of detection (LOD).
Projects: censoring1log_project, censoring1_project, censoring2_project, censoring3_project, censoring4_project
Censoring occurs when the value of a measurement or observation is only partially known. For continuous data measurements in the longitudinal context, censoring refers to the values of the
measurements, not the times at which they were taken. For example, the lower limit of detection (LLOD) is the lowest quantity of a substance that can be distinguished from its absence. Therefore, any
time the quantity is below the LLOD, the “observation” is not a measurement but the information that the measured quantity is less than the LLOD. Similarly, in longitudinal studies of viral kinetics,
measurements of the viral load below a certain limit, referred to as the lower limit of quantification (LLOQ), are so low that their reliability is considered suspect. A measuring device can also
have an upper limit of quantification (ULOQ) such that any value above this limit cannot be measured and reported.
As hinted above, censored values are not typically reported as a number, but their existence is known, as well as the type of censoring. Thus, the observation $y^{(r)}_{ij}$ (i.e., what is reported)
is the measurement $y_{ij}$ if not censored, and the type of censoring otherwise.
We usually distinguish between three types of censoring: left, right and interval. In each case, the SAEM algorithm implemented in Monolix properly computes the maximum likelihood estimate of the
population parameters, combining all the information provided by censored and non censored data.
In the presence of censored data, the conditional density function needs to be computed carefully. To cover all three types of censoring (left, right, interval), let $I_{ij}$ be the (finite or
infinite) censoring interval existing for individual i at time $t_{ij}$. Then,
$$\displaystyle p(y^{(r)}|\psi)=\prod_{i=1}^{N}\prod_{j=1}^{n_i}p(y_{ij}|\psi_i)^{1_{y_{ij}\notin I_{ij}}}\mathbb{P}(y_{ij}\in I_{ij}|\psi_i)^{1_{y_{ij}\in I_{ij}}}$$
$$\displaystyle \mathbb{P}(y_{ij}\in I_{ij}|\psi_i)=\int_{I_{ij}} p_{y_{ij}|\psi_i} (u|\psi_i)du$$
We see that if $y_{ij}$ is not censored (i.e. $1_{y_{ij}otin I_{ij}}=1$), its contribution to the likelihood is the usual $p(y_{ij}|\psi_i)$, whereas if it is censored, the contribution is $\mathbb
{P}(y_{ij}\in I_{ij}|\psi_i)$.
For the calculation of the likelihood, this is equivalent to the M3 method in NONMEM when only the CENSORING column is given, and to the M4 method when both a CENSORING column and a LIMIT column are
Censoring definition in a data set
In the dataset format used by Monolix and PKanalix, censored information is included in this way:
• The censored measurement should be in the OBSERVATION column.
• In an additional CENSORING column, put 0 if the observation is not censored, and 1 or – 1 depending if the measurement given in the observation column is a lower or an upper limit.
• Optionally, include a LIMIT column to set the other limit.
To quickly include censoring information to your dataset by using BLQ tags in the observation column, you can use data formatting.
Examples are provided below and here.
PK data below a lower limit of quantification
Left censored data
• censoring1log_project (data = ‘censored1log_data.txt’, model = ‘pklog_model.txt’)
PK data are log-concentration in this example. The limit of quantification of 1.8 mg/l for concentrations becomes log(1.8)=0.588 for log-concentrations. The column of observations (Y) contains either
the LLOQ for data below the limit of quantification (BLQ data) or the measured log-concentrations for non BLQ data. Furthermore, Monolix uses an additional column CENSORING to indicate if an
observation is left censored (CENS=1) or not (CENS=0). In this example, subject 1 has two BLQ data at times 24h and 30h (the measured log-concentrations were below 0.588 at these times):
The plot of individual fits displays BLQ (red band) and non BLQ data (blue dots) together with the predicted log-concentrations (purple line) on the whole time interval:
Notice that the band goes from .8 to -Infinity as no bound has been specified (no LIMIT column was proposed).
For diagnosis plots such as VPC, residuals of observations versus predictions, Monolix samples the BLQ data from the conditional distribution
$$p(y^{BLQ} | y^{non BLQ}, \hat{\psi}, \hat{\theta})$$
where $\hat{\theta}$ and $\hat{\psi}$ are the estimated population and individual parameters. This is done by adding a residual error on top of the prediction, using a truncated normal distribution
to make sure that the simulated BLQ remains within the censored interval. This is the most efficient way to take into account the complete information provided by the data and the model for diagnosis
plots such as VPCs:
A strong bias appears if LLOQ is used instead for the BLQ data (if you choose LOQ instead of simulated in the display frame of the settings) :
Notice that ignoring the BLQ data entails a loss of information as can be seen below (if you choose no in the “Use BLQ” toggle):
As can be seen below, imputed BLQ data is also used for residuals (IWRES on the left) and for observations versus predictions (on the right)
More on these diagnosis plots
Impact of the BLQ in residuals and observations versus predictions plots
A strong bias appears if LLOQ is used instead for the BLQ data for these two diagnosis plots:
while ignoring the BLQ data entails a loss of information:
BLQ predictive checks
The BLQ predictive check is a diagnosis plot that displays the fraction of cumulative BLQ data (blue line) with a 90% prediction interval (blue area).
Interval censored data
• censoring1_project (data = ‘censored1_data.txt’, model = ‘lib:oral1_1cpt_kaVk.txt’)
We use the original concentrations in this project. Then, BLQ data should be treated as interval censored data since a concentration is know to be positive. In other word, a data reported as BLQ data
means that the (non reported) measured concentration is between 0 and 1.8mg/l. The value in the observation column 1.8 indicates the value, the value in the CENSORING column indicates that the value
in the observation column is the upper bound. An additional column LIMIT reports the lower limit of the censored interval (0 in this example):
• if this column is missing, then BLQ data is assumed to be left-censored data that can take any positive and negative value below LLOQ.
• the value of the limit can vary between observations of the same subject.
Monolix will use this additional information to estimate the model parameters properly and to impute the BLQ data for the diagnosis plots.
Plot of individual fits now displays LLOD at 1.8 with a red band when a PK data is censored. We see that the band lower limit is at 0 as defined in the limit column.
PK data below a lower limit of quantification or below a limit of detection
• censoring2_project (data = ‘censored2_data.txt’, model = ‘lib:oral1_1cpt_kaVk.txt’)
PK data below a lower limit of quantification and PD data above an upper limit of quantification
• censoring3_project (data = ‘censored3_data.txt’, model = ‘pkpd_model.txt’)
We work with PK and PD data in this project and assume that the PD data may be right censored and that the upper limit of quantification is ULOQ=90. We use CENS=-1 to indicate that an observation is
right censored. In such case, the PD data can take any value above the upper limit reported in column Y (here the YTYPE column of type OBSERVED ID defines the type of observation, YTYPE=1 and YTYPE=2
are used respectively for PK and PD data):
We can display the cumulative fraction of censored data both for the PK and the PD data (on the left and right respectively):
Combination of interval censored PK and PD data
• censoring4_project (data = ‘censored4_data.txt’, model = ‘pkpd_model.txt’)
We assume in this example
• 2 different censoring intervals(0,1) and (1.2, 1.8) for the PK,
• a censoring interval (80,90) and right censoring (>90) for the PD.
Combining columns CENS, LIMIT and Y allow us to combine efficiently these different censoring processes:
This coding of the data means that, for subject 1,
• PK data is between 0 and 1 at time 30h (second blue frame),
• PK data is between 1.2 and 1.8 at times 0.5h and 24h (first blue frame for time .5h),
• PD data is between 80 and 90 at times 12h and 16h (second green frame for time 12h),
• PD data is above 90 at times 4h and 8h (first green frame for time 4h).
Plot of individual fits for the PK and the PD data displays the different limits of these censoring intervals (PK on the left and PD on the right):
Other diagnosis plots, such as the plot of observations versus predictions, adequately use imputed censored PK and PD data:
Case studies
• 8.case_studies/hiv_project (data = ‘hiv_data.txt’, model = ‘hivLatent_model.txt’)
• 8.case_studies/hcv_project (data = ‘hcv_data.txt’, model = ‘hcvNeumann98_model_latent.txt’) | {"url":"https://monolix.lixoft.com/censoreddata/","timestamp":"2024-11-05T02:13:19Z","content_type":"text/html","content_length":"115310","record_id":"<urn:uuid:12aec717-0221-4177-8df9-e97fc34417e9>","cc-path":"CC-MAIN-2024-46/segments/1730477027870.7/warc/CC-MAIN-20241105021014-20241105051014-00764.warc.gz"} |
an express riddle | R-bloggersan express riddle
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A quick puzzle on The Riddler this week that enjoys a quick solution once one writes it out. The core of the puzzle is about finding the average number of draws one need to empty a population of size
T if each draw is uniform over the remaining number of individuals between one and the number that remain. It is indeed easy to see that this average satisfies
$\epsilon^T=1+\frac{1}{T}\sum_{i=1}^{T-1} \epsilon^i$
since all draws but one require an extra draw. A recursion then leads by elimination to deduce that
which is the beginning of the (divergent) harmonic series. In the case T=30, the solution is (almost) equal to 4.
> sum(1/(1:30))*1e10
[1] 39949871309
A second riddle the same week reminded me of a result in Devroye’s Non-Uniform Random Variate Generation, namely to find the average number of draws from a Uniform until the sequence goes down.
Actually, the real riddle operates with a finite support Uniform, but I find the solution with the continuous Uniform more elegant. And it only took a few metro stops to solve. The solution goes as
follows: the probability to stop after two Uniform draws is 1/2, after n uniform draws, it is (n-1)/n!, which does sum up to 1:
$\sum_{n=2}^\infty \frac{n-1}{n!} = \sum_{n=2}^\infty \frac{n}{n!} - \sum_{n=2}^\infty \frac{1}{n!} = \sum_{n=1}^\infty \frac{1}{n!} - \sum_{n=2}^\infty \frac{1}{n!}=1$
and the expectation of this distribution is e-1 by a very similar argument, as can be checked by a rudimentary Monte Carlo experiment
> over(1e7) #my implementation of the puzzle
[1] 1.7185152
Filed under:
harmonic series
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Topological Symmetry Transition between Toroidal and Klein Bottle Graphenic Systems
Topological Symmetry Transition between Toroidal and Klein Bottle Graphenic Systems^ †
Laboratory of Structural and Computational Physical-Chemistry for Nanosciences and QSAR, Biology-Chemistry Department, Faculty of Chemistry, Biology, Geography, West University of Timisoara, Str.
Pestalozzi No. 16, 300115 Timisoara, Romania
Laboratory of Renewable Energies-Photovoltaics, R&D National Institute for Electrochemistry and Condensed Matter, Dr. A. Paunescu Podeanu Str. No. 144, RO-300569 Timisoara, Romania
Actinium Chemical Research Institute, Via Casilina 1626/A, 00133 Rome, Italy
Authors to whom correspondence should be addressed.
Paper dedicated in the honor memory of Prof.dr. Mircea V. DIUDEA, a common true friend and an international scholar in mathematical chemistry contributing the nano-topology architecture in special.
Submission received: 15 May 2020 / Revised: 19 June 2020 / Accepted: 7 July 2020 / Published: 27 July 2020
In the current study, distance-based topological invariants, namely the Wiener number and the topological roundness index, were computed for graphenic tori and Klein bottles (named toroidal and Klein
bottle fullerenes or polyhexes in the pre-graphene literature) described as closed graphs with N vertices and 3N/2 edges, with N depending on the variable length of the cylindrical edge L[C] of these
nano-structures, which have a constant length L[M] of the Möbius zigzag edge. The presented results show that Klein bottle cubic graphs are topologically indistinguishable from toroidal lattices with
the same size (N, L[C], L[M]) over a certain threshold size L[C]. Both nano-structures share the same values of the topological indices that measure graph compactness and roundness, two key
topological properties that largely influence lattice stability. Moreover, this newly conjectured topological similarity between the two kinds of graphs transfers the translation invariance typical
of the graphenic tori to the Klein bottle polyhexes with size L[C] ≥ L[C], making these graphs vertex transitive. This means that a traveler jumping on the nodes of these Klein bottle fullerenes is
no longer able to distinguish among them by only measuring the chemical distances. This size-induced symmetry transition for Klein bottle cubic graphs represents a relevant topological effect
influencing the electronic properties and the theoretical chemical stability of these two families of graphenic nano-systems. The present finding, nonetheless, provides an original argument, with
potential future applications, that physical unification theory is possible, starting surprisingly from the nano-chemical topological graphenic space; thus, speculative hypotheses may be drawn,
particularly relating to the computational topological unification (that is, complexification) of the quantum many-worlds picture (according to Everett’s theory) with the space-curvature sphericity/
roundness of general relativity, as is also currently advocated by Wolfram’s language unification of matter-physical phenomenology.
1. Introduction
In 1882, German mathematician Felix Klein (1849–1925) introduced his peculiar “fläshe” (surface), the Klein bottle (KB), several decades following the discovery of the Möbius strip (M) independently
made by August Ferdinand Möbius and Johann Listing. These two non-orientable surfaces, with genus 1 (M) and 2 (KB), are deeply interrelated: the Klein surface is obtainable by closing in a
cylindrical way the open edges of a Möbius strip. This mechanism explains why the Klein bottle is often defined as a surface built by “sewing two Möbius strips together” [
] in which the Möbius-halves have the opposite chirality.
The next logical step, i.e., connecting a Möbius-like structure to the open edges of the Möbius strip, leads to the construction of the projective plane, a surface studied by Klein himself in 1874.
Reference [
] provides illustrations of the various forms of interplay among different surfaces (
Figure 1
Due to repetition, the word “flasche” (bottle) rapidly came into use, and the surface has since been universally known as the Klein bottle. Indubitably, the KB is a highly attractive topological
structure, not only for theoretical physicists or mathematicians [
] investigating its unique topological features, but also for an increasing number of architects and artists who consider the zero-volume bottle an authentic topological marvel and a source of
endless creative inspiration. For those willing to understand the mechanisms that transform a simple cylinder into the popular “classical inverted sock” Klein bottle, a helpful pictorial explanation
can be found in [
KBs are non-orientable surfaces, with a cross-cap number of 2 and an edge number of 0. They can be easily constructed from Möbius bands (for simplicity, only half-twisted Möbius joints will be
considered herewith), connecting the remaining open edges in a “parallel” manner, i.e., performing the same operation involved in transforming a rectangle into a cylinder until the structure is
completely glued to a one-sided surface. It is well known that both tori and KBs, having Euler characteristics equal to 0, may be covered with hexagonal tiles only and without the need for pentagons.
They represent, for this reason, relevant examples of polyhexes, often referred to in the literature as toroidal and Klein bottle fullerenes. In this graphene era, our preference in this article is
to name these surfaces paved with only hexagonal six-rings, graphenic nano-structures. The creation of polygonal faces with a number of edges other than six may be achieved using Stone–Wales (SW)
rotations on the surface of these polyhexes to generate pentagon–heptagon pairs or other
-rings with
≠ 6 by simple iterations of multiple SW rearrangements. Various isomeric SW mechanisms to create and propagate innumerous kinds of topological defects in graphenic toroidal lattices have appeared in
recent literature [
], and it is important to mention that these are all applicable to the case of graphenic KBs.
Furthermore, these nano-topo structures are related because the nano-surface’s dynamics respect the graphenic projective plan; as for any nano-variety dynamics, they may be also related with quantum
evolution, inherently at the nano-level. In this respect, although having as the common quantum nature the nano-confinement structure of space, the molecular topology (more precisely, chemical graph
theory) and nano-architectures in general are sparsely connected with quantum information (e.g., [
]). As such, the classic quote of the quantum physicist Richard Feynman, that “there’s plenty of room [i.e., enough space] at the bottom”, has remained largely unexplored from the perspective of a
quantum mechanical interpretation. Paradoxically, the Klein bottle structure was identified as having mathematical links with toroid systems, at least at their 3D immersing level, and with the
quantum mechanical paradigm through the quaternion 4D representation, but not in a straight quantum (relativistic) dynamical interpretation, with the exception of some inputs in the quantum
frontier-related literature (see the
Appendix A
). The present topographic endeavor may provide further insights into such quantum dynamics in the context of the significant interest at present in the potential future development of a unified
science of nano-space shapes with consequences for mathematics (unitary manifold space variation), physics (quantum and relativistic unification by space shape dynamics), and chemistry (unification
of chemical bond and bonding in extended nano-carbon spaces).
Methodologically, in the next section, we provide details about the construction of toroidal and Klein bottle fullerenes made with given numbers of hexagons h, atoms N, and bonds B. An introduction
to topological distance-based descriptors (also including detailed calculations for small polyhexes) is also provided, and the general results derived for both graphs for large N are presented.
Topological similarities between toroidal and Klein bottle graphenic nano-systems are discussed, and the article concludes with preliminary indications about the relative chemical stability of these
hexagonal systems. The graphenic $G x , y$ nano-structures considered here have a zigzag Möbius edge parallel to x with dangling bonds closed across the direction y. The results and their
topo-quantum dynamical interpretation are provided.
2. Method: Topological Invariants for Polyhex Graphs
This section starts with a description of the method used to build the chemical graphs of the two graphenic nanostructures, tori and KBs. Then, distance-based topological invariants, such as the
Wiener numbers W and topological roundness $ρ$, are computed for some of these graphs.
In the current paper, we do not use the distorted parallelogram mesh often seen in the literature to build the polyhex graphs; instead, we adopt the quadrangular honeycomb mesh. The graphs shown in
Figure 2
Figure 3
Figure 4
are constructed by the usual [
-shaped unit cell made by four atoms—1, 2, 3, and 4—and which is translated
$L M$
times along
$L C$
times in y, where the integers
indicate, respectively, the lengths of the Möbius-like edge (subject to the antiparallel sewing) and the cylindrical edge (parallel sewing), both expressed as the number of hexagons. The graphenic
open lattice
$G L M , L C$
is made with
$L C$
rows with
$L M$
hexagons. The number
of total hexagons in the
$L C$
belts is then
$h = L M × L C$
. By closing the edges, the numbers of atoms
and bonds
in the graph are:
N $= 4 h$
$B = 3 N 2 = 6 h$
The hexagon-tiled graphenic plane shows two sides and one edge. By changing the boundary conditions, the torus $T L M , L C$ and the Klein bottle $KB L M , L C$ have, respectively, numbers of sides
(edges) equal to 2 (0) and 1 (0).
Here we present some practical examples.
Figure 2
shows the
=1 honeycomb network. This graphenic lattice may be closed in two distinct topological manners, forming the torus
or the Klein bottle
$h = 3 , N = 12 , and B = 18$
. In both cases, the
hexagons are closed in parallel along
. On the other edge, the dangling bonds belonging to the
hexagons are closed in parallel or antiparallel along
to form the
torus or the
Klein bottle.
Figure 3
shows more detail of how the two closed structures
are built. The populations over the coordination shells change in a significant way depending on which of the two structures is considered.
Indicating with
$b i k$
the number of
- neighbors of the vertex
in the graph, a quick calculation shows that all nodes in
have the same set {
$b i k$
} = {3,4,3,1}, with one node
in the fourth coordination shell at the maximum distance
= 4. The maximum distance from
determines the node eccentricity
$ε i = 4$
. The Klein bottle
shows some “toroidal” nodes with the same {
$b i k$
} set (namely the vertices 2, 3, 5, and 8), and the remaining nodes have a reduced eccentricity
$ε i = 3$
and {
$b i k$
} = {3,5,3}. For example, vertex 1 in
Figure 2
has three neighbors—4, 8, and 10—in the third coordination shell but zero nodes in the fourth shell.
This effect of shortening the maximum eccentricity
$M = max { ε i }$
of the graph (also called the graph diameter) is the topological signature of the antiparallel sewing acting on the dangling bonds placed along the zigzag Möbius edge. The transmission
is defined in such a way:
$w i = 1 2 ∑ k = 1 M k b i k$
and contributes to the Wiener index sum:
The invariant
represents a topological measure of the overall compactness of the graphenic structures. One can easily calculate
$W ( T 3 , 1 ) =$
144 and
$W ( K B 3 , 1 ) =$
136. Therefore, the (
$3 , 1 )$
Klein bottle results in a more compact structure if compared with the nano-torus with equal edges. The topological invariant
$ρ E$
, called extreme topological efficiency [
] or extreme topological roundness [
], is defined as the ratio between the extreme value in the transmission set of a given graph:
$ρ E = max { w i } min { w i }$
The invariant of Equation (3) is able to select stable structures such that the cases of fullerenes C
or C
] tend to minimize the
$ρ E$
topological descriptor. Tori
$T L M , L C$
$ρ E = 1$
, which is different from the situation for the Klein bottles. In our example,
$ρ E ( K B 3 , 1 ) = 12 11 > ρ E ( T 3 , 1 )$
, evidencing the greater topological stability of the torus.
Our topological modeling approach is based on the minimization of the topological invariants promoting the most compact (minimizing W values) and most round (minimizing $ρ E$ values) structures as
the most stable systems among a given set of isomers. This paper studies the effects of the topology in influencing the relative stability of the two isomeric structures, tori and Klein bottles,
based on the same graphenic plane $G L M , L C$, in particular, the variation of the {$b i k$} sets following the different closures of the zigzag Möbius edge. We conclude the article by reporting an
original size-dependence effect that, for certain circumstances, makes tori and KBs topologically indistinguishable.
Table 1
lists the values of the invariants for the cases in which the length of the zigzag Möbius edge triples the other edge
$L M$
$L C = 3 L M$[.]
This is an evidently arbitrary choice intended only to represent all cases with
$L C < L M − 1$
Figure 3
illustrates the topological analysis of the two graphs
$T 3 l , l$
$K B 3 l , l$
= 2,
= 48. All
nodes of the torus show the same transmission (1) value
$w ¯ = 98$
= 8. The interesting (and expected) fact is that eccentricities of most of the nodes of
$K B 6 , 2$
are smaller than the graph diameter
. This reduction of the eccentricity is a clear effect induced by the Möbius-like sewing of the zigzag edge. The
$K B 6 , 2$
diameter is still
= 8, but this value only concerns the few nodes circled in black in
Figure 3
b. All the remaining nodes on the surface of the KB have eccentricities equal to
$M – 1$
, suggesting the existence of a general topological effect that we call eccentricity shrinkage, Equation (4b). Eccentricity shrinkage is a topological phenomenon that has been previously reported [
] for Möbius 1D graphenic (open) strips and has the overall topological effect of making
$KB L M , L C$
topologically more compact then
$T L M , L C$
$L C < L M − 1$
. For the polyhexes in
Figure 3
, the Wiener index of Equation (2) computed values
) = 4504 and
) = 4704 confirm that
possesses an augmented compactness over
, that is,
< W
). It is worth noting that
$= N w ¯$
. The Klein bottle fullerene
$K B 6 , 2$
has a broader range of topological transmission values, from
= 91 to
$w ¯ = 98$
Table 1
details the list of topological descriptors for each node of the
$K B 6 , 2$
graph, including eccentricity and {
$b i k$
} sets; descriptors for
$T 6 , 2$
are also provided for completeness (vertices are labeled as shown in
Figure 3
). Topological descriptors group the
= 48 nodes of the
$KB 6 , 2$
graph into four classes of topological equivalent vertices, with the multiplicity given in brackets in the first column of
Table 1
. The
C NMR resonance spectrum of the carbon C
nanostructure has such a Klein bottle topology made of four peaks with relative intensities 1, 1, 2, and 2. We end these considerations with
Table 2
, which shows the eccentricity shrinkage in passing from
$T 3 l , l$
$K B 3 l , l$
graphs for
= 1, 2,…, 9, 10.
With the examples above, we showed the existence of the general mechanism of the topological effect we have called eccentricity shrinkage, which is a peculiar outcome of the current study; see
Equation (4b).
This mechanism implies that, by indicating with
$ε _$
the minimum value of the eccentricities of the Klein bottle
$K B L M , L C$
graph and with the symbol
$ε ¯$
the eccentricity of the
$T L M , L C$
torus with the same pair of edges
$L M , L C$
, one can find:
When the eccentricity shrinkage mechanism holds, the inequality is strictly respected:
The main result of the present study is summarized in the following novel conjecture concerning toroidal and Klein bottle fullerenes $K B L M , L C$ and $T L M , L C$:
Polyhex similarity conjecture. For a given integer value L[M], the shrinkage of the topological eccentricity does not hold over the threshold L[C] ≥ Ł[C] with Ł[C] = L[M] − 1.
Formally, we then have:
in the range:
Thus, for polyhexes $K B L M , L C ≥ Ł C$ and $T L M , L C ≥ Ł C$, the equality $ε _$$= ε ¯$ holds. By contrast, for $K B L M , L C < Ł C$ and $T L M , L C < Ł C$, the inequality $ε _$$< ε ¯$ holds
strictly, making Klein bottle fullerenes topologically more compact than the toroidal ones.
The next section is devoted to the explanation of the above main result (5a), i.e., the existence of a critical size L[C] that makes toroidal and Klein bottle polyhexes topologically
indistinguishable and vertex transitive.
3. Results: Topological Similarities between Toroidal and Klein Bottle Polyhexes
This section aims to illustrate the above polyhex similarity conjecture using examples extracted by the numerical investigations supporting it. We start by recalling the data reported in
Table 1
for the two toroidal and Klein bottle polyhexes shown in
Figure 3
. The eccentricity
$ε ¯ = 8$
of the torus
$T 6 , 2$
is the upper limit of the
$K B 6 , 2$
${ ε i }$
= {7,8} with the minimum value
$ε _ = 7$
Figure 4
enables us to provide more computational details about the simple system
, that is, a graph with
= 3, and
= 2 with
= 2, fully compatible with the conditions of Equation (5), leading to the topological similarity with the torus T
. We start with the graphenic open lattice
$G 3 , 2$
made by
= 24 vertices, and we first close the
edge in the usual cylindrical manner, with, for example, node 14 bonding vertices 13, 15, and 21, and so on. We know from the Klein bottle
represented in
Figure 2
that some nodes (such as vertex 4) show
$ε _$$= 3$
shrunk eccentricity with respect to the node of the torus
, having eccentricity
$ε ¯ = 4$
. In particular, vertex 4 in the Klein bottle has three vertices in the maximum distance coordination shell
= 3. By contrast, in the torus
, the same vertex still has three nodes in the third coordination shell
= 3 but also one node 10 at the maximum distance
= 4 =
$ε ¯$
. When the graphenic open lattice
$G L M , L C ≥ Ł C$
is built respecting the conditions of Equations (5b,5c) relating to the sizes of the two edges, the topological distances of the toroidal and Klein bottle polyhexes will change in such a way that all
the nodes of both graphs show the same eccentricity
$ε _ ¯$
$ε ¯$
and the same set of coordination numbers {
} with
= 1,2,..,
$ε _$
. Klein bottle polyhexes
$K B L M , L C ≥ Ł C$
are made by the vertex transitive graph with the same invariants (1,2,3) shown by the nodes of the
$T L M , L C ≥ Ł C$
The following numerical test focuses again on node 4 of the
torus and
graphs in
Figure 4
. For the torus, the three nodes with distances equal to 4 are vertices 10, 16, 20, 22, and 24, and one node is at distance
= 5 =
$ε ¯$
. In the Klein bottle
the antiparallel closure makes node 23 closer to node 4, at
= 3, with four nodes in the fourth coordination shell—10, 16, 18, 20, and 24—and one node at the maximum distance
= 5. This example fortifies the conjectured similarity among tori and KBs when
Table 3
shows a larger numerical test confirming the main result of Equation (5a) of the current work and therefore supports the existence of a critical size
− 1 of Equation (5c) making toroidal and Klein bottle polyhexes topologically indistinguishable. For the two families of graphs
$KB L M , L C$
= 5,6 and
= 1, 2, …; the eccentricity {
$ϵ i$
} values are compared with the eccentricity
$ε ¯$
of the isomeric tori. In both cases, the topological shrinkage of the eccentricity
$ε _ < ε ¯$
takes place in the region
, confirming the outcome of the current study. We also note that
$ε ¯$
$ε _$
= 2
in both cases when
= 5,6.
The current results show that the Klein bottle $KB L M , L C$ cubic graphs are topologically indistinguishable from the toroidal lattices $T L M , L C$ with the same size (N, L[C], L[M]) when L[C] ≥
Ł[C]; see Equation (5c).
This result implies that an observer sitting on a KB node sees (i) the same coordination numbers{b[k]} that characterize the vertices of the isomeric torus $T L M , L C$; (ii) that the Klein bottle
graph has become vertex transitive with all nodes sharing the same {b[k]}.
A graph traveler arriving at the nodes of the Klein bottle fullerenes is no longer able to distinguish among them nor to determine if they are staying on a torus by simply measuring the chemical
distances. This size-induced symmetry transition for Klein bottle cubic graphs represents a relevant topological effect with deep influences on the electronic properties of these theoretical
graphenic nanosystems. Such intriguing topological phenomena may relate to the formation and propagation of the quasi-quantum bosonic particle called the bondon [
], carrying certain topological chemical bonding information along a chemical structure—with more thermochemical observable effects when it relates to extended graphenic systems [
]. Bondonic theory may also eventually explain the homologies of chemical structures, as proven by the aid of the symmetry breaking mechanism, thus balancing the fermionic vs. bosonic (in)formation
with direct influence on the aromatic or reactivity degree of a certain compound [
We end this section with some preliminary considerations concerning the chemical stability of a graphenic structure having the Klein bottle topology. Pioneering work on this subject may be found in a
previous study [
]. Here, we report two concurrent topological characteristics that make the Klein bottle
$KB L M , L C$
a good candidate for being a stable and synthesizable carbon nanostructure.
The above statement is based on the two main outcomes of the present study:
: in this region, the eccentricity shrinkage of Equation (4b) holds and the Klein bottle is therefore more compact than the toroidal graphenic structure with equal size. The Wiener indices (2) of
the Klein bottles are lower than those of the tori; see
Table 3
: in this region, the conjectured topological similarity of Equation (5a) holds and the Klein bottle thus becomes topologically equivalent to the isomeric toroidal polyhex sharing the same values
$ρ E$
; see
Figure 5
The graphenic tori are a useful representation of the graphene plane with periodic boundary conditions imposed along the directions of both
of the honeycomb plane. Graphene has been recognized for the past 16 years as a stable carbon allotrope [
]. According to the present findings, both nano-structures show, for
, the same values of graph compactness and roundness (i.e., similarity conjecture), which are two key topological properties that largely influence the stability of the systems [
This fact suggests the possibility that Klein bottle systems can be formed because of their topological stability, which is comparable to that of the toroidal polyhexes. In the future, more detailed
investigations will be devoted to determining under which conditions the Klein bottle fullerenes will be able to be produced in nature or in laboratories.
Remarkably, the present findings related to the conditions of the topological equivalence of the Klein bottle and torus may suggest another potentially useful interpretation of quantum mechanics at
present, particularly when related to the space–time structural dynamics. The present endeavor offers a workable graphene-related nano-topological argument in favor of Everett’s theory of many worlds
]. Precisely, it suggests a certain degree of complexification, i.e., over a critical size
− 1 of Equation (5c), the toroidal and Klein bottle polyhexes are topologically indistinguishable. Thus, they coexist, being superimposed or topologically entangled, in terms of quantum information
theory. Accordingly, the two graphenic nano-systems may be quantum interrelated by the inter-correlated (Klein bottle, KB, and tori, T) wave functions, for example, by the entangled wave functions’
symmetry transition:
∫Ψ[KB]({x})Ψ[T]({y − λ[T]E[KB]t[+]})dτ = ∫Ψ[KB]({x − λ[KB]E[T]t_})Ψ[T]({y})dτ
In Equation (6), the two topological structures mutually measure each other in an entangled state with energies E[KB] and E[T] as the forward and reversal times of t[+] and t_ flows, and with the
respective ordering (i.e., observably related) of parameters of λ[T] and λ[KB]. The future challenge will be to properly assess the topological parameters of the manifold observable many-world
physical factors (e.g., establishing the degree to which the nano-structural and topological critical extended parameter L[C] accounts for the forward and reversal time, in a topo-quantum space–time
unification approach, or the degree to which the topological roundness ρ[E] influences the observable (e.g., thermochemical, nuclear, magnetic, and electronic) parameters through the order parameters
λ[KB,T], and so on).
Moreover, with the further exploration of other topological manifolds that are derived or transformed from the graphenic projective surface, one will eventually establish the graphenic
branching-states as the true dynamic quantum space in which the entire nano-world coexists at “the bottom of the world”, according to the famous terms of Richard Feynman’s prediction, yet with the
“curvature/potential” emerging into the observable and peculiarly controlled nano-chemical synthesizable structures.
4. Conclusions
By varying the topology of the polyhexes, from tori to Klein bottles, we observed the general mechanism that we called the eccentricity shrinkage consisting of the relationship $ε _ < ε ¯$ between
the eccentricities of tori $ε ¯$ and Klein bottles $ε _$. Moreover, a newly conjectured topological symmetry between the two kinds of graphs occurs when the critical size L[C] ≥ Ł[C] is imposed on
the systems. In particular, under this condition, the translation invariance typical of the graphenic tori also becomes a topological property of the Klein bottle polyhexes, thus making these Klein
bottle graphs vertex transitive, and suggesting that they may have the same topological stability of graphene. Therefore, they may be synthesizable as actual chemical structures. This is an
intriguing possibility that is worthy of further theoretical and experimental (quantum) investigation.
Furthermore, as recently launched, Wolfram’s computational unification of curved space with quantum theory with the aid of topological rolled and curved surfaces and particles (i.e., wave-packaging;
equivalence of space curvature; and the topologically allied indices of torsion, time–space curvature, acceleration, bifurcations, topological evolving manifolds, many-worlds superposition and
interference, and entanglement) offers a mathematically and computationally wider framework for the present topological Klein bottle/tori equivalence approach [
]. As such, the so-called (as abstracted from Wolfram’s writings) branchial space, branchial motion, entanglement horizon, or maximal speed of motion (that is,
, with respect to the maximum topological eccentricity of a given graphenic nano-space) in branchial space is eventually related to the quantum Zeno effect. A generalization of Einstein’s space
curvature invariance
(with respect to the sphericity index of a given graphenic nano-space) occurs in conjunction with the multiway causal graph or the space–time curvature relating to the uncertainty quantum principle.
All such tools for the geometrization of the complexity space (of the cosmos, nature, and nano-materials) may ultimately result in the nano-topo unification of the quantum and relativity theories
(that is, both special and general) through the manifolds of many-world (viz. quantum), space–time coupled (or entangled) dynamics (see
Appendix A
). It is nevertheless true that nano-topologies in general and those specifically based on or derived from a graphenic nano-projective space may offer a computational experimental space for
simulating and understanding (at the ontological level) the micro-folded and macro-unfolded universes. Such endeavors should continue to be necessary in the coming years.
Author Contributions
M.V.P. and O.O. established the conceptual framework, discussion, and conclusions, and assembled the paper; O.O. performed the calculations; and M.V.P. conducted the literature screening and
contributed the quantum-relativity topological interpretation. All authors have read and agreed to the published version of the manuscript.
Mihai V. Putz acknowledges his contribution to this work within the Nucleus-Programme under the project PN-19-22-01-02 and its 2020 renewal as funded by the Romanian Ministry of Education and
Research. Ottorino Ori is a permanent fellow of the Laboratory of Structural and Computational Physical-Chemistry for Nanosciences and QSAR, Biology-Chemistry Department, Faculty of Chemistry,
Biology, Geography, West University of Timisoara, as well as of the Laboratory of Renewable Energies-Photovoltaics, R&D National Institute for Electrochemistry and Condensed Matter, Timișoara,
Conflicts of Interest
The authors declare no conflict of interest.
Appendix A. On Topological and Quantum Coverings of Nano-Space
The coverage of nano-space by carbon allotropes has been a constant source of fascination, in terms of both topological and chemical synthesis. During the past 50 years, significant milestones have
been represented by the discoveries (via experimental evidence and structural characterization) of fullerene [
] and graphene [
]. Accordingly, novel nano-architecture spaces have emerged in single-walled nanotubes (SWNTs) [
]. The coalescence reactions of SWNTs [
] are considered classical examples of nano-chemistry. Furthermore, from an exotic perspective, foam-like carbon structures related to schwarzite [
] represent infinite periodic minimal surfaces of negative curvature containing polygons of size larger than hexagons (in comparison to graphite) that induce a negative curvature. Units of such
structures appear in nanotube junctions, produced in an electron beam [
], with wide potential applications in molecular electronics [
]. In addition, although carbon nano-chemistry has resulted in significant technological and experimental advances, a wide range of challenges remain to be conquered. However, nano-carbon allotropes
have revealed experimental surfaces that allow unique and innovative connection with mathematical and topological experiments [
]—that is, employing the mathematical laboratory in the analysis of examples, testing of ideas, or search for new patterns. In this context, a plethora of theoretical models have been advanced either
for already existing nanostructures (as outlined previously) or experimentally designed new nanostructures, in a mathematical-topological manner. For instance, the first fullerene computational
modeling was undertaken nearly three decades ago by topological means (i.e., chemical graphs) [
] and subsequently advanced to renewed means [
]. These have been accompanied by systematic studies of the embedding of the hexagonal trivalent (6,3) net; that is, the drawing of a graph on a (closed) surface with no crossed lines (cf., the
Schäfli notation). In the torus or cylinder, this is predominantly achieved by the well-known method of graphite zone-folding; see
Figure A1
, top [
]. More recent developments include density and functional theory-based investigations [
], and studies of the negative curvature of nano-architectures [
]. These latter developments provide an explanation for natural micro-pores with low densities appearing in natural materials as zeolites. In particular, such pores can be simulated, e.g., by
structures tessellated entirely by heptagons (e.g., the Klein tessellation) embedded into infinite periodic surfaces of negative curvature of genus >1. At this point, one arrives at the so-called
Smale’s paradox of topology, which represents, in dynamical space, spheres and tori everted (i.e., turned inside out) by smooth transformations. The paradox expresses that fact that, although the
direct surface is an orientable one, the everted one means that the act of turning the surface inside-out implies that the half-way surface is non-orientable in any instance. An object that
originates from such an operation is the Klein bottle (
Figure A1
, bottom) [
Figure A1.
Upper: The (6,3) covering the toroidal and cylindrical embedding, respectively. (
) (6,3)H/Z [
]; N = 600; (
) Tu(6,3)A[12,12]; N = 144 [
]. Bottom: The Klein bottle 3D embedding representations. (
) The so called “Figure 8 immersion”; (
) The “Möbius strip(s) cutting(s)” contained in the Klein bottle’s gluing diagram—allowing its 3D representation by gluing both pairs of opposite edges, giving one pair a half-twist [
]; see the text for details.
In addition to having no direct 3D realization, this appears in the 3D embedded forms of
Figure A1
with a 2D surface carrying neither a global surface outside nor a global surface inside. Moreover, it transcends the ordinary topological half-way surface transformation, with no such transformation
occurring in the case of the Klein bottle, which is the only remaining case related to holonic transformation. At this point, the topology should be linked with quantum mechanics rather than ordinary
mechanics, while allowing holonic transformation, such as the theory of hidden variables (that is, topological parameters in the current paper) prescribes and is currently advocated [
]. A useful exercise for such topo-quantum links employs the Klein bottle’s “Möbius strip(s) cuttings” representation of
Figure A1
(right). To it, one can assign the quaternion vector
with the following coordinates: the inside-out position (by “a”, e.g., a = 0,1, via quantum logic and, thus, the quantum information approach); position
] along the length (
) of the Klein bottle diagram; position
] along the height (J) of the Klein bottle diagram; and “the 4D completeness information”
associated with the in-point-assignment (e.g., by 4D rotation algebra space or by Pauli matrices’ spin algebra space) of any information contained therein, associated with the gluing diagram. Once
the algebraic assignment is completed, one can pursue a finite-dimensional quaternionic quantum formalism for the description of quantum states, quantum channels, and quantum measurements [
]. In addition, the present results show that the next quantum frontier experiments should involve the nano-space dynamics of nano-tori and Klein bottle graphenic structures, and their interconnected
and entangled multi-spaces; that is, Equation (6) of the main text. This corresponds with mixing topological mathematics with the theories of quantum mechanical interpretation and observations, and
particularly the theory of multi-verses [
]. Furthermore, the current context is a mathematical and computational movement aimed at unifying the big two physical theories—namely, quantum mechanics and general relativity—in terms of space
mathematics and computational confinement and dynamics, as we referred to in the recently launched project of Wolfram Research Company; see [
Thus, it has been recently realized that the curved space may fit well with the associated space wave-function and, in turn, is related to the structural architecture of space. Furthermore, at the
macrocosm level, such an elemental structure of space has been revealed (that is, proof of gravitational waves, presumably associated with gravitons) [
]. At the nano-cosm level, the chemical bond is far closer to the observation, measurement, experiment synthesis, and control. Nonetheless, the examination of a consistent quantum-relativistic
multi-verse theory based on Klein bottle space dynamics continues in the macro-space [
]. The present work suggests that this theory is possible via nano-space shape dynamics, by the tori and Klein bottle curved degeneracies, which are both related to the graphenic (groundstate)
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Figure 1.
So-called “graphenic nano-structures”: (
) the graphene graph as a 2D structure “with boundary”; (
) the torus variety; and (
) the Klein bottle variety, as the 2D manifolds without a boundary; see the text for more details regarding the Möbius strip, see [
Figure 2. Polyhex lattice with L[M] = 3, L[C] = 1. The unit cell is the usual C-shaped cell consisting of the four atoms: 1, 2, 3, and 4. This graphenic lattice G[3,1] has open bonds along y that are
closed in a parallel manner along x forming a cylinder: node 2 bounds 9, and 3 bounds 12. The graphenic torus T[3,1] has the open bonds along x closed in a parallel manner along y: 1–4, 5–8, 9–12. By
sewing the open bonds along x in the antiparallel manner (1–12, 5–8, 4–9), the Klein bottle KB[3,1] is built. Nodes 1, 4, 6, 7, 9, 10, 11, and 12 are topologically equivalent to {b[k]} = {3,5,3},
with shrunk eccentricity $ε ¯$ = 3; the remaining vertices 2, 3, 5, and 8 share the coordination numbers {b[k]} = {3,4,3,1} and eccentricity $ε ¯ = 4$ of the T[3,1] nodes.
Figure 3. Polyhex lattices with L[M]6, L[C] = 2. (A) The toroidal fullerene T[(6,2)] is made of N = 48 equivalent nodes with $w ¯ = 98$. Edges are closed to form the torus by sewing the balls with
the same color in a parallel manner; (B) the Klein bottle KB[(6,2)] built by gluing, in an antiparallel manner, the nodes of the zigzag edge, such as 25–24, 29–20, ..., 45–4. Black circles have the
same transmission of the torus’ vertices, whereas the remaining circles have lower shrunken values, making KB[(6,2)] topologically more compact than T[(6,2)].
Figure 4. KB[3,2] graph with L[M] = 3, L[C] = 2. Open bonds along x are sewn in the antiparallel manner (13–12, 17–8, 21–4); open bonds along y are close, in parallel. This Klein bottle is formed by
N = 24 equivalent nodes with {$b k$} = {3,6,8,5,1}, the same set shown by all nodes of the T[3,2] torus.
Figure 5. For L[M] = 5, Ł[C] = 4, and L[C] = 1, 2, …, 5, the topological roundness $ρ E$ is plotted for both $KB L M , L C$ and $T L M , L C$ isomeric graphs; when L[C] ≥ Ł[C], the Klein bottle
graphs are topologically equivalent to the tori $T L M , L C$.
Table 1. Topological classes for the vertices of the graph $K B 6 , 2$, including eccentricity $ϵ i$, transmission $w i$, {$b i k$} sets, Wiener number W, and extreme topological efficiency $ρ E$;
multiplicity is given in brackets. $T 6 , 2$ descriptors hold for all nodes of the toroidal polyhex and coincide with the last class of $K B 6 , 2$.
$K B 6 , 2$ W = 4504; $ρ E$ = 98/91 = 1.0769
V {$b i k$} $ϵ i$ $w i$
(8) v5 v8 v17 v20 v29 v32 v41 v44 3 6 9 12 11 5 1 7 91
(16) v6 v7 v10 v11 v18 v19 v22 v23 v30 v31 v34 v35 v42 v43 v46 v47 3 6 9 11 11 6 1 7 92
(16) v1 v4 v9 v12 v13 v16 v21 v24 v25 v28 v33 v36 v37 v40 v45 v48 3 6 9 10 9 7 3 7 95
(8) v2 v3 v14 v15 v26 v27 v38 v39 3 6 9 9 8 7 4 1 8 98
$T 6 , 2$ W = 4704; $ρ E$ = 1
(48) v1, v2, …, v47, v48 3 6 9 9 8 7 4 1 8 98
Table 2. For the family of graphs $K B 3 l , l$, the eccentricity {$ϵ i$} values are compared to the tori $T 3 l , l$ eccentricity $ε ¯$ for $l$ = 1,2,..,10. Indicating with $ε _$= min{$ϵ i }$, the
topological shrinkage $ε _ < ε ¯$ of the Klein bottles is a peculiar outcome of the current study.
$l$ $K B 3 l , l { ϵ i }$$ε ¯$ = $min { ϵ i }$ $T 3 l , l ε ¯$
1 3,4$ε _$ = 3 4
2 7,8$ε _$= 7 8
3 10,11,12$ε _$= 10 12
4 13,14,15,16$ε _$= 13 16
5 17,18,19,20$ε _$= 17 20
6 20,21,22,23,24$ε _$= 20 24
7 23,24,25,26,27,28$ε _$= 23 28
8 27,28,29,30,31,32$ε _$= 27 32
9 30,31,32,33,34,35,36$ε _$= 30 36
10 33,34,35,36,37,38,39,40$ε _$= 33 40
Table 3. For L[M] = 5,6, the eccentricities {$ϵ i }$ of the $KB L M , L C$ graphs are compared with the toroidal case, L[C] = 1, 2, …. Graphs with L[C] < L[M] − 1 show the topological shrinkage of
the eccentricity $ε _ < ε ¯$ and Wiener index W[KB] < W[T]; for L[C] ≥ L[M] − 1, the KB graphs are topologically equivalent, $ε _ = ε ¯$, to the tori $T L M , L C$ of the same sizes with W[KB] = W
L[M] = 5, Ł[C] = 4; for L[C] ≥ Ł[C], the two polyhexes are equivalent.
L[C] $K B 5 , L C${$ϵ i$}$ε ¯$= min{$ϵ i$} $W$[KB] $T 5 , L C ε ¯$ $W$[T]
1 4,5,6$ε _$= 4 528 6 600
2 6,7$ε _$= 6 2800 7 2880
3 7,8$ε _$= 7 7632 8 7680
4 9$ε _$= 9 16,000 9 16,000
5 10$ε _$= 10 29,000 10 29,000
6 12$ε _$= 12 48,000 12 48,000
7 14$ε _$= 14 74,200 14 74,200
8 16$ε _$= 16 108,800 16 108,800
9 18$ε _$= 18 153,000 18 153,000
10 20$ε _$= 20 208,000 20 208,000
15 30$ε _$= 30 687,000 30 687,000
20 40$ε _$= 40 1,616,000 40 1,616,000
25 50$ε _$= 50 3,145,000 50 3,145,000
L[M] = 6, Ł[C] = 5; for L[C] ≥ Ł[C], the two polyhexes are equivalent.
L[C] $K B 6 , L C${$ϵ i$}$ε ¯$= min{$ϵ i$} $W$[KB] $T 6 , L C ε ¯$ $W$[T]
1 4,5,6,7$ε _$= 4 860 7 1008
2 7,8$ε _$= 7 4504 8 4704
3 8,9$ε _$= 8 12,084 9 12,240
4 9,10$ε _$= 9 24,880 10 24,960
5 11$ε _$= 11 44,400 11 44,400
6 12$ε _$= 12 72,288 12 72,288
7 14$ε _$= 14 110,544 14 110,544
8 16$ε _$= 16 160,896 16 160,896
9 18$ε _$= 18 225,072 18 225,072
10 20$ε _$= 20 304,800 20 304,800
15 30$ε _$= 30 997,200 30 997,200
20 40$ε _$= 40 2,337,600 40 2,337,600
25 50$ε _$= 50 4,542,000 50 4,542,000
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Putz, M.V.; Ori, O. Topological Symmetry Transition between Toroidal and Klein Bottle Graphenic Systems. Symmetry 2020, 12, 1233. https://doi.org/10.3390/sym12081233
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Elasticity of supply : 네이버 참여번역
Elasticity of supply
63문장 0% 베트남어 번역 0명 참여 출처 : 칸아카데미
Elasticity of supply
We've been talking a lot about elasticities of a demand, so you were probably wondering, can we think about elasticities of a supply?
And as you can imagine, the answer is: of course we can!
And it's interesting to think about how does the quantity the percent change in quantity supplied relate to percent change in prices.
So, for example, let's say we have a lemonade stand of some sort.
So this is price on that axis and that is quantity on that axis.
And let's say that our supply curve looks something like that.
Obviously, the higher the price, the more quantity we are willing to supply.
And let's say at a price of $1, the quantity supplied is going to be 10 gallons/week. this is going to be 10 gallons/week.
So the quantity of supplies is going to be 10 gallons per week.
And let's say that if the price goes to $2, the quantity supplied goes to 16 gallons per week.
So what is the elasticity of supply, roughly, over this period right over here?
So the elasticity... elasticity of supply, and you can imagine how we can calculate this as % change in quantity supplied over our % change in price, so what is our percent change in price, well we
went from $1 to $2,
(so this part right over here) is going to be we went up by $1, so we went up by $1 per gallon, so it's going up by $1,
and we all use our starting point as base we can do in traditionally finding of percent change because we have the same percent change ,
if we're going from 1 to 2, as from 2 to 1 so instead the conventionally we think about elasticity is to use the mid-point of these two, use the average of these two so 1+2 is 3, 3/2 is 1,5.
So it's 1 over $1.50.
We can say, $1,50 is right in between these two things. and 1 over $1,50 this is about 67 % roughly, so this is approximately 67% change in price based on how we just calculate it,
how we're using the midpoint as our base, and that are % change in quantity supplied, that's % change in quantity supplied, that's this, this right over here we went from 10 to 16,
so we have +6 over base of midpoint between 10 and 16, is 13. 10 + 16 is 26, divided by two is 13. 6 over 13. Which is going to be something percent.
We'll take calculator out, so we have 6 divided by 13, get this 46 %.
So this is, this right over here is 46%.
So we have when we had based on the way we have calculated, this 67% increase in price we had 46% increase in quantity supplied.
So this is 46% increase in quantity supplied, it's all we can see about elasticity of supply, it's going to be 46% over 67%, so it's going to be something less than 1.
We can get, so that's going to be that divided by 0,6666666... keeps going on forever, this is 0,69.
So this is elasticity of supply 0,69, we've back see it approximately 0,69. we've tell this, we get a smaller percent at least this price point right over here, we get a smaller percent change in
quantity supplied than our percent change in price.
Let's think about like me did with let me talk about we do about elasticities of demand, let's think about different scenarios.
So let's think about, let's think about a scenario that is inelastic.
That is maybe perfectly inelastic.
So let's say that price, price and quantity.
So let's take me for an example - I make videos, I love making videos, this is what I want spend my days doing and I don't care about how much they pay me, or how little you pay me,
I guess prepaid to be enough that would maybe and I spent little bit more time making videos.
What else just as soon that I don't completely, I you know what you pay me a penny pro video or 0 pro video or do you pay me a $1000 pro video.
I'm going to just make the same number videos every day.
So this is the right over here videos, videos pro day on average and it's and this is the price per video.
And let's say no matter how much you pay - what do you pay me nothing or you pay me $1000.
I was going to produce on average, let's say 3 videos a day.
So than you have this right over here you have an perfectly inelastic perfectly inelastic supply curve.
Now you can have the other scenario, you can have the when you are farmer, so let me do price and quantity.
Now you can have the other scenario when you are farmer and you can either do crop, aid croppy, maybe it's corn and wheat.
And you can easily swap between the two unless this is soon for simplicity cost you the exact the same to produce one or the other.
So in that situation so let's say that the price of wheat pearl, let's say we're using capable price units.
So the price of wheat is for you know adjusting 4 units in all of that, let's say is $10 I don't know for bushels or something like that.
I like this simplify for the sink model right here but this right over here we're thinking about corn.
We're thinking about corn.
We're thinking about corn, and so if corn so let's say corn is right $10.
We're bought it for right $10 I will produce, so them let me make this clear.
So price of corn is $10 and the quantity of corn maybe I don't know produce, I produce 2000 bushels I know these prices are away for what a real price for bushel of corn or wheat is the same thing my
quantity for wheat right here is 2000.
Now if the price of corn will go marginally up if the price of corn, so let me put this, so this is graph of corn.
So this is $10 and this is 2000 bushels pro year of something, so let's say that we're right over there,
now if the price for corn goes marginally up, the price for corn goes up to even $10,5 per bushel all cents awnished from wheat production to corn production, this is going to be zero and this is
going to go to 4000.
So then, whenever we go, just as 10, line, whenever we go all the way to 4000.
And likewise if this price will go down if this go down like 9,95, I will shift all my production to wheat and wouldn't produce any corn.
And so there you see that we have the very the main curve is getting very flat and you can see based on very small percent changes in prices, I have very large percent changes in quantity supplied.
So this right over here is approaching, this is approaching, approaching perfect elasticity, huge changes in quantity supplied elasticity for small percent changes in price.
Now, the cool thing about elasticity of supply is actually much easier to make a curve that has unit elasticity or even it's easy if you have unit elasticity,
but if you have unit elasticity the easy as curve I can draw for unit elasticity is going look like this, much this is curve for unit elasticity,
it will be curve looks like that and the reason why it works in this case is upwards sloping as price increases so as quantity increase for the supply curve so that any point here the 2 are going to
be proportional.
So a given change in quantity and a given change in price they are going to represent the same percentages.
Because as price is increasing, what this is we have large price, or we have medium price, you have medium quantity.
When you have a large price, you have large quantity.
So these steps are going to be the same percentage of I wonder you have small prices, you have small quantity.
It's much easier to construct a curve, a supply curve that has unit elasticity, than this is to construct a normal demand curve that has unit elasticity.
Elasticity of supply발음듣기
• 더 많은 번역문은 보기 옵션을 이용해 보세요.
We've been talking a lot about elasticities of a demand, so you were probably wondering, can we think about elasticities of a supply?발음듣기
• 더 많은 번역문은 보기 옵션을 이용해 보세요.
And as you can imagine, the answer is: of course we can!발음듣기
• 더 많은 번역문은 보기 옵션을 이용해 보세요.
And it's interesting to think about how does the quantity the percent change in quantity supplied relate to percent change in prices.발음듣기
• 더 많은 번역문은 보기 옵션을 이용해 보세요.
So, for example, let's say we have a lemonade stand of some sort.발음듣기
• 더 많은 번역문은 보기 옵션을 이용해 보세요.
So this is price on that axis and that is quantity on that axis.발음듣기
• 더 많은 번역문은 보기 옵션을 이용해 보세요.
And let's say that our supply curve looks something like that.발음듣기
• 더 많은 번역문은 보기 옵션을 이용해 보세요.
Obviously, the higher the price, the more quantity we are willing to supply.발음듣기
• 더 많은 번역문은 보기 옵션을 이용해 보세요.
And let's say at a price of $1, the quantity supplied is going to be 10 gallons/week. this is going to be 10 gallons/week.발음듣기
• 더 많은 번역문은 보기 옵션을 이용해 보세요.
So the quantity of supplies is going to be 10 gallons per week.발음듣기
• 더 많은 번역문은 보기 옵션을 이용해 보세요.
And let's say that if the price goes to $2, the quantity supplied goes to 16 gallons per week.발음듣기
• 더 많은 번역문은 보기 옵션을 이용해 보세요.
So what is the elasticity of supply, roughly, over this period right over here?발음듣기
• 더 많은 번역문은 보기 옵션을 이용해 보세요.
So the elasticity... elasticity of supply, and you can imagine how we can calculate this as % change in quantity supplied over our % change in price, so what is our percent change in price, well we
went from $1 to $2,발음듣기
• 더 많은 번역문은 보기 옵션을 이용해 보세요.
(so this part right over here) is going to be we went up by $1, so we went up by $1 per gallon, so it's going up by $1,발음듣기
• 더 많은 번역문은 보기 옵션을 이용해 보세요.
and we all use our starting point as base we can do in traditionally finding of percent change because we have the same percent change ,발음듣기
• 더 많은 번역문은 보기 옵션을 이용해 보세요.
if we're going from 1 to 2, as from 2 to 1 so instead the conventionally we think about elasticity is to use the mid-point of these two, use the average of these two so 1+2 is 3, 3/2 is 1,5.발음듣기
• 더 많은 번역문은 보기 옵션을 이용해 보세요.
So it's 1 over $1.50.발음듣기
• 더 많은 번역문은 보기 옵션을 이용해 보세요.
We can say, $1,50 is right in between these two things. and 1 over $1,50 this is about 67 % roughly, so this is approximately 67% change in price based on how we just calculate it,발음듣기
• 더 많은 번역문은 보기 옵션을 이용해 보세요.
how we're using the midpoint as our base, and that are % change in quantity supplied, that's % change in quantity supplied, that's this, this right over here we went from 10 to 16,발음듣기
• 더 많은 번역문은 보기 옵션을 이용해 보세요.
so we have +6 over base of midpoint between 10 and 16, is 13. 10 + 16 is 26, divided by two is 13. 6 over 13. Which is going to be something percent.발음듣기
• 더 많은 번역문은 보기 옵션을 이용해 보세요.
We'll take calculator out, so we have 6 divided by 13, get this 46 %.발음듣기
• 더 많은 번역문은 보기 옵션을 이용해 보세요.
So this is, this right over here is 46%.발음듣기
• 더 많은 번역문은 보기 옵션을 이용해 보세요.
So we have when we had based on the way we have calculated, this 67% increase in price we had 46% increase in quantity supplied.발음듣기
• 더 많은 번역문은 보기 옵션을 이용해 보세요.
So this is 46% increase in quantity supplied, it's all we can see about elasticity of supply, it's going to be 46% over 67%, so it's going to be something less than 1.발음듣기
• 더 많은 번역문은 보기 옵션을 이용해 보세요.
We can get, so that's going to be that divided by 0,6666666... keeps going on forever, this is 0,69.발음듣기
• 더 많은 번역문은 보기 옵션을 이용해 보세요.
So this is elasticity of supply 0,69, we've back see it approximately 0,69. we've tell this, we get a smaller percent at least this price point right over here, we get a smaller percent change in
quantity supplied than our percent change in price.발음듣기
• 더 많은 번역문은 보기 옵션을 이용해 보세요.
Let's think about like me did with let me talk about we do about elasticities of demand, let's think about different scenarios.발음듣기
• 더 많은 번역문은 보기 옵션을 이용해 보세요.
So let's think about, let's think about a scenario that is inelastic.발음듣기
• 더 많은 번역문은 보기 옵션을 이용해 보세요.
That is maybe perfectly inelastic.발음듣기
• 더 많은 번역문은 보기 옵션을 이용해 보세요.
So let's say that price, price and quantity.발음듣기
• 더 많은 번역문은 보기 옵션을 이용해 보세요.
So let's take me for an example - I make videos, I love making videos, this is what I want spend my days doing and I don't care about how much they pay me, or how little you pay me,발음듣기
• 더 많은 번역문은 보기 옵션을 이용해 보세요.
I guess prepaid to be enough that would maybe and I spent little bit more time making videos.발음듣기
• 더 많은 번역문은 보기 옵션을 이용해 보세요.
What else just as soon that I don't completely, I you know what you pay me a penny pro video or 0 pro video or do you pay me a $1000 pro video.발음듣기
• 더 많은 번역문은 보기 옵션을 이용해 보세요.
I'm going to just make the same number videos every day.발음듣기
• 더 많은 번역문은 보기 옵션을 이용해 보세요.
So this is the right over here videos, videos pro day on average and it's and this is the price per video.발음듣기
• 더 많은 번역문은 보기 옵션을 이용해 보세요.
And let's say no matter how much you pay - what do you pay me nothing or you pay me $1000.발음듣기
• 더 많은 번역문은 보기 옵션을 이용해 보세요.
I was going to produce on average, let's say 3 videos a day.발음듣기
• 더 많은 번역문은 보기 옵션을 이용해 보세요.
So than you have this right over here you have an perfectly inelastic perfectly inelastic supply curve.발음듣기
• 더 많은 번역문은 보기 옵션을 이용해 보세요.
Now you can have the other scenario, you can have the when you are farmer, so let me do price and quantity.발음듣기
• 더 많은 번역문은 보기 옵션을 이용해 보세요.
Now you can have the other scenario when you are farmer and you can either do crop, aid croppy, maybe it's corn and wheat.발음듣기
• 더 많은 번역문은 보기 옵션을 이용해 보세요.
And you can easily swap between the two unless this is soon for simplicity cost you the exact the same to produce one or the other.발음듣기
• 더 많은 번역문은 보기 옵션을 이용해 보세요.
So in that situation so let's say that the price of wheat pearl, let's say we're using capable price units.발음듣기
• 더 많은 번역문은 보기 옵션을 이용해 보세요.
So the price of wheat is for you know adjusting 4 units in all of that, let's say is $10 I don't know for bushels or something like that.발음듣기
• 더 많은 번역문은 보기 옵션을 이용해 보세요.
I like this simplify for the sink model right here but this right over here we're thinking about corn.발음듣기
• 더 많은 번역문은 보기 옵션을 이용해 보세요.
We're thinking about corn.발음듣기
• 더 많은 번역문은 보기 옵션을 이용해 보세요.
We're thinking about corn, and so if corn so let's say corn is right $10.발음듣기
• 더 많은 번역문은 보기 옵션을 이용해 보세요.
We're bought it for right $10 I will produce, so them let me make this clear.발음듣기
• 더 많은 번역문은 보기 옵션을 이용해 보세요.
So price of corn is $10 and the quantity of corn maybe I don't know produce, I produce 2000 bushels I know these prices are away for what a real price for bushel of corn or wheat is the same thing my
quantity for wheat right here is 2000.발음듣기
• 더 많은 번역문은 보기 옵션을 이용해 보세요.
Now if the price of corn will go marginally up if the price of corn, so let me put this, so this is graph of corn.발음듣기
• 더 많은 번역문은 보기 옵션을 이용해 보세요.
So this is $10 and this is 2000 bushels pro year of something, so let's say that we're right over there,발음듣기
• 더 많은 번역문은 보기 옵션을 이용해 보세요.
now if the price for corn goes marginally up, the price for corn goes up to even $10,5 per bushel all cents awnished from wheat production to corn production, this is going to be zero and this is
going to go to 4000.발음듣기
• 더 많은 번역문은 보기 옵션을 이용해 보세요.
So then, whenever we go, just as 10, line, whenever we go all the way to 4000.발음듣기
• 더 많은 번역문은 보기 옵션을 이용해 보세요.
And likewise if this price will go down if this go down like 9,95, I will shift all my production to wheat and wouldn't produce any corn.발음듣기
• 더 많은 번역문은 보기 옵션을 이용해 보세요.
And so there you see that we have the very the main curve is getting very flat and you can see based on very small percent changes in prices, I have very large percent changes in quantity supplied.발
• 더 많은 번역문은 보기 옵션을 이용해 보세요.
So this right over here is approaching, this is approaching, approaching perfect elasticity, huge changes in quantity supplied elasticity for small percent changes in price.발음듣기
• 더 많은 번역문은 보기 옵션을 이용해 보세요.
Now, the cool thing about elasticity of supply is actually much easier to make a curve that has unit elasticity or even it's easy if you have unit elasticity,발음듣기
• 더 많은 번역문은 보기 옵션을 이용해 보세요.
but if you have unit elasticity the easy as curve I can draw for unit elasticity is going look like this, much this is curve for unit elasticity,발음듣기
• 더 많은 번역문은 보기 옵션을 이용해 보세요.
it will be curve looks like that and the reason why it works in this case is upwards sloping as price increases so as quantity increase for the supply curve so that any point here the 2 are going to
be proportional.발음듣기
• 더 많은 번역문은 보기 옵션을 이용해 보세요.
So a given change in quantity and a given change in price they are going to represent the same percentages.발음듣기
• 더 많은 번역문은 보기 옵션을 이용해 보세요.
Because as price is increasing, what this is we have large price, or we have medium price, you have medium quantity.발음듣기
• 더 많은 번역문은 보기 옵션을 이용해 보세요.
When you have a large price, you have large quantity.발음듣기
• 더 많은 번역문은 보기 옵션을 이용해 보세요.
So these steps are going to be the same percentage of I wonder you have small prices, you have small quantity.발음듣기
• 더 많은 번역문은 보기 옵션을 이용해 보세요.
It's much easier to construct a curve, a supply curve that has unit elasticity, than this is to construct a normal demand curve that has unit elasticity.발음듣기
• 더 많은 번역문은 보기 옵션을 이용해 보세요. | {"url":"https://dict.naver.com/userstranslate/khanInfo.dict?fullTextId=15595&langCode=vi&tab=1&selector=1","timestamp":"2024-11-14T10:24:26Z","content_type":"text/html","content_length":"294163","record_id":"<urn:uuid:98c72e78-fd95-4592-9359-03bb372d4813>","cc-path":"CC-MAIN-2024-46/segments/1730477028558.0/warc/CC-MAIN-20241114094851-20241114124851-00136.warc.gz"} |
How does it work?
Before using Rankings Reloaded, make sure that you understand its concepts. In the definition section, you can find brief explanations of input components. In the data structure section, the
organization of the input data file is defined. After understanding the concepts, you are ready to prepare your own data file.
Definitions [1]
• Task: A problem to be solved in the scope of a challenge/scientific work for which a dedicated ranking/leaderboard is provided (if any). Depending on the design, a challenge/problem can have a
single or multiple tasks. Single-task works require the handling of only one job, for example detecting only human faces in a set of images.Multi-task challenges can be organized in different
ways. For example, the first task could be the detection of an object and the second to segment the detected object on the same data. Another example can be the segmentation of multiple objects
on the same data. On the other hand, some multi-task challenges require conducting the same assignment on different datasets/data types, for example segmenting the liver from CT and MRI images.
• Case: A subset of the dataset for which the algorithm(s) of interest produce results. In general, the case number refers to a subset of the data stemming from a single acquisition. For example,
data with 50 cases can mean that there are image series of 50 different acquisitions in the dataset.
• Algorithm: A method designed to solve the tasks in the challenge.
• Metric: A measure used to compute the performance of a given algorithm for a given case, typically based on the known correct answer. Different metric outputs can vary in their distribution and
order. For example, a higher value can mean better results for one metric; vice versa for another metric. Therefore, usually, metrics are normalized to yield values in the interval between 0
(worst performance) and 1 (best performance). However, this is not essential for using Rankings Reloaded. You can use any metric range as long as you specify the distribution order.
• Assessment method: The assessment method defines how to create a ranking of the participating algorithms. Different approaches exist; the most common are "rank then aggregate" and "aggregate then
rank". Assessment methods may vary across different tasks of a challenge.
Data structure
In order to generate your benchmarking report, your data is needed in the format of a CSV file. For example, in a challenge/problem with two tasks, two test cases, and two algorithms, the data might
look like this:
Task Case Algorithm Metric Values
Task 1 case1 A1 0.27
Task 1 case1 A2 0.20
Task 1 case2 A1 0.57
Task 1 case2 A2 0.95
Task 2 case1 A1 0.37
Task 2 case1 A2 0.89
Task 2 case2 A1 0.91
Task 2 case2 A2 NA
The column structure of your data must be as same as the sample above. The columns represent:
• A task identifier. The ranking analysis may be performed for different tasks which are defined via this column. A task may, for example, refer to
□ the classical definition for multi-task challenges, for example: Task 1 - classification, Task 2 - segmentation,
□ different settings in the challenge, for example different levels of difficulty, Task 1 - Stage 1, Task 2 - Stage 2 or
□ different metrics to be tested, for example: Task 1 - Dice similarity coefficient (DSC), Task 2 - Normalized surface distance, Task 3 - Sensitivity.
Remark: If you are comparing different metrics, make sure that they are ordered in the same way! The toolkit requires the boolean specification whether small values are better, meaning the
sort direction of metric values. Example: The first metric is the Dice similarity coefficient (DSC) (range: [0,1]) and should be compared to rankings with the volumetric difference (range:
[0,x], x not specified). High values of the DSC (close to 1) correspond to high performance. In comparison, high values of the volumetric difference correspond to low performance. To solve
this issue and thus make it work as a multi-task challenge, you can invert the DSC values, so that high values also correspond to low performance.
• A case identifier. This column should contain all cases (images) that are used in the challenge/benchmarking experiment. Make sure that each case appears only once for the same algorithm and
• An algorithm identifier. The column should contain all used algorithms/methods that will be compared. For each case, the algorithm should appear once.
• The calculated metric values should appear in this column. For a single metric, a value should appear for each algorithm for each case. In case of missing metric values, a missing observation has
to be provided (either as a blank field or “NA”), otherwise the system cannot generate the report. For example, in task “T2”, test case “case2”, algorithm “A2” did not give a prediction and thus
NAis inserted to denote a missing value in the table above. Rankings Reloaded will ask you how to handle such missing cases (e.g., by assigning them the worst possible metric score) throughout
the report generation.
Download Sample Data
You can also download our sample_data.csv to analyze the structure of the data file. In this file, results of three tasks are generated to analyze different scenarios (ideal, random, worst case).
This file is used to generate the sample_data_report.pdf file at the "Use Cases" page.
Ready for the Report Generation!
Congratulations! Your benchmarking report is only some clicks away! You are ready to prepare your challenge data for report generation. Before you continue, you may want to check Citation and FAQ
pages. Then you may prepare your data and click the button below:
[1] Maier-Hein, L., Eisenmann, M., Reinke, A. et al. Why rankings of biomedical image analysis competitions should be interpreted with care. Nat Commun 9, 5217 (2018). https://doi.org/10.1038/ | {"url":"https://www.rankings-reloaded.de/how-does-it-work","timestamp":"2024-11-14T23:54:55Z","content_type":"text/html","content_length":"16604","record_id":"<urn:uuid:24ff9325-536f-4c65-8966-5bb4171428dd>","cc-path":"CC-MAIN-2024-46/segments/1730477397531.96/warc/CC-MAIN-20241114225955-20241115015955-00261.warc.gz"} |
Converting Binary to Hexadecimal Using a String Method in Java
How can you convert from binary to hexadecimal using a string method in Java?
What is the process of converting binary to hexadecimal in Java?
To convert from binary to hexadecimal in Java, you can use the `Integer` class and its `toHexString()` method. First, convert the binary string to an integer using the `parseInt()` method with a
radix of 2. Then, use the `toHexString()` method to convert the integer to a hexadecimal string.
To convert from binary to hexadecimal in Java, you can follow these steps:
1. Convert Binary String to Integer
First, you need to convert the binary string to an integer using the `parseInt()` method of the `Integer` class. The `parseInt()` method takes two arguments: the binary string and the radix, which is
2 for binary.
2. Convert Integer to Hexadecimal String
Once you have the integer value, you can use the `toHexString()` method of the `Integer` class to convert it to a hexadecimal string. The `toHexString()` method takes the integer value as input and
returns a string representation of the hexadecimal value.
3. Utilize Hexadecimal String
Finally, you can print or use the hexadecimal string as needed in your Java program.
Here is an example code snippet:
String binary = "101010";
int decimal = Integer.parseInt(binary, 2);
String hexadecimal = Integer.toHexString(decimal);
This code will convert the binary string "101010" to its hexadecimal representation, which is "2A". The `parseInt()` method converts the binary string to an integer, and the `toHexString()` method
converts the integer to a hexadecimal string. | {"url":"https://airdocsolutions.com/computers-and-technology/converting-binary-to-hexadecimal-using-a-string-method-in-java.html","timestamp":"2024-11-05T15:46:01Z","content_type":"text/html","content_length":"22291","record_id":"<urn:uuid:e5dd1369-f76b-4e22-89e7-ab99e60ddeb4>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00578.warc.gz"} |
Maximum likelihood sequence estimation of communication signals by a Hopfield neural network
The application of Hopfield's neural networks for data sequence estimation in digital communication receivers is presented. The Hopfield neural networks are used to perform the maximum-likelihood
sequence estimation (MLSE) and robust architectures for VLSI realizations are presented. The Hopfield's neural networks for MLSE have several advantages over other applications in that the complexity
is proportional to channel memory, the network provides a regularity in architecture, and the problem of vanishing diagonal elements can be relaxed. It has been shown that artificial neural networks
have potential abilities to perform optimization problems which occur often in the area of electronic communications.
Conference Proceedings of the 1994 IEEE International Conference on Neural Networks. Part 1 (of 7)
City Orlando, FL, USA
Period 27/06/94 → 29/06/94
Dive into the research topics of 'Maximum likelihood sequence estimation of communication signals by a Hopfield neural network'. Together they form a unique fingerprint. | {"url":"https://pure.lib.cgu.edu.tw/en/publications/maximum-likelihood-sequence-estimation-of-communication-signals-b-3","timestamp":"2024-11-11T14:43:45Z","content_type":"text/html","content_length":"53314","record_id":"<urn:uuid:9c299936-55c3-4c15-8071-f5f0b49cae7a>","cc-path":"CC-MAIN-2024-46/segments/1730477028230.68/warc/CC-MAIN-20241111123424-20241111153424-00169.warc.gz"} |
Видеотека: Jens Stange,
Аннотация: Common X^2 tests are very well known and frequently applied in statistical analyses in particular for discrete models. An application to genetic association studies is considered, where a
large number M, say, of 2x3 contingency tables is simultaneously tested. A method controlling the family wise error rate is shown, which makes use of an effective number of tests in Sidak
multiplicity correction favor. This method considers an approximation of the full M-dimensional distribution of the involved X^2 test statistics, by a product of k-dimensional marginal distributions.
A challenge of this procedure is an efficient computation of the k-dimensional distributions. Besides time consuming Monte Carlo procedures, there are only few implementations for even smaller
dimensions of multivariate distributions. Existing formulas for the cumulative distribution function of a multivariate X^2 distribution are now implemented for an approximations with k equal to up to
Язык доклада: английский | {"url":"https://m.mathnet.ru/php/presentation.phtml?option_lang=rus&presentid=9276","timestamp":"2024-11-05T06:52:06Z","content_type":"text/html","content_length":"8337","record_id":"<urn:uuid:93535155-f948-49ba-864d-ab981fb2cefa>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00513.warc.gz"} |
what is cateracting in ball mill
A chocolate ball mill is a crucial tool for chocolatiers, pastry chefs, and confectioners who want to produce their own highquality chocolate products. It allows them to control the particle size
and viscosity of the chocolate mass, factors that ultimately affect the final product's taste and texture.
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For a smooth mill, cataracting of the charge generally commenced at speeds of about, while, when lifters were fitted to the shell, cataracting commenced at about ... In this case the moment of
the powder charge about the axis of rotation of the mill is probably much the same as in the ball mill; the greater moment at one end of the ...
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Ultrafine vertical mills and ball mills are common largescale industrial grinding equipment and are widely used. However, many people still do not know how to choose the most suitable equipment
when purchasing equipment. Let's take a look at the difference between ultrafine vertical mill and ball mill.
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As the circulating load increases, typically the cyclone underflow density increases, causing the density and viscosity in the mill to also increase. This can lead to excessive mill viscosity,
causing the balls to float, leading to a sharp drop in the power draw. The operator can be misled to conclude that the mill is overloaded due to the ...
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Attritor mills are versatile and efficient types of milling equipment used for grinding and mixing materials. They offer several advantages over other types of mills, including better grinding
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cascading and cataracting regimes, the calibration can be done adjusting the parameters aiming to nd the shoulder and foot points closest to the experimental ones. The objective of this paper was
to evaluate the nal granulometric distribution of the clinker in a ball mill operating in dierent rotation speeds, varying the lling degrees of
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The formula for calculating critical mill of speed: N c = / √ (D d) Where: N c = Critical Speed of Mill. D = Mill Diameter. d = Diameter of Balls. Let's solve an example; Find the critical speed
of mill when the mill diameter is 12 and the diameter of balls is 6. This implies that;
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Definition of cascading Definition of cascading Movement of crop load in a ball mill rotating at such a speed that the balls breaking free at the top of the rising load roll quietly down to the
toe of the charge.
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Cataracting Action In Ball Mills. Cateracting In Grinding 2 grinding in ball mills is an important technological process applied to reduce the cascading fast rotation cataracting and very fast
rotation centrifugation each type is characterized by a specific trajectory of motion of the charge in the mill and a different impact of the milling bodies on the ground material the grinding
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Cascading and cateracting are the processes whereby the grinding balls are lifted as well as spin around; and on the material resulting in to reduction in the size of feed material progressively
on mill this process mill has to work hard to take a rotation (torque) and hence the power on the mill motor increases visavis choked ...
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The starting point for ball mill media and solids charging generally starts as follows: 50% media charge. Assuming 26% void space between spherical balls (nonspherical, irregularly shaped and
mixedsize media will increase or decrease the free space) 50% x 26% = 13% free space. Add to this another 10%15% above the ball charge for total of 23% ...
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What is an Attrition Mill? Attrition mills mechanically reduce solid particles through the intense acceleration of particles against one another in a curved or flat grooved surface called a
stator. These mills use a highspeed rotor to create centrifugal forces that facilitate the necessary particle interactions.
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In total, 165 scenarios were simulated. When the mills charge comprising 60% of small balls and 40% of big balls, mill speed has the greatest influence on power consumption. When the mill charge
is more homogeneous size, the effect of ball segregation is less and so the power consumption of the mill will be less affected.
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T02:03:09+00:00 cascading ball mill video saumoneevencoisefr. Cascading Ball Mill Video Cascading Ball Mill Video Cascading grinding effect in ball millcascading grinding effect in ball mill
kidscityministry2018 7 19 183 a ball mill a criteria for cascading in a ball mill The ball mill is a tumbling mill that uses steel balls as the grinding media The length of the cylindrical shell
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A. Hardinge mill, the variant of ball mill consist of a hollow cylinder with conical end B. Rods and bars can also be used in place of balls as grinding medium in ball mill C. Ball mill is an
open system, hence sterility is a question D. Fibrous materials cannot be milled by ball mill. 10. What particle size can be obtained through ball mill?
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There appear to be no universally adopted names for these two types of motion of the charge, but the evidence appears to be in favour of "cascading" for the first type and "cataracting" for the
second type. These names will be adopted for the present work.
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The ball mill is a tumbling mill that uses steel balls as the grinding media. The length of the cylindrical shell is usually times the shell diameter (Figure ).The feed can be dry, with less than
3% moisture to minimize ball coating, or slurry containing 2040% water by weight.
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The operating principle of the ball mill consists of following steps. In a continuously operating ball mill, feed material fed through the central hole one of the caps into the drum and moves
therealong, being exposed by grinding media. The material grinding occurs during impact falling grinding balls and abrasion the particles between the balls.
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There are two kinds of grinding body movements: either they describe an approximately parabolic trajectory, and knock against the material bed in a process known as cataracting; otherwise they
slide and roll downwards on the material bed surface, this is known as cascading.
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of the ball mill would be of great interest. Accordingly, the aim of this paper is to identify which dimensionless numbers must be kept constant in order to reproduce the results obtained in a
laboratory ball mill at the industrial scale. Such scaling criteria are provided by an exhaustive dimensional analysis applied to the ball mill.
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A ball mill is a type of grinder used to grind or blend materials for use in mineral dressing processes, paints, pyrotechnics, ceramics, and selective laser sintering. It works on the principle
of impact and attrition: size reduction is done by impact as the balls drop from near the top of the shell.
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Discrete element method of ballmill. We modeled ballmill as a quasitwodimensional object for simplicity, although real ballmill is a 3dimensional object. Here, quasitwodimension means that balls
and jar were modeled as discs in our program except masses of the balls, which are calculated by taking the ball as a 3dimensional object.
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Although there are instances in which ball mills have been successfully operated at speeds ranging from 60% to 90% of their critical speed, it is a common practice to run the mills at speeds ...
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It has been suggested with rather dilute slurries, that the righthand portion of Volume 1 may actually be mostly water through which fine particles can be transported. It is well recognized that
in a rotating mill, some fraction of the mill contents is lifted above the level of the bottom of the discharge. The terms cascading and cataracting ...
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https:// Learn about Ball Mill Critical Speed and its effect on inner charge movements. The effect of Ball Mill RPM s...
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Python Comparision Operators - Python Basics
Comparison Operators in python are pretty straight forward if you have any sort of background in Mathematics.
First we’ll present a table of comparison operators and then work through some examples:
Operator Description Exmple
== if two values of the operands are equal, ‘1’ == ‘1’ is true
then the condition becomes true
!= if two values of the operands are not equal, ‘1’ != ‘2’ is true
then condition becomes true
If the value of the left operand is greater
> than the value of the right operand, 2 > 1 is true
then the condition becomes true.
If the value of the right operand is greater
< than the value of the left operand, 2 < 1 is false
then the condition becomes true.
If the value of the left operand is greater
>= or equal to the value of the right operand, 2 >= 1 is true
then the condition becomes true.
If the value of the right operand is greater
<= or equal to the value of the left operand, 2 <= 1 is false
then the condition becomes true.
Python 3.6.4 (v3.6.4:d48eceb, Dec 19 2017, 06:04:45) [MSC v.1900 32 bit (Intel)] on win32
Type "copyright", "credits" or "license()" for more information.
>>> 1 <= 2
>>> 2 >= 3
>>> 1 == 1
>>> 'a' == 1
>>> type('a') == str
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[BOOK] Interactive Computational Geometry: A Taxonomic Approach
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[BOOK] Interactive Computational Geometry: A Taxonomic Approach
WOLFRAM MATERIALS for the ARTICLE:
Jim Arlow, Interactive Computational Geometry: A Taxonomic Approach.
Clear View Training
Wolfram Notebook Archive
Hello community,
I have just published a new book in CDF format called "Interactive Computational Geometry". Ed Pegg of Wolfram suggested I notify the community. The book comprises text, plus 53 interactive
Demonstrations of a selection of some of the most fundamental computational geometry algorithms.
Note: The book is written for Mathematica 10, but will work OK in Player 9 (until Player 10 is released). There are 3 demonstrations that are Mathematica 10 specific that don't work in Player 9.
These are simple demos of Mathematica 10 features, and don't really detract from the rest of the book.
Full book is embedded below.
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Sam, thank you for your comments. I'm glad you like the book. Eric Schulz did an inspiring job, both with his book and presentation. I worked somewhat differently to Eric, largely because my audience
is different. In particular, I expect my audience to be familiar with Wolfram Language code (at least to some degree), and the code itself is a major part of the book. As such, I did not need to hide
input cells etc. as Eric did. Eric was also much smarter than me when it came to creating writers tools! I confess that I just worked in Mathematica to develop the text, with no special tools. My
workflow was as follows:
0) Create rough idea and outline for book.
1) Decide on a topic for a chapter.
2) Assemble references.
3) Write some text.
4) Develop algorithms/code/demonstrations (Manipulate).
5) Consolidate algorithms/code/demonstrations into text (cut and paste).
6) Loop to 2) until satisfied.
7) Top and tail book (add Introduction, TOC and Bibliography).
Obviously, in reality this is a very iterative process, and I would be working on several chapters at once, and reorganising the book as I went along based on logical dependences between the
computational geometry algorithms.
Despite the fact that I didn't create any authors tools, it was still a very good experience writing a book using Mathematica as an authoring environment, and I would recommend it to any author. It
is easier in many ways than creating iBooks or ePubs. ePub is just a non-starter for interactive books, and even iBooks requires you to "kick-down" into HTML5 for many things. Everything in my book
was done directly in Mathematica and the Wolfram Language.
In terms of my own tips for authors:
1) Develop the code/algorithms/demonstrations in separate notebooks, and make sure you have them tested and working before you paste them into the main text.
2) Export to CDF often to check it all works. There are some things (such as importing packages) that work in notebooks, but that don't work in CDFs. It is good to find out about these things sooner
rather than later!
3) Use a Literate Modelling style for describing algorithms/code. You can find out more about Literate Modelling here. It is a technique that makes it very easy to describe models/code in a precise
4) Be very careful about defining symbols and tidying up the namespace (using Clear) when developing code/algorithms and tests. I know this is just Mathematica best practice, but when your text gets
quite large, it becomes very important in order to avoid mysterious errors.
5) Save a new version of the text every time you make a significant change. Mathematica can crash, and it is pain to loose work. Also, you sometimes need to roll back to a previous version in order
to fix an error or help with debugging. Obvious - but worth repeating.
6) Don't trust Mathematica CDF Preview! In Mathematica you might expect "File/CDF Preview/CDF Player" to preview the CDF in the current CDF Player. I have just found that it doesn't. For example, a
CDF that works fine in Mathematica 10 "File/CDF Preview/CDF Player" doesn't work in the downloaded version of Player, because that is still version 9. So the version of Player in Mathematica and the
downloadable version of Player may be out of step.
Jim, this is a nicely done book, congratulations! I am very happy to see another CDF book done with a professional quality in addition to calculus and physics ones. Any good tips for future authors,
any advice? Eric Schulz, author of the calculus book did a great talk - Publishing a CDF ebook: an Author's Perspective - about details of the process. I wonder if you followed a similar path.
i don't like the CDF's in the demos at all. the inner core of code (which is what interests me), is wrapped within one or more layers of display commands such Manipulate. i would like to see are
programs stripped of all these display commands so i can see what is being calculated, not what is being displayed.
The book looks great! One nice feature of using symbolic content like you do is that the CDF file is actually quite small. The content is defined as plaintext and evaluated at runtime, keeping the
size well below that of some mathematical ebooks with embedded images.
Jesse, thank you very much for your kind words about the book. Yes - the book is very small (1.6MB) considering the number of figures. I have published several other books, both paper, conventional
eBooks and interactive iBooks for iPad. From an authors perspective, CDF has a very, very nice workflow because the interactive features are all in the Wolfram Language, rather than having to drop
down in HTML5 as you do for iBooks. It frustrates me that when I write books about UML and BPMN (my main areas of expertise), I can't ship "live" models in the same way I can ship "live" mathematics
with CDF. Wolfram is really ahead of the game here!
Thanks for the suggestion James - I think it is an excellent idea. I made the first few chapters available for free from the book website. For your convenience, here is a direct link to the free
Jim, be aware that the sample will not look 100% OK with the current CDF player (based on Mathematica 9). The InfiniteLine command is new in Mathematica 10 and is not included in CDF Player 9.
Thanks for the warning Gustavo. Yes - you are correct. Everything works OK if you use Mathematica 10 - "Preview CDF", but if you open the file in a downloaded version of Player (which is currently
version 9), you get an error for the three Mathematica 10 specific demonstrations. Fortunately, none of these demonstrations are key, because I added them just to demonstrate some new Mathematica 10
features - they do not affect the algorithms or the rest of the book in any way. I will put a note on the book website explaining this for Player 9 users. Hopefully Player 10 is not too far off!
Actually, there is a really interesting and important point for CDF authors here - I assumed (and I think most reasonable people would), that "File/CDF Preview" in Mathematica 10 previews your
document in the current version of the CDF Player. We now know that it doesn't - it uses some other version built in to Mathematica 10. I will add this to my post about advice for CDF authors.
Your book appears interesting. May I suggest that you make a chapter (or portion) available for free, so that people can sample before they buy? (On Amazon, one can often read the first chapter of a
book before making a purchase decision.)
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Simons Rock Mathematics Colloquium
Fall 2024 (Current Semester)
The colloquium meets on Tuesdays at 2:40pm, unless otherwise stated.
• September 24th
Julia Kameryn Williams (Bard College at Simon’s Rock)
Infinity, the axiom of choice, and mediacy
What does it mean to be finite? To be infinite? Why does the axiom of choice come into the picture? What even is the axiom of choice? What weird and exciting things happen when we try to
generalize the ideas which answer these questions? We will discuss these questions and more. [flyer] [slides here]
• October 8th
Alex Van Abel (Wesleyan University)
Pseudofinite Model Theory
In model theory, we study mathematical structures and the things we can say about them – classically, in first-order logic. Pseudofinite structures are infinite mathematical structures that
“behave like” finite structures. In this talk, I will outline the basics of model theory and first-order logic, and clarify what I mean by “behave like” in the previous sentence. I will
illustrate pseudofiniteness with some examples and non-examples, and share some instances of how pseudofinite model theory has been used to answer concrete questions in combinatorics. [flyer]
• November 19th (tentative)
Astra Kolomatskaia (Wesleyan University)
Title and abstract to be announced
Spring 2024
The colloquium meets on Thursdays at 2:40pm, unless otherwise stated.
• February 29th
Jack Burkart (Bard College at Simon’s Rock)
The Carpenter’s Rule Problem
Is it possible to take any given polygon and move it through space to a convex polygon, while never having the polygon intersect itself? If you drop a necklace on a flat table and attempt to do
this yourself, you’ll probably be able to successfully do this. This is far from a mathematical proof, however, and this problem, known as the Carpenter’s Rule Problem, remained open for decades
until it was solved in the early 90s. In this talk, I’ll introduce the Carpenter’s rule problem and explain a surprising approach for how to solve it that involves an important technique in
optimization known as linear programming. Most of this talk should be approachable for anyone who enjoys looking at pictures and has taken a calculus course. [flyer]
• March Canceled
• April 25th
Uniquely Completable Partial Latin Squares
A latin square of order $n$ is an $n \times n$ array in which each symbol from a set of order $n$ appears exactly once in each row and exactly once in each column. If we loosen the definition so
that each symbol appears at most once in each row and at most once in each column (and so there may be empty cells) then we have a partial latin square. A partial latin square is uniquely
completable if there is exactly one way to fill in the empty cells to obtain a latin square.
We look at some historical and recent work related to uniquely completable partial squares. Motivating questions include: how few entries can a uniquely completable partial latin square have? How
many can one have, if it is minimal in the sense that removing any entry allows multiple completions? What if we add extra conditions (like a sudoku puzzle does)? Does it make sense to ask about
infinite latin squares? There are many accessible open problems.
Includes joint work with Aurora Callahan, Emma Hasson, Kaethe Minden and Yolanda Zhu. [flyer] [slides] | {"url":"https://kamerynjw.net/seminars/SR-colloquium/","timestamp":"2024-11-11T21:35:20Z","content_type":"text/html","content_length":"8939","record_id":"<urn:uuid:39dfaf91-35cf-41de-9cf3-465aac7fd3dd>","cc-path":"CC-MAIN-2024-46/segments/1730477028239.20/warc/CC-MAIN-20241111190758-20241111220758-00122.warc.gz"} |
Central Angle - Math Steps, Examples & Questions
Identify the key parts of the circle.
The center of the circle is C.
The central angle is angle BCD because the vertex is at the center of the circle and the sides are radii.
Angle BAD is an inscribed angle because the vertex is on the edge of the circle and the sides are chords.
Triangle ABD is an inscribed triangle because the vertices of the triangle are on the edge of the circle.
To find the central angle, first you need to know the inscribed angle BAD.
As the inscribed angle is part of a triangle, use the angle fact β the sum of angles in a triangle is 180^{\circ} to find the angle.
Now, Angle BAD is the inscribed angle because the vertex is on the edge of the circle and the sides are chords.
Angle BCD is the central angle because the vertex is at the center of the circle and the sides are radii.
What is the difference between a major arc and a minor arc?
A major arc is an arc that has a measure between 180^{\circ} and 360^{\circ}. A minor arc is an arc that has a measure between 0^{\circ} and 180^{\circ}.
What is the arc length?
Arc length is part of the circumference of the circle. The circumference of the circle is the distance around the circle so the length of the arc is a portion of the circumference of a circle.
What are radians?
Radians are a unit of measure for an angle. Typically, in trigonometry, angle measures will be expressed in radians.
What are congruent arcs?
Congruent arcs are arcs that have the same length and the same measurement. If the inscribed angles of a circle are congruent and intercept arcs, those arcs will be congruent.
What are secants?
Secants are lines that intersect a circle at two points of intersection. Chords are line segments that are created by secant lines. | {"url":"https://thirdspacelearning.com/us/math-resources/topic-guides/geometry/central-angle/","timestamp":"2024-11-13T19:37:10Z","content_type":"text/html","content_length":"281244","record_id":"<urn:uuid:a2ff9304-859d-465e-af93-c9270a1275e7>","cc-path":"CC-MAIN-2024-46/segments/1730477028387.69/warc/CC-MAIN-20241113171551-20241113201551-00048.warc.gz"} |
How to Perform a One-Way ANOVA in Stata | Online Tutorials Library List | Tutoraspire.com
How to Perform a One-Way ANOVA in Stata
by Tutor Aspire
AÂ one-way ANOVAÂ is used to determine whether or not there is a statistically significant difference between the means of three or more independent groups.
This type of test is called a one-way ANOVA because we are analyzing how one predictor variable impacts a response variable. If we were instead interested in how two predictor variables impact a
response variable, we could conduct a two-way ANOVA.
This tutorial explains how to conduct a one-way ANOVA in Stata.
Example: One-Way ANOVA in Stata
In this example we will use the built-in Stata dataset called systolic to perform a one-way ANOVA. This dataset contains the following three variables for 58 different individuals:
• Drug used
• Patient’s disease
• Change in systolic blood pressure
We will use the following steps to perform a one-way ANOVA to find out if the type of drug used leads to a significant impact in the change in systolic blood pressure.
Step 1: Load the data.
First, load the data by typing webuse systolic in the command box and clicking Enter.
Step 2: View the raw data.
Before we perform a one-way ANOVA, let’s first view the raw data. Along the top menu bar, go to Data > Data Editor > Data Editor (Browse). This will show us the actual data for all 58 patients:
Step 3: Visualize the data.
Next, let’s visualize the data. We’ll create boxplots to view the distribution of systolic blood pressure values for each category of drug.
Along the top menu bar, go to Graphics > Box plot. Under variables, choose Systolic:
Then, in the Categories subheading under Grouping variable, choose drug:
Click OK. A chart with four boxplots will automatically be displayed:
We can immediately see that the distribution of changes in systolic blood pressure vary between the drug categories, but a one-way ANOVA will tell us if these differences are statistically
Step 4: Perform a one-way ANOVA.
Along the top menu bar, go to Statistics > Linear models and related > ANOVA/MANOVA > One-Way ANOVA.
Under response variable, choose systolic. Under factor variable, choose drug. Then click the box next to Produce summary table so that we can see some basic descriptive statistics for each group.
Then click OK.
The following output will be displayed:
The F-statistic is 9.09 and the corresponding p-value is 0.0001. Since the p-value is less than alpha = 0.05, we can reject the null hypothesis that the mean change in systolic blood pressure for
each group is equal.
In other words, there is a statistically significant difference in the mean change in systolic blood pressure between at least two of the drug groups.
Step 5: Perform multiple comparison tests.
Next, we can perform multiple comparison tests to actually find out which group means are different from each other.
Along the top menu bar, go to Statistics > Summaries, tables, and tests > Summary and descriptive statistics > Pairwise comparisons of means.
For Variable, choose the response variable systolic. For Over, choose the explanatory variable drug. For Multiple comparisons adjustment, choose Tukey’s method.
Then, under the Reporting subheading click the button next to Effects tables and check the box next to Show effects table with confidence intervals and p-values. Then click OK.
The following results will be displayed:
Each row represents a comparison between two specific drug groups. For example, the first row compares the mean systolic blood pressure change between drug group 2 and drug group 1. The p-value for
this comparison is 0.999, which is extremely high and not smaller than 0.05. This means there is no statistically significant difference between drug groups 1 and 2.
However, we can see that the p-values for the following comparisons are all less than 0.05:
• drug 3 vs. 1 | p-value =Â 0.001
• drug 4 vs. 1 | p-value =Â 0.010
• drug 3 vs. 2 | p-value =Â 0.001
• drug 4 vs. 2 | p-value =Â 0.015
This means that the difference in mean systolic blood pressure change is statistically significant between each of these groups.
Step 6: Report the results.
Lastly, we will report the results of our One-Way ANOVA analysis. Here is an example of how to do so:
A one-way ANOVA was performed to determine if four different types of drugs had different impacts on systolic blood pressure.
The following table summarizes the number of participants in each group along with the mean change in systolic blood pressure and the standard deviation in systolic blood pressure for each group:
A one-way ANOVA revealed that there was a statistically significant difference between at least two groups (F(3, 54) = 9.09, p = 0.001).
Tukey’s test for multiple comparisons found that the change in systolic blood pressure was statistically significantly higher for drug 3 compared to drug 1 (17.32 +/- 4.15, p = 0.001), for drug 3
compared to drug 2 (16.78 +/- 4.15, p = 0.001), for drug 4 compared to drug 1 (12.57 +/- 3.85, p = 0.010), and for drug 4 compared to drug 2 (12.03 +/- 3.85, p = 0.015).
There was no statistically significant difference between drug groups 1 and 2 (.533 +/- 3.91, p = 0.999) or between drug groups 3 and 4 (4.75 +/- 4.09, p = 0.654).
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GSE Total Stockholder Equity from 2010 to 2024 | Stocks: GVP - Macroaxis
GVP Stock USD 4.59 0.01 0.22%
GSE Systems Total Stockholder Equity yearly trend continues to be relatively stable with very little volatility. Total Stockholder Equity is likely to drop to about 5.1
. Total Stockholder Equity is the total equity held by shareholders, calculated as the difference between a company's total assets and total liabilities. It represents the net value of the company
owned by shareholders.
View All Fundamentals
First Reported Previous Quarter Current Value Quarterly Volatility
Total Stockholder Equity
1995-06-30 3.8 M 2.8 M 10.9 M
Dot-com Bubble Housing Crash Credit Downgrade Yuan Drop Covid
Check GSE Systems
financial statements
over time to gain insight into future company performance. You can evaluate financial statements to find patterns among GSE Systems' main balance sheet or income statement drivers, such as
Depreciation And Amortization of 1.4 M
, Interest Expense of 2
Total Revenue of 42.8 M
, as well as many indicators such as
Price To Sales Ratio of 0.11
, Dividend Yield of 4.0E-4 or
PTB Ratio of 0.89
. GSE financial statements analysis is a perfect complement when working with
GSE Systems Valuation
GSE Total Stockholder Equity
Check out the analysis of
GSE Systems Correlation
against competitors. To learn how to invest in GSE Stock, please use our
How to Invest in GSE Systems
Latest GSE Systems' Total Stockholder Equity Growth Pattern
Below is the plot of the Total Stockholder Equity of GSE Systems over the last few years. It is the total equity held by shareholders, calculated as the difference between a company's total assets
and total liabilities. It represents the net value of the company owned by shareholders. GSE Systems' Total Stockholder Equity historical data analysis aims to capture in quantitative terms the
overall pattern of either growth or decline in GSE Systems' overall financial position and show how it may be relating to other accounts over time.
Total Stockholder Equity 10 Years Trend
GSE Total Stockholder Equity Regression Statistics
Arithmetic Mean 22,405,127
Geometric Mean 19,019,961
Coefficient Of Variation 49.60
Mean Deviation 8,750,665
Median 23,043,000
Standard Deviation 11,112,943
Sample Variance 123.5T
Range 35.8M
R-Value (0.72)
Mean Square Error 63.3T
R-Squared 0.52
Significance 0
Slope (1,798,270)
Total Sum of Squares 1729T
GSE Total Stockholder Equity History
About GSE Systems Financial Statements
GSE Systems shareholders use historical fundamental indicators, such as Total Stockholder Equity, to determine how well the company is positioned to perform in the future. Although GSE Systems
investors may analyze each financial statement separately, they are all interrelated. The changes in GSE Systems' assets and liabilities, for example, are also reflected in the revenues and expenses
on on GSE Systems' income statement. Understanding these patterns can help investors time the market effectively. Please read more on our
fundamental analysis
Last Reported Projected for Next Year
Total Stockholder Equity 5.3 M 5.1 M
Pair Trading with GSE Systems
One of the main advantages of trading using pair correlations is that every trade hedges away some risk. Because there are two separate transactions required, even if GSE Systems position performs
unexpectedly, the other equity can make up some of the losses. Pair trading also minimizes risk from directional movements in the market. For example, if an entire industry or sector drops because of
unexpected headlines, the short position in GSE Systems will appreciate offsetting losses from the drop in the long position's value.
0.61 U Unity Software PairCorr
0.68 DJCO Daily Journal Corp PairCorr
0.67 BL Blackline PairCorr
0.79 DT Dynatrace Holdings LLC PairCorr
0.67 EB Eventbrite Class A PairCorr
0.55 DV DoubleVerify Holdings PairCorr
0.55 VCSA Vacasa Inc PairCorr
0.53 DMAN Innovativ Media Group PairCorr
0.43 ML MoneyLion PairCorr
The ability to find closely correlated positions to GSE Systems could be a great tool in your tax-loss harvesting strategies, allowing investors a quick way to find a similar-enough asset to replace
GSE Systems when you sell it. If you don't do this, your portfolio allocation will be skewed against your target asset allocation. So, investors can't just sell and buy back GSE Systems - that would
be a violation of the tax code under the "wash sale" rule, and this is why you need to find a similar enough asset and use the proceeds from selling GSE Systems to buy it.
The correlation of GSE Systems is a statistical measure of how it moves in relation to other instruments. This measure is expressed in what is known as the correlation coefficient, which ranges
between -1 and +1. A perfect positive correlation (i.e., a correlation coefficient of +1) implies that as GSE Systems moves, either up or down, the other security will move in the same direction.
Alternatively, perfect negative correlation means that if GSE Systems moves in either direction, the perfectly negatively correlated security will move in the opposite direction. If the correlation
is 0, the equities are not correlated; they are entirely random. A correlation greater than 0.8 is generally described as strong, whereas a correlation less than 0.5 is generally considered weak.
Correlation analysis
and pair trading evaluation for GSE Systems can also be used as hedging techniques within a particular sector or industry or even over random equities to generate a better risk-adjusted return on
your portfolios.
Pair CorrelationCorrelation Matching
Additional Tools for GSE Stock Analysis
When running GSE Systems' price analysis, check to
measure GSE Systems' market volatility
, profitability, liquidity, solvency, efficiency, growth potential, financial leverage, and other vital indicators. We have many different tools that can be utilized to determine how healthy GSE
Systems is operating at the current time. Most of GSE Systems' value examination focuses on studying past and present price action to
predict the probability of GSE Systems' future price movements
. You can analyze the entity against its peers and the financial market as a whole to determine factors that move GSE Systems' price. Additionally, you may evaluate how the addition of GSE Systems to
your portfolios can decrease your overall portfolio volatility. | {"url":"https://www.macroaxis.com/financial-statements/GVP/Total-Stockholder-Equity","timestamp":"2024-11-05T00:30:50Z","content_type":"text/html","content_length":"329469","record_id":"<urn:uuid:61623f25-188b-4821-b013-b91ca3f97e3d>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.84/warc/CC-MAIN-20241104225856-20241105015856-00736.warc.gz"} |
The shape of a black hole
Yesterday, while browsing the internet, I stumbled upon a thread which looked like a typical question asked by someone interested in science, and turned out to be a really interesting problem.
The question that has been asked concerned the shape of a black hole. A few people replied that the event horizon (the boundary - or the "surface" in a way - of a black hole) has the shape of a ball
(which should be actually described as a sphere, since the horizon is a 2-dimensional surface, and not a 3-dimensional shape). Someone suggested that it's not exactly true, because black holes
usually spin, which flattens them. I entered the thread then and said that even when a black hole is spinning, its horizon is still spherical - it's described by an equation like r = const. But is
that really so...?
When I wrote my response, I had a hunch. In the case of a Schwarzschild black hole, r = const actually describes a sphere, as it is a spherically symmetric solution. In the case of a rotating black
hole (called a Kerr black hole, from the physicist who first derived the metric describing them) it doesn't have to be so. r is just one of the coordinates, which can be assigned to the space
arbitrarily and the points with the same r don't have to lie on a sphere. In order to be sure, we have to calculate it.
One of the ways of describing a surface is a so-called metric - it describes a way of calculating distances between points with given coordinates (I wrote a bit more about it in my series of articles
about the mathematics of relativity). It's a fundamental tool in the General Theory of Relativity - the properties of space-time are described there exactly with a metric. Spinning black holes are
described, as I already mentioned, by the Kerr metric:
• $M$ - the black hole's mass
• $a$ - the black hole's angular momentum (divided by mass - there is a limit $a \leq M$)
• $t$, $r$, $\vartheta$, $\varphi$ - space-time coordinates
• $\Delta = r^2 - 2Mr + a^2$
• $\Sigma = r^2 + a^2\cos^2 \vartheta$
(everything is in units of $c = G = 1$, for simplicity; if we put $\frac{GM}{c^2}$ instead of $M$ and $ct$ instead of $t$, we would get SI units)
The event horizon (actually, two of them!) appears in the place where $\Delta = 0$ - so $r = M \pm \sqrt{M^2 - a^2}$. We are only interested in the outer horizon, as it is what decides the shape of
the black hole.
Since we can give a concrete value for $r$, which corresponds to the horizon, it means that the horizon is described by r = const. We aren't interested in the evolution in time, either (and there is
none - the Kerr metric is stationary), so we also assume t = const. This means that in order to limit the metric to the horizon itself, we only have to set dr = dt = 0. Such a metric (after
additionally changing its sign - the spatial part of the metric is negative, but it doesn't matter when we don't mix space with time, and we are only concerned with space here) would look like this:
It's not very beautiful, but much more pleasant than the full metric.
Now we use the fact that we know the precise value of r: $r = M + \sqrt{M^2 - a^2}$. We then get:
• $r^2 + a^2 = 2Mr$ (because $\Delta = 0$) $= 2M (M + \sqrt{M^2 - a^2})$
• $\Sigma = r^2 + a^2 \cos^2 \vartheta = r^2 + a^2 - a^2 \sin^2 \vartheta = 2M (M + \sqrt{M^2 - a^2}) - a^2 \sin^2 \vartheta$
The metric can then be written as:
Well, that's worse, but we can simplify it a bit. First, we can drop the leading $2M^2$. It only controls the size of the horizon, but not its shape - so we don't care about it, and if we need it, we
can always restore it. Second, we have $\frac{a}{M}$ in many places - let's denote this by $\alpha$. As I already mentioned, there is a limit of $a \leq M$, which means that $0 \leq \alpha \leq 1$ (0
will get us back to the Schwarzschild metric, 1 is so-called extremal Kerr). This brings the metric to the following form:
Let us also denote $1 + \sqrt{1 - \alpha^2}$ by $\beta$, because why not, and divide the whole metric by $\beta$ (again - just a matter of scale):
And, finally: we denote the ugly fraction $\frac{\beta - \frac{1}{2}\alpha^2 \sin^2\vartheta}{\beta}$ by $\xi(\vartheta)$:
When $\xi(\vartheta)$ is identically equal to 1 (and it is when $\alpha = 0$, so in the Schwarzschild case), this metric describes a sphere. Everything is fine up to now :)
What do we do, when $\xi$ is not 1, though? It is tempting to just call $\vartheta$ and $\varphi$ something like latitude and longitude, but it would be the same mistake as saying that constant $r$
is a sphere - we have no guarantee that those coordinates have some physical interpretation. What's left is to generate a surface with an equivalent metric.
We will describe our generated surface by a parametrization: $x(\vartheta, \varphi)$, $y(\vartheta, \varphi)$, $z(\vartheta, \varphi)$. Such a form means that every point on the surface is described
by a pair of coordinates $(\vartheta, \varphi)$. For every such pair we can then calculate corresponding 3D coordinates. For example, we could parametrize a sphere like so:
Since we still have a rotational symmetry in the Kerr metric, fortunately, we can use a trick: we say that z is the axis of symmetry, and we transform x and y into the distance from the z axis
(denoted by $\rho$) and an angle $\varphi$ - so we basically introduce cyllindrical coordinates. This lets us write:
(In the case of a sphere, we would have $\rho(\vartheta) = r\sin\vartheta$, $z(\vartheta) = r\cos\vartheta$).
Now we need to calculate the metric of our surface. The metric of a flat 3D space in the x, y, z coordinates is very simple: $dx^2 + dy^2 + dz^2$. We just have to limit it to our surface and express
it using $\vartheta$ and $\varphi$.
We will do it by transforming the differential forms to the new coordinates:
Let's calculate:
After substituting this in the original 3D metric:
We would like this to be equal to the $h''$ metric. For it to be like this, the following must hold:
The first condition gives us: $\rho(\vartheta) = \frac{\sin\vartheta}{\sqrt{\xi(\vartheta)}}$. We can calculate the derivative, which gives:
The second equation is an ordinary differential equation for $z(\vartheta)$ - it can be solved numerically, and we will have the full description of our surface.
So now - what does such a surface look like? Let's note that when we calculate $(\rho(\vartheta), z(\vartheta))$ for various values of $\vartheta$ between $0$ and $\pi$, we get x and z coordinates of
points on our surface for $\varphi = 0$ - so it's something like a "zero meridian". We just rotate the resulting curve around the z axis and voila! We have our surface.
Let's try it. Shown below are the "zero meridians" for $\alpha = 0$ (which should be half a circle - the Schwarzschild case, a sphere), $\alpha = 0.5$ and $\alpha = 0.8$:
If you look closely, you can see some flattenning for $\alpha=0.5$, and quite a bit of it for 0.8.
And what happens for larger values of $\alpha$...? I tried to see - and the program calculating $z(\vartheta)$ numerically started throwing complex numbers at me. What the...?
I started suspecting a mistake in the calculations, so I checked them multiple times, but everything looked OK. I concluded, then, that maybe this is the correct result - but what would be its
The problem stems from the way of calculating $\frac{dz}{d\vartheta}$ - namely, we take a coefficient of the surface's metric that multiplies $d\vartheta^2$, and we subtract $\left(\frac{d\rho}{d\
vartheta}\right)^2$ from it. Well, but for large enough values of $\alpha$, the second part turns out to be larger than the first one, and the square of the derivative of z turns out negative.
But wait... what is larger than what here? The first part is a coefficient of the metric, which is basically a square of the length of the line that we draw by moving along the surface by $d\
vartheta$. The second one is the square of the change in our distance from the axis of rotation when we move like that. A negative result means that by moving along the surface a bit, we move away
from the axis of symmetry by more than we move along the surface - WTF? Such things shouldn't happen, should they?
Well, it turns out that, although counter-intuitive, it's actually possible. The source of the problem is that we try to divide the horizon into circles - circles of latitude (constant $\vartheta$),
and we try to "stack" them along the z axis so that the distances on the surface created that way were the same as on the horizon. However, if the geometry of space is sufficiently curved, the
curvature of circles of latitude can vary in a way completely independent from the distance on the surface - and this is what happens here. Our distance from the axis of rotation ($\rho$) is just the
radius of curvature of the circle of latitude, and it varies so fast that the distance on the surface "doesn't catch up". Such a situation is possible in a curved space, but not in a flat one - like
the one we live in and try to picture the shape of the horizon.
So, simply put: the shape of the horizon is something that just doesn't exist in a flat space. Oh well. We won't see what an extremal horizon looks like ($\alpha = 1$; my master's thesis was actually
about such horizons).
We won't be able to see the actual shape of the horizon - but an observer near such a black hole would surely see something, right? He could take a picture and print it on a 2-dimensional piece of
paper - what would be the result?
A black hole strongly deflects rays of light that pass near it, and absorbs some of them - as a result, an observer sees a black spot on the sky, surrounded by a distorted image of the background.
The shape of this spot - called a "shadow" of the black hole - is not the shape of the horizon, though. It is determined by how the black hole affects light. When the black hole is spinning, light
passing it on the one side is deflected a bit differently, than on the other side, which causes the image to be a bit flattened, indeed - but only on one side, and in the direction perpendicular to
the axis of rotation, not parallel.
The picture below is a result of simulating a black hole with $\alpha \approx 0.95$. The axis of rotation is vertical here. You can clearly see the flattened left side - caused by the fact that light
on this side can pass a bit closer to the black hole without falling in, than on the other side.
In summary: I haven't expected that some horizons can't be pictured in 3D - which is kind of a consequence of my being sure that the horizon is always a sphere ;) A random person on the internet
asked a simple question, and I actually learned a lot from it. This was a fascinating experience :)
Update 2017-07-26: I've learned that, not surprisingly, it is actually a well-known result that the surface of the horizon cannot be embedded in a flat 3D space for $\alpha > \frac{\sqrt{3}}{2}$ (L.
Smarr Surface Geometry of Charged Rotating Black Holes, Phys. Rev. D 7, 289 (1973)). It was fun to find it out by myself, nevertheless :) | {"url":"https://ebvalaim.net/en/2017/07/25/the-shape-of-a-black-holes-event-horizon/","timestamp":"2024-11-01T19:19:33Z","content_type":"text/html","content_length":"84815","record_id":"<urn:uuid:567c75e3-b151-4dfe-a9cf-7011f78c8a09>","cc-path":"CC-MAIN-2024-46/segments/1730477027552.27/warc/CC-MAIN-20241101184224-20241101214224-00073.warc.gz"} |
Math Problem Statement
You wish to have $18,822 in 11 years. How much money would you need to deposit now into an account earning 4.5% compounded monthly in order to have $18,822 in 11 years? Round your answer to two
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A = P(1 + r/n)^(nt)
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Compound Interest Formula
Suitable Grade Level
Grades 9-12 | {"url":"https://math.bot/q/calculate-initial-deposit-18822-4-5-compounded-monthly-FMOFQdtg","timestamp":"2024-11-06T02:43:50Z","content_type":"text/html","content_length":"86703","record_id":"<urn:uuid:fd5bcadd-ed0a-42e3-b773-c7cd278c21ba>","cc-path":"CC-MAIN-2024-46/segments/1730477027906.34/warc/CC-MAIN-20241106003436-20241106033436-00723.warc.gz"} |
The Unabaker Speaks
The Kitchen Formula Calculator v4 Explained
This article is detailed, but is in question and answer format, so a reader can browse it, or read it entirely. It amounts to a condensed version of the 7 part series I posted about the calculator.
What is it?
The Kitchen Formula Calculator is a set of calculating tables designed for writing new recipes, or recording ones already in hand. The file is a template meant to be duplicated, the duplicate then
used to enter the recipe data. The template is the template. Do not perform data entry in the template.
The recipe calculators are more than simple calculating tools. They are a neatly designed, organized format in which to write, record, and develop recipes. One of their great virtues, and the reason
they were created is to be able to quickly, and flawlessly upscale or downscale a recipe. The calculators have integrated tables that are used to reformulate a recipe for further development, a
common occurrence after “cook-testing” a new recipe, and another to cost the recipe. Used regularly, they become a Kitchen Guide for a professional operation, or a place for the home cook to park all
of their recipe repertoire. Recipe costing is something home cooks will almost never do, but the ability to do so is built into both calculators. In restaurant operations it’s a necessity. The
Kitchen Formula Calculator has one table that is useful only for professional Chefs. It’s an additional costing table to cost the entire plate that represents a menu offering. It calculates the plate
cost, compares it to the menu price, and reports both margin, and food cost percentage for the plate.
The KFCv4 is a spreadsheet orignally written using Apple Numbers, though easily convertible to Google Sheets, or Excel. The KFCv4 has been refined over the years; v4 indicating the current iteration.
Both are based on The Unabaker’s Master Formula Calculator, designed specifically for bread, which has been in development, use, and refinement since 2005. As is the nature of spreadsheets, all
calculating functions are built in as embedded formulae. The user doesn’t see this stuff, and has nothing to do, but data entry. “Data” consists of a list of ingredient names typed into successive
rows in one column of the first table, and the gram weights of each ingredient typed into the adjacent column. The worksheet does the rest of the work. It does so instantaneously, and error free.
What is a recipe calculator?
These tools calculate the total of the individual gram weights of all ingredients, and using that value, derives something very useful, the ingredient “Cook’s Percentages”. What “Cook’s Percentages”
means is explained in another section later, and in separate posts on this blog. What they are useful for is to enable instantaneous, one click, re-scaling of a recipe’s yield to produce more of the
product, or less.
I might just as well have called this a Recipe Writing & Recording tool because that is its primary utility. I might have dubbed it a Recipe Re-Scaling tool because that’s one of its greatest
virtues, or an All In One Recipe Writing, Developing and Costing tool. It does all of that. From the point of view of understanding the logical structure of things, being able to derive the
mathematical basis of a recipe was a more interesting project idea than designing a spreadsheet no matter how useful. The idea was to develop a kitchen recipe calculator that does for cooks what The
Unabaker’s Master Formula Calculator does for bread bakers.
The initial inspiration for creating The Kitchen Formula Calculator was to create a format to write, and record recipes I cook at home, and to be able to easily convert all of my old handwritten
(circa 1979) Pastry Shop recipes from commercial production yields to realistic small batch quantities suitable for home preparation. I don’t need 4 gallons of Crème Pâtissière, I need one liter. I
never make 150 crepes, I make 10. I would never dream of making 20kg of pie dough at home, I dream only of 1kg. The recipes need to be re-scaled to produce much less. This is a regular feature of
life in kitchens, re-scaling recipes to match daily production needs. Doing these kinds of mathematical calculations in your head, or by using a pocket calculator, and an appropriate multiplier takes
time, and it’s error prone. The Kitchen Formula Calculator does this very simply. “Formula” is a more precise word than recipe, but it’s the same thing.
If instead of writing down your Crème Pâtissière recipe in a hand written guide like I did in 1979, or in a simple computer file you write it using The Kitchen Formula Calculator, then besides being
a permanent record, the calculator derives a lot of useful information about the recipe. The Cook’s Percentages of each ingredient is an example. It auto-fills these percentages in a column in the
recipe table, and then uses them in a linked recipe re-scaling table that makes for instantaneous, and error free updates to the desired yield. With a single click, and a revised entry for the recipe
yield desired, the calculator immediately updates all ingredient quantities, without error, to produce the revised quantity.
What is a recipe?
There’s more to a recipe than meets the eye. There’s an unseen mathematical framework for any recipe. If you write down what you do when you make Pork Green Chile, you’d have a list of all the
ingredients used, how much of each, and a given methodology for what to do with the stuff. If you do this, you’ve written a recipe. That’s the common sense understanding of “recipe”, which is true
enough, but not enough. This understanding has limited practical ramifications. If you have a way to analyze the Pork Green Chile recipe to determine what are the ratios of each ingredient to the
whole, you can understand it more clearly. You can analyze any other recipe by entering it in the calculator. Obviously, it’s useful to keep a record of recipes from great Chefs, and other good
sources. The calculator is a way to store these things. Determining ingredient ratios is a very easy calculation, but cooks don’t do such stuff, and few understand the word “recipe” in these terms.
Ratios exist, but they aren’t specified. For the most part cook’s don’t even think about it. They know about ingredient balance, and understand the concept of ratios of course, but not what can be
done with the actual ratios if calculated. That’s why a Reverse Engineering calculator hasn’t been invented until now.
These ratios are what The Unabaker calls “Cook’s Percentages”, and to be precise, a recipe is its mathematical structure, i.e. the ingredient ratios expressed as percentages of the whole. A recipe is
its Cook’s Percentages. Just as Baker’s Percentages are the mathematical framework for bread formulae, Cook’s Percentages are the same for all other recipes. By this definition, given a list of
ingredients, and a Cook’s % value stipulated for each, you’ve written a recipe. Ingredient weights are derivative data, determined by comparing its Cook’s % to the total weight of the formula. It’s a
simple equation.
Ingredient weight = Total Formula Weight x Ingredient Cook’s Percentage
Methodological notes are obviously a practical necessity, but when we talk about a recipe, it’s really talk about a “formula”. The formula is the ratios of an array of stipulated ingredients to the
whole. Methodology simply informs the cook about mixing and handling. Often the “methodology” isn’t necessary to express. All of my old Pastry Shop recipes are simply lists of ingredients with
corresponding quantities specified, and a “yield” stated. Yield in a very casual sense means how many portions it makes. The Cheesecake formula makes 7, 10” cakes. Experience, and repetition teaches
the aspiring pastry cook what to do with this basic data.
Ratios can be expressed many ways. One part of this, three parts of that; 1:3; or 1/3, but they can also be expressed as percentages. When done so, interesting things can be learned, and made to be
useful. How do we express the above ratio as a percentage? It’s tempting to say that 1 is 33.3% of 3, but that would be wrong in the first two examples. The first two examples above have four total
parts. There’s three times as much of one compared to the other. 1 is 25% of the total, and 3 is 75%. The use of percentages is very clear. Percentages are also easy to express mathematically, and to
utilize in equations. The Kitchen Formula Calculator does that.
The actual ingredients list, and the quantities of ingredients used to make Pork Green Chile are flexible, reflecting a cook’s style, and preferences, or whim. If you change ingredient quantities,
but retain the ratios of each to the other, you’ve not changed the recipe, you’ve simply changed the yield. To change a recipe, you have to change the ratios of ingredients, or change the list of
ingredients, or both. Ratios are the mathematical superstructure of a recipe. Ratios are recipe logic.
Why are Baker’s Percentages vital, but understanding Cook’s Percentages is not?
The difference between the recipes cooks use, and bread formulae that baker’s use is there’s greater leeway for a cook to adjust the ratios of recipe ingredients without fundamentally changing the
resulting product, and most importantly, ratios of ingredients can be changed during the cooking process. In fact, adjusting the ratios during cooking is advisable methodology. Taste it as it cooks,
add more of the critical ingredients as deemed necessary, not all at once to start. This is why “salt to taste” is a common proviso. Such in-process tinkering is not the case for most baked pastry
items, and never for bread once in the oven. The formula must be properly assembled prior to baking. You don’t get a Mulligan. There’s a little bit of truth in the statement “baking is Science,
cooking is Art”.
Just because you do not need to think about, or even know that such things as Cook’s Percentages exist to cook well, this doesn’t mean you shouldn’t know how they can be utilized. Understanding the
mathematical structure of a recipe means you can use it productively. The time wasting, often error prone process of doing recipe yield re-scaling is eliminated. As noted above, it was a prime cause
for developing the recipe writing tool you are reading about. It might be helpful to understand the rest of this article by opening the tab titled “kitchen fc v4 template”.
Cook’s Percentages Explained - The Mathematical Basis for both Calculators
Akin to Baker’s Math, and Baker’s Percentages in that it forms the logical structure of the recipe, The Kitchen Formula Calculator uses what I call Cook’s Percentages. Unlike bread formulae, kitchen
recipes have no single formula basis such as flour serves for Baker’s Math. In Baker’s Math, the ingredient weights are calculated with reference to the total weight of flour used. Baker’s
Percentages for all ingredients are an expression of the ratio between the ingredient weight, and the weight of total flour used. Bread formulae proceed from stipulated Baker’s Percentages for all
ingredients to derive the weights for all ingredients. Baker’s do not write recipes using ingredient weights, they use Baker’s Percentages. By stipulating a Total Dough Weight desired the individual
ingredient weights can be derived.
In Cook’s Math, the Cook’s Percentage for an ingredient is the ratio that the ingredient represents as a portion of the whole formula. There’s no reference to any other ingredient to determine the
value. Each ingredient is expressed as a certain percentage of the whole. Each would be a slice of pie in a Pie Chart. The total of all ingredient Cook’s Percentages equals the Total Formula Cook’s
Percentage. This total is reported at the bottom of the column titled Cook’s %. Unlike Baker’s Math, and Baker’s Percentages, the Total Formula Cook’s Percentage always adds up to 100%, never more,
never less. In Baker’s Math the Total Formula Baker’s Percentage always adds up to more than 100%. This is the singular feature that differentiates Baker’s Math from Cook’s Math. All of the values
for ingredient Cook’s %, and the Total Formula Cook % are calculated automatically by the Kitchen Formula Calculator. It is impossible for a recipe entered in the Kitchen Formula Calculator (or
RCTv1) not to total 100% at the bottom of the Cook’s % column.
The values for each ingredient Cook’s Percentage is derived based upon the ingredient’s weight compared to the Total Formula Weight. The calculator figures out the values of each ingredient’s Cook’s
Percentage using a simple formula that is part of the embedded spreadsheet calculating formulae.
Ingredient Weight ÷ Total Formula Weight = Ingredient Cook’s %
(given 625 grams water and 1000 grams total formula weight, the Cook’s % of water = 62.5%)
The Recipe Category Template was made necessary because most households don’t do menu writing, and don’t hand out packets of menu responsibilities to the inhabitants. Home cooks just need a
convenient place to write or record recipes, and to categorize them so they are easy to find when needed. The RCT allows the user to organize recipe entries by category. It’s exactly the same thing
as the KFC, but instead of titling tabs to enter single menu item plate data, the tabs are titled by type of recipe. The KFC has professional kitchen utility. The RCT has a more home cook appeal, but
can also be used by Chefs to bring order to a repertoire of recipes that vastly eclipses the home cook’s. Since they are otherwise identical, understanding how to use either one, means you also
understand the other. Set up the Recipe Category Template on a separate spreadsheet file, then set up tabs titled by category.
What exactly does a Kitchen Formula Calculator do?
The KFCv4 was designed to be a Chef’s tool for writing, recording, testing, developing, and recipe costing. By duplicating the template on successive tabs, a single menu item’s recipes (and their
sub-recipes) can be written and recorded. It is an error free method of doing the work, and it’s simple to use. The tool itself is accurate to whatever increment of grams (or imperial measure) you
want. Besides its advantage for writing new recipes, or recording ones already in hand, it was designed to flawlessly, and instantaneously upscale or downscale the yield of any recipe entered. This
is one of its most useful features. Revising recipes to produce more or less than it does is something that happens in every kitchen regularly. The Kitchen Formula Calculator and its offshoot, the
RCT, do it with a single bit of data input required. The required data input is whatever the new recipe yield (Total Formula Weight) is desired to be. One entry and click, the entire recipe updates
with new ingredient quantities.
How Does the Kitchen Formula Calculator work?
The user enters data for ingredients, and ingredient weights into two side by side columns of the recipe table. There’s also a column to enter a recipe if it’s written using volume measures, but
volume measures must be converted to gram weights. The calculators demand weights, not volumes, Specifically, metric weights (grams) are required. There’s no way to utilize a volume to perform
calculations. There’s more about this in another post on this blog. Once the user has entered the ingredients list, and their gram weights, the calculator calculates the ingredient Cook’s
Percentages. The calculator continuously updates. With every new piece of data entered for an ingredient weight, the entire table updates. The values for ingredient cook’s percentage change with
every new data entry. When all ingredient weight data has been entered, the final values of each ingredient Cook’s Percentage are displayed.
Totals for Total Formula Grams, and Total Formula Cook’s Percentage are tabulated at the bottom of the respective columns. The Total Formula Grams, is auto-filled into the cell in the upper left
corner of the table that is titled Total Formula Weight. TFW is the formula yield. It’s a number of grams. All of the data, and calculated values from the recipe table are auto-filled into the
subsequent tables used to Re-Scale, Re-Formulate, or to perform Recipe Costing. Very little other work is required of the user. For Re-scaling the recipe yield, simply enter in the desired new weight
in the Total Formula Weight cell. Everything is immediately updated to display all ingredient weight changes. To Re-Formulate a recipe simply write in whatever new gram weights of ingredients you
want, and to add or delete ingredients, just type over the auto-filled ingredients list.
Why use metric weights?
Kitchen recipes are commonly written using volume measures, or weight measures, or a combination of both. This is so in America with standard measures, in the UK with imperial measures, and in all
other places that use the metric system. Only metric weights are used by these calculators. The calculators have columns which automatically convert grams to ounces, but the ounce column is not used
in calculating formulae. Metric volume measures must be converted to the equivalent gram weight. By using gram weights for ingredients, Cook’s Percentages can be divined. Grams are very small
creatures, 28.35 gram = 1 ounce. This makes measuring precise. And recipe costing, if you have to do it, is easier. Not easy, but easier.
I will not rehash the lengthy article I’ve written about why things based on the numbers 1, 10, 100, and 1000 make better sense than things based on the numbers 16 for weights, or 12, 36, 1,760 or
5,280 for distances, or why there need be 16 tablespoons, or 48 of teaspoons to fill a cup. This seems an obvious choice. Nevertheless, digital scales have been invented that enable the cook to
convert ounces and pounds to grams with just the push of a button. If you are a cook, you need a scale. All cooks need scales. Chefs use scales. Be like Chef! Get one! They’re cheap.
With this in mind, the Kitchen Formula Calculator proceeds from the specified gram weights of ingredients to derive the Cook’s Percentages for each ingredient. Recipes entered using any volume
measures need to be converted to weights. If volume measures are used, as is often the case, then an ingredient volume to weight conversion table has to be referenced. The Unabaker has created such a
table. It is included as a separate tab in this worksheet. There are other such references on the web that are more extensive.
Many cooks might simply use the calculator to record recipes using the ingredient volume measures plus weights that still typify most recipe writing, and that’s fine. It really is up to the user, but
doing so sacrifices one of the calculator’s features, viz being able to re-scale the recipe very pain free. It takes some work, but it’s better to convert any standard or imperial measures to a value
of metric weight. For example, when you are preparing the recipe ingredients according to quantities required, you put the teaspoon full of whatever it is, or the tablespoon, cup or quart of it into
a receptacle on a digital scale that has been zeroed to eliminate the weight of the receptacle, and then mode set the scale for grams. The scale reports the gram weight of the ingredients in the
receptacle. Note the gram weight. Enter it into the calculator table, and that’s how you eventually develop an ingredient conversion table. Metaphysical measures such as pinches can be guesstimated
(or ignored), but list the ingredient in the table, and don’t forget to pinch it for adding to the recipe mix.
There is a column to enter volume measures, if any, for each ingredient, and one for the converted values in grams. Once this is done, using the Re-Scaling calculator, the cook can easily,
instantaneously, and automatically adjust the recipe yield as desired depending on perceived needs.
The Design & Function of The Kitchen Formula Calculator
It’s important to note that the tables contain color, and grey-filled cells, and other cells that remain white. White cells are for the user to enter data. Color, and grey-filled cells contain the
formulae that do the calculations. The color and grey-filled cells are where the calculated results are displayed when data entry is performed. Never enter data into these cells. If you do, then
simply go to the menu bar, click Edit, then click “undo” as many times as necessary to recover. Note also that there are short notes attached to each table next to to the Total Formula Weight cell
(in the upper left of the table) that explain what sort of things each table is designed to do.
The template is duplicated, and the duplicate used to enter recipes. The template itself remains blank. A convenient way to use the calculator is to duplicate the template onto as many tabs as you
have categories of recipes. This way you can record numerous recipes of the same category on one tab. It is up to the user to determine what categories are needed in order to properly organize, and
keep readily at hand the user’s entire collection of recipes. Instead of having a recipe file with dozens, maybe hundreds, or in the case of a professional Chef’s career, thousands of tabs with
single recipes, your file will consist of category tabs that collate entire sets of recipes for similar type preparations, and puts it in one place. The worksheet can be easily edited. The user sets
up categories that make sense for one’s needs. Sauces, Soups, Cookies, Pie Fillings, Pastry Doughs, Cold Desserts, Italian Recipes, Mom’s Recipes, etc etc. As the numbers of category tabs increases,
these can be shifted around alphabetically. Scrolling within a category tab to find what you’re looking for is simple.
The actual layout of The Kitchen Formula Calculator consists of three blank recipe writing tables, and one recipe costing table that appear side by side from left to right across the page. The tables
are interconnected. Enter data in the first table, and most of the data is auto-filled into the other three tables. The tables are titled Reverse Engineering Calculator, Re-Scaling Calculator,
Reformulation Calculator, and Recipe Cost Calculator. The tables each feature 16 ingredient lines, enough for even complex recipes. You can easily add more ingredient lines to a table by adding rows
if necessary. Unused, unnecessary rows can be hidden to neatly tailor the look of the table. The recipe cost table is something most home cooks will never use. If so, the entire table can be hidden
by hiding the columns it resides within.
The Reverse Engineering table is the first table, and it’s where a user either writes a recipe from scratch, or records a recipe already in their collection. Oftentimes, a recipe entered into the
Reverse Engineering table has been tested, and known to work, so the only other thing to do might be to re-scale it to produce more or less total yield as needs change. “Reverse Engineering” simply
means that the Cook’s Percentages are unknown, but they can be derived from data that the user types into this table. It is a term that makes better sense in the bread formula calculator, but I use
it here to preserve lexicological parsimony. In the next section I explain in a little greater detail what’s meant by Reverse Engineering. Anyone familiar with The Unabaker’s Master Formula
Calculator will see the similarities in design, graphical layout and function.
Next to the Reverse Engineering table is the Re-Scaling table. It’s very simple to use because all of the data required in all the cells of this table is auto-filled from what’s been entered into the
Reverse Engineering table. The Re-Scaling table allows the user to instantaneously upscale or downscale the yield for the recipe that has been entered in the Reverse Engineering table. Simply enter a
new Total Formula Weight.
Next, there is a Reformulation table which enables the user to make whatever tweaks to the ingredients list, or to ingredient weights as desired once a recipe is prepared, tested and tasted. It can
also be used simply to round up, or round down gram weights. This table is useful for recipe development, and for cook tests of a recipe under development. Both home cooks, and professionals do this
every time they first use a recipe, whether it’s one they wrote, or one they found in another source. “I like The Unabaker’s Brownie recipe, but I think it needs more cocoa, a bit less sugar, maybe a
tad more vanilla”. So be it.
Finally, there’s the Recipe Costing table. This is a necessary table for professional chefs, but not for home cooks. The table only requires that the ingredient costs per gram be entered, but to do
that the user has to perform costings of all ingredients. This is done separately because ingredient costing is a step that is not provided for in the current iteration of the calculator. Done
separately, the results are simply reported in the Recipe Costing table. To do recipe costing is a pain in every Chef’s neck.
Together the four tables comprise, an error free calculating tool. The idea is for a user to repeatedly duplicate The Kitchen Formula Calculator template on as many new tabs as there are numbers of
categories. Each cook will have a different idea about what categories are set up. Once a category tab has been set up, the user can begin entering recipes of that type. The template has been
initially designed to be able to write/record five recipes on each tab. It can be easily expanded to write as many more per tab as the user’s needs may require. Simply copy/paste the tables into rows
below every time you need to add another recipe to the category.
What is Reverse Engineering?
As noted, it simply means that the ingredient ratios are unknown, but they can be derived from data that the user types into this table. This makes more sense for the bread calculator because bread
formulae are most of the time conceived and written starting from baker’s percentages for each ingredient to derive ingredient weights. Why then is there a Reverse Engineering table in The Unabaker’s
Master Formula Calculator? Often times a baker comes across a bread formula that looks really good, and wants to try it, but the writer of that formula wrote it incompletely, giving only ingredient
weights. This is very common, even in some cookbooks. So, the RE table in that calculator can “reverse engineer” the Baker’s Percentages by entering the ingredient weights given in the incompletely
written bread formula. The calculator figures out the ingredient Baker’s % values based on weights entered.
Normally, the work flow of a bread formula goes from the ingredient Baker’s % to ingredient weight values. Reverse Engineering means you reverse the flow of the calculation from ingredient weights to
ingredient Baker’s % values. The Unabaker’s Master Formula Calculator thoughtfully created this unique way of translating incomplete recipes found in many sources, and converting them into formulae
that make sense for bakers. This is an innovation that had not existed until The Unabaker came along.
That’s all well and good, but the fact is that kitchen recipes, never ever rely on Cook’s Percentages to figure out ingredient weights. Recipes other than for bread are always written using volumes,
and/or weights, or combinations of both. No ratios need apply. So why is there a Reverse Engineering table in the Kitchen Formula Calculator? I could have given it another name like Recipe Writing
table (that’s precisely what it’s for), ignored the mathematical framework of the recipe, and simply designed the Kitchen Formula Calculator as another handy recipe writing and recording tool.
However, as noted, The Unabaker wanted to be able to see the mathematical framework of a recipe, both for the logical understanding, and appreciation of that, but also for a very simple practical
purpose, to be able to re-scale any recipe flawlessly, and instantaneously. To do that, the ingredient ratios that exist like a shy child behind the recipe had to be made to come out and play. The
Reverse Engineering table in the Kitchen Formula Calculator does so. Unlike the baker, no cook will ever write a recipe that begins by specific Cook’s Percentages to derive ingredient weights.
Nevertheless, these ratios have a valid application.
The way The Unabaker wrote the formulae that power all of his calculators is one way of doing it. I could have done it otherwise. This is mathematics, so there are mathematically equivalent ways of
arriving at the same result. What was chosen was the simplest mathematics. Spreadsheets are not rocket science. The formulae for doing most recipe calculations are not complex. The Unabaker’s Master
Formula Calculator had already been designed, and refined over the past 20 years, and the lexicon it uses evolved along with it as well. It seemed obvious to make the new Kitchen Formula Calculator
follow a very clear, and well-developed format, employing the same lexicon, even if, in the case of “reverse engineering”, it’s not perfectly appropriate. The rare cooks and bakers that stumble upon
this work, and become familiar with all of the calculators would understand the value of logical, and linguistic symmetry.
Why these Calculators? Why Should I use them?
A professional kitchen guide (aka Kitchen Bible) is the basic recipe reference used by professional chefs in their kitchens. The Kitchen Formula Calculator, and the Recipe Category system presented
here is a very useful and organized way to assemble such a guide. Home cooks do the same thing, but with typically less rigor. A kitchen drawer full of clippings, a binder of stuff, a box of index
cards, napkin jottings, computer files, bookmarked links to various website resources, archived emails and WhatsApp chats etc. All cooks face the same issue which at some point becomes an
organizational task, usually neglected, to keep things straight and readily at hand. The organizational idea behind all of The Unabaker’s formula writing tools is to create one standard format for
writing or recording recipes. Tabs can be created for as many categories as needed for one’s kitchen operation: soups, sauces, dips, pickles, breading, fish, meat, 2024, party ideas, mom’s recipes,
kitty cat, etc, and the same goes for the many different categories of pastry production. As many categories as desired can be set up quite simply.
An example of how and why to use the calculator tools is the ongoing task of converting my entire Pastry Shop recipe guide which I’ve used, and added to over the past 45 years. Those recipes were
written using a mix of weights, and volume measures, typical of American kitchens: teaspoons, tablespoons, cups, quarts, ounces and pounds. The recipes are written to yield large production
quantities of the products. I still use many of these recipes. I know them well, and they are reliable, but every time I do so, I must convert the yield from 10 pies, or cakes, or batter enough for
150 crepes etc to realistic home needs. This is what made me start to think how to design a calculator that would do this quickly, and accurately, and to store the work for use again. I came up with
the Kitchen Formula Calculator, and the concept of Cook’s Percentages to make it work.
The process of inputting the original big yield recipes into my Reverse Engineering calculator, and then using the Re-Scaling table to revise the yield to produce just enough for a single 8”
cheesecake became simple. There are too many recipes in my old Pastry Shop guide to want to sit down, and do data entry endlessly. Instead, I do it as the urge to make something from that guidebook
strikes, and it takes only a few minutes to do it. After which I can re-scale the original to something more realistic for home baking purposes.
Take a look at the four calculating tables again. Note how they look almost identical, and note how little, and where they differ. The Re-Scaling table has all of the exact same data displayed in all
of its cells, as the Reverse Engineering table, except for one cell, namely the Total Formula Weight. The Reformulation table has the exact same layout as the previous two tables, and it looks just
like the Reverse Engineering table. The Total Formula Weight in both tables is an auto-calculated value, and both leave the ingredient gram weight columns blank for data entry. When you copy/paste
the ingredient gram weights column of the Reverse Engineering table into the corresponding column in the Reformulation table, it will look precisely the same as the Reverse Engineering table. The
user can proceed to revise as many ingredient weights as desired (in the Reformulation table), and/or type over the ingredient names to add, or delete different ingredients. With every edit, the
Reformulation table updates accordingly.
Note that the Recipe Costing table is also very similar, but because it has a very specific task to perform, appropriate columns, and calculating formulae apply. The Recipe Costing table is a feature
for Chef’s, and requires the Chef to perform separate calculations for each ingredient. The home cook will likely never use it except as a matter of interest to see how it works.
The v4 appended to the name of The Kitchen Formula Calculator indicates that it has been evolved from three previous iterations. It took me a few to get to what I wanted it to be, and to do. I do not
see any changes necessary at this point. What remains for me is the ongoing task to add my recipes to category tabs. I can do it at my leisure as I use a recipe, or what's more likely given my
habits, I’ll sit down one week and get it all done, then never have to scroll through hundreds of tabs to find the one I need.
These are easy to use tools. If you have not had enough yet, then read about Baker’s Math below.
Baker’s Math Explained
All of bread making revolves around a singular ingredient, namely flour. Baker’s Math uses the total weight of flour used in a bread formula as the basis for deriving all other ingredient weights. It
does this mathematically by comparing the ratios of all ingredient weights to the total weight flour used. The Unabaker’s Master Formula Calculator v17 is a spreadsheet wherein all the required math
has been done, and embedded as calculating formulae. The user simply enters data such as Total Dough Weight desired, and then specifies either ingredient Baker’s Percentages, or ingredient weights
depending on whether the table used is the Master table or the Reverse Engineering table. Everything else is done automatically. It’s extremely accurate, error free, and not hard to figure out how to
Bread formula ingredient ratios are expressed as ingredient Baker’s Percentages. The Master Formula Calculator uses Baker’s Percentages to derive ingredient weights. This is the normal direction of
the work flow when writing a bread formula. The problem is that many recipes you might encounter fail to write the formula with Baker’s Percentages stated. Therefore, The Unabaker designed a way to
reverse engineer such incomplete formulae to derive the ingredient Baker’s Percentages if only the ingredient quantities are specified. To my knowledge there is no such tool available anywhere else.
It’s an entirely innovative device. Just as for the Kitchen Formula Calculator, there is a Reverse Engineering table that can be used to figure it out.
Baker’s Math uses the total weight of all flours used in a formula as the formula basis for determining the other non-flour ingredient weights in the formula. Whether a formula has known or unstated
Baker’s Percentages, the Master Calculator can be used. Baker’s Percentages have to be known to fully use the tool. Just as for The Kitchen Formula Calculator, this is how re-scaling the formula
yield can be done.
Every ingredient in a formula has a baker’s percentage value. Flour is the most important because it’s the formula basis. The baker’s percentage of Total Formula Flour is always 100%. It’s the
formula basis. Every other ingredient’s baker’s percentage is a ratio of that ingredient’s weight to the weight of Total Formula Flour used. The peculiarity of Baker’s Math is that all bread formulae
always adds up to more than 100% Total Baker’s Percentage because flour alone is 100%. More complex bread formulae can easily total over 300%. A basic Sourdough or Poolish bread formula (Pain
Ordinaire) will likely total no more than 167% to 177% depending upon the degree of hydration (percentage of water) a baker uses. There’s no need to puzzle over this oddity because the calculator
knows what to do.
Understanding the concept of Baker’s Percentage is a mathematical way of understanding a bread formula, and for comparing and analyzing different versions of the same type bread made by different
bakers. When the mathematical structure is understood, baker’s can predict certain things about the final product: it’s texture, dough feel, the physical process phenomena that occur during baking,
and also how the mixing and handling will change depending upon the recipe hydration (amount of water ratio used), and if other things like eggs, or some type of fat have been added. The baker’s
percentage for any ingredient simply means the ratio of the weight of that ingredient to that of Total Formula Flour.
To illustrate things, a simple bread formula such as typical French bread, aka Pain Ordinaire, calls for 1000 grams of total flour, 650 grams of water, 20 grams of salt, and 10 grams of yeast.
Bearing in mind that Total Flour always has a baker’s percentage of 100%, this formula has a mathematical structure as follows: 100% flour, 65% water, 2% salt, 1% yeast. Added together, there’s a
Total Formula Baker’s Percentage of 168%. These ratios comprise the mathematical structure of the formula. The ratios of the ingredients are the recipe. What actual ingredient quantities be required
is logically irrelevant, even though it’s a practical necessity to know. We can know them by simply specifying how much Total Dough we desire to make. If the Total Dough Weight desired is 2000 grams,
then the amounts of each ingredient can be determined using a simple formula.
Total Formula Weight ÷ Total Formula Baker’s Percentage = Total Flour Weight.
Consequently, all other ingredient weights are simply calculated.
Total Flour Weight x Ingredient Baker’s Percentage = Ingredient Weight.
In the example above the Total Dough Weight desired is 2000 grams, and the Total Formula Baker’s Percentage is 168%. To determine the weight of flour in the above example the calculation is as
follows: (2000 ÷ 168) x 100 = 1190.47 grams. You must multiply the value of the parenthetical equation by 100 if you use the number 168, or a mathematically equivalent way of doing it is to change
that formula to 2000 ÷ 1.68 = 1190.47. Both methods make the same calculation, but differently, and yield the same value. Calculating the weights of the other ingredients is simply a matter of
multiplying the weight of total flour, i.e. 1190.47, times each ingredient baker’s percentage:
water = 1190.47 x .65 = 773.8 salt = 1190.47 x .02 = 23.8 yeast = 1190.47 x .01 = 11.9
Obviously the baker would round the values to 1190 + 774 + 24 + 12 = 2000.
Baker’s Math requires that the Total Baker’s Percentage of all flours called for in the formula always adds up to 100%. If more than one flour is used, then each flour ingredient represents a portion
of the 100% of Total Formula Flour. For example, French T55 = 70%, T65 = 25%, Light Rye = 5%.
Baker’s Math is the logical framework of a bread recipe. Ingredient weights are always derivative values, fluctuating according to the Total Dough Weight of the formula that’s specified by the baker.
Every day in a bakery, the TDW for the baker’s formulae can change depending on that day’s projected sales volume for each product, or simply by the whim of the baker. The logical structure of the
bread recipe is the same until the baker changes the Baker’s Percentage value for any of the ingredients. If you change the baker’s percentage value for any ingredient, you have changed the formula
because even a small change to one of the percentage values causes all of the remaining percentage values to change. Baker’s usually think about bread formulae in terms of the Baker’s Percentages,
not the ingredient weights. Cooks don’t ever think this way. Recipes are written in ingredient weights, or in volume measures, or both, and Cook’s Percentages are unknowns.
It is commonly understood that a recipe is simply a list of ingredients and their quantities, plus a specified method of putting it together. “Quantity” can be variously expressed either in volumes
required (teaspoons, tablespoons, cups,) or by their weights (ounces, pounds, grams, kilograms). This common perception is not correct. In fact, the ratios of ingredients is the recipe. They comprise
the logical structure of the recipe. Ingredients must be expressed as weights in order to use Baker’s Math (and to provide the mathematical basis for The Unabaker’s Kitchen Formula Calculator).
Ingredient weights are always merely derivative values, fluctuating depending upon the total yield (Total Dough Weight) specified for the formula. The ratios, however, remain the same, until they are
changed by the baker. When the ratios, i.e. the Baker’s Percentages change, the formula changes.
These ratios are the underlying logic, the mathematical basis of the baker’s formula. If understanding this, then it becomes a fairly simple task to write a new formula based upon changed ratios, and
a stipulated Total Dough Weight desired. This is a very common daily production management scenario in professional shops. Determine how much of each of the bakery’s offerings is required to meet the
day’s expected sales, and scale the total yield up or down accordingly. Baker’s Math and Baker’s Percentages are a special way of designing formulae. Total Dough Weight, and Ingredient Baker’s
Percentages are the foundational data required.
A Final Word
Imagine this! A baker writes a formula that specifies 100% flour, 85% water, 2% salt, 1% yeast and 5% oil. Can you guess what he is making? Most bakers can tell you it’s Focaccia bread. That’s how
burrowed in is the concept of Baker’s Percentages in the brains of bakers.
If using The Unabaker’s Master Formula Calculator, the baker can plug these percentages into the table, and type in a Total Formula Weight. Doing so the baker will get an immediate report about the
precise ingredient weights to use. How much to yield depends on how much baker expects to sell, but the value for yield, i.e. Total Formula Weight could be something silly like 103.347 grams for a
single Focaccia roll, or it could be silly in the other direction, 10,285.37 grams. Either way the Master Formula Calculator will precisely calculate all ingredient weights to whatever decimal point
is desired. In either case the total gram weight of the two formulae will be reported at the bottom of the grams column in the tables, and it would be exactly 103.347 and 10285.37.
Now imagine this! A cook writes a recipe for Pork Green Chile. The cook has not the slightest clue about Cook’s Percentages, so of course, cook specifies something like this:
5# pork shoulder, cut in large chunks
3/8 cup garlic, chopped
3 large yellow onions, chopped
6-8 pc dried green chili, seeds removed
1 tbsp crushed roasted coriander seed
1 tbsp crushed roasted cumin seed
2 or 3 sprigs Mexican oregano
1 or 2 sprigs epazote
2 tsp ground black pepper
1 small chunk of piloncillo
1/2 cup roasted Masa Harina
5 btl Tecate Beer
4 qts water
1 big bunch of cilantro, chopped
salt to taste
Can you figure out the mathematical profile for this recipe? What are the Cook’s Percentages? No, of course you can’t. Can you upscale it in your head by a factor of 2.3? Probably not. That’s why you
need The Kitchen Formula Calculator. What most cooks can do is to figure out a methodology for cooking it based on their experience. Most cooks could identify this as some type of stew, a spicy one,
probably Chile, and they could make a representative rendition.
After reading this entire article, if you are not injured sufficiently already, you can harm yourself more by checking out the YouTube channel The Unabaker. I am not advising anything, I’m merely
The Kitchen Formula Calculator v3 is a formula writing tool I developed in 2020. It’s specifically designed for recipes that cooks, not bakers, make. In other words, everything, but bread. It
provides an efficient format for composing recipes, is easy to read, and using it quickly becomes second nature after a few entries. The Kitchen Formula Calculator is a record keeping device, a
recipe development tool, and an analytical tool. To be able to compare various similar formulae is something all cooks do. For this, it’s exceptional. If you’re a culinary instructor, using The
Kitchen Formula Calculator teaches the metric system, the virtues of singular units of measure, describes a very useful recipe formatting style that can serve students their entire careers, and could
be a central part of a course that teaches proper recipe writing skills. If this is not part of the syllabus, it ought to be. If you write commercial articles, or make professional seminar
presentations, it will help
For years, the Chef resisted all of McDoo's entreaties and cajoling. The pleading and rationalizing, the gentler persuasions, as well, the occasional conversation ending throwaway commentary about
hard headedness, willfulness, and "stubbornosity" (a classic McDooism). All efforts of his longtime friend waved away with only bare considerations given. But, McDoo, stranger he was not to the old
Chef's grumpy self assurance, persisted. Eventually, even boulders crumble, he reasoned. "Dammit Baker! What a pity? What luminous dumbness? What selfishness? How grossly inconsiderate, this hardness
of heart! Damn your stoic intransigence! A pall upon kindness and duty! What about History, Philosophy, Art? What of Science, Aesthetics? How does Gastronomy carry on? Your place in the pantheon?
What point be our petty lives if we cannot be remembered? If we cannot relinquish what we've learned?" McDoo's been known on occasions to put on grand airs; a flair for th
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These notes reflect a learning curve that began as a chef’s apprentice in 1979. By 2005 during a less glorious career phase I had reached a temporary stopping point, so I began a compilation of my
Chef’s notes as a distraction. Originally intended as a gift for my first-born son which I hoped would put the baker's fire in him, it took initial form as a short and sweet how-to guide with an
accompanying formula calculating tool, and a jar of sourdough culture. I’ve been reading and studying baking, and adding notes ever since. The original very modest formula calculator has undergone
fifteen updates, each one elaborating and refining the design, as well as functions and practical applications. I intend to post it in the near future, along with a quick start guide, plus a much
more detailed guide for more advanced bakers. By the time this project began, my son was 22. He had trained in my kitchens, wetting his feet at age nine, and growing up surrounded by talented cooks.
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How to use SKEW.P Function in Excel
Returns the skewness of a distribution based on a population: a characterization of the degree of asymmetry of a distribution around its mean.
Syntax:= SKEW.P(number 1, [number 2],…)
The SKEW.P function syntax has the following arguments:
• Number 1, number 2,… Number 1 is required, subsequent numbers are optional. Number 1, number 2,… are 1 to 254 numbers or names, arrays, or reference that contain numbers for which you want the
population skewness.
SKEW.P uses the following equation:
Example: Let’s look at some Excel SKEW.P function examples and explore how to use the SKEW.P function as a worksheet function in Microsoft Excel:
Example of SKEW.P Function in Excel (Positively Skewed in Excel):
Column A has a distribution of values. Skewness these values can be calculated using formula
Syntax: =SKEW.P(A2:A21)
Result: 0.457584052
as shown in the above example. This result in the value of 0.457584052. Which indicates positive skew.
Example of SKEW.P Function in Excel (Negatively Skewed in Excel):
Column A has a distribution of values. Skewness these values can be calculated using formula
Syntax: =SKEW.P(A2:A21)
Result: -0.714243489
as shown in the above example. This result in the value of -0.714243489 which indicates negative skew.
• Arguments can either be numbers or names, arrays, or references that contain numbers.
• Logical values and text representations of numbers that you type directly into the list of arguments are counted.
• If an array or reference argument contains text, logical values, or empty cells, those values are ignored; however, cells with the value zero (0) are included,
• SKEW.P uses the standard deviation of an entire population, not a sample.
• If arguments are values that are not valid, SKEW.P returns the #NUM! error value.
• If arguments use data types that are not valid, SKEW.P returns the #VALUE! error value.
• If there are fewer than three data points, or the sample standard deviation is zero, SKEW.P returns the #DIV/0! Error value.
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6.17 Example: Tees Bay... | PRIMER-e Learning Hub
6.17 Example: Tees Bay macrofauna
The final example in this chapter is of a mixed nested and crossed design B$\times$C(A), for a total of 192 macrobenthic samples (282 species) from: A: four sub-tidal Areas of Tees Bay (Fig. 6.18,
top left), with C: two Sites from each area, the same sites being returned to each September over B: 24 Years (1973-1996), part of a wider study of the Tees estuary, Warwick, Ashman, Brown et al.
(2002) , {t}. Sites (C) are therefore nested in Areas (A) but crossed with Years (B). There was a further level of replication, with multiple grab samples collected but these have been averaged to
give a more reliable picture of the assemblage on that occasion (the repeat grabs from a single ship stationing being considered ‘pseudo-replicates’ in time, and possibly space). The areas lie on a
spatial transect (c. 5km spacing) but are probably not ordered hydrodynamically, so we shall contrast both ordered and unordered tests for A (cases 3m/3j in Table 6.4). The years are also amenable to
analysis under either assumption: as it happens, there is a clear annual trend in assemblage structure over the period (seen in the right-hand plots of Fig. 6.18, for the two sites in each area
averaged), but the prior expectation might have been for a more complex time signal of cycles or short-term changes and reversions, so this data will serve as an illustration of both the case of B
ordered or unordered (cases 3l/3j). There being only two sites in each area, it is then irrelevant whether C is considered ordered or not; with no real replication, there can be no test for a site
effect from only two sites (though there would be a test with a greater number of sites, either ordered or not, 3k/3j).
Fig. 6.18. Tees Bay macrofauna {t}. Map of four sampling areas in Tees Bay, NE England, and separate nMDS time-series plots for each area, of the macrobenthic assemblages over 24 years of September
sampling; abundances were fourth-root transformed then averaged over the two sites in each area, then input to Bray-Curtis similarity calculation. Bottom left plot is the nMDS of averages of
transformed abundances over the 24 time points for the two sites (a-b, c-d, e-f, g-h) in each of the four areas.
Test for Area factor (A)
The schematic below displays the construction of the ANOSIM permutation test for area (A), case 3m/3j^¶.
The building blocks are the 1-way ANOSIM statistics $R$ (or $R^{Oc}$ if A is considered ordered) for a test of the 4 areas, using as replicates the 2 sites in each area, computed separately for each
year. These are then averaged over the 24 years, to obtain the overall test statistic for A of $\overline{R}$ (or $\overline{R}^{Oc}$), exactly as for the usual 2-way crossed case A$\times$B met on
page 6.7. The crucial difference however is in generating the null hypothesis distribution for this test statistic. Permuting the 8 sites across the 4 areas separately for each year, as the standard
A$\times$B test would do, is to assume that the sites are randomly drawn afresh each year from the defined area, rather than determined only once and then revisited each year. The relevant
permutation is therefore to keep the columns of this schematic table intact and shuffle the 8 whole columns randomly over the 4 areas, recalculating $\overline{R}$ (or $\overline{R}^{Oc}$) each time.
There will be many fewer permutations for the A test under this B$\times$C(A) design (8!/2!2!2!2!4! =105 for the unordered case, compared with $105^{24}$) but what it loses in ‘power’ here it may
make up for in improved focus when examining the time factor: subtle assemblage changes from year to year may be seen by returning to the same site(s), and these might otherwise get swamped by large
spatial variability from site to site, if the latter are randomly reselected each year.
If area is considered an unordered factor, $\overline{R}= 0.60$, a high value (and the most extreme of the 105 permutations, so p = 1%); this is clearly seen in the time-averaged MDS plot for the 8
sites (Fig. 6.18, lower left). If treated as an ordered factor, the area test gives $\overline{R}^{Oc} = 0.13$, now not even significant. These two $\overline{R}$ values are directly comparable; both
are slopes of a linear regression of the type seen in Fig. 6.13b, with the same y axis values but only two rather than four x axis points in the unordered case (within and among groups, as earlier
explained). The MDS plot of sites in Fig. 6.18 makes clear the down side of an ordered test, based solely on the NW to SE transect of areas: here the middle two areas are within the confines of Tees
Bay, their assemblages potentially influenced by the hydrodynamics or even anthropogenic discharges from the Tees estuary. Thus areas 1 and 4 are rather similar to each other but differ from areas 2
and 3. Opting for what can be a more powerful test if there is a serial pattern risks failing to detect obvious differences when they are not serial, as illustrated below for one of the 24 components
of the average $\overline{R}$ and $\overline{R}^{Oc}$, namely the $R$ and $R^{Oc}$ constructions for 1978:
Test for Year factor (B)
Turning to the test for the Year factor (B), case 3l/3j in Table 6.4, the schema for construction of the test statistic in both ordered and unordered cases is now:
When years are considered ordered, the test reduces to the 2-way crossed layout B$\times$C (case 2d, Table 6.3) in which a 1-way ordered ANOSIM statistic without replicates ($R^{Os}$) is calculated
over years, separately for each of the 8 sites, and these values averaged to give $\overline{R}^{Os}$, exactly the test for trend seen in Fig. 6.14 for the Phuket coral reef data (though there the
trend was for spatial positions averaged over years, whilst here it is the opposite, of inter-annual trends averaged over sites). The appropriate permutation is the usual one of samples in each site
being randomly permuted across the years (since the null hypothesis specifies that there is no year effect, at any site). As Fig. 6.18 illustrates, this will be roundly rejected, with global $\
overline{R}^{Os} = 0.52$, which is significant at any fixed level, in effect, as shown by the null permutation distribution:
If it is considered unwise to test only for a time trend, rather than a more general pattern of annual changes, there is no replication which the test for B can exploit so the design falls back on an
indirect test of the type introduced in Fig. 6.9: evidence of differences among years is provided by a commonality of time patterns in space. A modified test statistic is needed here to cope with the
structuring of the spatial factors into a 2-way nested design of sites within areas. As shown in the above schematic diagram, a logical construction for the test statistic here is to use the matching
statistic $\rho_{av}$ among the sites within each area (though in this case there is only one $\rho$ since there are only 2 sites) and then average this across the areas, to give a doubly-averaged $\
overline{\rho}_ {av}$ statistic. If there are no annual differences this will, as usual, take the value zero, and the null hypothesis distribution is created by the same permutations as for the
ordered test. An inter-annual effect is therefore inferred from consistency in time patterns between sites. If (as might well be thought in this context) it is more appropriate to infer consistent
temporal change by noting commonality at the wider spatial scale of areas, then the sites should simply be averaged (see previous footnotes on how best to do this) to leave a 2-way A$\times$B design
with both factors unordered, and the B test uses the (singly-averaged) $\rho_{av}$ statistic of Fig. 6.9.
Generally one might expect the time pattern to be less consistent as the spatial scale widens, but here, based on sites, $ \overline{\rho}_ {av} = 0.62$ and on areas, $\rho_ {av} = 0.66$, perhaps
because averaging sites removes some of the variability in the sampling. Both $\rho$ statistics are again highly significant, though note that they cannot be compared with the $\overline{R}^{Os}$
value for the ordered case; the statistics are constructed differently.
Returning to the $\overline{R}^{Os}$ test for temporal trend, doubly averaging the statistics in that case, by site then area, could not actually change the previous value (0.52), though averaging
sites first and performing the 2-way design on areas $\times$ years does increase the value to $\overline{R}^{Os} = 0.60$, for the same reasons of reduction in sampling ‘noise’; it is this statistic
that reflects the overall trend seen in the four right-hand plots of Fig. 6.18. It would generally be of interest to ask whether the averaged $\overline{R}^{Os}$ hides a rather different trend for
each area, and the individual trend values $R^{Os}$ for each area (or site) could certainly be calculated and tested. The 4 areas here give the reasonably consistent values $R^{Os}$ = 0.67, 0.54,
0.50, 0.67 respectively (all p<<0.01%), though there is perhaps a suggestion here and in the plots that the wider regional trend seen in Areas 1 and 4, and for which there is evidence from other
North Sea locations (a potential result of changing hydrodynamics), is being impacted by more local changes within the Tees estuary, which will affect areas 2 and 3, within Tees Bay. This is a form
of interaction between Year and Area factors and we shall see later that limited progress can be made in exploring this type of interaction non-parametrically, through the definition of second-stage
MDS and tests (Chapter 16). These ask the question “does the assemblage temporal pattern change between areas, in contrast with its fluctuation within an area?”, and the comparison becomes one
between entire time sequences rather than between individual multivariate samples.
This raises the following important issue about the limitations of non-parametric tests in exploring the conventional interactions of additive linear models.
One crucial point needs to be made about all the 2- and 3-way tests of this chapter. They are fully non-parametric, being based only on the rank order of dissimilarities, which delivers great
robustness, but they cannot deliver the variance partitioning found in the semi-parametric methods of PERMANOVA+, the add-on routines to PRIMER ( Anderson, Gorley & Clarke (2008) ). PERMANOVA uses
the precise measurement scale of the dissimilarities to fit general linear models in the high-dimensional PCO ‘resemblance space’ and it is then able to partition effects of a factor into main
effects and 2-way (or 3-way or higher) interactions, each of which can then be tested. For some scientific questions, testing for the presence or absence of an interaction is the only form of
inference that will suffice: a good example would be for Before-After/ Control-Impact (BACI) study designs, and there are many further examples in Anderson, Gorley & Clarke (2008) and associated
papers. The non-parametric ANOSIM routine cannot (and could never) do this linear model variance-partitioning, of effects into main effects and interactions, because this form of interaction is a
purely metric concept. This is simply illustrated in the univariate case by a hypothetical 2-factor crossed design with two levels for both A and B (e.g. where the response variable y is clearance
rate of particles by a filter-feeding species under A1: low density and A2: high density of particulates, and B1: at night, B2: during the day), let us suppose with minimal variance in the
replicates, giving cell means of (left-hand side):
The data matrix for variable y demonstrates that there is significant interaction between particle density and day/night factors, because the means are not additive: the difference in clearance rate
between high and low density is not the same during the night (1) as during the day (4). But a simple log$_2$ transform of y gives the table to the right, in which there is now no interaction between
the factors: the difference between logged clearance rate at low and high particle density is the same during both day and night (1). Yet, both these tables are identical if viewed non-parametrically
, i.e. with the values replaced by their ranks.
This example is scarcely representative of the typical multivariate abundance matrix but it does illustrate that this simple form of interaction is essentially a parametric construction, based on
linear models of adding main effects, interactions and error. Though, as previously mentioned, ‘non-parametric interaction’ is not an altogether invalid concept (see Chapter 16), it cannot be
straightforwardly defined. The ANOSIM crossed designs are tests for the presence or otherwise of an effect of factor A; this may be a large effect at one level of another factor B, and smaller ones
at its other levels, or it may be a more consistent effect of A at all levels of B – these situations are not distinguished, and one way of viewing these $\overline{R}$ statistics is as combinations
of ‘main effects’ and ‘interactions’. What they tell you, robustly, is whether factor A has an overall effect, at least somewhere, having removed all contributions that the other crossed factor(s)
could possibly be having. They do not do this by subtracting some estimate under a general linear model of the effect of other terms. Their excision of other factors is more surgical than that: they
only ever compare the different levels of A under an identical level for all other combinations of factors. Therefore there can be no equivalent, for example, of the way that in linear models main
effects can apparently disappear because interactions ‘in different directions’ cancel them out. An $\overline{R}$ statistic is perfectly meaningful in the presence of interactions. Under the null
hypothesis, the component R values making up that average are all approximately zero; where there are effects some or all of those R values become positive. If enough of them do so (or one or two of
them do so enough), an effect is detected.
^¶ It is to be understood that each dot represents a sample of 282 species abundances (going into the page, if you like). Of course, data is not input into PRIMER in this (3-way) format but in the
usual species $\times$ (all) samples worksheet, with areas (1-4), years (73-96) and sites (a-h) identified in the associated factors sheet. | {"url":"https://learninghub.primer-e.com/books/change-in-marine-communities/page/617-example-tees-bay-macrofauna","timestamp":"2024-11-02T09:06:19Z","content_type":"text/html","content_length":"203465","record_id":"<urn:uuid:1cf3449d-b052-4a5d-832a-14ea286306cf>","cc-path":"CC-MAIN-2024-46/segments/1730477027709.8/warc/CC-MAIN-20241102071948-20241102101948-00107.warc.gz"} |
Re: Will the stock market
From: John Conover <john@email.johncon.com>
Subject: Re: Will the stock market
Date: 5 Feb 1999 19:33:36 -0000
Bill Terrell writes:
> conover@netcom.com wrote:
> > One of the advantages of using a random walk fractal as a first order
> > approximation for equity values is that the assumption can be
> > verified, emperically-just subtract yesterday's value from today's,
> > for all days, and assemble the results into a frequency
> > distribution. Its a very nice Gaussian distribution, as expected-for
> > all stocks through the century, by the day. The data for all stocks is
> > available on CD.
> There are those who intelligently argue a fractal model which is distinctly
> non-random-walk and non-normal in its distribution, including no less than Benoit B.
> Mandelbrot; see this month's (February) issue of Scientific American.
Oh, sure, Bill. But as a first order approximation, a random walk is
pretty good, (depending on who is telling the story, of course.) There
is a graph of the distributions of the daily marginal increments of
the DJIA, NYSE, and S&P 500, for 27 years at:
and it does seem to indicate, as you suggest, that there is a slight
persistence in equity indices. A random walk has statistically
independent increments-ie., a 50/50 chance that the next increment
will be like the previous increment. The indices, however, seem to
have about a 60% chance-meaning that there is a slight
"predictability" or "forecastabililty" from one day to the next. The
Hurst exponent, Fast Fourier Transform, and entropy studies of the
indices seem to support the contention, also, (the entropy is too low,
implying that the increments are not as random as a random walk would
The Hurst exponent, in addition, seems to indicate that there is a
four year "cyclic" phenomena in the indices-which could be interpreted
as the signature of a chaotic mechanism, possibly a strange attractor,
(but there could be structural reasons, too.) There are graphs of the
Hurst exponent for the NYSE at:
But as a first order approximation, a random walk fractal seems
adequate-at least as a conceptual description from a stochastic point
of view-and the mathematics of the statistics is simple, (ie.,
"bubbles" in the stock market follow a 1 / sqrt (t) frequency
distribution, etc.)
John Conover, john@email.johncon.com, http://www.johncon.com/
Copyright © 1999 John Conover, john@email.johncon.com. All Rights Reserved. Last modified: Fri Mar 26 18:52:52 PST 1999 $Id: 990205113415.1038.html,v 1.0 2001/11/17 23:05:50 conover Exp $ | {"url":"http://www.johncon.com/john/correspondence/990205113415.1038.html","timestamp":"2024-11-05T10:02:58Z","content_type":"text/html","content_length":"4519","record_id":"<urn:uuid:31fa8b74-eca0-49a0-9b7f-4221a139dd90>","cc-path":"CC-MAIN-2024-46/segments/1730477027878.78/warc/CC-MAIN-20241105083140-20241105113140-00717.warc.gz"} |
The wake-sleep algorithm for unsupervised neural networks
Before we start, let’s get some Python plumbing out of the way
import numpy as np
from itertools import islice
# I usually program in languages where this is built in :)
# https://stackoverflow.com/questions/6822725/rolling-or-sliding-window-iterator-in-python
def window(seq, n=2):
"Returns a sliding window (of width n) over data from the iterable"
" s -> (s0,s1,...s[n-1]), (s1,s2,...,sn), ... "
it = iter(seq)
result = tuple(islice(it, n))
if len(result) == n:
yield result
for elem in it:
result = result[1:] + (elem,)
yield result
… back to regular scheduled programming
The report starts here:
The wake-sleep algorithm for unsupervised neural networks
Paper by Geoffrey E Hinton, Peter Dayan, Brendan J Frey and Radford M Neal
Read, interpreted and Pythonized by Vadim Liventsev for Graphical Models of Statistical Inference course at Skoltech
What is this all about?
Artificial neural networks are typically used for supervised learning: given some training data with inputs and outputs create a model that maps inputs to outputs as closely as possible. This paper
introduces wake-sleep algorithm for unsupervised learning (training data contains only inputs) with neural networks and explains its theoretical underpinnings.
How does it work?
Wake-sleep algorithm mostly comes down to 2 clever tricks:
Clever trick 1: Probability - Entropy approach
A neural network can be seen as a coding scheme: the first layer takes an input vector and the last one outputs its representation (a code). If a reference set of “correct” representations is known,
one can minimize mean squared deviation of actual representations from reference representations. But it’s not necessary. Instead, one can take another approach inspired by coding theory: minimize
the complexity of representations. Set the weights of the connections in a neural network so that inputs are encoded in the most compact way possible.
Sounds simple, but developing this idea into an actual algorithm requires some preparation:
Meet Binary Stochastic Neuron
Binary Stochastic Neuron is a neuron that takes a vector of binary values and outputs 0 or 1 with probability defined by the logistic function
where s[v] is the output of neuron v, b[v] is the bias of neuron v, s[u] is the output of neuron u and w[uv] is the weight of the connection from neuron u to neuron v
or, in Python:
class StochasticNeuron():
def __init__(self, input_weights = np.array([]), bias = 0):
self.input_weights = np.array(input_weights)
self.bias = bias
def activation_probability(self, inputs):
inputs = np.array(inputs)
return 1 / (1 + np.exp(- self.bias - np.sum(self.input_weights * inputs)))
def activation_function(self, inputs):
return np.random.binomial(1, self.activation_probability(inputs))
def __call__(self, inputs):
return self.activation_function(inputs)
For example,
n = StochasticNeuron([0, -0.5, 1], 0)
n.activation_probability([False, True, False])
n.activation_probability([False, True, True])
And we can see that the neuron is stochastic by calculating it’s output several times in a row
[n([False, True, False]) for i in range(15)]
[1, 0, 1, 1, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1]
Stochastic neural networks
A layer of a neural network is just a tuple of neurons
# Unbiased, unadjusted layer
def create_layer(input_size, size):
return [StochasticNeuron(np.array([1.0 for i in range(input_size)]), 0) for i in range(size)]
layer = create_layer(3, 10)
inp = [1, 2, -4]
[neuron(inp) for neuron in layer]
[1, 1, 0, 0, 0, 0, 0, 0, 0, 0]
Once again, let’s infer the ouput of the layer several times and note the stochasticity
# Once again, it is stochastic
[[neuron(inp) for neuron in layer] for i in range(5)]
[[0, 0, 1, 0, 0, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 1, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 1, 1, 0, 0, 0],
[0, 0, 0, 0, 1, 1, 0, 0, 0, 0],
[0, 1, 0, 1, 0, 0, 0, 0, 0, 0]]
A multi-layer neural network is built by stacking several layers together with outputs of layer n as inputs of layer n+1
def create_network(layer_sizes):
return [create_layer(prev_size, size) for prev_size, size in window(layer_sizes, 2)]
Let’s build the network displayed below:
net = create_network([3, 4, 2])
The state of the entire network can be inferred by calculating neuron outputs layer-by-layer:
def infer(layers, inp):
state = [inp]
for layer in layers:
state.append([neuron(state[-1]) for neuron in layer])
return state
We can try to set the input vector to 1,0,0 and infer the state of the network
state = infer(net, [True, False, False])
[[True, False, False], [0, 1, 1, 1], [1, 1]]
OK, but what does it have to do with graphical models?
Great question! The image below is (probably) the most cited graphical model:
It is a Bayesian network describing the reasoning of a certain British detective who woke up late on a foggy British day to notice that his British lawn is wet. The arcs (edges, arrows) of the graph
define conditional probabilities of lawn being wet because of rain or because of a sprinkler.
Notice that there is little meaningful difference between nodes in a Bayesian network and our stochastic binary neurons. It is easy to show:
sprinkler_neuron = StochasticNeuron([-10], 0)
grass_neuron = StochasticNeuron([5,5], 0)
rain = True
print("Rain: " + str(bool(rain)))
sprinkler = sprinkler_neuron([rain])
print("Sprinkler active: " + str(bool(sprinkler)))
grass_wet = grass_neuron([rain, sprinkler])
print("Grass wet: " + str(bool(grass_wet)))
Rain: True
Sprinkler active: False
Grass wet: True
So, essentially, stochastic neural networks and graphical models are 2 languages used to discuss the same mathematical structure
But didn’t we set out to solve learning, not inference?
Yes! Statistical inference and learning are in a sense complementary problems: inference discusses “how would this neural network/graphical model behave?” and learning - “how to build a neural
network/graphical model that behaves the way we want?”. Hinton et al defined “the way we want” as minimizing the description length of the net’s output given the input.
Consider the following problem: you have n objects, some of which are more likely to occur than others, defined by some probability distribution. You have to come up with a code for every object so
that the expected length of the code will be as short as possible. This problem is less theoretical than it might seem: languages (natural and programming languages alike) attempt to solve whis exact
problem by describing concepts that occur often with short words (dog) and rare ones with long words (concatenation). Languages are far from perfect at it, but if a perfect language were to exist,
every word in it would be of length -logP where P is the probability of this word
-logP is called description length (Shannon coding theorem) and it is applicable to any probability distribution as every probability distribution, in theory, defines a coding scheme. Our stochastic
neural network is no exception: when an input vector and weights of all connections are fixed, every possible output of the net has a description length.
It is derived as follows:
The cost C of describing a single neuron is
where α is network state (aggregate of states of all enurons), s[j]^α is the state of neuron j given network state α and p[j]^α is the activation probability of neuron j (P(s[j]^α)=1)
Then the cost of describing state α in its entirety is simply the cost of describing all the hidden states in all the hidden layers plus the cost of describing the input vector given the hidden
where d is the input vector. Note that decomposition of C(α,d) into 2 addants is very meaningful. C(d|α) indicates how well the state of the network represents the input vector while C(α) corresponds
to the complexity of said state
And if we were to minimize this cost (we are) using gradient descent, we would (we will) use the following delta rule derived from the above formula:
Long time, no Python! Let’s implement it!
def learn(layers, state, rate):
for layer, (inputs, outputs) in zip(layers, window(state, 2)):
for neuron, output in zip(layer, outputs):
# Yes, I could greatly optimize it by caching probabilities
# As soon as I have some spare time
p = neuron.activation_probability(inputs)
neuron.input_weights -= np.array(
[rate * inpt * (output - p) for inpt, weight in zip(inputs, neuron.input_weights)]
Clever trick 2: using 2 reverse neural nets to train each other
There are several issues that haven’t been addressed yet:
• Our delta rules requires state α to already exist. And using some dummy state is a bad idea since that will jeopardize the quality of learning
• We have a network that outputs represenations of the input vector, but we don’t have a good way to reconstruct the input vector from its representations
Hinton et al proposed the following solution:
This architecture can be thought of as either a network where each pair of adjacent layers is connected by 2 sets of connections (one in each direction) or as 2 networks with shared state: one is
responsible for recognition: building a representation of input data, another for generation: reconstructing the input vector
Learning happens iteratively, each iteration consists of:
1. Recognition inference. Calculating state α.
2. Generative learning. Changing generative weights so that d is better described by α using the delta rule mentioned above.
3. Generative inference. Calculating a new (fantasy) d
4. Recognition learning. Changing recognition weights so that α is better described by d
English -> Python translation:
def wakesleep(layer_sizes, inputs, learning_rate, iterations):
recognition_connections = create_network(layer_sizes)
generative_connections = create_network(reversed(layer_sizes))
state = [inputs]
for i in range(iterations):
state = infer(recognition_connections, state[-1]) # That reverses state
learn(generative_connections, reversed(state), learning_rate)
state = infer(generative_connections, state[-1]) # That reverses it once again
learn(recognition_connections, reversed(state), learning_rate)
return reversed(state)
tuple(wakesleep([2, 10, 30], [-1, 1], 1, 100))
([1, 1],
[1, 1, 1, 1, 1, 1, 1, 1, 1, 1], | {"url":"https://vadim.me/publications/wakesleep/","timestamp":"2024-11-14T09:50:37Z","content_type":"text/html","content_length":"35077","record_id":"<urn:uuid:27ab1ddd-5e98-4e8a-8e05-1e999951766b>","cc-path":"CC-MAIN-2024-46/segments/1730477028558.0/warc/CC-MAIN-20241114094851-20241114124851-00074.warc.gz"} |
Berwald space
From Encyclopedia of Mathematics
A Berwald space Finsler space) such that its Berwald coefficients
Clearly, all Riemannian and locally Minkowski spaces are Berwald spaces (cf. also Riemannian space; Minkowski space). L. Berwald gave a complete characterization of such spaces. He used the frame
Berwald's theorem, slightly rephrased, reads as follows. If
Here Berwald connection).
Applications of Berwald spaces in biology, physics and stochastic processes can be found in [a1], [a2].
[a1] P.L. Antonelli, R.S. Ingarden, M. Matsumoto, "The theory of sprays and Finsler spaces with applications in physics and biology" , Kluwer Acad. Publ. (1993)
[a2] P.L. Antonelli, T. (eds.) Zastawniak, "Lagrange geometry, Finsler spaces and noise applied in biology and physics" Math. and Comput. Mod. (Special Issue) , 20 (1994)
How to Cite This Entry:
Berwald space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Berwald_space&oldid=12731
This article was adapted from an original article by P.L. Antonelli (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098.
See original article | {"url":"https://encyclopediaofmath.org/index.php?title=Berwald_space&oldid=12731","timestamp":"2024-11-12T22:20:02Z","content_type":"text/html","content_length":"20864","record_id":"<urn:uuid:f6c1d5f6-7435-4584-b50e-7ccd46cfc9c4>","cc-path":"CC-MAIN-2024-46/segments/1730477028290.49/warc/CC-MAIN-20241112212600-20241113002600-00357.warc.gz"} |
Renting vs Buying in context of future rent
25 Sep 2024
Title: The Economics of Renting vs Buying: A Comparative Analysis of Future Rental Income Streams
This study examines the financial implications of renting versus buying a property, with a focus on the future rental income streams that can be generated from an investment in real estate. We
develop a mathematical framework to compare the present value of future rental income with the cost of purchasing and owning a property. Our analysis highlights the importance of considering the time
value of money and the potential for capital appreciation when evaluating the merits of renting versus buying.
The decision to rent or buy a property is a complex one, influenced by various factors including personal preferences, financial circumstances, and market conditions. While buying a property can
provide a sense of ownership and potentially generate long-term capital gains, it also involves significant upfront costs and ongoing expenses such as mortgage payments, maintenance, and property
taxes. Renting, on the other hand, offers flexibility and freedom from these responsibilities, but may not provide the same level of financial returns.
Mathematical Framework:
Let’s denote the present value of future rental income streams as PV(R), the cost of purchasing a property as C, and the annual rental income as R. We can express the present value of future rental
income streams using the following formula:
PV(R) = ∑[R / (1 + i)^t]
where i is the annual interest rate and t represents the number of years.
The cost of purchasing a property can be represented by the following formula:
C = P + M + T
where P is the purchase price, M is the mortgage amount, and T is the total upfront costs (e.g., closing fees, inspections).
Comparative Analysis:
To compare the present value of future rental income streams with the cost of purchasing a property, we can use the following formula:
PV(R) / C = ∑[R / (1 + i)^t] / (P + M + T)
This ratio provides a quantitative measure of the relative merits of renting versus buying. A higher ratio indicates that renting is more financially advantageous, while a lower ratio suggests that
buying may be a better option.
The decision to rent or buy a property depends on various factors, including personal preferences, financial circumstances, and market conditions. Our analysis highlights the importance of
considering the time value of money and the potential for capital appreciation when evaluating the merits of renting versus buying. By using the mathematical framework presented in this study,
investors can make more informed decisions about their real estate investments.
• [1] Smith, J. (2020). The Economics of Renting vs Buying. Journal of Real Estate Finance, 10(2), 123-145.
• [2] Johnson, K. (2019). A Comparative Analysis of Renting and Buying in the Context of Future Rental Income Streams. Journal of Housing Economics, 45, 102754.
Note: The references provided are fictional and for demonstration purposes only.
Related articles for ‘future rent’ :
• Reading: Renting vs Buying in context of future rent
Calculators for ‘future rent’ | {"url":"https://blog.truegeometry.com/tutorials/education/73d33f4d972f2ad0971b9fcfebe1f0a2/JSON_TO_ARTCL_Renting_vs_Buying_in_context_of_future_rent.html","timestamp":"2024-11-05T07:33:30Z","content_type":"text/html","content_length":"17010","record_id":"<urn:uuid:bbe7cd38-4c16-48bd-a709-01bb71df8043>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00369.warc.gz"} |
In this post, a clear explanation on KVL and KCL has been provided. Kirchoff's Current Law and Kirchoff's Voltage Law can be applied only a closed path or loop. Kvl & Kcl Also, read: Equivalent
Resistance | Calculation Ohm's Law | example Kirchoff's Current Law : In the above circuit, the I1, I2, I3, I4, and I5 are the currents flowing through each terminal. … [Read more...] about
Kirchoff’s Current Law and Kirchoff’s Voltage Law | KVL and KCL | {"url":"https://www.zzoomit.com/page/818/","timestamp":"2024-11-09T03:10:36Z","content_type":"text/html","content_length":"53179","record_id":"<urn:uuid:b2bceec7-860e-415b-b06e-a2069be5a8e4>","cc-path":"CC-MAIN-2024-46/segments/1730477028115.85/warc/CC-MAIN-20241109022607-20241109052607-00033.warc.gz"} |
Vector Analysis
Rakennetyyppi: Opintojakso
Koodi: IZS9108
OPS: IT 2021
Taso: Insinööri (AMK)
Opiskeluvuosi: 3 (2023-2024)
Lukukausi: Kevät
Laajuus: 3 op
Vastuuopettaja: Mäkelä, Jarmo
Opetuskieli: Englanti
The basic problem of vector analysis is: How to differentiate and integrate vectors? This question raises in certain engineering applications of mathematics, especially when we consider a flow of
given substance, no matter whether that substance is, say, water flowing in a pipeline, or energy carried by a radio wave. During this course a student will learn the basic concepts and theorems of
vector analysis, and to apply them in the problems of mechanics, fluid mechanics, and electrical engineering. The student also learns to differentiate and integrate vectors in general curvelinear
coordinates, and not only in the familiar xyz coordinates.
Opiskelijan työmäärä
The total amount of student's work is 81 h, which contains 42 h of contact studies. The assessment of student’s own learning 1 h is included in contact lessons.
Edeltävät opinnot / Suositellut valinnaiset opinnot
Analysis: Differential- and Integral Calculus basics and Differential equations and series.
1. Summary of the basic vector calculus.
2. Vector fields.
3. Differentiation of a vector with respect to a parameter.
4. Line integrals of vector fields.
5. Gradient.
6. Divergence.
7. Curl.
8. Potential of the vector field.
9. Conservative vector field.
10. Surface integral.
11. Green’s theorem.
12. Two-dimensional surfaces embedded in three-dimensional space.
13. Curvelinear coordinates on a surface.
14. Coordinate curves and their tangent vectors.
15. Normal of a surface; orientable surfaces.
16. Calculation of the area of a general two-surface.
17. Flux of a vector field through a surface.
18. Stokes’s theorem.
19. Volume integral.
20. Gauss’s theorem.
21. Maxwell’s equations.
22. Continuity equation.
23. General curvelinear coordinates in three dimensions.
24. Surface- and volume integrals in the curvelinear coordinates: The Jacobi determinant.
25. Metric tensor.
26. Covariant and contravariant vector fields.
27. Christoffel symbol.
28. Differentation of vector fields in curvelinear coordinates.
Kreyszig, E: "Advanced Engineering Mathematics", John Wiley & Sons; the material prepared by the lecturer.
Opetusmuoto / Opetusmenetelmät
Theory, examples and exercises during the lectures. Homework exercises.
Grade 1: The student knows those subjects of the course, which are necessary for the forthcoming studies and working life.
Grade 3: The student is well-abled to utilize the course contents.
Grade 5: The student is able to apply creatively the contents of the course.
Homework exercises and an examination. | {"url":"https://ops.vamk.fi/fi/IT/2021/IZS9108/","timestamp":"2024-11-09T07:58:26Z","content_type":"text/html","content_length":"6502","record_id":"<urn:uuid:cb29f842-519a-419f-b8c5-3cbdb1693bb3>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.30/warc/CC-MAIN-20241109053958-20241109083958-00028.warc.gz"} |
Equivalence checking and Verified compilation
The use of safety-critical software applications is growing rapidly in aerospace, defence, automobile, transportation, cybersecurity, and energy sectors. Given the catastrophic costs of a software
bug in these sectors, it is not sufficient to use an off-the-shelf commodity compiler to compile the source code to executable code. A modern compiler has millions of lines of code, and tens of bugs
are discovered in popular compilers every month. A compiler bug may yield buggy executable code, even when the source code was correct.
These safety-critical sectors are usually governed by software safety standards, e.g., DO-178C (aerospace) requires the developer to verify the executable code, and not just the source code. Current
methods to verify executable code are ad hoc --- for example, developers often disable compiler optimizations and undertake laborious manual reviews of the executable code, as required for safety
Based on over eleven years of research at IIT Delhi, we have developed a formal equivalence checker that checks whether the executable code is equivalent to the source code. CompilerAI Labs Pvt. Ltd.
is a startup incubated at IIT Delhi that is commercializing this formal equivalence checker. Our equivalence checker can be used by the developers to obtain independently verifiable proofs of
correctness of the executable code (with respect to the source code) --- this enables a more systematic validation method for the correctness of a compilation than the current practice. You can
test-drive the equivalence checker on your browser by following the steps below.
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Install the equivalence checker on the browser
1. Go to https://vscode.dev
2. Click on the sixth icon from the top on the left pane, labeled "Extensions" (Ctrl+Shift+X).
3. Search for the "Eqchecker" extension by CompilerAI.
4. Check the version number of the "Eqchecker" extension. It should be at least 0.9.0.
5. Install the "Eqchecker" extension.
6. Wait for the extension to get installed.
Authenticate using your email address
1. Click on the second icon from the top on the left pane, labeled "Explorer" (Ctrl+Shift+E). Expand the "Equivalence Checks" and "Search Tree" panes by clicking on them.
2. Click on "Login" in the "Equivalence Checks" pane.
3. Enter your email address. Each email address is given a free quota of ten equivalence checks per month. We need to enforce this quota due to the compute intensive nature of an equivalence check.
After you enter your email address and press enter, a One-Time-Password (OTP) will be sent to your email address.
4. Enter the four digit OTP received on your email address.
Start an equivalence check
1. Click on the first icon from the top on the left pane to open a command menu. Open the files (or create new files) in the editor, for which you would like to compute equivalence or perform
verified compilation.
2. Click on "Start an Eqcheck" button to start an equivalence check. The button displays the email address of the current user and the number of remaining equivalence checks in her quota. Upon
clicking this button, you are presented a drop-down menu of potential equivalence checks (or verified compilations) that you can perform on the opened files.
If you choose a verified compilation (e.g., "Compile strlen_src.c"), the tool first compiles the C source code and then performs an equivalence check between the source code and the generated
32-bit x86 executable. If you choose an equivalence check (e.g., "strlen_src.c → strlen_dst.c"), then an equivalence check is performed between the two C programs, or a C program and an assembly
The equivalence checks are performed at function granularity. For two files to be compared for equivalence, they should have functions with the same names (but potentially different
implementations) in both files.
3. After an equivalence check (or verified compilation) begins, the progress of the compilation is shown as an eqcheck entry in the "Equivalence Checks" pane. The status of the corresponding entry
updates as the equivalence check proceeds.
If an input file contain multiple functions, a separate eqcheck entry is created for each function, after the initial processing of the input files.
Depending on the complexity of the transformations and the size of the input function, an equivalence check can take anywhere between a few seconds to a few hours. Our research continuously
strives to make this faster.
4. A successful equivalence check is represented by a green-coloured eqcheck pane, labeled "Found proof and safety".
View an equivalence proof and the search tree for a proof
1. For a successful equivalence check, you can right-click on the eqcheck entry to select "View Proof".
2. The visual proof is represented as a "Product Graph". Each edge of the product graph encodes the lockstep execution of the two programs being compared for equivalence. Each node of the product
graph represents the correlated PC (program counter) addresses of the two programs.
In addition to the product graph, the proof also includes panes that display the two programs (in source, assembly, and IR formats). You can click on an edge of the product graph to view the
correlated paths in these programs (in each format). The proof encodes the fact that the correlated paths behave identically and keep the two programs' states related.
This is a formal proof of equivalence: if our tool identifies two programs to be equivalent, then they are (in principle) guaranteed to behave identically for all possible legal inputs to the
3. Our equivalence checker constructs the equivalence proof incrementally. The proof construction is designed as a search algorithm. It is possible to view the search tree, both for an ongoing
equivalence check, and for a completed equivalence check. This can be done by right-clicking on an eqcheck entry and selecting "View Search Tree".
If the equivalence checker fails to successfully prove equivalence, the user may inspect the search tree to understand the reasons for the equivalence failure.
4. The search tree is displayed in the "Search Tree" pane in a tree representation. The tree represents all the different product-graphs explored by the equivalence checker before arriving at a
final proof.
You can click on any of the nodes of this search tree to view the (partial) product graph developed incrementally by the algorithm till that stage. Different branches of the tree may represent
different product graphs --- it is possible for the search algorithm to backtrack during this search for a proof.
Save and load a session
You can save a session (potentially with multiple ongoing equivalence checks) and load it later (potentially on a different machine). To access this feature, right-click on the "Start an Eqcheck"
Examples of successful equivalence checks with proofs
Each example entry shown below can be accessed by using the "Load Session" option in the equivalence checker with the corresponding session name (provided with each entry). We use the Clang-12
compiler for each compilation (which is validated by our equivalence checker).
• Testsuite for Vectorizing Compilers at O3 optimization: session name tsvc.
The compilations of these programs involve aggressive loop vectorizing transformations.
• The bzip2 compression utility at O1 optimization: session name bzip2.
These are larger functions with complex control flow and memory allocation patterns.
These examples demonstrate that the equivalence checker is able to compute equivalence for a large category of transformations on a large set of programs. If an equivalence check does not succeed
(e.g., it runs for a long time and gets terminated after a timeout), then this may be either because the programs were inequivalent or because our algorithm could not identify an equivalence proof.
The latter situation is due to the incompleteness of our equivalence checker. The equivalence checker is always sound --- if a formal equivalence proof is identified by the tool, the two input
programs are guaranteed to have equivalent runtime behaviour.
We are continuously improving the completeness of our equivalence checker by minimizing the cases where the equivalence checker is unable to identify an equivalence proof (when the programs were
indeed equivalent). If you use the equivalence checker to produce a verified compilation of a C program, you can make the equivalence proof search more tractable by dividing the input C program into
smaller individual functions.
Please try the equivalence checker for yourself. Please share your feedback with us at sorav@compiler.ai --- we very much appreciate your critical feedback. | {"url":"http://compiler.ai/","timestamp":"2024-11-05T03:17:48Z","content_type":"text/html","content_length":"15241","record_id":"<urn:uuid:2cadb3a2-ee75-47cb-8341-a1b8a8d09a6c>","cc-path":"CC-MAIN-2024-46/segments/1730477027870.7/warc/CC-MAIN-20241105021014-20241105051014-00021.warc.gz"} |