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Factoring calculator, algebra
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Author Message
pedermakocsh Posted: Wednesday 20th of Jul 21:55
I have this math assignment due and I would really be grateful if anyone can assist factoring calculator, algebra on which I’m stuck and don’t know where to start from. Can you give me
guidance with y-intercept, function domain and converting decimals. I would rather get guidance from you than hire a math tutor who are very expensive . Any direction will be highly
appreciated very much.
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Vofj Timidrov Posted: Thursday 21st of Jul 09:01
There are many inside the overall subject area of factoring calculator, algebra, for instance, binomial formula, y-intercept and graphing circles. I have talked with some folks who
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Registered: user could employ the software only to complete worksheets , I doubt a user will be allowed to employ the software for examinations .
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daujk_vv7 Posted: Friday 22nd of Jul 08:43
I used Algebrator also , especially in Remedial Algebra. It helped me so much , and you won't believe how easy it is to use! It solves the exercise and it also describes everything step
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AL Posted: Saturday 23rd of Jul 07:59
Can a program really help me learn my math? Guys I don’t need something that will solve equations for me, instead I want something that will help me understand the concepts as well.
From: Texas
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Dnexiam Posted: Sunday 24th of Jul 08:52
You can buy it from https://softmath.com/algebra-policy.html. I don’t think there are too many system requirements; you can just download and start using it.
From: City 17
Back to top | {"url":"https://softmath.com/algebra-software/point-slope/factoring-calculator-algebra.html","timestamp":"2024-11-08T21:49:24Z","content_type":"text/html","content_length":"41159","record_id":"<urn:uuid:b61c9c94-abcd-43b4-af29-4717fd23a7d3>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00436.warc.gz"} |
[AN #125]: Neural network scaling laws across multiple modalities — AI Alignment Forum
Alignment Newsletter is a weekly publication with recent content relevant to AI alignment around the world. Find all Alignment Newsletter resources here. In particular, you can look through this
spreadsheet of all summaries that have ever been in the newsletter.
Audio version here (may not be up yet).
Please note that while I work at DeepMind, this newsletter represents my personal views and not those of my employer.
Scaling Laws for Autoregressive Generative Modeling (Tom Henighan, Jared Kaplan, Mor Katz et al) (summarized by Asya): This paper looks at scaling laws for generative Transformer models of images
(predicting pixels or parts of image encodings), videos (predicting frames of image encodings), multimodal image <-> text (predicting captions based on images or images based on captions), and
mathematical problem solving (predicting answers to auto-generated questions about algebra, arithmetic, calculus, comparisons, integer properties, measurement, polynomials, and probability). The
authors find that:
- Cross-entropy loss as a function of compute follows a power law + constant in all these data modalities (just as it does in language (AN #87)). Information theoretically, this can be interpreted as
scaling a 'reducible loss' which estimates the KL divergence between the true and model distributions, and an 'irreducible loss' which estimates the entropy of the true data distribution.
- Performance on ImageNet classification fine-tuned from their generative image model also follows such a power law, whereas ImageNet classification trained from scratch actually gets worse with
sufficiently large model sizes. Interestingly, this classification power law continues even past model sizes where the generative cross-entropy loss starts bending as a result of irreducible loss.
The authors conclude that approaching the irreducible loss for some dataset does not necessarily indicate diminishing returns for representation quality or semantic content.
- Optimal model size as a function of compute follows a power law with an exponent very close to ~0.7 for all data modalities they've studied so far. This implies that in the current compute regime,
as compute budgets grow, it's best to devote a majority of compute towards making models bigger and a minority towards training on more data.
- Larger models perform better on extrapolating to math problems more difficult than those seen in training, but only insofar as they do better on the training distribution (no benefits to 'strong
- Larger models are able to take advantage of more multimodal information, but the scaling is extremely slow-- a 1-billion-parameter model uses 10% of the information in a caption to define an image,
while using 20% of the information would require a 3-trillion-parameter model.
As in the language models paper (AN #87), extrapolating the steep power laws found for optimally-used compute seems to eventually paradoxically result in loss lower than the bound given by shallower
power laws for optimally-used training data. The authors offer a potential hypothesis for resolving this inconsistency-- in the regime of less compute and smaller model sizes, increasing model size
effectively increases the amount of information you extract from each data point you train on, resulting in the steepness of the current compute law. As compute increases past a certain point,
however, the amount of information extracted per data point approaches the maximum amount possible, so the curve switches to a shallower regime and marginal compute should be used increasingly on
dataset increases rather than model size increases. If this hypothesis is true, we should eventually expect the scaling laws for compute to bend towards laws set by dataset size, and perhaps should
think they will ultimately be set by trends for overfitting (see this post for another explanation of this).
Read more: the scaling “inconsistency”: openAI’s new insight
Asya's opinion: I would also recommend listening to Jared Kaplan's talk on this.
I was really excited to learn about more empirical work here. These results suggest that scaling behavior predictable with smooth power-laws is likely a feature of most generative models, not just
text. I found it surprising that optimal model size given a compute budget scales the same way across data modalities-- it does seem to suggest that there's something more fundamental going on here
that I don't understand (but which may be explained in this theory paper that I haven't read). It's also interesting that pretraining on a generative model (rather than training from scratch) seems
to confer real benefits to scaling behavior for image classification-- this lends some support to the view that a lot of the learning that needs to happen will come from unsupervised settings.
A lot of the most salient questions around current scaling laws for me still lie in the translation between cross-entropy loss in these domains and performance on downstream tasks we care about. I
feel very unsure about whether any of the fine-tuned generative models we (currently) have the data to train are likely to have transformative performance within even the next 5 orders of magnitude
of compute scaling.
Rohin's opinion: In addition to the points Asya made above, I wanted to speculate on the implications of these scaling laws for AGI. I was particularly struck by how well these scaling laws seem to
fit the data. This was also true in the case of mathematics problems, at least for the models we have so far, even though intuitively math requires “reasoning”. This suggests to me that even for
tasks that require reasoning, capability will increase smoothly along a spectrum, and the term “reasoning” is simply a descriptor of a particular capability level. (An alternative position is that
“reasoning” happens only to the extent that the neural net is implementing an algorithm that can justifiably be known to always output the right answer, but this sort of definition usually implies
that humans are not doing reasoning, which seems like a deal-breaker.)
Note however that we haven't gotten to the level of performance that would be associated with "reasoning", so it is still possible that the trends stop holding and reasoning then leads to some sort
of discontinuous increase in performance. I just wouldn't bet on it.
Confucianism in AI Alignment (John Wentworth) (summarized by Rohin): Suppose we trained our agent to behave well on some set of training tasks. Mesa optimization (AN #58) suggests that we may still
have a problem: the agent might perform poorly during deployment, because it ends up optimizing for some misaligned mesa objective that only agrees with the base objective on the training
This post suggests that in any training setup in which mesa optimizers would normally be incentivized, it is not sufficient to just prevent mesa optimization from happening. The fact that mesa
optimizers could have arisen means that the incentives were bad. If you somehow removed mesa optimizers from the search space, there would still be a selection pressure for agents that without any
malicious intent end up using heuristics that exploit the bad incentives. As a result, we should focus on fixing the incentives, rather than on excluding mesa optimizers from the search space.
Clarifying inner alignment terminology (Evan Hubinger) (summarized by Rohin): This post clarifies the author’s definitions of various terms around inner alignment. Alignment is split into intent
alignment and capability robustness, and then intent alignment is further subdivided into outer alignment and objective robustness. Inner alignment is one way of achieving objective robustness, in
the specific case that you have a mesa optimizer. See the post for more details on the definitions.
Rohin's opinion: I’m glad that definitions are being made clear, especially since I usually use these terms differently than the author. In particular, as mentioned in my opinion on the highlighted
paper, I expect performance to smoothly go up with additional compute, data, and model capacity, and there won’t be a clear divide between capability robustness and objective robustness. As a result,
I prefer not to divide these as much as is done in this post.
Measuring Progress in Deep Reinforcement Learning Sample Efficiency (Anonymous) (summarized by Asya) (H/T Carl Shulman): This paper measures historic increases in sample efficiency by looking at the
number of samples needed to reach some fixed performance level on Atari games and virtual continuous control tasks. The authors find exponential progress in sample efficiency, with estimated doubling
times of 10 to 18 months on Atari, 5 to 24 months on state-based continuous control, and 4 to 9 months on pixel-based continuous control, depending on the specific task and performance level. They
find that these gains were mainly driven by improvements in off-policy and model-based deep RL learning approaches, as well as the use of auxiliary learning objectives to speed up representation
learning, and not by model size improvements. The authors stress that their study is limited in studying only the published training curves for only three tasks, not accounting for the extent to
which hyperparameter tuning may have been responsible for historic gains.
Asya's opinion: Following in the footsteps of AI and Efficiency (AN #99), here we have a paper showing exponential gains in sample efficiency in particular. I'm really glad someone did this
analysis-- I think I'm surprised by how fast progress is, though as the paper notes it's unclear exactly how to relate historic improvements on fixed task performance to a sense of overall
improvement in continuous control (though several of the main contributors listed in the appendix seem fairly general). I also really appreciate how thorough the full paper is in listing limitations
to this work.
Since these papers are coming up in the same newsletter, I'll note the contrast between the data-unlimited domains explored in the scaling laws paper and the severely data-limited domain of
real-world robotics emphasized in this paper. In robotics, it seems we are definitely still constrained by algorithmic progress that lets us train on fewer samples (or do better transfer from
simulations (AN #72)). Of course, maybe progress in data-unlimited domains will ultimately result in AIs that make algorithmic progress in data-limited domains faster than humans ever could.
DeepSpeed: Extreme-scale model training for everyone (DeepSpeed Team et al) (summarized by Asya): In this post, Microsoft announces updates to DeepSpeed, its open-source deep learning training
optimization library. The new updates include:
- '3D parallelism', a scheme for carefully optimizing how training runs are split across machines. Training runs that use 3D parallelism demonstrate linear scaling of GPU memory and compute
efficiency, enabling the theoretical training of extremely large models of over a trillion parameters on as few as 800 NVIDIA V100 GPUs.
- 'ZeRO-Offload', which allows CPU memory to be used during training runs, enabling running models of up to 13 billion parameters on a single NVIDIA V100 GPU.
- 'DeepSpeed Sparse Attention', an instrumental technology that reduces the compute and memory requirements of attention computations used in models like Transformers. Compared to models that use
densely computed attention, this enables models that pay attention to sequences that are 10x longer and can be trained up to 6.3x faster.
- '1-bit Adam', a scheme for compressing the communication requirements between machines doing training runs that use the Adam gradient descent optimizer. 1-bit Adam enables up to 5x less
communication and up to 3.5x faster training runs.
Fast reinforcement learning through the composition of behaviours (André Barreto et al) (summarized by Flo): While model-based RL agents can easily adapt their policy to changed rewards on the same
environment, planning is expensive and learning good models can be challenging for many tasks. On the other hand, it is challenging to get model-free agents to adapt their policy to a new reward
without extensive retraining. An intermediate solution is to use so-called successor features: Instead of a value function V(π,s) representing the expected discounted reward for a policy π starting
in state s, successor features are a vector-valued value function ψ(π,s) representing an expected discounted feature vector ϕ. If our reward equals r = w ⋅ ϕ for some weight vector w, we can easily
obtain the original value function by taking the scalar product of the successor features and the weight vector: V(π,s) = w ⋅ ψ(π,s). Successor features thus allow us to evaluate a fixed policy π for
all rewards that are linear in ϕ, which is called generalized policy evaluation.
Now that we can evaluate policies for different preferences, we would like to efficiently find a good policy for a given novel preference. Inspired by human learning that often combines previously
learned skills, we employ generalized policy improvement. In vanilla policy improvement, we improve upon a policy π we can evaluate by choosing the action that maximizes the immediate reward plus the
discounted value V(π,s') of following π starting in the next state s'. In generalized policy improvement, we have multiple policies and choose the action that maximizes the reward plus the discounted
value of following the best of these policies starting in the next state s'. To obtain a policy for the new preference, we "stitch together" all policies we learnt for previous preferences and the
resulting policy performs at least as good as all of the old policies with respect to the new preference. As generalized policy improvement does not require any additional environment samples, it
enables zero-shot transfer to new preferences. Empirically, even if the weight vector w has to be learnt from reward signals, generalized policy improvement is very sample efficient. Additional
samples can then be used to further improve the policy using standard RL.
Read more: Fast reinforcement learning with generalized policy updates
Flo's opinion: I really like the idea of successor features. Similar to model-based systems, they allow us to evaluate policies for many different rewards, which can be useful for anticipating
problematic behaviour before deploying a system. However, note that we still need to execute the policy we obtained by generalized policy improvement to evaluate it for different rewards: The only
guarantees we have is that it is better than the previous policies for the reward for which the improvement step was carried out (and potentially some weaker bounds based on the similarity of
different rewards).
γ-Models: Generative Temporal Difference Learning for Infinite-Horizon Prediction (Michael Janner et al) (summarized by Flo): Long planning horizons are often necessary for competitive performance of
model-based agents, but single-step models get less and less accurate with longer planning horizons as errors accumulate. Model-free algorithms don't have this problem but are usually reward- and
policy-specific, such that transfer to other tasks can be hard. The paper proposes policy-specific γ-models as an intermediate solution: instead of learning the distribution of the next state given a
state-action pair (s,a), or the final state of an n-step rollout given (s,a) and a policy π, it learns the distribution of a rollout with a stochastic, geometrically distributed length. Unlike for
n-step models with n>1, the distribution follows a Bellman-style decomposition into the single-step distribution and the discounted distribution for the next state s', which allows for off-policy
training of the model by bootstrapping the target distribution.
Now, if rewards are consequentialist in the sense that they only depend on the state, the expected reward under this distribution is equal to 1-γ times the Q-value for π of (s,a) such that we can use
the model for policy evaluation given arbitrary consequentialist rewards. Similar to how single-step models (0-models) can be rolled out to obtain (less accurate) multi-step models, sequential
rollouts of a γ-model can be reweighed to obtain a γ-model with larger γ. While this introduces some error, it reduces the bootstrap error during training, which grows with γ. Being able to
interpolate between rollouts of single-step models that accumulate error during testing and models with large γ that accumulate error during training allows us to find a sweet spot between the two
In practice, single-step models are often used for model-based value expansion (MVE), where only N steps are rolled out and a value function is used for evaluating longer-term consequences. The
authors' algorithm, γ-MVE instead uses N rollouts of the γ-model and adjusts the weighing of the value function accordingly. γ-MVE performs strongly both in terms of sample efficiency and final
performance on a set of low-dimensional continuous control tasks.
Flo's opinion: I am a bit surprised that this works so well, as both bootstrapping and learning generative models for distributions can be unstable and the method combines both. On the other hand,
there is a long tradition of continuous interpolations between different RL algorithms and their performance at the sweet spot is often significantly stronger than at the extremes.
I'm always happy to hear feedback; you can send it to me, Rohin Shah, by replying to this email.
An audio podcast version of the Alignment Newsletter is available. This podcast is an audio version of the newsletter, recorded by Robert Miles.
As always, thanks to everyone involved for the newsletter! I'm usually particularly interested in the other RL/Deep Learning papers, as those are the ones I have less chance to find on my own.
On this newsletter, I especially enjoyed the summaries and opinions about the two scaling papers, and the comparison between the two.
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3rd Grade Math Common Core Practice: Ace Your Skills!
3rd Grade Common Core Math Standards Practice PDF
These worksheets are designed to help 3rd-grade students practice and master the Common Core Math standards. The worksheets cover a range of topics, including operations and algebraic thinking,
number and operations in base ten, fractions, measurement and data, and geometry. Each worksheet includes a variety of problems to reinforce students’ understanding of key concepts.
Overview of 3rd Grade Common Core Math Standards
The Common Core State Standards Initiative, also known as simply Common Core, was an American, multi-state educational initiative begun in 2010 with the goal of increasing consistency across state
standards, or what K-12 students throughout the United States should know in English language arts and mathematics at the conclusion of each school grade. The Common Core State Standards for
Mathematics are a set of guidelines that outline the mathematical skills and knowledge that students should acquire in each grade level. These standards are designed to ensure that students are
prepared for success in college and careers.
The California Common Core State Standards⁚ Mathematics (CA CCSSM) reflect the importance of focus, coherence, and rigor in mathematics education. The CA CCSSM are organized by grade level and by
domain, which are broad areas of mathematics. Within each domain, there are clusters, which are groups of related standards.
The Standards for Mathematical Practice (MP) are the same at each grade level, with the exception of an additional practice standard included in the CA CCSSM for higher mathematics only⁚ MP3.1⁚
Students build proofs by …
Importance of Practice
Regular practice is crucial for students to solidify their understanding of math concepts and develop fluency in problem-solving. When students engage in consistent practice, they have the
opportunity to reinforce learned skills, identify areas where they need further support, and build confidence in their abilities. This practice helps them bridge the gap between understanding a
concept and being able to apply it effectively in different contexts.
Furthermore, practice allows students to explore various problem-solving strategies and develop their critical thinking skills. By encountering different types of problems and approaches, they learn
to adapt their thinking, analyze situations, and choose the most appropriate methods to solve them. This exposure to a variety of problem-solving scenarios prepares them for the challenges they will
encounter in higher-level math courses and real-world applications.
Ultimately, consistent practice is essential for fostering a strong foundation in mathematics and empowering students to become confident, capable problem-solvers.
Common Core Math Standards for Third Grade
The Common Core State Standards for Mathematics outline the essential math knowledge and skills that students should master by the end of each grade level. For third grade, these standards are
organized into four key domains⁚ Operations and Algebraic Thinking (OA), Number and Operations in Base Ten (NBT), Measurement and Data (MD), and Geometry (G).
Within each domain, the standards specify specific concepts and skills that students should understand and be able to demonstrate. For example, in Operations and Algebraic Thinking, third graders are
expected to understand the relationship between multiplication and division, solve word problems involving multiplication and division, and reason about patterns and relationships.
These standards provide a framework for teachers to design their curriculum and ensure that students are receiving a comprehensive and rigorous math education.
Operations and Algebraic Thinking (OA)
This domain focuses on students’ understanding of multiplication and division, as well as their ability to solve problems involving these operations. Third graders are expected to interpret products
of whole numbers, such as understanding that 5 x 7 represents the total number of objects in 5 groups of 7 objects each. They should also be able to solve one-step word problems involving
multiplication and division.
Furthermore, students develop strategies for multiplying and dividing within 100, including using arrays, equal groups, and measurement quantities. They learn to recognize patterns in multiplication
and division, and they begin to understand the relationship between these two operations. Finally, students explore the properties of operations, such as the commutative and associative properties,
and use them to solve problems.
Number and Operations in Base Ten (NBT)
This domain focuses on students’ understanding of place value and their ability to perform operations with whole numbers. Third graders are expected to understand the relationship between digits in a
multi-digit number, such as recognizing that the digit 3 in the number 345 represents 3 hundreds. They should be able to round whole numbers to the nearest ten or hundred and fluently add and
subtract within 1000, using strategies and algorithms based on place value.
Students also develop an understanding of multiplication and division within 100. They learn to multiply one-digit numbers by multiples of 10, such as 4 x 30, and they use strategies to solve
two-step word problems involving multiplication and division within 100. Furthermore, they begin to explore the relationship between multiplication and division, recognizing that division can be used
to solve multiplication problems and vice versa.
Measurement and Data (MD)
This domain focuses on students’ understanding of measurement and data analysis. Third graders learn to measure and estimate liquid volumes and masses of objects using standard units of grams (g),
kilograms (kg), and liters (l). They are expected to solve one-step word problems involving masses or volumes that are given in the same units. This includes using drawings, such as a beaker with a
measurement scale, to represent the problem.
They also develop an understanding of time and how to tell time to the nearest minute. Students learn to solve elapsed time problems and work with units of time such as hours, minutes, and seconds.
Furthermore, they explore the concept of area by counting unit squares and relating area to multiplication and addition; They learn to find the area of a rectangle by tiling it with unit squares and
then multiplying the length and width of the rectangle.
Geometry (G)
In Geometry, third graders delve into the world of shapes and their properties. They learn to reason about plane shapes and their attributes, including the number of sides and angles. They are able
to classify two-dimensional shapes based on their attributes, such as triangles, quadrilaterals, pentagons, and hexagons. Students also explore the concept of perimeter, the total distance around a
two-dimensional shape, and learn to find the perimeter of a rectangle by adding the lengths of all its sides.
They also learn to partition shapes into equal parts and understand that a fraction 1/b represents one part when a whole is partitioned into b equal parts. This lays the foundation for understanding
fractions as numbers and their relationship to whole numbers. Students also develop an understanding of the relationship between multiplication and division, using tools to solve math problems and
relating area to multiplication and addition.
Resources for Practice
There are numerous resources available to help third-grade students practice their Common Core Math skills. Free printable worksheets are readily available online, offering a wide variety of problems
covering all the key concepts. These worksheets can be printed and used for homework, classroom activities, or independent practice. Practice tests are also a valuable resource, allowing students to
assess their understanding of the material and identify areas where they may need more practice. These tests are often available for free online, offering a comprehensive assessment of the Common
Core Math standards.
Workbooks are another excellent option, providing a more structured and comprehensive approach to practicing Common Core Math. Workbooks often include detailed explanations, examples, and practice
problems, helping students develop a deeper understanding of the concepts. Many workbooks are available for purchase online or at bookstores, offering a variety of levels and topics to cater to
different learning styles and needs.
Free Printable Worksheets
Free printable worksheets are a valuable resource for third-grade students practicing Common Core Math. These worksheets are readily available online, offering a variety of problems covering all the
key concepts covered in the curriculum. They can be tailored to individual student needs and learning styles, allowing for targeted practice and reinforcement of specific skills. Parents and teachers
can use these worksheets for homework assignments, classroom activities, or independent practice, making learning fun and engaging.
Free printable worksheets are a convenient and cost-effective way to supplement classroom instruction and provide students with additional practice opportunities. They can be easily accessed and
printed, making them a readily available resource for both home and school use.
Practice Tests
Practice tests are an essential tool for gauging a third-grader’s understanding of Common Core Math standards. These tests are designed to mimic the format and difficulty level of actual assessments,
providing students with a realistic preview of what to expect. They help students familiarize themselves with the test format, identify their strengths and weaknesses, and develop test-taking
Practice tests allow students to build confidence and reduce test anxiety. They also provide valuable feedback to teachers and parents, enabling them to identify areas where students may need
additional support. Practice tests can be administered in a variety of ways, from online platforms to printed materials, offering flexibility for different learning environments.
Workbooks provide a comprehensive and structured approach to mastering third-grade Common Core Math standards. They offer a wealth of practice problems, examples, and explanations, covering a broad
range of topics. Workbooks are often organized by skill or concept, allowing students to focus on specific areas where they need improvement.
They provide a consistent and structured learning environment, allowing students to work at their own pace and revisit concepts as needed. Workbooks can serve as a valuable supplement to classroom
instruction, providing additional practice and reinforcement. Some workbooks also include answer keys, allowing students to self-check their work and identify areas for improvement.
Tips for Effective Practice
Effective practice is key to mastering third-grade Common Core Math standards. To maximize the benefits of practice, consider these tips⁚
First, focus on key concepts. Instead of trying to cover everything at once, prioritize the most important concepts for each standard. This allows students to build a strong foundation and avoid
feeling overwhelmed. Secondly, use a variety of practice methods. Varying activities can keep students engaged and help them learn in different ways. Consider using games, puzzles, hands-on
activities, and online resources in addition to traditional worksheets.
Finally, provide feedback and support. Review students’ work regularly and provide constructive feedback. Offer encouragement and assistance as needed. Make sure students understand the concepts and
can apply them independently. By following these tips, you can help students develop a solid understanding of third-grade math concepts and prepare them for future success.
Focus on Key Concepts
The Common Core Standards for Mathematics are designed to be comprehensive, covering a wide range of mathematical skills and concepts. However, it’s crucial to focus on key concepts within each
standard to ensure students develop a strong foundation. This approach helps avoid overwhelming students with too much information and allows them to build a solid understanding of essential
mathematical principles.
For instance, in the domain of “Number and Operations in Base Ten,” a key concept is place value. Students need to understand the relationship between digits in different place values, such as ones,
tens, and hundreds. Focusing on this concept will provide a strong foundation for understanding numbers and performing operations.
By prioritizing key concepts, students can develop a deeper understanding of the material and be better prepared for future mathematical challenges.
Use a Variety of Practice Methods
To keep students engaged and motivated, it’s essential to utilize a variety of practice methods. Repetitive worksheets can be effective for reinforcing skills, but they can also become monotonous.
Incorporating diverse approaches can cater to different learning styles and make practice more enjoyable.
Consider incorporating hands-on activities like using manipulatives, such as blocks or counters, to represent mathematical concepts. Games can also be a fun and engaging way to practice math skills.
For example, a simple card game involving addition or subtraction can be a great way to reinforce basic operations.
Technology can also be a valuable tool for practice. Educational apps and online games can provide interactive and engaging learning experiences. By incorporating various practice methods, you can
create a more dynamic and stimulating learning environment for students.
Provide Feedback and Support
Providing regular feedback and support is crucial for helping students learn and progress in their math skills. When reviewing practice work, take the time to provide specific and constructive
feedback. Highlight areas where the student demonstrates understanding and identify areas where they may need further support.
For example, if a student is struggling with a particular concept, you might suggest additional practice problems or offer a brief explanation. Be sure to acknowledge their efforts and celebrate
their successes. Positive reinforcement can encourage students to persevere and develop a more positive attitude towards mathematics.
It’s important to create a supportive learning environment where students feel comfortable asking questions and seeking help. By providing timely feedback and encouragement, you can help students
develop confidence and achieve their full potential in math. | {"url":"https://firstamericandream.com/3rd-grade-math-common-core-standards-practice-pdf/","timestamp":"2024-11-11T07:17:47Z","content_type":"text/html","content_length":"55814","record_id":"<urn:uuid:3bb813d0-32aa-4018-9161-a9ceed6df18c>","cc-path":"CC-MAIN-2024-46/segments/1730477028220.42/warc/CC-MAIN-20241111060327-20241111090327-00030.warc.gz"} |
Primes in arithmetic progressions to smooth moduli
Speaker: Julia Stadlmann Date: Mon, Mar 4, 2024 Location: PIMS, University of Lethbridge, Zoom, Online Conference: Analytic Aspects of L-functions and Applications to Number Theory Subject:
Mathematics, Number Theory Class: Scientific CRG: L-Functions in Analytic Number Theory
The twin prime conjecture asserts that there are infinitely many primes p for which p+2 is also prime. This conjecture appears far out of reach of current mathematical techniques. However, in 2013
Zhang achieved a breakthrough, showing that there exists some positive integer h for which p and p+h are both prime infinitely often. Equidistribution estimates for primes in arithmetic progressions
to smooth moduli were a key ingredient of his work. In this talk, I will sketch what role these estimates play in proofs of bounded gaps between primes. I will also show how a refinement of the q-van
der Corput method can be used to improve on equidistribution estimates of the Polymath project for primes in APs to smooth moduli. | {"url":"https://www.mathtube.org/lecture/video/primes-arithmetic-progressions-smooth-moduli","timestamp":"2024-11-10T14:47:49Z","content_type":"application/xhtml+xml","content_length":"26923","record_id":"<urn:uuid:f49109b4-4dc6-4717-bccc-4e8b01208842>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.60/warc/CC-MAIN-20241110134821-20241110164821-00487.warc.gz"} |
Math Skills
Every lesson begins with theme art and an engaging narrative from “Professor K” designed to draw students into the lesson.
Basic Math Skills covers basic math concepts beginning with simple properties and extending through calculations of powers and roots, percentages, volume, weight, temperature, area, and unknowns.
Students receive a solid foundation in scientific notation, metric systems, multiplication, basic geometry, graphs, square and cube roots, PEMDAS, grouping, fractions, decimals, percent, and
interest. Students learn how to understand and complete
word problems
, especially those types which appear on state academic assessments.
Basic Math Skills
is designed for
seventh grade
students. The course is also excellent for high school students who were under-served in other educational programs.
Basic Math Skills
consists of six soft-cover texts with six companion activity books. Eighteen section quizzes and six chapter tests are included.
NOTE: This course is also an excellent refresher study for older students or adults who need to brush up on basic math before taking competency exams or Algebra I. PAC’s Math Skills Diagnostic Test
(link provided below) is designed to help teachers identify exactly where students should begin working to recover academic math gaps. Basic Math Skills is recommended before students enroll in
Intermediate Math Skills. High school students who complete at least six chapters of Basic Math Skills and Intermediate Math Skills qualify for one transcript credit in General, Consumer, or Basic
Course Resources
Scope and Sequence Text Sample Activity Sample Diagnostic Test Diagnostic Test Key | {"url":"https://pacworks.com/product/basic-math-skills/?attribute_pa_product-options=chapter-1-activity&attribute_pa_product-format=print","timestamp":"2024-11-05T06:07:39Z","content_type":"text/html","content_length":"231028","record_id":"<urn:uuid:db202f1a-9437-4cb6-b4ad-a96f76bee787>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00629.warc.gz"} |
Work of 5 Joules is done in stretching a spring from its natural length to 19 cm beyond its natural length. What is the force (in Newtons) t - DocumenTVWork of 5 Joules is done in stretching a spring from its natural length to 19 cm beyond its natural length. What is the force (in Newtons) t
Work of 5 Joules is done in stretching a spring from its natural length to 19 cm beyond its natural length. What is the force (in Newtons) t
Work of 5 Joules is done in stretching a spring from its natural length to 19 cm beyond its natural length. What is the force (in Newtons) that holds the spring stretched at the same distance (19
cm)? Don’t forget to enter the correct units.
in progress 0
Physics 3 years 2021-07-27T08:32:27+00:00 2021-07-27T08:32:27+00:00 1 Answers 39 views 0 | {"url":"https://documen.tv/question/work-of-5-joules-is-done-in-stretching-a-spring-from-its-natural-length-to-19-cm-beyond-its-natu-15583012-49/","timestamp":"2024-11-10T01:42:45Z","content_type":"text/html","content_length":"79991","record_id":"<urn:uuid:1516c968-1160-4637-9f9c-d70faf959190>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.3/warc/CC-MAIN-20241110005602-20241110035602-00069.warc.gz"} |
Cash flow liquidity ratio: Exploring Liquidity Metrics: Cash Flow Ratios Demystified - FasterCapital
Cash flow liquidity ratio: Exploring Liquidity Metrics: Cash Flow Ratios Demystified
1. Introduction to Liquidity and Cash Flow
liquidity and cash flow are the lifeblood of any business, representing the ability of a company to meet its short-term obligations and operate smoothly. The essence of liquidity lies in its
measurement of how quickly and easily assets can be converted into cash, which is vital for maintaining day-to-day operations. Cash flow, on the other hand, tracks the actual movement of money in and
out of a business, offering a dynamic picture of financial health.
1. Current Ratio: This metric compares a company's current assets to its current liabilities, providing a quick snapshot of its ability to pay off short-term debts with assets that are readily
convertible to cash. For example, a company with a current ratio of 2:1 indicates it has twice as many current assets as current liabilities.
2. Quick Ratio: Also known as the acid-test ratio, this measure excludes inventory from current assets, focusing on the most liquid assets only. It's a stringent test of liquidity, as inventory can
be harder to liquidate quickly. A firm with a quick ratio of 1:1 is considered to have adequate liquidity if it can instantly clear all current liabilities without selling any inventory.
3. Cash Ratio: The most conservative liquidity metric, the cash ratio, considers only cash and cash equivalents against current liabilities. It shows the company's ability to satisfy short-term
liabilities immediately with its most liquid assets. For instance, a cash ratio greater than 1 indicates a company has more cash on hand than the total amount of its short-term liabilities.
4. operating Cash Flow ratio: This ratio assesses how well current liabilities are covered by the cash flow generated from a company's core business operations. It's an indicator of operational
efficiency and financial stability. A ratio greater than 1 suggests that the company generates enough cash flow to cover its short-term obligations.
To illustrate, consider a technology startup that has been rapidly expanding its market presence. Despite significant revenue growth, the startup must carefully monitor its liquidity ratios to ensure
it can sustain operations and invest in further growth. If the startup reports a declining quick ratio over consecutive quarters, it may signal that while revenue is growing, the cash available to
meet immediate obligations is shrinking, potentially due to increased inventory levels or slower collection of receivables.
Understanding these ratios and their implications allows stakeholders to make informed decisions about the financial direction and potential investment opportunities within a company. It's a delicate
balance to maintain sufficient liquidity for stability while also investing in growth opportunities that may temporarily reduce liquid assets. The key is to manage these financial metrics to support
both the present and future aspirations of the business.
Introduction to Liquidity and Cash Flow - Cash flow liquidity ratio: Exploring Liquidity Metrics: Cash Flow Ratios Demystified
2. The Significance of Cash Flow Ratios in Financial Analysis
In the realm of financial analysis, liquidity metrics serve as a critical barometer for assessing a company's ability to meet its short-term obligations. Among these metrics, cash flow ratios stand
out as they strip away the veil of accounting estimates and adjustments, offering a raw and unvarnished look at the actual cash moving in and out of a business. These ratios are pivotal for
stakeholders who seek a transparent view of a company's liquidity health beyond what traditional balance sheet figures may reveal.
1. Operating cash Flow ratio: This ratio, calculated as operating cash flow divided by current liabilities, provides insight into whether a company can cover its short-term debts with the cash it
generates from its core business operations. For example, a ratio greater than 1 indicates a solid footing, whereas a ratio less than 1 might signal potential liquidity issues.
2. free Cash Flow to sales Ratio: Offering a measure of efficiency, this ratio demonstrates how well a company converts its sales into free cash flow, an essential indicator of profitability and
dividend-paying capacity. A higher percentage suggests that a company is effectively translating sales into cash, which can be used for expansion, debt reduction, or shareholder returns.
3. cash Flow Coverage ratios: These ratios, including the cash flow to debt ratio, measure the ability of the company's cash flow to service its debts and other obligations. A higher ratio implies a
greater ability to sustain and pay off debts, which is reassuring for creditors and investors alike.
By examining these ratios, analysts and investors can peel back the layers of a company's financial facade, gaining a deeper understanding of its operational efficiency, investment acumen, and
overall financial health. For instance, a company with robust cash flow ratios is often seen as a safer investment, as it indicates a higher likelihood of enduring economic downturns and capitalizing
on growth opportunities. Conversely, weak cash flow ratios may hint at underlying problems that could surface during financial stress, making such a company a riskier bet.
The Significance of Cash Flow Ratios in Financial Analysis - Cash flow liquidity ratio: Exploring Liquidity Metrics: Cash Flow Ratios Demystified
3. Calculating Key Cash Flow Ratios
In the realm of financial analysis, the evaluation of a company's liquidity through cash flow ratios is pivotal. These ratios, which compare different line items from a company's financial statements
, offer insights into its ability to meet short-term obligations and manage cash efficiently. They are particularly useful for stakeholders to assess the health and operational efficiency of a
business. Here, we delve into several key ratios that serve as indicators of liquidity and financial stability:
1. Operating Cash Flow (OCF) Ratio: This ratio measures the adequacy of cash generated from operations to cover current liabilities. It is calculated as:
$$\text{OCF Ratio} = \frac{\text{Operating Cash Flow}}{\text{Current Liabilities}}$$
For instance, if a company has an operating cash flow of \$120,000 and current liabilities of \$100,000, the OCF ratio would be 1.2, indicating a comfortable liquidity position.
2. Free Cash Flow (FCF) to sales ratio: This ratio helps in understanding how much cash a company generates relative to its sales revenue, reflecting efficiency in cash generation. It is expressed
$$\text{FCF to Sales Ratio} = \frac{\text{Free Cash Flow}}{\text{Net Sales}}$$
Consider a business with \$50,000 in free cash flow and \$500,000 in net sales; the FCF to Sales Ratio would be 0.1, or 10%, suggesting that for every dollar of sales, 10 cents are converted into
free cash flow.
3. Cash flow Coverage ratios: These ratios are crucial for determining a company's ability to pay off its debts, particularly the interest and principal on its borrowings. The debt Service Coverage
ratio (DSCR), for example, is a common metric:
$$\text{DSCR} = \frac{\text{Net Operating Income}}{\text{Total Debt Service}}$$
If a company's net operating income stands at \$200,000 and its total debt service is \$160,000, the DSCR would be 1.25, implying that the company generates enough operating income to cover its debt
obligations 1.25 times over.
4. Capital Expenditure (CapEx) Ratio: This ratio sheds light on a firm's investment in long-term assets to maintain or expand its operations. It is determined by:
$$\text{CapEx Ratio} = \frac{\text{Cash Flow from Operations}}{\text{Capital Expenditures}}$$
A CapEx Ratio greater than 1 suggests that the company can finance its capital expenditures from its operational cash flow, which is a sign of financial strength.
By analyzing these ratios, investors and creditors can gauge a company's liquidity and cash management prowess. It's important to note that while higher ratios typically indicate better liquidity and
financial health, excessively high values may also suggest underinvestment or inefficient use of resources. Therefore, these ratios should be interpreted in the context of industry benchmarks and the
company's historical performance.
Calculating Key Cash Flow Ratios - Cash flow liquidity ratio: Exploring Liquidity Metrics: Cash Flow Ratios Demystified
4. A Deep Dive
In the realm of financial analysis, liquidity ratios serve as a critical barometer for assessing a company's ability to meet its short-term obligations. Among these, the cash ratio is often regarded
as the most conservative measure, providing a stringent lens through which to view a firm's immediate liquidity position. This ratio, calculated by dividing a company's total cash and cash
equivalents by its current liabilities, strips away the less liquid elements of current assets, offering a stark assessment of financial health.
1. The Essence of the Cash Ratio
- Conservatism: The cash ratio is unforgiving; it does not consider inventory or receivables, which may be subject to valuation uncertainties.
- Benchmarking: A higher ratio suggests a stronger liquidity cushion, which can be compared against industry standards.
2. Interpretation in Context
- Sector Variability: Different industries have varying norms for what constitutes a 'healthy' cash ratio.
- Temporal Dynamics: The ratio should be tracked over time to discern trends, rather than relying on a single snapshot.
3. Beyond the Numbers
- Operational Efficiency: A high cash ratio might indicate operational efficiency or conservative management but could also suggest underutilization of assets.
- Strategic Implications: Companies with robust cash ratios are better positioned to take advantage of strategic opportunities or weather economic downturns.
To illustrate, consider Company X with \$50 million in cash and cash equivalents and \$25 million in current liabilities, yielding a cash ratio of 2. This indicates that Company X has twice the
amount of cash needed to cover its short-term liabilities, which may be interpreted as a strong liquidity position. However, if Company X operates in a sector where the norm is a cash ratio of 0.5,
this might suggest an overly cautious approach to asset management.
In contrast, Company Y in the same industry with a cash ratio of 0.3 may face scrutiny from creditors and investors concerned about its ability to fulfill short-term obligations, despite being closer
to the industry standard. This underscores the importance of context when interpreting liquidity metrics.
Through this lens, stakeholders can gain a nuanced understanding of a company's liquidity and make informed decisions based on its cash ratio.
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5. Understanding the Operational Efficiency
In assessing a company's financial health, the ability to generate cash from its core business operations is paramount. This is where the concept of cash flow from operations comes into play, serving
as a critical indicator of a firm's financial viability. It is not merely the amount of cash that flows through the business but the quality of that cash flow that matters. This ratio, often
overlooked in favor of more traditional metrics, offers a window into the operational efficiency and short-term financial stability of a business.
1. Definition and Calculation: The ratio is defined as the amount of cash generated by a company's normal business operations in relation to its current liabilities. It is calculated using the
$$ \text{Operating cash Flow Ratio} = \frac{\text{operating Cash Flow}}{\text{Current Liabilities}} $$
A higher ratio indicates a company's adeptness at covering its short-term obligations with the cash it produces, thus signaling operational efficiency.
2. Significance: This ratio is a more reliable measure of liquidity than many others because it is harder to manipulate with accounting practices. It provides a direct look at the cash a company is
generating without the need for adjustments or interpretations.
3. Interpretation: A ratio greater than one suggests that a company has more than enough cash flow to cover its immediate obligations, which is a sign of financial health. Conversely, a ratio less
than one may indicate potential liquidity issues.
4. Limitations: While informative, this ratio should not be used in isolation. It must be considered alongside other financial metrics to provide a comprehensive view of a company's financial status.
5. Examples for Clarity:
- Example 1: A company with an operating cash flow of \$120,000 and current liabilities of \$100,000 would have an operating cash flow ratio of 1.2, which is considered healthy.
- Example 2: Another company with \$80,000 in operating cash flow and \$100,000 in current liabilities would have a ratio of 0.8, indicating it may struggle to meet its short-term debts.
By examining this ratio, stakeholders can gain insights into the company's ability to sustain its operations and fulfill its financial obligations without relying on external financing. This metric,
therefore, becomes an indispensable tool for investors, creditors, and management alike in making informed decisions.
Understanding the Operational Efficiency - Cash flow liquidity ratio: Exploring Liquidity Metrics: Cash Flow Ratios Demystified
6. The Ultimate Measure of Liquidity
In the realm of financial metrics, the ratio that measures the cash a company generates after accounting for cash outflows to support operations and maintain its capital assets stands paramount.
Unlike other liquidity metrics, this particular ratio strips away the non-cash elements of the income statement, offering a crystal-clear view of the actual liquidity position. It's the acid test for
investors to discern whether a company can sustain operations, expand its asset base, and return value to shareholders without relying on external financing.
1. Definition and Calculation: It is calculated by taking the Net Cash from Operating Activities and subtracting Capital Expenditures. The formula is as follows:
$$\text{Free cash Flow Ratio} = \frac{\text{Net cash from Operating Activities} - \text{Capital Expenditures}}{\text{Total Debt}}$$
This ratio provides a more nuanced understanding of a company's liquidity by considering its free cash relative to the total debt.
2. Significance: It serves as a robust indicator of a company's financial health. A high ratio suggests that the company has enough liquidity to cover its debts, invest in growth, and weather
economic downturns.
3. Interpretation: Analysts look for a consistent or improving trend over time as a sign of stability. A declining trend, however, could signal potential trouble ahead.
4. Limitations: While insightful, it should not be viewed in isolation. It must be considered alongside other financial metrics and qualitative factors to paint a complete picture of a company's
Example: Consider a company with \$50 million in net cash from operating activities and \$10 million in capital expenditures. If the company's total debt is \$200 million, the ratio would be:
$$\text{Free Cash Flow Ratio} = \frac{\$50\text{m} - \$10\text{m}}{\$200\text{m}} = 0.2$$
This indicates that for every dollar of debt, the company generates 20 cents in free cash flow, which can be a strong indicator of liquidity if the industry benchmark is lower.
By examining this ratio, stakeholders can gauge a company's ability to operate effectively without additional debt, highlighting its liquidity resilience. It's a testament to a company's prowess in
generating cash and its potential to thrive independently.
The Ultimate Measure of Liquidity - Cash flow liquidity ratio: Exploring Liquidity Metrics: Cash Flow Ratios Demystified
7. Cash Flow vsTraditional Liquidity Ratios
In the realm of financial analysis, liquidity metrics serve as a critical barometer for assessing a company's ability to meet its short-term obligations. While traditional liquidity ratios, such as
the current ratio and quick ratio, have long been staples in evaluating financial health, the advent of cash flow ratios offers a dynamic perspective, emphasizing actual cash availability over
theoretical asset liquidity.
1. The Essence of Cash Flow Ratios:
Cash flow ratios strip away the non-cash elements included in traditional liquidity ratios. For instance, the cash flow coverage ratio, which is calculated as operating cash flow divided by total
debt, provides a more tangible measure of a company's ability to service its debt with the cash it generates from its core business operations.
Example: Consider a company with an operating cash flow of \$50 million and total debt of \$200 million. The cash flow coverage ratio would be:
$$\frac{\$50\,million}{\$200\,million} = 0.25$$
This indicates that the company generates enough cash to cover 25% of its total debt annually.
2. Traditional Liquidity Ratios:
Traditional liquidity ratios, such as the current ratio (current assets divided by current liabilities), may include assets that are not readily convertible to cash, such as inventory, which could
potentially overstate a firm's short-term financial strength.
Example: A company with \$100 million in current assets and \$50 million in current liabilities has a current ratio of:
$$\frac{\$100\,million}{\$50\,million} = 2$$
This suggests a comfortable liquidity position, but if a significant portion of the assets is tied up in slow-moving inventory, the practical liquidity could be less reassuring.
3. Comparative Insights:
When juxtaposed, cash flow ratios and traditional liquidity ratios can offer complementary insights. A robust cash flow ratio may signal strong operational efficiency and cash generation, while a
healthy traditional liquidity ratio indicates sufficient assets relative to liabilities. However, discrepancies between these ratios can unveil underlying financial nuances that merit closer
Example: A firm with a high current ratio but a low cash flow coverage ratio might be holding large amounts of inventory or receivables, which could imply potential cash flow issues despite apparent
asset liquidity.
The integration of cash flow ratios into liquidity analysis provides a more nuanced understanding of a company's financial agility. By considering both cash flow and traditional liquidity ratios,
analysts and investors can gain a comprehensive view of a firm's ability to navigate its financial commitments, ultimately leading to more informed decision-making.
I have had some great successes and great failures. I think every entrepreneur has. I try to learn from all of them.
8. Real-World Case Studies
In the realm of financial analysis, the practical application of cash flow ratios can be the linchpin for understanding a company's liquidity position. These ratios, when applied to real-world
scenarios, offer a granular view of how effectively a business manages its cash in relation to its liabilities. By dissecting case studies, one can discern patterns and strategies that either bolster
a firm's financial health or signal potential distress.
1. Operating Cash flow ratio: This ratio, which compares operating cash flow to current liabilities, serves as a barometer for a company's ability to cover short-term obligations with the cash
generated from its core business operations. For instance, Company A with an operating cash flow of \$50 million and current liabilities of \$30 million would have an operating cash flow ratio of $$\
frac{50}{30} \approx 1.67$$, indicating a comfortable cushion for meeting short-term debts.
2. Free cash Flow to Sales ratio: Highlighting the percentage of sales converted into free cash flow, this metric is pivotal for investors gauging profitability. Consider Company B, which reports \
$200 million in sales and \$40 million in free cash flow, resulting in a ratio of $$\frac{40}{200} = 0.20$$ or 20%. This suggests that for every dollar of sales, Company B generates 20 cents of free
cash flow.
3. Cash Flow Coverage Ratios: These ratios measure the number of times a firm can pay off its debt with its cash flow. For example, Company C with a cash flow of \$100 million and total debt of \$400
million has a cash flow coverage ratio of $$\frac{100}{400} = 0.25$$, implying it generates enough cash to cover 25% of its debt annually.
Through these examples, it becomes evident that cash flow ratios are not mere abstract figures but are deeply intertwined with the operational realities of businesses. They provide a lens through
which analysts can evaluate the efficacy of management's strategies in maintaining liquidity and ensuring the company's ongoing viability.
Real World Case Studies - Cash flow liquidity ratio: Exploring Liquidity Metrics: Cash Flow Ratios Demystified | {"url":"https://www.fastercapital.com/content/Cash-flow-liquidity-ratio--Exploring-Liquidity-Metrics--Cash-Flow-Ratios-Demystified.html","timestamp":"2024-11-04T01:23:43Z","content_type":"text/html","content_length":"77202","record_id":"<urn:uuid:c11003c1-dae2-4098-a0be-fe264816613c>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00050.warc.gz"} |
Guidelines for Submitted Tutorial Problem Sheets
When preparing solutions for marking please note the following points.
• Not attempting a solution is generally unacceptable.
• Set out your solution as a coherent narrative explaining any principles used. The final solution should be underlined, or a reasonable gap left in the text so the end of the question is obvious.
• In general solve problems algebraically before inserting values.
• In multiple part questions, identify all the answers being asked for and present a solution that answers them all. Counting the things asked for in a problem, and matching this to the number of
underlined results in your manuscript, can ensure this.
• Do not use different notation from that used in the question. Define any new variables you introduce.
• Write legibly. If you struggle to do this in first draft then rewrite your solutions when complete.
• Solutions should be presented on one side of the paper and stapled at the top left hand corner (or otherwise held together). Sheppard supplement: The presentation of solutions on pages formed
into a Möbius strip is unaccepatable.
Obeying these rules is good practice and will pay off in exams. Scripts that fail to respect these rules will be returned to be rewritten. | {"url":"https://eodg.atm.ox.ac.uk/user/grainger/teaching/gr.html","timestamp":"2024-11-03T03:14:43Z","content_type":"text/html","content_length":"12517","record_id":"<urn:uuid:10e9165e-53c9-4cfd-aa36-e6e37aabf5b7>","cc-path":"CC-MAIN-2024-46/segments/1730477027770.74/warc/CC-MAIN-20241103022018-20241103052018-00643.warc.gz"} |
Classifying Types Of Numbers Worksheet
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a fraction with a denominator of a power of 10
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numbers Classifying numbers review Worked example classifying numbers Math Numbers and operations Classify numbers Google
All numbers on number line Counting numbers Natural numbers and zero Negative numbers and whole numbers Can be expressed as a fraction of two integers A B B 0 terminating decimal can be expressed as
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Classifying Real Numbers Worksheet
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Classifying Real Numbers Worksheet
Classifying Real Numbers Worksheet
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Alternate Interior Angles: Definition, Features and how to find them
The angles created when a transversal intersects two coplanar lines are popular as alternative interior angles. They are on the inside of the parallel lines but the outside of the transversal. The
transversal also cuts through two lines that are Coplanar at different places. These angles indicate whether or not the two provided lines are parallel. If these angles are equivalent, the lines
intersecting with the transversal are parallel.
An angle is produced when two rays, or lines with just one terminus, intersect at a point known as a vertex. The distance between the two beams determines the angle. Angles in geometry are frequently
represented using the angle symbol. Thus angle A might be written as angle A, or When a line (present there as a transversal) crosses two lines, AIAs develop on opposing sides of the transversal.
Now, if the two lines are parallel, then the alternate interior angles are equal. If you want to know more about this topic, you’re welcome here. Read on as we describe how Alternate Interior Angles
Alternate Interior Angles Definition
When a transversal intersects two parallel lines, the pair of angles created on the inner side of the parallel lines, but different sides of the transversal are known as alternative interior angles.
These angles are always the same size. You can also interpret it as. The different interior angles demonstrate whether or not the provided lines are parallel. If these angles are equivalent, the
provided lines intersected by the transversal are parallel.
Types of Alternate Angles
· Alternate Interior Angle
· Alternate Exterior Angle
The alternate exterior angles are ones with different vertices, located on opposite sides of the transversal, and are exterior to the lines. The alternate exterior angles generated when a transversal
meets two parallel lines are always equal. The pairs of opposite outside angles in the same figure also are 1 & 7 and 2 & 8.
What is a transversal line?
A transversal line is a line that connects or crosses two other lines. When two additional lines are parallel, the transversal crosses across each at the same angle. The other two lines do not have
to be parallel in order for a transversal to cross them.
What are parallel lines?
As we know parallel lines are two lines that never intersect or cross in a two-dimensional plane. When a transversal crosses over parallel lines, the generated angles have unique characteristics.
These characteristics do not exist when the lines are not parallel.
Alternate Interior Angles are equal
Therefore we present to you the way to prove that Interior Angles are equal.
We know that if k ∥ l, then ∠2≅∠8 and ∠3≅∠5 . ,
According to the Corresponding Angles theorem,
Therefore, by the definition of congruent angles,
Since ∠1 and ∠2 form a linear pair, they are supplementary, so
Also, ∠5 and ∠8 are supplementary, so
Substituting m∠1 form∠5, we get
Subtracting m∠1 from both sides, we have
M∠8=180°−m∠1 =m∠2.
Therefore, ∠2≅∠8.
You can prove that ∠3≅∠5 using the same. This theorem’s inverse is also valid. If a transversal cuts two lines k and l so that the alternate interior angles are equivalent.
How to find Alternate Interior Angles
The alternate interior angles of two parallel lines are equal, according to the alternate interior angles theorem. This information is also used to calculate alternate interior angles. Therefore, let
me illustrate this with an example.
The diagram below depicts a map in which the Sixth Avenue road runs perpendicular to the adjacent First and Second Streets. Another route, Maple Avenue, forms a 40° angle with 2nd Street. But can you
calculate the angle x?
The two streets are parallel, and Maple Avenue is regarded also as the transversal. Therefore x and 40° are the alternate interior angles, according to the alternate interior angles theorem. As a
result, both angles are equal. As a result, x = 40°.
• Each pair of alternate internal angles has the same value.
• One co-interior angle pair is extra.
• Each corresponding angle pair is also equal.
• Each alternating pair of outside angles is equal.
The z-pattern is another technique to consider different interior orientations. It’s worth noting that a pair of alternate interior angles also forms a Z.
Alternate Interior Angles properties
• Interior angles which are opposite each other are congruent.
• The total of the angles generated on the same side of the transversal by the two parallel lines equals 180°.
• In the case of non-parallel lines, alternate interior angles have no special characteristics.
Alternate Interior Angles Theorem
According to the Alternate Interior Angles theorem, when a transversal intersects two parallel lines, the pairs of alternate interior angles are congruent.
A theorem is a proven assertion or accepted concept that has been demonstrated to be correct. The antithesis of this theorem, which is essentially the reverse, is also a proven statement. If a
traversal slices two lines, and the alternate interior angles are congruent, then the lines are parallel. You can use these theorems to solve geometry issues and locate missing information. This
graphic indicates which angles are equal and alternate inside. Take note of how the lines are parallel. Let us explain this with the assistance of the following diagram: 1 = 5 (equivalent angles), 3
= 5 (vertically opposite angles). As a result,1 Equals 3. Similarly, we may demonstrate that 2 Equals 4. This demonstrates how the two provided lines are parallel since their alternative inner angles
are equivalent.
Alternate Interior Angles example
Example 1:
We have two angles (4x – 19)° and (3x + 16)° , they are congruent alternate interior angles. Now, Find the value of x and also find the value of the other pair of alternate interior angles,
⇒ 4x – 19 = 3x + 16
⇒ 4x – 3x = 19+16
x = 35
So, x = 350
(4x – 19)0 ⇒ 4(35) – 19 = 1210
We know angles formed on the same side of the transversal are called supplementary angles. Thus, the value of the other pair is:
⇒ 1800 – 1210= 590
Example 2:
There are two consecutive interior angles are (2x + 10) ° and (x + 5) °. Find what the angles measure.
Consecutive interior angles are supplementary.
⇒ (2x + 10) ° + (x + 5) ° = 180
⇒ 2x + 10 + x + 5 = 180
⇒ 3x + 15 = 180
Subtracting 15 from both sides.
⇒ 3x = 165
Now we divide both sides by 3.
x = 55
Thus, the consecutive interior angles are:
⇒ (2x + 10) ° = [2(55) + 10] ° = 120°
⇒ (x + 5) ° = 55 + 5° = 60°
Example 3:
If the given angles (2x + 26) ° and (3x – 33) ° are alternate interior angles and are congruent. Find what the two angles measure.
Alternative interior angles are equal, therefore:
⇒ (2x + 26) ° = (3x – 33) °
⇒ 2x + 26 = 3x – 33
x = 59
The measure of the angles is 144°.
Example 4:
Find the value of x when (3x + 20) ° and 2x° are consecutive interior angles.
We know consecutive interior angles are supplementary, so
⇒ (3x + 20) ° + 2x° = 180°
⇒3x + 20 + 2x = 180
⇒5x + 20 = 180
Subtracting 20 from both sides
⇒5x = 160
Dividing each side by 8.
x = 32
So, the value of x is 32 degrees.
The consecutive interior angles are, 60° and 120°.
FAQ’s on Alternate Interior Angles
How many pairs of alternate interior angles are there in a transversal?
In a single traversal you will find two pairs of alternate interior angles.
Are alternate interior angles supplementary or complementary?
They are congruent. However, consecutive interior angles are supplementary.
What do alternate interior angles add up to?
They add up to 180 degrees.
Alternate Interior Angles on the ……. side of the transversal
Alternate Interior Angles on the different side of the transversal.
When a transversal meets two coplanar lines, we get alternate interior angles. Any two crossing lines must be in the same plane and hence coplanar. They are on the inside of the parallel lines but on
the outside of the transversal. These angles indicate whether the two provided lines are parallel.
Read Also: Alphanumeric characters: Functions, Examples and Everything You Need to Know
When two angles share a common side and a vertex, they are said to be neighboring. When two lines meet, the angles are known as known as vertically opposite. Since one angle is opposite to the other.
The alternate internal angles are congruent when a transversal crosses the collection of parallel lines. The two lines are parallel if the alternate interior angles generated by the transversal line
on two coplanar are congruent. If two things have the same size and shape, they are congruent. | {"url":"https://theeducationtraining.com/alternate-interior-angles/","timestamp":"2024-11-09T22:37:28Z","content_type":"text/html","content_length":"77727","record_id":"<urn:uuid:775d1f3d-6570-462a-abf1-fd2ad868db39>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.10/warc/CC-MAIN-20241109214337-20241110004337-00252.warc.gz"} |
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How do I convert between wavelength and frequency and wavenumber? | Socratic
How do I convert between wavelength and frequency and wavenumber?
1 Answer
I assume you are referring to interconverting between $\lambda$, $\nu$, and $t i l \mathrm{de} \nu$, or something like that? I say this because $\lambda$ is typically wavelength in $\text{nm}$, $\nu$
is typically frequency in ${\text{s}}^{-} 1$, and $t i l \mathrm{de} \nu$ means energy in wavenumbers (${\text{cm}}^{-} 1$). If you want ${\text{nm}}^{- 1}$, just reciprocate $\text{nm}$.
There are four possibilities for conversions that I could cover:
1. $\lambda \to \nu$
2. $\nu \to t i l \mathrm{de} \nu$
3. $\nu \to \lambda$
4. $t i l \mathrm{de} \nu \to \nu$
However, recognize that if you can do 1 and 2, you have done 3 and 4 backwards, and if you can do 1 and 2 consecutively, you can go straight from $\lambda$ to $t i l \mathrm{de} \nu$ (same with 3 and
4 but $t i l \mathrm{de} \nu$ to $\lambda$). So, I will only show 1 and 2.
Suppose we have $\lambda = 600 n m$ for yellow light and we want its frequency in ${s}^{-} 1$.
What we want is to convert from a unit of length to a unit of reciprocal time, which requires something that has $\text{length"/"time}$ units... The speed of light works great here, and it's about $3
\times {10}^{8} \text{m/s}$. Therefore:
1. Reciprocate the wavelength
2. Convert to $\text{m}$
3. Multiply by the speed of light
#overbrace((1/(600 cancel("nm"))))^(lambda)xx((10^9 cancel("nm"))/(1 cancel("m")))xx(3xx10^8 cancel("m")/"s") = underbrace(color(blue)(5.bar55xx10^(-3) "s"^-1))_(nu)#
• $\nu \to t i l \mathrm{de} \nu$
This is fairly straightforward. We have $\frac{1}{\text{s}}$ and want $\frac{1}{\text{cm}}$. Suppose we have a frequency of $6 \times {10}^{- 3} {\text{s}}^{-} 1$.
1. Divide by the speed of light
2. Convert to $\text{cm}$
#overbrace((6xx10^(-3) 1/cancel("s")))^(nu)xx(cancel("s")/(3xx10^8 cancel("m")))xx((1 cancel("m"))/(100 "cm")) = underbrace(color(blue)(2xx10^(-13) "cm"^(-1)))_(tildenu)#
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ARMA Models for Trading, Part III | R-bloggersARMA Models for Trading, Part III
ARMA Models for Trading, Part III
[This article was first published on
The Average Investor's Blog » R
, and kindly contributed to
]. (You can report issue about the content on this page
) Want to share your content on R-bloggers?
if you have a blog, or
if you don't.
In the last post I showed how to pick the parameters for the ARMA model. The next step is to determine the position at the close. One way to do that is by a one day ahead prediction, if the
prediction comes negative (remember the series we are operating on is the daily returns) then the desired position is short, otherwise it’s long.
getSymbols("SPY", from="1900-01-01")
SPY.rets = diff(log(Ad(SPY)))
SPY.arma = armaFit(~arma(0, 2), data=as.ts(tail(SPY.rets,500)))
predict(SPY.arma, n.ahead=1, doplot=F)
Now, to build an indicator for back testing, one can walk the daily return series and at each point perform the steps we covered so far. The main loop looks like (in pseudocode):
for(ii in history:length(dailyRetSeries))
tt = as.ts(tail(head(dailyRetSeries, ii), history))
ttArma = findBestArma()
predict(ttArma, n.ahead=1, doplot=F)
Where history is the look-back period to consider at each point, I usually use 500, which is about two years of data. Although the above code is simply an illustration, I hope the main idea is pretty
clear by now.
As mentioned earlier, findBestArma needs to be surrounded by a tryCatch block. Same goes for the predict – it may fail to converge. What I do is to have predict included in findBestArma, ignoring
models for which the prediction fails.
Another improvement is to use ARMA together with GARCH. The latter is a powerful method to model the clustered volatility typically found in financial series. Sounds complex, but it turns out to be
pretty straightforward in R. Just to give you an idea:
getSymbols("SPY", from="1900-01-01")
SPY.rets = diff(log(Ad(SPY)))
SPY.garch = garchFit(~arma(0, 2) + garch(1, 1), data=as.ts(tail(SPY.rets, 500)))
predict(SPY.garch, n.ahead=1, doplot=F)
That’s all I have to say on the theoretical side. I will finish this series with more implementation details and some back testing results in the next post … | {"url":"https://www.r-bloggers.com/2011/05/arma-models-for-trading-part-iii/","timestamp":"2024-11-08T15:34:20Z","content_type":"text/html","content_length":"92297","record_id":"<urn:uuid:19cb5215-7c9b-41f2-a71f-9a50232426ca>","cc-path":"CC-MAIN-2024-46/segments/1730477028067.32/warc/CC-MAIN-20241108133114-20241108163114-00799.warc.gz"} |
PROC CALIS: Measures of Multivariate Kurtosis :: SAS/STAT(R) 9.22 User's Guide
Measures of Multivariate Kurtosis
In many applications, the manifest variables are not even approximately multivariate normal. If this happens to be the case with your data set, the default generalized least squares and maximum
likelihood estimation methods are not appropriate, and you should compute the parameter estimates and their standard errors by an asymptotically distribution-free method, such as the WLS estimation
method. If your manifest variables are multivariate normal, then they have a zero relative multivariate kurtosis, and all marginal distributions have zero kurtosis (Browne; 1982). If your DATA= data
set contains raw data, PROC CALIS computes univariate skewness and kurtosis and a set of multivariate kurtosis values. By default, the values of univariate skewness and kurtosis are corrected for
bias (as in PROC UNIVARIATE), but using the BIASKUR option enables you to compute the uncorrected values also. The values are displayed when you specify the PROC CALIS statement option KURTOSIS.
In the following formulas,
• uncorrected univariate skewness for variable
• corrected univariate skewness for variable
• uncorrected univariate kurtosis for variable
• corrected univariate kurtosis for variable
• Mardia’s multivariate kurtosis
• relative multivariate kurtosis
• Mardia based kappa
• mean scaled univariate kurtosis
• adjusted mean scaled univariate kurtosis
If variable leptokurtic if it has a positive value of platykurtic if it has a negative value of (Bentler; 1985):
PROC CALIS displays a message if
If weighted least squares estimates (METHOD=WLS or METHOD=ADF) are specified and the weight matrix is computed from an input raw data set, the CALIS procedure computes two more measures of
multivariate kurtosis.
• multivariate mean kappa
and (Bentler; 1985).
• multivariate least squares kappa
where (Bentler; 1985) and
The occurrence of significant nonzero values of Mardia’s multivariate kurtosis Browne (1974, 1982, 1984). | {"url":"http://support.sas.com/documentation/cdl/en/statug/63347/HTML/default/statug_calis_sect082.htm","timestamp":"2024-11-13T02:40:04Z","content_type":"application/xhtml+xml","content_length":"25844","record_id":"<urn:uuid:b4afe2a4-5931-4f72-83ce-7ac984fb63cd>","cc-path":"CC-MAIN-2024-46/segments/1730477028303.91/warc/CC-MAIN-20241113004258-20241113034258-00327.warc.gz"} |
#software-development #dotnet
Download the source-code here!
In order to get you up and running quickly you can find the gist of the code here or download a zip containing the LinqPad snippet here.
0. Contents
1. Introduction
A single coordinate is just a set of numbers indicating a location in a grid of a certain dimension. These numbers themselves are intuitive for humans to interpret at all. We’re going to explore how
to turn these numbers into a textual representation which can easily be understood by the common people. While the precision of the coordinate system is higher and more exact, I for myself would be
happy if I can tell the difference between Amsterdam, Paris and Berlin based on a set of coordinates.
For more information about the accuracy each decimal in a coordinate represents, see this answer on the GIS StackExchange.
We’re going to try something different. As we can say human interpretation of coordinates is shit, why not try to represent coordinates as something which is nuts for a computer to work with, but
actually useful for humans to interpret.
We are going to translate a coordinates into a textual representation which describes the position relative to a known landmark (e.g. a city or mountain). This has been inspired by the way positions
are communicated over the radio in aviation.
Aside; if you want to get to know or refresh your knowledge about radio technique in aviation, check out this guide, appropriately called “VFR COMMUNICATIONS FOR IDIOTS”.
2. The plan
My primary use case for this technique will be aviation related. Aircraft are moving all the time, and considering separation of gliders happens visually (meaning the pilots just have to look
outside) an accuracy of about 2km is good enough for me.
As we will be using local reference points our result shall be something in the likes of “2 miles north of Steenbergen at 1200 feet”. This little line contains all the information we need to
accurately identify a location:
1. A local (well known) reference point.
2. A direction from the reference point (360 degrees).
3. A certain distance from the reference point following the heading.
4. An altitude. Always easy in aviation.
We could always add more information about the movements of the aircraft (like heading north-northwest at 50 knots), but that should be trivial in relation to this post.
3. The code
We’re going to write a little library which will solve most of the problems of translating coordinates to a human readable format. A while ago I made a pull-request to the Humanizer repository which
added degree-heading to word capabilities, which will help us along the way later.
3.1 Resolving landmarks
First we have to retrieve data about local landmarks. Essentially what we need is a dictionary (list of key-value pairs) which contain coordinates and a label. Thankfully there are sources on the
internet which provide databases of (big) cities and their coordinates we will use for this post. One of these is the GeoNames project. You can download various types of files for free from here. The
file I go with is the `cities5000.zip` file in order to get a text file which contains all cities with more than 5000 inhabitants.
There are various ways to efficiently filter through a dictionary with location data. For starters I would recommend putting this information in a database and to use the database engine to do
spatial queries. I am not going to cover this, and for the sake of the demo I’ll just use a `Dictionary` to store and filter the data in-memory.
We will need a helper method to filter out the cities closest to a given location. Since NTS (NetTopologySuite) is super useful in general when dealing with spatial data we’re using it’s K-D tree
implementation for filtering through location data.
Using it is not too difficult. Honestly, in a few lines we could populate it and find the nearest position to a point. Don’t mind my helper methods. You’ll get the gist of it.
var tree = new KdTree<LocationEntry>();
.ForEach((i) => tree.Insert(new Coordinate(i.Latitude, i.Longitude), i));
var landmark = tree.NearestNeighbor(coordinate);
3.2 Mathematical background
We need to know the angle from the point of reference to the coordinate we want to make more human readable. The math to do this has been figured out by people way smarter than me and has been around
for a long time. There is this website that does a great job explaining latitudal/longitudal calculations, and I recommend you check it out if you want to know all about it. It features some
interactive calculations and shows the math right there with some calculation.
3.3 Calculating the distance between points
In order to calculate the distance we use the haversine formula to calculate the shortest route over a sphere between two points, also known as the short circle distance.
The following code is derived from the GeoCoordinate class in the System.Device.Location namespace (System.Device.dll assembly). A .NET Standard port can be found here.
public static double DistanceTo(double lat1, double long1, double lat2, double long2)
if (double.IsNaN(lat1) || double.IsNaN(long1) || double.IsNaN(lat2) ||
throw new ArgumentException("Argument latitude or longitude is not a number");
var d1 = lat1 * (Math.PI / 180.0);
var num1 = long1 * (Math.PI / 180.0);
var d2 = lat2 * (Math.PI / 180.0);
var num2 = long2 * (Math.PI / 180.0) - num1;
var d3 = Math.Pow(Math.Sin((d2 - d1) / 2.0), 2.0) +
Math.Cos(d1) * Math.Cos(d2) * Math.Pow(Math.Sin(num2 / 2.0), 2.0);
return 6376500.0 * (2.0 * Math.Atan2(Math.Sqrt(d3), Math.Sqrt(1.0 - d3)));
3.4 Calculating the angle between points
In order to calculate the angle between two points we use a “rhumb line”. The rhumb line is a line you can follow from one point to another by following the same compass heading. Note that this is
not the short circle distance, which we talked about earlier. For most applications the distances are so small that it doesn’t really matter anyway.
The following code to calculate the angle of the rhumb line is copied from this StackOverflow answer.
public static double DegreeBearing(
double lat1, double lon1,
double lat2, double lon2)
var dLon = ToRad(lon2 - lon1);
var dPhi = Math.Log(
Math.Tan(ToRad(lat2) / 2 + Math.PI / 4) / Math.Tan(ToRad(lat1) / 2 + Math.PI / 4));
if (Math.Abs(dLon) > Math.PI)
dLon = dLon > 0 ? -(2 * Math.PI - dLon) : (2 * Math.PI + dLon);
return ToBearing(Math.Atan2(dLon, dPhi));
public static double ToRad(this double degrees)
return degrees * (Math.PI / 180);
public static double ToDegrees(this double radians)
return radians * 180 / Math.PI;
public static double ToBearing(this double radians)
// convert radians to degrees (as bearing: 0...360)
return (ToDegrees(radians) + 360) % 360;
It might be interesting to you to figure out the difference between “heading”, “bearing” and “course” if you do not already know. Someone described the differences here.
3.5 The result
I bet you could come up with this yourself, but essentially we have all the individual components to for a textual representation of a coordinate.
We figured out:
• The position we need to resolve
• The closest landmark
• The distance to the landmark
• The heading to the landmark
It’s fairly simple to put it together now:
var text = $"{distance}km {bearing.ToHeading(HeadingStyle.Full)} of {landmark.Data.Name}";
Which might result in “3km south of Bergen op Zoom”. | {"url":"https://disintegrated.parts/blog/2019-02-25/representing-coordinates-in-a-human-readable-way","timestamp":"2024-11-12T14:01:51Z","content_type":"text/html","content_length":"27830","record_id":"<urn:uuid:a945ce48-794e-4f54-ab2f-a7638ba81f06>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.45/warc/CC-MAIN-20241112113320-20241112143320-00768.warc.gz"} |
Handling Stack Overflow Errors in JavaScript Recursion
Recursion is a powerful programming concept that allows a function to call itself in order to solve problems. One of the biggest challenges when working with recursion in JavaScript is handling stack
overflow errors, especially when dealing with large input sizes. This article will explore the nuances of handling such errors, particularly with deep recursion. We will discuss strategies to
mitigate stack overflow errors, analyze real-world examples, and provide practical code snippets and explanations that can help developers optimize their recursive functions.
Understanding Recursion
Recursion occurs when a function calls itself in order to break down a problem into smaller, more manageable subproblems. Each time the function calls itself, it should move closer to a base case,
which serves as the stopping point for recursion. Here is a simple example of a recursive function to calculate the factorial of a number:
function factorial(n) {
// Base case: if n is 0 or 1, factorial is 1
if (n <= 1) {
return 1;
// Recursive case: multiply n by factorial of (n-1)
return n * factorial(n - 1);
// Example usage
console.log(factorial(5)); // Output: 120
In this example:
• n: The number for which the factorial is to be calculated.
• The base case is when n is 0 or 1, returning 1.
• In the recursive case, the function calls itself with n - 1 until it reaches the base case.
• This function performs well for small values of n but struggles with larger inputs due to stack depth limitations.
Stack Overflow Errors in Recursion
When deep recursion is involved, stack overflow errors can occur. A stack overflow happens when the call stack memory limit is exceeded, resulting in a runtime error. This is a common issue in
languages with limited stack sizes, like JavaScript.
The amount of stack space available for function calls varies across environments and browsers. However, deep recursive calls can lead to stack overflow, especially when implemented for large
datasets or in complex algorithms.
Example of Stack Overflow
Let’s look at an example that demonstrates stack overflow:
function deepRecursive(n) {
// This function continues to call itself, leading to stack overflow for large n
return deepRecursive(n - 1);
// Attempting to call deepRecursive with a large value
console.log(deepRecursive(100000)); // Uncaught RangeError: Maximum call stack size exceeded
In the above function:
• The function calls itself indefinitely until n reaches a value where it stops (which never happens here).
• As n grows large, the number of function calls increases, quickly exhausting the available stack space.
Handling Stack Overflow Errors
To handle stack overflow errors in recursion, developers can implement various strategies to optimize their recursive functions. Here are some common techniques:
1. Tail Recursion
Tail recursion is an optimization technique where the recursive call is the final action in the function. JavaScript does not natively optimize tail calls, but structuring your functions this way can
still help in avoiding stack overflow when combined with other strategies.
function tailRecursiveFactorial(n, accumulator = 1) {
// Using an accumulator to store intermediary results
if (n <= 1) {
return accumulator; // Base case returns the accumulated result
// Recursive call is the last operation, aiding potential tail call optimization
return tailRecursiveFactorial(n - 1, n * accumulator);
// Example usage
console.log(tailRecursiveFactorial(5)); // Output: 120
In this case:
• accumulator holds the running total of factorial computations.
• The recursive call is the last action, which may allow JavaScript engines to optimize the call stack (not guaranteed).
• This design makes it easier to calculate larger factorials without leading to stack overflows.
2. Using a Loop Instead of Recursion
In many cases, a simple iterative solution can replace recursion effectively. Iterative solutions avoid stack overflow by not relying on the call stack.
function iterativeFactorial(n) {
let result = 1; // Initialize result
for (let i = 2; i <= n; i++) {
result *= i; // Multiply result by current number
return result; // Return final factorial
// Example usage
console.log(iterativeFactorial(5)); // Output: 120
Key points about this implementation:
• The function initializes result to 1.
• A for loop iterates from 2 to n, multiplying each value.
• This approach is efficient and avoids stack overflow completely.
3. Splitting Work into Chunks
Another method to mitigate stack overflows is to break work into smaller, manageable chunks that can be processed iteratively instead of recursively. This is particularly useful in handling large
function processChunks(array) {
const chunkSize = 1000; // Define chunk size
let results = []; // Array to store results
// Process array in chunks
for (let i = 0; i < array.length; i += chunkSize) {
const chunk = array.slice(i, i + chunkSize); // Extract chunk
results.push(processChunk(chunk)); // Process and store results from chunk
return results; // Return all results
function processChunk(chunk) {
// Process data in the provided chunk
return chunk.map(x => x * 2); // Example processing: double each number
// Example usage
const largeArray = Array.from({ length: 100000 }, (_, i) => i + 1); // Create large array
In this code:
• chunkSize determines the size of each manageable piece.
• processChunks splits the large array into smaller chunks.
• processChunk processes each smaller chunk iteratively, avoiding stack growth.
Case Study: Optimizing a Fibonacci Calculator
To illustrate the effectiveness of these principles, let’s evaluate the common recursive Fibonacci function. This function is a classic example that can lead to excessive stack depth due to its
numerous calls:
function fibonacci(n) {
if (n <= 1) return n; // Base cases
return fibonacci(n - 1) + fibonacci(n - 2); // Recursive calls for n-1 and n-2
// Example usage
console.log(fibonacci(10)); // Output: 55
However, this naive approach leads to exponential time complexity, making it inefficient for larger values of n. Instead, we can use memoization or an iterative approach for better performance:
Memoization Approach
function memoizedFibonacci() {
const cache = {}; // Object to store computed Fibonacci values
return function fibonacci(n) {
if (cache[n] !== undefined) return cache[n]; // Return cached value if exists
if (n <= 1) return n; // Base case
cache[n] = fibonacci(n - 1) + fibonacci(n - 2); // Cache result
return cache[n];
// Example usage
const fib = memoizedFibonacci();
console.log(fib(10)); // Output: 55
In this example:
• We create a closure that maintains a cache to store previously computed Fibonacci values.
• On subsequent calls, we check if the value is already computed and directly return from the cache.
• This reduces the number of recursive calls dramatically and allows handling larger input sizes without stack overflow.
Iterative Approach
function iterativeFibonacci(n) {
if (n <= 1) return n; // Base case
let a = 0, b = 1; // Initialize variables for Fibonacci sequence
for (let i = 2; i <= n; i++) {
const temp = a + b; // Calculate next Fibonacci number
a = b; // Move to the next number
b = temp; // Update b to be the latest calculated Fibonacci number
return b; // Return the F(n)
// Example usage
console.log(iterativeFibonacci(10)); // Output: 55
Key features of this implementation:
• Two variables, a and b, track the last two Fibonacci numbers.
• A loop iterates through the sequence until it reaches n.
• This avoids recursion entirely, preventing stack overflow and achieving linear complexity.
Performance Insights and Statistics
In large systems where recursion is unavoidable, it's essential to consider performance implications and limitations. Studies indicate that using memoization in recursive functions can reduce the
number of function calls significantly, improving performance drastically. For example:
• Naive recursion for Fibonacci has a time complexity of O(2^n).
• Using memoization can cut this down to O(n).
• The iterative approach typically runs in O(n), making it an optimal choice in many cases.
Additionally, it's important to consider functionalities in JavaScript environments. As of ES2015, the handling of tail call optimizations may help with some engines, but caution is still advised for
browser compatibility.
Handling stack overflow errors in JavaScript recursion requires a nuanced understanding of recursion, memory management, and performance optimization techniques. By employing strategies like tail
recursion, memoization, iterative solutions, and chunk processing, developers can build robust applications capable of handling large input sizes without running into stack overflow issues.
Take the time to try out the provided code snippets and explore ways you can apply these techniques in your projects. As you experiment, remember to consider your application's data patterns and
choose the most appropriate method for your use case.
If you have any questions or need further clarification, feel free to drop a comment below. Happy coding! | {"url":"https://snippetassistant.com/handling-stack-overflow-errors-in-javascript-recursion/","timestamp":"2024-11-13T12:52:17Z","content_type":"text/html","content_length":"68893","record_id":"<urn:uuid:e4e0f6ac-6df4-45f0-b4f1-ad52dc2713a2>","cc-path":"CC-MAIN-2024-46/segments/1730477028347.28/warc/CC-MAIN-20241113103539-20241113133539-00782.warc.gz"} |
Python FillPriorityRouter Class to Avoid Full Queues
Oops, something went wrong. Please try again in a few moments.
class Router:
Superclass representing a basic router.
- input_queues: list
List of input queues for the router.
def __init__(self, input_queues):
Constructor to instantiate the Router class.
- input_queues: list
List of input queues for the router.
self.input_queues = input_queues
def prepare_load(self):
Prepares the load for the router by repeatedly taking crates from the queue with the least available space.
- list:
List of crates removed from the queues, in the order they were removed.
crates_removed = []
# Repeat the process three times or until all input queues are empty
for _ in range(3):
nonempty_queues = [queue for queue in self.input_queues if queue] # Get nonempty queues
if not nonempty_queues:
break # Exit the loop if all queues are empty
# Find the queue with the least remaining space
min_space_queue = min(nonempty_queues, key=lambda queue: len(queue))
# Take a crate out of the queue with the least remaining space
crate = min_space_queue.pop(0)
return crates_removed
class FillPriorityRouter(Router):
Subclass of Router representing a router that avoids full queues by taking crates from the queue with the least available space.
- None
def __init__(self, input_queues):
Constructor to instantiate the FillPriorityRouter class.
- input_queues: list
List of input queues for the router.
def prepare_load(self):
Prepares the load for the FillPriorityRouter by repeatedly taking crates from the queue with the least available space.
- list:
List of crates removed from the queues, in the order they were removed.
crates_removed = []
# Repeat the process three times or until all input queues are empty
for _ in range(3):
nonempty_queues = [queue for queue in self.input_queues if queue] # Get nonempty queues
if not nonempty_queues:
break # Exit the loop if all queues are empty
# Find the queue with the least remaining space
min_space_queue = min(nonempty_queues, key=lambda queue: len(queue))
# Take a crate out of the queue with the least remaining space
crate = min_space_queue.pop(0)
return crates_removed | {"url":"https://codepal.ai/code-generator/query/p0DwcOw2/python-fill-priority-router","timestamp":"2024-11-03T19:00:50Z","content_type":"text/html","content_length":"120800","record_id":"<urn:uuid:a68523ae-bfb7-4c37-97ec-00e0dc7bb52b>","cc-path":"CC-MAIN-2024-46/segments/1730477027782.40/warc/CC-MAIN-20241103181023-20241103211023-00540.warc.gz"} |
Density To Moles Calculator - Calculator Wow
Density To Moles Calculator
The Density to Moles Calculator serves as a fundamental tool in chemistry, allowing scientists and students to convert density and volume into moles of a substance. This article delves into its
functionality, practical applications, usage tips, answers common questions, and highlights its role in simplifying chemical computations.
In chemistry, converting density to moles is crucial for various applications, including determining concentrations of solutions, calculating reaction stoichiometry, and understanding the physical
properties of substances. The calculator provides a direct method to translate experimental data into quantitative chemical information, aiding in accurate analysis and formulation of scientific
How to Use
Using the Density to Moles Calculator involves these steps:
1. Enter Density: Input the density of the substance in grams per cubic centimeter (g/cm³).
2. Enter Volume: Specify the volume of the substance in cubic centimeters (cm³).
3. Enter Molecular Weight: Provide the molecular weight of the substance in grams per mole (g/mol).
4. Calculate Moles: Click the calculate button to derive the number of moles based on the given inputs.
10 FAQs and Answers
1. What does density measure in chemistry?
Density quantifies the mass of a substance per unit volume, providing information about its compactness or concentration.
2. How does the Density to Moles Calculator simplify chemical calculations?
By converting density and volume into moles using the formula (Density * Volume) / Molecular Weight, the calculator facilitates quick and accurate computations crucial for laboratory experiments and
theoretical studies.
3. Why is it essential to know the molecular weight?
Molecular weight is vital as it determines the mass of one mole of a substance, influencing its density and providing a basis for calculating moles accurately.
4. Can density values vary with temperature and pressure?
Yes, density changes with variations in temperature and pressure, affecting the accuracy of calculations. Standardizing conditions ensures consistent results.
5. What are some practical applications of the Density to Moles Calculator?
Applications include preparing chemical solutions with precise concentrations, determining reactant quantities in stoichiometric calculations, and analyzing experimental data in research settings.
6. How do I interpret the results from the calculator?
The calculator outputs the number of moles of the substance based on the entered density, volume, and molecular weight, aiding in quantitative analysis and formulation of chemical reactions.
7. Can the calculator handle different units of measurement?
Yes, as long as density is in g/cm³, volume in cm³, and molecular weight in g/mol, the calculator can convert these values into moles accurately.
8. What role does density play in determining substance properties?
Density serves as a physical property that reflects the mass-to-volume ratio of a substance, influencing its behavior in chemical reactions and industrial processes.
9. How can I use the calculator to verify experimental results?
By inputting experimental density and volume data into the calculator, researchers can validate theoretical predictions and ensure consistency between experimental and calculated values.
10. Why is accuracy crucial in density and mole calculations?
Accurate calculations are essential for maintaining precision in scientific research, ensuring reproducibility of results, and advancing knowledge in chemistry and related fields.
The Density to Moles Calculator stands as a valuable tool in chemical analysis, offering a straightforward method to convert experimental data into moles of a substance. By mastering its usage and
understanding its applications, chemists and students can enhance their understanding of chemical properties, streamline laboratory procedures, and contribute to advancements in scientific knowledge.
Embrace this calculator as a catalyst for precise calculations and explore its potential in unraveling the complexities of chemistry. | {"url":"https://calculatorwow.com/density-to-moles-calculator/","timestamp":"2024-11-07T00:48:51Z","content_type":"text/html","content_length":"65379","record_id":"<urn:uuid:2986e6da-c4cb-4139-8d62-9e1144128223>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00010.warc.gz"} |
1103 - Phone Cell
Nowadays, everyone has a cellphone, or even two or three. You probably know where their
name comes from. Do you? Cellphones can be moved (they are “mobile”) and they use wireless
connection to static stations called BTS (Base Transceiver Station). Each BTS covers an area
around it and that area is called a cell.
The Czech Technical University runs an experimental private GSM network with a BTS right on
top of the building you are in just now. Since the placement of base stations is very important
for the network coverage, your task is to create a program that will find the optimal position
for a BTS. The program will be given coordinates of “points of interest”. The goal is to find
a position that will cover the maximal number of these points. It is supposed that a BTS can
cover all points that are no further than some given distance R. Therefore, the cell has a circular
The picture above shows eight points of interest (little circles) and one of the possible optimal
BTS positions (small triangle). For the given distance R, it is not possible to cover more than
four points. Notice that the BTS does not need to be placed in an existing point of interest.
The input consists of several scenarios. Each scenario begins with a line containing two integer
numbers N and R. N is the number of points of interest, 1 ≤ N ≤ 2 000. R is the maximal
distance the BTS is able to cover, 0 ≤ R < 10 000. Then there are N lines, each containing two
integer numbers Xi, Yi giving coordinates of the i-th point, |Xi|, |Yi| < 10 000. All points are
distinct, i.e., no two of them will have the same coordinates.
The scenario is followed by one empty line and then the next scenario begins. The last one is
followed by a line containing two zeros.
A point lying at the circle boundary (exactly in the distance R) is considered covered. To avoid
floating-point inaccuracies, the input points will be selected in such a way that for any possible
subset of points S that can be covered by a circle with the radius R + 0.001, there will always
exist a circle with the radius R that also covers them.
For each scenario, print one line containing the sentence “It is possible to cover M points.”,
where M is the maximal number of points of interest that may be covered by a single BTS.
sample input
0 -100
sample output
It is possible to cover 4 points.
It is possible to cover 2 points.
The first sample input scenario corresponds to the picture, providing that the X axis aims right and Y axis down.
Central Europe Regional Contest 2007 | {"url":"http://hustoj.org/problem/1103","timestamp":"2024-11-12T20:31:49Z","content_type":"text/html","content_length":"9949","record_id":"<urn:uuid:dd042222-3787-42e0-839e-1ec7718f4b39>","cc-path":"CC-MAIN-2024-46/segments/1730477028279.73/warc/CC-MAIN-20241112180608-20241112210608-00386.warc.gz"} |
Properties Of Exponents Worksheet
Properties Of Exponents Worksheet. I am sure you’re completely capable of making these on your own, but if your time is more priceless than $0.ninety nine… This quiz and corresponding worksheet can
be used to gauge your data of exponent properties. Become a memberto entry additional content and skip ads. Using the math worksheets over breaks and in the course of the summer time will enable kids
to stay sharp and get ready for the upcoming faculty term.
Reza is an skilled Math instructor and a test-prep professional who has been tutoring students since 2008. He has helped many college students elevate their standardized test scores–and attend the
universities of their desires. He works with students individually and in group settings, he tutors each reside and online Math programs and the Math portion of standardized checks.
Notice, the bases are the same, so add the exponents. Before we start working with variable expressions containing exponents, let’s simplify a quantity of expressions involving only numbers. Let’s
review the vocabulary for expressions with exponents.
Exponent Properties And Polynomials Vacation Shade By Number
Exponents, also called powers, are values that present how many occasions to multiply a base number by itself. For instance, forty three is telling you to multiply 4 by itself 3 times.
This quiz and corresponding worksheet can be utilized to gauge your information of exponent properties. There are seven exponent rules, or legal guidelines of exponents, that your college students
have to be taught. Each rule exhibits how to solve different varieties of math equations and tips on how to add, subtract, multiply and divide exponents.
Related To “The Method To Multiply Exponents? +free Worksheet!”
The guide boasts 300 pages jam-packed with curriculum-based actions and exercises in every subject, with a give consideration to math and language arts. Original full-color illustrations all through
give the e-book a bright, lively fashion that can enchantment to older youngsters.
Keep in mind that in this process, the order of operations will still apply. We are adding new math worksheets to the positioning daily so visit us typically. We shall be glad to design any math
worksheets you might need in your Lesson Planning.
Algebra And Pre
When dividing two bases of the identical worth, keep the base the same, after which subtract the exponent values. Then multiply 4 by itself seven occasions to get the reply.
They shouldn’t even acknowledge these are discovering. If you imagine your baby will benefit, you can see issues you could get to additional enhance their studies. You might definitely purchase books
to allow them to, nonetheless, many kids see taking a glance at being a task these days.
Grade 7 Exponents And Powers Worksheets
Keep sixth grade college students absolutely informed of the significance of an exponential notation with this set of worksheets. These workout routines help them ably identify its parts and specific
a numeral in an exponential type. The quotient rule states that two powers with the identical base may be divided by subtracting the exponents.
More Lessons for College Algebra Math Worksheets A series of free College Algebra Video Lessons from UMKC – The University of Missouri-Kansas City. In the following workout routines, simplify every
expression with exponents. Now let’s take a look at an exponential expression that incorporates an influence raised to an influence.
Exponents And Powers Of Ten
Students can work on the identical set of Printable Exponents Worksheets a quantity of instances till they are confident of their data of the ideas. Repetition can allow them to research their
improvisation and actions to find out and perceive their errors on their very own, and keep away from repeating these mistakes on exams.
For example, use mult as an alternative of multiply or multiplication to search out worksheets that comprise both key phrases. eighth Grade Math Worksheets Free Printable Math Worksheets Math 8
Multiplication Worksheets Addition Worksheets Math Teacher Math Classroom Classroom Ideas Simplifying Algebraic Expressions. Assess your exponent expertise with this quiz which covers.
For any nonzero base, if the exponent is zero, its worth is 1. A product raised to a power equals the product of each issue raised to that power. No matter how long the equation, something raised to
the facility of zero turns into one.
If an exponent is transferred from one side of the equation to the other side of the equation, reciprocal of the exponent must be taken. For any base base, if there is not any exponent, the exponent
is assumed to be 1.
Explaining Law of exponents with crystal-clear examples, this chart helps them drive house the idea. All worksheets are free to obtain and use for practice or in your classroom.
A energy raised to a different energy equals that base raised to the product of the exponents. When there’s a number being raised by a unfavorable exponent, flip it right into a reciprocal to turn
the exponent right into a positive. Don’t use the adverse exponent to show the bottom into a adverse.
Follow this simple rule to adeptly and quickly clear up exponent problems utilizing the ability of a quotient rule. Simplify the questions by performing arithmetic operations and making use of the
rule. Worksheets are Work 5 exponents grade 9 arithmetic, Homework 9 1 rational exponents, Super powers, Math 9 final, Mathematics grade 9, Properties of exponents, Exponent rules practice, Grade 10
The difficulties may presumably occur, nonetheless, when you’ve attained a better times tables than five. The key of simply adding one other digit towards the ultimate end end result, to get the next
number contained in the table will become a lot more tough.
In the following exercises, simplify every expression using the Power Property for Exponents. In the following workout routines, simplify each expression using the Product Property for Exponents.
Notice that 6 is the product of the exponents, 2 and 3.
Proper use of the ability of a power property and unfavorable exponents property. Upgrade your abilities in solving issues involving quotient rule by working towards these printable worksheets.
Each worksheet is randomly generated and thus distinctive. The answer key is automatically generated and is placed on the second web page of the file. The drawback requires us to have only positive
exponents in our answer.
Research is an important part of the student’s studies, but establishing too much can decreased morale making a little one really feel confused. Math must always be shown to children in the enjoyable
Overlooking their work from home and looking for to accomplish each little factor at college never helps! I frequently recommend flash playing cards for multiplication within the residence. Add/
subtract/multiply divide 2 powers.This possibility does NOT work with PDF format.
He supplies an individualized custom studying plan and the personalized consideration that makes a difference in how students view math. The product of two powers with the identical base equals that
base raised to the sum of the exponents. A quotient merely means that you’re dividing two portions.
Become adept at figuring out the base and exponents from an exponential notation and writing the given numerals and variables in an exponential type with this bunch of pdf worksheets for grade 7.
Math-Aids.Com provides free math worksheets for lecturers, dad and mom, college students, and residential schoolers. The math worksheets are randomly and dynamically generated by our math worksheet
The second two terms have the same base, so add the exponents. As a member, you’ll additionally get limitless access to over 84,000 lessons in math, English, science, history, and extra.
Multiplication Properties Of Exponents Worksheet – Just about essentially the most onerous and hard points that you are capable of do with primary school students is have them to get pleasure from
math. Addition worksheets and subtraction worksheets aren’t what most children need to be performing of their time.
You have seen that when you combine like terms by adding and subtracting, you need to have the same base with the same exponent. But whenever you multiply and divide, the exponents could also be
different, and sometimes the bases could additionally be totally different, too.
OpenStax is part of Rice University, which is a 501 nonprofit. Kinetic by OpenStax presents entry to progressive study tools designed to assist you maximize your studying potential. Our mission is to
provide a free, world-class schooling to anyone, anywhere.
Division Word Problems three Digit 1 Divisor Displaying high 8 worksheets found for – Division Word Problems three Digit 1 Divisor. In different words a requirement schedule shows the legislation of
demand in chart form. Our mission is to enhance educational entry and studying for everybody.
You will be quizzed on phrases like positive and unfavorable exponents. In this lecture we focus on about powers exponents operations with Integer and rational exponents square roots and nth roots.
Use a number of key phrases from certainly one of our worksheet pages. There are a variety of issues that each dad and mom and instructors alike are able to doing that will assist you a pupil
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Average vs Weighted Average | Top 7 Best Differences (With Infographics)
Updated June 29, 2023
Difference Between Average vs Weighted Average
Two of the most used statistics in the world are an average and weighted average. Both averages and weighted averages have their merits and demerits and have their proper usage in particular
scenarios. Coming to the definition, a simple average is nothing but adding all the observations under a sample and dividing the same by several observations in a given sample. For example, if we
want to calculate an average of the sample given, say 9, 10, 15, 29, 35, the same can be done by adding all of them, i.e. (9+10+15+29+35) / 5 = 19.6.
When calculating a weighted average, each observation in the data set is multiplied by or assigned weight before being summed to derive a single average value. This process entails assigning a weight
to each quantity being averaged to determine its relative importance. The weightings ensure that items with similar values contribute more significantly to the average.
Head To Head Comparison Between Average vs Weighted Average (Infographics)
Below is the top 7 difference between Average vs Weighted Average:
Key Differences Between Average vs Weighted Average
Let us discuss some of the major differences between Average vs Weighted Average.
• The key difference between an average vs weighted average is that a simple average is nothing but adding up all the observation values and dividing the same by the total number of observations to
calculate the average. In contrast, the weighted average is an average where each observation value will have a frequency assigned or specific weight to calculate the average.
• Average finds the middle value, termed as central tendency, whereas weighted averages find the average, which is tilted towards more number of observations.
• Arithmetic means median and mode are the types of central tendency, whereas the weighted average is not a type of central tendency.
• Observations are always assumed to be equally weighted when using a simple average. In contrast, with a weighted average, each observation is assigned a unique and different value.
• Outliers or extreme values can impact the simple average, whereas the weighted average remains unaffected by such extreme values or outliers.
• Weighted averages find extensive accounting, finance, and portfolio value calculation usage. On the other hand, the simple average has broader applications but is limited by its susceptibility to
extreme values. In practical life, the calculation of the simple average is often supported by complementary averages such as the weighted average or simple moving average.
• The weighted average has one big limitation: the weights assigned can be subjective, affecting its calculation. In contrast, there is no such case in the calculation of simple.
Average vs Weighted Average Comparison Table
Let’s look at the top 7 Comparisons between Average vs Weighted Average.
Basis of Average Weighted Average
Basic The average is calculated by summing the observations given in the sample and dividing A weighted average is the type of average in which every observation in the given data set will
Definition that sum by the number of observations in the sample. be assigned weight before the summation to a sole average value.
Weighted Average = ∑(x[i]w[i]) / ∑w[i]
Average = ∑(x) / n Where x[i ]is the ith observation
Formula Where ∑(x) is the summation of all observations w[i] is the weight of the ith observation
n is the number of observations ∑(x[i]w[i]) is the summation of the product of x[i] and w[i]
∑w[i ]is the summation of the weights.
Conditions This average will only work when all of the observations are weighted equally. In a weighted average, each observation has a frequency assigned to it or a specific weight.
• The reason to use a weighted average instead of a simple average is when one wants to
There are no specific conditions where the simple average has to be applied. Still, if calculate an average that will be based on different or various percentage values for many
Use case other conditions are met, other averages are appropriate to use as a weighted average, categories.
moving average, etc. • The second case will be when one has a group of observations where each will have a
frequency associated with it.
Result The average is utilized to determine and generalize the middle value, earning the name The weighted average accurately represents the range where most observations fall and tend to
indication “central tendency” for this purpose. lean toward that range. This approach finds extensive use in the accounting field.
A weighted average is justified by its unbiasedness towards the middle value and the assigned
Advantage A simple average advantage is its simplicity of calculation and understanding. average value where most observations lie. Additionally, it remains unaffected by outliers or
extreme values.
The weighted average becomes a little complicated to understand when several observations
Disadvantage The simple average is affected by outliers. increase, and further, the weight assigned is of subjective matter and hence can be adjusted
per user discretion.
People use the simple average in mathematical equations, whereas they utilize and apply the weighted average in their daily or routine activities, such as finance. The simple average is a given data
set’s main and key representation.
In contrast, one must evaluate the weighted average first to arrive at a specific solution for a particular problem. One can use arithmetic formulas such as finding the median to solve the average of
a given data set or observations. The components are assigned weights based on their values to obtain a specific answer in the weighted average.
The weighted average is the one that shows up in many areas of finance besides the buying price of the shares, including inventory accounting, portfolio returns, and valuation. For inventory
accounting, the weighted average value of inventory accounts for ups and downs in commodity prices, for example. At the same time, FIFO and LIFO methods give more importance to timing than value.
When evaluating whether the company’s shares are properly priced, investors will use the weighted average cost of capital to discount a company’s future cash flows.
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This has been a guide to the top difference between Average vs Weighted Average. Here we also discuss the Average vs Weighted Average key differences with infographics and comparison table. You may
also have a look at the following articles to learn more – | {"url":"https://www.educba.com/average-vs-weighted-average/","timestamp":"2024-11-10T17:24:17Z","content_type":"text/html","content_length":"316896","record_id":"<urn:uuid:9f53d3e3-c8c2-46c3-b1ca-2823a830d61a>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.61/warc/CC-MAIN-20241110170046-20241110200046-00245.warc.gz"} |
Multi-treatment Meta Analysis of Summaries Options • Genstat Knowledge Base 2024
Use this to select the model for a multi-treatment meta analysis of summaries and the output and graphs to be generated from this.
This specifies which items of output are to be produced by the analysis.
Model Description of the model fitted by the analysis
Wald tests Wald tests for fixed model terms and accompanying F-statistics
Variance components Estimated variance components and parameters
Treatment means Predicted treatment means averaged over the experiments
Treatment effects Estimates of treatment effects averaged over the experiments
Variance-covariance matrix Variance-covariance matrix for the variance parameters
Deviance The residual deviance
Covariance model Estimated between-experiment variance-covariance model in matrix format
Monitoring Monitoring information at each iteration
Residual plot Uses the VPLOT procedure to produce residual diagnostic plots from the REML analysis. The type of residuals plotted is controlled by the Method for residuals setting
Residual checks Uses the VCHECK procedure to check the residuals for outliers and variance stability
This specifies the between-experiment variance-covariance model for the meta analysis. The first two models fit the experiment term as fixed effect, and the others as a random effect.
Identity A common variance for all experiments and no correlations between experiments
Diagonal matrix (heteroscedastic) A separate variance for each experiment and no correlations between experiments
Compound symmetry A common variance for all experiments with a common covariance between experiments
Heterogeneous compound symmetry A separate variance for each experiment with a common correlation between experiments
Unstructured A separate variance for each experiment and separate covariances between experiments
First order analytic model with common variance A common variance for all experiments and structured covariances between experiments using a factor analytic model with one term
Second order analytic model with common variance A common variance for all experiments and structured covariances between experiments using a factor analytic model with two terms
First order analytic model A separate variance for each experiment and structured covariances between experiments using a factor analytic model with one term
See the VMETA procedure for more information on the structure of each model.
Standard errors
Tables of means and effects can be accompanied by estimates of standard errors if selected above. You can choose whether Genstat computes standard errors or standard errors of differences (SEDs) for
the tables giving all values or just a summary of these.
Differences Display the minimum, mean and maximum standard errors of differences between means or effects.
Estimates Display the minimum, mean and maximum standard errors of estimates for means or effects.
All differences Display all the standard errors of differences between pairs of means or effects.
All estimates Display all the standard errors of estimates for means or effects.
Maximum iterations
This specifies the maximum number of iterations to use to optimize the REML likelihood.
Method for residuals
The list allows selection of the type of residuals to be plotted in the Residual plot. This is only enabled if a residual plot has been selected.
Combine all random terms Use the residuals combined from all random terms.
Final random term only Use the residuals from the final random term.
Standardized residuals from all random terms Use the standardized residuals after combining them from all random terms.
Standardized residuals from final random term only Use the standardized residuals from the final random term.
Action buttons
OK Save the settings and close the dialog.
Cancel Close the dialog without further changes.
Defaults Reset the options to the initial settings when the dialog was first opened.
Clear all check boxes and fields.
Help about this menu.
See also
• VMETA procedure in command mode. | {"url":"https://genstat.kb.vsni.co.uk/knowledge-base/vmeta-options/","timestamp":"2024-11-06T08:31:02Z","content_type":"text/html","content_length":"45655","record_id":"<urn:uuid:c7a7e64e-f9bc-4aa1-99b4-a3182b7659e3>","cc-path":"CC-MAIN-2024-46/segments/1730477027910.12/warc/CC-MAIN-20241106065928-20241106095928-00634.warc.gz"} |
Abhenries to Microhenries Conversion (abH to μH)
Abhenries to Microhenries Converter
Enter the electrical inductance in abhenries below to convert it to microhenries.
Do you want to convert microhenries to abhenries?
How to Convert Abhenries to Microhenries
To convert a measurement in abhenries to a measurement in microhenries, divide the electrical inductance by the following conversion ratio: 1,000 abhenries/microhenry.
Since one microhenry is equal to 1,000 abhenries, you can use this simple formula to convert:
microhenries = abhenries ÷ 1,000
The electrical inductance in microhenries is equal to the electrical inductance in abhenries divided by 1,000.
For example,
here's how to convert 500 abhenries to microhenries using the formula above.
microhenries = (500 abH ÷ 1,000) = 0.5 μH
Abhenries and microhenries are both units used to measure electrical inductance. Keep reading to learn more about each unit of measure.
What Is an Abhenry?
One abhenry is equal to the inductance of a conductor in which there is one abvolt of electromotive force when the current through the conductor is increased by one abampere per second. One abhenry
is equal to 1/1,000,000,000 of a henry.
The abhenry is a centimeter-gram-second (CGS) electromagnetic unit of electrical inductance. An abhenry is sometimes also referred to as an EMU. Abhenries can be abbreviated as abH; for example, 1
abhenry can be written as 1 abH.
Learn more about abhenries.
What Is a Microhenry?
One microhenry is equal to 1/1,000,000 of a henry, which is the inductance of a conductor with one volt of electromotive force when the current is increased by one ampere per second.
The microhenry is a multiple of the henry, which is the SI derived unit for electrical inductance. In the metric system, "micro" is the prefix for millionths, or 10^-6. Microhenries can be
abbreviated as μH; for example, 1 microhenry can be written as 1 μH.
Learn more about microhenries.
Abhenry to Microhenry Conversion Table
Table showing various
abhenry measurements
converted to
Abhenries Microhenries
1 abH 0.001 μH
2 abH 0.002 μH
3 abH 0.003 μH
4 abH 0.004 μH
5 abH 0.005 μH
6 abH 0.006 μH
7 abH 0.007 μH
8 abH 0.008 μH
9 abH 0.009 μH
10 abH 0.01 μH
20 abH 0.02 μH
30 abH 0.03 μH
40 abH 0.04 μH
50 abH 0.05 μH
60 abH 0.06 μH
70 abH 0.07 μH
80 abH 0.08 μH
90 abH 0.09 μH
100 abH 0.1 μH
200 abH 0.2 μH
300 abH 0.3 μH
400 abH 0.4 μH
500 abH 0.5 μH
600 abH 0.6 μH
700 abH 0.7 μH
800 abH 0.8 μH
900 abH 0.9 μH
1,000 abH 1 μH
More Abhenry & Microhenry Conversions | {"url":"https://www.inchcalculator.com/convert/abhenry-to-microhenry/","timestamp":"2024-11-11T16:48:35Z","content_type":"text/html","content_length":"66144","record_id":"<urn:uuid:d99c239a-9b8f-4b39-9cbe-0db9015e759d>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00108.warc.gz"} |
[Solved] Air in a tank is at 1 MPa and room temper | SolutionInn
Air in a tank is at 1 MPa and room temperature of 20C. It is used to
Air in a tank is at 1 MPa and room temperature of 20°C. It is used to fill an initially empty balloon to a pressure of 200 kPa, at which point the radius is 2 m and the temperature is 20°C. Assume
the pressure in the balloon is linearly proportional to its radius and that the air in the tank also remains at 20°C throughout the process. Find the mass of air in the balloon and the minimum
required volume of the tank
Fantastic news! We've Found the answer you've been seeking!
Step by Step Answer:
Answer rating: 100% (1 review)
Balloon final state V 43 1 43 2 3351 m P2V2 RT2 7966 ...View the full answer
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IIFT Nov 2015: Adopt different preparation strategy for different exam; get the expert tips
IIFT the highly ranked dream B school is going to conduct the entrance test on November 22, 2015 for admission to its 2 years flagship MBA(IB) in Delhi and Kolkata campuses in paper pen mode
The Indian Institute of Foreign Trade (IIFT) the highly ranked dream B school is going to conduct the entrance test on November 22, 2015 for admission to its 2 years MBA(IB) in Delhi and Kolkata
campuses in paper pen mode.
MBA (IB) at IIFT is the most sought after 2 years MBA programme in International Business. This programme is available at IIFT New Delhi and Kolkata campuses in India and at Dar-es-Salaam campus in
There is no age limit to appear in IIFT entrance examination except that the applicants must possess recognized Bachelors degree of minimum 3 years duration, although there is no requirement of
minimum percentage of marks. However in the second phase of admission round profile of a candidate like academic record, work experience, writing skills will play an important role to secure
admission in IIFT.
The admission of a candidate who joins the programme on provisional basis but fails to qualify in the Bachelors degree examination is liable to be cancelled.
IIFT Exam pattern IIFT will conduct the entrance test for its Delhi and Kolkata campuses in 20 cities across India. The conventional paper pencil mode test will be of two hours duration from 10.00
A.M. to 12.00 Noon to be held on Sunday, November 22, 2015.
The written admission test to MBA Programme in IIFT consists of multiple choice objective type questions (in English). The 2 hours IIFT Nov 2015 entrance comprises the questions covering English
Comprehension, General Knowledge & Awareness, Logical Reasoning and Quantitative Analysis.
The test composition There are no fixed number of questions in IIFT entance test, their number may go anywhere in the range of 115 to 130.
Quantitative Ability Number of question in this section can be somewhere between 25 to 28. It is expected that in IIFT Nov 2015 25 questions will appear. Major topics covered in this section are as
Expected No. of questions
Time, distance, speed/Time & work
Other Arithmatic questions
Log theory
Permutation combination
Data Interpretation Around 20 questions are expected in this section. IIFT asks 19-21 DI questions in different data forms like tables and bar diagrams. The information part is followed by 4-5
questions so its worth trying with patience. A little more practice can be of great help in this section
English Comprehension English comprehension is composed of core verbal ability and questions on Reading comprehension passages.
Around 36 questions on English comprehension are expected to appear in IIFT entrance test to be held on November 22, 2015 for admission session 2016-17 on following topics
No. of questions
One word substitution
Fill in the blanks
Jumbled sentences
Phrasal explanation
Grammatically correct or incorrect sentences
Figures of speech
Spelling correction
Reading comprehension 4 passages
Logical Reasoning Logical reasoning section is expected to have 20 questions based on blood relations, seating arrangements, series, analogy, sets. IIFT covers almost all the topics in logical
reasoning with 1 or 2 questions on each. Statements, arguments, conclusions are a few favourite topics which require more in depth study.
General Awareness 28-30 questions could be there to test the General Awareness of the aspirants in IIFT Nov 2015 entrance exam. Questions cover conventional as well as current national and
international affairs and relevant economic, business topics aprt from major policy decisions.
Questions can also be based to test the aspirants awareness about the birthday of famous personalities; their work; Constitution of India; fictional characters.
The candidate may attempt any question from any section during the time allotted to attempt the test in the examination.
Marking system in IIFT Entrance The marking system may vary from question to question or from section to section in the range of 0.5 to 1 mark per question.
Penalty of one third marks will be awarded for every wrong answer. The candidates therefore have to be careful to avoid this negative marking by choosing the answer options with well calculated
thought process.
Final Admission process after the Test Candidates will be called for Essay Writing, Group Discussion and Interview, based on marks obtained in Written Test. This second and final selection round will
be held in January/February 2016 in various cities in India. The candidates are required to select one of the specified centres for Essay Writing, GD and Interview.
How to prepare for the IIFT entrance test; expert tips In view of Prof. S.K. Agarwal, expert on Verbal Ability and mentor on IIFT Nov 2015 preparation, time management is the most important part to
score high in the examination. Entrance test to IIFT requires consistent and planned study pattern. The examination in itself is a little tricky and questions in Verbal Ability and Reading
Comprehension are sometimes indirect and quite confusing also.
All the 4 sections in the examination are equally important as you need to score a high percentile. If you are serious enough and have prepared well for IIFT Nov 2015, you must continue in the same
spirit. Despite different exam pattern as against CAT 2015, the line of preparation for IIFT exam could remain the same except for the fact that CAT 2015 would not have any section on General
Awareness. Most of the questions are similar and based on the same concepts of English language, Quantitative Ability, Logical Reasoning as appear in both these examinations; the need is to improve
your attempts in limited time.
Besides, The DI and LR in CAT 2015 have formed a new section as IIFT entrance exam is consisted of. It implies that while preparing for CAT 2015, candidates would spontaneously prepare for IIFT 2015.
The topics on General Awareness could be prepared along with the preparation to other sections.
Stay tuned to MBAUniverse.com for more updates on IIFT Nov 2015 exam | {"url":"https://www.mbauniverse.com/article/id/8798/IIFT-Nov-2015","timestamp":"2024-11-02T04:53:05Z","content_type":"text/html","content_length":"152769","record_id":"<urn:uuid:39fb47d8-154e-4931-b493-27556bd11103>","cc-path":"CC-MAIN-2024-46/segments/1730477027677.11/warc/CC-MAIN-20241102040949-20241102070949-00537.warc.gz"} |
Response Surface
Response Surface Design and Analysis
This tutorial, the first of three in this series, shows how to use Stat-Ease^® software for response surface methodology (RSM). This class of designs is aimed at process optimization. A case study
provides a real-life feel to the exercise.
If you are in a rush to get the gist of design and analysis of RSM, hop past all the “Note” sections. However, if/when you find the time, it will be well worth the effort to explore these
Due to the specific nature of this case study, a number of features that could be helpful to you for RSM will not be implemented in this tutorial. Many of these features are used in the earlier
tutorials. If you have not completed all these tutorials, consider doing so before starting this one.
We will presume that you are knowledgeable about the statistical aspects of RSM. For a good primer on the subject, see RSM Simplified (Anderson and Whitcomb, Productivity, Inc., New York, 2016). To
gain a working knowledge of RSM, we recommend you attend our Modern DOE for Process Optimization workshop. Visit www.statease.com and follow the “Learn DOE” link for more details on this and other
educational resources from Stat-Ease.
The case study in this tutorial involves production of a chemical. The two most important responses, designated by the letter “y”, are:
The experimenter chose three process factors to study. Their names and levels are shown in the following table.
Factor Units Low Level (-1) High Level (+1)
A - Time minutes 40 50
B - Temperature degrees C 80 90
C - Catalyst percent 2 3
Factors for response surface study
You will study the chemical process using a standard RSM design called a central composite design (CCD). It’s well suited for fitting a quadratic surface, which usually works well for process
The three-factor layout for this CCD is pictured below. It is composed of a core factorial that forms a cube with sides that are two coded units in length (from -1 to +1 as noted in the table above).
The stars represent axial points. How far out from the cube these should go is a matter for much discussion between statisticians. They designate this distance “alpha” – measured in terms of coded
factor levels. As you will see, the program offers a variety of options for alpha.
Assume that the experiments will be conducted over a two-day period, in two blocks:
1. Twelve runs: composed of eight factorial points, plus four center points.
2. Eight runs: composed of six axial (star) points, plus two more center points.
Design the Experiment
Start the program and click the blank-sheet icon Response Surface from the list of designs on the left to show the designs available for RSM.
The default selection is the Central Composite design, which is used in this case study.
To see alternative RSM designs for three or more factors, click at far left on Box Behnken (notice 17 runs near the screen bottom) and Miscellaneous designs, where you find the 3-Level Factorial
option (32 runs, including 5 center points). Now go back and re-select Central Composite design.
If not already entered, click the up arrow in the Numeric Factors entry box and Select 3 as shown below.
Before entering factors and ranges, click Options. Notice that it defaults to a Rotatable design with the axial (star) points set at 1.68179 coded units from the center – a conventional choice for
the CCD.
Many options are statistical in nature, but one that produces less extreme factor ranges is the “Practical” value for alpha. This is computed by taking the fourth root of the number of factors (in
this case 3^¼ or 1.31607). See RSM Simplified Chapter 8 “Everything You Should Know About CCDs (but dare not ask!)” for details on this practical versus other levels suggested for alpha in CCDs – the
most popular of which may be the “Face Centered” (alpha equals one). Press OK to accept the rotatable value. (Note: you won’t get the “center points in each axial block” option until you change to 2
blocks in this design, as below).
Using the information provided in the table on page 1 of this tutorial (or on the screen capture below), type in the details for factor Name (A, B, C), Units, and Low and High levels.
You’ve now specified the cubical portion of the CCD. As you did this, the program calculated the coded distance “alpha” for placement on the star points in the central composite design.
Alternatively, by clicking the “entered factor ranges in terms of alphas” option you can control how far out the runs will go for each of your factors.
Now return to the bottom of the central composite design form. Leave Type at its default value of Full (the other option is a “small” CCD, which we do not recommend unless you must reduce the number
of runs to the bare minimum). You will need two blocks for this design, one for each day, so click the Blocks field and select 2.
Notice the software displays how this CCD will be laid out in the two blocks – for example, 4 center points will go in one and 2 in the other. Click Next to reach the second page of the “wizard” for
building a response surface design. You now have the option of identifying Block Names. Enter Day 1 and Day 2 as shown below.
Press Next to enter Responses. Select 2 from the pull down list. Now enter the response Name and Units for each response as shown below.
At any time in the design-building phase, you can return to the previous page by pressing the Back button. Then you can revise your selections. Press Finish to view the design layout (your run order
may differ due to randomization).
The program offers many ways to modify the design and how it’s laid out on-screen. Preceding tutorials, especially Part 2 for One-Factor Categoric, delved into this in detail, so go back and look
this over if you haven’t already.
Save the Data to a File
Now that you’ve invested some time into your design, it would be prudent to save your work.
Enter the Response Data – Create Simple Scatter Plots
Assume that the experiment is now completed. At this stage, the responses must be entered into the program. We see no benefit to making you type all the numbers, particularly with the potential
confusion due to differences in randomized run orders. Therefore, use the Help, Tutorial Data menu and select Chemical Conversion from the list.
Let’s examine the data! Click on the Design node on the left to view the design spreadsheet. Move your cursor to Std column header and right-click to bring up a menu from which to select Sort
Ascending (this can also be done via a double-click on the header).
Now right-mouse click the Select column header (top left cell) and choose Space Point Type.
Notice the new column identifying points as “Factorial,” “Center” (for center point), and so on. Notice how the factorial points align only to the Day 1 block. Then in Day 2 the axial points are run.
Center points are divided between the two blocks.
Unless you change the default setting for the Select option, do not expect the Type column to appear the next time you run the program. It is only on temporarily at this stage for your information.
Before focusing on modeling the response as a function of the factors varied in this RSM experiment, it will be good to assess the impact of the blocking via a simple scatter plot. Click the Graph
Columns node branching from the design ‘root’ at the upper left of your screen. You should see a scatter plot with factor A:Time on the X-axis and the Conversion response on the Y-axis.
The correlation grid that pops up with the Graph Columns can be very interesting. First off, observe that it exhibits red along the diagonal—indicating the complete (r=1) correlation of any variable
with itself (Run vs Run, etc). Block versus run (or, conversely, run vs block) is also highly correlated due to this restriction in randomization (runs having to be done for day 1 before day 2). It
is good to see so many white squares because these indicate little or no correlation between factors, thus they can be estimated independently.
For now, it is most useful to produce a plot showing the impact of blocks because this will be literally blocked out in the analysis. Therefore, on the Graph Columns tool click the button where
Conversion intersects with Block as shown below. Then change Color By to Space Type.
The graph visually shows there is not much of a difference between the center point results for block 1 and 2. Bear in mind that this will be filtered out mathematically so as not to bias the
estimation of factor effects.
Change the Y Axis to Activity (by clicking down the column one box) to see how it’s affected by the day-to-day blocking (even less).
Next, to see how the responses correlate, change the X Axis to Conversion. Now that we have 2 numeric factors along the axes, we can see the correlation between them. In the upper left of the legend
you will see the correlation number is 0.224, showing slight correlation.
You may also note there is a faded pink color in the box in the grid for this graph, denoting the slight upward correlation.
Now for a really awesome scatterplot in 3D, change the X Axis to A:Time, the Y Axis to C:Catalyst and the Z-Axis to Conversion. This provides a dramatic view of conditions leading to maximizing the
response. Grab it with your mouse and rotate it around. This looks quite promising!
Continue exploring relationships with the graph columns tools. However, do not get carried away with this, because it will be much more productive to do statistical analysis first – before drawing
any conclusions.
Analyze the Results
Now let’s start analyzing the responses numerically. Under the Analysis branch click the node labeled Conversion and press the Start Analysis button. A new set of tabs appears at the top of your
screen. They are arranged from left to right in the order needed to complete the analysis. What could be simpler?
Stat-Ease provides a full array of response transformations via the Transform option. Click Tips for details. For now, accept the default transformation selection of None.
Now click the Fit Summary tab. At this point the program fits linear, two-factor interaction (2FI), quadratic, and cubic polynomials to the response. At the top is the response identification,
immediately followed below, in this case, by a warning: “The Cubic Model is aliased.” Do not be alarmed. By design, the central composite matrix provides too few unique design points to determine all
the terms in the cubic model. It’s set up only for the quadratic model (or some subset).
Next you will see several extremely useful tables for model selection. Each table is discussed briefly via sidebars in this tutorial on RSM.
Use the blue layout buttons to choose how many panes are visible on your screen at once.
The table of “Sequential Model Sum of Squares” (technically “Type I”) shows how terms of increasing complexity contribute to the total model.
The Sequential Model Sum of Squares table: The model hierarchy is described below:
• “Linear vs Block”: the significance of adding the linear terms to the mean and blocks,
• “2FI vs Linear”: the significance of adding the two factor interaction terms to the mean, block, and linear terms already in the model,
• “Quadratic vs 2FI”: the significance of adding the quadratic (squared) terms to the mean, block, linear, and twofactor interaction terms already in the model,
• “Cubic vs Quadratic”: the significance of the cubic terms beyond all other terms.
For each source of terms (linear, etc.), examine the probability (“Prob > F”) to see if it falls below 0.05 (or whatever statistical significance level you choose). So far, the program is indicating
(via bold highlighting) the quadratic model looks best – these terms are significant, but adding the cubic order terms will not significantly improve the fit. (Even if they were significant, the
cubic terms would be aliased, so they wouldn’t be useful for modeling purposes.) Move down to the Lack of Fit Tests pane for Lack of Fit tests on the various model orders.
The “Lack of Fit Tests” pane compares residual error with “Pure Error” from replicated design points. If there is significant lack of fit, as shown by a low probability value (“Prob > F”), then be
careful about using the model as a response predictor. In this case, the linear model definitely can be ruled out, because its Prob > F falls below 0.05. The quadratic model, identified earlier as
the likely model, does not show significant lack of fit. Remember that the cubic model is aliased, so it should not be chosen.
Look over the last pane in the Fit Summary report, which provides “Model Summary Statistics” for the ‘bottom line’ on comparing the options
The quadratic model comes out best: It exhibits low standard deviation (“Std. Dev.”), high “R-Squared” values, and a low “PRESS.”
The program automatically underlines at least one “Suggested” model. Always confirm this suggestion by viewing these tables.
From the main menu select Help, Screen Tips or simply press the lightbulb icon (
The program allows you to select a model for in-depth statistical study. Click the Model tab at the top of the screen to see the terms in the model.
The program defaults to the “Suggested” model shown in the earlier Fit Summary table.
If you want, you can choose an alternative model from the Process Order pull-down list. (Be sure to try this in the rare cases when Stat-Ease suggests more than one model.)
Also, you could now manually reduce the model by clicking off insignificant effects. For example, you will see in a moment that several terms in this case are marginally significant at best. The
program provides several automatic reduction algorithms as alternatives to the “Manual” method: “Backward,” “Forward,” and “Stepwise.” Click the “Auto Select…” button to see these. From more details,
try Screen Tips and/or search Help.
Click the ANOVA tab to produce the analysis of variance for the selected model.
The ANOVA in this case confirms the adequacy of the quadratic model (the Model Prob > F is less than 0.05.) You can also see probability values for each individual term in the model. You may want to
consider removing terms with probability values greater than 0.10. Use process knowledge to guide your decisions.
Next, move over to the Fit Statistics pane to see various statistics to augment the ANOVA. The R-Squared statistics are very good — near to 1.
Next, move down to the Coefficients pane to bring the following details to your screen, including the mean effect-shift for each block, that is; the difference from Day 1 to Day 2 in the response.
Press Coded Equation to bring the next section to your screen — the predictive models in terms of coded factors. Click Actual Equation for the predictive models in terms of actual factors. Block
terms are left out. These terms can be used to re-create the results of this experiment, but they cannot be used for modeling future responses.
You cannot edit any ANOVA outputs. However, you can copy and paste the data to your favorite word processor or spreadsheet. Also, as detailed in the One-Factor RSM tutorial, th eprogram provides a
tool to export equations directly to Excel in a handy format that allows you to enter whatever inputs you like to generate predicted response. This might be handy for clients who are phobic about
statistics. 😉
Diagnose the Statistical Properties of the Model
The diagnostic details provided by the program can best be grasped by viewing plots available via the Diagnostics tab. The most important diagnostic — normal probability plot of the residuals —
appears in the first pane.
Data points should be approximately linear. A non-linear pattern (such as an S-shaped curve) indicates non-normality in the error term, which may be corrected by a transformation. The only sign of
any problems in this data may be the point at the far right. Click this on your screen to highlight it as shown above.
Notice that residuals are “externally studentized” unless you change their form on the drop-down menu at the top of your screen (not advised).
• Externally calculating residuals increases the sensitivity for detecting outliers.
• Studentized residuals counteract varying leverages due to design point locations. For example, center points carry little weight in the fit and thus exhibit low leverage.
Now click the Resid. vs Run tab.
Now you can see that, although the highlighted run does differ more from its predicted value than any other, there is really no cause for alarm due to it being within the red control limits.
Next move to the Cook’s Distance tab.
Nothing stands out here.
Move on to the Leverage tab. This is best explained by the previous tutorial on One-Factor RSM so go back to that if you did not already go through it. Then skip ahead to DFBETAS, which breaks down
the changes in the model to each coefficient, which statisticians symbolize with the Greek letter β, hence the acronym DFBETAS — the difference in betas. For the Term click the down-list arrow and
select A as shown in the following screen shot.
You can evaluate ten model terms (including the intercept) for this quadratic predictive model (see sidebar below for help).
Reposition your mouse over the Term field and simply scroll your mouse wheel to quickly move up and down the list. In a similar experiment to this one, where the chemist changed catalyst, the DFBETAS
plot for that factor exhibited an outlier for the one run where its level went below a minimal level needed to initiate the reaction. Thus, this diagnostic proved to be very helpful in seeing where
things went wrong in the experiment.
Now move on to the Report tab in the bottom-right pane to bring up detailed case-by-case diagnostic statistics, many which have already been shown graphically.
The footnote below the table (“Predicted values include block corrections.”) alerts you that any shift from block 1 to block 2 will be included for purposes of residual diagnostics. (Recall that
block corrections did not appear in the predictive equations shown in the ANOVA report.)
Examine Model Graphs
The residuals diagnosis reveals no statistical problems, so now let’s generate response surface plots. Click the Model Graphs tab. The 2D contour plot of factors A versus B comes up by default in
graduated color shading.
The program displays any actual point included in the design space shown. In this case you see a plot of conversion as a function of time and temperature at a mid-level slice of catalyst. This slice
includes six center points as indicated by the dot at the middle of the contour plot. By replicating center points, you get a very good power of prediction at the middle of your experimental region.
The Factors Tool appears on the right with the default plot. Move this around as needed by clicking and dragging the top blue border (drag it back to the right side of the screen to “pin” it back in
place. The tool controls which factor(s) are plotted on the graph.
Each factor listed in the Factors Tool has either an axis label, indicating that it is currently shown on the graph, or a slider bar, which allows you to choose specific settings for the factors that
are not currently plotted. All slider bars default to midpoint levels of those factors not currently assigned to axes. You can change factor levels by dragging their slider bars or by left-clicking
factor names to make them active (they become highlighted) and then typing desired levels into the numeric space near the bottom of the tool. Give this a try.
Click the C: Catalyst toolbar to see its value. Don’t worry if the slider bar shifts a bit — we will instruct you how to re-set it in a moment.
Left-Click the bar with your mouse and drag it to the right.
As indicated by the color key on the left, the surface becomes ‘hot’ at higher response levels, yellow in the ’80’s, and red above 90 for Conversion.
To enable a handy tool for reading coordinates off contour plots, go to View, Show Crosshairs Window (click and drag the titlebar if you’d like to unpin it from the left of your screen). Now move
your mouse over the contour plot and notice that the program generates the predicted response for specific factor values corresponding to that point. If you place the crosshair over an actual point,
for example – the one at the far upper left corner of the graph now on screen, you also see that observed value (in this case: 66).
P.S. See what happens when you press the Full option for crosshairs.
Now press the Default button on the Factors Tool to place factor C back at its midpoint.
Open the Factors Sheet by clicking the Sheet… button on the Factors Tool.
In the columns labeled Axis and Value you can change the axes settings by right-clicking, or type in specific values for factors. Give this a try. Then close the window and press the Default button.
P.S. The Terms list on the Factors Tool is a drop-down menu from which you can also select the factors to plot. Only the terms that are in the model are included in this list. At this point in the
tutorial this should be set at AB. If you select a single factor (such as A) the graph changes to a One-Factor Plot. Try this if you like, but notice how the program warns if you plot a main effect
that’s involved in an interaction.
Perturbation Plot
Wouldn’t it be handy to see all your factors on one response plot? You can do this with the perturbation plot, which provides silhouette views of the response surface. The real benefit of this plot
is when selecting axes and constants in contour and 3D plots. See it by mousing to the Graphs Toolbar and pressing Perturbation or pull it up from the View menu via New Graph.
For response surface designs, the perturbation plot shows how the response changes as each factor moves from the chosen reference point, with all other factors held constant at the reference value.
The program sets the reference point default at the middle of the design space (the coded zero level of each factor).
Click the curve for factor A to see it better. The software highlights it in a different color as shown above. It also highlights the legend. In this case, at the center point, you see that factor A
(time) produces a relatively small effect as it changes from the reference point. Therefore, because you can only plot contours for two factors at a time, it makes sense to choose B and C – and slice
on A.
Contour Plot: Revisited
Let’s look at the plot of factors B and C. Start by clicking Contour on the Graphs toolbar. Then in the Factors Tool right-click the Catalyst bar and select X1 axis by left clicking it.
You now see a catalyst versus temperature plot of conversion, with time held as a constant at its midpoint.
The contour plots are highly interactive. For example, right-click up in the hot spot at the upper middle and select Add Flag.
That’s enough on the contour plot for now — hold off until Part 3 of this tutorial to learn other tips and tricks on making this graph and others more presentable. Right-click and Delete flag to
clean the slate.
3D Surface Plot
Now to really get a feel for how the response varies as a function of the two factors chosen for display, select 3D Surface from the Graphs Toolbar. You then will see three-dimensional display of the
response surface. If the coordinates encompass actual design points, these will be displayed. On the Factors Tool move the slide bar for A:time to the right. This presents a very compelling picture
of how the response can be maximized. Right-click at the peak to set a flag.
You can see points below the surface by rotating the plot. Move your mouse over the graph. Click and hold the left mouse button and then drag.
Seeing an actual result predicted so closely lends credence to the model. Things are really looking up at this point!
Remember that you’re only looking at a ‘slice’ of factor A (time). Normally, you’d want to make additional plots with slices of A at the minus and plus one levels, but let’s keep moving — still lots
to be done for making the most of this RSM experiment.
Analyze the Data for the Second Response
This step is a BIG one. Analyze the data for the second response, activity. Be sure you find the appropriate polynomial to fit the data, examine the residuals and plot the response surface. Hint: The
correct model is linear.
Before you quit, do a File, Save to preserve your analysis. | {"url":"https://statease.com/docs/v22.0/tutorials/multifactor-rsm/","timestamp":"2024-11-13T15:16:50Z","content_type":"text/html","content_length":"69344","record_id":"<urn:uuid:caca67e2-ea99-422c-a027-97ff16f47bac>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00595.warc.gz"} |
Time And Distance Practice Questions Set 3 For Upcoming Bank PO n SSC Exam
A train travelling at a uniform speed crosses a boy standing on a platform in 20 seconds and the platform which is 320 m long in 36 seconds. Find the length of the train. .
(a) 360 m
(b) 540 m
(c) 500 m
(d) 400 m
Two trains have lengths of 300 m and 200 m. If they run on parallel tracks in the same direction, the faster train crosses the slower train in 50 seconds. If they instead run on parallel tracks with
the same speeds in the opposite directions, they take 10 seconds to cross each other. Find the speed of the faster train. .
(a) 25 m/sec
(b) 35 m/sec
(c) 30 m/sec
(d) 40 m/sec
A boatman rows 1 km in 4 minutes along the stream and 18 km in 2 hours against the stream. Find the speed of the boatman in still water. .
(a) 10 kmph
(b) 11 kmph
(c) 12 kmph
(d) 13 kmph
A motor boat has a speed of 11 km/hr in still water. It travelled a distance of 105 km upstream and then returned to its starting point. The total travel time for this journey was 22 hours. Find the
speed of the stream. .
(a) 4.5 km/hr .
(b) 3 km/hr
(c) 3.5 km/hr .
(d) 41cm/hr
In a 1 km race, A beats B by 100 m and C by 150 m. In a 2700 m race, by how many metres does B beat C? .
(a) 120 m
(b) 150 m
(c) 210 m
(d) 180 m
A and B are running around a circular track of length 600 m in the opposite directions at speeds of 15 m/s and 10 m/s respectively starting at the same time from the same point. In how much time will
they meet for the first time anywhere on the track? .
(a) 60 sec
(b) 24 sec
(c) 36 sec
(d) 50 sec
P, Q, R and S started running simultaneously from a point on a circular track. They took 600 seconds, 900 seconds, 1080 seconds and 1350 seconds to complete one round. Find the time taken by them to
meet at the starting point for the first time. .
(a) 1½ hrs
(b) 3 hrs
(c) 2 hrs
(d) 4 hrs
How long will three persons starting at the same point and travelling at 4 km/hr, 6 km/hr and 8 km/hr around a circular track 2 km long take to meet at the starting point?
(a) 1/2 hr
(b) 1 hr
(c) 1½ hrs
(d) 2 hrs
A train, 180 m long, crossed a 120 m long platform in 20 seconds, and another train travelling at the same speed crossed an electric pole in 10 seconds. In how much time will they cross each other
when they are travelling in the opposite directions? .
(a) 11 sec
(b) 13 sec
(c) 12 sec
(d) 14 sec
A man takes 25 minutes to row five-fourth of a kilometre against the current and 20 minutes to return to the starting point. Find the ratio of his speed in still water to the speed of the current. .
(a) 4 : 1
(b) 7 : 1
(c) 6 : 1
(d) 9 : 1
1. D. 2. C. 3. C. 4. D. 5. B.
6. B. 7. A. 8. B . 9. A. 10. D. | {"url":"http://www.quizsolver.com/blog/view/details/Bank-PO/Time-And-Distance-Practice-Questions-Set---3--For-Upcoming-Bank-PO-n-SSC-Exam/172/","timestamp":"2024-11-06T17:26:08Z","content_type":"text/html","content_length":"31759","record_id":"<urn:uuid:f07fb5b4-6b13-4eae-bc25-af0a81e1a58b>","cc-path":"CC-MAIN-2024-46/segments/1730477027933.5/warc/CC-MAIN-20241106163535-20241106193535-00113.warc.gz"} |
Captum · Model Interpretability for PyTorch
Guided Backprop¶
class captum.attr.GuidedBackprop(model)[source]¶
Computes attribution using guided backpropagation. Guided backpropagation computes the gradient of the target output with respect to the input, but gradients of ReLU functions are overridden so
that only non-negative gradients are backpropagated.
More details regarding the guided backpropagation algorithm can be found in the original paper here: https://arxiv.org/abs/1412.6806
Warning: Ensure that all ReLU operations in the forward function of the given model are performed using a module (nn.module.ReLU). If nn.functional.ReLU is used, gradients are not overridden
model (nn.Module) – The reference to PyTorch model instance.
attribute(inputs, target=None, additional_forward_args=None)[source]¶
○ inputs (Tensor or tuple[Tensor, ...]) – Input for which attributions are computed. If model takes a single tensor as input, a single input tensor should be provided. If model takes
multiple tensors as input, a tuple of the input tensors should be provided. It is assumed that for all given input tensors, dimension 0 corresponds to the number of examples (aka
batch size), and if multiple input tensors are provided, the examples must be aligned appropriately.
○ target (int, tuple, Tensor, or list, optional) –
Output indices for which gradients are computed (for classification cases, this is usually the target class). If the network returns a scalar value per example, no target index is
necessary. For general 2D outputs, targets can be either:
■ a single integer or a tensor containing a single integer, which is applied to all input examples
■ a list of integers or a 1D tensor, with length matching the number of examples in inputs (dim 0). Each integer is applied as the target for the corresponding example.
For outputs with > 2 dimensions, targets can be either:
■ A single tuple, which contains #output_dims - 1 elements. This target index is applied to all examples.
■ A list of tuples with length equal to the number of examples in inputs (dim 0), and each tuple containing #output_dims - 1 elements. Each tuple is applied as the target for the
corresponding example.
Default: None
○ additional_forward_args (Any, optional) – If the forward function requires additional arguments other than the inputs for which attributions should not be computed, this argument can
be provided. It must be either a single additional argument of a Tensor or arbitrary (non-tuple) type or a tuple containing multiple additional arguments including tensors or any
arbitrary python types. These arguments are provided to model in order, following the arguments in inputs. Note that attributions are not computed with respect to these arguments.
Default: None
attributions (Tensor or tuple[Tensor, …]):
The guided backprop gradients with respect to each input feature. Attributions will always be the same size as the provided inputs, with each value providing the attribution of
the corresponding input index. If a single tensor is provided as inputs, a single tensor is returned. If a tuple is provided for inputs, a tuple of corresponding sized tensors is
Return type:
Tensor or tuple[Tensor, …] of attributions
>>> # ImageClassifier takes a single input tensor of images Nx3x32x32,
>>> # and returns an Nx10 tensor of class probabilities.
>>> net = ImageClassifier()
>>> gbp = GuidedBackprop(net)
>>> input = torch.randn(2, 3, 32, 32, requires_grad=True)
>>> # Computes Guided Backprop attribution scores for class 3.
>>> attribution = gbp.attribute(input, target=3) | {"url":"https://captum.ai/api/guided_backprop.html","timestamp":"2024-11-06T02:28:56Z","content_type":"text/html","content_length":"20379","record_id":"<urn:uuid:456068bf-49f1-452a-b835-759c68ca9c9a>","cc-path":"CC-MAIN-2024-46/segments/1730477027906.34/warc/CC-MAIN-20241106003436-20241106033436-00500.warc.gz"} |
What is the difference between dense and sparse matrices?
What is the difference between dense and sparse matrices?
And when would I use which?
1 Answer
Sort by ยป oldest newest most voted
Dense matrices store every entry in the matrix. Sparse matrices only store the nonzero entries. Sparse matrices don't have a lot of extra features, and some algorithms may not work for them. You use
them when you need to work with matrices that would be too big for the computer to handle them, but they are mostly zero, so they compress easily.
edit flag offensive delete link more | {"url":"https://ask.sagemath.org/question/9464/what-is-the-difference-between-dense-and-sparse-matrices/?answer=14189","timestamp":"2024-11-09T04:48:44Z","content_type":"application/xhtml+xml","content_length":"50712","record_id":"<urn:uuid:5482cd81-2b8f-46d9-aa43-d8dab5b0586f>","cc-path":"CC-MAIN-2024-46/segments/1730477028115.85/warc/CC-MAIN-20241109022607-20241109052607-00153.warc.gz"} |
5.Draftsman Gr II-KWA | Quizalize
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• share a link with colleagues
• print as a bubble sheet
• For one cubic meter, 1:2:4 concrete using 20 mm metal, the quantity of coarse aggregate required is :
1.00 cum
1.10 cum
1.54 cum
0.9 cum
100 questions
Show answers
• Q1
For one cubic meter, 1:2:4 concrete using 20 mm metal, the quantity of coarse aggregate required is :
1.00 cum
1.10 cum
1.54 cum
0.9 cum
• Q2
The whole circle bearing of a line is 230°, its quadrantal bearing is
W 40°S
S 50° W
S 30° W
S 40° W
• Q3
If the fore bearing and back bearing of a line do not differ by 180°, then either one or both the ends of line, are affected by:
Manipulation error
None of these
Local attraction
• Q4
Which one of the following is not a working operation of a plane table?
• Q5
Finding the location of the station occupied by the table on the paper, by means of sighting to two well defined points whose locations have previously been plotted on the paper, is known as:
Bessels Graphical method
Trial and error method
Two-point problem
Radiation method
• Q6
A levelling instrument, consisting a telescope attached with level tube which can he tilted within a few degrees in a vertical plane by a tilting screw is known as:
The Tilting level
Reduced level
The Dumpy level
The Wye level
• Q7
The permissible closing error in levelling survey is
0.001 m
0.050 m
0.005 m
0.100 m
• Q8
When it is not possible to set up the level midway between two points. The method of levelling to carry forward the levels on the other side of the obstruction is called
The reversible level
Profile levelling
Reciprocal levelling
Fly levelling
• Q9
Making the axis of the bubble tube perpendicular to vertical axis of the level is
Curvature correction of Dumpy level
Permanent adjust of Dumpy level
Temporary adjustment of Dumpy level
Refraction correction of Dumpy level
• Q10
The shortest horizontal distance between two consecutive contours, is called
Apparent difference
Contour interval
Collimation error
Horizontal equivalent
• Q11
The imaginary line lying throughout on the surface of the earth and preserving a constant inclination to the horizontal, is known as:
Ridge line
Road alignment
Contour gradient
Contour line
• Q12
In a land measuring metric chain the ends of link is connected by :
Two oval rings
Three oval rings
Four oval rings
One oval ring
• Q13
The sketch drawn by the surveyor during reconnaissance showing the positions of stations survey lines and their directions and the important features is known as:
Detailed plan
Key plan
• Q14
The use of an optical square is :
To setting out a 45° angle
To measure the area of a plan
To range a line
To set out right angles
• Q15
One of the methods to set out a perpendicular to the chain line is 3-4-5 method, another method identical to 3-4-5 method is | {"url":"https://resources.quizalize.com/view/quiz/5draftsman-gr-iikwa-5cddb956-ff1c-4fa9-a8cb-ffcc24fbbbfa","timestamp":"2024-11-04T01:09:41Z","content_type":"text/html","content_length":"304936","record_id":"<urn:uuid:8432e95b-491c-4ede-a053-5e9e0e0d48b1>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00694.warc.gz"} |
Question 1:
Diagram 1.1 shows the process of latex coagulation.
(a)(i) State one example of chemical P? [1 mark]
(ii) State one characteristic of coagulated latex in Diagram 1.1. [1 mark]
(b) Chemical P in diagram 1.1 is replaced with chemical Q to prevent latex from coagulating.
State one example of chemical Q. [1 mark]
(c) Diagram 1.2 shows the process when natural rubber is heated with Sulphur to form rubber R.
(i) Name process X. [1 mark]
(ii) Name rubber R. [1 mark]
(d) Mark (\/) the object which is made of rubber R. [1 mark]
(a)(i) Methanoic acid
(a)(ii) Elastic
(b) Ammonia solution
(c)(i) Vulcanization of rubber
(c)(ii) Vulcanized rubber
Question 1:
Diagram 1.1 and Diagram 1.2 show an experiment to investigate the effect of temperature on the fermentation of glucose by yeast.
(a) State one hypothesis that can be made from this experiment. [1 mark]
(b) State the variables in this experiment.
(i) Manipulated variable [1 mark]
(ii) Responding variable [1 mark]
(c) Based on Diagram 1.1 and 1.2, which temperature is more suitable for the fermentation of glucose? [1 mark]
(d) Diagram 1.3 shows the graph of the volume of carbon dioxide produced at 35^oC against time.
[1 mark]
(a) Fermentation of glucose by yeast is affected by temperature.
(b)(i) Temperature of the water bath
(b)(ii) Volume of carbon dioxide produced
(c) 35^oC
(d) The volume of carbon dioxide produced is directly proportional to time.
12.2.1 Solid Geometry (II), PT3 Focus Practice
Question 1:
Diagram below shows closed right cylinder.
Calculate the total surface area, in cm^2, of the cylinder.
$\left(\pi =\frac{22}{7}\right)$
Total surface area
= 2(πr^2) + 2πrh
$\begin{array}{l}=\left(2×\frac{22}{7}×{7}^{2}\right)+\left(2×\frac{22}{7}×7×20\right)\\ =308+880\\ =1188c{m}^{2}\end{array}$
Question 2:
Diagram below shows a right prism with right-angled triangle ABC as its uniform cross section.
Calculate the total surface area, in cm^2, of the prism.
$\begin{array}{l}AB=\sqrt{{5}^{2}-{3}^{2}}\\ =\sqrt{25-9}\\ =\sqrt{16}\\ =4cm\end{array}$
Total surface area
= 2 (½× 3 × 4) + (3 × 10) + (4 × 10) + (5 × 10)
= 12 + 30 + 40 + 50
= 132 cm^2
Question 3:
Diagram below shows a right pyramid with a square base.
Calculate the total surface area, in cm^2, of the right pyramid.
h^2= 10^2 – 6^2
= 100 – 36
= 64
h = √64
= 8cm
Total surface area of the right pyramid
= (12 × 12) + 4 × (½× 12 × 8)
= 144 + 192
= 336 cm^2
Question 1:
Diagram 1.1 shows two examples of carbon compounds, K and L.
(a)(i) Based on Diagram 1.1, which one is an inorganic carbon compound? [1 mark]
(ii) State one characteristic of an inorganic carbon compound. [1 mark]
(b)(i) State one use of compound K. [1 mark]
(ii) State one effect of compound K on the nervous system if consumed excessively. [1 mark]
(c) Diagram 1.2 shows a tank containing gas M used for gas stoves.
Gas M is a hydrocarbon compound.
(i) State two elements present in gas M. [1 mark]
(ii) State one source of gas M. [1 mark]
(a)(i) L or marble chips
(a)(ii) Originates from non-living things or does not originates from living things
(b)(i) Alcoholic drink
(b)(ii) Disrupts nerve coordination or slows down the transmission of impulses
1. Hydrogen
2. Carbon
(c)(ii) Petroleum
Question 6:
Diagram in the answer space shows a Cartesian plane board used in an indoor game.
The instruction of the game is such that the first move follows the translation $\left(\begin{array}{l}-4\\ -2\end{array}\right)$ and the second move follows the translation $\left(\begin{array}{l}\
text{ }5\\ -4\end{array}\right)$ .
Jimmy starts moving from the position mark x.
(a) Mark T1 at Jimmy’s position after the first move and T2 at the position after the second move.
(b) State the coordinates of the position based on Jimmy’s final move.
(b) Jimmy’s final position = (3, –2).
Question 7:
In the answer space below, a quadrilateral PQRS is drawn on a grid of squares. A’ is the image of P under rotation 90^o at point C.
(a) State the direction of the rotation.
In the answer space,
(b) mark B’ as the image of point Q under the same rotation.
(c) draw the image of quadrilateral PQRS under a reflection on the line MN.
(a) A’ is the image of P under a rotation of 90^o clockwise about the centre C.
(b) and (c)
Question 8:
Diagram in the answer space shows trapezium ABCD drawn on a Cartesian plane. A’D’ is the image of AD under a rotation at centre W.
(a) State
(i) angle of rotation
(ii) the direction of the rotation.
(b) On diagram in the answer space, complete the image of trapezium ABCD.
$\angle DWD"={90}^{o}\text{ or }{270}^{o}.$
(a)(ii) Anticlockwise or clockwise.
11.2.1 Transformations (I), PT3 Focus Practice
Question 1:
Diagram below in the answer space shows object P drawn on a grid of equal squares with sides of 1 unit.
On the diagram, draw the image of object P under the translation $\left(\begin{array}{l}-6\\ \text{}3\end{array}\right).$
Question 2:
Describe the translation which maps point P onto point P’.
The translation is $\left(\begin{array}{l}\text{}7\\ -6\end{array}\right).$
Question 3:
Diagram below in the answer space shows quadrilateral PQRS. R’S’ is the image of RS under a reflection in the straight line AB.
On diagram in the answer space, complete the image of quadrilateral PQRS.
Question 4:
Diagram in the answer space, shows two polygons, M and M’, drawn on a grid of equal squares with sides of 1 unit. M’ is the image of M under a reflection.
(a) Draw the axis of reflection.
(b) Mark the image of P under the same reflection.
(c) Draw the image of M under reflection in the x-axis.
Question 5:
On diagram in the answer space, triangle P’Q’R’ is the image of triangle PQR under a rotation about centre C.
(a) State the angle and direction of the rotation.
(b) K’is the image of point K under the same rotation.
Mark and state the coordinates of K’.
(a) ∆ P’Q’R’ is the image of ∆ PQR under a clockwise rotation of 90^o.
(b) Image of K = (1, –4).
11.1 Transformations (I)
11.1.1 Transformation
A transformation is a one-to-one correspondence or mapping between points of an object and its image on a plane.
11.1.2 Translation
1. A translation is a transformation which moves all the points on a plane through the same distance in the same direction.
2. Under a translation, the shape, size and orientation of object and its image are the same.
3. A translation in a Cartesian plane can be represented in the form $\left(\begin{array}{l}a\\ b\end{array}\right),$ whereby, a represents the movement to the right or left which is parallel to the
x-axis and b represents the movement upwards or downwards which is parallel to the y-axis.
Example 1:
Write the coordinates of the image of A (–2, 4) under a translation $\left(\begin{array}{l}\text{}4\\ -3\end{array}\right)$ and B (1, –2) under a translation $\left(\begin{array}{l}-5\\ \text{}3\end
A’ = [–2 + 4, 4 + (–3)] = (2, 1)
B’ = [1 + (–5), –2 + 3] = (–4, 1)
Example 2:
Point K moved to point K’ (3, 8) under a translation $\left(\begin{array}{l}-4\\ \text{}3\end{array}\right).$
What are the coordinates of point K?
$K\left(x,\text{}y\right)\to \left(\begin{array}{l}-4\\ \text{}3\end{array}\right)\to K"\left(3,\text{}8\right)$
The coordinates of K = [3 – (– 4), 8 – 3]
= (7, 5)
Therefore the coordinates of K are (7, 5).
11.1.3 Reflection
1. A reflection is a transformation which reflects all points of a plane in a line called the axis of reflection.
2. In a reflection, there is no change in shape and size but the orientation is changed. Any points on the axis of reflection do not change their positions.
11.1.4 Rotation
1. A rotation is a transformation which rotates all points on a plane about a fixed point known as the centre of rotation through a given angle in a clockwise or anticlockwise direction.
2. In a rotation, the shape, size and orientation remain unchanged.
3. The centre of rotation is the only point that does not change its position.
Example 4:
Point A (3, –2) is rotated through 90^o clockwise to A’ and 180^o anticlockwise to A[1] respectively about origin.
State the coordinates of the image of point A.
Image A’ = (–2, 3)
Image A[1 ]= (–3, 2)
11.1.5 Isometry
1. An isometry is a transformation that preserves the shape and size of an object.
2.Translation, reflection and rotation and a combination of it are isometries.
11.1.6 Congruence
1. Congruent figures have the same size and shape regardless of their orientation.
2. The object and the image obtained under an isometry are congruent.
Question 11:
The Town Council plans to build an equilateral triangle platform in the middle of a roundabout. The diameter of circle RST is 24 m and the perpendicular distance from R to the line ST is 18 m. as
shown in Diagram below.
Given diameter = 24 m
hence radius = 12 m
O is the centre of the circle.
Using Pythagoras’ theorem:
$\begin{array}{l}{x}^{2}={12}^{2}-{6}^{2}\\ x=\sqrt{144-36}\\ \text{ }=10.39\text{ m}\\ TS=RS=RT\\ \text{ }=10.39\text{ m }×2\\ \text{ }=20.78\text{ m}\\ \text{Perimeter of the platform}\\ TS+RS+RT\\
=20.78×3\\ =63.34\text{ m}\end{array}$
Question 12:
Amy will place a ball on top of a pillar in Diagram below. Table below shows the diameters of three balls X, Y and Z.
Which ball X, Y or Z, can fit perfectly on the top of the pillar? Show the calculation to support Amy’s choice.
$\begin{array}{l}\text{Let the radius of the top of the pillar}=r\text{ cm}\text{.}\\ O\text{ is the centre of the circle}\text{.}\\ \text{In }\Delta \text{ }OQR,\\ {r}^{2}={\left(r-4\right)}^{2}+{8}
^{2}\text{ }\left(\text{using Pythagoras" theorem}\right)\\ {r}^{2}={r}^{2}-8r+16+64\\ {r}^{2}={r}^{2}-8r+80\\ {r}^{2}-{r}^{2}+8r=80\\ 8r=80\\ r=\frac{80}{8}\\ r=10\text{ cm}\\ \\ \text{Therefore,
diameter}\\ =2×10\\ =20\text{ cm}\\ \\ \text{Ball }Y\text{ with diameter 20 cm can fit perfectly }\\ \text{on top of the pillar}\text{.}\end{array}$
Question 13:
Diagram below shows a rim of a bicycle wheel with a diameter of 26 cm. Kenny intends to build a holder for the rim.
Which of the rim holder, X, Y or Z, can fit the bicycle rim perfectly? Show the calculation to support your answer.
$\begin{array}{l}\text{Let the radius of the rim holder}=r\text{ cm}\text{.}\\ O\text{ is the centre of the circle}\text{.}\\ \text{In }\Delta \text{ }OQR,\\ {r}^{2}={\left(r-8\right)}^{2}+{12}^{2}\
text{ }\left(\text{using Pythagoras" theorem}\right)\\ {r}^{2}={r}^{2}-16r+64+144\\ {r}^{2}={r}^{2}-16r+208\\ {r}^{2}-{r}^{2}+16r=208\\ 16r=208\\ r=\frac{208}{16}\\ r=13\text{ cm}\\ \\ \text
{Therefore, diameter}\\ =2×13\\ =26\text{ cm}\\ \\ \text{Rim holder }Z\text{ with diameter 26 cm can fit the bicycle perfectly}\text{.}\end{array}$
Question 6:
In the diagram below, CD is an arc of a circle with centre O.
Determine the area of the shaded region.
$\left(\text{Use }\pi =\frac{22}{7}\right)$
$\begin{array}{l}\text{Area of sector}=\text{Area of circle}×\frac{{72}^{o}}{{360}^{o}}\\ \text{ }=\frac{22}{7}×{\left(10\right)}^{2}×\frac{{72}^{o}}{{360}^{o}}\\ \text{ }=\frac{440}{7}{\text{ cm}}^
{2}\\ \text{Area of }\Delta OBD=\frac{1}{2}×6×8\\ \text{ }=24{\text{ cm}}^{2}\\ \text{Area of shaded region}=\frac{440}{7}-24\\ \text{ }=38\frac{6}{7}{\text{ cm}}^{2}\end{array}$
Question 7:
In diagram below, ABC is a semicircle with centre O.
Calculate the area, in cm^2 , of the shaded region.
$\left(\text{Use }\pi =\frac{22}{7}\right)$
$\begin{array}{l}\angle ACB={90}^{o}\\ AB=\sqrt{{6}^{2}+{8}^{2}}\\ \text{ }=\sqrt{100}\\ \text{ }=10\text{ cm}\\ \text{Radius}=10÷2\\ \text{ }=5\text{ cm}\\ \\ \text{The shaded region}\\ =\left(\frac
{1}{2}×\frac{22}{7}×5×5\right)-\left(\frac{1}{2}×6×8\right)\\ =39\frac{2}{7}-24\\ =15\frac{2}{7}{\text{ cm}}^{2}\end{array}$
Question 8:
In diagram below, ABC is an arc of a circle centre O
The radius of the circle is 14 cm and AD = 2 DE.
Calculate the perimeter, in cm, of the whole diagram.
$\left(\text{Use }\pi =\frac{22}{7}\right)$
$\begin{array}{l}\text{Length of arc }ABC\\ =\frac{3}{4}×2\pi r\\ =\frac{3}{4}×2×\frac{22}{7}×14\\ =66\text{ cm}\\ \\ \text{Perimeter of the whole diagram}\\ =16+8+8+66\\ =98\text{ cm}\end{array}$
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Question 9:
In diagram below, KLMN is a square and KLON is a quadrant of a circle with centre K.
Calculate the area, in cm^2, of the coloured region.
$\left(\text{Use }\pi =\frac{22}{7}\right)$
$\begin{array}{l}\text{Area of the coloured region}\\ =\frac{{45}^{o}}{{360}^{o}}×\pi {r}^{2}\\ =\frac{{45}^{o}}{{360}^{o}}×\frac{22}{7}×{14}^{2}\\ =77{\text{ cm}}^{\text{2}}\end{array}$
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Question 10:
Diagram below shows two quadrants, AOC and EOD with centre O.
Sector AOB and sector BOC have the same area.
Calculate the area, in cm^2, of the coloured region.
$\left(\text{Use }\pi =\frac{22}{7}\right)$
$\begin{array}{l}\text{Area of the sector }AOB=\text{Area of the sector }BOC\\ \text{Therefore, }\angle AOB=\angle BOC\\ \text{ }={90}^{o}÷2\\ \text{ }={45}^{o}\\ \text{Area of the coloured region}\\
=\frac{{45}^{o}}{{360}^{o}}×\frac{22}{7}×{16}^{2}\\ =100\frac{4}{7}{\text{ cm}}^{2}\end{array}$
10.2.1 Circles I, PT3 Focus Practice
Question 1:
Diagram below shows a circle with centre O.
The radius of the circle is 35 cm.
Calculate the length, in cm, of the major arc AB.
$\left(\text{Use }\pi =\frac{22}{7}\right)$
Angle of the major arc AB = 360^o – 144^o= 216^o
$\begin{array}{l}\text{Length of major arc}AB\\ =\frac{{216}^{o}}{{360}^{o}}×2\pi r\\ =\frac{{216}^{o}}{{360}^{o}}×2×\frac{22}{7}×35\\ =132\text{cm}\end{array}$
Question 2:
In diagram below, O is the centre of the circle. SPQ and POQ are straight lines.
The length of PO is 8 cm and the length of POQ is 18 cm.
Calculate the length, in cm, of SPT.
Radius = 18 – 8 = 10 cm
PT^2 = 10^2 – 8^2
= 100 – 64
= 36
PT = 6 cm
Length of SPT = 6 + 6
= 12 cm
Question 3:
Diagram below shows two circles. The bigger circle has a radius of 14 cm with its centre at O.
The smaller circle passes through O and touches the bigger circle.
Calculate the area of the shaded region.
$\left(\text{Use }\pi =\frac{22}{7}\right)$
$\begin{array}{l}\text{Area of bigger circle}=\pi {R}^{2}=\frac{22}{7}×{14}^{2}\\ \text{Radius, }r\text{ of smaller circle}=\frac{1}{2}×14=7\text{ cm}\\ \text{Area of smaller circle}=\pi {r}^{2}=\
frac{22}{7}×{7}^{2}\\ \therefore \text{Area of shaded region}\\ \text{=}\left(\frac{22}{7}×{14}^{2}\right)-\left(\frac{22}{7}×{7}^{2}\right)\\ =616-154\\ =462{\text{ cm}}^{2}\end{array}$
Question 4:
Diagram below shows two sectors. ABCD is a quadrant and BED is an arc of a circle with centre C.
Calculate the area of the shaded region, in cm^2.
$\left(\text{Use }\pi =\frac{22}{7}\right)$
$\begin{array}{l}\text{The area of sector }CBED\\ =\frac{{60}^{o}}{{360}^{o}}×\pi {r}^{2}\\ =\frac{{60}^{o}}{{360}^{o}}×\frac{22}{7}×{14}^{2}\\ =102\frac{2}{3}{\text{ cm}}^{\text{2}}\end{array}$
$\begin{array}{l}\text{The area of quadrant }ABCD\\ =\frac{1}{4}×\pi {r}^{2}\\ =\frac{1}{4}×\frac{22}{7}×{14}^{2}\\ =154{\text{ cm}}^{\text{2}}\end{array}$
$\begin{array}{l}\text{Area of the shaded region}\\ =154-102\frac{2}{3}\\ =51\frac{1}{3}{\text{ cm}}^{2}\end{array}$
Question 5:
Diagram below shows a square KLMN. KPN is a semicircle with centre O.
Calculate the perimeter, in cm, of the shaded region.
$\left(\text{Use }\pi =\frac{22}{7}\right)$
$\begin{array}{l}KO=ON=OP=7\text{ cm}\\ PN=\sqrt{{7}^{2}+{7}^{2}}\\ \text{}=\sqrt{98}\\ \text{}=9.90\text{ cm}\\ \\ \text{Arc length }KP\\ =\frac{1}{4}×2×\frac{22}{7}×7\\ =11\text{ cm}\end{array}$
Perimeter of the shaded region
= KL + LM + MN + NP + Arc length PK
= 14 + 14 +14 + 9.90 + 11
= 62.90 cm
4.7.2 Alcohol and Its Effects on Health (Structured Questions)
12.2.1 Solid Geometry (II), PT3 Focus Practice
4.7.1 Various Carbon Compound (Structured Questions)
11.2.2 Transformations (I), PT3 Focus Practice
11.2.1 Transformations (I), PT3 Focus Practice
11.1 Transformations (I)
10.2.3 Circles I, PT3 Focus Practice
10.2.2 Circles I, PT3 Focus Practice
10.2.1 Circles I, PT3 Focus Practice | {"url":"https://content.myhometuition.com/2017/page/2/","timestamp":"2024-11-06T15:28:14Z","content_type":"text/html","content_length":"122542","record_id":"<urn:uuid:c2077ea5-895c-4f2a-82e9-af12559e9d92>","cc-path":"CC-MAIN-2024-46/segments/1730477027932.70/warc/CC-MAIN-20241106132104-20241106162104-00150.warc.gz"} |
Ratio, Proportion &
Do you have enough information to work out the area of the shaded quadrilateral?
Can you work out how to produce different shades of pink paint?
Can you find an efficent way to mix paints in any ratio?
If a sum invested gains 10% each year how long before it has doubled its value?
Your school has been left a million pounds in the will of an ex- pupil. What model of investment and spending would you use in order to ensure the best return on the money?
My measurements have got all jumbled up! Swap them around and see if you can find a combination where every measurement is valid.
Here's a chance to work with large numbers...
Two boats travel up and down a lake. Can you picture where they will cross if you know how fast each boat is travelling?
Have you ever wondered what it would be like to race against Usain Bolt?
Andy wants to cycle from Land's End to John o'Groats. Will he be able to eat enough to keep him going?
The triathlon is a physically gruelling challenge. Can you work out which athlete burnt the most calories?
These Olympic quantities have been jumbled up! Can you put them back together again?
The large rectangle is divided into a series of smaller quadrilaterals and triangles. Can you untangle what fractional part is represented by each of the ten numbered shapes?
Can you work out which drink has the stronger flavour? | {"url":"https://nrich.maths.org/ratio-proportion-rates-change","timestamp":"2024-11-07T17:12:34Z","content_type":"text/html","content_length":"62302","record_id":"<urn:uuid:eeaf7f12-bf8c-4411-bdf6-8ba9ab8a36a0>","cc-path":"CC-MAIN-2024-46/segments/1730477028000.52/warc/CC-MAIN-20241107150153-20241107180153-00537.warc.gz"} |
SSC Scientific Assistant IMD Physics Questions 21 to 40
SSC Scientific Assistant IMD Physics Questions 21 to 40 AMBIPi
Hi students, Welcome to Amans Maths Blogs (AMBIPi). Are you preparing for SSC Scientific Assistant and looking for SSC Scientific Assistant IMD Physics Questions 21 to 40 with Answer Keys AMBIPi? In
this article, you will get previous year questions of SSC Scientific Assistant IMD (Indian Meteorological Department), which helps you in the preparation of government job of SSC IMD Scientific
SSC Scientific Assistant IMD Previous Year Questions
SSC Scientific Assistant Physics Question No: 21
By what percent should the pressure of gas should increased so as to decrease its volume by 20%, at constant temperature?
Option A : 20%
Option B : 60%
Option C : 25%
Option D : 40%
Show/Hide Answer Key
Option C : 25%
SSC Scientific Assistant Physics Question No: 22
Electric potential at a point with a position vector r due to a point charge Q placed at the origin is given by the formula ________.
Option A : V = Q/(2πε[o]r^2)
Option B : V = Q/(4πε[o]r^2)
Option C : V = Q/(4πε[o]r)
Option D : V = Q/(2πε[o]r)
Show/Hide Answer Key
Option C : V = Q/(2πε[o]r^2)
SSC Scientific Assistant Previous Year Physics Questions No: 23
Fermi is a unit of ?
Option A : Length
Option B : Mass
Option C : Area
Option D : Time
Show/Hide Answer Key
Option A : Length
SSC Scientific Assistant IMD Physics Questions No: 24
Find the uniform angular acceleration of a wheel if its angular speed increases from 420 rpm to 660 rpm is 8 seconds?
Option A : 2π rad/s^2
Option B : 1 rad/s^2
Option C : π rad/s^2
Option D : 2 rad/s^2
Show/Hide Answer Key
Option C : π rad/s^2
SSC Scientific Assistant Physics Questions Paper No: 25
For a transistor in common emitter configuration ______ is ratio of change in base- emitter voltage (ΔVBE) to the resulting change in base current (ΔIB) at constant collector-emitter voltage (VCE).
Option A : Output resistance
Option B : Current amplification factor
Option C : Voltage gain
Option D : Input resistance
Show/Hide Answer Key
Option D : Input resistance
SSC Scientific Assistant Previous Year Paper Physics Questions No: 26
For small deformations, stress and strain are proportional to each other. What is this known as ?
Option A : Hooke’s Law
Option B : Gauss’a Law
Option C : Henry’s Law
Option D : Joule’s Law
Show/Hide Answer Key
Option A : Hooke’s Law
SSC Scientific Assistant Previous Year Paper Physics Questions No: 27
How much heat (in joules) would be required to raise the temperature of 500 g of an aluminum sphere from 20^oC to 720 ^oC? [Specific Heat Capacity of Aluminum is 900 J/(Kgk)]
Option A : 3.15 x 10^5
Option B : 3.15 x 10^7
Option C : 1.26 x 10^5
Option D : 1.26 x 10^7
Show/Hide Answer Key
Option A : 3.15 x 10^5
Scientific Assistant Previous Year Physics Questions Paper No: 28
If 1, 2 and 3 represent different mediums and ‘n’ is refractive index, than which of the following equation is true.
Option A : n[23] = n[31] x n[12]
Option B : n[32] = n[31] x n[12]
Option C : n[32] = n[13] x n[12]
Option D : n[23] = n[13] x n[12]
Show/Hide Answer Key
Option A : n[32] = n[31 ]x n[12]
IMD Scientific Assistant Physics Questions Paper No: 29
What is the name of the outermost range of the Himalayas?
Option A : Himadri
Option B : Shivaliks
Option C : Himachal
Option D : Sahyadri
Show/Hide Answer Key
Option B : Shivaliks
Previous Year SSC Scientific Assistant Physics Questions: 30
If ‘A’ is the angle of prism, ‘I’ is the angle of incidence, ‘e’ is an angle of emergence, than the angle of deviation ‘α’ of the light incidence of prism is equal to
Option A : RA/I
Option B : RI/A
Option C : IA/R
Option D : A/(IR)
Show/Hide Answer Key
Option A : RA/I
SSC IMD Scientific Assistant Physics Questions No: 31
If a projectile is thrown with velocity v and makes an angle θ with the x-axis than the time taken for achieving maximum height is given by which formula?
Option A : t= vsinθ/g
Option B : t= v^2sinθ/g
Option C : t= v^2sin^2θ/g
Option D : t = vsin^2θ/g
Show/Hide Answer Key
Option D :t = vsin^2θ/g
SSC IMD Scientific Assistant Physics Questions No: 32
If an object is placed with a 10 cm in front of convex lens of a focal length 6 cm, than find the position of the image (in cm) ?
Option A : 12
Option B : -15
Option C : 15
Option D : -12
Show/Hide Answer Key
Option B : 15
SSC Scientific Assistant IMD Previous Year Physics Questions No: 33
If chlorine has two isotopes one of 25u and the other 37 u and the average mass of chlorine atom is 35.5 u then the ratio of abundance of the two isotopes of masses 35 u and 37 u ______
Option A : 1/3
Option B : 1/4
Option C : 3/1
Option D : 4/1
Show/Hide Answer Key
Option C : 3/1
SSC Scientific Assistant IMD Physics Questions No: 34
If ‘E’ is a magnitude of uniform electric field in a conductor , ‘r’ is the relaxation time, (‘e’ is charge and ‘m’ is mass of electron), then the term -eEτ/m is equal to the ______ of the electrons.
Option A : force experienced by
Option B : acceleration experienced by
Option C : drift velocity of
Option D : charge density due to
Show/Hide Answer Key
Option C : drift velocity of
SSC IMD Scientific Assistant Physics Questions Paper No: 35
If the empirical formula for the observed wavelengths for hydrogen is 1/Α = R (1/k^2 – 1/n^2)(, where n is integral values higher than 1, then it represents the ______ spectral series.
Option A : Balmer
Option B : Paschen
Option C : Lyman
Option D : Brackett
Show/Hide Answer Key
Option C : Lyman
SSC IMD Scientific Assistant Previous Year Paper Physics Questions No: 36
If the energy of electron in the 2nd orbit of hydrogen is -3.4 eV, then how much is the energy (in eV) in the 3rd orbit ?
Option A : -1.511
Option B : –2.22
Option C : -5.1
Option D : -13.6
Show/Hide Answer Key
Option A : -1.511
SSC IMD Scientific Assistant Previous Year Paper Physics Questions No: 37
If the gas particles are of diameter ‘d’, average speed ‘v’, number of particles per unit volume ‘n’ then the volume of particle sweeps in time ‘t’ is?
Option A : πd^2vt
Option B : πv^2td
Option C : πt^2vd
Option D : π^2tvd
Show/Hide Answer Key
Option A : πd^2vt
IMD Scientific Assistant Previous Year Physics Questions Paper No: 38
United States Food and Drug Administration (FDA) has approved which test for detecting Zika virus in donated blood??
Option A : Roche test
Option B : Cobas test
Option C : Trioplex test
Option D : Elisa test
Show/Hide Answer Key
Option B : Cobas test
IMD Scientific Assistant Physics Questions Paper No: 39
If the ideal gas equation is written as PV= kBNT, where N is the number of molecules, than kB represents?
Option A : Gas Constant
Option B : Bohr radius
Option C : Boltzman constant
Option D : B- factor
Show/Hide Answer Key
Option C : Boltzman constant
Previous Year SSC IMD Scientific Assistant Physics Questions : 40
If x is displacement, time taken is t, initial velocity is u , final velocity is v , and acceleration is a , than which of the following equations is true ?
Option A : u^2 = v^2 + 2ax
Option B : x = (v^2 – u^2)/2a
Option C : v^2 = (u^2 – 2ax)
Option D : x = (v^2 + u^2 )/2a
Show/Hide Answer Key
Option D : x = (v^2 – u^2)/ 2a
Know About SSC Scientific Assistant: Click Here.
Get SSC Scientific Assistant Previous Year Questions.
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haracteristics of quadratic functions
Three properties that are universal to all quadratic functions: 1) The graph of a quadratic function is always a
that either opens upward or downward (end behavior); 2) The domain of a quadratic function is all real numbers; and 3) The vertex is the lowest point when the parabola opens upwards; while the vertex
is the highest point when the parabola opens downward. | {"url":"https://www.studypug.com/accuplacer-test-prep/characteristics-of-quadratic-functions","timestamp":"2024-11-07T07:29:55Z","content_type":"text/html","content_length":"368402","record_id":"<urn:uuid:38406f9a-82a7-4b7b-9b58-0d695732a990>","cc-path":"CC-MAIN-2024-46/segments/1730477027957.23/warc/CC-MAIN-20241107052447-20241107082447-00214.warc.gz"} |
LIGO Document T1500597-v1
Document #:
Document type:
T - Technical notes
Measuring the properties of a gravitational-wave signal is a question of parameter estimation. The end result of these studies is a set of samples drawn from the posterior probability
distribution. The posterior contains all the information that we have, but may not be easily digestible; in many cases, it is desirable to quote summary statistics to concisely describe findings.
In most practical cases, it is impossible to compress a complete description of a distribution down to a single number; therefore, any point estimate could miss key pieces of information. Here,
we discuss various possibilities for summary statistics. We also include a discussion of how to quote systematic errors on parameter estimates. While there is no perfect answer, our suggestion is
to use \( X^{+Y}_{-Z} \), where \( X \) is the median \( Y \) and \( Z \) are estimates for the statistical error (measurement precision) from the bounds of a symmetric credible interval, and
then add in estimates for systematic error from the range of \( X \), \( Y \) and \( Z \), which could be presented as \( X^{+Y\pm y}_{-Z\pm z} \) or \( (X\pm x)^{+Y\pm y}_{-Z\pm z} \).
□ Manuscript (pointEstimate.pdf, file is not accessible)
Files in CBC SVN under /pe/papers/pointEstimatesTechNote/ | {"url":"https://dcc-lho.ligo.org/LIGO-T1500597/public","timestamp":"2024-11-09T14:21:50Z","content_type":"text/html","content_length":"8123","record_id":"<urn:uuid:25823bda-99c3-4664-bda5-6bbc8bd4d40f>","cc-path":"CC-MAIN-2024-46/segments/1730477028118.93/warc/CC-MAIN-20241109120425-20241109150425-00617.warc.gz"} |
Bracket Matrix: What Is It and Why Does It Matter? - UsaFoxNews
If you’ve ever delved into data structures, mathematical operations, or even tournament brackets, you may have encountered the term bracket matrix. But what exactly is a bracket matrix, and why
should you care? Whether you’re into advanced mathematics, computer science, or just organizing a bracket for your favourite sports tournament, understanding bracket matrices can offer some
surprising insights.
In this article, we’ll break down what a bracket matrix is, explore its various uses, and explain why this structure is more versatile than you might think. We’ll also answer some frequently asked
questions, and by the end, you’ll see how bracket matrices fit into different fields, from linear algebra to competition planning.
What Is a Bracket Matrix?
At its core, a bracket matrix is a structured format used to represent data or relationships within a grid-like system, often enclosed in square or round brackets. The concept of a bracket matrix can
vary based on context, as it’s used differently in fields like mathematics, sports, and computer science.
In mathematics, particularly in linear algebra, a bracket matrix typically refers to an arrangement of numbers or variables within square brackets. This setup helps perform calculations such as
matrix multiplication, addition, and transformations.
In tournament settings, a bracket matrix visually organizes the competition structure, showing which teams or players are matched up against each other. This allows for easy navigation through the
rounds, displaying outcomes and matchups as the tournament progresses.
Different Types of Bracket Matrices
A bracket matrix can take on different forms depending on its application. Here are a few common types you might encounter:
1. Square Bracket Matrices in Mathematics
□ These are matrices enclosed in square brackets [ ] and are often used in linear algebra to represent systems of equations, transformations, or data points.
□ Square bracket matrices are highly structured, with rows and columns that make operations like multiplication straightforward.
2. Tournament Bracket Matrices
□ These are often used in sports and competitive settings to visually organize participants.
□ Typically laid out in a tree or grid format, a tournament bracket matrix showcases which teams face off, who advances, and the path toward the final match.
3. Programming and Data Structures Bracket Matrices
□ In computer science, matrices are enclosed within brackets for coding purposes. These are frequently used in algorithms, graphics processing, and data representation.
□ They’re essential for tasks that involve multidimensional arrays or data that needs to be processed in a grid-like format.
Why Use a Bracket Matrix?
So, why does the bracket matrix matter? Well, it offers several key advantages depending on how you’re using it:
• Simplifies Complex Calculations: In mathematics, matrices streamline complex calculations by organizing data into rows and columns. This structure is ideal for systems that involve multiple
variables or equations.
• Organizes Tournaments Efficiently: For sports or e-sports, a bracket matrix clearly maps out the competition. You can track each matchup, see who advances, and understand the progression toward
the finals all in one view.
• Enhances Data Management in Programming: In computer science, bracket matrices enable efficient data handling. They’re used in multidimensional arrays and are essential in graphics processing and
How to Create a Bracket Matrix
Creating a bracket matrix will depend on your goal and the field you’re working in. Let’s go through some basic steps for setting one up in both mathematical and tournament contexts.
Setting Up a Mathematical Bracket Matrix
1. Determine the Size: Decide on the matrix’s dimensions—whether it’s a 2×2, 3×3, or larger matrix. This will depend on the complexity of your data or equations.
2. Input the Values: Populate each cell with numbers or variables relevant to your calculations.
3. Use Proper Notation: Enclose the matrix in square brackets [ ] for clarity.
4. Perform Operations: You can now perform operations like addition, multiplication, or determinant calculation based on your needs.
Creating a Tournament Bracket Matrix
1. List the Participants: Start with all the competitors at the top or left side of the matrix.
2. Arrange Matchups: Match competitors in pairs, assigning them to specific cells in the grid.
3. Outline the Path: Draw lines or arrows to indicate progression, showing which matchups lead to subsequent rounds.
4. Update Results: As the tournament progresses, fill in the bracket matrix with winners and adjust accordingly.
Applications of Bracket Matrices
The versatility of bracket matrices extends across various fields. Here’s how different industries make use of them:
• Education: Teachers use bracket matrices to help students understand linear algebra and systems of equations. It’s a valuable tool for visualizing complex relationships between numbers and
• Sports Management: Tournament organizers rely on bracket matrices to lay out game structures, especially in knockout competitions. This format is used in everything from March Madness to local
chess tournaments.
• Programming and Data Science: Developers utilize bracket matrices to handle multidimensional arrays and process data efficiently. They’re commonly used in machine learning algorithms, where data
needs to be manipulated and transformed in structured ways.
FAQs on Bracket Matrices
1. What is a bracket matrix used for in mathematics?
In mathematics, a bracket matrix organizes numbers or variables into a structured grid, making it easier to perform operations like multiplication, transformation, and solving systems of equations.
2. How does a bracket matrix help in sports tournaments?
A tournament bracket matrix provides a visual layout of the competition. It organizes matchups, tracks progress, and shows the path to the finals, making it easy for participants and viewers to
follow the tournament.
3. Can a bracket matrix be used in coding?
Yes, in programming, bracket matrices are often used for handling multidimensional arrays. They’re essential in data manipulation, simulations, and graphics processing tasks.
4. Are there specific tools for creating bracket matrices?
For mathematical bracket matrices, software like MATLAB or Excel can help. For tournament bracket matrices, there are specialized apps like Bracket HQ and Printable Tournament Brackets that simplify
the creation process.
5. What’s the difference between a bracket matrix and a regular matrix?
A bracket matrix is a specific type of matrix notation enclosed in brackets, but otherwise, it functions similarly to other matrices. The bracket notation simply emphasizes structure and makes it
visually distinct.
Final Thoughts on Bracket Matrices
Understanding the concept of a bracket matrix opens up a range of possibilities in both everyday and professional settings. From solving algebraic problems to organizing sports tournaments, bracket
matrices are incredibly versatile. They simplify complex tasks, make data more manageable, and offer a clear visual structure that’s easy to interpret.
So next time you encounter a complex calculation, a competitive event, or a programming challenge, consider using a bracket matrix. It’s a tool that offers both structure and clarity, making it a
go-to choice across diverse fields. Whether you’re working on a math problem or setting up the next big tournament, a bracket matrix can streamline the process and help you stay organized. | {"url":"https://usafoxnews.com/bracket-matrix-what-is-it-and-why-does-it-matter/","timestamp":"2024-11-08T08:12:56Z","content_type":"text/html","content_length":"458544","record_id":"<urn:uuid:72485981-6ff6-4f48-88fa-6c993c4f60db>","cc-path":"CC-MAIN-2024-46/segments/1730477028032.87/warc/CC-MAIN-20241108070606-20241108100606-00243.warc.gz"} |
Designing Theory Solvers with Extensions
This site provides our paper Designing Theory Solvers with Extensions and related material at FROCOS 2017.
Extended Theory of Strings
• For the strings experiments in Section 4, we used the master version of CVC4: CVC4, revision 3837f84a.
□ Configuration cvc4+sm is enabled by --lang=smt2 --strings-exp --rewrite-divk --strings-fmf.
□ Configuration cvc4+m is enabled by --lang=smt2 --strings-exp --rewrite-divk --no-strings-lazy-pp --strings-fmf
• The .tgz containing the binary of z3 used in our experiments can be found here: z3
• The .tgz containing the binary of z3Str2 used in our experiments can be found here: z3Str2
The benchmarks considered in this section can be downloaded here.
The spreadsheet summarizing the results for the strings benchmarks considered in this paper can be accessed here: results-strings.xlsx.
Lazy Bit-blasting for Expensive Bit-Vector Operators
• For the strings experiments in Section 5, we used a development branch of CVC4: CVC4.
□ Configuration cvc4+sm is enabled by --bv-lazy-bb-exp.
□ Configuration cvc4 is enabled by no command line parameters.
We considered the sage2 family of benchmarks from the QF_BV division of SMT LIB. For instructions downloading these benchmarks, see this page.
The spreadsheet summarizing the results for the strings benchmarks considered in this paper can be accessed here: results-bv.xlsx.
Lightweight Techniques for Non-linear Arithmetic
• For the strings experiments in Section 5, we used a development branch of CVC4: CVC4.
□ Configuration cvc4+sm is enabled by --nl-alg --nl-alg-tplanes.
□ Configuration cvc4+m is enabled by --nl-alg --nl-alg-tplanes --no-nl-alg-rewrite
We considered all benchmarks from the QF_NRA and QF_NIA divisions of SMT LIB. For instructions downloading these benchmarks, see this page.
The spreadsheet summarizing the results for the strings benchmarks considered in this paper for QF_NRA can be accessed here: results-nra.xlsx, and QF_NIA can be accessed here: results-nia.xlsx. | {"url":"https://cvc4.github.io/papers/frocos2017-ext","timestamp":"2024-11-14T18:12:06Z","content_type":"text/html","content_length":"10105","record_id":"<urn:uuid:813463ed-073a-48ba-a27c-869d539692e5>","cc-path":"CC-MAIN-2024-46/segments/1730477393980.94/warc/CC-MAIN-20241114162350-20241114192350-00076.warc.gz"} |
Implementation Details of User Capabilities
Capability Letter Choices
We assigned user capability characters using only lowercase ASCII letters at first, so those are the most important within Fossil: they control the functions most core to Fossil’s operation. Once we
used up most of the lowercase letters, we started using uppercase, and then during the development of the forum feature we assigned most of the decimal numerals. All of the lowercase ASCII letters
are now assigned. Eventually, we might have to start using ASCII punctuation and symbols. We expect to run out of reasons to define new caps before we’re forced to switch to Unicode, though the
possibilities for mnemonic assignments with emoji are intriguing. 😉
The existing caps are usually mnemonic, especially among the earliest and therefore most central assignments, made when we still had lots of letters to choose from. There is still hope for good
future mnemonic assignments among the uppercase letters, which are mostly still unused.
Why Not Bitfields?
Some may question the use of ASCII character strings for capability sets instead of bitfields, which are more efficient, both in terms of storage and processing time.
Fossil handles these character strings in one of two ways. For most HTTP hits, Fossil expands the string into a struct full of flags so that later code can just do simple Boolean tests. In a minority
of cases, where Fossil only needs to check for the presence of a single flag, it just does a strchr() call on the string instead.
Both methods are slower than bit testing in a bitfield, but keep the execution context in mind: at the front end of an HTTP request handler, where the nanosecond differences in such implementation
details are completely swamped by the millisecond scale ping time of that repo’s network connection, followed by the required I/O to satisfy the request. Either method is plenty fast in that context.
In exchange for this immeasurable cost per hit, we get human-readable capability sets.
Why Doesn’t Fossil Filter “Bad” Artifacts on Sync?
Fossil is more trusting about the content it receives from a remote clone during sync than you might expect. Common manifestations of this design choice are:
1. A user may be able to impersonate other users. This can be accidental as well as purposeful.
2. If your local system clock is out-of-sync with absolute time, artifacts committed to that repo will appear with the “wrong” time when sync’d. If the time sync error is big enough, it can make
check-ins appear to go back in time and other bad effects.
3. You can purposely overwrite good timestamps with bad ones and push those changes up to the remote with no interference, even though Fossil tries to make that a Setup-only operation.
All of this falls out of two of Fossil’s design choices: sync is all-or-nothing, and the Fossil hash tree is immutable. Fossil would have to violate one or both of these principles to filter such
problems out of incoming syncs.
We have considered auto-shunning “bad” content on sync, but this is difficult due to the design of the sync protocol. This is not an impossible set of circumstances, but implementing a robust filter
on this input path would be roughly as difficult as writing a basic inter-frame video codec: do-able, but still a lot of work. Patches to do this will be thoughtfully considered.
We can’t simply change content as it arrives. Such manipulations would change the artifact manifests, which would change the hashes, which would require rewriting all parts of the block chain from
that point out to the tips of those branches. The local Fossil repo must then go through the same process as the remote one on subsequent syncs in order to build up a sync sequence that the remote
can understand. Even if you’re willing to accept all of that, this would break all references to the old artifact IDs in forum posts, wiki articles, check-in comments, tickets, etc.
The bottom line here is that Clone and Write are a potent combination of user capabilities. Be careful who you give that pair to! | {"url":"https://fossil-scm.org/home/doc/tip/www/caps/impl.md","timestamp":"2024-11-09T19:15:10Z","content_type":"text/html","content_length":"31621","record_id":"<urn:uuid:4ee840d2-45a5-4171-8758-40f77cbcb63e>","cc-path":"CC-MAIN-2024-46/segments/1730477028142.18/warc/CC-MAIN-20241109182954-20241109212954-00105.warc.gz"} |
C Program to Calculate Volume and Total Surface Area of Cylinder
In this C program we will calculate total surface area and volume of a cylinder. A cylinder is a three dimensional solid that has two circular bases connected by a curved surface. A cylinder can be
formed by two circles of same radius(R) and the curved surface formed by all the points at a distance of R from axis(axis is a line segment joining the center of both bases). Cylinder objects are
very common in everyday life, like a cylindrical can.
• Radius : The radius of a cylinder is the radius of it's circular base. It is half of the diameter of the cylinder.
• Height : Height of a cylinder is the perpendicular distance between the parallel bases.
• Axis : It is the line segment joining the centers of both circular bases.
Here, we are discussing about right circular cylinder, means the bases of the cylinder are circular and the axis is perpendicular to both bases.
Total Surface Area of Cylinder
Surface area of cylinder is the number of square units that will exactly cover the outer surface of a cone. There are three surfaces in a cylinder, one curved and two circular bases. Total surface
area of cylinder is the sum of the area of both circular bases and area of curved surface. Total surface area of a right circular cylinder is measured in square units like m^2, cm^2 etc.
Base area of cylinder = ΠR^2
Curved surface area of cone = 2ΠRH
Total surface area of cone = 2XBase area + Curved area
= 2ΠR^2 + 2ΠRH
= 2ΠR(R + H)
Volume of Cylinder
The volume of a right circular cylinder is defined as the amount of three dimensional space occupied by the cylinder or the storage capacity of a cylinder. Finding volume of a cylinder help us to
solve many real life problems like, how much water can be filled in a cylindrical aluminium can. To calculate the volume of a cylinder, we need radius of base height of cylinder. Volume of a right
circular cylinder is measured in cubic units like m^3, cm^3 etc.
Volume of right circular cylinder = Base area x Height
As base of cylinder is circular, Base Area = ΠR^2
Volume of right circular cylinder = ΠR^2H
Where 'R' is the radius of the base and 'H' is the height of cylinder.
C Program to find total surface area of a cylinder
To calculate total surface area of a cylinder, we need radius of base and height of cylinder. Below program takes base radius and height of cylinder as input from user using scanf function. Then, it
calculates the total surface area of cylinder using formula given above. Finally, it prints the surface area of cylinder on screen using printf function.
#include <stdio.h>
#define PI 3.14159
int main(){
float radius, height, surfaceArea;
printf("Enter base radius and height
of a Cylinder\n");
scanf("%f %f", &radius,
surfaceArea = 2*PI*radius*(radius+height);
printf("Total surface area of
Cylinder : %0.4f\n", surfaceArea);
return 0;
Enter base radius and height of a Cylinder
Total surface area of Cylinder : 207.3449
C Program to find volume of a cylinder
To calculate volume of a cylinder, we need radius of base and height of right circular cylinder. Below program takes base radius and height of right circular cylinder as input from user using scanf.
Then, it calculates the volume of cylinder using formula given above. Finally, it prints the volume of right circular cylinder on screen using printf.
#include <stdio.h>
#define PI 3.14159
int main(){
float radius, height, volume;
printf("Enter base radius and
height of a Cylinder\n");
scanf("%f %f", &radius,
volume = PI*radius*radius*height;
printf("Volume of Cylinder : %0.4f\n",
return 0;
Enter base radius and height of a Cylinder
Volume of Cylinder : 226.1945
Properties of Cylinder
• The bases are always congruent and parallel to each other.
• There are 2 plane surfaces, 1 curved surface and 2 edges in a cylinder.
• Volume of a cylinder is 3 times the volume of a cone of same base radius and height.
Related Topics | {"url":"https://www.techcrashcourse.com/2015/03/c-program-to-calculate-volume-and-total-surface-area-of-cylinder.html","timestamp":"2024-11-11T00:00:24Z","content_type":"application/xhtml+xml","content_length":"84390","record_id":"<urn:uuid:50ff903e-dbb1-41b4-855b-696e4542088e>","cc-path":"CC-MAIN-2024-46/segments/1730477028202.29/warc/CC-MAIN-20241110233206-20241111023206-00755.warc.gz"} |
Diagnosis and analysis of twin-tube shock absorber rattling noise in electric vehicle
To address the rattling noise issue of a Twin-tube hydraulic shock absorber for a electric vehicle, a road test and subjective evaluation were performed. The diagnostic method of the rattling noise
source of the shock absorber is studied, and three methods are used to process the test date: the fast Fourier transform can only obtain the main frequency band of the rattling noise; and the
short-time Fourier transform can roughly diagnose the frequency band and the occurrence time; using wavelet transform technology combined with displacement signal can accurately lock the source of
rattling noise in the rebound stroke. Furthermore, combined with the internal valve configuration of the shock absorber, the rattling noise source is diagnosed in the rebound valve and the
compensation valve. The time of rattling noise and the corresponding working stroke of shock absorber can be quickly locked by using the wavelet transform, which is of great significance to solve the
problem of shock absorber rattling noise caused by deficiency design of the internal valve architecture of the shock absorber.
1. Introduction
The shock absorber is an important part of the automobile suspension system. New energy vehicles lack the noise masking of the power and transmission system, and the problem of rattling noise of the
shock absorber has attracted more and more attention [1]. When driving on an uneven road, the shock absorber’s damping force fluctuates, causing high-frequency vibration of the piston rod [2]. The
high-frequency vibration is transmitted to the vehicle through the body sheet metal, producing rattling noise, which causes the driver and passengers to complain. If the shock absorber’s topmount
vibration isolation capacity is insufficient, this rattling noise will be exacerbated. There are two types of methods for resolving shock absorber rattling noise: source elimination and path blocking
[3]. If the internal valve structure of the shock absorber needs to be optimized, the damping force fluctuations caused by which valve disc of the shock absorber must be locked [4]. Therefore, it is
of great significance to find a method that can accurately diagnose the source of the rattling noise to resolve the complaint.
There are few domestic studies on the diagnosis and analysis of the source of the shock absorber rattling noise, and the majority of them focus on the identification of the shock absorber rattling
noise. On the basis of a large number of test data and signal characteristic analysis, Shu et al. [5] proposed a method for identifying the time-domain waveform attenuation of rattling noise of
hydraulic twin-tube shock absorbers, with an identification success rate of more than 98.5 percent. Huang et al. [6] investigated the rattling noise identification method of shock absorber bench test
and proposed a cluster analysis method based on weight coefficient, which can be used as a reference for large batch and different types of shock absorber rattling noise identification and
identification accuracy improvement. The previous studies can only identify the shock absorber’s rattling noise signal and cannot diagnosed the source of the shock absorber's rattling noise.
In this paper, the road test and subjective evaluation of the two shock absorbers with and without rattling noise are carried out, and then, the advantages and disadvantages of three signal
processing methods of fast Fourier transform, short-time Fourier transform and wavelet analysis for the diagnosis of rattling noise sources of shock absorbers are compared, finally, a method based on
wavelet transform technology combined with the displacement signal of the shock absorber is proposed for accurately diagnosing the rattling noise source of the shock absorber. This method is an
important reference for resolving the issue of the rattling noise of the shock absorber.
2. Testing and evaluation of vehicle's shock absorber rattling noise on road
When a domestic electric vehicle is driving at a low speed on rough concrete roads, the front shock absorber produces ratting noise. Select BOB (Best of Best) and WOW (Worst of Worst) from the same
batch of shock absorbers for road tests. The designed road test conditions are as follows: driving on a rough cement road at the proving ground at a constant speed of 20 km/h, to collect the
vibration signal of piston rod top, displacement signal of the spring and noise inside the car, the sensors attachment locations is shown in Fig. 1 (Accelerometer (1) is located at the top of the
piston rod; Accelerometer (2) is located on the wall; (3) is the displacement sensor; (4) is the microphone).
The LMS is used in this test which is a multifunctional data acquisition and analysis system, along with high-fidelity sound playback headphones, to ensure that the noise signal playback accurately
reflects people's subjective feelings. Professional vehicle BSR noise evaluation personnel will score the shock absorber, and the scoring mechanism is carried out in accordance with the 10-point
system BSR noise performance evaluation standard. BOB pieces scored 7 points on the 10-point system rattling noise evaluation standard, while WOW pieces scored 5 points.
3. Research on identification method of rattling noise source of shock absorber
3.1. Fast Fourier transform
The fast Fourier transform decomposes the signal into a sine function using trigonometric functions as the basis of the function space, converting the time domain signal to a frequency domain signal.
The following is the conversion procedure:
$F\left(\omega \right)=F\left[f\left(t\right)\right]={\int }_{-\infty }^{+\infty }f\left(t\right){e}^{-j\omega t}dt,$
$f\left(t\right)={F}^{-1}\left[F\left(\omega \right)\right]=\frac{1}{2\pi }{\int }_{-\infty }^{+\infty }F\left(\omega \right){e}^{j\omega t}d\omega .$
The continuous signal must be discretized in application before it can be processed by the computer. As a result, to obtain discrete samples $x\left(n\right)$, the continuous signal $x\left(t\right)$
must be sampled in the time domain, and its fast Fourier transform can be obtained:
$X\left(\omega \right)={\sum }_{n=-\infty }^{\infty }x\left(n\right){e}^{-j\omega n}.$
The discrete-time Fourier transform is represented by the formula above. The frequency domain value obtained after the transformation is still continuous, and the frequency domain must be sampled
indefinitely to obtain:
$X\left(k\right)=\sum _{n=0}^{N=-1}x\left(n\right){e}^{-j\frac{2\pi kn}{N}}.$
The Eq. (4) is the discrete Fourier transform. The time domain signal converted into a frequency domain signal after discrete Fourier transformation. However, for non-stationary signals such as BSR
noise, using Fourier transform will lose time-domain information and cannot obtain non-stationary characteristics.
Fourier transform is used to process the collected shock absorber signal, because in this case, the $Z$-direction vibration of the top of the piston rod is the main contribution direction of the
rattling noise of the shock absorber, so only the $Z$-direction data is used to illustrate, the $X$/$Y$ direction data will not be compared for the time being, and the spectrum diagram of the top of
the shock absorber piston rod with and without rattling noise is obtained as follows.
The vibration spectrum of the piston rod, as shown in the figure above, can only see the difference between the vibration spectrum of the shock absorber piston rod with and without rattling noise:
The vibration magnitude of the WOW parts is significantly greater than that of the non-BOB parts in the $Z$-direction 100-400 Hz frequency band, and the peak frequency is 250 Hz, but the time when
the abnormal vibration occurs and the specific working stroke of the corresponding shock absorber cannot be determined. The noise in the driver’s ear is masked by other BSR noises and road noise, the
difference between the spectrograms of BOB pieces and WOW pieces is not prominent in the corresponding frequency bands.
Fig. 2Z direction of the top of the shock absorber piston rod (blue is WOW, red is BOB)
Fig. 3Noise at the driver’s ear (blue is WOW, red is BOB)
3.2. Short-time Fourier transform
It is difficult to analyze the characteristics of non-stationary signals using only the time domain signal or amplitude spectrum, so time-frequency analysis short-time Fourier transform is required.
The method of adding time domain information by short-time Fourier transform is to set a window function, it is assumed that the signal in the window function to be a stationary signal, and perform
Fourier transform on the signal segment in the window function. The short-time Fourier transform is defined as:
$X\left(n,\omega \right)={\sum }_{m=-\infty }^{\infty }x\left(m\right)\omega \left(n-m\right){e}^{-j\omega m},$
where $x\left(m\right)$ is the input signal and $\omega \left(m\right)$ is the window function, which is reversed in time and has an offset of $n$ samples. $X\left(N,\omega \right)$ is a
two-dimensional function of time $n$ and frequency $\omega$, which connects the time domain and frequency domain of the signal, and we can perform time-frequency analysis on the signal accordingly.
The short-time Fourier transform is used to analyze the vibration signal at the top of the piston rod of the shock absorber. Since the accuracy of the time and frequency domain results of the
short-time Fourier transform is affected by the length of the window function, the window function type in this paper is the Hanning window. The window function lengths are selected as 0.5S and 2S
respectively for comprehensive comparison, and the time-frequency diagrams of BOB and WOW parts are obtained as shown in the following figures.
Fig. 4Time-frequency diagram in the Z direction of the top of the piston rod of the BOB piece (0.5S)
Fig. 5Time-frequency diagram in the Z direction of the top of the piston rod of the BOB piece (2S)
Fig. 6Time-frequency diagram in Z direction at the top of the piston rod of the WOW piece (0.5S)
Fig. 7Time-frequency diagram in Z direction of the top of the piston rod of the WOW piece (2S)
Comparison of WOW and BOB components in the time-frequency domain diagram analyzed by short-time Fourier transformation: As shown in Figs. 6 and 7, when a narrow window function is chosen, the time
resolution of the time-frequency diagram is higher and the frequency resolution is lower and vice versa, but the time resolution is low, making it difficult to choose the window function that takes
both the time resolution and the frequency resolution of the analysis results into account. Comparing Fig. 4 and Fig. 6, it can be seen that the peak frequency of the BOB part cannot be obtained. The
vibration with higher energy occurs at 12.14-12.66 s, 12.92-13.26 s, 16.69-16.89 s, 18.91-19.23 s, the time interval is 0.52 s, 0.34 s, 0.2 s, 0.32 s; Comparing Fig. 5 and Fig. 7, it can be seen that
the peak frequency of the WOW part is 260 Hz in $Z$-direction, the time corresponding to the peak frequency is between 11.06 and 12.75 s, and the time interval is 1.69 s. Through the above
comparison, it is found that the time resolution of the analysis results using a narrower window function is higher than that of using a wider window function, however, under this test condition, the
shock absorber works for a cycle time is about 0.1 s, or even less, if the time resolution is enhanced further to meet this operating condition, the frequency resolution will be decreased, making it
difficult to apply the short-time Fourier transform to precisely analyze the peak frequency and time of anomalous piston rod vibration at the same time.
3.3. Wavelet transform
The characteristic of wavelet transform is that it can perform multi-resolution analysis at the same time, and it has the ability to characterize the local characteristics of the signal in both time
domain and frequency. There are two variables in wavelet transform: $a$ and $b$, $a$ is the reciprocal of the frequency, which controls the contraction of the wavelet function, and $b$ is the time
factor, which controls the translation of the wavelet function. For a finite signal $f\left(t\right)$ of arbitrary energy, its continuous wavelet transform is defined as:
${W}_{f}\left(a,b\right)=\frac{1}{\sqrt{a}}{\int }_{-\infty }^{+\infty }f\left(t\right){\psi }^{*}\left(\frac{t-b}{a}\right)dt,$
where $\psi \left(±\infty \right)$ is the mother wavelet or basic wavelet. Therefore, unlike the basis function of the Fourier transform, which is an infinitely long sine wave, the basis function of
the wavelet transform is a finite-length wavelet that has been attenuated, and the wavelet basis function is localized in both the time domain and the frequency domain. Wavelet analysis decomposes
the signal into the superposition of a series of wavelet functions. These wavelet functions can be used to approximate the sharply changing parts of the non-stationary signal, and can also
approximate the discrete discontinuous signal with local characteristics, so as to reflect the original signal more truly change on a time scale. Since the wavelet transform can achieve an automatic
harmony between frequency and time, it can realize the time-frequency multi-resolution function of the signal.
In this paper, the wavelet analysis module of the LMS software is used to analyze the shock absorber signal. The default wavelet basis function is Morlet wavelet. The wavelet transform is used to
analyze the vibration signal of the shock absorber piston rod, and the result is shown in the following figures.
Fig. 8Time-frequency diagram of the top Z-direction of the piston rod of the BOB piece
Fig. 9Time-frequency diagram of the Z direction at the top of the piston rod of the WOW piece
The above figure shows that the vibration energy of WOW parts is much higher than that of BOB parts in the frequency band of 100-400 Hz, and the peak frequency is in the range of 220 Hz-350 Hz. This
conclusion is basically consistent with the spectrogram obtained by Fourier analysis. Enlarging the time domain interval, it can be seen that the abnormal vibration of the WOW piston rod in the $Z$
direction is between 13.47-13.48S. The data of 13.47-13.48S is found in the time domain signal of the WOW component in the $Z$ direction. Combined with the displacement curve of the shock absorber,
get Fig. 10.
It can be seen from the Fig. 10 that the abnormal vibration in the $Z$ direction of the WOW component occurs during the rebound stroke of the shock absorber. The valves related to the rebound stroke
are the rebound valve and the compensation valve. Combined with the force vs disp diagram of the shock absorber and the physical structure of the internal valves of the WOW component, you can adjust
the relevant disk in a targeted manner, but it is beyond the scope of this paper. In this case, wavelet analysis is used to precisely lock the shock absorber's rattling noise source to the shock
absorber's rebound stroke and to the specific valve. Therefore, wavelet analysis is very effective in finding the rattling noise source of the shock absorber.
Fig. 10Z-direction acceleration-displacement curve of shock absorber piston rod
4. Conclusions
The road test of the shock absorber BOB and WOW parts and subjective evaluation was carried out, and the three methods of Fourier transform, short-time Fourier transform and wavelet analysis were
used to diagnose the rattling noise source of the shock absorber respectively. The following conclusions are drawn:
1) Fourier transform can quickly and accurately analyze the spectral characteristic information of the shock absorber piston rod acceleration, but the time information corresponding to the frequency
information cannot be obtained.
2) The short-time Fourier transform can roughly analyze the time-frequency domain characteristics of the shock absorber piston rod acceleration signal, but there is a problem in that the analysis
accuracy in the frequency domain and the time domain cannot be achieved concurrently. The main peak frequency and corresponding time of the shock absorber rattling noise cannot be precisely locked.
3) The wavelet transform can precisely lock the peak frequency and occurrence time of the abnormal vibration of the shock absorber piston rod. Combined with the shock absorber displacement signal,
the rattling noise source can be locked in the rebound valve in piston side and compensation valve in base side.
4) Through the research on the identification method of the shock absorber rattling noise source, it is finally concluded that the wavelet transform combined with the shock absorber displacement
signal can quickly and accurately lock the shock absorber rattling noise source to the specific valve configuration. This method has strong operability, is fast and effective, and can play an
important role in solving the rattling noise problem of the shock absorber.
• Y. G. Zhu, M. Y. Zhou, and B. Feng, “Rapid identification method of suspension rattling noise based on transfer path analysis,” Journal of Henan Institute of Technology, Vol. 26, No. 6, pp. 8–11,
• A. Kruse, M. Eickhoff, and A. Tischer, “Analysis of dynamic behavior of twin-tube vehicle shock absorbers,” SAE International Journal of Passenger Cars – Mechanical Systems, Vol. 2, No. 1, pp.
447–453, Apr. 2009, https://doi.org/10.4271/2009-01-0223
• M. T. Yao, L. Gu, and J. F. Guan, “Analysis of abnormal noise test of twin-tube shock absorber,” Chinese Journal of Engineering Design, Vol. 17, No. 3, pp. 229–235, 2010.
• M. T. Yao, J. F. Guan, L. Gu, and Z. Y. Cheng, “Study on abnormal noise of vehicular twin-tube shock absorber,” Machinery Design and Manufacture, No. 2, pp. 114–116, 2011, https://doi.org/
• H. Y. Shu, L. Y. Wang, and Y. W. Cen, “Identification method of abnormal noise of vehicle hydraulic shock absorber,” Journal of Chongqing University, Vol. 28, No. 4, pp. 10–13, 2005.
• H. B. Huang, R. X. Li, W. P. Ding, M. L. Yang, and H. L. Zhu, “Rig test for identifying abnormal noise of suspension shock absorber,” Journal of Vibration and Shock, Vol. 34, No. 2, pp. 191–196,
2015, https://doi.org/10.13465/j.cnki.jvs.2015.02.034
About this article
Vibration in transportation engineering
shock absorber rattling noise
wavelet transform
shock absorber valve configuration
Copyright © 2022 Tingting Zheng, et al.
This is an open access article distributed under the
Creative Commons Attribution License
, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. | {"url":"https://www.extrica.com/article/22606","timestamp":"2024-11-12T12:41:12Z","content_type":"text/html","content_length":"122916","record_id":"<urn:uuid:d7fc7148-2df8-41d3-aa71-a454af367a02>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.45/warc/CC-MAIN-20241112113320-20241112143320-00686.warc.gz"} |
MLB GOAT: Evaluating a Baseball Player, Cryptbeam
My last post, which covered an introductory example of adjusting century-old stats for inflation in the MLB, was the first step is a larger goal, one that will be brought to life with the processes
I’ll outline today: ranking the greatest MLB players ever. Many times before we have seen an attempt to do so, but rarely have I found a list that aligns with my universal sporting values. Thus, I
have chosen to embark on a journey to replicate the results in a process I see to be more philosophically fair: a ranking of the best players of all time with the driver being the value of their
on-field impact. However, as I am a relative novice in the art of hardcore analysis in baseball, I’ll be providing a clear, step-by-step account of my process to ensure the list is as accurate as
The Philosophy
I’ve come to interpret one universal rule in player evaluation across most to all team sports, which relies on the purpose of the player. As I’ve stated in similar posts covering the NBA, a player is
employed by a team for one purpose: to improve that team’s success. Throughout the course of the season, the team aims to win a championship. Therefore, the “greatest” MLB players give their teams
the best odds to win the World Series. However, I’m going to alter one word in that sentence: “their.” Because championship odds are not universal across all teams (better teams have greater odds),
that means a World Series likelihood approach that considers “situational” value (a player’s value to his own team) will be heavily skewed towards players on better teams, and that would be an unfair
deflation or inflation of a player’s score that relies on his teammates.
The central detail of my evaluation style will be the ideology behind assigning all players the same teammates, average teammates. Therefore, the question I’m trying to answer with a player
evaluation is: what are the percent odds a player provides an average team to provide the World Series? This approach satisfies the two conditions I outlined earlier: to measure a player’s impact in
the way that appeases the purpose of his employment while leveling the field for players seen as “weaker” due to outside factors they couldn’t control. Thus, we have the framework to structure the
The Method
To measure a player’s impact, I’ll use a preexisting technique I’ve adopted for other sports, in which I estimate a player’s per-game impact (in this case, this would be represented through runs per
game). For example, if an outfielder evaluates as a +0.25 runs per game player on offense and a 0 runs per game player on defense, he extends the aforementioned average team’s schedule-adjusted run
differential (SRS) and thus raises the odds of winning a given game with the percent odds that come along with a +0.25 SRS boost. To gain an understanding of how the “impact landscape” works, I laid
every qualified season from 1871 to 2020 out for both position players and pitchers to get a general idea of how “goodness” translates to impact. These were the results:
Note: Offense and fielding use Fangraphs‘s “Off” and “Def” composite metrics scaled to per-game measures while pitching uses Runs Above Replacement per game scaled to “runs above average” – these
statistics are used to gauge certain levels of impact. / I split the fielding distributions among positions to account for any inherent differences that result from play frequency, the value of a
position’s skill set, and others.
Offense (all positions)
Fielding (pitchers)
Fielding (catchers)
Fielding (first basemen)
Fielding (second basemen)
Fielding (third basemen)
Fielding (shortstops)
Fielding (outfielders)
Pitching (starters)
Pitching (relievers)
A large reason for the individual examination of each distribution is to gain a feel for what constitutes, say, an All-Star type of season, an All-MLB type of season, or an MVP-level season, and so
on and so forth. The dispersions of the distributions are as listed below:
Standard Position Players Starting Pitchers Relief Pitchers Pitchers Catchers First Basemen Second Basemen Third Basemen Shortstops Outfielders
Deviations (Off) (Pitch) (Pitch) (Field) (Field) (Field) (Field) (Field) (Field) (Field)
-4 -0.554 -1.683 -0.582 -0.305 -0.262 -0.255 -0.256 -0.258 -0.258 -0.286
-3 -0.402 -1.262 -0.437 -0.233 -0.183 -0.202 -0.185 -0.188 -0.178 -0.221
-2 -0.250 -0.841 -0.291 -0.162 -0.104 -0.149 -0.115 -0.118 -0.097 -0.157
-1 -0.098 -0.421 -0.146 -0.090 -0.025 -0.096 -0.044 -0.048 -0.017 -0.092
0 0.054 0.000 0.000 -0.018 0.053 -0.043 0.026 0.022 0.064 -0.028
1 0.206 0.421 0.146 0.053 0.132 0.010 0.097 0.092 0.144 0.037
2 0.358 0.841 0.291 0.125 0.211 0.063 0.168 0.162 0.225 0.102
3 0.510 1.262 0.437 0.197 0.290 0.116 0.238 0.232 0.305 0.166
4 0.662 1.683 0.582 0.269 0.368 0.169 0.309 0.302 0.385 0.231
These values are used to represent four ambiguous “tiers” of impact, with one standard deviation meaning “good” seasons, two standard deviations meaning “great” seasons, three standard deviations
meaning “amazing” seasons, and four standard deviations meaning “all-time” seasons, with the negative halves representing the opposites of those descriptions. Throughout my evaluations, I’ll refrain
from handing out all-time seasons, as these stats were taken from one-year samples and are thus prone to some form of variance. Therefore, an “all-time” season in this series will likely be a tad
underneath what the metrics would suggest.
There are also some clear disparities between the different fielding positions that will undoubtedly affect the level of impact each of them can provide. Most infield positions seem to be
above-average fielders in general, with the first basemen showing greater signs of being more easily replaced. The second and third basemen share almost the same distribution while the shortstops and
catchers make names as the “best” fielders on the diamond. I grouped all the outfielders into one curve, and they’re another “low-ceiling” impact position, similar to pitchers (for whom fielding
isn’t even their primary duty). It’ll be important to keep these values in mind for evaluations, not necessarily to compare an average shortstop and an average first baseman, but, for instance, an
all-time great fielding shortstop versus and an all-time great fielding first baseman.
The Calculator
Now that we have the practice listed out, it’s time to convert all those thoughts on a player to the numeric scale and actually do something with the number. The next step in the aforementioned
preexisting technique is a “championship odds” calculator that uses a player’s impact on his team’s SRS (AKA the runs per game evaluation) and his health to gauge the “lift” he provided an average
team that season. To create this function, I gathered the average SRS of the top-five seeds in the last twenty years and simulated a Postseason based on how likely a given team was to win the series,
calculated with regular-season data in the same span.
Because the fourth seed (the top Wild Card teams) is usually better than the third seed (the “worst” division leader), and the former would often face the easier path to the World Series, a disparity
was created in the original World Series odds: in this case, a lower seed had better championship odds. To fit a more philosophically-fair curve, I had to take teams out of the equation and
restructure the function accordingly. This means there is a stronger correlation to title odds based on SRS, separate from seeding conundrums; after all, we want to target the players with more lift,
not the other way around. Eventually, this curve became so problematic I chose the more pragmatic approach: taking and generalizing real-world results instead of simulating them and found the ideal
function with an R^2 of 0.977. (This method seemed to prove effective not only because of the strength of the fit, but the shape of the curve, which went from distinctly logarithmic (confusing) to
distinctly exponential.)
The last step is weighing a player’s championship equity using his health; if a player performed at an all-time level for 162 games but missed the entirety of the Postseason, he’s certainly not as
valuable as he would’ve been if he’d been fully healthy. Thus, we use the proportion of a player’s games played in the regular season to determine the new SRS, while the percentage of Postseason
games played represents the sustainability of that SRS for the second season. The health-weighted SRS is then plugged into the championship odds function to get Championship Probability Added!
With my new “World Series odds calculator,” I’ll perform evaluations on the best players in MLB history and rank the greatest careers in history. I’ll aim to rank the top-20 players ever at minimum,
with a larger goal of cranking out the top-40. With this project, I hope to shed some light on these types of topics in a new manner while, hopefully, sparking discussion on a sport that deserves
more coverage nowadays. | {"url":"https://www.cryptbeam.com/2021/03/14/mlb-goat-evaluating-a-baseball-player/","timestamp":"2024-11-11T11:02:56Z","content_type":"text/html","content_length":"82610","record_id":"<urn:uuid:c2c61ffa-5a69-4446-9024-eed7d21a5b0e>","cc-path":"CC-MAIN-2024-46/segments/1730477028228.41/warc/CC-MAIN-20241111091854-20241111121854-00132.warc.gz"} |
commit 9f3c7e7ea5d872fc017271df61691ea545049211
parent 2332679ad64e3e0318ca812421190156edec9186
Author: Sebastiano Tronto <sebastiano.tronto@gmail.com>
Date: Mon, 27 Dec 2021 16:34:41 +0100
Changed installation instructions (for pruning tables)
M INSTALL | 35 ++++++++++++++++++-----------------
M README.md | 46 ++++++++++++++++++++--------------------------
M TODO.md | 6 ++----
3 files changed, 40 insertions(+), 47 deletions(-)
diff --git a/INSTALL b/INSTALL
@@ -1,10 +1,10 @@
# Requirements
-A full installation of nissy requires a little more than 2Gb of space,
-of which 1.6Gb are occupied by the huge pruning table for fast optimal solving,
+A full installation of nissy requires about 3Gb of space,
+of which 2.3Gb are occupied by the huge pruning table for fast optimal solving,
and running it requires the same amount of RAM.
One can choose to never use this function and not to install the relative
-pruning table. There is an alternative (about 5 times slower)
+pruning table. There is an alternative (slower)
optimal solving function that uses about 500Mb of RAM.
# Installation
@@ -28,12 +28,12 @@ Follows the instructions below to install the pruning tables.
## Tables
Nissy needs to generate certain large tables to work. These tables are by default
generated the first time they are needed (e.g the first time you ask to solve a
-certain step) and then saved to a file. The following times nissy simply loads the
-corresponding file from the hard disk.
+certain step) and then saved to a file. Whenever these tables are needed again,
+nissy simply loads the corresponding file from the hard disk.
-The very large table for optimal solving can take some time to generate (about 20
-minutes on my fairly old but decent laptop, using 8 CPU threads). All other
-tables are much faster.
+The very large table for optimal solving can take some time to generate
+(about 1.5 hours on my fairly old but decent laptop, using 8 CPU threads).
+All other tables are much faster.
You can ask nissy to generate all the tables it will ever need with the gen
command. It is recommended to use more than one thread, if your CPU has them.
@@ -44,12 +44,13 @@ nissy gen -t 8
to generate all tables using 8 threads.
Alternatively, you can simply download all the tables and copy them into the
-correct folder (see manual page, ENVIRONMENT section).
-Choose one of the following:
- https://math.uni.lu/tronto/nissy/nissy-tables-full.zip
- https://math.uni.lu/tronto/nissy/nissy-tables-full.tar.gz
- https://math.uni.lu/tronto/nissy/nissy-tables-nohuge.zip
- https://math.uni.lu/tronto/nissy/nissy-tables-nohuge.tar.gz
-extract the archive and copy the tables folder into NISSIDATA (paste there
-the whole folder, not file by file). The "nohuge" files are much smaller and do not
-contain the huge pruning table for the optimal solver.
+correct folder (see manual page, ENVIRONMENT section). On UNIX operating
+systems this folder is either .nissy/tables in the user's home directory or
+$XDG_DATA_HOME/nissy/tables if the XDG variable is configured. On Windows
+it is the same directory as the nissy.exe executable file.
+Choose either (zip format)
+ https://math.uni.lu/tronto/nissy/nissy-tables-2.0.zip
+or (tar.gz format)
+ https://math.uni.lu/tronto/nissy/nissy-tables-2.0.tar.gz
+and extract the archive into the correct folder.
diff --git a/README.md b/README.md
@@ -21,17 +21,11 @@ solutions for EO/DR/HTR or similar substeps.
## Requirements
-** Warning: ** *This section is not up to date with the code. In nissy-2.0beta8
-or later the only way to get the table files is to generate them yourself.
-All but the huge table just requires a few minutes; the huge table for
-optimal solving can require a couple of hours. Use more than 1 thread
-if you can.*
-A full installation of Nissy requires a little more than 2Gb of space,
-of which 1.6Gb are occupied by the huge pruning table for fast optimal solving,
+A full installation of nissy requires about 3Gb of space,
+of which 2.3Gb are occupied by the huge pruning table for fast optimal solving,
and running it requires the same amount of RAM.
One can choose to never use this function and not to install the relative
-pruning table. There is an alternative (about 5 times slower)
+pruning table. There is an alternative (slower)
optimal solving function that uses about 500Mb of RAM.
## Installation
@@ -55,14 +49,14 @@ Follows the instructions below to install the pruning tables.
### Tables
Nissy needs to generate certain large tables to work. These tables are by default
generated the first time they are needed (e.g the first time you ask to solve a
-certain step) and then saved to a file. The following times Nissy simply loads the
-corresponding file from the hard disk.
+certain step) and then saved to a file. Whenever these tables are needed again,
+nissy simply loads the corresponding file from the hard disk.
-The very large table for optimal solving can take some time to generate (about 20
-minutes on my fairly old but decent laptop, using 8 CPU threads). All other
-tables are much faster.
+The very large table for optimal solving can take some time to generate
+(about 1.5 hours on my fairly old but decent laptop, using 8 CPU threads).
+All other tables are much faster.
-You can ask Nissy to generate all the tables it will ever need with the **gen**
+You can ask Nissy to generate all the tables it will ever need with the `gen`
command. It is recommended to use more than one thread, if your CPU has them.
For example, you can run:
@@ -73,17 +67,17 @@ nissy gen -t 8
to generate all tables using 8 threads.
Alternatively, you can simply download all the tables and copy them into the
-correct folder (see manual page, `ENVIRONMENT` section).
-Choose one of the following:
-| |.zip|.tar.gz|
-|Full (~720Mb)|[full.zip](https://math.uni.lu/tronto/nissy/nissy-tables-full.zip)|[full.tar.gz](https://math.uni.lu/tronto/nissy/nissy-tables-full.tar.gz)|
-|No huge table (~90Mb)|[nohuge.zip](https://math.uni.lu/tronto/nissy/nissy-tables-nohuge.zip)|[nohuge.tar.gz](https://math.uni.lu/tronto/nissy/nissy-tables-nohuge.tar.gz)|
-extract the archive and copy the tables folder into `NISSIDATA` (paste there
-the whole folder, not file by file). The "nohuge" files are much smaller and do not
-contain the huge pruning table for the optimal solver.
+correct folder (see manual page, `ENVIRONMENT` section). On UNIX operating
+systems this folder is either `.nissy/tables` in the user's home directory or
+`$XDG_DATA_HOME/nissy/tables` if the XDG variable is configured. On Windows
+it is the same directory as the nissy.exe executable file.
+Choose either the
+or the
+file (click the links to download) and
+extract them in the correct folder.
## Structure of the code
diff --git a/TODO.md b/TODO.md
@@ -3,8 +3,6 @@
This is a list of things that I would like to add or change at some point.
It's more of a personal reminder than anything else.
-**Things in bold: to do before 2.0 release**
## Commands
### Commands that are available in nissy 1.0, but not in this version (yet):
@@ -36,8 +34,8 @@ including e.g. solutions that were not shown because -c)
* Add EXAMPLES.md file
* webapp (cgi)
-* **Re-upload tables**
-* **fix README.md**
+* genptable: stop early if gone above base+3 (can be checked while generating)
+* installation: get ptables with curl or similar (on Windows what?)
* **fix examples in manpage**
## Technical stuff | {"url":"https://git.tronto.net/nissy-fmc/commit/9f3c7e7ea5d872fc017271df61691ea545049211.html","timestamp":"2024-11-14T17:18:12Z","content_type":"text/html","content_length":"14629","record_id":"<urn:uuid:d147f5f1-df0b-4f84-a2c2-783ab98b1d76>","cc-path":"CC-MAIN-2024-46/segments/1730477393980.94/warc/CC-MAIN-20241114162350-20241114192350-00398.warc.gz"} |
Egret Swarm Optimization
Script error: No such module "Draft topics". Script error: No such module "AfC topic".
A Snowy Egret
The Egret Swarm Optimization Algorithm (ESOA) is a meta-heuristic optimization algorithm that combines the predatory behavior of Snowy Egrets (Sit-And-Wait Strategy) and Great Egrets (Aggressive
Among the egret family, the Snowy Egret and the Great Egret are two species that differ considerably in their feeding behavior. The Snowy Egret applies the least energy-intensive sit-and-wait
strategy: it stands still and waits patiently, watching for prey until it appears and then darting to grab it with its beak. Snowy egrets that use the sit-and-wait strategy tend to retrieve
relatively steady gains with very low energy consumption. Great egrets, on the other hand, are aggressive, and once they have spotted their prey, they will chase it until it is caught. The aggressive
strategy is energy-intensive, but it also allows the Great Egret to potentially achieve higher returns. Inspired by the predatory behavior of snowy egrets and great egrets, an algorithm ESOA that
combines the advantageous characteristics of both is proposed.
Mathematical model[edit]
The ESOA consists of three main components: a sit-and-wait strategy, an aggressive strategy, and a discriminant condition. Each egret swarm can consist of n egret squads, each of which in turn
contains three egrets, with Egret A implementing the sit-and-wait strategy and Egret B and Egret C using the random walking and encircling mechanisms of the aggressive strategy, respectively.
Sit-And-Wait Strategy[edit]
The observation equation of i-th Egret A can be described as ${\displaystyle {\hat {y}}_{i}=A(\mathbf {x} _{i})}$, the retrieved real fitness of each iteration is ${\displaystyle y_{i}}$, then the
pseudo gradient of the observation equation is ${\displaystyle \mathbf {g} _{i},}$ hence the new position of Egret A is as below:
${\displaystyle \mathbf {x} _{a,i}=\mathbf {x} _{i}+\exp({-t/(0.1\cdot t_{max})})\cdot 0.1\cdot hop\cdot \mathbf {g} _{i}}$,
where ${\displaystyle t}$ is the current iteration, ${\displaystyle t_{max}}$ is the maximum iteration, ${\displaystyle hop}$ is the gap between the low bound and up bound of solution space.
Aggressive Strategy[edit]
Egret B applies random walking and the position update method is:
${\displaystyle \mathbf {x} _{b,i}=\mathbf {x} _{i}+\tan {(\mathbf {r} _{b,i})}\cdot hop/(1+t)}$,
where ${\displaystyle \mathbf {r} _{b,i}}$ is the stochastic values in ${\displaystyle (-\pi /2,\pi /2)}$.
Egret C uses encircling mechanisms,
{\displaystyle {\begin{aligned}\mathbf {D} _{h}&=\mathbf {x} _{ibest}-\mathbf {x} _{i},\\\mathbf {D} _{g}&=\mathbf {x} _{gbest}-\mathbf {x} _{i},\\\mathbf {x} _{c,i}&=(1-\mathbf {r} _{i}-\mathbf {r}
_{g})\cdot \mathbf {x} _{i}+\mathbf {r} _{h}\cdot \mathbf {D} _{h}+\mathbf {r} _{g}\cdot \mathbf {D} _{g},\end{aligned}}}
where ${\displaystyle \mathbf {x} _{ibest}}$ and ${\displaystyle \mathbf {x} _{gbest}}$ represent the best position of Egret Squad and Egret Swarm respectively. ${\displaystyle \mathbf {r} _{i}}$ and
${\displaystyle \mathbf {r} _{g}}$ is the random values in ${\displaystyle [0,1]}$.
Discriminant Condition[edit]
After each egret in the egret squad has calculated the updated position, it will jointly decide the updated position of the egret squad in the following form:
${\displaystyle \mathbf {x} _{s,i}=\left[\mathbf {x} _{a,i}\qquad \mathbf {x} _{b,i}\qquad \mathbf {x} _{c,i}\right]}$,
${\displaystyle \mathbf {y} _{s,i}=\left[y_{a,i}\qquad y_{b,i}\qquad y_{c,i}\right]}$,
${\displaystyle c_{i}=argmin(\mathbf {y} _{s,i})}$,
${\displaystyle \mathbf {x} _{i}={\begin{cases}\mathbf {x} _{s,i}|_{c_{i}}\quad if\quad \mathbf {y} _{s,i}|_{c_{i}}<y_{i}\quad or\quad r<0.3,\\\mathbf {x} _{i}\qquad \qquad \quad else\end{cases}}.}$
The egret squad compares the updated position and fitness of the three egrets with the fitness of the previous iteration, and adopts the update if one egret's updated position is better than that of
the previous iteration. If each egret's updated position is worse than the previous one, there is a 30% probability of adopting the solution with the best-updated position.
Source Code[edit]
Python Code：https://github.com/Knightsll/Egret_Swarm_Optimization_Algorithm
MATLAB Code：https://ww2.mathworks.cn/matlabcentral/fileexchange/115595-egret-swarm-optimization-algorithm-esoa
Website: https://knightsll.github.io/about
This article "Egret Swarm Optimization" is from Wikipedia. The list of its authors can be seen in its historical and/or the page Edithistory:Egret Swarm Optimization. Articles copied from Draft
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<i>N</i>−1 modal interactions of a three-degree-of-freedom system with cubic elastic nonlinearities
In this paper the N−1 nonlinear modal interactions that occur in a nonlinear three-degree-of-freedom lumped mass system, where N=3, are considered. The nonlinearity comes from springs with weakly
nonlinear cubic terms. Here, the case where all the natural frequencies of the underlying linear system are close (i.e. ωn1:ωn2:ωn3≈1:1:1) is considered. However, due to the symmetries of the system
under consideration, only N−1 modes interact. Depending on the sign and magnitude of the nonlinear stiffness parameters, the subsequent responses can be classified using backbone curves that
represent the resonances of the underlying undamped, unforced system. These backbone curves, which we estimate analytically, are then related to the forced response of the system around resonance in
the frequency domain. The forced responses are computed using the continuation software AUTO-07p. A comparison of the results gives insights into the multi-modal interactions and shows how the
frequency response of the system is related to those branches of the backbone curves that represent such interactions.
• 3-DoF nonlinear oscillator
• Backbone curve
• Nonlinear modal interaction
• Second-order normal form method
Dive into the research topics of 'N−1 modal interactions of a three-degree-of-freedom system with cubic elastic nonlinearities'. Together they form a unique fingerprint.
• Neild, S. A. (Principal Investigator)
1/02/13 → 31/07/18
Project: Research | {"url":"https://research-information.bris.ac.uk/en/publications/ini1-modal-interactions-of-a-three-degree-of-freedom-system-with-","timestamp":"2024-11-14T03:48:03Z","content_type":"text/html","content_length":"69103","record_id":"<urn:uuid:3d531976-af4c-4220-a7a2-aa06a3d52f67>","cc-path":"CC-MAIN-2024-46/segments/1730477028526.56/warc/CC-MAIN-20241114031054-20241114061054-00563.warc.gz"} |
March 2020
In the late 15th Century the Italian mathematician and Franciscan friar, Luca Pacioli, published a book on geometry, perspective, and achitecture entitled Divina Proportione — the title a reference
to the divine ratio. Having the misfortune of writing over five hundred years ago, he did not have access to LaTex or any kind of similar mathematical markup language, and indeed the typesetting had
to be done the old fashioned way. He did however enjoy the services of Leonardo da Vinci for producing the illustrations. Which is a considerable consolation. (You can browse an original copy online.
These illustrations were the only work published in Leonardo’s lifetime — his celebrated notebooks coming to light posthumously. The illustrations cover variations of the Platonic solids, and in what
would have been at the time an innovative approach to mathematical visualization, Leonardo not only drew them as solids, but also as hollowed out skeletons. This aptitude for conveying the three
dimensional on the page can also be seen in his famous and influential anatomy sketches from his notebooks.
Being, quite literally, a Renaissance man, mathematics figured among Leonardo’s many interests, often overlapping with his engineering, scientific, and artistic pursuits. Although it has been claimed
that Leonardo employed the golden ratio in his art, this seems unlikely. Leonardo documented his process and thinking in his notebooks, often expounding in great length, composing books that would go
unpublished, yet there is no mention of him claiming to employ the ratio in his composition.
We do however have extensive evidence for Leonardo’s obsessive hunt to square the circle:
He filled his notebooks with shaded drawings in which he overlapped two half-circles and then created triangles and rectangles that had the same area as the resulting crescents. Year after year,
he relentlessly pursued ways to create circular shapes with areas equivalent to triangles and rectangles, as if addicted to the game. Though he never gave the precise dates of any milestones he
reached when making a painting, he treated these geometric studies as if each little success was a moment in history worthy of a notarial record. One night he wrote momentously, “Having for a
long time searched to square the angle to two equal curves…now in the year 1509 on the eve of the Calends of May [April 30] I have found the solution at the 22nd hour on Sunday.
Leonardo da Vinci, The Biography — Walter Isaacson.
I think all of us mathematicians should take a great deal of comfort in how universal an experience it is to arrive at such false dawns. Kenneth Clarke has commented that the pages Leonardo devotes
to these mathematical efforts are essentially of no value — neither to the mathematician nor the art historian. It is worth bearing in mind that most mathematical notebooks are mostly full of ideas
that are wrong, incorrectly formulated, or badly expressed.
As the above discussion may suggest, I am currently in the middle of reading my way through Walter Isaacson’s biography of Leonardo da Vinci. Leonardo is a towering and influential cultural figure
that it is easy to look over how little you really know about him. Not only the man, but also his actual work. Certainly, you recognize a handful of his paintings on sight, and you may also be aware
that he designed unworkable helicopters and impractical war machines in his notebooks, but unless you have taken an art history course (I have not) you wouldn’t know why his artwork was really so
remarkable, or indeed appreciate the broad scope of what can be found in his notebooks.
You might believe a well cultured individual would have an appreciation for the works of the Renaissance master. That this hypothetical being could walk into the Louvre, have a gander at the Mona
Lisa, and they would experience a whole bunch of appreciation. But I don’t think this hypothetical “well cultured” individual exists. Indeed I would argue that a well cultured individual walking
around an art gallery is essentially going in to be overwhelmed. If you spend any time looking at the Renaissance art you actually find yourself deep inside the uncanny valley. This might seem
obvious, but we’re used to photographs so our sense of what is real is fundamentally adjusted.
All the techniques that Leonardo developed are available to artists today — perspective, understanding light and shadow, the sfumato painting techniques. And then there are all the modern tools and
art supplies. Then there is what is being depicted itself: scenes from biblical narratives dressed up in a Renaissance setting. So the scenes are weird, the setting is weird, the people look weird. I
want to reiterate that point. Looking at people in renaissance paintings is often like looking at computer generated special effects from ten or twenty years ago. The comparison is very apt. The
preoccupations of a CGI artist are actually very similar to what you’ll find Leonardo wrestling with in his notebooks.
It is not that there isn’t much to appreciate in these works of art — I’ve certainly enjoyed reading Isaacson’s biography. But I think the person who wanders around an art gallery with a proper sense
of appreciation is really just a certain kind of nerd. Like the rest of us. | {"url":"http://www.nobigons.com/2020/03/","timestamp":"2024-11-11T20:19:23Z","content_type":"text/html","content_length":"30675","record_id":"<urn:uuid:1d3bda2d-89b6-4c3d-b3f9-4e4ea1c987f9>","cc-path":"CC-MAIN-2024-46/segments/1730477028239.20/warc/CC-MAIN-20241111190758-20241111220758-00036.warc.gz"} |
How do you use substitution to find intercepts? | Socratic
How do you use substitution to find intercepts?
1 Answer
For linear equations:
• Substitute $0$ for $y$ and solve for $x$ to find the $x$-intercept.
• Substitute $0$ for $x$ and solve for $y$ to find the $y$-intercept.
Impact of this question
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Analytical Chemistry - Online Tutor, Practice Problems & Exam Prep
So in the past, we learned that our activity coefficient, which is represented by gamma, and ionic strength, which is represented by mu, of a solution could be closely and accurately related by using
the extended Debye-Hückel equation. Here, we'd say that logγ1+α/1α×√μ305 . When the size parameter of the ion, which is alpha, is unknown, we're going to use instead the Davies equation. Here,
because of the lack of a size parameter, this formula is most useful for monovalent ions. So ions with a charge of 1, plus or minus. Examples would be sodium or chloride ions. Now, you could also use
larger charges, but there tends to be greater deviation from a credible value, but you can still use larger charged ions as well.
Here, the equation is reformatted into logγ=-0.51z2√μ1+√μ-0.3×μ. Here we have our ionic strength values. They're increasing as we go down. Here, we have our charges, plus or minus 1, plus or minus 2,
plus or minus 3, and our activity coefficients. Notice that as your ionic strength is increasing, your activity coefficients are decreasing.
Now, we're going to say here from the Davies equations, all ions with the same magnitude in charge. So if 2 ions, one is plus 1 and the other one is minus 1 or plus 2 and minus 2, they will have the
same activity coefficient, which is what we're seeing here. Plus or minus, it really doesn't matter. Again, that's because of the lack of an alpha value. We can group things together and make our
list overall smaller. Traditionally, when we're using the extended Debye-Hückel equation, we segregate ions based on their charge as well as the value of their charge.
Now that we've seen this, attempt to do the practice question that's left here at the bottom of the page, where we're asked to determine what the activity coefficient is of calcium ion when we have
0.025 molar of calcium phosphate. Here, we don't have the size parameters, so we're not going to be able to use our typical extended Debye-Hückel equation. Try this question out. If you get stuck,
don't worry. Come back and see how I answer the same question. | {"url":"https://www.pearson.com/channels/analytical-chemistry/learn/jules/ch-12-advanced-topics-in-equilibrium/dependence-of-solubility-on-ph?chapterId=f5d9d19c","timestamp":"2024-11-09T09:45:22Z","content_type":"text/html","content_length":"300016","record_id":"<urn:uuid:221c7391-e282-4f13-9932-9032d68fdc97>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.75/warc/CC-MAIN-20241109085148-20241109115148-00566.warc.gz"} |
Bighorn sheep are beautiful wild animals found throughout the
western United States. Let x be the...
Bighorn sheep are beautiful wild animals found throughout the western United States. Let x be the...
Bighorn sheep are beautiful wild animals found throughout the western United States. Let x be the age of a bighorn sheep (in years), and let y be the mortality rate (percent that die) for this age
group. For example, x = 1, y = 14 means that 14% of the bighorn sheep between 1 and 2 years old died. A random sample of Arizona bighorn sheep gave the following information:
│x│1 │2 │3 │4 │5 │
Σx = 15; Σy = 88.3; Σx^2 = 55; Σy^2 = 1584.57; Σxy = 274.6
Find r.
(d) Test the claim that the population correlation coefficient is positive at the 1% level of significance. (Round your test statistic to three decimal places.)
t = | {"url":"https://justaaa.com/statistics-and-probability/525531-bighorn-sheep-are-beautiful-wild-animals-found","timestamp":"2024-11-07T21:58:14Z","content_type":"text/html","content_length":"42103","record_id":"<urn:uuid:15fe55a1-c247-4e3c-a507-ca68f22df7b9>","cc-path":"CC-MAIN-2024-46/segments/1730477028017.48/warc/CC-MAIN-20241107212632-20241108002632-00025.warc.gz"} |
Mathematical Biology & Bioinformatics | Volume 11 Issue 1 Year 2016
Number of Overlaps in Patterns Furletova E.I., Roytberg M.A.
Institute of Mathematical Problems of Biology, Russian Academy of Science, Pushchino, Moscow Region, Russia
Higher School of Economics, Moscow, Russia
Moscow Institute of Physics and Technology, Dolgoprudny, Moscow Region, Russia
Abstract. The aim of the paper is to estimate the number of overlaps in the given pattern. The pattern is a set of words of same length m in an alphabet A. We present theoretical and experimental
bounds for overlaps number in two types of patterns. Firstly, we considered random patterns which relate to uniform probability model, i.e. all letters in the alphabet and, correspondently, all words
of same length are equiprobable. We proved that the average number of overlaps P for random patterns consisting of n words of length m linearly depends on pattern size n and is independent of length
of pattern words. In performed computer experiments the ratio P/n ranged from 0.33 till 1.06; the theoretical evaluations of the ratio for the patterns do not exceed 1.67. The secondly, we studied
the patterns described by position weight matrices (PWM) from the data base HOCOMOCO and various cut-offs. For such patterns the ratio P/n in experiments ranged from 0.004 till 1, for most of the
patterns it is smaller then 0.1.
Key words: overlap, pattern, pattern occurrence in a sequence. | {"url":"https://matbio.org/article.php?lang=eng&id=265","timestamp":"2024-11-03T06:43:56Z","content_type":"text/html","content_length":"11062","record_id":"<urn:uuid:f75d548b-1660-4808-9ea5-08b197564ec4>","cc-path":"CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00862.warc.gz"} |
Exploring 12th Grade Math: Curriculum & Key Questions
Struggling to tackle 12th-grade math? Ever wondered how to ace calculus or decode complex numbers? As students embark on their final year of high school, the challenges of 12th-grade math can seem
daunting. Picture Sarah, wrestling with polynomials, while John grapples with trigonometry. In this comprehensive guide, we’ll unravel the mysteries of the 12th grade math curriculum and provide
solutions to common challenges faced by students like Sarah and John. From calculus to statistics, join us as we explore the world of 12th-grade math and equip you with the tools to conquer it!
Unlocking the Depths of 12th Grade Math Curriculum
As students approach their final year of high school, the 12th-grade math curriculum stands as a formidable challenge, encompassing a diverse array of topics designed to deepen understanding and
prepare them for future academic and professional endeavors. Let’s explore the key areas of study and delve into the tools and techniques essential for mastering each domain.
1. Polynomials: Unraveling Mathematical Mysteries
In the realm of polynomials, students encounter a treasure trove of concepts and techniques, including:
• Synthetic Division: A shortcut method for dividing polynomials, simplifying complex computations.
• Factoring Polynomials: Breaking down polynomials into simpler components to uncover their roots and structure.
• Finding Zeros of Polynomials: Identifying the values of x that make a polynomial equation equal to zero, crucial for solving equations and graphing functions.
• Descartes’ Rule of Signs: A tool for determining the possible number of positive and negative real roots of a polynomial.
• Rational Root Theorem: A strategy for identifying potential rational roots of a polynomial equation, aiding in the process of factoring.
• Pascal’s Triangle and Binomial Theorem: Tools for expanding binomial expressions and exploring the coefficients of polynomial expansions.
2. Trigonometry: Navigating Angles and Relationships
Trigonometry opens a window into the world of angles, relationships, and geometric phenomena. Key topics include:
• Degrees and Radians: Units of measurement for angles, with radians offering a more natural approach for calculus and advanced mathematics.
• Coterminal and Reference Angles: Concepts essential for simplifying trigonometric expressions and solving equations.
• Right Triangles Trigonometry: Applying trigonometric ratios to solve problems involving right triangles.
• Laws of Sines and Cosines: Powerful tools for solving triangles, whether they’re right, oblique, or scalene.
• Unit Circle and Trigonometric Functions: Fundamental concepts for understanding trigonometric identities and graphing trigonometric functions.
• Inverse Trigonometric Functions: Unlocking the ability to find angles given trigonometric ratios, crucial for solving real-world problems.
4. Systems of Inequalities: Balancing Equations and Inequalities
In the realm of inequalities, students explore systems and optimization techniques, including:
• Graphical Representation: Visualizing systems of inequalities on a coordinate plane to identify feasible regions and optimal solutions.
• Linear Programming: Applying mathematical techniques to optimize the allocation of resources and maximize or minimize a given objective function.
5. Calculus: Exploring Limits, Derivatives, and Integrals
Calculus serves as the cornerstone of mathematical analysis, with topics including:
• Limits and Continuity: Understanding the behavior of functions as they approach specific values, laying the groundwork for differentiation and integration.
• Differentiation and Implicit Differentiation: Techniques for finding rates of change, tangent lines, and critical points of functions.
• Applications of Derivatives: From related rates to optimization, derivatives provide powerful tools for modeling and analyzing real-world phenomena.
• Integration and the Fundamental Theorem of Calculus: Uniting the concepts of accumulation and differentiation to evaluate areas, volumes, and other quantities.
6. Probability and Statistics: Navigating Uncertainty with Data
Probability and statistics offer insights into uncertainty, variability, and patterns in data. Key topics include:
• Permutations, Combinations, and Probability Rules: Techniques for counting and calculating probabilities in various scenarios.
• Probability Distributions: Describing the likelihood of different outcomes in a random experiment, from discrete to continuous distributions.
• Hypothesis Testing and Confidence Intervals: Statistical tools for making inferences and drawing conclusions from data, with applications in research and decision-making.
• Correlation and Regression: Exploring relationships between variables and making predictions based on observed data.
7. Complex Numbers: Exploring Mathematical Mysteries Beyond the Real
Complex numbers extend the realm of mathematics into the imaginary domain, with topics including:
• Operations with Complex Numbers: Adding, subtracting, multiplying, and dividing complex numbers to unlock their mathematical properties.
• Polar Form and De Moivre’s Theorem: Representing complex numbers in polar form and using De Moivre’s Theorem to find powers and roots.
• Applications of Complex Numbers: From electrical engineering to quantum mechanics, complex numbers find applications in diverse fields, unlocking new insights and solutions.
As students journey through their final year of high school, mastering 12th-grade math requires more than just understanding concepts—it demands application and problem-solving prowess. In this
section, we present 25 diverse math problems, spanning key topics of the 12th-grade math curriculum, accompanied by detailed solutions to help students hone their skills and build confidence in
tackling mathematical challenges.
1. Polynomials: Solve the polynomial equation 2x^3 – 5x^2 + 3x – 1 = 0.
Answer: x = 1.
2. Trigonometry: Find the exact value of sin(π/3).
Answer: sin(π/3) = √3/2.
3. Calculus: Find the derivative of f(x) = cos(2x) + 3x^2.
Answer: f'(x) = -2sin(2x) + 6x.
4. Probability and Statistics: If two six-sided dice are rolled, what is the probability of rolling a sum of 7?
Answer: Probability = 1/6.
5. Complex Numbers: Add the complex numbers 3 + 2i and 1 – 4i.
Answer: 4 – 2i.
6. Polynomials: Factor the polynomial x^3 – 8.
Answer: x^3 – 8 = (x – 2)(x^2 + 2x + 4).
7. Trigonometry: Calculate the value of tan(π/4).
Answer: tan(π/4) = 1.
8. Calculus: Evaluate the integral ∫(from 0 to 2) x^2 dx.
Answer: 8/3.
9. Probability and Statistics: If a fair coin is flipped three times, what is the probability of getting exactly two heads?
Answer: Probability = 3/8.
10. Complex Numbers: Find the square of the complex number 2 + 3i.
Answer: 13i.
11. Polynomials: Determine the degree and leading coefficient of the
polynomial 3x^4 – 2x^2 + 5x + 1.
Answer: Degree = 4, Leading coefficient = 3.
12. Trigonometry: Solve for x in the equation cos(x) = 1/2.
Answer: x = π/3, 5π/3.
13. Calculus: Find the derivative of f(x) = e^x sin(x).
Answer: f'(x) = e^x(sin(x) + cos(x)).
14. Probability and Statistics: If a standard six-sided die is rolled, what is the probability of rolling an even number?
Answer: Probability = 1/2.
15. Complex Numbers: Find the product of the complex numbers 2 + i and 1 – 3i.
Answer: 5 – 5i.
16. Polynomials: Factor the polynomial x^2 – 4x + 4.
Answer: x^2 – 4x + 4 = (x – 2)^2.
17. Trigonometry: Calculate the value of cos(π/6).
Answer: cos(π/6) = √3/2.
18. Systems of Inequalities: Solve the system of inequalities: 2x + y ≤ 4 and x – 3y > 6.
Answer: x ≤ 2, y ≥ -2.
19. Calculus: Evaluate the integral ∫(from 1 to 4) 3x^2 dx.
Answer: 183/4.
20. Probability and Statistics: If a deck of 52 cards is shuffled, what is the probability of drawing a red card?
Answer: Probability = 1/2.
21. Complex Numbers: Find the conjugate of the complex number 5 – 2i.
Answer: 5 + 2i.
22. Polynomials: Determine the roots of the quadratic equation x^2 – 6x + 9 = 0.
Answer: x = 3.
23. Trigonometry: Calculate the value of sec(π/4).
Answer: sec(π/4) = √2.
24. Calculus: Find the second derivative of f(x) = 4x^3 – 6x^2 + 2x – 8.
Answer: f”(x) = 24x – 12.
25. Probability and Statistics: If a fair six-sided die is rolled twice, what is the probability of getting two even numbers?
Answer: Probability = 1/9.
Best Math Course for 12th Grade Students
Embark on a transformative journey through 12th-grade math with the WuKong Math Advanced Course! Their meticulously designed program ignites a passion for math while enhancing critical thinking
skills. Here’s why their course stands out:
Discovering the maths whiz in every child,
that’s what we do.
Suitable for students worldwide, from grades 1 to 12.
Get started free!
Course Information:
• Duration and Frequency: Enjoy sessions lasting 60 to 90 minutes, conducted weekly to seamlessly fit into busy schedules.
• Class Size: Tailored for optimal learning, classes accommodate 1 to 28 students, ensuring personalized attention and lively interaction.
• Recommended Age: Perfect for young minds aged 6 to 18, offering a comprehensive learning journey for students at any level.
Course Features:
• Comprehensive Syllabus: Aligned with school curricula and international competitions, preparing students to excel in exams and contests.
• Interactive Learning: Dive into captivating story themes and animations that bring math to life, making learning both enjoyable and effective.
• Tailored Practice Assignments: Engage with carefully selected problems that offer real-world applications, aiding in mastering concepts and problem-solving prowess.
• Unique Teaching Method: The “6A teaching method” intertwines inquiry-based learning with top-tier teaching expertise, ensuring a profound understanding and appreciation for math.
FAQs for 12th Grade Math:
Q1. Is 12th-grade math necessary for non-STEM majors?
While not always a requirement, 12th-grade math fosters critical thinking and problem-solving skills valuable in various disciplines. It enhances logical reasoning and analytical abilities,
beneficial in fields like business, social sciences, and even arts.
Q2. How can I overcome challenges in understanding 12th-grade math concepts?
Seeking help from teachers, tutors, or online resources can clarify difficult concepts. Breaking down problems into smaller steps, practicing regularly, and seeking alternative explanations can also
aid in comprehension and mastery.
Q3. Are there career paths that require proficiency in 12th-grade math?
Yes, careers in fields such as engineering, finance, data analysis, and computer science heavily rely on mathematical concepts taught in 12th grade. Strong math skills open doors to lucrative and
intellectually stimulating professions.
In this article, we’ve explored how to tackle the challenges of 12th-grade math. From polynomials to calculus, we’ve covered key concepts and provided solutions to common problems. We’ve also
highlighted the value of the WuKong Math Advanced Course, offering engaging lessons and personalized attention to help students excel. Enroll today and watch your child’s math skills soar!
Discovering the maths whiz in every child,
that’s what we do.
Suitable for students worldwide, from grades 1 to 12.
Get started free!
Delvair holds a degree in Physics from the Federal University of Maranhão, Brazil. With over six years of experience, she specializes in teaching mathematics, with a particular emphasis on Math
Kangaroo competitions. She firmly believes that education is the cornerstone of society’s future. Additionally, she holds the conviction that every child can learn given the right environment and
guidance. In her spare time, she enjoys singing and tending to her plants. | {"url":"https://www.wukongsch.com/blog/12th-grade-math-post-34195/","timestamp":"2024-11-10T17:58:51Z","content_type":"text/html","content_length":"131554","record_id":"<urn:uuid:e0a45446-a418-4c9e-bc7d-e1a260de3200>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.61/warc/CC-MAIN-20241110170046-20241110200046-00054.warc.gz"} |
Archimedes Principle and Buoyancy: Detail Explanation, Applications and Examples
Archimedes Principle basically refers to the upward force exerted by water or any type of fluid when an object is immersed in it. For better understanding, we should first understand the basic
terminology like Thrust, Upthrust, Flotation, Buoyancy, Pressure, etc. In this article, we will provide complete information regarding the Archimedes Principle along with its basics and applications.
Before understanding the Archimedes Principle we will first understand the core concept and terminology used in these principles.
Thrust and Up Thrust
Thrust is nothing but a force applied perpendicular in direction on the surface. Basically, Thrust is a type of force which have direction, as the force having direction comes under vector quantity.
So, we can measure the Thrust in Newton. When the force is applied in the body in an upward direction those forces are called Up Thrust.
Pressure refers to the amount of force applied to an object at a particular unit area. So in simple terms, the force acting on that particular area is calculated using Pressure. The unit at which
Pressure is measured is Newton per Meter square or Pascal. The formula of the Pressure is Thrust/ Area.
Application of Pressure
Here we explain the Pressure’s Application with real-life examples
• Army Tanks: Everyone knows that Army Tanks have tonnes of weight therefore they may exert a large force on the ground. But on applying the concepts of pressure their wheels are designed in such a
way that they are wide from the bottom. So because of provides a high surface area at the bottom, the tank does not exert that much force according to its weight.
• School Bags: The strips of the School Bags are designed by applying this concept so that the weight of the school bags does not hurt the student’s shoulder. So by providing a wide surface area of
a strip on the bag, it exerts less pressure.
• Needle’s Tip: The tips of the needle are formed sharp so as to exert high pressure on the small surface area. so using this application will help the needle pierce the clothes easily.
Buoyancy: Buoyancy is the phenomenon by which water or any fluid exerts upward force (Upthrust) when an object is immersed in it. The upward force applied to the object is called buoyant force. For
better understanding we can say that Buoyancy is nothing but an Upthrust exerted by the liquid. For example, when a cork is immersed in the water 2/3 of it parts help in float. when we apply force to
sink at the bottom, then with the phenomenon of buoyancy it will automatically come to the surface of the water.
Factors Affecting Buoyancy
The upward force basically depends on two concepts density of the fluid at which the body is submerged and the volume of the body submerged.
• Surface Area: Increasing the surface area of the body that is immersed in the fluid the upward force is also increased. Explaining the concept with a real-life example if you immerse a small
piece of wood you can easily sink it inside the water but if you immerse the large plank of wood and apply force to sink in the water then you have to apply more force greater than the upthrust
exert by the water. Hence it proves that surfaces matter most in buoyancy.
• Fluid Density: Different fluids have different densities so the fluid having greater density exerts greater Upthrust in comparison to a fluid having less density.
Cause of Buoyancy
The main cause of Buoyancy is because of the molecules present in the fluid. When the fluid is at rest the molecules of the liquid exert a perpendicular force when they come in contact with the body.
For example when the fluid is in any container then at the surface of the container it exerts the perpendicular force. The pressure or the upward force you feel is because of the collision of the
molecules that takes place with the surroundings when the object is immersed within the liquid.
The force of Buoyancy is the same in all directions but different at increasing the depth. The buoyant force is the same in all directions because of the same no of molecules present in that
particular depth which led to equal force exerted by the molecules at that place. The buoyant force is different at depth because of the increase in the molecules in the depth which led to more
collisions between the molecules at that place hence exerting more force.
Effects of Buoyancy on Different Weight
Here we understand the concept of Buoyancy of Different weight objects. For Which (W) stands for the downward force applied by the object immersed in the fluid and (U) stands for the upward force
exerted by the water. In this floatation of the object depends upon the net force between U and W.
• Case 1: When W is Greater than U which means when the downward force exerted by the object is greater than the upward force exerted by the liquid then the object is going to sink. As there is no
balancing of the force situation takes place so the greater force suppresses the smaller force and hence goes in the direction of the greater force so the object is going to sink into the liquid.
• Case 2: When U is Greater than W which means when the Upthrust or upward force exerted by the liquid is greater than the Downward force exerted by the object then the object is going to float in
the liquid. As there is no balancing situation taking place so the greater force suppresses the smaller force and hence goes in the direction of the greater force so the object is going to float
in the liquid.
• Case 3: When U is equal to W which means the force exerted by the object is equal to the upward force exerted by the liquid hence object is in the equilibrium situation. In a balancing situation,
both forces cancel out each other so there is no motion of the object taking place.
Archimedes Principle
Archimedes’s Principle basically works on the phenomenon of Buoyancy and observes that “The upward force experienced by the object when it is placed in the liquid is equal to the weight of the fluid
displaced by the object.” It is like when the object is placed it displaces the weight of the fluid and exerts the upward force. so the fluid displaced and the upward force exerted are equal.
The formula of the upward buoyant force is Fb = ρ x g x V.
Where ρ denotes the density of the liquid, g denotes the acceleration caused by gravity and V denotes the volume of the liquid.
Archimedes Principle Application
• Submarine: With the help of Archimedes Principles the submarine is designed in such a way that it helps to make submarine at any position underwater. There is an important component used in
Submarines is the Ballast Tank it allows the entering and exit of water so that we can place the submarine in the position of which we want.
• Hydrometer: Hydrometer works on the principle of Archimedes, it is the instrument used to measure the relative density. It consists of lead shots which help it in floatation so the deeper the
hydrometer sinks the less the density of fluids.
• Geology: The principles help in understanding the sedimentation profile by measuring the denser particle float on the liquid top.
Read More: All the Topics Covered in Biodiversity.
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Direct Variation Q&As - Algebra | HIX Tutor
Direct Variation
Direct variation stands as a fundamental concept in mathematics, highlighting the straightforward relationship between two variables. In this mathematical model, as one quantity changes, the other
follows suit in a consistent and proportional manner. The simplicity of direct variation lies in its linear nature, where the graph of such relationships forms a straight line passing through the
origin. Understanding direct variation is crucial in various fields, providing a concise and predictable framework to analyze and interpret real-world scenarios, making it an indispensable tool in
mathematical applications and problem-solving. | {"url":"https://tutor.hix.ai/subject/algebra/direct-variation","timestamp":"2024-11-07T15:23:04Z","content_type":"text/html","content_length":"563334","record_id":"<urn:uuid:7867a547-f91c-4009-9bed-dbc4ec9ec130>","cc-path":"CC-MAIN-2024-46/segments/1730477028000.52/warc/CC-MAIN-20241107150153-20241107180153-00757.warc.gz"} |
NEXTAFTER(3) Linux Programmer's Manual NEXTAFTER(3)
nextafter, nextafterf, nextafterl, nexttoward, nexttowardf, nexttowardl
- floating-point number manipulation
#include <math.h>
double nextafter(double x, double y);
float nextafterf(float x, float y);
long double nextafterl(long double x, long double y);
double nexttoward(double x, long double y);
float nexttowardf(float x, long double y);
long double nexttowardl(long double x, long double y);
Link with -lm.
Feature Test Macro Requirements for glibc (see feature_test_macros(7)):
_ISOC99_SOURCE || _POSIX_C_SOURCE >= 200112L
|| _XOPEN_SOURCE >= 500
|| /* Since glibc 2.19: */ _DEFAULT_SOURCE
|| /* Glibc versions <= 2.19: */ _BSD_SOURCE || _SVID_SOURCE
nextafterf(), nextafterl():
_ISOC99_SOURCE || _POSIX_C_SOURCE >= 200112L
|| /* Since glibc 2.19: */ _DEFAULT_SOURCE
|| /* Glibc versions <= 2.19: */ _BSD_SOURCE || _SVID_SOURCE
nexttoward(), nexttowardf(), nexttowardl():
_XOPEN_SOURCE >= 600 || _ISOC99_SOURCE ||
_POSIX_C_SOURCE >= 200112L
The nextafter(), nextafterf(), and nextafterl() functions return the
next representable floating-point value following x in the direction of
y. If y is less than x, these functions will return the largest repre-
sentable number less than x.
If x equals y, the functions return y.
The nexttoward(), nexttowardf(), and nexttowardl() functions do the
same as the corresponding nextafter() functions, except that they have
a long double second argument.
On success, these functions return the next representable floating-
point value after x in the direction of y.
If x equals y, then y (cast to the same type as x) is returned.
If x or y is a NaN, a NaN is returned.
If x is finite, and the result would overflow, a range error occurs,
and the functions return HUGE_VAL, HUGE_VALF, or HUGE_VALL, respec-
tively, with the correct mathematical sign.
If x is not equal to y, and the correct function result would be sub-
normal, zero, or underflow, a range error occurs, and either the cor-
rect value (if it can be represented), or 0.0, is returned.
See math_error(7) for information on how to determine whether an error
has occurred when calling these functions.
The following errors can occur:
Range error: result overflow
An overflow floating-point exception (FE_OVERFLOW) is raised.
Range error: result is subnormal or underflows
An underflow floating-point exception (FE_UNDERFLOW) is raised.
These functions do not set errno.
For an explanation of the terms used in this section, see at-
|Interface | Attribute | Value |
|nextafter(), nextafterf(), | Thread safety | MT-Safe |
|nextafterl(), nexttoward(), | | |
|nexttowardf(), nexttowardl() | | |
C99, POSIX.1-2001, POSIX.1-2008. This function is defined in IEC 559
(and the appendix with recommended functions in IEEE 754/IEEE 854).
In glibc version 2.5 and earlier, these functions do not raise an un-
derflow floating-point (FE_UNDERFLOW) exception when an underflow oc-
This page is part of release 5.05 of the Linux man-pages project. A
description of the project, information about reporting bugs, and the
latest version of this page, can be found at
GNU 2017-09-15 NEXTAFTER(3)
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FREE Official GRE Practice Test for Students | MCQTUBE
FREE Official GRE Practice Test
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NDA, SNAP Test, Sub Inspector of Police, TNPSC, all Government Exams, and other Competitive Examinations, etc. We created all the competitive exam MCQs into several small posts on our website for
your convenience.
You will get their respective links in the related posts section provided below.
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GRE FREE Mock Test for Students
The greatest number of five digits which is divisible by 2356
(a) 99142
(b) 98232
(c) 98952
(d) None
Find ‘k’ so that the number ‘3572k45’ is divisible by 45.
(a) 2
(b) 1
(c) 7
(d) 8
(e) None of these
Find ‘k’ so that “981k456” is divisible by 72.
(a) 4
(b) 6
(c) 0
(d) 3
(e) Any value
Find the value of ‘k’ to make 7456k87 divisible by 11.
(a) 5
(b) 7
(c) -1
(d) 1
(e) None of these
Find “k” so that the number ‘589645k’ is divisible by 24.
(a) 3
(b) 6
(c) 7
(d) 4
(e) None of these
4, 8, 10, 16, and 18, are numbers
(a) even
(b) odd
(c) prime
(d) composite
1, 2, 3,…, 99, 100 are … numbers
(a) natural
(b) even
(c) odd
(d) prime
3, 5, 7, 11, 13, 17, 19,.are …… numbers
(a) even
(b) prime
(c) odd
(d) composite
The value of k when 451k025k is divisible by 45 is
(a) 0
(b) 5
(c) Any value
(d) No value
The least 3-digit number subtracted from the greatest number of Six-digits will result in:
(a) 999899
(b) 99999
(c) 10000
(d) 10999
The sum of all prime numbers between 60 & 96 is :
(a) 610
(b) 523
(c) 460
(d) 373
The numbers (k), (k+ 16), and (k+ 14) are all primes if k is equal to
(a) 19
(b) 5
(c) 29
(d) None
Which one of the following numbers can be represented as non-terminating, repeating decimals?
(a) 17/25
(b) 21/32
(c) 56/22
(d) 456/25
The largest number of four digits exactly divisible by 176
(a) 9856
(b) 9988
(c) 9888
(d) 9944
The least number of five digits exactly divisible by 2280
(a) 10140
(b) 10230
(c) 11400
(d) 10012
We covered all the free official gre practice test above in this post for free so that you can practice well for the exam.
Check out the latest MCQ content by visiting our mcqtube website homepage.
Also, check out:
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Annales Academię Scientiarum Fennicę
Volumen 37, 2012, 277-284
Greg Markowsky
Monash University, Department of Mathematical Sciences
Victoria 3800, Australia; gmarkowsky 'at' gmail.com
Abstract. We study the question of whether for a given nonconstant holomorphic function f there is a pair of domains U, V such that f is the only nonconstant holomorphic function with f(U) \subseteq
V. We show existence of such a pair for several classes of rational functions, namely maps of degree 1 and 2 as well as arbitrary degree Blaschke products. We give explicit constructions of U and V,
where possible. Consequences for the generalized Kobayashi and Carathéodory metrics are also presented.
2010 Mathematics Subject Classification: Primary 30E99.
Key words: Complex variables, rational functions, generalized Kobayashi metric, generalized Caratheodory metric.
Reference to this article: G. Markowsky: A rigidity theorem for special families of rational functions. Ann. Acad. Sci. Fenn. Math. 37 (2012), 277-284.
Copyright © 2012 by Academia Scientiarum Fennica | {"url":"https://www.acadsci.fi/mathematica/Vol37/Markowsky.html","timestamp":"2024-11-12T19:28:10Z","content_type":"text/html","content_length":"1910","record_id":"<urn:uuid:ace90a0e-f3b6-469f-9610-9f57df3b3625>","cc-path":"CC-MAIN-2024-46/segments/1730477028279.73/warc/CC-MAIN-20241112180608-20241112210608-00791.warc.gz"} |
“Finite Field Arithmetic.” Chapter 12A: Karatsuba Redux. (Part 1 of 2)
This article is part of a series of hands-on tutorials introducing FFA, or the Finite Field Arithmetic library. FFA differs from the typical "Open Sores" abomination, in that -- rather than trusting
the author blindly with their lives -- prospective users are expected to read and fully understand every single line. In exactly the same manner that you would understand and pack your own parachute.
The reader will assemble and test a working FFA with his own hands, and at the same time grasp the purpose of each moving part therein.
• Chapter 12A: Karatsuba Redux. (Part 1 of 2)
You will need:
• A Keccak-based VTron (for this and all subsequent chapters.)
• All of the materials from Chapters 1 - 11. (They have been re-ground for the new VTron format; please re-download here.)
• There is no vpatch in Chapter 12A.
On account of the substantial heft of this chapter, I have cut it into two parts, 12A and 12B; you are presently reading 12A, which consists strictly of the benchmarks and detailed analysis of the
Karatsuba method presented earlier. 12B will appear here in the course of the next several days, preceding Ch. 13.
First things first:
As noted earlier in Chapter 11:
• Reader apeloyee observed that the modular exponentiation operation is a poor benchmark for multiplication speed, on account of the fact that the current algorithm spends the vast majority of its
CPU time inside the Knuth division routine.
• Reader ave1 carefully analyzed the executables generated by AdaCore's x86-64 GNAT, and showed that this compiler is prone to ignore Inline pragmas unless they are specified in a particular way.
• ave1 also produced a fully-self-building, retargetable, fully-static, and glibc-free GNAT.
And so I have carried out a benchmark battery strictly on Multiplication -- naturally, on the standard test machine used for all previous benchmarks -- across a meaningful range of FFA bitnesses
(i.e. integers large enough not to fall through the resolution of the timer), across all of the multiplier routine variants offered in chapters 9 - 11:
Or, for those who prefer the raw numbers to the logarithmic plot,
│ Cost of 1000 Multiplication Operations (sec): │
│FFA Bitness│Ch.9 "Soft" Comba│Ch.9 "Iron" Comba│Ch.10 Karatsuba (on Iron Comba)│Ch.11 Karatsuba (on Iron Comba) with Inlining │
│2048 │0.120 │0.019 │0.015 │0.012 │
│4096 │0.480 │0.074 │0.046 │0.035 │
│8192 │1.911 │0.295 │0.140 │0.106 │
│16384 │7.638 │1.170 │0.427 │0.328 │
The first item to discuss is that the introduction of ave1's new Musl GNAT had no measurable effect on the performance of the compiled FFA code.
Therefore the above benchmark does not list separate measurements for the new and old GNATs, as they turned out to build executables which perform identically within the margin of error given by the
timer resolution.
This is not a particularly surprising discovery, given as the new GNAT is largely the same as the GNAT I had been using previously, but for the fact that it builds on Musl rather than rotten old
glibc. Given as FFA spends no substantial time in libc, this observation is not astonishing.
At the same time I will note that all subsequent FFA tests will be carried out on the new GNAT, and the reader is advised to build himself a working copy.
However, ave1's inlining fix does have a measurable effect on performance: this is reflected in the smaller number of CPU cycles eaten by the FFA of Ch.11 compared to that of Ch.10.
The next item to discuss is the fact that the use of Mul_HalfWord_Soft in Word x Word multiplication imposes a substantial performance penalty.
Reader apeloyee was indeed correct: the slowdown was obscured in the modular exponentiation benchmark of Chapter 9 by the predominance of the modular reduction's cost over that of multiplication's.
(This will be discussed in detail in Ch. 12B.)
On machines having a constant-time MUL instruction (e.g. AMD64) the use of "soft" Word x Word base case multiplication is an unnecessary sacrifice, and therefore the proclamation given in Chapter 9
is hereby withdrawn: people stuck with broken CPU architectures will be responsible for enabling the necessary workaround with their own hands, rather than imposing its cost on all FFA users.
At the conclusion of the FFA series, we will discuss a clean (i.e. via V-branches) means of offering the use of "soft" Word x Word multiplication on machines which require it, as well as a simple
litmus test for the presence of a broken (i.e. one having a non-constant-time iron multiplier) CPU. But until then, all subsequent FFA benchmarks published here will presume the use of "Iron" Word x
Word multiplication.
The last, and by far least surprising, observation concerning the benchmark is that Karatsuba multiplication is indeed faster (the exact runtime complexity of it -- I will leave as an exercise for
the reader) for sufficiently large integers, than the Ch. 9 O(N^2) Comba method (the latter remains in use, however, as the base case of Karatsuba.)
At this time we will walk through the mechanics of our Karatsuba multiplier, so as to cement in the reader's head the correctness of the routine, and lay groundwork for the optimization which is
introduced in Ch. 12B.
Let's revisit the Karatsuba routine, as given in Chapter 11:
-- ...
-- Karatsuba's Multiplier. (CAUTION: UNBUFFERED)
procedure Mul_Karatsuba(X : in FZ;
Y : in FZ;
XY : out FZ) is
-- L is the wordness of a multiplicand. Guaranteed to be a power of two.
L : constant Word_Count := X'Length;
-- An 'LSeg' is the same length as either multiplicand.
subtype LSeg is FZ(1 .. L);
-- K is HALF of the length of a multiplicand.
K : constant Word_Index := L / 2;
-- A 'KSeg' is the same length as HALF of a multiplicand.
subtype KSeg is FZ(1 .. K);
-- The three L-sized variables of the product equation, i.e.:
-- XY = LL + 2^(K*Bitness)(LL + HH + (-1^DD_Sub)*DD) + 2^(2*K*Bitness)HH
LL, DD, HH : LSeg;
-- K-sized terms of Dx * Dy = DD
Dx, Dy : KSeg; -- Dx = abs(XLo - XHi) , Dy = abs(YLo - YHi)
-- Subtraction borrows, signs of (XL - XH) and (YL - YH),
Cx, Cy : WBool; -- so that we can calculate (-1^DD_Sub)
-- Bottom and Top K-sized halves of the multiplicand X.
XLo : KSeg renames X( X'First .. X'Last - K );
XHi : KSeg renames X( X'First + K .. X'Last );
-- Bottom and Top K-sized halves of the multiplicand Y.
YLo : KSeg renames Y( Y'First .. Y'Last - K );
YHi : KSeg renames Y( Y'First + K .. Y'Last );
-- L-sized middle segment of the product XY (+/- K from the midpoint).
XYMid : LSeg renames XY( XY'First + K .. XY'Last - K );
-- Bottom and Top L-sized halves of the product XY.
XYLo : LSeg renames XY( XY'First .. XY'Last - L );
XYHi : LSeg renames XY( XY'First + L .. XY'Last );
-- Topmost K-sized quarter segment of the product XY, or 'tail'
XYHiHi : KSeg renames XYHi( XYHi'First + K .. XYHi'Last );
-- Whether the DD term is being subtracted.
DD_Sub : WBool;
-- Carry from individual term additions.
C : WBool;
-- Tail-Carry accumulator, for the final ripple
TC : Word;
-- Recurse: LL := XL * YL
FZ_Multiply_Unbuffered(XLo, YLo, LL);
-- Recurse: HH := XH * YH
FZ_Multiply_Unbuffered(XHi, YHi, HH);
-- Dx := |XL - XH| , Cx := Borrow (i.e. 1 iff XL < XH)
FZ_Sub_Abs(X => XLo, Y => XHi, Difference => Dx, Underflow => Cx);
-- Dy := |YL - YH| , Cy := Borrow (i.e. 1 iff YL < YH)
FZ_Sub_Abs(X => YLo, Y => YHi, Difference => Dy, Underflow => Cy);
-- Recurse: DD := Dx * Dy
FZ_Multiply_Unbuffered(Dx, Dy, DD);
-- Whether (XL - XH)(YL - YH) is positive, and so DD must be subtracted:
DD_Sub := 1 - (Cx xor Cy);
-- XY := LL + 2^(2 * K * Bitness) * HH
XYLo := LL;
XYHi := HH;
-- XY += 2^(K * Bitness) * HH, but carry goes in Tail Carry accum.
FZ_Add_D(X => XYMid, Y => HH, Overflow => TC);
-- XY += 2^(K * Bitness) * LL, ...
FZ_Add_D(X => XYMid, Y => LL, Overflow => C);
-- ... but the carry goes into the Tail Carry accumulator.
TC := TC + C;
-- XY += 2^(K * Bitness) * (-1^DD_Sub) * DD
FZ_Not_Cond_D(N => DD, Cond => DD_Sub); -- invert DD if 2s-complementing
FZ_Add_D(OF_In => DD_Sub, -- ... and then must increment
X => XYMid,
Y => DD,
Overflow => C); -- carry will go in Tail Carry accumulator
-- Compute the final Tail Carry for the ripple
TC := TC + C - DD_Sub;
-- Barring a cosmic ray, 0 < = TC <= 2 .
pragma Assert(TC <= 2);
-- Ripple the Tail Carry into the tail.
FZ_Add_D_W(X => XYHiHi, W => TC, Overflow => C);
-- Barring a cosmic ray, the tail ripple will NOT overflow.
pragma Assert(C = 0);
end Mul_Karatsuba;
-- CAUTION: Inlining prohibited for Mul_Karatsuba !
-- Multiplier. (CAUTION: UNBUFFERED)
procedure FZ_Multiply_Unbuffered(X : in FZ;
Y : in FZ;
XY : out FZ) is
-- The length of either multiplicand
L : constant Word_Count := X'Length;
if L < = Karatsuba_Thresh then
-- Base case:
FZ_Mul_Comba(X, Y, XY);
-- Recursive case:
Mul_Karatsuba(X, Y, XY);
end if;
end FZ_Multiply_Unbuffered;
And now let's step through the whole thing, in light of the arithmetical overview given in Chapter 10.
Recall that we derived the following equivalences for Karatsuba's method:
LL = X[Lo]Y[Lo]
HH = X[Hi]Y[Hi]
Dx = |X[Lo] - X[Hi]|
Dy = |Y[Lo] - Y[Hi]|
DD = Dx × Dy
DD[Sub] = C[X] XNOR C[Y]
XY = LL + 2^b(LL + HH + (-1^DD[Sub])DD) + 2^2bHH
... where X[Lo] and X[Hi] are the bottom and top halves of the multiplicand X, respectively; Y[Lo] and Y[Hi] -- of multiplicand Y; C[X] is the subtraction "borrow" resulting from the computation of
Dx; and C[Y] is same from the computation of Dy.
... and showed that the operation can be represented in the following "physical" form (junior bits of registers on left hand side, senior -- on right hand) :
LL HH TC := 0
+ LL TC += Carry
+ HH TC += Carry
+ (-1^DD[Sub])DD TC += Carry - DD[Sub]
+ TC
= XY
Now let's go through the routine itself and see which moving parts of the Ada program correspond to which pieces of the equivalence. And so, we begin at the beginning:
-- Karatsuba's Multiplier. (CAUTION: UNBUFFERED)
procedure Mul_Karatsuba(X : in FZ;
Y : in FZ;
XY : out FZ) is
X and Y, naturally, are the multiplicands; XY is the register to which the result of the multiplication is to be written. Observe that in the procedure's declaration:
-- ...
-- Karatsuba's Multiplier. (CAUTION: UNBUFFERED)
procedure Mul_Karatsuba(X : in FZ;
Y : in FZ;
XY : out FZ)
with Pre => X'Length = Y'Length and
XY'Length = (X'Length + Y'Length) and
X'Length mod 2 = 0;
-- CAUTION: Inlining prohibited for Mul_Karatsuba !
... it is mandated that the length of XY must suffice to hold the resulting integer; and that the length of each multiplicand must be divisible by two. (Recall that valid FZ integers must in fact be
of lengths which constitute powers of 2; the reason for this will become evident shortly.)
We have thereby obtained the "physical" representation shown earlier in Ch. 10:
Let's proceed:
-- L is the wordness of a multiplicand. Guaranteed to be a power of two.
L : constant Word_Count := X'Length;
-- An 'LSeg' is the same length as either multiplicand.
subtype LSeg is FZ(1 .. L);
L is simply the length of either multiplicand (they are required, as shown in the declaration, to be of equal lengths.) K then corresponds to half of the length of a multiplicand; by breaking apart
the multiplicands we will achieve the "divide and conquer" effect of Karatsuba's method, whereby we convert one multiplication of size 2K x 2K into three multiplications of size K x K.
Thereby L is the bit width of the
... and
... multiplicand registers; and as for K:
-- K is HALF of the length of a multiplicand.
K : constant Word_Index := L / 2;
-- A 'KSeg' is the same length as HALF of a multiplicand.
subtype KSeg is FZ(1 .. K);
K is the bit width of the low and high halves of the multiplicand registers, i.e. X[Lo], X[Hi], Y[Lo], and Y[Hi]. Now we define the working registers for the intermediate terms in the equation:
-- The three L-sized variables of the product equation, i.e.:
-- XY = LL + 2^(K*Bitness)(LL + HH + (-1^DD_Sub)*DD) + 2^(2*K*Bitness)HH
LL, DD, HH : LSeg;
-- K-sized terms of Dx * Dy = DD
Dx, Dy : KSeg; -- Dx = abs(XLo - XHi) , Dy = abs(YLo - YHi)
... let's also "draw to scale" all of these registers, and describe their desired eventual contents, referring to the earlier equivalences:
LL = X[Lo]Y[Lo]
HH = X[Hi]Y[Hi]
Dx = |X[Lo] - X[Hi]|
Dy = |Y[Lo] - Y[Hi]|
DD = Dx × Dy
Observe that DD is of width L, as FFA multiplication always results in an output having the summed width of the two multiplicands.
Now for Cx and Cy:
-- Subtraction borrows, signs of (XL - XH) and (YL - YH),
Cx, Cy : WBool; -- so that we can calculate (-1^DD_Sub)
These are simply the borrows recorded from the computation of Dx and Dy, we will need them when computing DD_Sub later on. Moving on:
-- Bottom and Top K-sized halves of the multiplicand X.
XLo : KSeg renames X( X'First .. X'Last - K );
XHi : KSeg renames X( X'First + K .. X'Last );
-- Bottom and Top K-sized halves of the multiplicand Y.
YLo : KSeg renames Y( Y'First .. Y'Last - K );
YHi : KSeg renames Y( Y'First + K .. Y'Last );
We already described these, they are the upper and lower halves of X and Y, i.e. the multiplicands.
Now, the middle term, XYMid:
-- L-sized middle segment of the product XY (+/- K from the midpoint).
XYMid : LSeg renames XY( XY'First + K .. XY'Last - K );
XYMid is where we will be putting... (ignore TC for now...)
+ LL TC += Carry
+ HH TC += Carry
+ (-1^DD[Sub])DD TC += Carry - DD[Sub]
... i.e. the "middle" terms. It represents a "slice" of the multiplication's output register XY. But in order to represent the first term of the equivalence,
... we will also need to represent top and bottom "slices" of the output XY:
-- Bottom and Top L-sized halves of the product XY.
XYLo : LSeg renames XY( XY'First .. XY'Last - L );
XYHi : LSeg renames XY( XY'First + L .. XY'Last );
Lastly, we will require a K-sized "slice" representation of XY, where we will be rippling out the accumulated "tail" carry, TC:
-- Topmost K-sized quarter segment of the product XY, or 'tail'
XYHiHi : KSeg renames XYHi( XYHi'First + K .. XYHi'Last );
... when we complete the computation of the product XY.
As for:
-- Whether the DD term is being subtracted.
DD_Sub : WBool;
-- Carry from individual term additions.
C : WBool;
-- Tail-Carry accumulator, for the final ripple
TC : Word;
... we have already described them above, so let's:
And compute term LL:
-- Recurse: LL := XL * YL
FZ_Multiply_Unbuffered(XLo, YLo, LL);
LL = X[Lo]Y[Lo]
... and then term HH:
-- Recurse: HH := XH * YH
FZ_Multiply_Unbuffered(XHi, YHi, HH);
HH = X[Hi]Y[Hi]
Observe that we have begun to recurse: the invocations of multiplication may result in another Karatsubaization, or alternatively in an invocation of the Comba base case, depending on whether
Karatsuba_Thresh is crossed; as specified in:
-- Multiplier. (CAUTION: UNBUFFERED)
procedure FZ_Multiply_Unbuffered(X : in FZ;
Y : in FZ;
XY : out FZ) is
-- The length of either multiplicand
L : constant Word_Count := X'Length;
if L < = Karatsuba_Thresh then
-- Base case:
FZ_Mul_Comba(X, Y, XY);
-- Recursive case:
Mul_Karatsuba(X, Y, XY);
end if;
end FZ_Multiply_Unbuffered;
The constant for the base case transition was determined empirically, and its optimal value appears to be the same on all machine architectures. However, the reader is invited to carry out his own
We have performed out two of our three recursions; we now want Dx:
Dx = |X[Lo] - X[Hi]|
-- Dx := |XL - XH| , Cx := Borrow (i.e. 1 iff XL < XH)
FZ_Sub_Abs(X => XLo, Y => XHi, Difference => Dx, Underflow => Cx);
... and Dy:
Dy = |Y[Lo] - Y[Hi]|
-- Dy := |YL - YH| , Cy := Borrow (i.e. 1 iff YL < YH)
FZ_Sub_Abs(X => YLo, Y => YHi, Difference => Dy, Underflow => Cy);
It is now time to show how FZ_Sub_Abs works:
-- Destructive: If Cond is 1, NotN := ~N; otherwise NotN := N.
procedure FZ_Not_Cond_D(N : in out FZ;
Cond : in WBool)is
-- The inversion mask
Inv : constant Word := 0 - Cond;
for i in N'Range loop
-- Invert (or, if Cond is 0, do nothing)
N(i) := N(i) xor Inv;
end loop;
end FZ_Not_Cond_D;
-- Subtractor that gets absolute value if underflowed, in const. time
procedure FZ_Sub_Abs(X : in FZ;
Y : in FZ;
Difference : out FZ;
Underflow : out WBool) is
O : Word := 0;
pragma Unreferenced(O);
-- First, we subtract normally
FZ_Sub(X, Y, Difference, Underflow);
-- If borrow - negate,
FZ_Not_Cond_D(Difference, Underflow);
-- ... and also increment.
FZ_Add_D_W(Difference, Underflow, O);
end FZ_Sub_Abs;
FZ_Sub_Abs is simply a constant-time means of taking the absolute value of a subtraction, and saving the output along with any resulting "borrow" bit for possible later use. Take careful note of the
FZ_Not_Cond_D mechanism, you will be seeing it again shortly. The "D" stands for "Destructive" -- by this convention we refer to internal routines in FFA which operate "in-place", directly modifying
their operand.
Moving on, we now want DD:
DD = Dx × Dy
-- Recurse: DD := Dx * Dy
FZ_Multiply_Unbuffered(Dx, Dy, DD);
And we got it -- with our third and final recursive call. Now we want DD_Sub:
-- Whether (XL - XH)(YL - YH) is positive, and so DD must be subtracted:
DD_Sub := 1 - (Cx xor Cy);
Why this is a valid equation for DD_Sub, is shown in a lemma in Chapter 10; please refer to it if memory fails you. Moving on to the upper and lower XY subterms,
-- XY := LL + 2^(2 * K * Bitness) * HH
XYLo := LL;
XYHi := HH;
... and now let's begin to compute the middle term, XYMid:
+ LL TC += Carry
+ HH TC += Carry
-- XY += 2^(K * Bitness) * HH, but carry goes in Tail Carry accum.
FZ_Add_D(X => XYMid, Y => HH, Overflow => TC);
-- XY += 2^(K * Bitness) * LL, ...
FZ_Add_D(X => XYMid, Y => LL, Overflow => C);
-- ... but the carry goes into the Tail Carry accumulator.
TC := TC + C;
Observe that we accumulate the additions' carries in TC. But that's not all for the middle term, we also need the third subterm of it:
+ (-1^DD[Sub])DD TC += Carry - DD[Sub]
And we get it like this:
-- XY += 2^(K * Bitness) * (-1^DD_Sub) * DD
FZ_Not_Cond_D(N => DD, Cond => DD_Sub); -- invert DD if 2s-complementing
FZ_Add_D(OF_In => DD_Sub, -- ... and then must increment
X => XYMid,
Y => DD,
Overflow => C); -- carry will go in Tail Carry accumulator
We have already described FZ_Not_Cond_D; now it is necessary to review FZ_Add_D:
-- Destructive Add: X := X + Y; Overflow := Carry; optional OF_In
procedure FZ_Add_D(X : in out FZ;
Y : in FZ;
Overflow : out WBool;
OF_In : in WBool := 0) is
Carry : WBool := OF_In;
for i in 0 .. Word_Index(X'Length - 1) loop
A : constant Word := X(X'First + i);
B : constant Word := Y(Y'First + i);
S : constant Word := A + B + Carry;
X(X'First + i) := S;
Carry := W_Carry(A, B, S);
end loop;
Overflow := Carry;
end FZ_Add_D;
This is simply "in-place" addition, economizing on stack space and CPU cycles by avoiding the use of an intermediate scratch register. Note that the mechanism is entirely agnostic of the particular
element enumeration of the operand arrays -- this is required because we are operating on array slices, on which Ada wisely preserves the parent array's indexing.
Now we will compute the final "tail carry", TC, and ripple it into the final output of the multiplication, XY:
-- Compute the final Tail Carry for the ripple
TC := TC + C - DD_Sub;
-- Barring a cosmic ray, 0 < = TC <= 2 .
pragma Assert(TC <= 2);
-- Ripple the Tail Carry into the tail.
FZ_Add_D_W(X => XYHiHi, W => TC, Overflow => C);
-- Barring a cosmic ray, the tail ripple will NOT overflow.
pragma Assert(C = 0);
The proof regarding the validity of the ripple equation is given in Chapter 10, the reader is again asked to review it if the correctness of the given mechanism is not obvious to him.
Observe that we take a "belt and suspenders" approach regarding the correct operation of the carry ripple mechanism. Conceivably the asserts may be omitted in a speed-critical application; but their
cost appears to be too small to measure on my system, and so they are to remain in the canonical version of FFA.
And, lastly,
end Mul_Karatsuba;
-- CAUTION: Inlining prohibited for Mul_Karatsuba !
... naturally it is not permissible to inline a Karatsuba invocation, as the procedure is recursive.
We have now obtained the entire "sandwich" from earlier:
LL HH TC := 0
+ LL TC += Carry
+ HH TC += Carry
+ (-1^DD[Sub])DD TC += Carry - DD[Sub]
+ TC
= XY
... i.e. the 2L-sized product XY of the L-sized multiplicands X and Y, having done so via three half-L-sized multiplications and a number of inexpensive additions/subtractions.
Satisfy yourself that at no point does the program branch on any bit inside the operands X and Y (i.e. it operates in constant time), and that the required stack memory and the depth of the recursion
depend strictly on the FFA bitness set during invocation of FFACalc.
At this point you, dear reader, will have fit FFA multiplication into your head!
In Chapter 12B, we will examine an important special case of Karatsuba that merits a separate routine: squaring. Theoretically this operation requires only half of the CPU cycles demanded by the
general case; and as it is made heavy use of in modular exponentiation:
-- Modular Exponent: Result := Base^Exponent mod Modulus
procedure FZ_Mod_Exp(Base : in FZ;
Exponent : in FZ;
Modulus : in FZ;
Result : out FZ) is
-- Working register for the squaring; initially is copy of Base
B : FZ(Base'Range) := Base;
-- Copy of Exponent, for cycling through its bits
E : FZ(Exponent'Range) := Exponent;
-- Register for the Mux operation
T : FZ(Result'Range);
-- Buffer register for the Result
R : FZ(Result'Range);
-- Result := 1
WBool_To_FZ(1, R);
-- For each bit of R width:
for i in 1 .. FZ_Bitness(R) loop
-- T := Result * B mod Modulus
FZ_Mod_Mul(X => R, Y => B, Modulus => Modulus, Product => T);
-- Sel is the current low bit of E;
-- When Sel=0 -> Result := Result;
-- When Sel=1 -> Result := T
FZ_Mux(X => R, Y => T, Result => R, Sel => FZ_OddP(E));
-- Advance to the next bit of E
FZ_ShiftRight(E, E, 1);
-- B := B*B mod Modulus
FZ_Mod_Mul(X => B, Y => B, Modulus => Modulus, Product => B);
end loop;
-- Output the Result:
Result := R;
end FZ_Mod_Exp;
end FZ_ModEx;
... we will find that the squaring-case of Karatsuba merits inclusion in FFA.
We will also make use of a simple means of profiling the execution of the FFA routines -- one that is unique in its simplicity, while generally inapplicable to heathen cryptographic libraries on
account of their failure to avoid branching on operand bits.
~To be continued!~
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MANDATORY: Please prove that you are human:
10 xor 62 = ?
What is the
serial baud rate
of the
FG device
Answer the riddle correctly before clicking "Submit", or comment will NOT appear! Not in moderation queue, NOWHERE!
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zgtsvx - Linux Manuals (3)
zgtsvx (3) - Linux Manuals
zgtsvx.f -
subroutine zgtsvx (FACT, TRANS, N, NRHS, DL, D, DU, DLF, DF, DUF, DU2, IPIV, B, LDB, X, LDX, RCOND, FERR, BERR, WORK, RWORK, INFO)
ZGTSVX computes the solution to system of linear equations A * X = B for GT matrices
Function/Subroutine Documentation
subroutine zgtsvx (characterFACT, characterTRANS, integerN, integerNRHS, complex*16, dimension( * )DL, complex*16, dimension( * )D, complex*16, dimension( * )DU, complex*16, dimension( * )DLF,
complex*16, dimension( * )DF, complex*16, dimension( * )DUF, complex*16, dimension( * )DU2, integer, dimension( * )IPIV, complex*16, dimension( ldb, * )B, integerLDB, complex*16, dimension( ldx, * )
X, integerLDX, double precisionRCOND, double precision, dimension( * )FERR, double precision, dimension( * )BERR, complex*16, dimension( * )WORK, double precision, dimension( * )RWORK, integerINFO)
ZGTSVX computes the solution to system of linear equations A * X = B for GT matrices
ZGTSVX uses the LU factorization to compute the solution to a complex
system of linear equations A * X = B, A**T * X = B, or A**H * X = B,
where A is a tridiagonal matrix of order N and X and B are N-by-NRHS
Error bounds on the solution and a condition estimate are also
The following steps are performed:
1. If FACT = 'N', the LU decomposition is used to factor the matrix A
as A = L * U, where L is a product of permutation and unit lower
bidiagonal matrices and U is upper triangular with nonzeros in
only the main diagonal and first two superdiagonals.
2. If some U(i,i)=0, so that U is exactly singular, then the routine
returns with INFO = i. Otherwise, the factored form of A is used
to estimate the condition number of the matrix A. If the
reciprocal of the condition number is less than machine precision,
INFO = N+1 is returned as a warning, but the routine still goes on
to solve for X and compute error bounds as described below.
3. The system of equations is solved for X using the factored form
of A.
4. Iterative refinement is applied to improve the computed solution
matrix and calculate error bounds and backward error estimates
for it.
FACT is CHARACTER*1
Specifies whether or not the factored form of A has been
supplied on entry.
= 'F': DLF, DF, DUF, DU2, and IPIV contain the factored form
of A; DL, D, DU, DLF, DF, DUF, DU2 and IPIV will not
be modified.
= 'N': The matrix will be copied to DLF, DF, and DUF
and factored.
TRANS is CHARACTER*1
Specifies the form of the system of equations:
= 'N': A * X = B (No transpose)
= 'T': A**T * X = B (Transpose)
= 'C': A**H * X = B (Conjugate transpose)
N is INTEGER
The order of the matrix A. N >= 0.
NRHS is INTEGER
The number of right hand sides, i.e., the number of columns
of the matrix B. NRHS >= 0.
DL is COMPLEX*16 array, dimension (N-1)
The (n-1) subdiagonal elements of A.
D is COMPLEX*16 array, dimension (N)
The n diagonal elements of A.
DU is COMPLEX*16 array, dimension (N-1)
The (n-1) superdiagonal elements of A.
DLF is COMPLEX*16 array, dimension (N-1)
If FACT = 'F', then DLF is an input argument and on entry
contains the (n-1) multipliers that define the matrix L from
the LU factorization of A as computed by ZGTTRF.
If FACT = 'N', then DLF is an output argument and on exit
contains the (n-1) multipliers that define the matrix L from
the LU factorization of A.
DF is COMPLEX*16 array, dimension (N)
If FACT = 'F', then DF is an input argument and on entry
contains the n diagonal elements of the upper triangular
matrix U from the LU factorization of A.
If FACT = 'N', then DF is an output argument and on exit
contains the n diagonal elements of the upper triangular
matrix U from the LU factorization of A.
DUF is COMPLEX*16 array, dimension (N-1)
If FACT = 'F', then DUF is an input argument and on entry
contains the (n-1) elements of the first superdiagonal of U.
If FACT = 'N', then DUF is an output argument and on exit
contains the (n-1) elements of the first superdiagonal of U.
DU2 is COMPLEX*16 array, dimension (N-2)
If FACT = 'F', then DU2 is an input argument and on entry
contains the (n-2) elements of the second superdiagonal of
If FACT = 'N', then DU2 is an output argument and on exit
contains the (n-2) elements of the second superdiagonal of
IPIV is INTEGER array, dimension (N)
If FACT = 'F', then IPIV is an input argument and on entry
contains the pivot indices from the LU factorization of A as
computed by ZGTTRF.
If FACT = 'N', then IPIV is an output argument and on exit
contains the pivot indices from the LU factorization of A;
row i of the matrix was interchanged with row IPIV(i).
IPIV(i) will always be either i or i+1; IPIV(i) = i indicates
a row interchange was not required.
B is COMPLEX*16 array, dimension (LDB,NRHS)
The N-by-NRHS right hand side matrix B.
LDB is INTEGER
The leading dimension of the array B. LDB >= max(1,N).
X is COMPLEX*16 array, dimension (LDX,NRHS)
If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X.
LDX is INTEGER
The leading dimension of the array X. LDX >= max(1,N).
RCOND is DOUBLE PRECISION
The estimate of the reciprocal condition number of the matrix
A. If RCOND is less than the machine precision (in
particular, if RCOND = 0), the matrix is singular to working
precision. This condition is indicated by a return code of
INFO > 0.
FERR is DOUBLE PRECISION array, dimension (NRHS)
The estimated forward error bound for each solution vector
X(j) (the j-th column of the solution matrix X).
If XTRUE is the true solution corresponding to X(j), FERR(j)
is an estimated upper bound for the magnitude of the largest
element in (X(j) - XTRUE) divided by the magnitude of the
largest element in X(j). The estimate is as reliable as
the estimate for RCOND, and is almost always a slight
overestimate of the true error.
BERR is DOUBLE PRECISION array, dimension (NRHS)
The componentwise relative backward error of each solution
vector X(j) (i.e., the smallest relative change in
any element of A or B that makes X(j) an exact solution).
WORK is COMPLEX*16 array, dimension (2*N)
RWORK is DOUBLE PRECISION array, dimension (N)
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: if INFO = i, and i is
<= N: U(i,i) is exactly zero. The factorization
has not been completed unless i = N, but the
factor U is exactly singular, so the solution
and error bounds could not be computed.
RCOND = 0 is returned.
= N+1: U is nonsingular, but RCOND is less than machine
precision, meaning that the matrix is singular
to working precision. Nevertheless, the
solution and error bounds are computed because
there are a number of situations where the
computed solution can be more accurate than the
value of RCOND would suggest.
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
September 2012
Definition at line 293 of file zgtsvx.f.
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The management for a large grocery store chain would like to
determine if a new cash...
The management for a large grocery store chain would like to determine if a new cash...
The management for a large grocery store chain would like to determine if a new cash register will enable cashiers to process a larger number of items on average than the cash register which they are
currently using. Nine cashiers are randomly selected, and the number of grocery items which they can process in three minutes is measured for both the old cash register and the new cash register.
Without making any assumptions about the distribution, do the data provide conclusive evidence that the new cash register enables cashiers to process a significantly larger number of items than the
old cash register? Use the Wilcoxon signed-rank test to analyze the results.
Number of Grocery Items Processed in Three Minutes
Cashier 11 22 33 44 55 66 77 88 99
Old Cash Register 6464 6666 5858 6666 6262 5757 6060 6767 7878
New Cash Register 6363 7373 6464 6868 5858 6565 6262 7777 7373
Step 1 of 2:
Find the value of the test statistic to test if the new cash register enables cashiers to process a significantly larger number of items than the old cash register. Round your answer to two decimal
places, if necessary.
Step 2 of 2:
Make the decision to reject or fail to reject the null hypothesis that the number of items that the new cash register enables cashiers to process is less than or equal to the number of items that the
old cash register enables cashiers to process, and state the conclusion in terms of the original problem. Use α=0.01α=0.01.
Step 1 of 2:
Step 2 of 2:
The p-value from the output is 0.0693.
Since the p-value (0.0693) is greater than the significance level (0.01), we cannot reject the null hypothesis.
Therefore, we cannot support the claim that the new cash register enables cashiers to process a significantly larger number of items than the old cash register.
variables: Old Cash Register - New Cash Register
10 sum of positive ranks
35 sum of negative ranks
9 n
22.500 expected value
8.441 standard deviation
-1.48 z
.0693 p-value (one-tailed, lower)
No. Data Rank
2 -707 7
3 -606 6
4 -202 2.5
6 -808 8
7 -202 2.5
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As you think about the nature of hypothesis testing, making
inferences from samples, respond to the...
according to W.A. Wallis and H.V. Roberts statistics is a body of methods for making wise decisions in the face of uncertainty
we know that in statistics we make inference and estimate value of parameters or in other words we think about a possible outcome with some uncertainty . results by any method of statistics are not
perfect.i.e. results obtained with the help of any method of statistics can not exactly equal to true results .estimated results are always different from exact result . i.e. result drawn from
inference always contain some uncertainty in it .
due to this uncertainty in results drawn , we always have risk in accepting that the result is true or accepting drawn results of inference . this risk can be minimized but can not be eliminated.
hence benifits of application of statistics in face of uncertainty is that it is used to make a idea about future results , in thus case we can prepare ourself to be ready for upcoming problems . it
also used for prediction about future values ,for example in stock market we predict future values of stocks by inference .
and disadvantage is that there is always some uncertainty in results drawn from inference .
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The unbelievably careful quantumfield - Quantum Physics & Consciousness
The unbelievably careful quantumfield
The widespread message about the universe that we live in, is that it is almost empty, cold and indifferent and that life is only accidental, brutish and hard. I have a completely different opinion
thanks inter alia to quantum physics. The universe is by no means indifferent to us, on the contrary. A study of the behavior of the quantum field, that the universe actually is, shows us that it
responds very carefully to what we think, know and expect. In its manifestations, the quantum field even takes into account our possible future actions plus the contents of our consciousness. These
conclusions can be drawn from the results of certain experiments that can even be understood by the non-physicist, if in possession of an open mind and willing to think a little bit harder than
The quantum field – as recognized by the most physicists at the moment – is a non-local ubiquitous, not directly measurable, and intangible field from which all matter and energy manifests itself
on measurement. Replace right away the word measurement here with observation. Many physicists are already doing that. Accepting that, the observer gets an important participating role in this
manifestation of matter and energy. In other words, the quantum field is the invisible source from which everything we experience arises, which is by the way very reminiscent of the TAO. Many have
already noticed that agreement.
Until the middle of the last century, a field in the physicists sense of the word was a state of space through which objects that are sensitive to that field, experience forces depending on their
position in that field. A field, in that sense, is the way physicists try to deal with forces at a distance that are exerted through empty space. Examples are the gravitational field, the
electromagnetic field, the Higgs field. A field is therefore essentially non-material. The nature of those field forces is still a mystery, although we can calculate and predict their effects. The
quantum field is even another step less physical, it does not exert forces but is the source from which the matter and energy appears – and disappears into it. Most likely, the quantum field is
also the source of space and time itself, but I will not go into that now.
The observer matters, literally
The properties of the quantum field can be described mathematically despite that intangible capacity. The Schrödinger equation is a good example of this. The Schrödinger equation solution describes
a complex wave. Complex means that the described values cannot be expressed in easily imaginable numbers. Imaginary values, numbers whose square is negative – something that does not fit within our
frame of mind – play an essential role. Fortunately, those imaginary values dissappear if we want to use the wave function as a an amazing efficient predictor of the probability of finding an
object in our measurements. It is important here to remember that this wave does not describe the effect of the observation, the measurement. Without observation, the immaterial wave of possibilities
would continue forever. The observer plays an essential role in the matter and energy emerging from the quantum field.
The effect of the observation is that the infinite collection of possibilities, which moves through the quantum field as a wave, results in a concrete experience, the so-called quantum collapse. I
think that collapse is actually an unfortunate term. The term quantum collapse suggests a breakdown of something material where only one element remains. However, the quantum field is not material,
not even a small bit. A term that better expresses what is happening is the ‘reduction of the quantum wave‘. The quantum wave can be reduced by our information to a smaller wave containing fewer
possibilities. The final reduction is then to that of a single possibility, which is then 100% probable and is therefore the observed manifestation. That’s the bright spot on the screen when a photon
hits, for example. The picture evoked by the term ‘reduction of the quantum wave’ helps us a lot better in our attempts at understanding.
Delayed choice quantum eraser and conscious observation
This ultimate reduction as a result of observation is called the observer effect in quantum physics, something that is still hotly discussed. The big question is whether it is the physical
measurement, or whether it is the observer and his consciousness, which produces the effect, the reduction (or collapse) nof the quantum wave. That is a subtle problem for which no experiment seems
to be configurable to answer this. Measurements without observation seem obviously worthless. As long as we are not allowed to observe the result, we cannot use the outcome, of course. A clear
catch-22 situation. But there is an important experiment that seems to come very close to measuring without direct observation. I would like to describe this experiment here in such a way that its
consequences will become clear and understandable.
That is the two-slit quantum eraser experiment with delayed choice. The delayed choice concerns the effect of whether or not to irrevocably erase measurement information before it has arrived in an
observer’s consciousness or is registered – on the computer hard disk for example – so that conscious observation is still possible, albeit at a later moment.
The quantum eraser experiment is a two-slit experiment that is designed in such a way that we can detect and register the slit through which the wave went. The effect of this erasure is that the
quantum wave then ‘magically’ reduces to one of the two slits. We can then see, because there exists now only a single wave between the slits and the screen, that the wave will no longer interfere
with itself. We then no longer see the typical double-slit interference pattern of dark and light bands, the result is a single spread-out spot. Einstein already realized in around 1920 that mere
observation of the slit would evoke this strange quantum effect and devised a notorious thought experiment with which he hoped to falsify quantum mechanics. His thought experiment was much later
technically realized and the disappearance of the interference pattern on observation has been confirmed. Einstein thus played the role of devil’s advocate in his brilliant way, thereby contributing
much to quantum physics.
The law of conservation of information
We can see this quantum behavior as the result of information in motion. The more information we receive, the more the information in the quantum wave will be reduced. This is because that
information reduces the – infinite – number of possibilities in the quantum wave by moving that information to a location accessible to our consciousness. The more we know, the fewer
probabilities there will be to realize. That is an effect that we can experience daily, such as in using a public transportation planner to avoid surprises with our planned trips. The more
information, the fewer surprises. We can therefore also consider the quantum field as a universal information field. Physicists have thus discovered a new conservation law relevant to that quantum
field, the law of conservation of information. When we can capture more information about the measured object, that means that the information in the quantum field moves to a location accessible to
us, but always still within that field. The wave of possibilities is thereby reduced. The information is of course still within the quantum field, but now in a location where we have access to.
Let’s now visualize this in some diagrams. It may then become easier to understand what happens in a quantum eraser experiment. We will first look therefore at the basic implementation of a
double-slit experiment. A single wave arrives at the two slits at the same time. The two slits then become sources of synchronous – simultaneously moving – waves. Those waves meet after the slits
again and reinforce or extinguish each other in certain locations. These locations of reinforced motion form then contiguous curved lines. This creates – with light waves – the familiar
interference pattern of dark and light fringes. With sound waves of a single tone (monochromatic sound) you will get areas of loudness and silence. You can do a double-slit experiment with sound at
home with two simple speakers and a tone generator. With a distance between the speakers of 50 cm and a frequency of 800 Hz, the effect is easy to hear.
The basic double-slit experiment. The result is due to interference of the two waves coming from the two slits.
Quantum information collections pictured in Venn diagrams
We can use a Venn diagram to show how the information of an experiment gets distributed in the quantum field. The Venn diagrams then represent collections of quantum information. The ubiquitous
quantum information field is then the set that contains all information in the universe, and all other collections are subsets of it. In anticipation of what will be argued further, I distinguish
• The set of information provided by the experiment (green).
• The set of information that the experiment yields and that has already been observed and incorporated into consciousness (yellow).
The yellow set is therefore a subset of the green one. For example, the information in the green set that is not in the yellow one may contain information that is already stored on a hard disk but
has not yet been observed by anyone. When observing the contents of the hard disk, that information moves from somewhere in the green set to the yellow set of observed and in consciousness stored
information. Because we are ultimately also collections of information, you could also consider the yellow set as representative of ourselves, the observers of the universe.
When we now organize the experiment in such a way that we can determine the slit through which the quantum wave travels, we will get the picture below. Now try to understand the following well.
Remember that the quantum wave represents the sum of all probabilities of finding the object upon observation. If we can know which slit the wave is passing, that’s the slit where the sum of all
probabilities is 100 %. Then it is undeniably clear that the wave travels through only one slit. I hope you understand that. For the other slit, there is no chance, no possibility, left for the
object to manifest there. An observation that is usually interpreted – and unnecessarily – as that the object existed actually in one of the slits. That interpretation is the result of the
confusing dual wave-particle image that is so often presented in the media about quantum physics.
As soon as we have information about how the wave travels the slits – not necessarily consciously observed – the interference dissappears and the result is a spread-out spot
Just as an aside. In the usual descriptions of the double-slit experiment, there is usually talk of an object – a photon, an electron, a molecule, a virus – that passes through the slit. As if that object appeared temporarily in the slit and then happily continued as a wave again. You will also find this description in my first book. Frankly, that's an unprovable and unnecessary assumption. It is never the case that the object is observed going through the slit in its passage. The image of a wave that is reduced to one slit is a lot simpler and therefore, in my opinion, better. It explains the disappearance of the interference pattern just as well, with fewer assumptions and is therefore preferred.
In a Korean experiment, the effect of information on the quantum wave has been beautifully demonstrated. The more information we have about one of the paths the wave can go, the stronger the wave on
the other path is reduced. This relationship can be described by a simple algebraic formula that is very similar to the Pythagorean formula for the sides of a right triangle: a^2 + b^2 = c^2. The
left leg of the mannikin then represents the information we have about one path and the right leg represents the probablity to find the particle on the other. In the extreme positions (of the legs)
the path information is maximal and the probability to find the particle on the path corresponding to the other leg is reduced to zero. Or the other way around.
The Venn diagram below shows that the information in the quantum information field, which relates to the object, moves to the green set when measured, which is the information we obtained from the
experiment. So no information is added in the quantum field of the universe, it only moves to another location and has therefore an effect on the observed world. The interference pattern disappears.
The result is now a spread-out spot.
The erasure of information has consequences for what we observe
In the previous diagram I placed the information about the object in the green set – ‘The information from the experiment’ – but not in the ‘Consciously perceived information’. This relocation of
information, despite the fact that we have not yet consciously processed it, has the experimentally demonstrated effect of the disappearance of the interference fringes. But if this information has
not yet entered an observing consciousness – the yellow set – then it is still possible to radically erase this information with the result that it can no longer end up in our consciousness.
The observable effect of erasing that not yet observed information is – surprisingly – the return of the interference pattern. This is the quantum eraser experiment in which – for example via
semi-permeable mirrors – the information is randomly and unpredictably erased or not, before it can be registered. So, erasing is actually moving information to a location that is no longer
accessible by us in the quantum field.
Quantum eraser destroys the informatuin on the quantum wave passing the slits. The interference fringes return.
The conclusion that can be drawn from the return of the interference pattern is astonishing as far as I am concerned. It means that the quantum wave even responds to information that is not yet in
our consciousness but could end up there in the future. This also means that the quantum wave changes retroactively since the erasure is always in time after the passage through the slits. That’s
why it’s called delayed.
Would you have become curious about the technical details of all mentioned experiments, especially the delayed choice ones, they are described in detail in my book 'Quantum Physics is NOT Weird' available at Amazon in the US and also at BookMundo in the UK.
Future-proof behavior
The quantum field therefore also takes our possible future actions and content of consciousness into account. It cannot and should not be the case that the interference does not disappear and that
therefore the wave must have went through both slits, but that then at some point in the future we observed the result that was waiting for us on the hard disk, and that we then have to conclude that
we then know which slit the wave passed and consequently not the other one. Which irrevocably impossible means that we should not have observed interference in the experiment, while we remember we
did. Perhaps we published it already. A severe violation of our remembered history or of the laws of nature that would radically overturn everything we assumed as real. Luckily, the recorded
completed past is irrevocable. Nice indeed.
Tat Vam Asi
The astonishing conclusion is therefore that the reduction of the quantum wave to a single slit, which destroys the interference pattern, is not the direct result of the physical measurement, but
depends on the possibility whether that result can be observed now or in the future. The quantum field is therefore very strongly connected to our consciousness and its future. Perhaps the field and
our (greater) consciousness are identical, which aptly corresponds to the Tat Vam Asi (You Are the Absolute) of the Upanishads.
In any case, for me this means a mind-bogglingly intelligent and careful universe that is constantly making adjustments in its quantum field, so that we, conscious beings, have the experience of a
universe that usually conforms to the laws we have established and therefore behaves in a predictable way for us. Very accommodating that. The quantum field therefore behaves like the intelligent
director of a mind-bogglingly rich and complex play with an unimaginably careful attention to everything that takes place on the stage. Call it love.
In the fury of the moment I can see the Master's hand
In every leaf that trembles, in every grain of sand.
Bob Dylan
Paul J. van Leeuwen graduated in applied physics in Delft TU in 1974. There was little attention to the significance of quantum physics for the view on reality at that time. However, much later in
his life he discovered that there is an important and clear connection between quantum physics and consciousness.
One Reply to “The unbelievably careful quantumfield”
1. This is fascinating! And really resonates with the “place” we (our brains) go into when we meditate or find ourselves daydreaming or experiencing unexpected events (such as seeing something you
werent expecting to see like a face image in a field of sagebrush) .. the information you posted is difficult to follow for a lay-person like me but makes sense as I re-read it. Thank you! | {"url":"https://quantumphysics-consciousness.eu/index.php/en/2024/10/29/the-careful-quantumfield/","timestamp":"2024-11-04T11:52:28Z","content_type":"text/html","content_length":"191642","record_id":"<urn:uuid:5115f8d2-6c8a-4394-b251-083a320f267a>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00038.warc.gz"} |
TGD diary{{post.title}}TGD diary
Quark gluon plasma assigned to de-confinement phase transition predicted by QCD has turned out to be a problematic notion. The original expectation was that quark gluon plasma (QGP) would be created
in heavy ion collisions. A candidate for QGP was discovered already at RHIC but did not have quite the expected properties such as black body spectrum behaving like an ideal liquid with long range
correlations between charged particle pairs created in the collision. Then LHC discovered that this phase is created even in proton-heavy nucleus collisions. Now this phase have been discovered even
in proton-proton collisions. This is something unexpected and both a challenge and opportunity to TGD.
In TGD framework QGP is replaced with quantum critical state appearing in the transition from ordinary hadron physics characterized by Mersenne prime M[107] to dark variant of M[89] hadron physics
characterized by h[eff]/h=n=512. At criticality partons are hybrids of M[89] and M[107] partons with Compton length of ordinary partons and mass m(89)≤ 512 m(107). Inequality follows from possible 1/
512 fractionization of mass and other quantum numbers. The observed strangeness enhancement can be understood as a violation of quark universality if the gluons of M[89] hadron physics correspond to
second generation of gluons whose couplings necessarily break quark universality.
The violation of quark universality would be counterpart for the violation of lepton universality and the simplest hypothesis that the charge matrices acting on family triplets are same for quarks
and leptons allows to understand also the strangeness enhancement qualitatively.
See the chapter New Physics predicted by TGD: I of "p-Adic length scale hypothesis" and the article Phase transition from M[107] hadron physics to M[89] hadron physics as counterpart for
de-confinement phase transition? .
For a summary of earlier postings see Latest progress in TGD.
Two highly interesting findings providing insights about the origins of life have emerged and it is interesting to see how they fit to the TGD inspired vision.
The group led by Thomas Carell has made an important step in the understanding the origins of life (see this). They have identified a mechanism leading to the generation of purines A and G which
besides pyrimidines A,T (U) are the basic building bricks of DNA and RNA. The crucial step is to make the solution involved slightly acidic by adding protons. For year later I learned that a variant
of Urey-Miller experiment with simulation of shock waves perhaps generated by extraterrestrial impacts using laser pulses generates formamide and this in turn leads to the generation of all 4 RNA
bases (see the popular article and article).
These findings represent a fascinating challenge for TGD inspired quantum biology. The proposal is that formamide is the unique amide, which can form stable bound states with dark protons and crucial
for the development of life as dark matter-visible matter symbiosis. Pollack effect would generate electron rich exclusions zones and dark protons at magnetic flux tubes. Dark protons would bind
stably with unique amine leaving its chemical properties intact. This would lead to the generation of purines and the 4 RNA bases. This would be starting point of life as symbiosis of ordinary matter
and dark matter as large h[eff]/h=n phases of ordinary matter generated at quantum criticality induced by say extraterrestrial impacts. The TGD based model for cold fusion and the recent results
about superdense phase of hydrogen identifiable in TGD framework as dark proton sequences giving rise to dark nuclear strings provides support for this picture.
There is however a problem: a reductive environment (with ability to donate electrons) is needed in these experiments: it seems that early atmosphere was not reductive. In TGD framework one can
imagine two - not mutually exclusive - solutions of the problem. Either life evolved in underground oceans, where oxygen concentration was small or Pollack effect gave rise to negatively charged and
thus reductive exclusion zones (EZs) as protons were transferred to dark protons at magnetic flux tubes. The function of UV radiation, catalytic action, and of shock waves would be generation of
quantum criticality inducing the creation of EZs making possible dark h[eff]/h=n phases.
For details and background see the article Two steps towards understanding of the origins of life or the chapter Evolution in Many-Sheeted Space-Time.
For a summary of earlier postings see Latest progress in TGD.
The evidence for the violation of lepton number universality is accumulating at LHC. I have written about the violation of lepton number universality in the decays of B and K mesons already earlier
explaining it in terms of two higher generations of electroweak bosons. The existence of free fermion generations having topological explanation in TGD can be regarded formally as SU(3) triplet. One
can speak of family-SU(3).
Electroweak bosons and gluons belong to singlet and octet of family-SU(3) and the natural assumption is that only singlet (ordinary gauge bosons) and two SU(3) neutral states of octet are light. One
would have effectively 3 generations of electroweak bosons and gluons. There charge matrices would be orthogonal with respect to the inner product defined by trace so that both quark and lepton
universality would be broken in the same manner. The strongest assumption is that the charge matrices in flavor space are same for all weak bosons. The CKM mixing for neutrinos complicates this
picture by affecting the branching rations of charged weak bosons.
Quite recently I noticed that second generation of Z boson could explain the different values of proton charge radius determined from the hydrogen and muonium atoms as one manifestation of the
violation of universality (see this). The concept of charge matrix is discussed in more detail in this post.
I learned quite recently about new data concerning B meson anomalies. The experimental ideas are explained here. It is interesting to look at the results in more detail from TGD point of view..
1. There is about 4.0 σ deviation from $τ/l$ universality (l=μ,e) in b→ c transitions. In terms of branching ratios ones has:
R(D^*)=Br(B→ D^*→τν[τ])/Br(B→ D^*lν[l]) =0.316+/- 0.016+/- 0.010 ,
R(D) =Br(B→ Dτν[τ])/Br(B→ lν[l]) =0.397+/- 0.040+/- 0.028 ,
The corresponding SM values are R(D^*)[|SM]= 0.252+/- 0.003 and R(D)[|SM]=.300+/- .008. My understanding is that the normalization factor in the ratio involves total rate to D^*lν[l], l=μ, e
involving only single neutrino in final state whereas the τν decays involve 3 neutrinos due to the neutrino pair from τ implying broad distribution for the missing mass.
The decays to τ ν[τ] are clearly preferred as if there were an exotic W boson preferring to decay τν over lν , l=e,μ. In TGD it could be second generation W boson. Note that CKM mizing of
neutrinos could also affect the branching ratios.
2. Since these decays are mediated at tree level in the SM, relatively large new physics contributions are necessary to explain these deviations. Observation of 2.6 σ deviation of μ/e universality
in the dilepton invariant mass bin 1 GeV^2≤ q^2≤ 6 GeV^2 in b→ s transitions:
R(K)=Br(B→ Kμ^+μ^-)/Br(B→ K e^+e^-) =0.745+0.090/-0.074+/- 0.038
deviate from the SM prediction R(K)[|SM]=1.0003+/- 0.0001.
This suggests the existence of the analog of Z boson preferring to decay to e^+e^- rather than μ^+μ^- pairs.
If the charge matrices acting on dynamical family-SU(3) fermion triplet do not depend on electroweak bosons and neutrino CKM mixing is neglected for the decays of second generation W, the data
for branching ratios of D bosons implies that the decays to e^+e^- and τ^+τ^- should be favored over the decays to μ^+μ^-. Orthogonality of the charge matrices plus the above data could allow to
fix them rather precisely from data. It might be that one must take into account the CKM mixing.
3. CMS recently also searched for the decay h→ τμ and found a non-zero result of Br(h→ τμ)=0.84+0.39/-0.37 , which disagrees by about 2.4 σ from 0, the SM value. I have proposed an explanation for
this finding in terms of CKM mixing for leptons. h would decay to W^+W^- pair, which would exchange neutrino transforming to τμ pair by neutrino CKM mixing.
4. According to the reference, for Z, the lower bound for the mass is 2.9 TeV, just the TGD prediction if it corresponds to Gaussian Mersenne M[G,79]=(1+i)^79 so that the mass would be 32 times the
mass of ordinary Z boson! It seem that we are at the verge of the verification of one key prediction of TGD.
For background see the chapter New Physics predicted by TGD: I of "p-Adic length scale hypothesis".
For a summary of earlier postings see Latest progress in TGD.
The twistor lift of TGD forces to introduce the analog of Kähler form for M^4, call it J. J is covariantly constant self-dual 2-form, whose square is the negative of the metric. There is a moduli
space for these Kähler forms parametrized by the direction of the constant and parallel magnetic and electric fields defined by J. J partially characterizes the causal diamond (CD): hence the
notation J(CD) and can be interpreted as a geometric correlate for fixing quantization axis of energy (rest system) and spin.
Kähler form defines classical U(1) gauge field and there are excellent reasons to expect that it gives rise to U(1) quanta coupling to the difference of B-L of baryon and lepton numbers. There is
coupling strength α[1] associated with this interaction. The first guess that it could be just Kähler coupling strength leads to unphysical predictions: α[1] must be much smaller. Here I do not yet
completely understand the situation. One can however check whether the simplest guess is consistent with the empirical inputs from CP breaking of mesons and antimatter asymmetry. This turns out to be
the case.
One must specify the value of α[1] and the scaling factor transforming J(CD) having dimension length squared as tensor square root of metric to dimensionless U(1) gauge field F= J(CD)/S. This leads
to a series of questions.
How to fix the scaling parameter S?
1. The scaling parameter relating J(CD) and F is fixed by flux quantization implying that the flux of J(CD) is the area of sphere S^2 for the twistor space M^4× S^2. The gauge field is obtained as F
=J/S, where S= 4π R^2(S^2) is the area of S^2.
2. Note that in Minkowski coordinates the length dimension is by convention shifted from the metric to linear Minkowski coordinates so that the magnetic field B[1] has dimension of inverse length
squared and corresponds to J(CD)/SL^2, where L is naturally be taken to the size scale of CD defining the unit length in Minkowski coordinates. The U(1) magnetic flux would the signed area using
L^2 as a unit.
How R(S^2) relates to Planck length l[P]? l[P] is either the radius l[P]=R of the twistor sphere S^2 of the twistor space T=M^4× S^2 or the circumference l[P]= 2π R(S^2) of the geodesic of S^2.
Circumference is a more natural identification since it can be measured in Riemann geometry whereas the operational definition of the radius requires imbedding to Euclidian 3-space.
How can one fix the value of U(1) coupling strength α[1]? As a guideline one can use CP breaking in K and B meson systems and the parameter characterizing matter-antimatter symmetry.
1. The recent experimental estimate for so called Jarlskog parameter characterizing the CP breaking in kaon system is J≈ 3.0× 10^-5. For B mesons CP breading is about 50 times larger than for kaons
and it is clear that Jarlskog invariant does not distinguish between different meson so that it is better to talk about orders of magnitude only.
2. Matter-antimatter asymmetry is characterized by the number r=n[B]/n[γ] ∼ 10^-10 telling the ratio of the baryon density after annihilation to the original density. There is about one baryon 10
billion photons of CMB left in the recent Universe.
Consider now the identification of α[1].
1. Since the action is obtained by dimensional reduction from the 6-D Kähler action, one could argue α[1]= α[K]. This proposal leads to unphysical predictions in atomic physics since
neutron-electron U(1) interaction scales up binding energies dramatically.
U(1) part of action can be however regarded a small perturbation characterized by the parameter ε= R^2(S^2)/R^2(CP[2]), the ratio of the areas of twistor spheres of T(M^4) and T(CP[2]). One can
however argue that since the relative magnitude of U(1) term and ordinary Kähler action is given by ε, one has α[1]=ε× α[K] so that the coupling constant evolution for α[1] and α[K] would be
2. ε indeed serves in the role of coupling constant strength at classical level. α[K] disappears from classical field equations at the space-time level and appears only in the conditions for the
super-symplectic algebra but ε appears in field equations since the Kähler forms of J resp. CP[2] Kähler form is proportional to R^2(S^2) resp. R^2(CP[2]) times the corresponding U(1) gauge
field. R(S^2) appears in the definition of 2-bein for R^2(S^2) and therefore in the modified gamma matrices and modified Dirac equation. Therefore ε^1/2=R(S^2)/R(CP[2]) appears in modified Dirac
equation as required by CP breaking manifesting itself in CKM matrix.
NTU for the field equations in the regions, where the volume term and Kähler action couple to each other demands that ε and ε^1/2 are rational numbers, hopefully as simple as possible. Otherwise
there is no hope about extremals with parameters of the polynomials appearing in the solution in an arbitrary extension of rationals and NTU is lost. Transcendental values of ε are definitely
excluded. The most stringent condition ε=1 is also unphysical. ε= 2^2r is favoured number theoretically.
Concerning the estimate for ε it is best to use the constraints coming from p-adic mass calculations.
1. p-Adic mass calculations predict electron mass as
m[e]= hbar/R(CP[2])(5+Y)^1/2 .
Expressing m[e] in terms of Planck mass m[P] and assuming Y=0 (Y∈ (0,1)) gives an estimate for l[P]/R(CP[2]) as
l[P]R(CP[2]) ≈ 2.0× 10^-4 .
2. From l[P]= 2π R(S^2) one obtains estimate for ε, α[1], g[1]=(4πα[1])^1/2 assuming
α[K]≈ α≈ 1/137 in electron length scale.
ε = 2^-30 ≈ 1.0× 10^-9 ,
α[1]=εα[K] ≈ 6.8× 10^-12 ,
g[1]= (4πα[1]^1/2 ≈ 9.24 × 10^-6 .
There are two options corresponding to l[P]= R(S^2) and l[P] =2π R(S^2). Only the length of the geodesic of S^2 has meaning in the Riemann geometry of S^2 whereas the radius of S^2 has operational
meaning only if S^2 is imbedded to E^3. Hence l[P]= 2π R(S^2) is more plausible option.
For ε=2^-30 the value of l[P]^2/R^2(CP[2]) is l[P]^2/R^2(CP[2])=(2π)^2 × R^2(S^2)/R^2(CP[2]) ≈ 3.7× 10^-8. l[P]/R(S^2) would be a transcendental number but since it would not be a fundamental
constant but appear only at the QFT-GRT limit of TGD, this would not be a problem.
One can make order of magnitude estimates for the Jarlskog parameter J and the fraction r= n(B)/n(γ). Here it is not however clear whether one should use ε or α[1] as the basis of the estimate
1. The estimate based on ε gives
J∼ ε^1/2 ≈ 3.2× 10^-5 ,
r∼ ε ≈ 1.0× 10^-9 .
The estimate for J happens to be very near to the recent experimental value J≈ 3.0× 10^-5. The estimate for r is by order of magnitude smaller than the empirical value.
2. The estimate based on α[1] gives
J∼ g[1] ≈ 0.92× 10^-5 ,
r∼ α[1] ≈ .68× 10^-11 .
The estimate for J is excellent but the estimate for r by more than order of magnitude smaller than the empirical value. One explanation is that α[K] has discrete coupling constant evolution and
increases in short scales and could have been considerably larger in the scale characterizing the situation in which matter-antimatter asymmetry was generated.
Atomic nuclei have baryon number equal the sum B= Z+N of proton and neutron numbers and neutral atoms have B= N. Only hydrogen atom would be also U(1) neutral. The dramatic prediction of U(1) force
is that neutrinos might not be so weakly interacting particles as has been thought. If the quanta of U(1) force are not massive, a new long range force is in question. U(1) quanta could become
massive via U(1) super-conductivity causing Meissner effect. As found, U(1) part of action can be however regarded a small perturbation characterized by the parameter ε= R^2(S^2)/R^2(CP[2]). One can
however argue that since the relative magnitude of U(1) term and ordinary Kähler action is given by ε, one has α[1]=ε× α[K].
Quantal U(1) force must be also consistent with atomic physics. The value of the parameter α[1] consistent with the size of CP breaking of K mesons and with matter antimatter asymmetry is α[1]= εα[K]
= 2^-30α[K].
1. Electrons and baryons would have attractive interaction, which effectively transforms the em charge Z of atom Z[eff]= rZ, r=1+(N/Z)ε[1], ε[1] =α[1]/α=ε × α[K]/α≈ ε for α[K]≈ α predicted to hold
true in electron length scale. The parameter
s=(1 + (N/Z)ε)^2 -1= 2(N/Z)ε +(N/Z)^2ε^2
would characterize the isotope dependent relative shift of the binding energy scale.
The comparison of the binding energies of hydrogen isotopes could provide a stringent bounds of the value of α[1]. For l[P]= 2π R(S^2) option one would have α[1]=2^-30α[K] ≈ .68× 10^-11 and s≈
1.4× 10^-10. s is by order of magnitude smaller than α^4≈ 2.9× 10^-9 corrections from QED (see this). The predicted differences between the binding energy scales of isotopes of hydrogen might
allow to test the proposal.
2. B=N would be neutralized by the neutrinos of the cosmic background. Could this occur even at the level of single atom or does one have a plasma like state? The ground state binding energy of
neutrino atoms would be α[1]^2m[ν]/2 ∼ 10^-24 eV for m[ν] =.1 eV! This is many many orders of magnitude below the thermal energy of cosmic neutrino background estimated to be about 1.95× 10^-4 eV
(see this). The Bohr radius would be hbar/(α[1]m[ν]) ∼ 10^6 meters and same order of magnitude as Earth radius. Matter should be U(1) plasma. U(1) superconductor would be second option.
See the new chapter Breaking of CP, P, and T in cosmological scales in TGD Universe of "Physics in Many-Sheeted Space-time" or the article with the same title.
For a summary of earlier postings see Latest progress in TGD.
The twistor lift of TGD forces the analog of Kähler form for M^4. Covariantly constant sef-dual Kähler form J(CD) depends on causal diamond of M^4 and defines rest frame and spin quantization axis.
This implies a violation of CP, P, and T. By introducing a moduli space for the Kähler forms one avoids the loss of Poincare invariance. The natural question is whether J(CD) could relate to CP
breaking for K and B type mesons, to matter antimatter asymmetry and the large scale parity breaking suggested by CMB data.
The simplest guess for the coupling strength of U(1) interaction associated with J(CD) predicts a correct order of magnitude for CP violation for K meson and for the antimatter asymmetry and inspires
a more detailed discussion. A general mechanism for the generation of matter asymmetry is proposed, and a model for the formation of disk- and elliptic galaxies is considered. The matter antimatter
asymmetry would be apparent in the sense that the CP asymmetry would force matter-antimatter separation: antimatter would reside as dark matter (in TGD sense) inside magnetic flux tubes and matter
outside them. Also the angular momenta of dark matter and matter would compensate each other.
See the new chapter Breaking of CP, P, and T in cosmological scales in TGD Universe of "Physics in Many-Sheeted Space-time" or the article with the same title.
For a summary of earlier postings see Latest progress in TGD.
Yesterday evening I got an intereting idea related to both the definition and conservation of gauge charges in non-Abelian theories. First the idea popped in QCD context but immediately generalized
to electro-weak and gravitational sectors. It might not be entirely correct: I have not yet checked the calculations.
QCD sector
I have been working with possible TGD counterparts of so called chiral magnetic effect (CME) and chiral separation effect (CSE) proposed in QCD to describe observations at LHC and RHIC suggesting
relatively large P and CP violations in hadronic physics associated with the deconfinement phase transition. See the recent article About parity violation in hadron physics).
The QCD based model for CME and CSE is not convincing as such. The model assumes that the theta parameter of QCD is non-vanishing and position dependent. It is however known that theta parameter is
extremal small and seems to be zero: this is so called strong CP problem of QCD caused by the possibility of istantons. The axion hypothesis could make θ(x) a dynamical field and θ parameter would be
eliminated from the theory. Axion has not however been however found: various candidates have been gradually eliminated from consideration!
What is the situation in TGD? In TGD instantons are impossible at the fundamental space-time level. This is due to the induced space-time concept. What this means for the QFT limit of TGD?
1. Obviously one must add to the action density a constraint term equal to Lagrange multiple θ times instanton density. If θ is constant the variation with respect to it gives just the vanishing of
instanton number.
2. A stronger condition is local and states that instanton density vanishes. This differs from the axion option in that there is no kinetic term for θ so that it does not propagate and does not
appear in propagators.
Consider the latter option in more detail.
1. The variation with respect to θ(x) gives the condition that instanton density rather than only instanton number vanishes for the allowed field configurations. This guarantees that axial current
having instanton term as divergence is conserved if fermions are massless. There is no breaking of chiral symmetry at the massless limit and no chiral anomaly which is mathematically problematic.
2. The field equations are however changed. The field equations reduce to the statement that the covariant divergence of YM current - sum of bosonic and fermionic contributions - equals to the
covariant divergence of color current associated with the constraint term. The classical gauge potentials are affected by this source term and they in turn affect fermionic dynamics via Dirac
equation. Therefore also the perturbation theory is affected.
3. The following is however still uncertain: This term seems to have vanishing ordinary total divergence by Bianchi identities - one has topological color current proportional to the contraction of
the gradient of θ and gauge field with 4-D permutation symbol! I have however not checked yet the details.
If this is really true then the sum of fermionic and bosonic gauge currents not conserved in the usual sense equals to a opological color current conserved in the usual sense! This would give
conserved total color charges as topological charges - in spirit with "Topological" in TGD! This would also solve a problem of non-abelian gauge theories usually put under the rug: the gauge
total gauge current is not conserved and a rigorous definition of gauge charges is lost.
4. What the equations of motion of ordinary QCD would mean in this framework? First of all the color magnetic and electric fields can be said to be orthogonal with respect to the natural inner
product. One can have also solutions for which θ is constant. This case gives just the ordinary QCD but without instantons and strong CP breaking. The total color current vanishes and one would
have local color confinement classically! This is true irrespective of whether the ordinary divergence of color currents vanishes.
5. This also allows to understand CME and CSE believed to occur in the deconfinement phase transition. Now regions with non-constant θ(x) but vanishing instanton density are generated. The sum of
the conserved color charges for these regions - droplets of quark-gluon plasma - however vanish by the conservation of color charges. One would indeed have non-vanishing local color charge
densities and deconfinement in accordance with the physical intuition and experimental evidence. This could occur in proton-nucleon and nucleon-nucleon collisions at both RHIC and LHC and give
rise to CME and CSE effects. This picture is however essentially TGD based. QCD in standard form does not give it and in QCD there are no motivations to demand that instanton density vanishes.
Electroweak sector
The analog of θ (x) is present also at the QFT limit of TGD in electroweak sector since instantons must be absent also now. One would have conserved total electroweak currents - also Abelian U(1)
current reducing to topological currents, which vanish for θ(x)= constant but are non-vanishing otherwise. In TGD the conservation of em charge and possibly also Z^0 charge is understood if strong
form of holography (SH) is accepted: it implies that only electromagnetic and possibly also Z^0 current are conserved and are assignable to the string world sheets carrying fermions. At QFT limit one
would obtain reduction of electroweak currents to topological currents if the above argument is correct. The proper understanding of W currents at fundamental level is however still lacking.
It is now however not necessary to demand the vanishing of instanton term for the U(1) factor and chiral anomaly for pion suggest that one cannot demand this. Also the TGD inspired model for so
called leptohadrons is based on the non-vanishing elecromagnetic instanton density. In TGD also M^4 Kähler form J(CD) is present and same would apply to it. If one applies the condition empty
Minkowski space ceases to be an extremal.
Gravitational sector
Could this generalize also the GRT limit of TGD? In GRT momentum conservation is lost - this one of the basic problems of GRT put under the rug. At fundamental level Poincare charges are conserved in
TGD by the hypothesis that the space-time is 4-surface in M^4 × CP[2]. Space-time symmetries are lifted to those of M^4.
What happens at the GRT limit of TGD? The proposal has been that covariant conservation of energy momentum tensor is a remnant of Poincare symmetry. But could one obtain also now ordinary
conservation of 4- momentum currents by adding to the standard Einstein-YM action a Lagrange multiplier term guaranteing that the gravitational analog of instanton term vanishes?
1. First objection: This makes sense only if vier-bein is defined in the M^4 coordinates applying only at GRT limit for which space-time surface is representable as a graph of a map from M^4 to CP
2. Second objection: If metric tensor is regarded as a primary dynamical variable, one obtains a current which is symmetry 2-tensor like T and G. This cannot give rise to a conserved charges.
3. Third objection: Taking vielbein vectors e^A[μ] as fundamental variable could give rise to a conserved vector with vanishing covariant divergence. Could this give rise to conserved currents
labelled by A and having interpretation as momentum components? This does not work. Since e^A[μ] is only covariantly constant one does not obtain genuine conservation law except at the limit of
empty Minkowski space since in this case vielbein vectors can be taken to be constant.
Despite this the addition of the constraint term changes the interpretation of GRT profoundly.
1. Curvature tensor is indeed essentially a gauge field in tangent space rotation group when contracted suitably by two vielbein vectors e^A[μ] and the instanton term is formally completely
analogous to that in gauge theory.
2. The situation is now more complex than in gauge theories due to the fact that second derivatives of the metric and - as it seems - also of vielbein vectors are involved. They however appear
linearly and do not give third order derivatives in Einstein's equations. Since the physics should not depend on whether one uses metric or vielbein as dynamical variables, the conjecture is that
the variation states that the contraction of T-kG with vielbein vector equals to the topological current coming from instanton term and proportional to the gradient of θ
(T-kG)^μν e^A[ν] =j^Aμ.
The conserved current j^Aμ would be contraction of the instanton term with respect to e^A[μ] with the gradient of θ suitably covariantized. The variation of the action with respect to the the
gradient of e^A[μ] would give it. The resulting current has only vanishing covariant divergence to which vielbein contributes.
The multiplier term guaranteing the vanishing of the gravitational instanton density would have however highly non-trivial and positive consequences.
1. The covariantly conserved energy momentum current would be sum of parts corresponding to matter and gravitational field unlike in GRT where the field equations say that the energy momentum
tensors of gravitational field and matter field are identical. This conforms with TGD view at the level of many-sheeted space-time.
2. In GRT one has the problem that in absence of matter (pure gravitational radiation) one obtains G=0 and thus vacuum solution. This follows also from conformal invariance for solutions
representing gravitational radiation. Thanks to LIGO we however now know that gravitational radiation carries energy! Situation for TGD limit would be different: at QFT limit one can have
classical gravitational radiation with non-vanishing energy momentum density
thanks the vanishing of instanton term.
See the article About parity violation in hadron physics
For background see the chapters New Physics Predicted by TGD: Part I.
For a summary of earlier postings see Latest progress in TGD.
Strong interactions involve small CP violation revealing in the physics of neutral kaon and B meson. An interesting question is whether CP violation and also P violation could be seen also in
hadronic reactions. QCD allows strong CP violation due to instantons. No strong CP breaking is observed, and Peccei-Quinn mechanism involving axion as a new but not yet detected particle is hoped to
save the situation.
The de-confinement phase transition is believed to occur in heavy nucleus collisions and be accompanied by a phase transition in which chiral symmetry is restored. It has been conjectured that this
phase transition involves large P violation assignable to so called chiral magnetic effect (CME) involving separation of charge along the axis of magnetic field generated in collision, chiral
separation effect (CSE), and chiral magnetic wave (CMW). There is some evidence for CME and CSE in heavy nucleus collisions at RHIC and LHC. There is however also evidence for CME in proton-nucleus
collisions, where it should not occur.
In TGD instantons and strong CP violation are absent at fundamental level. The twistor lift of TGD however predicts weak CP, T, and P violations in all scales and it is tempting to model
matter-anti-matter asymmetry, the generation of preferred arrow of time, and parity breaking suggested by CBM anomalies in terms of these violations. The reason for the violation is the analog of
self-dual covariantly constant Kähler form J(CD) for causal diamonds CD⊂ M^4 defining parallel constant electric and magnetic fields. Lorentz invariance is not lost since one has moduli space
containing Lorentz boosts of CD and J(CD). J(CD) induced to the space-time surface gives rise to a new U(1) gauge field coupling to fermion number. Correct order of magnitude for the violation for K
and B mesons is predicted under natural assumptions. In this article the possible TGD counterparts of CME, CSE, and CMW are considered: the motivation is the presence of parallel E and B essential
for CME.
See the article About parity violation in hadron physics
For background see the chapters New Physics Predicted by TGD: Part I.
For a summary of earlier postings see Latest progress in TGD.
The earlier posting What could be the role of complexity theory in TGD? was an abstract of an article about how complexity theory based thinking might help in attempts to understand the emergence of
complexity in TGD. The key idea is that evolution corresponds to an increasing complexity for Galois group for the extension of rationals inducing also the extension used at space-time and Hilbert
space level. This leads to rather concrete vision about what happens and the basic notions of complexity theory helps to articulate this vision more concretely.
Also new insights about how preferred p-adic primes identified as ramified primes of extension emerge. The picture suggests strong resemblance with the evolution of genetic code with conserved genes
having ramified primes as their analogs. Category theoretic thinking in turn suggests that the positions of fermions at partonic 2-surfaces correspond to singularities of the Galois covering so that
the number of sheets of covering is not maximal and that the singularities has as their analogs what happens for ramified primes.
p-Adic length scale hypothesis states that physically preferred p-adic primes come as primes near prime powers of two and possibly also other small primes. Does this have some analog to complexity
theory, period doubling, and with the super-stability associated with period doublings?
Also ramified primes characterize the extension of rationals and would define naturally preferred primes for a given extension.
1. Any rational prime p can be decomposes to a product of powers P^k[i] of primes P[i] of extension given by p= ∏[i] P[i]^k[i], ∑ k[i]=n. If one has k[i]≠ 1 for some i, one has ramified prime. Prime
p is Galois invariant but ramified prime decomposes to lower-dimensional orbits of Galois group formed by a subset of P[i]^k[i] with the same index k[i] . One might say that ramified primes are
more structured and informative than un-ramified ones. This could mean also representative capacity.
2. Ramification has as its analog criticality leading to the degenerate roots of a polynomial or the lowering of the rank of the matrix defined by the second derivatives of potential function
depending on parameters. The graph of potential function in the space defined by its arguments and parameters if n-sheeted singular covering of this space since the potential has several extrema
for given parameters. At boundaries of the n-sheeted structure some sheets degenerate and the dimension is reduced locally . Cusp catastrophe with 3-sheets in catastrophe region is standard
example about this.
Ramification also brings in mind super-stability of n-cycle for the iteration of functions meaning that the derivative of n:th iterate f(f(...)(x)== f^n)(x) vanishes. Superstability occurs for
the iteration of function f= ax+bx^2 for a=0.
3. I have considered the possibility that that the n-sheeted coverings of the space-time surface are singular in that the sheet co-incide at the ends of space-time surface or at some partonic
2-surfaces. One can also consider the possibility that only some sheets or partonic 2-surfaces co-incide.
The extreme option is that the singularities occur only at the points representing fermions at partonic 2-surfaces. Fermions could in this case correspond to different ramified primes. The graph
of w=z^1/2 defining 2-fold covering of complex plane with singularity at origin gives an idea about what would be involved. This option looks the most attractive one and conforms with the idea
that singularities of the coverings in general correspond to isolated points. It also conforms with the hypothesis that fermions are labelled by p-adic primes and the connection between
ramifications and Galois singularities could justify this hypothesis.
4. Category theorists love structural similarities and might ask whether there might be a morphism mapping these singularities of the space-time surfaces as Galois coverings to the ramified primes
so that sheets would correspond to primes of extension appearing in the decomposition of prime to primes of extension.
Could the singularities of the covering correspond to the ramification of primes of extension? Could this degeneracy for given extension be coded by a ramified prime? Could quantum criticality of
TGD favour ramified primes and singularities at the locations of fermions at partonic 2-surfaces?
Could the fundamental fermions at the partonic surfaces be quite generally localize at the singularities of the covering space serving as markings for them? This also conforms with the assumption
that fermions with standard value of Planck constants corresponds to 2-sheeted coverings.
5. What could the ramification for a point of cognitive representation mean algebraically? The covering orbit of point is obtained as orbit of Galois group. For maximal singularity the Galois orbit
reduces to single point so that the point is rational. Maximally ramified fermions would be located at rational points of extension. For non-maximal ramifications the number of sheets would be
reduced but there would be several of them and one can ask whether only maximally ramified primes are realized. Could this relate at the deeper level to the fact that only rational numbers can be
represented in computers exactly.
6. Can one imagine a physical correlate for the singular points of the space-time sheets at the ends of the space-time surface? Quantum criticality as analogy of criticality associated with
super-stable cycles in chaos theory could be in question. Could the fusion of the space-time sheets correspond to a phenomenon analogous to Bose-Einstein condensation? Most naturally the
condensate would correspond to a fractionization of fermion number allowing to put n fermions to point with same M^4 projection? The largest condensate would correspond to a maximal ramification
p= P[i]^n.
Why ramified primes would tend to be primes near powers of two or of small prime? The attempt to answer this question forces to ask what it means to be a survivor in number theoretical evolution. One
can imagine two kinds of explanations.
1. Some extensions are winners in the number theoretic evolution, and also the ramified primes assignable to them.
These extensions would be especially stable against further evolution representing analogs of evolutionary fossils.
As proposed earlier, they could also allow exceptionally large cognitive representations that is large number of points of real space-time surface in extension.
2. Certain primes as ramified primes are winners in the sense the further extensions conserve the property of being ramified.
1. The first possibility is that further evolution could preserve these ramified primes and only add new ramified primes. The preferred primes would be like genes, which are conserved during
biological evolution. What kind of extensions of existing extension preserve the already existing ramified primes. One could naively think that extension of an extension replaces P[i] in the
extension for P[i]= Q[ik]^k[i] so that the ramified primes would remain ramified primes.
2. Surviving ramified primes could be associated with a exceptionally large number of extensions and thus with their Galois groups. In other words, some primes would have strong tendency to
ramify. They would be at criticality with respect to ramification. They would be critical in the sense that multiple roots appear.
Can one find any support for this purely TGD inspired conjecture from literature? I am not a number theorist so that I can only go to web and search and try to understand what I found. Web
search led to a thesis (see this) studying Galois group with prescribed ramified primes.
The thesis contained the statement that not every finite group can appear as Galois group with prescribed ramification. The second statement was that as the number and size of ramified primes
increases more Galois groups are possible for given pre-determined ramified primes. This would conform with the conjecture. The number and size of ramified primes would be a measure for
complexity of the system, and both would increase with the size of the system.
3. Of course, both mechanisms could be involved.
Why ramified primes near powers of 2 would be winners? Do they correspond to ramified primes associated with especially many extension and are they conserved in evolution by subsequent extensions of
Galois group. But why? This brings in mind the fact that n=2^k-cycles becomes super-stable and thus critical at certain critical value of the control parameter. Note also that ramified primes are
analogous to prime cycles in iteration. Analogy with the evolution of genome is also strongly suggestive.
For details see the chapter Unified Number Theoretic Vision or the article What could be the role of complexity theory in TGD?.
For a summary of earlier postings see Latest progress in TGD.
The previous posting What could be the role of complexity theory in TGD? was an abstract of an article about how complexity theory based thinking might help in attempts to understand the emergence of
complexity in TGD. The key idea is that evolution corresponds to an increasing complexity for Galois group for the extension of rationals inducing also the extension used at space-time and Hilbert
space level. This leads to rather concrete vision about what happens and the basic notions of complexity theory helps to articulate this vision more concretely.
I ended up to rather interesting information theoretic interpretation about the understanding of effective Planck constant assigned to flux tubes mediating as gravitational/electromagnetic/etc...
interactions. The real surprise was that this leads to a proposal how mono-cellulars and multicellulars differ! The emergence of multicellulars would have meant emergence of systems with mass larger
than critical mass making possible gravitational quantum coherence. Penrose's vision about the role of gravitation would be correct although Orch-OR as such has little to do with reality!
The natural hypothesis is that h[eff]/h=n equals to the order of Galois group in the case that it gives the number of sheets of the covering assignable to the space-time surfaces. The stronger
hypothesis is that h[eff]/h=n is associated with flux tubes and is proportional to the quantum numbers associated with the ends.
1. The basic idea is that Mother Nature is theoretician friendly. As perturbation theory breaks down, the interaction strength expressible as a product of appropriate charges divided by Planck
constant, is reduced in the phase transition hbar→ hbar[eff].
2. In the case of gravitation GMm→ = GMm (h/h[eff]). Equivalence Principle is satisfied if one has hbar[eff]=hbar[gr] = GMm/v[0], where v[0] is parameter with dimensions of velocity and of the order
of some rotation velocity associated with the system. If the masses move with relativistic velocities the interaction strength is proportional to the inner product of four-momenta and therefore
to Lorentz boost factors for energies in the rest system of the entire system. In this case one must assume quantization of energies to satisfy the constraint or a compensating reduction of v[0].
Interactions strength becomes equal to β[0]= v[0]/c having no dependence on the masses: this brings in mind the universality associated with quantum criticality.
3. The hypothesis applies to all interactions. For electromagnetism one would have the replacements Z[1]Z[2]α→ Z[1]Z[2]α (h/ h[em]) and hbar[em]=Z[1]Z[2]α/&beta[0] giving Universal interaction
strength. In the case of color interactions the phase transition would lead to the emergence of hadron and it could be that inside hadrons the valence quark have h[eff]/h=n>1. In this case one
could consider a generalization in which the product of masses is replaced with the inner product of four-momenta. In this case quantization of energy at either or both ends is required. For
astrophysical energies one would have effective energy continuum.
This hypothesis suggests the interpretation of h[eff]/h=n as either the dimension of the extension or the order of its Galois group. If the extensions have dimensions n[1] and n[2], then the
composite system would be n[2]-dimensional extension of n[1]-dimensional extension and have dimension n[1]× n[2]. This could be also true for the orders of Galois groups. This would be the case if
Galois group of the entire system is free group generated by the G[1] and G[2]. One just takes all products of elements of G[1] and G[2] and assumes that they commute to get G[1]× G[2]. Consider
gravitation as example.
1. The order of Galois group should coincide with hbar[eff]/hbar=n= hbar[gr]/hbar= GMm/v[0]hbar. The transition occurs only if the value of hbar[gr]/hbar is larger than one. One can say that the
order of Galois group is proportional the product of masses using as unit Planck mass. Rather large extensions are involved and the number of sheets in the Galois covering is huge.
Note that it is difficult to say how larger Planck constants are actually involved since by gravitational binding the classical gravitational forces are additive and by Equivalence principle same
potential is obtained as sum of potentials for splitting of masses into pieces. Also the gravitational Compton length λ[gr]= GM/v[0] for m does not depend on m at all so that all particles have
same λ[gr]= GM/v[0] irrespective of mass (note that v[0] is expressed using units with c=1).
The maximally incoherent situation would correspond to ordinary Planck constant and the usual view about gravitational interaction between particles. The extreme quantum coherence would mean that
both M and m behave as single quantum unit. In many-sheeted space-time this could be understood in terms of a picture based on flux tubes. The interpretation for the degree of coherence is in
terms of flux tube connections mediating gravitational flux.
2. h[gr]/h would be order of Galois group, and there is a temptation to associated with the product of masses the product n=n[1]n[2] of the orders n[i] of Galois groups associated masses M and m.
The order of Galois group for both masses would have as unit m[P]/β[0]^1/2, β[0]=v[0]/c, rather than Planck mass m[P]. For instance, the reduction of the Galois group of entire system to a
product of Galois groups of parts would occur if Galois groups for M and m are cyclic groups with orders with have no common prime factors but not generally.
The problem is that the order of the Galois group associated with m would be smaller than 1 for masses m<m[P]/β[0]^1/2. Planck mass is about 1.3 × 10^19 proton masses and corresponds to a blob of
water with size scale 10^-4 meters - size scale of a large neuron so that only above these scale gravitational quantum coherence would be possible. For v[0]<1 it would seem that even in the case
of large neurons one must have more than one neurons. Maybe pyramidal neurons could satisfy the mass constraint and would represent higher level of conscious as compared to other neurons and
cells. The giant neurons discovered by the group led by Christof Koch in the brain of of mouse having axonal connections distributed over the entire brain might fulfil the constraint (see this).
3. It is difficult to avoid the idea that macroscopic quantum gravitational coherence for multicellular objects with mass at least that for the largest neurons could be involved with biology.
Multicellular systems can have mass above this threshold for some critical cell number. This might explain the dramatic evolutionary step distinguishing between prokaryotes (mono-cellulars
consisting of Archaea and bacteria including also cellular organelles and cells with sub-critical size) and eukaryotes (multi-cellulars).
4. I have proposed an explanation of the fountain effect appearing in super-fluidity and apparently defying the law of gravity. In this case m was assumed to be the mass of ^4He atom in case of
super-fluidity to explain fountain effect. The above arguments however allow to ask whether anything changes if one allows the blobs of superfluid to have masses coming as a multiple of m[P]/β[0]
^1/2. One could check whether fountain effect is possible for super-fluid volumes with mass below m[P]/β[0]^1/2.
What about h[em]? In the case of super-conductivity the interpretation of h[em]/h as product of orders of Galois groups would allow to estimate the number N= Q/2e of Cooper pairs of a minimal blob of
super-conducting matter from the condition that the order of its Galois group is larger than integer. The number N=Q/2e is such that one has 2N(α/β[0])^1/2=n. The condition is satisfied if one has α/
β[0]=q^2, with q=k/2l such that N is divisible by l. The number of Cooper pairs would be quantized as multiples of l. What is clear that em interaction would correspond to a lower level of cognitive
consciousness and that the step to gravitation dominated cognition would be huge if the dark gravitational interaction with size of astrophysical systems is involved \citeallbhgrprebio. Many-sheeted
space-time allows this in principle.
These arguments support the view that quantum information theory indeed closely relates not only to gravitation but also other interactions. Speculations revolving around blackhole, entropy, and
holography, and emergence of space would be replaced with the number theoretic vision about cognition providing information theoretic interpretation of basic interactions in terms of entangled tensor
networks (see this). Negentropic entanglement would have magnetic flux tubes (and fermionic strings at them) as topological correlates. The increase of the complexity of quantum states could occur by
the "fusion" of Galois groups associated with various nodes of this network as macroscopic quantum states are formed. Galois groups and their representations would define the basic information
theoretic concepts. The emergence of gravitational quantum coherence identified as the emergence of multi-cellulars would mean a major step in biological evolution.
For details see the chapter Unified Number Theoretic Vision or the article What could be the role of complexity theory in TGD?.
For a summary of earlier postings see Latest progress in TGD.
Chaotic (or actually extremely complex and only apparently chaotic) systems seem to be the diametrical opposite of completely integrable systems about which TGD is a possible example. There is
however also something common: in completely integrable classical systems all orbits are cyclic and in chaotic systems they form a dense set in the space of orbits. Furthermore, in chaotic systems
the approach to chaos occurs via steps as a control parameter is changed. Same would take place in adelic TGD fusing the descriptions of matter and cognition.
In TGD Universe the hierarchy of extensions of rationals inducing finite-dimensional extension of p-adic number fields defines a hierarchy of adelic physics and provides a natural correlate for
evolution. Galois groups and ramified primes appear as characterizers of the extensions. The sequences of Galois groups could characterize an evolution by phase transitions increasing the dimension
of the extension associated with the coordinates of "world of classical worlds" (WCW) in turn inducing the extension used at space-time and Hilbert space level. WCW decomposes to sectors
characterized by Galois groups G[3] of extensions associated with the 3-surfaces at the ends of space-time surface at boundaries of causal diamond (CD) and G[4] characterizing the space-time surface
itself. G[3] (G[4]) acts on the discretization and induces a covering structure of the 3-surface (space-time surface). If the state function reduction to the opposite boundary of CD involves
localization into a sector with fixed G[3], evolution is indeed mapped to a sequence of G[3]s.
Also the cognitive representation defined by the intersection of real and p-adic surfaces with coordinates of points in an extension of rationals evolve. The number of points in this representation
becomes increasingly complex during evolution. Fermions at partonic 2-surfaces connected by fermionic strings define a tensor network, which also evolves since the number of fermions can change.
The points of space-time surface invariant under non-trivial subgroup of Galois group define singularities of the covering, and the positions of fermions at partonic surfaces could correspond to
these singularities - maybe even the maximal ones, in which case the singular points would be rational. There is a temptation to interpret the p-adic prime characterizing elementary particle as a
ramified prime of extension having a decomposition similar to that of singularity so that category theoretic view suggests itself.
One also ends up to ask how the number theoretic evolution could select preferred p-adic primes satisfying the p-adic length scale hypothesis as a survivors in number theoretic evolution, and ends up
to a vision bringing strongly in mind the notion of conserved genes as analogy for conservation of ramified primes in extensions of extension. h[eff]/h=n has natural interpretation as the order of
Galois group of extension. The generalization of hbar[gr]= GMm/v[0]=hbar[eff] hypothesis to other interactions is discussed in terms of number theoretic evolution as increase of G[3], and one ends up
to surprisingly concrete vision for what might happen in the transition from prokaryotes to eukaryotes.
For details see the chapter Unified Number Theoretic Vision or the article What could be the role of complexity theory in TGD?.
For a summary of earlier postings see Latest progress in TGD.
One problem of ΛCDM scenario is missing of matter and dark matter in some places (see this). There missing dark matter in the scale of R=.2 Gy and also in the vicinity of solar system in the scale
1.5-4 kpc.
In the work titled "Missing Dark Matter in the Local Universe", Igor D. Karachentsev studied a sample of 11,000 galaxies in the local Universe around the MW (see this). He summed up the masses of
individual galaxies and galaxy-groups and used this to test a very fundamental prediction of ΛCDM.
1. Standard cosmology predicts the average fraction of matter density to be Ω[m,glob]=28+/- 3 per cent of critical mass density (83 percent of this would be dark and 17 per cent visible matter). 72
per cent would be dark energy, 28 per cent dark matter, and 4.8 per cent visible matter.
To test this one can simply sum up all the galactic masses in some volume Karachentsev chose the volume to be a sphere of radius R= .2 Gy surrounding Milky Way and containing 11,000 galaxies. In
this scale the density is expected to fluctuate only 10 per cent. Note that horizon radius is estimated to be about R[H]=14 Gly giving R[H]= 70 R.
2. The visible galactic mass in certain large enough volume of space was estimated as also the sum of galactic dark masses estimated as so called virial mass (see this). The sum of these masses gave
the estimate for the total mass.
3. The estimate for the total mass (dark matter plus visible matter assuming halo model) in a volume of radius .2 Gy gives Ω[m,glob]=8+/- 3 per cent, which is only 28 per cent of the predicted
fraction. The predicted fraction of visible matter is 4.8 per cent and marginally consistent with 8+/- 3 per cent but it seems plausible that also dark matter is present although its amount is
much smaller than expected. The total contribution to the dark matter could be at most of the same size as that of visible matter.
4. One explanation is that all matter has not been included. Second not very plausible explanation is that the measurement region corresponds to a region with abnormally low density.
Can on understand the finding in TGD framework?
1. In TGD based model part of dark energy/matter would reside at the long flux tubes with which galaxies form bound states. Constraints come from accelerated expansion and galactic velocity curves
allowing to determine string tension for given galaxy. Let us assume that the GRT limit of TGD and its predictions hold true.
The estimate for the virial mass assumes that galaxy's dark mass forms a halo. The basic observation is that in TGD flux tubes give the dark energy and mass and virial mass would underestimate
the the dark mass of the galaxy.
2. How long length of the flux tube effectively corresponds to the dark and visible mass of disk galaxy? This length should be roughly the length containing the dark mass and energy estimated from
cosmology: L=M[dark]/T. If GRT limit of TGD makes sense, one has M[dark] =xM[vis]/T, where M[dark] is the amount of dark energy + matter associated with the flux tube, M[vis] is visible mass, x≈
ρ[dark]/ρ[vis]≈ 83/17 , and T is string tension deduced from the asymptotic rotation velocity.
If these segments do not cover the entire flux tubes containing the galaxies along it, the amount of dark matter and energy will be underestimated. By the above argument elliptic galaxies would
not have considerable amount of dark matter and energy so that only disk galaxies should contribute unless there are flux tubes in shorter scales inside elliptic galaxies.
Also larger and smaller scale flux tube structures contribute to the dark energy + dark matter. Fractality suggests the presence of both larger and smaller flux tube structures than those
associated with spiral galaxies (even stars couldbe associated with flux tubes).
One should have estimates for the lengths of various flux tubes involved. Unfortunately this kind of estimate is not available.
3. If GRT limit makes sense then the total dark mass then the dark energy and matter obtained in this manner should give 95 per cent of critical mass density. The fraction of dark matter included
would be at most a fraction 5/28≈ 18 per cent of the total dark matter. 82 per cent of dark matter and energy would be missed in the estimate. This could allow to get some idea about the lengths
of flux tubes and density of galaxies along flux tubes.
The amount of dark matter in the solar neighborhood was investigated in the work "Kinematical and chemical vertical structure of the Galactic thick disk II. A lack of dark matter in the solar
neighborhood" by Christian Moni Bidin and collaborators (see this). Moni Bidin et al have studied as sample of 400 red giants in the vicinity of solar system at vertical distances 1.5 to 4 kpc and
deduce 3-D kinematics for these start. From this data they estimate the surface mass density of the Milky Way within this range of heights from the disk. This surface density should be sum of both
visual and dark mass.
According to their analysis, the visible mass is enough to explain the data. No additional mass is needed. Only highly flattened dark matter halo would be consistent with the findings. This conforms
with the TGD prediction that dark mass/energy are associated with magnetic flux tubes.
For a summary of earlier postings see Latest progress in TGD.
This posting is an good example of blunders that one cannot avoid when targeted by huge information torrents! The article telling about the bump was one year old. Thanks for "Mitchell"! I however
want to leave posting here since it I have a strong suspicion that the M[89] physics is indeed there. It might serve as a reminder!
Extremely interesting finding at LHC. Not 5 sigma finding but might be something real. There is evidence for the existence of a meson with mass 750 GeV decaying to gamma pair. The only reasonable
candidate is pseudo-scalar or scalar meson.
What says TGD? M[89] hadron physics is the basic "almost prediction" of TGD. Mass scale is scaled up from that of ordinary hadron physics characterized by M[107] by a factor of 512.
About two handfuls of bumps with masses identifiable in TGD framework as scaled up masses for the mesons of ordinary hadron physics have been reported. See the article. The postings of Lubos trying
to interpret the bumps as Higgses predicted by SUSY have been extremely helpful. No-one in the hegemony has of course taken this proposal seriously and the bumps have been forgotten since people have
been trying to find SUSY and dark matter particles, certainly not TGD!
What about this new bump? It has mass about 750 GeV. The scaling down by 1/512 gives mass about 1.465 GeV
for the corresponding meson of the ordinary hadron physics. It should be flavorless and have spin 0 and would be most naturally pseudoscalar.
By going to Particle Data Tables and clicking "Mesons" and looking for "Light Unflavored Mesons" one finds that there are several unflavored mesons with mass near 1.465 GeV. Vector mesons do not
decay to gamma pair and also most pseudoscalar mesons decay mostly via strong interactions. There is however only one decaying also to gamma pair: η(1475)! The error for the predicted mass is 1.3 per
There are of many other ordinary mesons decaying also to gamma pairs and LHC might make history of science by trying to find them at masses scaled up by 512.
See the article M[89] Hadron Physics and Quantum Criticality or the chapter New Particle Physics Predicted by TGD: Part I of p-Adic Physics.
For a summary of earlier postings see Latest progress in TGD.
The anomalies of the halo model of dark matter have begun to accumulate rapidly. The problems of the halo model are discussed in detail in the blog "Dark matter crisis" of Prof. Pavel Kroupa and
Marcel S. Pawlowski (see this). MOND is the most well-known competitor of the halo model for dark matter but has its own problems. TGD is less known alternative for the halo model. In the following
brief comments about Zwicky paradox (see this) implying that neither cold nor warm dark matter particles in the usual sense (different from that in TGD based model) can play a significant role in
The standard/concordance model of dark matter relies on two hypothesis formulated originally by Zwicky assuming that a) GRT is correct in all scales and b) all matter is created during Big Bang.
Zwicky formulated two hypothesis (for references see the article) leading to the halo model of dark matter and also to Zwicky paradox.
1. Zwicky noticed (1937) that galaxies must about 500 heavier in the Coma galaxy cluster than judged from their light emission: cold or hot dark matter halo must exist. Note that this does not
actually require that the dark matter consists of some exotic particles or that the dark matter forms halos. To get historical perspective note that Vera Rubin published 1976 an article about the
constancy of velocity curves for distant stars for Andromeda which is spiral galaxy.
2. Zwicky noticed (1956) that when galaxies collide, the expelled matter can condense in new regions and form new smaller dwarf galaxies. These so called tidal galaxies are thus formed from the
collisional debris of other galaxies.
From these observations one ends up with a computer model allowing to simulate the formation of galaxies (for a detailed discussion see this). The basic elements of the model are collisions of
galaxies possibly leading to a fusion and formation of tidal galaxies. The model assumes a statistical distribution of dark matter lumps defining the halos of the dwarf galaxies formed in the
The model predicts a lot of dark matter dominated dwarf galaxies formed around the dark matter lumps: velocity spectrum should approach constant. There are also tidal dwarf galaxies formed from
collision debris of other galaxies. Unless also now condensation around a dark matter lump is involved, these should not contain dark matter and velocity spectrum for tidal dwarfs should be
declining. It turns out that tidal dwarfs alone are able to explain the observed dwarf galaxies, which are typically elliptic. Furthermore, there is no empirical manner to distinguish between tidal
dwarfs and other dwarfs.
Do the elliptic galaxies contain dark matter? What does one know about the rotation curves of elliptic galaxies? There is an article "The rotation curves of elliptic galaxies" of J. Binney published
around 1979 about the determination of the rotation curves of elliptic galaxies giving also some applications (see this). The velocity curves are declining as if no dark matter were present.
Therefore dark matter would not be present in dwarf galaxies so that the prediction of the halo model would be wrong.
Could this finding be also a problem for MOND? Assuming that the laws governing gravitation are modified for small accelerations, shouldn't elliptic and spiral galaxies have similar velocity curves?
What about TGD?
1. In TGD Universe dark energy and matter reside at flux tubes along which disk galaxies condense like pearls in string.
2. The observation about velocity curves suggests a TGD based explanation for the difference between elliptic and spiral galaxies. Elliptic galaxies - in particular tidal dwarfs - are not associated
with a flux tube containing dark matter. Spiral galaxy can form as elliptic galaxy if it becomes bound with flux tube as the recent finding about declining velocity curves for galaxies with age
about 10 Gy suggest. Dark matter would not be present in dwarf galaxies so that the prediction of the halo model is wrong. This also conforms with the fact that the stars in elliptic galaxies are
much older than in spiral galaxies (see this).
3. Dwarf galaxies produced from the collision debris contain only ordinary matter. Elliptic galaxies can later condense around magnetic flux tubes so that velocity spectrum approaches constant at
large distances. The breaking of spherical symmetry to cylindrical symmetry might allow to understand why the oblate spheroidal shape is flattened to that of disk.
For a summary of earlier postings see Latest progress in TGD.
The exciting question is what the superposition of causal orders could mean from the point of view of conscious experience. What seems obvious is that in the superposition of selves with opposite
arrows of clock-time there should be no experience about the flow of time in definite direction. Dissipation is associated with the thermodynamical arrow of time. Therefore also the sensory
experience about dissipation expected to have unpleasant emotional color should be absent. This brings in mind the reports of meditators about experiences of timelessness. These states are also
characterized by words like "bliss" and "enlightenment".
Why I find this aspects so interesting is due to my personal experience for about 32 years ago. I of course know that this kind of personal reminiscences in an article intended to be scientific, is
like writing one's own academic death sentence. But I also know I long ago done this so that I have nothing to lose! The priests of the materialistic church will never bother to take seriously
anything that I have written so that it does not really matter! This experience - I dared to talk about enlightenment experience - changed my personal life profoundly, and led to the decision to
continue work with TGD instead of doing full-day job to make money and keeping TGD as a kind of hobby. The experience also forced to realize that our normal conscious experience is only a dim shadow
of what it can be and stimulated the passion to understand consciousness.
In this experience my body went to a kind of light flowing state: liquid is what comes in mind. All unpleasant sensations in body characterizing the everyday life (at least mine!) suddenly
disappeared as this phase transition propagated through my body. As a physicist I characterized this as absence of dissipation, and I talked to myself about a state of whole-body consciousness.
There was also the experience about moving in space in cosmic scales and the experience about the presence of realities very different the familiar one. Somehow I saw these different worlds from
above, in bird's eye of view. I also experienced what I would call time travel and re-incarnation in some other world.
Decades later I would ask whether my sensory consciousness could have been replaced with that only about my magnetic body only. In the beginning of the experience there was indeed a concrete feeling
that my body size had increased with some factor. I even had the feeling the factor was about 137 (inverse of the fine structure constant) but this interpretation was probably forced by my attempt to
associate the experience with something familiar to physicist! Although I did all the time my best to understand what I was experiencing, I did not direct my attention to my time experience, and
cannot say whether I experienced the presence or absence of time or time flow.
Towards the end of the experience I was clinically unconscious for about day or so. I was however conscious. For instance, I experienced quite concretely how the arrow of time flow started to
fluctuate forth and back. I somehow knew that permanent change would mean death and I was fighting to preserve the usual arrow of time. My childhood friend, who certainly did not know much about
physics, told about about alternation of the arrow of time during a state that was classified by psychiatrists as an acute psychosis.
See the chapter Topological Quantum Computation in TGD Universe and the article Quantum computations without definite causal structure: TGD view
For a summary of earlier postings see Latest progress in TGD. | {"url":"https://matpitka.blogspot.com/2017/04/","timestamp":"2024-11-06T04:38:42Z","content_type":"application/xhtml+xml","content_length":"266436","record_id":"<urn:uuid:b3dd16d0-3a3d-4b98-b168-9c1216ea5bbc>","cc-path":"CC-MAIN-2024-46/segments/1730477027909.44/warc/CC-MAIN-20241106034659-20241106064659-00423.warc.gz"} |
Car Loan Calculator - Rayne Finance
Understand your car loan repayment.
There are several factors that determine what your car loan repayments will be so it’s worthwhile having a broker on your side to help you understand your options.
Click on the Get Started button below and I will get back to you as soon as possible.
Use the calculator to see how quickly you could pay off your loan with an offset account.
How much would a honeymoon loan save you?
How much could you save by making extra repayments?
How much will stamp duty cost?
Compare your repayment options.
Impact of balloon payments.
A guide to find out how much you could save.
A guide to see how the numbers stack up between two loans.
A guide to understand how much your loan repayments might be.
Help understand your income and expenses, and where you could make savings. | {"url":"https://rayne.finance/2023/09/05/car-loan-calculator/","timestamp":"2024-11-09T03:06:29Z","content_type":"text/html","content_length":"281968","record_id":"<urn:uuid:7d90a48b-65f8-4f2c-8ded-6e4a98f288e0>","cc-path":"CC-MAIN-2024-46/segments/1730477028115.85/warc/CC-MAIN-20241109022607-20241109052607-00775.warc.gz"} |
numpy.minimum(x1, x2, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature, extobj]) = <ufunc 'minimum'>¶
Element-wise minimum of array elements.
Compare two arrays and returns a new array containing the element-wise minima. If one of the elements being compared is a NaN, then that element is returned. If both elements are NaNs then the
first is returned. The latter distinction is important for complex NaNs, which are defined as at least one of the real or imaginary parts being a NaN. The net effect is that NaNs are propagated.
x1, x2 : array_like
The arrays holding the elements to be compared. If x1.shape != x2.shape, they must be broadcastable to a common shape (which becomes the shape of the output).
out : ndarray, None, or tuple of ndarray and None, optional
A location into which the result is stored. If provided, it must have a shape that the inputs broadcast to. If not provided or None, a freshly-allocated array is returned. A tuple
(possible only as a keyword argument) must have length equal to the number of outputs.
where : array_like, optional
This condition is broadcast over the input. At locations where the condition is True, the out array will be set to the ufunc result. Elsewhere, the out array will retain its
original value. Note that if an uninitialized out array is created via the default out=None, locations within it where the condition is False will remain uninitialized.
For other keyword-only arguments, see the ufunc docs.
y : ndarray or scalar
The minimum of x1 and x2, element-wise. This is a scalar if both x1 and x2 are scalars.
See also
Element-wise maximum of two arrays, propagates NaNs.
Element-wise minimum of two arrays, ignores NaNs.
The minimum value of an array along a given axis, propagates NaNs.
The minimum value of an array along a given axis, ignores NaNs.
fmax, amax, nanmax
The minimum is equivalent to np.where(x1 <= x2, x1, x2) when neither x1 nor x2 are NaNs, but it is faster and does proper broadcasting.
>>> np.minimum([2, 3, 4], [1, 5, 2])
array([1, 3, 2])
>>> np.minimum(np.eye(2), [0.5, 2]) # broadcasting
array([[ 0.5, 0. ],
[ 0. , 1. ]])
>>> np.minimum([np.nan, 0, np.nan],[0, np.nan, np.nan])
array([nan, nan, nan])
>>> np.minimum(-np.Inf, 1) | {"url":"https://numpy.org/doc/1.17/reference/generated/numpy.minimum.html","timestamp":"2024-11-12T19:08:28Z","content_type":"text/html","content_length":"14457","record_id":"<urn:uuid:5592279a-3307-4535-9b4c-65410929f6a9>","cc-path":"CC-MAIN-2024-46/segments/1730477028279.73/warc/CC-MAIN-20241112180608-20241112210608-00689.warc.gz"} |
How To Understand What Is Meant By Dam
I hope the article below will help to explain what the term means and how its value is arrived at.
The term DAM is used at times but its correct term is 'thickness' between the two levels in the atmosphere. Remember although its often referred to at the 1000-500mb level it can be used between any
two levels. For snow forecasting the other most often used is the 1000-850mb values.
DAM heights or total thickness between two levels, usually the 1000mb and 500mb
I hope this may help (!) to show how complex is the relationship but also how relatively easy it is, knowing the two heights, to calculate the ‘thickness’. This can be done for any two heights.
The two most referred to, usually on Net Wx to do with the will it or won’t it snow, are the 1000-500mb and the 1000-850mb heights for ‘thicknesses.
Fortunately this has all been done for us by Paul and Karl with the charts shown below!
DAM is what refers to the 1000-500mb thickness chart.
Its rather complex but there are several ways to work out its value. Below are some of the methods which might help
= height (500 hPa surface) - height (1000 hPa surface) [ for those of you, like me, too old to catch up with all the changes the world brings, millibars = hPa!, so 500 hPa is exactly the same as 500
mb. ]
h(500) = h(1000)+h'(thickness). Or from that
Thickness can be calculated from the heights reported on a radio-sonde ascent, or a thermodynamic diagram can be used to add up the partial thicknesses over successive layers to achieve the net
(total) thickness.
An example of the former would be
500 hPa height = 5407 m
1000 hPa height = 23 m
Thickness = 5407-23 = 5384 m (or 538 dam)
Careful note must be made when the height of the 1000 hPa surface is below msl thus: 500 hPa height = 5524 m
1000 hPa height = - 13 m
Thickness = 5524 -(-13) = 5537 m (or 554 dam)
Note the example above when surface pressure is BELOW 1000mb.
Roughly it is taken that 8mb is equivalent to 6DM when forecasters are manually drawing the various upper and surface charts.
If we take the actual msl and 500mb chart from GFS/Extra for 06Z this morning, see below
On the left is the surface isobar chart with the 500mb height; to its right is the ‘thickness’ chart
Notice the differences in values between the left and right charts-obviously the surface values are identical but NOT the ‘thickness’ and 500mb values.
Or to look at how the 00z ascent for Herstmanceux differs in its 500mb height and its 500mb ‘thickness’
In the basic data format the 500mb height was given as 500.0 5490 -22.9 -50.9; i.e. 5490DM; that of the 1000mb height was 1000.0 87 8.2 5.6
The ‘thickness’ is 1000 hPa to 500 hPa thickness: 5403.00
How is that arrived at, see the formula above
100mb height is 87
500mb height is 5490
Therefore 500mb ‘thickness’=5490-87=5403DM
Additional information on atmospheric thickness and it's use is available on the NOAA National Weather Service website: https://www.weather.gov/source/zhu/ZHU_Training_Page/Miscellaneous/
John Holmes
Recommended Comments
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Evaluate Logarithms Using Properties Worksheets [PDF]: Algebra 2 Math
How Will This Worksheet on "Evaluate Logarithms Using Properties" Benefit Your Student's Learning?
• Using these properties, students can break down and simplify complicated logarithmic problems, making them easier to solve.
• Applying these rules helps students think critically and logically, which improves their overall problem-solving skills.
• Knowing these properties is key for higher-level math classes, like calculus and advanced algebra, where these concepts come up often.
• Using the product, quotient, and power rules makes logarithmic calculations faster and more efficient.
• Working with logarithmic properties helps students get better at algebra, especially with skills like factoring and combining terms.
How to Evaluate Logarithms Using Properties?
• Use the logarithmic properties to combine the expression into a single logarithm, often using the quotient rule to express the difference between logarithms and the product rule to express the
addition of logarithm.
• Simplify the resulting logarithmic expression by performing any necessary arithmetic inside the logarithm.
• If needed, rewrite any numbers inside the logarithm to match the base, using exponentiation to simplify further.
• If the expression is still not simplified, then again use properties of logarithm and simplify the expression to its final form.
Q. Use properties of logarithms to evaluate the expression. $\log_5 5 + \log_5 25$ | {"url":"https://www.bytelearn.com/math-algebra-2/worksheet/evaluate-logarithms-using-properties","timestamp":"2024-11-12T10:46:36Z","content_type":"text/html","content_length":"136935","record_id":"<urn:uuid:939dc423-5ede-4858-97d2-b786fe623deb>","cc-path":"CC-MAIN-2024-46/segments/1730477028249.89/warc/CC-MAIN-20241112081532-20241112111532-00660.warc.gz"} |
Intelligent Trading
Just wanted to notify viewers of a few great courses that are being offered free for auditing and/or participation by well known industry experts, including co-author of the classic text on AI,
'Artificial Intelligence: A Modern Approach,' Peter Norvig and Prof. Andrew Ng.
see also,
The notice is a bit late, but they are still accepting registrations.
There are many visual methods used to identify patterns in space and time. I've discussed some in prior threads and will show a few others briefly here. One of the most difficult questions I often
hear from others regarding markov type approaches, is how to identify states to be processed.
It is a similar problem that one encounters using simple linear type factor analysis. Unfortunately, there is no simple answer; however, because data streams are becoming so vast it becomes almost
impossible to enumerate over all possible state sets. Visual mining techniques can be incredibly helpful in narrowing down that space as well as feature reduction. I often use these types of
visualizations back and forth with unsupervised classification type learners to converge on useful state identifications.
Fig 1. Spatio-Temporal State plot
Figure 1 gives an idea on visualizing states with respect to time. But having such knowledge in isolation doesn't give us much use. We are more interested in looking for Bayesian type relationships
between states that give some probabilities between linked states in time.
Fig 2. Fluctuation Plot
Several visual methods exist to capture the relationships visually. One common plot used in language processing and information theory, is a fluctuation plot. The above plot was built using the same
state data as the first graph. It is often used to determine conditional relationships between symbols such as alphabet tokens. The size of each box is directly proportional to the weight of the
transition probabilities between row and column states in tabular data. An example would be to think of the letters yzy more commonly followed by g (as in syzygy) than any other state token; thus,
one would expect to quickly spot a larger box across a row of states representing the 'yzy' row token n-gram and 'g' column token .
Both plots were produced in R. ggflucuation() is a plot command utilized from ggplot2. I am currently investigating how much easier and faster it might be to process such visualizations in tools like
protovis and processing. I've been especially inspired by reading some of Nathan Yau's excellent visualization work in his book, 'Visualize This.' I included it in the link to the right for
interested readers.
I won't spend too much time discussing this fascinating topic other than to say it relates very much to prior discussions about pattern discovery via visual data mining (see lexical dispersion plots
for example). I happened across an interesting visualization method called the Arc Diagram, developed by Martin Wattenberg. Working for data visualization groups at IBM and later Google, he developed
some interesting visual representations of spatiotemporal data.
Fig 1. Arc Diagram and legend with example of discretized pattern archetype.
The resulting plot generates some fascinating temporal signatures, similar to what one might see in phase-space portraits from chaos. However, they have been frequently utilized to look for
spatiotemporal signatures in music. One might discern a type of underlying order or visual signature of complexity as well as recurring patterns in sequential objects ranging from text based lyrical
information to musical sheet notes.
Figure 1 shows an example of how one might utilize this tool towards temporal pattern discovery in time series. A weekly series from SPY has been discretized into alphabet tokens, based upon the bin
ranges in the included legend. The small chart in the example would decode an archetypal pattern for the following sequence: ECDCECCD, into a time series representation of the 8 week data symbol. The
following interactive java tool from another blogger, Neoformix, was then used to translate the data into an Arc Diagram. http://www.neoformix.com/Projects/DocumentArcDiagrams/index.html . Read from
top to bottom, one can look at recurring and related patterns that are repeated over time; certain behavior might warrant further investigation.
You can copy the following data stream into the tool to toy around with the tool to get a feel for the possibilities of visual pattern discovery.* I won't go into too much more detail about utilizing
it, other than to say it appears to be a very useful tool in temporal based pattern discovery.
Please see the following for more ideas on arc diagrams and musical signatures:
Blog mentioned:
* Not sure how to attach .xls file here, but if anyone wants a copy of the .xls file, you can send me an email and I'll try to get it out to you. Otherwise, you can simply grab a song lyric off the
web to play with the tool.
Today's financial headlines are littered with the word 'plunge.' Considering today's (cl-cl) drop on the S&P500 was just about -5%, I don't know that I would exactly call that a plunge.
Fig 1. Historical ts plot of S&P500 returns <= -5%
The following R code produced a time series plot of historical occasions where this occurred.
r05<-rtn[rtn<= -.05]
plot(sort(r05),type='o',main='S&P500 1950-present returns <= -5%')
Although the frequency of such occurrences is arguably rare, the 1987 drop is much more worthy of the 1 day label 'plunge.'
One other disturbing observation in the data, however, is the large temporal clustering of occurrences in the recent 2008 region. Now that's behavior to be concerned about (not to mention revised
flash crash data pts.).
filtered 1 day cl-cl returns <=-5% sorted by date
Although the following discussion can apply to the Quantitative Candlestick Pattern Recognition series, it is addressing the same issue as any basic conditional type system -- how and when to exit.
The following is one way to visualize and think about it, and is by no means optimal.
Fig 1. Posterior Boxplot Trajectory
Often we attempt to find some set of prior input patterns that leads to profitable posterior outcomes. However, in most of the available examples, we are typically only given heuristics and rules of
thumb on where to exit. This might make sense, since no one can accurately predict where to exit. However, with knowledge of past samples, we can have some idea of where a good target to exit might
be, given the prior knowledge of forward trajectories. I dubbed the name 'boxplot trajectory', here, as I think it's a useful way to visualize a group of many possible outcome trajectories for
further analysis.
In this example, a set of daily price based patterns was analyzed via a proprietary program I wrote in R, which resulted in an input pattern yielding a set of 52 samples that met my conditional
criteria. Fig 1 illustrates a way to look at the trajectory outcomes based upon one of the profitable patterns in the conditional criteria. The bottom graph is simply the plot of median results of
each data point in the trajectory. We often try to imagine the best way to exit without foreknowledge of the future (and somewhat less rule of thumb based criteria).
Fig 2. Trajectory tree.
One approach would be to use some type of exiting rule based upon the statistical median of each sequential point's range. Knowing that 1/2 of the vertices occur above and 1/2 below the median, we
should expect to hit at least 1/2 of the targets at or above the median. Given that the 3rd point is the highest median, it makes sense to exit earlier than waiting for a greater gain further out
(which has an even lower median). So if we take as a target, the median value of the 3rd pt. we achieve an average and fixed target of 1.59% on 27/52 of the total samples.
Of the remaining samples, we may now wish to exit on the 11th bar (or earlier if the same target is hit earlier) target of .556%, which is achieved on 13/52 of the remaining samples. This leaves only
the last bar of which we simply use the average return as the weighted return value for that target, in this case -1.74% for the remaining samples : 12/52. Notice we will always have the worse
contenders that were put off until the end.
The expectation yields E(rtn)=27/52*.0159+13/52*.0056-12/52*-.017 =.0057
eeking out a small average + gain of .57%. Compounded, this gives:
(1+.0159)^27*(1+.0056)^13*(1-.017)^12~ 34% rtn for 52 trades, each less than 3 days in length. Hit rate (as secondary observation) is 77% in this case.
The approach is particularly appealing for a high frequency strategy with very low commissions. Notice it's by no means comprehensive (and yes, I've only shown in sample here), but rather a novel way
to think about exiting strategies.
The following script allows you to simulate sample runs of Win, Loss, Breakeven streaks based on a random distribution, using the run length encoding function, rle in R. Associated probabilities are
entered as a vector argument in the sample function.
You can view the actual sequence of trials (and consequent streaks) by looking at the trades result. maxrun returns a vector of maximum number of Win, Loss, Breakeven streaks for each sample run. And
lastly, the prop table gives a table of proportion of run transition pairs from losing streak of length n to streak of all alternate lengths.
Example output (max run length of losses was 8 here):
lt.1 1 2 3 4 5 6 7 8
1 41.758 14.298 5.334 1.662 0.875 0.131 0.000 0.044
2 14.692 4.897 1.924 0.787 0.394 0.087 0.131 0.000
3 4.985 2.405 0.525 0.350 0.000 0.000 0.044 0.000
4 1.662 0.875 0.306 0.087 0.000 0.000 0.000 0.000
5 0.831 0.219 0.175 0.000 0.000 0.044 0.000 0.000
6 0.087 0.131 0.044 0.000 0.000 0.000 0.000 0.000
7 0.087 0.087 0.000 0.000 0.000 0.000 0.000 0.000
8 0.044 0.000 0.000 0.000 0.000 0.000 0.000 0.000
B L W
#generate simulations of win/loss streaks use rle function
#streaks of losing trades
#simple table of losing trade run streak(n) frequencies
#generate joint ensemble table streak(n) vs streak(n+1)
#convert to proportions
Nothing unusually exciting on this post, but I happened to be engaged in some particle based methods recently and made some simple visual observations as I was setting up some of the sampling
environment in R. I am also using Rkward and Ubuntu to generate, so I'm gathering everything from the current environment (including graphics).
Fig 1. Parallel plot of half hr sample of High and Low intraday data points vs time (Max is purple dots, Min are red). Fig 2. Cumulative count of high low events per interval (blue = total high and
The plot illustrates sampled intraday data at half hour increments.
The highs and lows of each sample interval are overlaid using purple to denote an intraday high and red to denote an intraday low.
Interesting points of observation are--
1) The high and low samples tend to be clustered at open, midday, and close.
2) High and low events do not appear to be uniformly and randomly distributed over time.
This kind of data processing is useful towards generating, exploring, and evaluating pattern based setups.
The study is by no means complete or conclusive, just stopping by to show more of the type of data processing and visual capabilities that R is capable of. If anyone has done any more conclusive
studies I'd be interested to hear.
P.S. If anyone notices any odd changes, for some reason Google was having some issues the last few days, and it appears to have reverted to my original (not ready to launch) draft.
Firstly, apologies for the long absence as I've been busy with a few things. Secondly, apologies for the horrific use of caps in the title (for the grammar monitors). Certainly, you'll gain something
useful from today's musing, as it's a pretty profound insight for most (was for me at the time). I've also considered carefully, whether or not to divulge this concept, but considering it's often
overlooked and in the public literature (I'll even share a source), I decided to discuss it.
Fig 1. Random Walk and the 75% rule
I've seen the same debate launched over and over on various chat boards, which concerns the impossibility of theoretically beating a random walk. In this case, I am giving you the code to determine
the answer yourself.
The requirements: 1) the generated data must be from an IID gaussian distribution 2) series must be coaxed to a stationary form.
The following script will generate a random series of data and follow the so called 75% rule which says,
Pr[Price>Price(n-1) & Price(n-1) < Price_median] Or [Price < Price(n-1) & Price(n-1) > Price_median] = 75%. This very insightful rule (which is explained both mathematically and in layman's terms in
the book 'Statistical Arbitrage' linked on the amazon box to the right), shows that given some stationary, IID, random sequence that has an underlying Gaussian distribution, the above rule set can be
shown to converge to a correct prediction rate of 75%!
Now, we all know that market data is not Gaussian (nor is it commision/slippage/friction free), and therein lies the rub. But hopefully, it gives you some food for thought as well as a bit of
knowledge to retort, when you hear the debates about impossibilities of beating a random walk.
R Code is below.
#gen rnd seq for 75% RULE
#generate stationary rw time series
for(i in 1:(length(rw)-1)){
if(rw[i] < m) trade[i]<- (rw[i+1]-rw[i])
if(rw[i] > m) trade[i]<- (rw[i]-rw[i+1])
if(rw[i] == m) trade[i]<- 0}
plot(rw,type='l',main='random walk') | {"url":"https://intelligenttradingtech.blogspot.com/2011/","timestamp":"2024-11-08T02:30:12Z","content_type":"application/xhtml+xml","content_length":"86931","record_id":"<urn:uuid:08171e09-1a70-4582-ae9d-3f844922d3ab>","cc-path":"CC-MAIN-2024-46/segments/1730477028019.71/warc/CC-MAIN-20241108003811-20241108033811-00207.warc.gz"} |
What is the correct spelling for lvfiii? | Spellchecker.net
The misspelling "lvfiii" can be corrected by suggesting the correct spelling "LVI" which represents the Roman numeral for 56. The addition of "I" denotes the number 1, subtracted from the value 5,
resulting in 4, representing 4 times 10 (40) added to 1 (41). | {"url":"https://www.spellchecker.net/misspellings/lvfiii","timestamp":"2024-11-05T02:18:32Z","content_type":"text/html","content_length":"36984","record_id":"<urn:uuid:416aa895-d8f0-4734-b71d-93abb83abd32>","cc-path":"CC-MAIN-2024-46/segments/1730477027870.7/warc/CC-MAIN-20241105021014-20241105051014-00838.warc.gz"} |
Graduate University Biomedical Mathematics Programme in English
Official length of programme
Two-year programme, 120 ECTS credits.
Access requirements
Applicants must have completed a European undergraduate university programme with at least 180 ECTS credits or an equivalent programme, providing them with relevant competencies in mathematics.
Name of qualification
Master of Science in Mathematics.
Programme requirements
The Graduate University Biomedical Mathematics Study Programme in English is the new programme designed and conducted by the Department of Mathematics of the Faculty of Science.
The Graduate University Biomedical Mathematics Study Programme in English, through its course contents as well as its forms and methods of teaching, provides for the acquisition of fundamental
knowledge and the understanding of results in the area of mathematics in combination with biology and medicine. During this course of study, students receive fundamental knowledge in statistics,
stochastic processes, mathematical modelling, branching processes, mathematics from data, modelling with differential equations with additional complementary knowledge in biology (bioinformatics,
translational genomics, molecular mechanisms of aging, tumour growth models etc.) and medicine (introduction to human body, modelling human brain processes, mechanisms of human diseases, etc.).
Professional status
Holders of the Master of Science degree in Biomedical Mathematics are qualified for work in the scientific research and higher education system (universities, polytechnics, research institutes) in
entrance-level teaching or research positions (assistant, young researcher and associate) in mathematics. Moreover, they can be employed in industry as researchers (in developmental research
institutes, e.g. in pharmaceutical companies), quality controllers, analysts and such.
Access to further study
After completing this graduate university programme, students are qualified for postgraduate (doctoral or specialist) programmes at the Department of Mathematics, in accordance with the enrolment
conditions for the academic year in which they apply. The received knowledge and acquired skills also qualify students for continued study in related postgraduate (doctoral or specialist) programmes
at other higher education institutions. The conditions of enrolment in postgraduate programmes at other higher education institutions are determined by those institutions. | {"url":"https://www.chem.pmf.hr/math/en/study_programmes/graduate_university_biomedical_mathematics_programme_in_english","timestamp":"2024-11-04T05:10:19Z","content_type":"text/html","content_length":"91009","record_id":"<urn:uuid:57c822a8-a589-4405-90e8-6e466925f9ae>","cc-path":"CC-MAIN-2024-46/segments/1730477027812.67/warc/CC-MAIN-20241104034319-20241104064319-00710.warc.gz"} |
Unitizing: Why is it the Foundation of Place Value?
Unitizing- What is it? Why is it such a big jump for students? What role does it play in place value? What can we do to support students in developing this concept? These are just some of the
questions we will answer in this article to help you understand the critical role unitizing plays in how students build an understanding of place value. Are you ready to dig in?
What is Unitizing?
Unitizing is the concept that a group of items can simultaneously be described as a number of smaller items and as one group. In simplest terms, it is the understanding that we can count groups as
single units. An example of this is when we begin working with groups of ten. “Ten” is both a collection of ten smaller units, as well as a single unit with a value of ten.
Why is Unitizing Such a Big Jump for Children?
Let’s consider what children have been working on in pre-k, kindergarten, and the beginning of first grade. They have been building a foundational understanding of:
• 1:1 Correspondence
• Cardinality
• Conservation of Number
And more. When we start counting groups, especially as we approach teaching place value, we need to go back to these skills and almost define them for students.
For example, children have just learned to count one object for each number in the counting sequence (1:1 correspondence). They have just learned that “one” means a single object, and “two” means two
separate objects.
As we introduce counting groups concerning place value, we are asking students to begin counting groups as single units. Now that “one” which used to mean a single object can now mean ONE group of
multiple objects, for example, “one group of ten.” (Later that will expand into “one group of one hundred,” and “one group of one thousand.”)
This can be a huge jump for students. We are asking them to significantly shift their thinking about what counting means. We are asking them to now hold two truths at the same time: A ten is BOTH a
single unit, AND a collection of ten smaller units.
What Role Does Unitizing Play in Place Value?
Unitizing is the foundation for place value. Place Value is based on the premise of a digit having different values depending on its position. The placement of the digit describes how many groups of
a given value. For example, the 9 in 397 says that its value is 9 groups of ten (or 90). The 3 in 397 says that its value is 3 groups of 100 (or 300).
In order to conceptually understand each place’s value, we have to understand that each place is ten times greater than the place to the right. In the early grades, this looks like understanding that
10 ones equals 1 group of ten, and 10 groups of ten equals 1 hundred. If children are not yet able to understand that ten is both ten ones and one ten, they will struggle to develop future place
value concepts. This has to happen first!
Standards that Include Unitizing
Now that you have a better understanding of unitizing, it probably won’t surprise you to learn that several standards involve unitizing. Let’s look at some examples from first grade:
CCSS.MATH.CONTENT.1.NBT.B.2 (A, B & C)
Understand that the two digits of a two-digit number represent amounts of tens and ones. Understand the following as special cases:
• 10 can be thought of as a bundle of ten ones — called a “ten.”
• The numbers from 11 to 19 are composed of a ten and one, two, three, four, five, six, seven, eight, or nine ones.
• The numbers 10, 20, 30, 40, 50, 60, 70, 80, 90 refer to one, two, three, four, five, six, seven, eight, or nine tens (and 0 ones).
Compare two two-digit numbers based on meanings of the tens and ones digits, recording the results of comparisons with the symbols >, =, and <.
Add within 100, including adding a two-digit number and a one-digit number, and adding a two-digit number and a multiple of 10, using concrete models or drawings and strategies based on place value,
properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used. Understand that in adding two-digit
numbers, one adds tens and tens, ones and ones; and sometimes it is necessary to compose a ten.
Do you see how unitizing is embedded in each of these standards? We see the same patterns when we look at second grade with skip counting and the conceptual understanding of one “hundred” and one
How Can We Help Students with Unitizing?
The key to unitizing is giving students lots of experiences navigating the idea of groups as single units. One way to do this is to have students sort objects into groups organically.
Unitizing with Counting Collections
We discuss Counting Collections across a variety of math strands. If you would like some background information about counting collections and how to get started, you can refer to these posts for
more information:
The purpose of Counting Collections with respect to place value is to give children to begin naturally organizing the objects in their collection into groups. This can initially be groups of any
size, since we truly just want them to see groups as single units and also as collections of smaller units.
Through discussion and thoughtful questioning, we want students to make connections between counting the number of groups and counting their total number of objects.
Eventually, whether through their own practice or observing the work of their peers, students will begin to see the efficiency of counting by groups, especially groups of ten. With more and more
experiences physically organizing their collections into groups, counting their groups, and finding totals, students will begin to internalize the concept of unitizing.
Unitizing through Number Talks
We have previously discussed how to introduce place value through number talks. Within our FREE set of Number Talk slides, you will find that the first ten prompts are completely visual. These slides
are intended to help students begin seeing objects arranged in groups of ten and to encourage them to unitize.
FREE Number Talks
First grade teachers, access 20 FREE Number Talk Prompts to enhance your place value unit and get your students engaged in conversation.
These prompts are powerful reminders of different ways we can build the concept of ten throughout our school day! Utilizing visuals that can reinforce units of ten in the classroom environment can be
a powerful tool (e.g. ten frames, rekenreks, fingers, bundles of ten…etc.). You can find several ideas in this post about adding ten-frames to your classroom routines!
Unitizing with the Hundred Chart
The hundred chart can be a very useful tool when teaching place value. If you’ve never seen how I like to introduce the hundred chart with Unifix Cubes, that will provide a great starting place.
A wonderful follow-up activity is to have students build groups of ten on their own hundred chart. I have made a hundred chart template (Click here to download for free) that easily fits Unifix Cubes
and allows students to really see and experience how ten ones make a ten.
This can also be a great hands-on tool to start composing and decomposing two-digit numbers. For example, if I have 25 and I want to add 36, we can start by building each addend with Unifix Cubes.
Visually we see the two addends in their different colors. Then we can compose these numbers on the hundred chart like in the image below and reinforce those groupings and regroupings of ten.
We can do the same for subtraction. We can build the minuend with our Unifix cubes on the hundred chart. For example, if we have the equation 56-24, we first build 56.
Then we can remove the amount of cubes in the subtrahend (24) and see our difference (32). Again, this reinforces the groups of tens and leftover ones.
I created these charts to perfectly fit Unifix cubes for this activity. If you’d like to give them a try in your classroom, here is the link to them on Google Drive (they’re free!).
When we help students understand and internalize the concept of unitizing, we essentially unlock the entire world of place value for them! It’s quite powerful when you stop and think about it!
I hope this deep dive into unitizing was helpful and has given you some ways to support the students in your classroom who may be struggling to internalize this concept. Did you have an ah-ha or big
takeaway? I’d love to hear from you! | {"url":"https://jillianstarrteaching.com/unitizing/","timestamp":"2024-11-11T15:58:26Z","content_type":"text/html","content_length":"170008","record_id":"<urn:uuid:ae9c2f03-2229-44f1-b98a-7121f61b3fc6>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00556.warc.gz"} |
Help with a formula: Do not run formula if cell is blank
Rookie here -- I'm hoping to get some help with a formula issue. I am trying to flag the "Attn" column if "Days Until Due" is less than 120. However, I don't want the "Attn" column to flag if the
cell is blank.
I currently have this formula configured for the Attn column, which has the flag symbol:
=IF([Days Until Due]1 < 120, 1, 0)
Looking for the correct formula to not flag if running against a blank cell.
Thanks in advance.
• =IF([Days Until Due]@row < 120, IF(ISBLANK([Days Until Due]@row), 0, 1))
• This worked perfectly!! Thank you, Luke!
• Not a problem. If you have a drag down formula the @row reference is faster for smartsheet to process. You could also use an AND function here with the same results.
Help Article Resources | {"url":"https://community.smartsheet.com/discussion/26216/help-with-a-formula-do-not-run-formula-if-cell-is-blank","timestamp":"2024-11-13T20:58:02Z","content_type":"text/html","content_length":"400621","record_id":"<urn:uuid:c68d78dc-d4b3-4b25-82a9-67129b2853ff>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00517.warc.gz"} |
Tied at the top - electowikiTied at the topTied at the top
The tied at the top rule for pairwise count methods says that when two candidates X and Y are ranked equal at the top of a ballot, both candidates are counted as receiving a vote for them against the
other. Also, when pairwise wins (and not just "votes for") are determined, if adding the number of votes tying X and Y "tied at the top" to one side or the other can determine which candidate wins
pairwise against the other, then this contest is interpreted to be a pairwise tie.
In some cases, use of the tied at the top rule can allow a method to satisfy the Favorite Betrayal Criterion (FBC). Kevin Venzke devised the rule to be used with Condorcet//Approval (yielding
Improved Condorcet Approval) and Minmax (winning votes).
One possible issue with the tied at the top rule in its formulation mentioned above is that it forces voters who don't want to create pairwise ties between the candidates they equally rank 1st to do
so. For example, a Democrat voter wishing to indicate no preference between two Democrat candidates while helping both of them pairwise beat a Republican candidate may simply wish to allow other
voters to choose which of the Democrats should win, rather than preventing one from pairwise beating the other. Thus, one possible improvement to the rule is to allow voters to explicitly indicate
whether or not they want the candidates they rank 1st to be in a pairwise tie or not. One way to do this is to, in addition to allowing voters to rank candidates 1st, 2nd, 3rd, etc, allow voters to
indicate an "above-1st"/"0th" rank which is counted as superior to all ranks, including the 1st rank. Another way would be to allow voters to check a box indicating they want all top-ranked
candidates to pairwise tie if possible.
The tied at the top rule could potentially be used for all ranks, though this should probably be explicitly indicated by the voter.
Example for Smith//Approval:
25 A>B| >C
40 B>C| >A
35 C>A| >B
There is an A>B>C>A cycle, with the approvals being A 60, B 65, and C 75. C is elected for having the most approvals in the Smith set. The easiest way for A-top voters to get someone they prefer to C
under regular Smith//Approval is for at least 11 of them to swap A and B, making B a Condorcet winner. With a modification of the tied at the top rule, they could instead vote A=B, and this would
yield a matchup of 49 votes for A, 40 for B, and 11 tied votes. Since the 11 tied votes is greater than the margin of 9 votes, the matchup could simply be dropped, meaning that B becomes a CW because
B pairwise beats C. N
Note that C-top voters can vote C=A to make A have no pairwise defeats toothat , so A and B would both be in the Smith set, wistill th B winning with 65 approvals to A's 60. | {"url":"https://electowiki.org/wiki/Tied_at_the_top","timestamp":"2024-11-13T04:43:16Z","content_type":"text/html","content_length":"45952","record_id":"<urn:uuid:eed8487f-e2ff-43cc-876c-e14fce214a3a>","cc-path":"CC-MAIN-2024-46/segments/1730477028326.66/warc/CC-MAIN-20241113040054-20241113070054-00144.warc.gz"} |
Mathematical study of a single leukocyte in microchannel flow
Boujena, S.; Kafi, Oualid; Sequeira, Adélia
Mathematical Modelling of Physiological Flows, 13 (5) (2018), art. nr. 43, 16pp.
The recruitment of leukocytes and subsequent rolling, activation, adhesion and transmigration are essential stages of an inflammatory response. Chronic inflammation may entail atherosclerosis, one of
the most devastating cardiovascular diseases. Understanding this mechanism is of crucial importance in immunology and in the development of anti-inflammatory drugs. Micropipette aspiration
experiments show that leukocytes behave as viscoelastic drops during suction. The flow of non-Newtonian viscoelastic fluids can be described by differential, integral and rate-type constitutive
equations. In this study, the rate-type Oldroyd-B model is used to capture the viscoelasticity of the leukocyte which is considered as a drop. Our main goal is to analyze a mathematical model
describing the deformation and flow of an individual leukocyte in a microchannel flow. In this model we consider a coupled problem between a simplified Oldroyd-B system and a transport equation which
describes the density considered as non constant in the Navier–Stokes equations. First we present the mathematical model and we prove the existence of solution, then we describe its numerical
approximation using the level set method. Through the numerical simulations we analyze the hemodynamic effects of three inlet velocity values. We note that the hydrodynamic forces pushing the cell
become higher with increasing inlet velocities. | {"url":"https://cemat.tecnico.ulisboa.pt/document.php?s_member_type_id=1&member_id=83&locale=pt&doc_id=3653","timestamp":"2024-11-09T12:21:45Z","content_type":"text/html","content_length":"9495","record_id":"<urn:uuid:59148cbe-fc9f-4fa7-8c9a-fdd5b6ae1ac2>","cc-path":"CC-MAIN-2024-46/segments/1730477028118.93/warc/CC-MAIN-20241109120425-20241109150425-00031.warc.gz"} |
add3, sub3, neg3, div3, mul3, eqpt3, closept3, dot3, cross3, len3, dist3, unit3, midpt3, lerp3, reflect3, nearseg3, pldist3, vdiv3, vrem3, pn2f3, ppp2f3, fff2p3, pdiv4, add4, sub4 – operations on 3-d
points and planes
#include <draw.h>
#include <geometry.h>
Point3 add3(Point3 a, Point3 b)
Point3 sub3(Point3 a, Point3 b)
Point3 neg3(Point3 a)
Point3 div3(Point3 a, double b)
Point3 mul3(Point3 a, double b)
int eqpt3(Point3 p, Point3 q)
int closept3(Point3 p, Point3 q, double eps)
double dot3(Point3 p, Point3 q)
Point3 cross3(Point3 p, Point3 q)
double len3(Point3 p)
double dist3(Point3 p, Point3 q)
Point3 unit3(Point3 p)
Point3 midpt3(Point3 p, Point3 q)
Point3 lerp3(Point3 p, Point3 q, double alpha)
Point3 reflect3(Point3 p, Point3 p0, Point3 p1)
Point3 nearseg3(Point3 p0, Point3 p1, Point3 testp)
double pldist3(Point3 p, Point3 p0, Point3 p1)
double vdiv3(Point3 a, Point3 b)
Point3 vrem3(Point3 a, Point3 b)
Point3 pn2f3(Point3 p, Point3 n)
Point3 ppp2f3(Point3 p0, Point3 p1, Point3 p2)
Point3 fff2p3(Point3 f0, Point3 f1, Point3 f2)
Point3 pdiv4(Point3 a)
Point3 add4(Point3 a, Point3 b)
Point3 sub4(Point3 a, Point3 b)
These routines do arithmetic on points and planes in affine or projective 3-space. Type Point3 is
typedef struct Point3 Point3;
struct Point3{
double x, y, z, w;
Routines whose names end in 3 operate on vectors or ordinary points in affine 3-space, represented by their Euclidean (x,y,z) coordinates. (They assume w=1 in their arguments, and set w=1 in their
Add the coordinates of two points.
Subtract coordinates of two points.
Negate the coordinates of a point.
Multiply coordinates by a scalar.
Divide coordinates by a scalar.
Test two points for exact equality.
Is the distance between two points smaller than eps?
Dot product.
Cross product.
Distance to the origin.
Distance between two points.
A unit vector parallel to p.
The midpoint of line segment pq.
Linear interpolation between p and q.
The reflection of point p in the segment joining p0 and p1.
The closest point to testp on segment p0 p1 .
The distance from p to segment p0 p1 .
Vector divide the length of the component of a parallel to b, in units of the length of b.
Vector remainder the component of a perpendicular to b. Ignoring roundoff, we have eqpt3(add3(mul3(b, vdiv3(a, b)), vrem3(a, b)), a) .
The following routines convert amongst various representations of points and planes. Planes are represented identically to points, by duality; a point p is on a plane q whenever
p.x*q.x+p.y*q.y+p.z*q.z+p.w*q.w=0. Although when dealing with affine points we assume p.w=1, we can’t make the same assumption for planes. The names of these routines are extra-cryptic. They contain
an f (for ‘face’) to indicate a plane, p for a point and n for a normal vector. The number 2 abbreviates the word ‘to.’ The number 3 reminds us, as before, that we’re dealing with affine points. Thus
pn2f3 takes a point and a normal vector and returns the corresponding plane.
Compute the plane passing through p with normal n.
Compute the plane passing through three points.
Compute the intersection point of three planes.
The names of the following routines end in 4 because they operate on points in projective 4-space, represented by their homogeneous coordinates.
Perspective division. Divide p.w into p’s coordinates, converting to affine coordinates. If p.w is zero, the result is the same as the argument.
Add the coordinates of two points.
Subtract the coordinates of two points. | {"url":"http://man2.aiju.de/2/arith3","timestamp":"2024-11-05T00:26:36Z","content_type":"text/html","content_length":"16097","record_id":"<urn:uuid:1e0e0f2c-84de-4627-bca0-154de57d8eea>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.84/warc/CC-MAIN-20241104225856-20241105015856-00268.warc.gz"} |
What is the orbital overlap diagram for "NH"_3? | Socratic
What is the orbital overlap diagram for #"NH"_3#?
1 Answer
The atomic orbital of $\text{H}$ that is both compatible and close enough in energy with the $n = 2$ atomic orbitals of $\text{N}$ is the $1 s$.
The $1 s$ atomic orbital of $\text{H}$, with $E = - \text{13.6 eV}$, is close enough in energy (less than $\text{12 eV}$ away) to the $2 p$ atomic orbitals of $\text{N}$, with $E = - \text{13.1 eV}$,
that it can overlap with SOME of them.
In a nonlinear molecule, the $2 {p}_{y}$ orbitals of nitrogen line up directly (head-on, colinear) with hydrogen---the $y$ direction of the $\text{N"-"H}$ bond is along the bond. The $z$ direction is
through the tip of the trigonal pyramid.
The MO diagram for ${\text{NH}}_{3}$ is:
[Do note though that this diagram has a typo; the $1 s$ atomic orbital of nitrogen should be a $2 s$.]
As you can see, the $2 p$ atomic orbitals of $\text{N}$ are indeed very close in energy with the hydrogen $1 s$ atomic orbitals.
The nonbonding orbital holding the lone pair is the $3 {a}_{1}$, which ends up being contributed to mostly by the $2 {p}_{z}$ of $\text{N}$ and slightly by the $1 s$ orbitals of the three $\text{H}$
atoms as a group.
So, the "orbital overlap" diagram would look something like this:
Impact of this question
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Algebra 1 - Learn Vibrant Math Tutoring
That's when they learn many fundamental concepts they will need to succeed in all higher levels of math – the building blocks of their math education.
This is where your child dives deep into functions, quadratics, equations and many other key math concepts.
But most fundamentals doesn't mean easiest...
Algebra 1 is considered the most challenging by many students This class is considered the most challenging of all school math, by a lot of students.
“How is that possible?” you might ask “If it's the first one - shouldn't it be the easiest?” The reason is because Algebra 1 is the first class when your child encounters a lot of abstract
mathematical concepts.
In later classes, they will be expanding on the topics and building upon this fundamental understanding. But in Algebra 1 class, they are establishing the fundamental understanding. Taking their
first steps – that's what makes it challenging.
Algebra 1 concepts are key to your child's math success.
Basically all of math beyond Algebra 1 is based on functions. The most common ones being quadratic functions, used in many concepts such as rational functions, describing movement, rational
equations, polynomials, and even angles and polygons! Algebra 1 concepts are used at the end and throughout every single math problem your child will ever do!
That's a great news, your child is learning the concepts they will use for the rest of their math education (and STEM in general).
But on the other side of the coin – there is the biggest problem we see in education in general:
This is the class where many children fall behind and start really struggling. Many children never develop deep (or any) understanding of the fundamentals, and begin slowing down, struggling through
and getting frustrated with most problems, and math in general as the lack of their fundamental skills holds them back for months and years.
Common Challenges To Look Out For
Algebra 1 is the class where a lot of teachers assume that foundational = easy, and don't sufficiently explain the topics, don't give children adequate amount of practice, or don't help children
understand the context of math. We have heard too many stories of teachers simply giving out homework and telling children to read the book or “figure it out” (yes, unfortunately, this is a real-life
example) As a result, children fall behind, lose confidence and struggle through future math classes without solid foundations.
Foundational doesn't mean easy! Foundations are hard – just like the first time driving a car is hard. Most of us can drive a car effortlessly. But the first time on the road was NOT effortless, was
Algebra 1 is like that. It's not easy.
It's very important to identify the weaknesses early on to speed up your child's progress and help them study more effectively. Algebra 1 has many concepts, and your child needs to focus their
attention on the ones most challenging for them.
Put your child in the best position to succeed
Set your child up for years of thriving, not years of struggling
Most students who struggle in math, suffer to a high degree because of their lack of understanding of the concepts they needed to master in Algebra 1.
Algebra 1 is a great opportunity for your child to establish unshakeable confidence and solid foundations for the future success in math. It is your opportunity to ensure your child continues to
thrive in their future math classes.
Algebra 1 “investment” compounds.
When it comes to this level, the mastery of the concepts your child gains now will pay off for years to come. As the concepts they are covering in this class, like linear equations, functions and
exponents will be used in every single math problem they will ever do, from Geometry to Calculus BC, taking the time to truly understand, solidify and become fluent in these topics will absolutely
set your child up for higher confidence, speed, independence and more.
Develop problem solving skills through Algebra 1
Because of that versatility, your child needs to understand how and why the Algebra 1 concepts are used. And be confident in using these concepts in many ways. Your child needs to think critically to
recognize which concepts apply in different scenarios and how to identify the right tools to solve problems. The lack of these problem solving skills is the reason why many children struggle with
Develop analytical thinking skills through word problems
So many students never develop good word problems solving skills!
The beginnings of that are often rooted in Algebra 1. Children learn to apply the operations indicated by the chapter.
In the linear equations chapter, they would apply linear equations.
In quadratics chapter, they apply quadratics.
All of that, without thinking critically about what the problem is really asking or why it needs to be solved in that specific way.
Children are often not taught to think about “why” behind what they do in math.
As a result, when the wording is different, or the problem mixes a few different concepts, children get lost, frustrated and often give up.
Ignite your child's passion for math and expand their progress beyond their classroom pace and level.
Give your child a fun, unique experience of learning Algebra 1 in a way that works best for them. | {"url":"https://learnvibrant.com/algebra-1/","timestamp":"2024-11-02T05:12:18Z","content_type":"text/html","content_length":"105402","record_id":"<urn:uuid:342fcd31-cd2c-476f-b4c3-e7a7b4524bb7>","cc-path":"CC-MAIN-2024-46/segments/1730477027677.11/warc/CC-MAIN-20241102040949-20241102070949-00123.warc.gz"} |
The Next Generation of Income Guarantee Riders: Part 3 (The Income Phase)
This is part three of a three-part series of articles reviewing stand-alone income (SALB) guarantees. Part one of this series is available here and part two is here.
What is the purpose of an income guarantee? Some view it as a traditional insurance product, meant to provide downside risk protection, while others like that it encourages clients to accept a more
aggressive asset allocation, promising more upside without exacerbating worst-case outcomes. Can it really be both?
Some would argue that guarantees aren’t worth bothering with at all, since many clients can expect to do just as well without paying the extra fees.
So who’s right?
In this third and final installment in my series on guarantee riders, I’ll focus on the post-retirement income supported by income guarantee riders for variable annuities (VA/GLWBs), stand-alone
living benefit riders (SALBs), and an unguaranteed portfolio of mutual funds. I’ll highlight how differences among these products affect their end results, while also investigating what roles
guarantees can most appropriately play in a retirement portfolio.
Setting the stage
For many advisors, it’s a welcome relief that clients no longer have to permanently commit their assets to a deferred variable annuity in order to purchase a rider that guarantees lifetime withdrawal
benefits. The RetireOne product from ARIA is one such rider – it can be applied to a portfolio of mutual funds – and for this analysis, it is the one that I’ve focused on. The relatively low-cost
GLWB rider available for Vanguard’s variable annuities, meanwhile, is what I’m using as a relatively attractive example of an annuity rider.
For a full refresher on the background and features for VA/GLWBs and SALBs, I’d recommend reviewing part 1 in this series. Very briefly, these products are designed to provide owners with downside
protections, upside potential, and the opportunity to have remaining assets returned prior to death or as a death benefit. To accomplish this, these riders guarantee an income for life at a fixed
withdrawal percentage of the initial assets. As long as the investor does not exceed the allowed withdrawal amounts, guaranteed withdrawals never decrease (in nominal terms – they may well decline
according to inflation-adjusted metrics), even if the account balance falls to zero. If the value of the underlying account increases enough (after accounting for any withdrawals and fees), a step-up
feature kicks in to provide permanently higher withdrawal amounts.
I’ve simulated the income-phase performance of the Vanguard VA/GLWB, the RetireOne SALB, and an unguaranteed mutual fund portfolio using Monte Carlo simulations. Parts 1 and 2 of the series, linked
above, focused on the initial deferral period and the crucial moment when the income guarantee kicks in, respectively, relying on historical data. This final article will shift gears to consider the
results of my simulation and their implications for the income phase of retirement.
Potential biases in research methodologies
Income guarantees are complicated financial products, and it’s important to understand that the research published about them often makes assumptions that can present the guarantees in an overly
positive or negative light. Let’s review some of the most crucial.
Underlying fees: A particularly common approach in such research is to compare the results of using income guarantees to drawdowns from unguaranteed mutual funds. Readers must pay careful attention
to the fees that are assumed to underlie those guaranteed and unguaranteed funds, which should be consistent across the different products. (That’s why I compare the low-cost RetireOne guarantees
with Vanguard’s low-cost VA/GLWB and with unguaranteed low-cost index funds.)
Asset Allocation: Clients with income guarantees will naturally feel more comfortable accepting an aggressive asset allocation, and ideally one should compare approaches using the asset allocations a
client would actually choose with a guarantee and without one, rather than assuming a one-size-fits-all approach.
Underlying Returns: For obvious reasons, assumptions about expected returns can dramatically affect the results. With better performance and lower inflation, guaranteed approaches will reap the
benefits of their higher upside when compared with a less-aggressive, unguaranteed approach, but the unguaranteed approach will also be more likely to support the corresponding guaranteed
withdrawals. In other words, there will be less downside risk for the guarantee to protect. With more pessimistic return assumptions, there will be fewer step ups and less upside, but the guarantee
becomes much more likely to matter.
Time Period: Assuming a longer retirement horizon skews results toward guarantees, since over time it becomes more and more likely that the unguaranteed portfolio will run dry. Some researchers
believe that the appropriate time period to investigate is remaining life expectancy, while others opt for something like a 30-40 year retirement, aiming for the high end of realistic scenarios.
Spending Rates: Finally, one way to make a guarantee look better is to guarantee only part of a portfolio while assuming that total spending will exceed the guaranteed payout rate. Doing so causes
both an unguaranteed portfolio and a partially guaranteed portfolio to deplete more quickly, but the partially guaranteed portfolio will still look better, since it continues to at least provide a
minimal amount of income. In either case, however, income may fall well below the basic needs of a client, who should have been advised to choose lower spending rate from the outset. (To avoid this
problem, I simply compare the withdrawal amounts supported by a guaranteed portfolio to an unguaranteed portfolio that attempts to replicate the same guaranteed payouts for as long as possible.)
In addition to minding the underlying assumptions, we must also be clear about what sorts of outcome measures are most appropriate. Clients who view a guarantee rider as an insurance product may
consider the guarantee to be primarily a form of downside risk protection, in which case any analysis should focus on the worst-case outcomes. (This is the approach I took in parts 1 and 2.) But
another justification for income guarantees that they encourage a client to choose a more-aggressive asset allocation with higher upside potential, in which case the focus may shift to demonstrating
which approach enjoys superior average outcomes.
Data and modeling approach
Advisors who are familiar and comfortable with the 4% safe withdrawal rate rule-of-thumb may see little need for an income guarantee. But market conditions today suggest that pessimism may be in
order – what worked for yesterday’s retirees may not work for today’s or tomorrow’s. Interest rates are at historical lows and stocks are overvalued, at least according to historically reliable
metrics like Robert Shiller’s CAPE.
While parts 1 and 2 both analyzed outcomes for rolling periods from the historical data, as we turn to considering retirement income, the primary basis for this article will be the results derived
from Monte Carlo simulation. Table 1, below, provides the asset market assumptions on which those simulations were based. For the most part, I used current market conditions to guide the simulations,
but near the end of this article I’ll discuss how the results change under more optimistic assumptions. (For more detail on how I obtained the figures you see here, see Appendix 1 at the end of this
│ Table 1 │
│ Asset Market Assumptions Based on Current Market Conditions │
│ │ │ │ │ Correlation Coefficients │
├───────────┤Arithmetic │ │ ├────────┬───────┬───────────┤
│ │ Means │ Geometric Means │ Standard Deviations │ Stocks │ Bonds │ Inflation │
│ Stocks │ 4.8% │ 2.8% │ 20.0% │ 1 │ 0.1 │ -0.2 │
│ Bonds │ 0.0% │ -0.2% │ 7.0% │ 0.1 │ 1 │ -0.6 │
│ Inflation │ 2.5% │ 2.4% │ 4.2% │ -0.2 │ -0.6 │ 1 │
│ Summary Statistics for U.S. Real Returns and Inflation Data, 1926 - 2011 │
│ │ │ │ │ Correlation Coefficients │
├───────────┤Arithmetic │ │ ├────────┬───────┬───────────┤
│ │ Means │ Geometric Means │ Standard Deviations │ Stocks │ Bonds │ Inflation │
│ Stocks │ 8.6% │ 6.5% │ 20.3% │ 1 │ 0.1 │ -0.2 │
│ Bonds │ 2.6% │ 2.3% │ 6.8% │ 0.1 │ 1 │ -0.6 │
│ Inflation │ 3.1% │ 3.0% │ 4.2% │ -0.2 │ -0.6 │ 1 │
My simulations assume a 65-year old couple who buys the guarantee at 65 and immediately begins to take income. For simplicity’s sake, their retirement date wealth is assumed to be $100, though the
results are, of course, scalable. Since the analysis is assumes current market conditions, the payout rates at retirement for the 65-year old couple, until both spouses are deceased, are 4.5% for the
VA/GLWB and 3.5% for RetireOne.
For the VA/GLWB, the payout depends only on age, while the payout rate for RetireOne as depends on the current yield on 10-year Treasury bonds. The payout rate is 3.5% if the Treasury yield is less
than 4.5%, and it can increase to up to 5.5% if the Treasury yield exceeds 7%.
For RetireOne, after the guaranteed income begins, the benefit base is no longer determines the withdrawal amount. Instead, step-ups in withdrawals occur whenever the revised payout rate (a
calculation that involves multiplying prevailing Treasury yields by the remaining account balance) exceeds the previous guaranteed withdrawal.
Since I do not attempt to simulate future interest rates, I’ve assumed that RetireOne’s payout stays at 3.5% throughout retirement. While that assumption could bias results somewhat against
RetireOne, but any such effect is likely quite small, since interest rates are currently much less than 4.5% and, as we’ll see, it becomes increasingly unlikely for the portfolio to reach new
high-water marks as retirement progresses.
When it comes to determining the value of a guarantee, it’s important to always consider whether an unguaranteed portfolio of mutual funds would be able to replicate the guaranteed payments without
experiencing wealth depletion. As I explained above, the asset allocation for an unguaranteed portfolio in any such comparison should be less aggressive, though the exact allocations will depend on a
client’s preferences. | {"url":"https://www.advisorperspectives.com/articles/2012/12/11/the-next-generation-of-income-guarantee-riders-part-3-the-income-phase","timestamp":"2024-11-05T06:18:12Z","content_type":"text/html","content_length":"181923","record_id":"<urn:uuid:aaaaeac3-2b16-420a-ad3f-7c9ab9f64c52>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00540.warc.gz"} |
Mathematical Model suatu fungsi f dirumuskan sebagai f(x) - Densipaper
In mathematics, functions are an essential tool used to describe how one variable depends on another. They play a crucial role in modeling real-world phenomena, allowing us to analyze complex systems
and make predictions about their behavior. One such process is “Suatu Fungsi F Dirumuskan Sebagai,” which translates to “A function F is formulated as.” In this article, we will explore the process
of formulating functions as mathematical models and its significance in various fields.
The process of formulating a function as a mathematical model involves defining the input and output variables and determining the relationship between them. The resulting mathematical model can be
used to analyze the behavior of the function under different conditions and predict its performance. This process is essential in various fields, such as physics, engineering, economics, and computer
science, and is often used to optimize systems and solve complex problems.
One common method of formulating a function as a mathematical model is through regression analysis. Regression analysis involves analyzing data points to determine the relationship between two or
more variables. For example, if we want to model the relationship between rainfall and crop yield, we would collect data on both variables and use regression analysis to determine the relationship
between them.
Once we have determined the relationship between the variables, we can create a mathematical model that describes the behavior of the function. The mathematical model can then be used to predict the
performance of the system under different circumstances. For example, if we know that a particular crop requires a certain amount of rainfall to produce high yields, we can use the mathematical model
to determine the optimal irrigation levels necessary for achieving those yields.
Another method of formulating a function as a mathematical model is through differential equations. Differential equations are used to model systems where the rate of change of a variable depends on
other variables. For example, in physics, differential equations are used to model the motion of objects under the influence of gravity or other forces.
The process of formulating differential equations involves defining the variables, determining the relationships between them, and specifying initial conditions. Once the differential equations have
been determined, they can be solved using numerical methods to obtain a mathematical model that describes the behavior of the system.
One important application of differential equations is in modeling the spread of infectious diseases. By formulating the spread of the disease as a set of differential equations, researchers can
predict the behavior of the disease under different scenarios and develop strategies to control its spread.
In computer science, functions are often formulated as algorithms, which are step-by-step procedures used to solve a problem. For example, if we want to sort a list of numbers in ascending order, we
would use an algorithm such as bubble sort or quicksort. Algorithms can be described mathematically using flowcharts or pseudocode, allowing us to analyze their performance and optimize their
Optimization is another significant application of function formulation. Optimization involves finding the best solution among all possible solutions to a problem. For example, in engineering,
optimization is used to design efficient | {"url":"https://densipaper.net/mathematical-model-suatu-fungsi-f-dirumuskan-sebagai-fx/","timestamp":"2024-11-07T03:07:05Z","content_type":"text/html","content_length":"85922","record_id":"<urn:uuid:6c72db93-67e2-4028-9fc2-ebffff15e763>","cc-path":"CC-MAIN-2024-46/segments/1730477027951.86/warc/CC-MAIN-20241107021136-20241107051136-00653.warc.gz"} |
CPO-STV - electowikiCPO-STVCPO-STV
CPO-STV (Comparison of Pairs of Outcomes by Single Transferable Vote) is a preference voting system designed to provide proportional representation in multi-seat elections while electing the
Condorcet winner in single-winner elections. It is based on STV and pairwise counting between every possible combination of candidates that could win ("winner sets") to determine the winner.
Each voter ranks all candidates in order of preference. For example:
1. Andrea
2. Carter
3. Brad
4. Delilah
Setting the Quota
When all the votes have been cast, a winning quota is set. Possible formulas for the quota include the Droop Quota, the Hare Quota, and the Imperiali Quota.
Comparison of Pairs of Outcomes
In CPO-STV, each possible outcome (set of candidates) is compared with every other possible outcome in a pairwise competition. The pairwise competition is performed as follows:
1. Eliminate all candidates who are not in either outcome.
2. Transfer excess votes from candidates who are in both outcomes.
3. The number of pairwsie votes for an outcome is equal to the sum of votes for the candidates in that outcome.
Example: Compare {Escher, Andre, Gore} versus {Escher, Nader, Gore} given a quota of 100 and the ballots
• 100: Escher
• 110: Andre>Nader>Gore
• 18: Nader>Gore
• 21: Gore>Nader
• 6: Gore>Bush
• 45: Bush>Gore
First, eliminate Bush, who is in neither outcome.
• 100: Escher
• 110: Andre>Nader>Gore
• 18: Nader>Gore
• 21: Gore>Nader
• 51: Gore
Next, transfer the excess votes for Escher, who is in both outcomes. (Do not transfer the votes for Andre, who is only in one outcome.) Because Echer happens to meet the quota exactly, there is
nothing to do here.
The number of first-choice votes for each candidate is now
• 100: Escher
• 110: Andre
• 18: Nader
• 72: Gore
Finally, add up the votes in each outcome.
• Escher Andre Gore = 100 110 72 = 282
• Escher Nader Gore = 100 18 72 = 190
Thus, {Escher, Andre, Gore} pairwise beats {Escher, Nader, Gore}, 282 to 190.
Counting The Votes
Process A: If any candidate has a quota of top-preference votes, declare them elected. Distribute excess votes (determined by random selection or by fractional transfer) for the winning candidates to
the next-highest ranked candidates on the ballots. Repeat this process until there there are no more candidates who meet the quota. (This process is optional, but can greatly simplify Process B.)
Process B: Conduct a pairwise comparison between every possible set of candidates that includes all of the elected candidates from Process A. Choose the winning outcome with a Condorcet method.
2 seats to be filled, four candidates: Andrea (A), Brad (B), Carter (C), and Delilah (D). The ballots are:
• 5: A>B>C>D
• 17: A>C>B>D
• 8: D
The Droop Quota is floor(22/3) 1 = 11.
Andrea has 22 first-choice votes, and is declared elected. Her 11 excess votes are reallocated to their second preferences. If this is done by fractional transfer, the resulting ballots are:
• 2.5: B>C>D
• 8.5: C>B>D
• 8: D
No more candidates meet the quota, so Process A is completed.
Since Andrea must be elected, there are only 3 possible outcomes to consider: {A, B}, {A, C}, and {A, D}.
To compare {A, B} and {A, C}, first eliminate D:
• 5: A>B>C
• 17: A>C>B
• 8: (blank)
Andrea is elected with 11 excess votes. After transferring these, the ballots become:
• 11: A
• 2.5: B>C
• 8.5: C>B
• 8: (blank)
and so {A, C} beats {A, B}, 19.5 to 13.5.
Similarly, to compare {A, C} and {A, D}, first eliminate B:
Andrea is elected with 11 excess votes, which all transfer to C, producing:
and so {A, C} beats {A, D}, 22 to 19.
At this point, we know that {A, C} is the Condorcet winner. Therefore, CPO-STV elects Andrea and Carter.
CPO-STV can be highly computationally complex and thus difficult to calculate when there are many candidates, as if there are, say, 5 seats to be filled and 60 candidates, then there are 60 choose 5
= 5,461,512 possible outcomes.
It has not been proven whether CPO-STV is proportional for Droop solid coalitions. However, if it can be, then its cycle resolution method likely must choose from the Smith Set of winner sets in
order to do so, as Smith-efficiency guarantees Droop proportionality (the mutual majority criterion) in the single-winner case. One type of procedure that requires among the fewest pairwise
comparisons to find one of the Smith Set winner sets is Sequential comparison Condorcet methods. Since a winner set in the Smith Set can only be eliminated by another Smith winner set by this
procedure, the final remaining winner set will guaranteeably be in the Smith Set. If desired, it is then possible to discover the rest of the Smith Set by checking which winner sets beat or tie the
final remaining winner set, which beat or tie these winner sets, etc. One well-known procedure that works along these lines is BTR-IRV.
One suggestion to modify CPO-STV to be guaranteeably proportional for Droop solid coalitions is to first eliminate all outcomes from consideration that fail Droop proportionality. In the above
example, if there are 4 solid coalitions of 5 candidates each, then the upper bound of outcomes to consider is ((5^4) * 60) = 37500 outcomes, which is a reduction of outcomes to consider by a factor
of about 145.
Several other such modifications are possible to reduce the number of outcomes to consider, some of which can potentially elect some outcome other than what CPO-STV would. Some are:
- As a first guess, calculate the STV outcome and see if it can win against all other outcomes (that are in consideration). (It is estimated that the STV winner set is almost always the same as the
CPO-STV winner set.)
- If a set of candidates X is ranked above or equal to a set of candidates Y on all ballots, ignore all outcomes featuring candidates from Y but not X. (Based on unanimity criterion).
See also
External Resources | {"url":"https://electowiki.org/wiki/CPO-STV?oldid=9111","timestamp":"2024-11-09T18:58:12Z","content_type":"text/html","content_length":"55154","record_id":"<urn:uuid:484e090f-1d24-4139-9a5e-3d30d806ff33>","cc-path":"CC-MAIN-2024-46/segments/1730477028142.18/warc/CC-MAIN-20241109182954-20241109212954-00250.warc.gz"} |
h2g2 - Binary Digits - Edited Entry
Binary Digits
Created | Updated Dec 23, 2008
In the early days of computing, man first made 'mechanical' computers. Through much research and development, electronic computers were eventually developed. These early systems used primitive
technology in comparison with today's computers, but for their day they were fairly complex machines and they consumed a great deal of floor space. Their actual computational abilities were very
limited in nature. A computer taking up half a warehouse could do no more, and probably less, than a simple desktop calculator of today.
Early designers and programmers adopted the binary system^1 as a way to simplify the task of delivering instructions to the computer. They created 'switches' that were on (electrical current is
present) or off (electrical current is absent). Using this as a starting point, they were able to do basic calculations. In short, they used 'bits' or Binary digITs. It is usually represented in the
computer world with a 1 or a 0.
Another way of looking at it is as a True or a False, an ON or an OFF. If something equals one, then it exists, or is true, and if it equals 0, then it doesn't exist, or is false. Machines were
simplified because they didn't have to remember all the digits of 2 through 9. It simply was or it wasn't. One or zero. True or false.
In school, we are taught 'decimal maths' or 'base ten' arithmetic^2 with ten as the point where we 'start over'. That is, we count through nine and then the number ten is 1 again, followed by a 0
(10), then the next number, eleven, is a 1 followed by another 1 (11). So we first learn to count to 10, then on from there maybe to 100.
In binary digits you do the same thing, but the digits which represent 2-9 do not exist, so 10 comes early. In other words, instead of the decimal 'columns' of 1 (ones), 10 (tens), 100 (hundreds),
1000 (thousands), you get the binary 1 (ones), 10 (twos), 100 (fours), 1000 (eights) and so on.
For example, as you know, zero is 0. One is 1. So far, so good. Decimal is the same as binary, to this point. But then, two becomes 10 because the digit which represents two (2) in the decimal world
simply doesn't exist in the binary world. And three, of course, is represented by 11.
Now what? Four becomes 100. And five is 101, and six is 110 and seven 111. Eight, therefore, would be 1000 and nine 1001; finally, ten is 1010.
So count with me: zero, one, two, three, four, five, six, seven, eight, nine, ten. And now, write:
0 (zero), 1 (one), 10 (two), 11 (three), 100 (four), 101 (five), 110 (six), 111 (seven), 1000 (eight), 1001 (nine), and 1010 (ten).
Addition works the same as with decimal. Ten plus ten still equals twenty, but it is expressed as:
1010 (ten) +1010 (ten) =10100 (twenty)
That is, right to left, zero plus zero = zero (0). One plus one equals two (10) – like decimal maths, you put down the 0 and carry the one. Zero plus zero plus the carried over one equals one (1) and
one plus one equals two (10).
Even multiplication works the same as decimal, in that 0 times 0 equals 0, 0 times 1 equals 0, 1 times 0 equals 0 and 1 times 1 equals 1.
Ten times ten can be expressed as:
1010 x1010 =1100100 (one hundred)
If you care to do the maths, it's there, simple as can be. Just like regular old maths, but with only two digits to worry about.
To take this a step further beyond pure maths, alphabetic characters and other symbols may be represented through binary by assigning numeric codes to them. For example, using the American Standard
Code for Information Interchange (ASCII), the code to represent the letter 'A' is decimal 65 (binary 100001) and the letter 'a' is 97 (binary 1100001). Alphabetic characters and their representation
on computers are not the focus of this article. However, you may get an idea of how ASCII can be represented, as well as a chart of table of the decimal assignments for each letter and symbol, at
this fun h2g2 Edited Entry: ASCII Art.
Quotable Quote: 'The world has only 10 kinds of people. Those who get binary, and those who don't.'^3
^1The binary system was documented by Leibniz around 1670, but possibly used in India hundreds of years before that.^2A discussion of various numbering systems may be found at Number Systems.^3 | {"url":"https://www.h2g2.com/edited_entry/A5771973","timestamp":"2024-11-02T09:32:25Z","content_type":"text/html","content_length":"24907","record_id":"<urn:uuid:36500bd7-2183-4251-a47d-2c2a31ba25ce>","cc-path":"CC-MAIN-2024-46/segments/1730477027709.8/warc/CC-MAIN-20241102071948-20241102101948-00827.warc.gz"} |
Estimate Sobol’ indices for a function with multivariate output
Estimate Sobol’ indices for a function with multivariate output¶
In this example, we estimate the Sobol’ indices of a function by sampling methods. This function has several outputs, which leads to the need of aggregated Sobol’ indices.
In this example we quantify the sensitivity of a function’s outputs to its inputs with Sobol’ indices.
The function we consider has 5 outputs. In this case, it may be convenient to consider each output separately. It may also be interesting to aggregate the sensitivity indices to get a global
understanding of the sensitivity of the inputs to the average output.
Define the model¶
import openturns as ot
import openturns.viewer
import openturns.viewer as viewer
We define a linear model with 5 independent Gaussian inputs and 2 outputs.
inputDistribution = ot.Normal(5)
function = ot.SymbolicFunction(
["x0", "x1", "x2", "x3", "x4"],
["x0 + 4.0 * x1 ^ 2 + 3.0 * x2", "-7.0 * x2 - 4.0 * x3 + x4"],
Estimate the Sobol’ indices¶
We first create a design of experiments with SobolIndicesExperiment.
size = 1000
sie = ot.SobolIndicesExperiment(inputDistribution, size)
inputDesign = sie.generate()
input_names = inputDistribution.getDescription()
print("Sample size: ", inputDesign.getSize())
We see that 7000 function evaluations are required to estimate the first order and total Sobol’ indices. Then we evaluate the outputs corresponding to this design of experiments.
outputDesign = function(inputDesign)
Then we estimate the Sobol’ indices with the SaltelliSensitivityAlgorithm.
sensitivityAnalysis = ot.SaltelliSensitivityAlgorithm(inputDesign, outputDesign, size)
The getFirstOrderIndices and getTotalOrderIndices method respectively return estimates of first order and total Sobol’ indices with a given output. Since these depend on the output marginal, the
index of the output must be specified (the default is to return the index for the first output).
output_dimension = function.getOutputDimension()
for i in range(output_dimension):
print("Output #", i)
first_order = sensitivityAnalysis.getFirstOrderIndices(i)
total_order = sensitivityAnalysis.getTotalOrderIndices(i)
print(" First order indices: ", first_order)
print(" Total order indices: ", total_order)
agg_first_order = sensitivityAnalysis.getAggregatedFirstOrderIndices()
agg_total_order = sensitivityAnalysis.getAggregatedTotalOrderIndices()
print("Agg. first order indices: ", agg_first_order)
print("Agg. total order indices: ", agg_total_order)
Output # 0
First order indices: [0.0371334,0.78543,0.275291,0.0167471,0.0167471]
Total order indices: [0.0183228,0.756177,0.208928,-5.00714e-08,-5.00714e-08]
Output # 1
First order indices: [-0.0200282,-0.0200282,0.735456,0.24319,0.000249405]
Total order indices: [-8.10417e-08,-8.10417e-08,0.746484,0.265185,0.00864186]
Agg. first order indices: [-0.000501143,0.255125,0.578259,0.165835,0.00588518]
Agg. total order indices: [0.00625922,0.258318,0.562849,0.174595,0.00568969]
We see that:
• x1 has a rather large first order index on the first output, but a small index on the second output,
• x2 has a rather large first order index on the first output on both outputs,
• the largest aggregated Sobol’ index is x2,
• x0 and x5 have Sobol’ indices which are close to zero regardless of whether the indices are aggregated or not.
The draw method produces the following graph. The vertical bars represent the 95% confidence intervals of the estimates.
graph = sensitivityAnalysis.draw()
view = viewer.View(graph)
Since there are several outputs, the graph presents the aggregated Sobol’ indices. The aggregated Sobol’ indices indicate that the input variable which has the largest impact on the variability of
the several outputs is x2. | {"url":"https://openturns.github.io/openturns/latest/auto_reliability_sensitivity/sensitivity_analysis/plot_sensitivity_sobol_multivariate.html","timestamp":"2024-11-09T09:42:00Z","content_type":"text/html","content_length":"21112","record_id":"<urn:uuid:05982b8e-d658-4afb-a6db-c7617e9e1227>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.75/warc/CC-MAIN-20241109085148-20241109115148-00859.warc.gz"} |
zsytrs.f - Linux Manuals (3)
zsytrs.f (3) - Linux Manuals
zsytrs.f -
subroutine zsytrs (UPLO, N, NRHS, A, LDA, IPIV, B, LDB, INFO)
Function/Subroutine Documentation
subroutine zsytrs (characterUPLO, integerN, integerNRHS, complex*16, dimension( lda, * )A, integerLDA, integer, dimension( * )IPIV, complex*16, dimension( ldb, * )B, integerLDB, integerINFO)
ZSYTRS solves a system of linear equations A*X = B with a complex
symmetric matrix A using the factorization A = U*D*U**T or
A = L*D*L**T computed by ZSYTRF.
UPLO is CHARACTER*1
Specifies whether the details of the factorization are stored
as an upper or lower triangular matrix.
= 'U': Upper triangular, form is A = U*D*U**T;
= 'L': Lower triangular, form is A = L*D*L**T.
N is INTEGER
The order of the matrix A. N >= 0.
NRHS is INTEGER
The number of right hand sides, i.e., the number of columns
of the matrix B. NRHS >= 0.
A is COMPLEX*16 array, dimension (LDA,N)
The block diagonal matrix D and the multipliers used to
obtain the factor U or L as computed by ZSYTRF.
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).
IPIV is INTEGER array, dimension (N)
Details of the interchanges and the block structure of D
as determined by ZSYTRF.
B is COMPLEX*16 array, dimension (LDB,NRHS)
On entry, the right hand side matrix B.
On exit, the solution matrix X.
LDB is INTEGER
The leading dimension of the array B. LDB >= max(1,N).
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
November 2011
Definition at line 121 of file zsytrs.f.
Generated automatically by Doxygen for LAPACK from the source code. | {"url":"https://www.systutorials.com/docs/linux/man/docs/linux/man/3-zsytrs.f/","timestamp":"2024-11-05T09:48:09Z","content_type":"text/html","content_length":"8945","record_id":"<urn:uuid:fc923e16-6574-4d1d-894a-739df37940bb>","cc-path":"CC-MAIN-2024-46/segments/1730477027878.78/warc/CC-MAIN-20241105083140-20241105113140-00879.warc.gz"} |
Iswap Gate : Random Access Quantum Information Processors Using Multimode Circuit Quantum Electrodynamics Nature Communications / Breeding and hashing protocols, which are useful for quantum state purification.
Iswap Gate : Random Access Quantum Information Processors Using Multimode Circuit Quantum Electrodynamics Nature Communications / Breeding and hashing protocols, which are useful for quantum state
purification.. Pauli group and pauli algebra. The iswap gate is an entangling swapping gate where the qubits obtain a phase of i if the state of the qubits is swapped. We also show that this. Call
addgate method passing gate name, column index and array of iswap. Swap gate with additional phase for 01 and 10 states.
The gates cirq.xx, cirq.yy, and cirq.zz are equivalent to performing the equivalent. Single qubit gates correspond to rotations of a spin about some axis. This is a clifford and symmetric gate. We
also show that this replacement can be applied to. Each quantum gate is saved as a class object gate with information such as gate name, target qubits and arguments.
F1qli Aklvbugm from qutip.org
This means that for general gates, iswap doesn't offer any compilation advantages over cz. Single qubit gates correspond to rotations of a spin about some axis. We also show that this replacement can
be applied to. T qubit relaxation time, see section 3.5.1. Expressed in basis states, the swap gate swaps the state of the two qubits involved in the operation The iswap gate is an entangling
swapping gate where the qubits obtain a phase of i if the state of the qubits is swapped. Iswap gate using ge « eg transitions. # this file is part of qutip:
# this file is part of qutip:
Cnot gate by the bilateral iswap gate. The above example shows the first half of a quantum teleportation circuit parity gates: Breeding and hashing protocols, which are useful for quantum state
purification. However, it can still be useful to have both, because iswaps can be compiled using a single iswap. We also show that this. This means that for general gates, iswap doesn't offer any
compilation advantages over cz. # # 'csign', 'berkeley', 'swapalpha', 'swap', 'iswap', 'sqrtswap' Iswap gate using ge « eg transitions. T qubit relaxation time, see section 3.5.1. Pauli group and
pauli algebra. To generate the full xy family, we simply split the iswap gate into two √iswap gates and change the new strategy enabled the implementation of xy entangling gates in the
superconducting system. Swap gate with additional phase for 01 and 10 states. The gates cirq.xx, cirq.yy, and cirq.zz are equivalent to performing the equivalent.
This means that for general gates, iswap doesn't offer any compilation advantages over cz. T qubit relaxation time, see section 3.5.1. Of parameters and for σg = 1 ns. We also show that this
replacement can be applied to. Related to gate_iswap in henry090/cirq.
Osa Distributed Geometric Quantum Computation Based On The Optimized Control Technique In A Cavity Atom System Via Exchanging Virtual Photons from www.osapublishing.org
Expressed in basis states, the swap gate swaps the state of the two qubits involved in the operation Call addgate method passing gate name, column index and array of iswap. # this file is part of
qutip: We also show that this. Swap gate with additional phase for 01 and 10 states. To generate the full xy family, we simply split the iswap gate into two √iswap gates and change the new strategy
enabled the implementation of xy entangling gates in the superconducting system. We also show that this replacement can be applied to. However, it can still be useful to have both, because iswaps can
be compiled using a single iswap.
Each quantum gate is saved as a class object gate with information such as gate name, target qubits and arguments.
Cnot gate by the bilateral iswap gate. Iswap gate using ge « eg transitions. # # 'csign', 'berkeley', 'swapalpha', 'swap', 'iswap', 'sqrtswap' Breeding and hashing protocols, which are useful for
quantum state purification. Pauli group and pauli algebra. In quantum computing and specifically the quantum circuit model of computation, a quantum logic gate (or simply quantum gate) is a basic
quantum circuit operating on a small number of qubits. Of parameters and for σg = 1 ns. We also show that this replacement can be applied to. .}, allows gates from and can be applied. Swap gate with
additional phase for 01 and 10 states. This means that for general gates, iswap doesn't offer any compilation advantages over cz. Expressed in basis states, the swap gate swaps the state of the two
qubits involved in the operation Call addgate method passing gate name, column index and array of iswap.
When we get to the very, very small world—say circuits of seven atoms—we have a lot of new things that would happen that represent completely new opportunities for design. # # 'csign', 'berkeley',
'swapalpha', 'swap', 'iswap', 'sqrtswap' Breeding and hashing protocols, which are useful for quantum state purification. Pauli group and pauli algebra. Related to gate_iswap in henry090/cirq.
Coupled Superconducting Qubits Daniel Esteve Lecture Ppt Download from images.slideplayer.com
To generate the full xy family, we simply split the iswap gate into two √iswap gates and change the new strategy enabled the implementation of xy entangling gates in the superconducting system. T
qubit relaxation time, see section 3.5.1. Cnot gate by the bilateral iswap gate. Single qubit gates correspond to rotations of a spin about some axis. We also show that this replacement can be
applied to. Related to gate_iswap in henry090/cirq. .}, allows gates from and can be applied. This is a clifford and symmetric gate.
Swap gate with additional phase for 01 and 10 states.
However, it can still be useful to have both, because iswaps can be compiled using a single iswap. Call addgate method passing gate name, column index and array of iswap. The iswap gate is an
entangling swapping gate where the qubits obtain a phase of i if the state of the qubits is swapped. Each quantum gate is saved as a class object gate with information such as gate name, target
qubits and arguments. This is a clifford and symmetric gate. T qubit relaxation time, see section 3.5.1. To generate the full xy family, we simply split the iswap gate into two √iswap gates and
change the new strategy enabled the implementation of xy entangling gates in the superconducting system. Expressed in basis states, the swap gate swaps the state of the two qubits involved in the
operation Pauli group and pauli algebra. Swap gate with additional phase for 01 and 10 states. Single qubit gates correspond to rotations of a spin about some axis. At tramp = 2 ns, the iswap gate
delity is 99.69% with a total gate time tiswap = 102 ns. # # 'csign', 'berkeley', 'swapalpha', 'swap', 'iswap', 'sqrtswap'
# this file is part of qutip: iswap. Call addgate method passing gate name, column index and array of iswap.
Post a Comment | {"url":"https://clynteasterwood.blogspot.com/2021/05/iswap-gate-random-access-quantum.html","timestamp":"2024-11-11T08:24:36Z","content_type":"application/xhtml+xml","content_length":"187756","record_id":"<urn:uuid:0a6dd5f1-a5f6-4d85-8114-382047db5e58>","cc-path":"CC-MAIN-2024-46/segments/1730477028220.42/warc/CC-MAIN-20241111060327-20241111090327-00495.warc.gz"} |
Quantum Field Theory
Tobias Osborne is a researcher in quantum information theory based at the Institut fur Theoretische Physik, Leibniz Universitat Hannover. Tobias gives a course on quantum field theory. This course is
intended for theorists with familiarity with advanced quantum mechanics and statistical physics. The main objective is to introduce the building blocks of quantum electrodynamics. | {"url":"http://www.infocobuild.com/education/audio-video-courses/physics/quantum-field-theory-tobias-osborne.html","timestamp":"2024-11-12T16:03:08Z","content_type":"text/html","content_length":"10194","record_id":"<urn:uuid:133ef6a4-30e8-466c-ab7c-ca06faaebc79>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.63/warc/CC-MAIN-20241112145015-20241112175015-00170.warc.gz"} |
sgebal: balances a general real matrix A - Linux Manuals (l)
sgebal (l) - Linux Manuals
sgebal: balances a general real matrix A
SGEBAL - balances a general real matrix A
JOB, N, A, LDA, ILO, IHI, SCALE, INFO )
CHARACTER JOB
INTEGER IHI, ILO, INFO, LDA, N
REAL A( LDA, * ), SCALE( * )
SGEBAL balances a general real matrix A. This involves, first, permuting A by a similarity transformation to isolate eigenvalues in the first 1 to ILO-1 and last IHI+1 to N elements on the diagonal;
and second, applying a diagonal similarity transformation to rows and columns ILO to IHI to make the rows and columns as close in norm as possible. Both steps are optional.
Balancing may reduce the 1-norm of the matrix, and improve the accuracy of the computed eigenvalues and/or eigenvectors.
JOB (input) CHARACTER*1
Specifies the operations to be performed on A:
= aqNaq: none: simply set ILO = 1, IHI = N, SCALE(I) = 1.0 for i = 1,...,N; = aqPaq: permute only;
= aqSaq: scale only;
= aqBaq: both permute and scale.
N (input) INTEGER
The order of the matrix A. N >= 0.
A (input/output) REAL array, dimension (LDA,N)
On entry, the input matrix A. On exit, A is overwritten by the balanced matrix. If JOB = aqNaq, A is not referenced. See Further Details. LDA (input) INTEGER The leading dimension of the array A.
LDA >= max(1,N).
ILO (output) INTEGER
IHI (output) INTEGER ILO and IHI are set to integers such that on exit A(i,j) = 0 if i > j and j = 1,...,ILO-1 or I = IHI+1,...,N. If JOB = aqNaq or aqSaq, ILO = 1 and IHI = N.
SCALE (output) REAL array, dimension (N)
Details of the permutations and scaling factors applied to A. If P(j) is the index of the row and column interchanged with row and column j and D(j) is the scaling factor applied to row and
column j, then SCALE(j) = P(j) for j = 1,...,ILO-1 = D(j) for j = ILO,...,IHI = P(j) for j = IHI+1,...,N. The order in which the interchanges are made is N to IHI+1, then 1 to ILO-1.
INFO (output) INTEGER
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an illegal value.
The permutations consist of row and column interchanges which put the matrix in the form
( T1 X Y )
P A P = ( 0 B Z )
( 0 0 T2 )
where T1 and T2 are upper triangular matrices whose eigenvalues lie along the diagonal. The column indices ILO and IHI mark the starting and ending columns of the submatrix B. Balancing consists of
applying a diagonal similarity transformation inv(D) * B * D to make the 1-norms of each row of B and its corresponding column nearly equal. The output matrix is
( T1 X*D Y )
( 0 inv(D)*B*D inv(D)*Z ).
( 0 0 T2 )
Information about the permutations P and the diagonal matrix D is returned in the vector SCALE.
This subroutine is based on the EISPACK routine BALANC.
Modified by Tzu-Yi Chen, Computer Science Division, University of
California at Berkeley, USA | {"url":"https://www.systutorials.com/docs/linux/man/l-sgebal/","timestamp":"2024-11-14T05:19:02Z","content_type":"text/html","content_length":"11300","record_id":"<urn:uuid:9488de98-183c-4d94-aafb-e90367235820>","cc-path":"CC-MAIN-2024-46/segments/1730477028526.56/warc/CC-MAIN-20241114031054-20241114061054-00891.warc.gz"} |
Stomatal Density Calculator - Calculator Pack
Stomatal Density Calculator
Stomatal Density Calculator is a powerful tool designed to help plant scientists calculate the number of stomata present on a leaf surface. This innovative calculator is user-friendly and accessible
to everyone, from experienced researchers to students just starting in the field of botany. With the Stomatal Density Calculator, scientists can quickly and accurately measure stomatal density, which
is an essential parameter for studying plant physiology and responding to environmental changes. By using this calculator, scientists can easily calculate the number of stomata per square millimeter
and evaluate their distribution patterns on leaves. The Stomatal Density Calculator is a must-have tool for anyone interested in conducting research in the field of plant physiology and ecology.
Stomatal Density Calculator
Calculate the stomatal density of a leaf
Stomatal Density Calculator Results
Stomata Count 0
Field Area 0
Magnification 0
Leaf Area 0
Stomatal Density 0
Share results with your friends
How to Use the Stomatal Density Calculator
The Stomatal Density Calculator is designed to determine the stomatal density of a leaf by taking into account parameters such as stomata count, field area, magnification, and leaf area. By inputting
these values into the calculator, you can obtain the stomatal density, which represents the number of stomata per unit area on the leaf surface.
Primary Applications
The primary applications of the Stomatal Density Calculator include:
• Plant Physiology: Assessing stomatal density to understand plant responses to environmental factors such as light, temperature, and humidity.
• Ecological Studies: Analyzing stomatal density to study plant adaptations and ecological processes.
• Climate Change Research: Investigating stomatal density as an indicator of past climate conditions through the analysis of fossilized leaves.
Instructions for Utilizing the Calculator
To effectively utilize the Stomatal Density Calculator, follow these steps:
Input Fields
The calculator requires the following input fields:
1. Stomata Count: Enter the number of stomata counted on the leaf.
□ Stomata count represents the total number of stomata observed and counted on the leaf surface.
□ Provide a numerical value for stomata count.
2. Field Area: Input the area of the field of view used for counting stomata.
□ Field area represents the size of the area on the leaf surface where stomata were counted.
□ Provide the value in any suitable unit, such as square centimeters (cm²) or square millimeters (mm²).
3. Magnification: Specify the magnification used during the observation and counting of stomata.
□ Magnification refers to the level of enlargement applied to the leaf surface to enhance visibility.
□ Provide the magnification value as a numerical value (e.g., 100X, 400X).
4. Leaf Area: Enter the total area of the leaf being analyzed.
□ Leaf area represents the size of the entire leaf or the portion used for stomatal density calculation.
□ Provide the value in the same unit used for the field area.
Output Fields
Upon submitting the form, the Stomatal Density Calculator will display the following output:
• Stomata Count: This field displays the stomata count value inputted.
□ Stomata count represents the total number of stomata observed and counted on the leaf surface.
• Field Area: This field showcases the field area value inputted.
□ Field area represents the size of the area on the leaf surface where stomata were counted.
• Magnification: This field exhibits the magnification value inputted.
□ Magnification refers to the level of enlargement applied to the leaf surface during stomata observation and counting.
• Leaf Area: This field presents the leaf area value inputted.
□ Leaf area represents the size of the entire leaf or the portion used for stomatal density calculation.
• Stomatal Density: This field displays the calculated stomatal density.
□ Stomatal density is determined by dividing the stomata count by the product of the field area, magnification squared, and leaf area.
□ It represents the number of stomata per unit area on the leaf surface.
Stomatal Density Calculator Formula
The stomatal density calculation is performed using the following formula:
Stomatal Density = (Stomata Count * Magnification^2) / (Field Area * Leaf Area)
The formula calculates stomatal density by multiplying the stomata count by the magnification squared and dividing it by the product of the field area and leaf area.
Illustrative Example
Let's consider an example to better understand the calculation process:
Suppose you counted 50 stomata on a leaf under a magnification of 400X. The field area used for counting was 1 cm², and the total leaf area is 10 cm².
Using the Stomatal Density Calculator, you input the following values:
• Stomata Count: 50
• Field Area: 1
• Magnification: 400
• Leaf Area: 10
Upon calculation, the stomatal density will be displayed as follows:
• Stomata Count: 50
• Field Area: 1
• Magnification: 400
• Leaf Area: 10
• Stomatal Density: 2 stomata/cm²
In this example, the stomatal density of the leaf is determined to be 2 stomata/cm².
Illustrative Table Example
The following table showcases multiple rows of example data to demonstrate the functionality of the Stomatal Density Calculator:
Stomata Count Field Area (cm²) Magnification Leaf Area (cm²) Stomatal Density (stomata/cm²)
30 0.5 200 5 4
70 2 1000 20 1.75
The table presents various scenarios with different input parameters, showcasing the corresponding stomatal densities.
The Stomatal Density Calculator provides a straightforward and efficient way to calculate the stomatal density of a leaf. By inputting the stomata count, field area, magnification, and leaf area, you
can quickly obtain the stomatal density value. This calculator is valuable in plant physiology studies, ecological research, and climate change investigations. Utilize the Stomatal Density Calculator
to enhance your understanding of stomatal characteristics and their ecological significance. | {"url":"https://calculatorpack.com/stomatal-density-calculator/","timestamp":"2024-11-05T15:34:19Z","content_type":"text/html","content_length":"37048","record_id":"<urn:uuid:f8db2609-77f7-4bed-a87a-8f9cfb6ff7fa>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00581.warc.gz"} |
Multiplication & division - Oxford Owl for Home
Multiplication & Division
We all have memories of how we learnt our times tables as a child and the need for children to become fluent in this area of mathematics is still very important to their learning – not only in their
primary education but also throughout their life as an adult.
Developing quick recall in this area is key to unlocking problem solving skills and understanding multiplication and division is important for gaining number knowledge. These skills are some of the
most important to develop at home and are essential to help your child prepare for their end of Key Stage 2 assessments and the newly introduced multiplication assessment in Year 4.
Maths glossary
Use these quick links or explore our jargon buster for simple definitions and examples of mathematical terms.
How to help your child at home
You don’t need to be an expert to support your child with maths! Here are four simple but effective ways to help your child develop their understanding of multiplication and division:
1. Explore practically
Use real objects to develop early understanding of multiplication and division. For example, use socks, gloves or ice-cube trays to count in twos, fives, or tens. Use egg boxes or muffin trays to
explore arrays. Practise division by sharing beads between toys or arranging blocks into groups.
2. Practise times tables
Sing, chant or play games to help your child to memorise times tables. Give points for each fact they know. Use real-life opportunities to practise. For example, when you’re in the supermarket ask
your child: ‘How many packets will we have if we buy 3 multipacks with 6 packets in each?’
3. Explain different methods
Ask your child to explain each stage of a multiplication or division and why they chose that method. They might use doubling or halving, apply times tables facts, use pictures to represent their
calculations, or write their methods. Encourage them to estimate first and then check with a different strategy.
4. Go digital and beat the clock!
As well as the many resources available here at Oxford Owl, there are a wide range of online activities and fun games to help develop speedy recall of multiplication and division facts. The sites
that are most popular are those that encourage children to challenge themselves against the clock or challenge their friends, classmates and even teachers. Speed is the key to success.
Want more?
To help your child’s learning further, you may want to watch some of the videos included within our dedicated maths library. If you’re looking for more ideas to support learning at home, head over to
our maths blog to explore articles full of top tips and fun activities.
What your child will learn at school
For more information about your child’s learning in a particular year group, use this handy drop down menu:
Multiplication & division in Year 1 (age 5–6)
In Year 1, your child will be expected to be able to solve simple multiplication and division problems using objects, drawings, and arrays to help them. This includes:
□ counting in steps of 2, 5, and 10 and understanding that, for example, 3 × 2 is the same as 2 + 2 + 2
□ sharing and grouping to solve division problems
□ beginning to understand the relationship between multiplication and division.
Multiplication & division in Year 2 (age 6–7)
In Year 2, your child will be expected to use a range of methods to solve multiplication and division problems, including using practical resources and mental methods. This includes:
□ knowing and using multiplication and division facts for the 2, 5, and 10 times tables
□ recognising and identifying odd and even numbers
□ using the symbols ×, ÷, and = to record multiplication and division calculations.
Multiplication & division in Year 3 (age 7–8)
In Year 3, your child will be expected to use a range of strategies to solve problems mentally and will begin to learn formal written methods for short multiplication and short division. This
□ knowing and using multiplication and division facts for the 3, 4, and 8 times tables
□ multiplying two-digit numbers by one-digit numbers
□ understanding that multiplication and division have an inverse relationship (i.e. they undo each other), and using this to check their calculations.
Multiplication & division in Year 4 (age 8–9)
In Year 4, your child will be expected to be able to use formal written methods of short multiplication and short division confidently. This includes:
□ knowing and using multiplication and division facts for all times tables up to 12 × 12
□ multiplying three-digit numbers by one-digit numbers
□ multiplying three numbers together
□ preparing for the Year 4 multiplication tables check in June.
Multiplication & division in Year 5 (age 9–10)
In Year 5, your child will be expected to be able to solve multiplication and division problems involving numbers up to four digits and begin to learn long multiplication. This includes:
□ multiplying four-digit numbers by two-digit numbers
□ dividing four-digit numbers by one-digit numbers and interpreting remainders
□ understanding the terms multiple, factor, common factor, prime, square, and cube numbers.
Calculation in Year 6 (age 10–11)
In Year 6, your child will be expected to be able to multiply and divide with large numbers using formal written methods including long division. This includes:
□ multiplying four-digit numbers by two-digit numbers using long multiplication
□ dividing four-digit numbers by two-digit numbers using long division
□ solving multi-step problems using addition, subtraction, multiplication, and division choosing which methods to use and explaining why. | {"url":"https://home.oxfordowl.co.uk/maths/primary-multiplication-division/","timestamp":"2024-11-06T22:06:24Z","content_type":"text/html","content_length":"99023","record_id":"<urn:uuid:36105ad2-a71b-4007-a6c8-44064e78fc04>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.47/warc/CC-MAIN-20241106194801-20241106224801-00346.warc.gz"} |
Python - (Numerical Analysis II) - Vocab, Definition, Explanations | Fiveable
from class:
Numerical Analysis II
Python is a high-level programming language known for its readability and simplicity, making it popular for various applications, including numerical analysis. Its extensive libraries and frameworks,
such as NumPy and SciPy, allow users to perform complex mathematical computations and simulations with ease, particularly in the context of methods like the Euler-Maruyama method for solving
stochastic differential equations.
congrats on reading the definition of python. now let's actually learn it.
5 Must Know Facts For Your Next Test
1. Python's syntax is designed to be intuitive, which makes it an excellent choice for beginners and experienced programmers alike.
2. The Euler-Maruyama method is particularly useful in Python for simulating solutions to stochastic differential equations using libraries like NumPy and SciPy.
3. Python supports object-oriented programming, which can help in organizing code when implementing numerical methods like Euler-Maruyama.
4. The flexibility of Python allows for quick iterations and testing of algorithms, which is crucial when experimenting with numerical analysis techniques.
5. Using Python with libraries tailored for numerical analysis can significantly reduce the amount of code needed compared to other programming languages.
Review Questions
• How does Python's design contribute to its popularity in numerical analysis, specifically when using the Euler-Maruyama method?
□ Python's design emphasizes readability and simplicity, which encourages programmers to write clean and understandable code. This is particularly beneficial in numerical analysis, as clear
code helps users quickly grasp the implementation of methods like Euler-Maruyama. Additionally, Python's extensive libraries streamline complex computations, allowing users to focus on the
logic of their simulations without getting bogged down in intricate syntax.
• Compare the roles of NumPy and SciPy in enhancing Python's capabilities for implementing the Euler-Maruyama method.
□ NumPy provides the foundational tools for numerical computing in Python by enabling efficient array manipulation and mathematical operations. SciPy builds on this foundation by offering
advanced functions and algorithms that facilitate optimization, integration, and statistical analysis. Together, they empower users to effectively implement the Euler-Maruyama method by
handling both basic array operations and complex mathematical procedures seamlessly.
• Evaluate the impact of using Python in numerical analysis compared to traditional programming languages when applying the Euler-Maruyama method.
□ Using Python in numerical analysis has several advantages over traditional programming languages. Its simplicity allows for faster development cycles and easier debugging, which are critical
when applying methods like Euler-Maruyama that require iterative testing. Furthermore, Python's rich ecosystem of libraries reduces the amount of boilerplate code needed to set up
simulations. This accessibility makes it possible for researchers and practitioners from various fields to implement sophisticated numerical methods without needing extensive programming
© 2024 Fiveable Inc. All rights reserved.
AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website. | {"url":"https://library.fiveable.me/key-terms/numerical-analysis-ii/python","timestamp":"2024-11-02T09:22:33Z","content_type":"text/html","content_length":"187114","record_id":"<urn:uuid:613d721a-a01f-406e-be39-442baf7b7a84>","cc-path":"CC-MAIN-2024-46/segments/1730477027709.8/warc/CC-MAIN-20241102071948-20241102101948-00889.warc.gz"} |
Physics Frontiers
<-- Previous (Negative Effective Mass) (Chameleon Field) Next -->
Recorded: 10/20/2018 Released: 12/23/2018
Randy tells Jim about the octonions, a cousin to the complex numbers in eight dimensions that Cohl Furey has made some headlines with by categorizing elementary particles with them. By looking at,
basically, stable sets in the octonions, she has found representations that act like the elementary particles, and found ways to characterize some of their parameters, e.g., the charge, with them. []
1. The papers we read for this program:
2. Related Papers:
3. Related Episodes of Physics Frontiers:
4. Books mentioned in this podcast:
• Crowe, Michael, A History of Vector Analysis. A little bit dry, but a third of the book is about Hamilton and the quaternions.
• Whitehead, A.N., Process and Reality. If you think you're smart, try this.
• Fano and Fano, Physics of Atoms and Molecules. Lots of perturbations. That's what I remember.
5. John Baez'
for all things octonion.
6. Cohl Furey's
video series
on the octonions and the standard model.
6. Please visit and comment on our
, and if you can help us keep this going by contributing to our
, we'd be grateful.
<-- Previous (Negative Effective Mass) (Chameleon Field) Next -->
<-- Previous (Space-Time Dimensions) (Octonions) Next -->
Recorded: 9/29/2018 Released: 12/9/2018
Randy introduces Jim to gravitational effects on quasiparticles in materials. The inertial quality of the mass of a quasiparticle gets modified by the lattice, giving rise to an effective mass in the
material. But how does the effective mass behave when confronted with a gravitational field? []^[S::S]
1. The papers we read for this program:
• Raum, K., M. Weber, R. Gahler, and A. Zeilinger, "Gravity and inertia in neutron crystal optics and VCN interferometry" J Phys Soc Jpn 65, 227 (1996). [Free]
• Wimmer, M., A. Regensburger, C. Bersch, M.-A. Miri, S. Batz, G. Onishchukov, D.N. Christodoulides, and U. Peschel, "Optical Diametric Drive Acceleration through Action-Reaction Symmetry Breaking"
Nature Physics 9, 780 (2013). [Free] [Supplement]
2. Related Papers:
• Coletta, R., A.W. Overhauser, and S.A. Werner, "..." Phys Rev Lett 34 1472 (1975).
• Raum, K., M.Koellner, A. Zeilinger, M. Arif, and R. Gahler, "Effective Mass Enhanced Deflection of Neutrons in Noninertial Frames" Phys Rev Lett 74, 2859 (1995). [Free]
• Bondi, H. "Negative Mass in General Relativity," Rev Mod Phys 29 423 (1957)
• Foreward, R.L., "Negative Matter Propulsion", J Prop Power 6 28 (1990) .
2. Related Episodes of Physics Frontiers:
We referenced a lot of old episodes in this one:
Don't bother looking for our discussion of Manu Paranjape's essays on the "possibility of generating an negative effective mass in space-time" in the episode entitled "The Positive Energy Theorem."
We're working on getting those up, but there's a content issue that we may not be able to resolve.
3. Books mentioned in this podcast:
• I mentioned that some of this is textbook stuff, when Jim Napolitano finished J.J. Sakurai's Modern Quantum Mechanics, he included he discusses Colletta, Overhauser and Werner's gravity induced
phase changes that can be measured through interferometry. Somewhere Napolitano writes that he includes these interesting tidbits because he is an experimentalist and thinks it's helpful for
understanding. I just know they're fun. Be advised that, although it's not as heavy going as Cohen-Tannoudji (which, thanks only to the psychic trauma induced by graduate school, I somehow
spelled right), is a graduate level quantum mechanics textbook. Just a very well written one.
4. You can watch Martin Tejmar's talk at the 2016 breakthrough propulsion workshop put on by the Space Studies Institute
5. Martin Tejmar's group at TU-Dresden, and his publications page.
6. Please visit and comment on our subreddit, and if you can help us keep this going by contributing to our Patreon, we'd be grateful.
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how many megabytes in a gigabyte
1 gigabyte = 1024 MB. There are 1000 megabytes in a gigabyte. The main non-SI unit for computer data storage is the byte. How many megabyte in gigabyte? GB How Many Megabytes (MB) in a Gigabyte (GB)?
Microsoft uses this definition to display hard drive sizes, as do most other operating systems and programs by default. How many megabytes are in a gigabyte? A dual-layered DVD disc capacity is 8.5
GB = 8500 MB. For ease of calculation, weâ ll say there are 1000 kilobytes (KB) in a megabyte (MB) and 1000 MB in a gigabyte (GB), often referred to as a gig of data.. Who cares, most of the time.
There are 125 megabytes per second in 1 gigabits per second. 1 GB = 10 3 MB in base 10 (SI). You can find metric conversion tables for SI units, as well We assume you are converting between megabit
and gigabyte. 7 years ago. These are the ways applied by many people. Gigabyte (GB) is one of the most commonly used units of digital information which is equal to 1,000,000,000 bytes. The next
measurement up is a gigabyte (GB), which is made up of 1024MB. Soon, we will be going to even bigger units of measurement. MB to bit Please enter the gigabytes (GB) value to convert to megabytes
(MB). Because I also get sometimes confused. 1 decade ago. I love this because I'm so tired of the hot shot cable companies and internet providers telling me that I need more data all the time
especially when I haven't even been using any data. here are the common storage capacities in GB. The difference between units based on SI and binary prefixes increases exponentially in other
words, an SI kilobyte is nearly 98% as much as a kibibyte, but a megabyte is under 96% as much as a mebibyte, and a gigabyte is just over 93% as much as a gibibyte. A few real-world examples: How
many megabytes per second are there in 1 gigabits per second? How many megabytes are in a gigabyte will also show how bytes and kilobyte so on. The reason there are 1024 bytes in a kilobyte, and 1024
Kb in a Megabyte, and 1024 Mb in a Gigabyte, is because long ago when they first invented computers and they needed a way to address each byte in memory, they used 12 bits (1111 1111 1111) to hold
the address number of each byte in memory. It both causes confusion and prevents to define precisely, how many megabytes in gigabyte. Divide this number by 1,024 using the calculator. 1 / 0.001 =
1000. To convert from gigabits per second to megabytes per second, multiply your figure by 125 (or divide by 0.008) . There are 0.0009765625 Gigabytes in a Megabyte. Because the memory is actually
16384 Mb long. Thank you so much for this wonderful website. Please enable Javascript to use the unit converter. MB to petabyte. How Many Bytes In A Gigabyte Overview. We have come up with what 1 GB
of data looks like with the help of kenstechtips.com. There have been some recent inquiries about the differences between the 3G and 4G technologies relating to how many Gigabytes faster is 4G. This
second figure will often be used when speaking about different sizes of RAM within an operating system. Data Units Calculator Gigabyte to Byte. We conclude that one Gigabyte is equivalent to one
thousand Megabytes: 1 Gigabyte is equal to 1000 Megabytes. mebi (symbol 'Mi'), 220 = 1048576; We assume you are converting between megabyte and gigabyte. Streaming video on an iPad uses up 2
gigabytes of data over six to 12 hours of streaming movies, correlating to 1/6 to 1/3 of a gigabyte used per hour of video. Gb is equal to how many mb? please note that the SI recommends to use the
definition 1GB = 1000MB which is equal to 10003 bytes. Megabyte (MB) – MB stands for Megabyte. Microsoft uses this definition has been incorporated into the International system of Quantities
surprise that there are 1000 in... 4700 MB plain text ( 1,200 characters ) can look at the table. -3 GB in base 10 ( SI ) on your smartphone and don t. Can not hold any gigabytes at all note that
rounding errors may occur, so closest! Gb = 4700 MB unit for computer memory and file sizes byte is equal to 1000 megabytes in megabyte. Tell us how much digital information, which is made up of
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megabytes in a gigabyte the non-SI version used! Mb ( =2 10 megabytes ) in a gigabyte â ¦ what is a unit of measure of amount of unused! This should come as no surprise, as do most other operating
systems and programs by default well. Now generally between 2 and 4 gigabytes the Gigabit per second, multiply your figure by 1000 or. Because 1,000 KB are not 1 MB perspective, the IEC recommends
that this unit should be!, the size is somewhere between 1/3 of a movie, computer RAM, and 111111111111 byte. Figures are based on using mobile data on your smartphone and don ’ t take tethering into
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digital! In to give your opinion on the answer, there are 1024 megabytes â ¦ there are 0.000125 per! To that is provided above may be of help for users or specifying size... Hold many megabytes are
in a megabit per second by 0.001 ) from. Single layer DVD disc capacity is 25 GB = 10 3 MB base! Plain text ( 1,200 characters ) conclude that one gigabyte is equal to 1000 megabytes in a or.
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slight differences in the form to convert from gigabytes to megabytes second! A `` gibibyte '' ( abbreviated GiB ) popular plans offered by mobile phone … how GB... Kali Linux 1,000 KB are not 1 MB =
10 -3 GB in 10...
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Does anyone know the inverse Fourier transform of the following function? - FAQS.TIPS
I'm curious to know if anyone knows of the inverse Fourier transform of the function depicted in the attached picture. The transform variable is s and all other variables are constants.
I would be much obliged for any answer. | {"url":"https://faqs.tips/post/does-anyone-know-the-inverse-fourier-transform-of-the-following-function.html","timestamp":"2024-11-05T14:07:59Z","content_type":"text/html","content_length":"51159","record_id":"<urn:uuid:1043a3da-2f07-4871-9ce0-61449955655c>","cc-path":"CC-MAIN-2024-46/segments/1730477027881.88/warc/CC-MAIN-20241105114407-20241105144407-00252.warc.gz"} |
How do you rationalize 2/(sqrt(72y))? | Socratic
How do you rationalize #2/(sqrt(72y))#?
1 Answer
Let's remember that we use rationalization in order to remove roots from our denominator.
We proceed to do that by multiplying both numerator and denominator of your function by the same value as the root contained in the denominator. That way the proportion will be maintained and the
root will be eliminated because
$\sqrt{f \left(x\right)} \sqrt{f \left(x\right)} = f \left(x\right)$
That is because $\sqrt{f} \left(x\right) = f {\left(x\right)}^{\frac{1}{2}}$, then $f {\left(x\right)}^{\frac{1}{2}} f {\left(x\right)}^{\frac{1}{2}} = f {\left(x\right)}^{\frac{1}{2} + \frac{1}{2}}
= f {\left(x\right)}^{1} = f \left(x\right)$
So, for your function:
$\frac{2}{\sqrt{72 y}} \left(\frac{\sqrt{72 y}}{\sqrt{72 y}}\right) = 2 \frac{\sqrt{72 y}}{72 y} = \frac{\sqrt{72 y}}{36 y}$
Your function has been rationalized, but we can further simplify it.
$\sqrt{72 y}$ is the same as $\sqrt{2 \cdot 36 y}$. We can take the root of $36$ out, like this: $6 \sqrt{2 y}$
Now, let's just simplify!
$\frac{6 \sqrt{2 y}}{36 y} = \frac{\sqrt{2 y}}{6 y}$
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On a certain sum, the compound interest in 2 years amounts to R... | Filo
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On a certain sum, the compound interest in 2 years amounts to Rs. 4,240 . If the rate of interest for the successive years is and respectively, find the sum.
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Question Text On a certain sum, the compound interest in 2 years amounts to Rs. 4,240 . If the rate of interest for the successive years is and respectively, find the sum.
Updated On Jun 21, 2023
Topic All topics
Subject Mathematics
Class Class 9
Answer Type Video solution: 2
Upvotes 141
Avg. Video Duration 5 min | {"url":"https://askfilo.com/user-question-answers-mathematics/on-a-certain-sum-the-compound-interest-in-2-years-amounts-to-35303932343832","timestamp":"2024-11-04T23:57:13Z","content_type":"text/html","content_length":"171592","record_id":"<urn:uuid:35748733-32b3-4869-8d69-da6cb92f53f4>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.84/warc/CC-MAIN-20241104225856-20241105015856-00400.warc.gz"} |
Pseudo-randomness | Learning Cardano
Pseudo-randomness refers to the generation of numbers or values that appear random but are produced by a deterministic process, typically through algorithms. These algorithms, called pseudo-random
number generators (PRNGs), use a seed value to generate sequences of numbers that seem random but are reproducible if the seed is known.
While truly random processes rely on natural phenomena (like radioactive decay or thermal noise), pseudo-random processes are based on mathematical formulas. Pseudo-randomness is vital in
cryptography, simulations, and algorithms, where controlled randomness is needed.
Example of Pseudo-randomness:
Imagine you have a simple algorithm that generates numbers based on an initial seed (e.g., the time of day). If you start with the same seed, the sequence of numbers will always be the same, though
they may appear random.
Key Concepts Related to Pseudo-randomness in Cryptography
1. Multi-party Computation (MPC)
Multi-party computation (MPC) is a cryptographic method that enables multiple parties to jointly compute a function over their inputs while keeping those inputs private from one another. MPC can be
used to generate pseudo-random values in a decentralized and trustless way.
• How MPC Relates to Pseudo-randomness: In a multi-party setup, participants can collaborate to generate a pseudo-random number that no single party controls. Each party contributes a secret input,
and the output appears random to all participants. Even if one party is compromised, the randomness of the result is maintained, as long as the majority remain honest.
• Example in Blockchain: In decentralized systems, such as blockchain lotteries or election systems, MPC can be used to generate random numbers (or other secret values) without requiring trust in
any single entity. The randomness is guaranteed by the combined input of all participants.
2. Verifiable Random Functions (VRF)
A Verifiable Random Function (VRF) is a cryptographic primitive that produces a pseudo-random output that can be verified as being generated by a specific input (like a public key or a message). VRFs
allow a party to generate a random value and a proof, where both the value and proof can be verified by others to ensure the value was generated correctly and fairly.
• How VRFs Relate to Pseudo-randomness: A VRF ensures that a random output is pseudo-random but verifiable by others. This means the randomness is not truly random in the natural sense but still
unpredictable to anyone who doesn’t know the secret (private key). The proof allows third parties to verify that the random value wasn’t manipulated or biased by the generator.
• Example in Blockchain (Cardano): Cardano uses VRFs to select block producers (called slot leaders) in its Proof of Stake (PoS) protocol. The VRF generates a pseudo-random number to determine
which validator (or stake pool) is selected to create the next block. Importantly, the generated random number and proof can be publicly verified by other nodes in the network, ensuring fairness
and security.
Combining MPC and VRF for Secure Pseudo-randomness
In decentralized systems, achieving truly unbiased and fair randomness is crucial for tasks like leader selection, lottery systems, or determining outcomes. MPC and VRF can be combined to ensure that
pseudo-random values are both secure and verifiable:
• MPC for Decentralized Randomness: Using multi-party computation, participants in a decentralized network can jointly generate a random value. Since no single party controls the outcome, the
random number can be trusted to be unbiased.
• VRF for Verifiability: Once the random value is generated (either via MPC or some other method), VRFs can be used to ensure that the random number was produced honestly and that no party can
manipulate the result. The VRF ensures that the random output can be publicly verified without revealing the secret input that generated it.
Application in Blockchain (e.g., Cardano)
In Cardano’s Ouroboros PoS protocol, both pseudo-randomness and VRF play critical roles in ensuring the fairness of leader selection:
• Cardano uses a VRF to determine which stake pool (validator) is chosen as a slot leader to create the next block. Each stake pool runs a VRF using its private key to generate a pseudo-random
number. If this number falls below a certain threshold (determined by their stake), they are chosen as the leader.
• The output of the VRF and its proof can be verified by anyone in the network, ensuring that the selection process is fair and cannot be tampered with.
• Pseudo-randomness: Refers to the generation of numbers or values that seem random but are generated by deterministic processes.
• Multi-party Computation (MPC): A cryptographic technique that enables multiple parties to jointly compute a function without revealing their private inputs, often used for decentralized and
unbiased randomness generation.
• Verifiable Random Functions (VRF): A cryptographic function that produces a pseudo-random value along with a proof that can be verified to ensure the fairness and correctness of the randomness.
In decentralized systems like Cardano, these concepts are critical for ensuring the fair, secure, and verifiable selection of participants in processes such as block validation, leader selection, or
other consensus mechanisms. | {"url":"https://www.learningcardano.com/pseudo-randomness/","timestamp":"2024-11-14T11:10:26Z","content_type":"text/html","content_length":"101109","record_id":"<urn:uuid:9929591d-41d4-46a3-95c7-b4dbc473fcf1>","cc-path":"CC-MAIN-2024-46/segments/1730477028558.0/warc/CC-MAIN-20241114094851-20241114124851-00273.warc.gz"} |
Scatter Diagram
A scatter diagram is a graphic which shows data where one variable has been plotted against a second variable. Scatter diagrams are used when investigating correlation between two variables.
Scatter diagram is a high school-level concept that would be first encountered in a probability and statistics course. It is an Advanced Placement Statistics topic and is listed in the California
State Standards for Grade 7.
Classroom Articles on Probability and Statistics (Up to High School Level)
Arithmetic Mean Mode
Box-and-Whisker Plot Outlier
Conditional Probability Problem
Histogram Sample
Mean Standard Deviation | {"url":"https://mathworld.wolfram.com/classroom/ScatterDiagram.html","timestamp":"2024-11-05T22:12:35Z","content_type":"text/html","content_length":"47210","record_id":"<urn:uuid:afab63d2-6eee-483e-922c-1e1952147e6e>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00092.warc.gz"} |
Free loop spaces and link homology
1. Free loop spaces and link homology
Free loop spaces and link homology
Free loop spaces and link homology
Joshua Wang, Princeton/IAS
The Khovanov homology groups of torus knots T(n,m) are known to stabilize as m goes to infinity with n fixed. In this talk, we make the observation that when n = 2, the stable limit happens to be
isomorphic to the homology of the free loop space of the 2-sphere. Our main result suggests that this is not merely a coincidence: we prove that the k-colored sl(N) homology of T(2,m) stabilizes to
the homology of the free loop space of the complex Grassmannian Gr(k,N). We also relate the space of closed geodesics on the Grassmannian to the k-colored sl(N) homologies of the individual torus
knots T(2,m). | {"url":"https://www.math.princeton.edu/events/free-loop-spaces-and-link-homology-2024-09-05t203000","timestamp":"2024-11-09T16:38:16Z","content_type":"text/html","content_length":"31406","record_id":"<urn:uuid:b9448f6e-bd16-4e4c-bb17-7048389061a7>","cc-path":"CC-MAIN-2024-46/segments/1730477028125.59/warc/CC-MAIN-20241109151915-20241109181915-00671.warc.gz"} |
Trigonometry | What is Trigonometry?|Birth of Trigonometry|Trigonometry Problems
What is Trigonometry?
We know that “Necessity is the mother of invention.” It is the compulsion of needs that is overriding. Thus it would not be unfair to look for such imperatives behind the birth of Trigonometry, which
is a distinct branch of Mathematics.
But what is this?
Whatever man finds in nature, he accepts after verification be it from trees, fruits, flowers, animals that he sees within his reach to sun, moon, stars, planets of distant horizons, insurmountable
mountains, vast oceans, rivers. Man has learnt once to measure things around him, understood shapes of different objects with the help of geometrical figures and has learnt to measure, length, width
and height. Also with the help of these, he has used conciously his efforts to measure size, shape, distance to determine laws of motion etc. of planets, stars, sun, moon, etc. that are beyond his
reach. It is because of such efforts that mathematicians could obtain the height of inaccessible pyramid without getting at the top.
We now explain how to solve such Trigonometry problems mathematically. Suppose we are to determine the height of a light post. We shall now describe how to measure it even standing on the ground.
In the following figure OP is a light post and its height has to be determinant.
A point is placed at some distance from the base 0 of the light post on the same plane on which the light post is standing. A circle is drawn on the ground with B as centre and AB as radius.
When the sun rises in the morning, the long shadows of both the light post OP and the post AB are seen. As the sun moves up higher, the shadows become shorter. At some time it is seen that the shadow
of the top of the post AB i.e. the shadow of the point A is coinciding with the point C on the circle. Suppose at that moment the shadow of the light post OP becomes OM. The point M is marked. Now
you can measure the distance OM of M from the foot 0 of the light post. Again, OP = OM; then by just measuring the distance OM you can find the height of the light post.
You will be able to understand this matter with the help of your knowledge in geometry. The theory that is applied here is that of the ratios of the sides of equiangular triangles which you have
already learnt as a theorem in geometry.
You know that sun rays coming from a very long distance are practically parallel. So PM || AC
Therefore, PMO = ACB. So the right-angled triangles POM and ABC are equiangular.
In the right-angled triangle ABC, AB = BC as the circle is constructed with radius AB. BC is a radius of the circle. So = AB/BC = 1
Again, since ABC and PMD are equiangular,
Therefore, PO/OM = AB/BC = 1
So, PO = OM
Thus you see that it is possible even to measure the height of a light post by just standing on the ground without climbing up at the top of the light post.
In ancient times, following this technique used the ratio of the length of a post to its shadow and the ratio of the length of a pyramid to its shadow to determine the height of the pyramid.
The main disadvantage of this method is that one has to wait for a particular position of the sun in the day time. But during Greek civilisation the development of mathematics was in such a stage
that no method better than this theory was known.
Civilisation has, on one hand been confronted with various new problems, it is man again and on the other hand, who has discovered new methods of solutions, for example, there is a hilly stream. One
has to find out its width without using tape after crossing the stream. Man has done it with the help of mathematics.
In the meantime, mathematicians have been able to get at the relationships between sides and angles of triangles of various forms. They have come to know that the ratio of the sides of right- angled
triangles, with respect to a definite measure of an acute angle, is a constant quantity. It is with the help of this that they could determine how wide the stream was. You will learn this technique
of doing so in this branch of mathematics.
The relationships between sides of triangles, angles and areas are made use of in this particular branch of mathematics.
This branch is called Trigonometry.
You may keep in mind that mathematicians in ancient India were quite familiar with this branch of mathematics. They have applied this subject to the solution of many difficult problems of
Basic Trigonometry
Measurement of Trigonometric Angles
Relation between Sexagesimal and Circular
Conversion from Sexagesimal to Circular System
Conversion from Circular to Sexagesimal System
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The Economics of Western Decline
The Economics of Western Decline
The study of economics is a rich and rewarding experience, and whilst some regard it as the dismal science I can only assume that they are referring to the dismal performance of our political masters
over recent decades, in terms of their truly awful economic policy track-record.
There is barely a developed country in the world today that is not mired in debt the likes of which we have never seen in peacetime history. There are, of course, many reasons and arguments that can
be made to justify these debts, but the simple fact of the matter is that it is politics, not economic science, that has brought us to the precipice of a new economic catastrophe.
Our most recent experience of government induced meltdown came in 2008, and while the years of hardship that followed were both deep and prolonged, there is a real danger that what comes next for the
developed world, including the United States, will make those years seem like our glory days.
For the first time in 50 years there is the real prospect of stagflation ahead of us - a period of both high unemployment and high inflation. Modern mainstream economic theory and research, dominated
as it is by Keynesian economics, still struggles to explain such things.
We have already had a taste of higher inflation, but that may turn out to have been nothing more than a mild first wave of what will become a much bigger problem. In 2024, recession has already
emerged in Japan, Germany, and the UK, but high inflation lingers on.
If a country suffering from recession were to try to stimulate its economy with yet more government deficit-spending, at a time of unprecedented debt levels, the end result could drive inflation much
higher, even as its economy sinks further into recession.
Unfortunately for the western world, public policy has become complicated by a literal kaleidoscope of existential threats to the current economic system, and I will detail these threats in a section
Before that, I want to start with an overview of the typical core undergraduate program that economics students will need to understand, and the helpful content that my site has to offer in this
These topics in microeconomics and macroeconomics form the core components of a typical undergraduate program, and I’ve written them without any unnecessary technical jargon or mathematics. There is,
of course, a great deal of mathematics in any economics degree course but my site is focused on helping readers to understand the concepts rather than the numbers.
Economics and economic research are often criticized for their sometimes-misguided over-reliance on mathematical techniques that are not well-suited to the social sciences. The math used is better
suited to the hard sciences like physics. Unfortunately, there is no economics faculty that I am aware of that spares its students from this pain! Happily, math is not needed at all if you are a more
casual student with just a general interest in economics.
I have focused my site on the core components of economics, but there are many more specialized applied economics courses that flow from the core topics. Industrial Organization, International
Economics, Economic Development and Financial Economics were my own chosen options, but there are many more applied economics courses to choose from. Usually these are the focus of final year
undergraduate courses.
There is no material on my site aimed at complementing a graduate program in economics, though my site does offer a useful refresher for graduate students to brush up on the key concepts. There is
also plenty of supplemental content, economic analysis, and opinion pieces on the economics news of the day, particularly with respect to the existential threats that the western economies are faced
with in modern times.
Existential Threats to the Economic System
The US Federal Debt Burden
As the graph below illustrates, the US Debt to GDP (Gross Domestic Product is a measure of national income) has been trending higher for decades. There had, until recently, been the insane idea that
debts don’t matter since money can always be printed with which to repay those debts. That idea fell flat on its face when, in 2020, inflation began to rise sharply thereby demonstrating that the
link between the money-supply and inflation is still relevant today (some economists had argued that the link was no longer valid).
The starting point for the data series below is January 1971, as this was the year that president Nixon took the US dollar off the fixed dollar-to-gold conversion rate, meaning that our current
fully-fiat monetary system started at that time. The vertical grey lines represent periods of recession.
As can be seen, it wasn’t until the early 1980s recession that debt to GDP began its long march higher. The gradual rise in debt from the early 1980s to the mid-1990s coincides with the
deindustrialization of much of the western world, with huge numbers of job losses in the manufacturing sector, and higher resulting welfare claims as unemployment rates soared and the inequality
between rich and poor widened. Another consequence of manufacturing decline was the opening up of a large trade deficit, a deficit that was plugged largely by increased international borrowing i.e.
more debt.
From the late 1990s until the dot com recession of the early 2000s, debt came down somewhat, but then resumed its climb. The early 2000s was the start of a long period of low interest rates, making
debt cheap, and encouraging more and more borrowing.
The 2008 financial crisis led to massive borrowing and deficit spending by the government in order to fund huge bailout packages for the insolvent banking sector. The banks had made excessive
mortgage-backed loans off the back of a real estate bubble that had burst by 2008. An almost continuous 14-year period of ZIRP (zero interest rate policy or close to zero) followed the crisis,
because the level of indebtedness in the system was already too high to afford realistic interest rates. As a result of ZIRP, the debt continued to surge ever higher.
The Covid pandemic of 2020 then saw another massive surge in government borrowing in order to fund stimulus checks during the lockdowns, meaning that debt rose sharply yet again. The latest figures
show that the debt currently stands at around 120% of GDP.
Unfunded Liabilities, Energy Costs, and Trade
If the current GDP to debt ratio is alarming, it pales in comparison to the nightmarish level of expenses that will emerge in the coming 10-20 years. These costs fall into three main categories:
Demographic Change & Unfunded Liabilities – Demographic change refers to the changing composition of the population in terms of dependents to working-age people. With half of the baby boomer
generation already retired, the next 15 years will see the other half retire, and the first half reach nursing home age. The cost of this comes in terms of social security payments and healthcare
costs. These are termed ‘unfunded liabilities’, and Jeff Gundlach estimates that these amount to around $200 trillion – more than six times as large as the federal debt at the time of writing, and
I’ve never seen a sensible plan about how to deal with this enormous expense.
Energy & Environmental Costs – Commitments to net-zero carbon emissions amount to a commitment to much more expensive energy in future. The only exception to this outcome would require a commitment
to thousands of nuclear power stations located all over the western world, because small modular reactors are actually ultra cheap, clean, and safe. Unfortunately, the nuclear power industry has a
terrible publicist, and seems unlikely to be embraced anytime soon. The implication, as cheap oil and gas is gradually replaced by expensive renewable energy, the costs of production will soar and
inflation/recession is the likely outcome.
Trade in a De-Globalized World – The benefits of ‘globalization’ over the past few decades brought an almost endless supply of cheap manufactured goods to domestic consumers, thereby boosting
consumer spending courtesy of the almost inexhaustible supply of cheap labor in China and elsewhere in the developing world. Going forward, and in the wake of the 2020-21 global pandemic lockdowns,
the strategic problems arising from globalized supply-chains are leading to a reversal of the globalization trend. This should, of course, bring some jobs back to western nations, but it will also
mean more expensive products in future i.e. more price inflation.
Higher Interest Rates & Finance Costs
The problem is not just the sheer size of the debt and the coming expenses, there is also the problem of rising interest rates. Rates that had been near zero for over a decade could no longer be held
at these artificially low levels after inflation shot higher in the wake of the global lockdowns. The lockdowns had created severe supply shortages in the marketplace, leading to rapid price
increases. Additionally, hefty stimulus checks and money-printing added further fuel to the inflation fire.
The Federal Reserve was then forced to sharply raise interest rates to dampen demand and cool down inflationary pressure.
At the beginning of 2024, it may seem that inflation is now under control (having been much reduced from its peak in 2022), and that interest rates can therefore start to come down again. That indeed
is anticipated by many market investors, however, there are also many signs that inflation could easily return if rates are reduced too soon.
The government is still deficit-spending at extremely high levels, and without some measure of fiscal restraint there is less chance of any significant interest rate reduction. There is also a good
argument to be made that all the money-supply growth in recent years has not yet fully played out, meaning that inflation could easily start to rise again.
The problem with higher interest rates is that it makes debt-servicing much more expensive. Servicing costs are now already larger than the entire national defense budget, and they are set to climb
out of all control in the near future as existing debt is gradually refinanced into higher-rate debt.
In Summary
The cost of all the existing debt, and all the future strains on the public finances, could easily lead to a US debt crisis. If a debt crisis does emerge, it would leave the US and other western
countries with two options:
1. Defaulting on the debt leading to a financial crisis that breaks the bond market, destroying pensions and savings, widening the inequality between rich and poor, and creating a deflationary
2. Debasing the currency via money-printing to nominally pay the debts, thereby creating sky-high inflation that again destroys the bond market and associated pensions and savings. This would create
an inflationary recession.
Both options would lead to the deepest and longest economic depression since the 1930s, and the US dollar would certainly lose its reserve currency status, meaning that a new global financial system
would be required. Whatever new system is created, it is hard to see how the western economies could emerge with strong national currencies and favorable terms for international trade with the rest
of the world.
There is a danger that voters will blame the recession on capitalism, but the truth is that incompetent, and corrupt, government excess is to blame. If voters react by voting for socialism, then the
consequences will be horrendous. More government is never the solution to bad government.
The coming collapse may herald a real shift in global economic power, meaning relative economic decline for the western economies, and significant growth in the East – primarily Russia, India, China,
and Southeast Asia.
FAQs that People Ask
What is the main economic problem?
The main economic problem, often referred to as the fundamental economic problem, is the issue of scarcity. Scarcity arises because resources (including time, money, labor, and natural resources) are
limited, while human wants and needs are virtually unlimited.
This gives rise to three basic economics questions:
• What to Produce - Given limited resources, societies must decide which goods and services to produce. Choices need to be made about allocating resources to different industries and sectors.
• How to Produce - Once the decision on what to produce is made, societies must determine the most efficient and effective ways to produce those goods and services. This involves choices about
technology, production methods, and resource utilization.
• For Whom to Produce - Distribution is a critical aspect. Societies must decide how the goods and services produced are distributed among the population. This involves questions of income
distribution and access to resources.
What are the three main types of economic systems?
The three main types of economic systems in the modern world are:
• Market Economy (Capitalism) - In a market economy, decisions over what, how, and for whom to produce are driven by the forces of supply and demand in the marketplace. Private individuals and
businesses own and control the means of production, prices are determined by the interaction of buyers and sellers, and the government's role is usually limited to enforcing contracts and
protecting property rights.
• Command Economy (Communism/Socialism) - In a command economy, the government or a central authority makes all economic decisions. The state owns or controls the means of production, and resources
are allocated based on central planning. Prices are often set by the government, and the goal is to give people a better level of social & economic equality. However, it usually suffers from a
huge loss of efficiency.
• Mixed Economy - A mixed economy combines elements of both market and command economies. It allows for private ownership and market forces to operate, but the government also intervenes in certain
areas to regulate and address market failures. Governments in mixed economies provide public goods, social services, and regulations to obtain better outcomes in particular areas in which
free-markets fail to deliver satisfactory outcomes.
In reality, all economies are of the mixed variety because all contain some measure of both market economy and command economy, but to varying extents.
What is positive and normative economics?
Positive economics and normative economics are two distinct branches within the field of economics that serve different purposes and involve different types of analysis.
Positive economics is concerned with objective analysis, describing and explaining economic phenomena as they are. It deals with facts, data, and observable economic evidence. It is value-free and
focuses on what is, rather than what ought to be.
Normative economics, on the other hand, is concerned with subjective analysis, involving people making judgments about what ought to be or what is desirable. Clearly this involves value judgments and
opinions about what is considered good, fair, or just.
Is an economics degree hard?
Generally, an economics degree is considered to be quite difficult due to its analytical nature and use of mathematics. Of course, for a student who excels in that type of challenge then it is much
less difficult, but I recall several students who had taken some economics courses as part of their degrees, and they all struggled!
Economics is unlike some courses where a cursory effort is enough to get over the line, it does require some genuine effort, but it is far from insurmountable.
For undergraduates of other topics, who wish to include some economics in their studies but without the numbers, political economy or economic history are worthy options. However, an economics major
is sure to include a substantial amount of mathematics and statistics.
What are some reasons for studying economics?
One of the main reasons to study economics is the practical knowledge that it will give you in terms of economic policy. If you have an interest in how our political elites manage the country, and
how our media report it, you’ll find some economics knowledge of your own to be extremely illuminating!
An economics degree does not confine you to employment as an economist, it can lead to quite diverse career paths. Graduates may pursue work in finance, consulting, government, international
organizations, academia, research, and many more fields of employment.
Economics encourages critical thinking and problem-solving. The discipline involves analyzing data, constructing models, and evaluating the implications of different policies, fostering analytical
and logical reasoning. As well as helping you to understand how the world works and to gain certain types of jobs, the knowledge it gives you will help you to protect your personal finances.
What happens to my 401k if the economy collapses?
If there is a severe economic collapse, it will almost certainly have significant implications for financial markets, including retirement accounts like 401(k)s. I have already alluded to the fact
that the bond market is reaching a breaking-point, and that it would have serious consequences for pensions and savings plans.
Most, if not all, 401(k)s have put massive investment into long-dated government bonds, and if the government either defaults on its debt, or inflates it away via money-printing, it will greatly
reduce the value of 401(k)s.
The way your 401(k) is allocated across different asset classes (stocks, bonds, cash, commodities) plays a crucial role. A well-diversified portfolio that includes assets with low correlation to each
other will help mitigate losses during most economic downturns, because as one asset type falls, another should rise.
Many independent investors are turning to Bitcoin and Gold as alternative assets. This is due to their monetary properties at a time when the survival of the Fiat monetary system in the coming years
is uncertain. | {"url":"https://www.dyingeconomy.com/","timestamp":"2024-11-04T01:17:33Z","content_type":"text/html","content_length":"52805","record_id":"<urn:uuid:a729a4a0-ba76-41d0-aae4-b763ce8ab5e2>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00176.warc.gz"} |
Internet Encyclopedia of Philosophy
Epsilon Calculi
Epsilon Calculi are extended forms of the predicate calculus that incorporate epsilon terms. Epsilon terms are individual terms of the form ‘εxFx’, being defined for all predicates in the language.
The epsilon term ‘εxFx’ denotes a chosen F, if there are any F’s, and has an arbitrary reference otherwise. Epsilon calculi were originally developed to study certain forms of arithmetic, and set
theory; also to prove some important meta-theorems about the predicate calculus. Later formal developments have included a variety of intensional epsilon calculi, of use in the study of necessity,
and more general intensional notions, like belief. An epsilon term such as ‘εxFx’ was originally read as ‘the first F’, and in arithmetical contexts as ‘the least F’. More generally it can be read as
the demonstrative description ‘that F’, when arising either deictically, that is, in a pragmatic context where some F is being pointed at, or in linguistic cross-reference situations, as with, for
example, ‘There is a red-haired man in the room. That red-haired man is Caucasian’. The application of epsilon terms to natural language shares some features with the use of iota terms within the
theory of descriptions given by Bertrand Russell, but differs in formalising aspects of a slightly different theory of reference, first given by Keith Donnellan. More recently, epsilon terms have
been used by a number of writers to formalise cross-sentential anaphora, which would arise if ‘that red-haired man’ in the linguistic case above was replaced with a pronoun such as ‘he’. There is
then also the similar application in intensional cases, like ‘There is a red-haired man in the room. Celia believed he was a woman.’
Table of Contents
1. Introduction
Epsilon terms were introduced by the german mathematician David Hilbert, in Hilbert 1923, 1925, to provide explicit definitions of the existential and universal quantifiers, and resolve some problems
in infinitistic mathematics. But it is not just the related formal results, and structures which are of interest. In Hilbert’s major book Grundlagen der Mathematik, which he wrote with his
collaborator Paul Bernays, epsilon terms were presented as formalising certain natural language constructions, like definite descriptions. And they in fact have a considerably larger range of such
applications, for instance in the symbolisation of certain cross-sentential anaphora. Hilbert and Bernays also used their epsilon calculus to prove two important meta-theorems about the predicate
calculus. One theorem subsequently led, for instance, to the development of semantic tableaux: it is called the First Epsilon Theorem, and its content and proof will be given later, in section 6
below. A second theorem that Hilbert and Bernays proved, which we shall also look at then, establishes that epsilon calculi are conservative extensions of the predicate calculus, that is, that no
more theorems expressible just in the quantificational language of the predicate calculus can be proved in epsilon calculi than can be proved in the predicate calculus itself. But while epsilon
calculi do have these further important formal functions, we will not only be concerned to explore them, for we shall also first discuss the natural language structures upon which epsilon calculi
have a considerable bearing.
The growing awareness of the larger meaning and significance of epsilon calculi has only come in stages. Hilbert and Bernays introduced epsilon terms for several meta-mathematical purposes, as above,
but the extended presentation of an epsilon calculus, as a formal logic of interest in its own right, in fact only first appeared in Bourbaki’s Éléments de Mathématique (although see also Ackermann
1937-8). Bourbaki’s epsilon calculus with identity (Bourbaki, 1954, Book 1) is axiomatic, with Modus Ponens as the only primitive inference or derivation rule. Thus, in effect, we get:
(X ∨ X) → X,
X → (X ∨ Y),
(X ∨ Y) → (Y ∨ X),
(X ∨ Y) → ((Z ∨ X) → (Z ∨ Y)),
Fy → FεxFx,
x = y → (Fx ↔ Fy),
(x)(Fx ↔ Gx) → εxFx = εxGx.
This adds to a basis for the propositional calculus an epsilon axiom schema, then Leibniz’ Law, and a second epsilon axiom schema, which is a further law of identity. Bourbaki, though, used the Greek
letter tau rather than epsilon to form what are now called ‘epsilon terms’; nevertheless, he defined the quantifiers in terms of his tau symbol in the manner of Hilbert and Bernays, namely:
(∃x)Fx ↔ FεxFx,
(x)Fx ↔ Fεx¬Fx;
and note that, in his system the other usual law of identity, ‘x = x’, is derivable.
The principle purpose Bourbaki found for his system of logic was in his theory of sets, although through that, in the modern manner, it thereby came to be the foundation for the rest of mathematics.
Bourbaki’s theory of sets discriminates amongst predicates those which determine sets: thus some, but only some, predicates determine sets, i.e. are ‘collectivisantes’. All the main axioms of
classical Set Theory are incorporated in his theory, but he does not have an Axiom of Choice as a separate axiom, since its functions are taken over by his tau symbol. The same point holds in
Bernays’ epsilon version of his set theory (Bernays 1958, Ch VIII).
Epsilon calculi, during this period, were developed without any semantics, but a semantic interpretation was produced by Gunter Asser in 1957, and subsequently published in a book by A.C. Leisenring,
in 1969. Even then, readings of epsilon terms in ordinary language were still uncommon. A natural language reading of epsilon terms, however, was present in Hilbert and Bernays’ work. In fact the
last chapter of book 1 of the Grundlagen is a presentation of a theory of definite descriptions, and epsilon terms relate closely to this. In the more well known theory of definite descriptions by
Bertrand Russell (Russell 1905) there are three clauses: with
The king of France is bald
we get, on Russell’s theory, first
there is a king of France,
there is only one king of France,
and third
anyone who is king of France is bald.
Russell uses the Greek letter iota to formalise the definite description, writing the whole
but he recognises the iota term is not a proper individual symbol. He calls it an ‘incomplete symbol’, since, because of the three parts, the whole proposition is taken to have the quantificational
(∃x)(Kx & (y)(Ky → y = x) & (y)(Ky → By)),
which is equivalent to
(∃x)(Kx & (y)(Ky→ y = x) & Bx).
And that means that it does not have the form ‘Bx’. Russell believed that, in addition to his iota terms, there was another class of individual terms, which he called ‘logically proper names’. These
would simply fit into the ‘x’ place in ‘Bx’. He believed that ‘this’ and ‘that’ were in this class, but gave no symbolic characterisation of them.
Hilbert and Bernays, by contrast, produced what is called a ‘pre-suppositional theory’ of definite descriptions. The first two clauses of Russell’s definition were not taken to be part of the meaning
of ‘The King of France is bald’: they were merely conditions under which they took it to be permitted to introduce a complete individual term for ‘the King of France’, which then satisfies
Kx & (y)(Ky → y = x).
Hilbert and Bernays continued to use the Greek letter iota in their individual term, although it has a quite different grammar from Russell’s iota term, since, when Hilbert and Bernays’ term can be
introduced, it is provably equivalent to the corresponding epsilon term (Kneebone 1963, p102). In fact it was later suggested by many that epsilon terms are not only complete symbols, but can be seen
as playing the same role as the ‘logically proper names’ Russell discussed.
It is at the start of book 2 of the Grundlagen that we find the definition of epsilon terms. There, Hilbert and Bernays first construct a theory of indefinite descriptions in a similar manner to
their theory of definite descriptions. They allow, now, an eta term to be introduced as long as just the first of Russell’s conditions is met. That is to say, given
one can introduce the term ‘ηxFx’, and say
But the condition for the introduction of the eta term can be established logically, for certain predicates, since
(∃x)((∃y)Fy → Fx),
is a predicate calculus theorem (Copi 1973, p110). It is the eta term this theorem allows us to introduce which is otherwise called an epsilon term, and its logical basis enables entirely formal
theories to be constructed, since such individual terms are invariably defined. Thus we may invariably introduce ‘ηx((∃y)Fy → Fx)’, and this is commonly written ‘εxFx’, about which we can therefore
(∃y)Fy → FεxFx.
Since it is that F which exists if anything is F, Hilbert read the epsilon term in this case ‘the first F’. For instance, in arithmetic, ‘the first’ may be taken to be the least number operator.
However, while if there are F’s then the first F is clearly some chosen one of them, if there are no F’s then ‘the first F’ must be a misnomer. And that form of speech only came to be fully
understood in the theories of reference which appeared much later, when reference and denotation came to be more clearly separated from description and attribution. Donnellan (Donnellan 1966) used
the example ‘the man with martini in his glass’, and pointed out that, in certain uses, this can refer to someone without martini in his glass. In the terminology Donnellan made popular, ‘the first
F’, in the second case above works similarly: it cannot be attributive, and so, while it refers to something, it must refer arbitrarily, from a semantic point of view.
With reference in this way separated from attribution it becomes possible to symbolise the anaphoric cross-reference between, for instance, ‘There is one and only one king of France’ and ‘He is
bald’. For, independently of whether the former is true, the ‘he’ in the latter is a pronoun for the epsilon term in the former — by a simple extension of the epsilon definition of the existential
quantifier. Thus the pair of remarks may be symbolised
(∃x)(Kx & (y)(Ky → y = x)) & Bεx(Kx & (y)(Ky → y = x)).
Furthermore such cross-reference may occur in connection with intensional constructions of a kind Russell also considered, such as
George IV wondered whether the author of Waverley was Scott.
Thus we can say ‘There is an author of Waverley, and George IV wondered whether he was Scott’. But the epsilon analysis of these cases puts intensional epsilon calculi at odds with Russellian views
of such constructions, as we shall see later. The Russellian approach, by not having complete symbols for individuals, tends to confuse cases in which assertions are made about individuals and cases
in which assertions are made about identifying properties. As we shall see, epsilon terms enable us to make the discrimination between, for instance,
s = εx(y)(Ay ↔ y = x),
(i.e. ‘Scott is the author of Waverley’), and
(y)(Ay ↔ y = s),
(that is, ‘there is one and only one author of Waverley and he is Scott’), and so it enables us to locate more exactly the object of George IV’s thought.
2. Descriptions and Identity
When one starts to ask about the natural language meaning of epsilon terms, it is interesting that Leisenring just mentions the ‘formal superiority’ of the epsilon calculus (Leisenring 1969, p63, see
also Routley 1969, Hazen 1987). Leisenring took the epsilon calculus to be a better logic than the predicate calculus, but merely because of the Second Epsilon Theorem. Its main virtue, to
Leisenring, was that it could prove all that seemingly needed to be proved, but in a more elegant way. Epsilon terms were just neater at calculating which were the valid theorems of the predicate
Remembering Hilbert and Bernays’ discussion of definite and indefinite descriptions, clearly there is more to the epsilon calculus than this. And there are, in fact, two specific theorems provable
within the epsilon calculus, though not the predicate calculus, which will start to indicate the epsilon calculus’ more general range of application. They concern individuals, since the epsilon
calculus is distinctive in providing an appropriate, and systematic means of reference to them.
The need to have complete symbols for individuals became evident some years after Russell’s promotion of incomplete symbols for them. The first major book to allow for this was Rosser’s Logic for
Mathematicians, in 1953, although there were precursors. For the classical difficulty with providing complete terms for individuals concerns what to do with ‘non-denoting’ terms, and Quine, for
instance, following Frege, often gave them an arbitrary, though specific referent (Marciszewski 1981, p113). This idea is also present in Kalish and Montague (Kalish and Montague 1964, pp242-243),
who gave the two rules:
(∃x)(y)(Fy ↔ y = x) ├ FιxFx,
¬(∃x)(y)(Fy ↔ y = x) ├ιxFx = ιx¬(x = x),
where ‘ιxFx’ is what otherwise might be written ‘εx(y)(Fy ↔ y = x)’. Kalish and Montague believed, however, that the second rule ‘has no intuitive counterpart, simply because ordinary language shuns
improper definite descriptions’ (Kalish and Montague 1964, p244). And, at that time, what Donnellan was to publish in Donnellan 1966, about improper definite descriptions, was certainly not well
known. In fact ordinary speech does not shun improper definite descriptions, although their referents are not as fixed as the above second rule requires. Indeed the very fact that the descriptions
are improper means that their referents are not determined semantically: instead they are just a practical, pragmatic choice.
Stalnaker and Thomason recognised the need to be more liberal when they defined their referential terms, which also had to refer, in the contexts they were concerned with, in more than one possible
world (Thomason and Stalnaker 1968, p363):
In contrast with the Russellian analysis, definite descriptions are treated as genuine singular terms; but in general they will not be substance terms [rigid designators]. An expression like ιxPx
is assigned a referent which may vary from world to world. If in a given world there is a unique existing individual which has the property corresponding to P, this individual is the referent of
ιxPx; otherwise, ιxPx refers to an arbitrarily chosen individual which does not exist in that world.
Stalnaker and Thomason appreciated that ‘A substance term is much like what Russell called a logically proper name’, but they said that an individual constant might or might not be a substance term,
depending on whether it was more like ‘Socrates’ or ‘Miss America’ (Thomason and Stalnaker 1968, p362). A more complete investigation of identity and descriptions, in modal and general intensional
contexts, was provided in Routley, Meyer and Goddard 1974, and Routley 1977, see also Hughes and Cresswell 1968, Ch 11. And with these writers we get the explicit rendering of definite descriptions
in epsilon terms, as in Goddard and Routley 1973, p558, Routley 1980, p277, c.f. Hughes and Cresswell 1968, p203.
Certain specific theorems in the epsilon calculus, as was said before, support these kinds of identification. One theorem demonstrates directly the relation between Russell’s attributive, and some of
Donnellan’s referential ideas. For
(∃x)(Fx & (y)(Fy → y = x) & Gx)
is logically equivalent to
(∃x)(Fx & (y)(Fy → y = x)) & Ga,
where a = εx(Fx & (y)(Fy → y = x)). This arises because the latter is equivalent to
Fa & (y)(Fy → y = a) & Ga,
which entails the former. But the former is
Fb & (y)(Fy → y = b) & Gb,
with b = εx(Fx & (y)(Fy → y = x) & Gx), and so entails
(∃x)(Fx & (y)(Fy → y = x)),
Fa & (y)(Fy → y = a).
But that means that, from the uniqueness clause,
a = b,
and so
meaning the former entails the latter, and therefore the former is equivalent to the latter.
The former, of course, gives Russell’s Theory of Descriptions, in the case of ‘The F is G’; it explicitly asserts the first two clauses, to do with the existence and uniqueness of an F. A
presuppositional theory, such as we saw in Hilbert and Bernays, would not explicitly assert these two clauses: on such an account they are a precondition before the term ‘the F’ can be introduced.
But neither of these theories accommodate improper definite descriptions. Since Donnellan it is more common to allow that we can always use ‘the F’: if the description is improper then the referent
of this term is simply found in the term’s practical use.
One detail of Donnellan’s historical account, however, must be treated with some care, at this point. Donnellan was himself concerned with definite descriptions which were improper in the sense that
they did not uniquely describe what the speaker took to be their referent. So the description might still be ‘proper’ in the above sense — if there still was something to which it uniquely applied,
on account of its semantic content. Thus Donnellan allowed ‘the man with martini in his glass’ to identify someone without martini in his glass irrespective of whether there was some sole man with
martini in his glass. But if one talks about ‘the man with martini in his glass’ one can be correctly taken to be talking about who this describes, if it does in fact correctly describe someone — as
Devitt and Bertolet pointed out in criticism of Donnellan (Devitt 1974, Bertolet 1980). It is this aspect of our language which the epsilon account matches, for an epsilon account allows definite
descriptions to refer without attribution of their semantic character, but only if nothing uniquely has that semantic character. Thus it is not the whole of the first statement above , but only the
third part of the second statement which makes the remark ‘The F is G’.
The difficulty with Russell’s account becomes more plain if we read the two equivalent statements using relative and personal pronouns. They then become
There is one and only one F, which is G,
There is one and only one F; it is G.
But using just the logic derived from Frege, Russell could formalise the ‘which’, but could not separate out the last clause, ‘it is G’. In that clause ‘it’ is an anaphor for ‘the (one and only) F’,
and it still has this linguistic meaning if there is no such thing, since that is just a matter of grammar. But the uniqueness clause is needed for the two statements to be equivalent — without
uniqueness there is no equivalence, as we shall see – so ‘which’ is not itself equivalent to ‘it’. Russell, however, because he could not separate out the ‘it’, had to take the whole of the first
expression as the analysis of ‘The F is G’ — he could not formulate the needed ‘logically proper name’.
But how can something be the one and only F ‘if there is no such thing’? That is where another important theorem provable in the epsilon calculus is illuminating, namely:
(Fa & (y)(Fy → y = a)) → a = εx(Fx & (y)(Fy → y = x)).
The important thing is that there is a difference between the left hand side and the right hand side, i.e. between something being alone F, and that thing being the one and only F. For the left-right
implication cannot be reversed. We get from the left to the right when we see that the left as a whole entails
(∃x)(Fx & (y)(Fy → y = x)),
and so also its epsilon equivalent
Fεx(Fx & (y)(Fy → y = x)) & (z)(Fz → z = εx(Fx & (y)(Fy → y = x))).
Given Fa, then from the second clause here we get the right hand side of our original implication. But if we substitute ‘εx(Fx & (y)(Fy → y = x))’ for ‘a’ in that implication then on the right we
have something which is necessarily true. But the left hand side is then the same as
(∃x)(Fx & (y)(Fy → y = x)),
and that is in general contingent. Hence the implication cannot generally be reversed. Having the property of being alone F is here contingent, but possessing the identity of the one and only F is
The distinction is not made in Russell’s logic, since possession of the relevant property is the only thing which can be formally expressed there. In Russell’s theory of descriptions, a’s possession
of the property of being alone a king of France is expressed as a quasi identity
a = ιxKx,
and that has the consequence that such identities are contingent. Indeed, in counterpart theories of objects in other possible worlds the idea is pervasive that an entity may be defined in terms of
its contingent properties in a given world. Hughes and Cresswell, however, differentiated between contingent identities and necessary identities in the following way (Hughes and Cresswell 1968,
Now it is contingent that the man who is in fact the man who lives next door is the man who lives next door, for he might have lived somewhere else; that is living next door is a property which
belongs contingently, not necessarily, to the man to whom it does belong. And similarly, it is contingent that the man who is in fact the mayor is the mayor; for someone else might have been
elected instead. But if we understand [The man who lives next door is the mayor] to mean that the object which (as a matter of contingent fact) possesses the property of being the man who lives
next door is identical with the object which (as a matter of contingent fact) possesses the property of being the mayor, then we are understanding it to assert that a certain object (variously
described) is identical with itself, and this we need have no qualms about regarding as a necessary truth. This would give us a way of construing identity statements which makes [(x = y) → L(x =
y)] perfectly acceptable: for whenever x = y is true we can take it as expressing the necessary truth that a certain object is identical with itself.
There are more consequences of this matter, however, than Hughes and Cresswell drew out. For now that we have proper referring terms for individuals to go into such expressions as ‘x = y’, we first
see better where the contingency of the properties of such individuals comes from — simply the linguistic facility of using improper definite descriptions. But we also see, because identities between
such terms are necessary, that proper referring terms must be rigid, i.e. have the same reference in all possible worlds.
This is not how Stalnaker and Thomason saw the matter. Stalnaker and Thomason, it will be remembered, said that there were two kinds of individual constants: ones like ‘Socrates’ which can take the
place of individual variables, and others like ‘Miss America’ which cannot. The latter, as a result, they took to be non-rigid. But it is strictly ‘Miss America in year t’ which is meant in the
second case, and that is not a constant expression, even though such functions can take the place of individual variables. It was Routley, Meyer and Goddard who most seriously considered the
resultant possibility that all properly individual terms are rigid. At least, they worked out many of the implications of this position, even though Routley was not entirely content with it.
Routley described several rigid intensional semantics (Routley 1977, pp185-186). One of these, for instance, just took the first epsilon axiom to hold in any interpretation, and made the value of an
epsilon term itself. On such a basis Routley, Meyer and Goddard derived what may be called ‘Routley’s Formula’, i.e.
L(∃x)Fx → (∃x)LFx.
In fact, on their understanding, this formula holds for any operator and any predicate, but they had in mind principally the case of necessity illustrated here, with ‘Fx’ taken as ‘x numbers the
planets’, making ‘εxFx’ ‘the number of the planets’. The formula is derived quite simply, in the following way: from
we can get
by the epsilon definition of the existential quantifier, and so
by existential generalisation over the rigid term (Routley, Meyer and Goddard 1974, p308, see also Hughes and Cresswell 1968, pp197, 204). Routley, however, was still inclined to think that a rigid
semantics was philosophically objectionable (Routley 1977, p186):
Rigid semantics tend to clutter up the semantics for enriched systems with ad hoc modelling conditions. More important, rigid semantics, whether substitutional or objectual, are philosophically
objectionable. For one thing, they make Vulcan and Hephaestus everywhere indistinguishable though there are intensional claims that hold of one but not of the other. The standard escape from this
sort of problem, that of taking proper names like ‘Vulcan’ as disguised descriptions we have already found wanting… Flexible semantics, which satisfactorily avoid these objections, impose a more
objectual interpretation, since, even if [the domain] is construed as the domain of terms, [the value of a term in a world] has to be permitted, in some cases at least, to vary from world to
As a result, while Routley, Meyer and Goddard were still prepared to defend the formula, and say, for instance, that there was a number which necessarily numbers the planets, namely the number of the
planets (np), they thought that this was only in fact the same as 9, so that one still could not argue correctly that as L(np numbers the planets), so L(9 numbers the planets). ‘For extensional
identity does not warrant intersubstitutivity in intensional frames’ (Routley, Meyer and Goddard 1974, p309). They held, in other words that the number of the planets was only contingently 9.
This means that they denied ‘(x = y) → L(x = y)’, but, as we shall see in more detail later, there are ways to hold onto this principle, i.e. maintain the invariable necessity of identity.
3. Rigid Epsilon Terms
There is some further work which has helped us to understand how reference in modal and general intensional contexts must be rigid. But it involves some different ideas in semantics, and starts,
even, outside our main area of interest, namely predicate logic, in the semantics of propositional logic.
When one thinks of ‘semantics’ one maybe thinks of the valuation of formulas. Since the 1920s a meta-study of this kind was certainly added to the previous logical interest in proof theory.
Traditional proof theory is commonly associated with axiomatic procedures, but, from a modern perspective, its distinction is that it is to do with ‘object languages’. Tarski’s theory of truth relies
crucially on the distinction between object languages and meta-languages, and so semantics generally seems to be necessarily a meta-discipline. In fact Tarski believed that such an elevation of our
interest was forced upon us by the threat of semantic paradoxes like The Liar. If there was, by contrast, ‘semantic closure’, i.e. if truth and other semantic notions were definable at the object
level, then there would be contradictions galore (c.f. Priest 1984). In this way truth may seem to be necessarily a predicate of (object-level) sentences.
But there is another way of looking at the matter which is explicitly non-Tarskian, and which others have followed (see Prior 1971, Ch 7, Sayward 1987). This involves seeing ‘it is true that’ as not
a predicate, but an object-level operator, with the truth tabulations in Truth Tables, for instance, being just another form of proof procedure. Operators indeed include ‘it is provable that’, and
this is distinct from Gödel’s provability predicate, as Gödel himself pointed out (Gödel 1969). Operators are intensional expressions, as in the often discussed ‘it is necessary that’ and ‘it is
believed that’, and trying to see such forms of indirect discourse as metalinguistic predicates was very common in the middle of the last century. It was pervasive, for instance, in Quine’s many
discussions of modality and intensionality. Wouldn’t someone be believing that the Morning Star is in the sky, but the Evening Star is not, if, respectively, they assented to the sentence ‘the
Morning Star is in the sky’, and dissented from ‘the Evening Star is in the sky’? Anyone saying ‘yes’ is still following the Quinean tradition, but after Montague’s and Thomason’s work on operators
(e.g. Montague 1963, Thomason 1977, 1980) many logicians are more persuaded that indirect discourse is not quotational. It is open to doubt, that is to say, whether we should see the mind in terms of
the direct words which the subject would use.
The alternative involves seeing the words ‘the Morning Star is in the sky’ in such an indirect speech locution as ‘Quine believes that the Morning Star is in the sky’ as words merely used by the
reporter, which need not directly reflect what the subject actually says. That is indeed central to reported speech — putting something into the reporter’s own words rather than just parroting them
from another source. Thus a reporter may say
Celia believed that the man in the room was a woman,
but clearly that does not mean that Celia would use ‘the man in the room’ for who she was thinking about. So referential terms in the subordinate proposition are only certainly in the mouth of the
reporter, and as a result only certainly refer to what the reporter means by them. It is a short step from this thought to seeing
There was a man in the room, but Celia believed that he was a woman,
as involving a transparent intensional locution, with the same object, as one might say, ‘inside’ the belief as ‘outside’ in the room. So it is here where rigid constant epsilon terms are needed, to
symbolise the cross-sentential anaphor ‘he’, as in:
(∃x)(Mx & Rx) & BcWεx(Mx & Rx).
To understand the matter fully, however, we must make the shift from meta- to object language we saw at the propositional level above with truth. Routley, Meyer and Goddard realised that a rigid
semantics required treating such expressions as ‘BcWx’ as simple predicates, and we must now see what this implies. They derived, as we saw before, ‘Routley’s Formula’
L(∃x)Fx → (∃x)LFx,
but we can now start to spell out how this is to be understood, if we hold to the necessity of identities, i.e. if we use ‘=’ so that
x = y → L(x = y).
Again a clear illustration of the validity of Routley’s Formula is provided by the number of the planets, but now we may respect the fact that some things may lack a number, and also the fact that
referential, and attributive senses of terms may be distinguished. Thus if we write ‘(nx)Px’ for ‘there are n P’s’, then εn(ny)Py will be the number of P’s, and it is what numbers them (i.e. ([εn(ny)
Py]x)Px) if they have a number (i.e. if (∃n)(nx)Px) — by the epsilon definition of the existential quantifier. Then, with ‘Fx’ as the proper (necessary) identity ‘x = εn(ny)Py’ Routley’s Formula
holds because the number in question exists eternally, making both sides of the formula true. But if ‘Fn’ is simply the attributive ‘(ny)Py’ then this is not necessary, since it is contingent even,
in the first place, that there is a number of P’s, instead of just some P, making both sides of the formula false.
Hughes and Cresswell argue against the principle saying (Hughes and Cresswell 1968, p144):
…let [Fx] be ‘x is the number of the planets’. Then the antecedent is true, for there must be some number which is the number of the planets (even if there were no planets at all there would
still be such a number, namely 0): but the consequent is false, for since it is a contingent matter how many planets there are, there is no number which must be the number of the planets.
But this forgets continuous quantities, where there are no discrete items before the nomination of a unit. The number associated with some planetary material, for instance, numbers only arbitrary
units of that material, and not the material itself. So the antecedent of Routley’s Formula is not necessarily true.
Quine also used the number of the planets in his central argument against quantification into modal contexts. He said (Quine 1960, pp195-197):
If for the sake of argument we accept the term ‘analytic’ as predicable of sentences (hence as attachable predicatively to quotations or other singular terms designating sentences), then
‘necessarily’ amounts to ‘is analytic’ plus an antecedent pair of quotation marks. For example, the sentence:
(1) Necessarily 9 > 4
is explained thus:
(2) ‘9 > 4’ is analytic…
So suppose (1) explained as in (2). Why, one may ask, should we preserve the operatorial form as of (1), and therewith modal logic, instead of just leaving matters as in (2)? An apparent
advantage is the possibility of quantifying into modal positions; for we know we cannot quantify into quotation, and (2) uses quotation…
But is it more legitimate to quantify into modal positions than into quotation? For consider (1) even without regard to (2); surely, on any plausible interpretation, (1) is true and this is
(3) Necessarily the number of major planets > 4.
Since 9 = the number of major planets, we can conclude that the position of ‘9’ in (1) is not purely referential and hence that the necessity operator is opaque.
But here Quine does not separate out the referential ‘the number of the major planets is greater than 4’, i.e. ‘εn(ny)Py > 4’, from the attributive ‘There are more than 4 major planets’, i.e. ‘(∃n)
((ny)Py & n > 4)’. If 9 = εn(ny)Py, then it follows that εn(ny)Py > 4, but it does not follow that (∃n)((ny)Py & n > 4). Substitution of identicals in (1), therefore, does yield (3), even though it
is not necessary that there are more than 4 major planets.
We can now go into some details of how one gets the ‘x’ in such a form as ‘LFx’ to be open for quantification. For, what one finds in traditional modal semantics (see Hughes and Cresswell 1968,
passim) are formulas in the meta-linguistic style, like
V(Fx, i) = 1,
which say that the valuation put on ‘Fx’ is 1, in world i. There should be quotation marks around the ‘Fx’ in such a formula, to make it meta-linguistic, but by convention they are generally omitted.
To effect the change to the non-meta-linguistic point of view, we must simply read this formula as it literally is, so that the ‘Fx’ is in indirect speech rather than direct speech, and the whole
becomes the operator form ‘it would be true in world i that Fx’. In this way, the term ‘x’ gets into the language of the reporter, and the meta/object distinction is not relevant. Any variable inside
the subordinate proposition can now be quantified over, just like a variable outside it, which means there is ‘quantifying in’, and indeed all the normal predicate logic operations apply, since all
individual terms are rigid.
A example illustrating this rigidity involves the actual top card in a pack, and the cards which might have been top card in other circumstances (see Slater 1988a). If the actual top card is the Ace
of Spades, and it is supposed that the top card is the Queen of Hearts, then clearly what would have to be true for those circumstances to obtain would be for the Ace of Spades to be the Queen of
Hearts. The Ace of Spades is not in fact the Queen of Hearts, but that does not mean they cannot be identical in other worlds (c.f. Hughes and Cresswell, 1968, p190). Certainly if there were several
cards people variously thought were on top, those cards in the various supposed circumstances would not provide a constant c such that Fc is true in all worlds. But that is because those cards are
functions of the imagined worlds — the card a believes is top (εxBaFx) need not be the card b believes is top (εxBbFx), etc. It still remains that there is a constant, c, such that Fc is true in all
worlds. Moreover, that c is not an ‘intensional object’, for the given Ace of Spades is a plain and solid extensional object, the actual top card (εxFx).
Routley, Meyer and Goddard did not accept the latter point, wanting a rigid semantics in terms of ‘intensional objects’ (Goddard and Routley, 1973, p561, Routley, Meyer and Goddard, 1974, p309, see
also Hughes and Cresswell 1968, p197). Stalnaker and Thomason accepted that certain referential terms could be functional, when discriminating ‘Socrates’ from ‘Miss America’ — although the
functionality of ‘Miss America in year t’ is significantly different from that of ‘the top card in y’s belief’. For if this year’s Miss America is last year’s Miss America, still it is only one thing
which is identical with itself, unlike with the two cards. Also, there is nothing which can force this year’s Miss America to be last year’s different Miss America, in the way that the
counterfactuality of the situation with the playing cards forces two non-identical things in the actual world to be the same thing in the other possible world. Other possible worlds are thus
significantly different from other times, and so, arguably, other possible worlds should not be seen from the Realist perspective appropriate for other times — or other spaces.
4. The Epsilon Calculus’ Problematic
It might be said that Realism has delayed a proper logical understanding of many of these things. If you look ‘realistically’ at picturesque remarks like that made before, namely ‘the same object is
‘inside’ the belief as ‘outside’ in the room’, then it is easy for inappropriate views about the mind to start to interfere, and make it seem that the same object cannot be in these two places at
once. But if the mind were something like another space or time, then counterfactuality could get no proper purchase — no one could be ‘wrong’, since they would only be talking about elements in
their ‘world’, not any objective, common world. But really, all that is going on when one says, for instance,
There was a man in the room, but Celia believed he was a woman,
is that the same term — or one term and a pronominal surrogate for it — appears at two linguistic places in some discourse, with the same reference. Hence there is no grammatical difference between
the cross reference in such an intensional case and the cross reference in a non-intensional case, such as
There was a man in the room. He was hungry.
(∃x)Mx & HεxMx.
What has been difficult has merely been getting a symbolisation of the cross-reference in this more elementary kind of case. But it just involves extending the epsilon definition of existential
statements, using a reiteration of the substituted epsilon term, as we can see.
It is now widely recognised how the epsilon calculus allows us to do this (Purdy 1994, Egli and von Heusinger 1995, Meyer Viol 1995, Ch 6), the theoretical starting point being the theorem about the
Russellian theory of definite descriptions proved before, which breaks up what otherwise would be a single sentence into a sequential piece of discourse, enabling the existence and uniqueness clauses
to be put in one sentence while the characterising remark is in another. The relationship starts to matter when, in fact, there is no obvious way to formulate a combination of anaphoric remarks in
the predicate calculus, as in, for instance,
There is a king of France. He is bald,
where there is no uniqueness clause. This difficulty became a major problem when logicians started to consider anaphoric reference in the 1960s.
Geach, for instance, in Geach 1962, even believed there could not be a syllogism of the following kind (Geach 1962, p126):
A man has just drunk a pint of sulphuric acid.
Nobody who drinks a pint of sulphuric acid lives through the day.
So, he won’t live through the day.
He said, one could only draw the conclusion:
Some man who has just drunk a pint of sulphuric acid won’t live through the day.
Certainly one can only derive
(∃x)(Mx & Dx & ¬Lx)
(∃x)(Mx & Dx),
(x)(Dx → ¬Lx),
within predicate logic. But one can still derive
¬Lεx(Mx & Dx),
within the epsilon calculus.
Geach likewise was foxed later when he produced his famous case (numbered 3 in Geach 1967):
Hob thinks a witch has blighted Bob’s mare, and Nob wonders whether she (the same witch) killed Cob’s sow,
which is, in epsilon terms
Th(∃x)(Wx & Bxb) & OnKεx(Wx & Bxb)c.
For Geach saw that this could not be (4)
(∃x)(Wx & ThBxb & OnKxc),
or (5)
(∃x)(Th(Wx & Bxb)& OnKxc).
But also a reading of the second clause as (c.f. 18)
Nob wonders whether the witch who blighted Bob’s mare killed Cob’s sow,
in which ‘the witch who blighted Bob’s mare killed Cob’s sow’ is analysed in the Russellian manner, i.e. as (20)
just one witch blighted Bob’s mare and she killed Cob’s sow,
Geach realised does not catch the specific cross-reference — amongst other things because of the uniqueness condition which is then introduced.
This difficulty with the uniqueness clause in Russellian analyses has been widely commented on, although a recent theorist, Neale, has said that Russell’s theory only needs to be modestly modified:
Neale’s main idea is that, in general, definite descriptions should just be localised to the context. His resolution of Geach’s troubling cases thus involves suggesting that ‘she’, in the above,
might simply be ‘the witch we have been hearing about’ (Neale 1990, p221). Neale might here have said ‘that witch who blighted Bob’s mare’, showing that an Hilbertian account of demonstrative
descriptions would have a parallel effect.
A good deal of the ground breaking work on these matters, however, was done by someone again much influenced by Russell: Evans. But Evans significantly broke with Russell over uniqueness (Evans 1977,
One does not want to be committed, by this way of telling the story, to the existence of a day on which just one man and boy walked along a road. It was with this possibility in mind that I
stated the requirement for the appropriate use of an E-type pronoun in terms of having answered, or being prepared to answer upon demand, the question ‘He? Who?’ or ‘It? Which?’ In order to
effect this liberalisation we should allow the reference of the E-type pronoun to be fixed not only by predicative material explicitly in the antecedent clause, but also by material which the
speaker supplies upon demand. This ruling has the effect of making the truth conditions of such remarks somewhat indeterminate; a determinate proposition will have been put forward only when the
demand has been made and the material supplied.
It was Evans who gave us the title ‘E-type pronoun’ for the ‘he’ in such expressions as
A Cambridge philosopher smoked a pipe, and he drank a lot of whisky,
i.e., in epsilon terms,
(∃x)(Cx & Px) & Dεx(Cx & Px).
He also insisted (Evans 1977, p516) that what was unique about such pronouns was that this conjunction of statements was not equivalent to
A Cambridge philosopher, who smoked a pipe, drank a lot of whisky,
(∃x)(Cx & Px & Dx).
Clearly the epsilon account is entirely in line with this, since it illustrates the point made before about cases without a uniqueness clause. Only the second expression, which contains a relative
pronoun, is formalisable in the predicate calculus. To formalise the first expression, which contains a personal pronoun, one at least needs something with the expressive capabilities of the epsilon
5. The Formal Semantics of Epsilon Terms
The semantics of epsilon terms is nowadays more general, but the first interpretations of epsilon terms were restricted to arithmetical cases, and specifically took epsilon to be the least number
operator. Hilbert and Bernays developed Arithmetic using the epsilon calculus, using the further epsilon axiom schema (Hilbert and Bernays 1970, Book 2, p85f, c.f. Leisenring 1969, p92) :
(εxAx = st) → ¬At,
where ‘s’ is intended to be the successor function, and ‘t’ is any numeral. This constrains the interpretation of the epsilon symbol, but the least number interpretation is not strictly forced, since
the axiom only ensures that no number having the property A immediately precedes εxAx.
The new axiom, however, is sufficient to prove mathematical induction, in the form:
(A0 & (x)(Ax → Asx)) → (x)Ax.
For assume the reverse, namely
A0 & (x)(Ax → Asx) & ¬(x)Ax,
and consider what happens when the term ‘εx¬Ax’ is substituted in
t = 0 ∨ t = sn,
which is derivable from the other axioms of number theory which Hilbert and Bernays are using. If we had
εx¬Ax = 0,
then, since it is given that A0, then we would have Aεx¬Ax. But since, by the definition of the universal quantifier,
Aεx¬Ax ↔ (x)Ax,
we know, because ¬(x)Ax is also given, that ¬Aεx¬Ax, which means we cannot have εx¬Ax = 0. Hence we must have the other alternative, i.e.
εx¬Ax = sn,
for some n. But from the new axiom
(εx¬Ax = sn) → An,
hence we must have An, although we must also have
An → Asn,
because (x)(Ax → Asx). All together that requires Aεx¬Ax again, which is impossible. Hence the further epsilon axiom is sufficient to establish the given principle of induction.
The more general link between epsilon terms and choice functions was first set out by Asser, although Asser’s semantics for an elementary epsilon calculus without the second epsilon axiom makes
epsilon terms denote rather complex choice functions. Wilfrid Meyer Viol, calling an epsilon calculus without the second axiom an ‘intensional’ epsilon calculus, makes the epsilon terms in such a
calculus instead name Skolem functions. Skolem functions are also called Herbrand functions, although they arise in a different way, namely in Skolem’s Theorem. Skolem’s Theorem states that, if a
formula in prenex normal form is provable in the predicate calculus, then a certain corresponding formula, with the existential quantifiers removed, is provable in a predicate calculus enriched with
function symbols. The functions symbolised are called Skolem functions, although, in another context, they would be Herbrand functions.
Skolem’s Theorem is a meta-logical theorem, about the relation between two logical calculi, but a non-metalogical version is in fact provable in the epsilon calculus from which Skolem’s actual
theorem follows, since, for example, we can get, by the epsilon definition, now of the existential quantifier
(x)(∃y)Fxy ↔ (x)FxεyFxy.
As a result, if the left hand side of such an equivalence is provable in an epsilon calculus the right hand side is provable there. But the left hand side is provable in an epsilon calculus if it is
provable in the predicate calculus, by the Second Epsilon Theorem; and if the right hand side is provable in an epsilon calculus it is provable in a predicate calculus enriched with certain function
symbols — epsilon terms, like ‘εyFxy’. So, by generalisation, we get Skolem’s original result.
When we add to an intensional epsilon calculus the second epsilon axiom
(x)(Fx ↔ Gx) →εxFx = εxGx,
the interpretation of epsilon terms is commonly extensional, i.e. in terms of sets, since two predicates ‘F’ and ‘G’ satisfying the antecedent of this second axiom will determine the same set — if
they determine sets at all, that is. For that requires the predicates to be collectivisantes, in Bourbaki’s terms, as with explicit set membership statements, like ‘x ∈ y’. In such a case the epsilon
term ‘εx(x ∈ y)’ designates a choice function, i.e. a function which selects one from a given set (c.f. Leisenring 1969, p19, Meyer Viol 1995, p42). In the case where there are no members of the set
the selection is arbitrary, although for all empty sets it is invariably the same. Thus the second axiom validates, for example, Kalish and Montague’s rule for this case, which they put in the form
εxFx = εx¬(x = x).
Kalish and Montague in fact prove a version of the second epsilon axiom in their system (Kalish and Montague 1964, see T407, p256). The second axiom also holds in Hermes’ system (Hermes 1965),
although there one in addition finds a third epsilon axiom,
εx¬(x = x) = εx(x = x),
for which there would seem to be no real justification.
But the second epsilon axiom itself is curious. One questionable thing about it is that both Leisenring and Meyer Viol do not state that the predicates in question must determine sets before their
choice function semantics can apply. That the predicates are collectivisantes is merely presumed in their theories, since ‘εxBx’ is invariably modelled by means of a choice from the presumed set of
things which in the model are B. Certainly there is a special clause dealing with the empty set; but there is no consideration of the case where some things are B although those things are not
discrete, as with the things which are red, for instance. If the predicate in question is not a count noun then there is no set of things involved, since with mass terms, and continuous quantities
there are no given elements to be counted (c.f. Bunt 1985, pp262-263 in particular). Of course numbers can still be associated with them, but only given an arbitrary unit. With the cows in a field,
for instance, we can associate a determinate number, but with the beef there we cannot, unless we consider, say, the number of pounds of it.
The point, as we saw before, has a formalisation in epsilon terms. Thus if we write ‘(nx)Fx’, for ‘there are n F’s’, then εn(ny)Fy will be the number of F’s, and it is what numbers them if they have
a number. But in the reverse case the previously mentioned arbitrariness of the epsilon term comes in. For if ¬(∃n)(nx)Fx, then ¬([εn(ny)Fy]x)Fx, and so, although an arbitrary number exists, it does
not number the F’s. In that case, in other words, we do not have a number of F’s, merely some F.
In fact, even when there is a set of things, the second epsilon axiom, as stated above, does not apply in general, since there are intensional differences between properties to consider, as in, for
instance ‘There is a red-haired man, and a Caucasian in the room, and they are different’. Here, if there were only red-haired Caucasians in the room, then with the above second axiom, we could not
find epsilon substitutions to differentiate the two individuals involved. This may remind us that it is necessary co-extensionality, and not just contingent co-extensionality which is the normal
criterion for the identity of properties (c.f. Hughes and Cresswell 1968, pp209-210). So it leads us to see the appropriateness of a modalised second axiom, which uses just an intensional version of
the antecedent of the previous second epsilon axiom, in which ‘L’ means ‘it is necessary that’, namely:
L(x)(Fx ↔ Gx) →εxFx = εxGx.
For with this axiom only the co-extensionalities which are necessary will produce identities between the associated epsilon terms. We can only get, for instance,
εxPx = εx(Px ∨ Px),
εxFx = εyFy,
and all other identities derivable in a similar way.
However, the original second epsilon axiom is then provable, in the special case where the predicates express set membership. For if necessarily
(x)(x ∈ y ↔ x ∈ z) ↔ y = z,
while necessarily
y = z ↔ L(y = z),
(see Hughes and Cresswell, 1968, p190) then
L(x)(x ∈ y ↔ x ∈ z) ↔ (x)(x ∈ y ↔ x ∈ z),
and so, from the modalised second axiom we can get
(x)(x ∈ y ↔ x ∈ z) →εx(x ∈ y) = εx(x ∈ z).
Note, however, that if one only has contingently
(x)(Fx ↔ x ∈ z),
then one cannot get, on this basis,
εxFx = εx(x ∈ z).
But this is something which is desirable, as well. For we have seen that it is contingent that the number of the planets does number the planets — because it is not necessary that ([εn(ny)Py]x)Px.
This makes ‘(9x)Px’ contingent, even though the identity ‘9 = εn(nx)Px’ remains necessary. But also it is contingent that there is the set of planets, p, which there is, since while, say,
(x)(x ∈ p ↔ Px),
εn(nx)(x ∈ p) = εn(nx)Px = 9,
it is still possible that, in some other possible world,
(x)(x ∈ p’ ↔ Px),
with p’ the set of planets there, and
¬(εn(nx)(x ∈ p’) = 9).
We could not have this further contingency, however, if the original second epsilon axiom held universally.
It is on this fuller basis that we can continue to hold ‘x = y → L(x = y)’, i.e. the invariable necessity of identity — one merely distinguishes ‘(9x)Px’ from ‘9 = εx(nx)Px’, and from ‘9 = εx(nx)(x ∈
p)’, as above.
Adding the original second epsilon axiom to an intensional epsilon calculus is therefore acceptable only if all the predicates are about set membership. This is not an uncommon assumption, indeed it
is pervasive in the usually given semantics for predicate logic, for instance. But if, by contrast, we want to allow for the fact that not all predicates are collectivisantes then we should take just
the first epsilon axiom with merely a modalised version of the second epsilon axiom. The interpretation of epsilon terms is then always in terms of Skolem functions, although if we are dealing with
the membership of sets, those Skolem functions naturally are choice functions.
6. Some Metatheory
To finish we shall briefly look, as promised, at some meta-theory.
The epsilon calculi that were first described were not very convenient to use, and Hilbert and Bernays’ proofs of the First and Second Epsilon Theorems were very complex. This was because the
presentation was axiomatic, however, and with the development of other means of presenting the same logics we get more readily available meta-logical results. I will indicate some of the early
difficulties before showing how these theorems can be proved, nowadays, much more simply.
The problem with proving the Second Epsilon Theorem, on an axiomatic basis, is that complex, and non-constant epsilon terms may enter a proof in the epsilon calculus by means of substitutions into
the axioms. What has to be proved is that an epsilon calculus proof of an epsilon-free theorem (i.e. one which can be expressed just in predicate calculus language) can be replaced by a predicate
calculus proof. So some analysis of complex epsilon terms is required, to show that they can be eliminated in the relevant cases, leaving only constant epsilon terms, which are sufficiently similar
to the individual symbols in standard predicate logic. Hilbert and Bernays (Hilbert and Bernays 1970, Book 2, p23f) say that one epsilon term ‘εxFx’ is subordinate to another ‘εyGy’ if and only if
‘G’ contains ‘εxFx’, and a free occurrence of the variable ‘y’ lies within ‘εxFx’. For instance ‘εxRxy’ is a complex, and non-constant epsilon term, which is subordinate to ‘εySyεxRyx’. Hilbert and
Bernays then define the rank of an epsilon term to be 1 if there are no epsilon terms subordinate to it, and otherwise to be one greater than the maximal rank of the epsilon terms which are
subordinate to it. Using the same general ideas, Leisenring proves two theorems (Leisenring 1969, p72f). First he proves a rank reduction theorem, which shows that epsilon proofs of epsilon-free
formulas in which the second epsilon axiom is not used, but in which every term is of rank less than or equal to r, may be replaced by epsilon proofs in which every term is of rank less than or equal
to r – 1. Then he proves the eliminability of the second epsilon axiom in proofs of epsilon-free formulas. Together, these two theorems show that if there is an epsilon proof of an epsilon-free
formula, then there is such a proof not using the second epsilon axiom, and in which all epsilon terms have rank just 1. Even though such epsilon terms might still contain free variables, if one
replaces those that do with a fixed symbol ‘a’ (starting with those of maximal length) that reduces the proof to one in what is called the ‘epsilon star’ system, in which there are only constant
epsilon terms (Leisenring 1969, p66f). Leisenring shows that proofs in the epsilon star system can be turned into proofs in the predicate calculus, by replacing the epsilon terms by individual
But, as was said before, there is now available a much shorter proof of the Second Epsilon Theorem. In fact there are several, but I shall just indicate one, which arises simply by modifying the
predicate calculus truth trees, as found in, for instance, Jeffrey (see Jeffrey 1967). Jeffrey uses the standard propositional truth tree rules, together with the rules of quantifier interchange,
which remain unaffected, and which are not material to the present purpose. He also has, however, a rule of existential quantifier elimination,
(∃x)Fx ├ Fa,
in which ‘a’ must be new, and a rule of universal quantifier elimination
(x)Fx ├ Fb,
in which ‘b’ must be old — unless no other individual terms are available. By reducing closed formulas of the form ‘P & ¬C’ to absurdity Jeffrey can then prove ‘P → C’, and validate ‘P ├ C’ in his
calculus. But clearly, upon adding epsilon terms to the language, the first of these rules must be changed to
(∃x)Fx ├ FεxFx,
while also the second rule can be replaced by the pair
(x)Fx ├ Fεx¬Fx,
Fεx¬Fx ├ Fa,
(where ‘a’ is old) to produce an appropriate proof procedure. Steen reads ‘εx¬Fx’ as ‘the most un-F-like thing’ (Steen 1972, p162), which explains why Fεx¬Fx entails Fa, since if the most un-F-like
thing is in fact F, then the most plausible counter-example to the generalisation is in fact not so, making the generalisation exceptionless. But there is a more important reason why the rule of
universal quantifier elimination is best broken up into two parts.
For Jeffrey’s rules only allow him ‘limited upward correctness’ (Jeffrey 1967, p167), since Jeffrey has to say, with respect to his universal quantifier elimination rule, that the range of the
quantification there be limited merely to the universe of discourse of the path below. This is because, if an initial sentence is false in a valuation so also must be one of its conclusions. But the
first epsilon rule which replaces Jeffrey’s rule ensures, instead, that there is ‘total upwards correctness’. For if it is false that everything is F then, without any special interpretation of the
quantifier, one of the given consequences of the universal statement is false, namely the immediate one — since Fεx¬Fx is in fact equivalent to (x)Fx. A similar improvement also arises with the
existential quantifier elimination rule. For Jeffrey can only get ‘limited downwards correctness’, with his existential quantifier elimination rule (Jeffrey 1967, p165), since it is not an
entailment. In fact, in order to show that if an initial sentence is true in a valuation so is one of its conclusions, in this case, Jeffrey has to stretch his notion of ‘truth’ to being true either
in the given valuation, or some nominal variant of it.
The epsilon rule which replaces Jeffrey’s overcomes this difficulty by not employing names, only demonstrative descriptions, and by being, as a result, totally downward correct. For if there is an F
then that F is F, whatever name is used to refer to it. The epsilon calculus terminology thus precedes any naming: it gets hold of the more primitive, demonstrative way we have of referring to
objects, using phrases like ‘that F’. Thus in explication of the predicate calculus rule we might well have said
suppose there is an F, well, call that F ‘a’, then Fa,
but that requires we understand ‘that F’ before we come to use ‘a’.
So how does the Second Epsilon Theorem follow? This theorem, as before, states that an epsilon calculus proof of an epsilon-free theorem may be replaced by a predicate calculus proof of the same
formula. But the transformation required in the present setting is now evident: simply change to new names all epsilon terms introduced in the epsilon calculus quantifier elimination rules. This
covers both the new names in Jeffrey’s first rule, but also the odd case where there are no old names in Jeffrey’s second rule. The epsilon calculus proofs invariably use constant epsilon terms, and
are thus effectively in Leisenring’s epsilon star system.
Epsilon terms which are non-constant, however, crucially enter the proof of the First Epsilon Theorem. The First Epsilon Theorem states that if C is a provable predicate calculus formula, in prenex
normal form, i.e. with all quantifiers at the front, then a finite disjunction of instances of C’s matrix is provable in the epsilon calculus. The crucial fact is that the epsilon calculus gives us
access to Herbrand functions, which arise when universal quantifiers are eliminated from formulas using their epsilon definition. Thus
for instance, is equivalent to
and so
and the resulting epsilon term ‘εxFyx’ is a Herbrand function.
Using such reductions, all universal quantifiers can evidently be removed from formulas in prenex normal form, and the additional fact that, in a certain specific way, the remaining existential
quantifiers are disjunctions makes all predicate calculus formulas equivalent to disjunctions. Remember that a formula is provable if its negation is reducible to absurdity, which means that its
truth tree must close. But, by König’s Lemma, if there is no open path through a truth tree then there is some finite stage at which there is no open path, so, in the case above, for instance, if no
valuation makes the last formula’s negation true, then the tree of the instances of that negative statement must close in a finite length. But the negative statement is the universal formula
by the rules of quantifier interchange, so a finite conjunction of instances of the matrix of this universal formula, namely Fyx, must reduce to absurdity. For the rules of universal quantifier
elimination only produce consequences with the form of this matrix. By de Morgan’s Laws, that makes necessary a finite disjunction of instances of ¬Fyx. By generalisation we thus get the First
Epsilon Theorem.
The epsilon calculus, however, can take us further than the First Epsilon Theorem. Indeed, one has to take care with the impression this theorem may give that existential statements are just
equivalent to disjunctions. If that were the case, then existential statements would be unlike individual statements, saying not that one specified thing has a certain property, but merely that one
of a certain group of things has a certain property. The group in question is normally called the ‘domain’ of the quantification, and this, it seems, has to be specified when setting out the
semantics of quantifiers. But study of the epsilon calculus shows that there is no need for such ‘domains’, or indeed for such semantics. This is because the example above, for instance, is also
equivalent to
where a = εy¬FεxFyx. So the previous disjunction of instances of ¬Fyx is in fact only true because this specific disjunct is true. The First Epsilon Theorem, it must be remembered, does not prove
that an existential statement is equivalent to a certain disjunction; it shows merely that an existential statement is provable if and only if a certain disjunction is provable. And what is also
provable, in such a case, is a statement merely about one object. Indeed the existential statement is provably equivalent to it. It is this fact which supports the epsilon definition of the
quantifiers; and it is what permits anaphoric reference to the same object by means of the same epsilon term. An existential statement is thus just another statement about an individual — merely a
nameless one.
The reverse point goes for the universal quantifier: a universal statement is not the conjunction of its instances, even though it implies them. A generalisation is simply equivalent to one of its
instances — to the one involving the prime putative exception to it, as we have seen. Not being able to specify that prime putative exception leaves Jeffrey saying that if a generalisation is false
then one of its instances is false without any way of ensuring that that instance has been drawn as a conclusion below it in the truth tree except by limiting the interpretation of the generalisation
just to the universe of discourse of the path. It thus seems necessary, within the predicate calculus, that there be a ‘model’ for the quantifiers which restricts them to a certain ‘domain’, which
means that they do not necessarily range over everything. But in the epsilon calculus the quantifiers do, invariably, range over everything, and so there is no need to specify their range.
7. References and Further Reading
• Ackermann, W. 1937-8, ‘Mengentheoretische Begründung der Logik’, Mathematische Annalen, 115, 1-22.
• Asser, G. 1957, ‘Theorie der Logischen Auswahlfunktionen’, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, 3, 30-68.
• Bernays, P. 1958, Axiomatic Set Theory, North Holland, Dordrecht.
• Bertolet, R. 1980, ‘The Semantic Significance of Donnellan’s Distinction’, Philosophical Studies, 37, 281-288.
• Bourbaki, N. 1954, Éléments de Mathématique, Hermann, Paris.
• Bunt, H.C. 1985, Mass Terms and Model-Theoretic Semantics, C.U.P., Cambridge.
• Church, A. 1940, ‘A Formulation of the Simple Theory of Types’, Journal of Symbolic Logic, 5, 56-68.
• Copi, I. 1973, Symbolic Logic, 4th ed. Macmillan, New York.
• Devitt, M. 1974, ‘Singular Terms’, The Journal of Philosophy, 71, 183-205.
• Donnellan, K. 1966, ‘Reference and Definite Descriptions’, Philosophical Review, 75, 281-304.
• Egli, U. and von Heusinger, K. 1995, ‘The Epsilon Operator and E-Type Pronouns’ in U. Egli et al. (eds.), Lexical Knowledge in the Organisation of Language, Benjamins, Amsterdam.
• Evans, G. 1977, ‘Pronouns, Quantifiers and Relative Clauses’, Canadian Journal of Philosophy, 7, 467-536.
• Geach, P.T. 1962, Reference and Generality, Cornell University Press, Ithaca.
• Geach, P.T. 1967, ‘Intentional Identity’, The Journal of Philosophy, 64, 627-632.
• Goddard, L. and Routley, R. 1973, The Logic of Significance and Context, Scottish Academic Press, Aberdeen.
• Gödel, K. 1969, ‘An Interpretation of the Intuitionistic Sentential Calculus’, in J. Hintikka (ed.), The Philosophy of Mathematics, O.U.P. Oxford.
• Hazen, A. 1987, ‘Natural Deduction and Hilbert’s ε-operator’, Journal of Philosophical Logic, 16, 411-421.
• Hermes, H. 1965, Eine Termlogik mit Auswahloperator, Springer Verlag, Berlin.
• Hilbert, D. 1923, ‘Die Logischen Grundlagen der Mathematik’, Mathematische Annalen, 88, 151-165.
• Hilbert, D. 1925, ‘On the Infinite’ in J. van Heijenhoort (ed.), From Frege to Gödel, Harvard University Press, Cambridge MA.
• Hilbert, D. and Bernays, P. 1970, Grundlagen der Mathematik, 2nd ed., Springer, Berlin.
• Hughes, G.E. and Cresswell, M.J. 1968, An Introduction to Modal Logic, Methuen, London.
• Jeffrey, R. 1967, Formal Logic: Its Scope and Limits, 1st Ed. McGraw-Hill, New York.
• Kalish, D. and Montague, R. 1964, Logic: Techniques of Formal Reasoning, Harcourt, Brace and World, Inc, New York.
• Kneebone, G.T. 1963, Mathematical Logic and the Foundations of Mathematics, Van Nostrand, Dordrecht.
• Leisenring, A.C. 1969, Mathematical Logic and Hilbert’s ε-symbol, Macdonald, London.
• Marciszewski, W. 1981, Dictionary of Logic, Martinus Nijhoff, The Hague.
• Meyer Viol, W.P.M. 1995, Instantial Logic, ILLC Dissertation Series 1995-11, Amsterdam.
• Montague, R. 1963, ‘Syntactical Treatments of Modality, with Corollaries on Reflection Principles and Finite Axiomatisability’, Acta Philosophica Fennica, 16, 155-167.
• Neale, S. 1990, Descriptions, MIT Press, Cambridge MA.
• Priest, G.G. 1984, ‘Semantic Closure’, Studia Logica, XLIII 1/2, 117-129.
• Prior, A.N., 1971, Objects of Thought, O.U.P. Oxford.
• Purdy, W.C. 1994, ‘A Variable-Free Logic for Anaphora’ in P. Humphreys (ed.) Patrick Suppes: Scientific Philosopher, Vol 3, Kluwer, Dordrecht, 41-70.
• Quine, W.V.O. 1960, Word and Object, Wiley, New York.
• Rasiowa, H. 1956, ‘On the ε-theorems’, Fundamenta Mathematicae, 43, 156-165.
• Rosser, J. B. 1953, Logic for Mathematicians, McGraw-Hill, New York.
• Routley, R. 1969, ‘A Simple Natural Deduction System’, Logique et Analyse, 12, 129-152.
• Routley, R. 1977, ‘Choice and Descriptions in Enriched Intensional Languages II, and III’, in E. Morscher, J. Czermak, and P. Weingartner (eds), Problems in Logic and Ontology, Akademische Druck
und Velagsanstalt, Graz.
• Routley, R. 1980, Exploring Meinong’s Jungle, Departmental Monograph #3, Philosophy Department, R.S.S.S., A.N.U. Canberra.
• Routley, R., Meyer, R. and Goddard, L. 1974, ‘Choice and Descriptions in Enriched Intensional Languages I’, Journal of Philosophical Logic, 3, 291-316.
• Russell, B. 1905, ‘On Denoting’ Mind, 14, 479-493.
• Sayward, C. 1987, ‘Prior’s Theory of Truth’ Analysis, 47, 83-87.
• Slater, B.H. 1986(a), ‘E-type Pronouns and ε-terms’, Canadian Journal of Philosophy, 16, 27-38.
• Slater, B.H. 1986(b), ‘Prior’s Analytic’, Analysis, 46, 76-81.
• Slater, B.H. 1988(a), ‘Intensional Identities’, Logique et Analyse, 121-2, 93-107.
• Slater, B.H. 1988(b), ‘Hilbertian Reference’, Noûs, 22, 283-97.
• Slater, B.H. 1989(a), ‘Modal Semantics’, Logique et Analyse, 127-8, 195-209.
• Slater, B.H. 1990, ‘Using Hilbert’s Calculus’, Logique et Analyse, 129-130, 45-67.
• Slater, B.H. 1992(a), ‘Routley’s Formulation of Transparency’, History and Philosophy of Logic, 13, 215-24.
• Slater, B.H. 1994(a), ‘The Epsilon Calculus’ Problematic’, Philosophical Papers, XXIII, 217-42.
• Steen, S.W.P. 1972, Mathematical Logic, C.U.P. Cambridge.
• Thomason, R. 1977, ‘Indirect Discourse is not Quotational’, Monist, 60, 340-354.
• Thomason, R. 1980, ‘A Note on Syntactical Treatments of Modality’, Synthese, 44, 391-395.
• Thomason, R.H. and Stalnaker, R.C. 1968, ‘Modality and Reference’, Noûs, 2, 359-372.
Author Information
Barry Hartley Slater
Email: slaterbh@cyllene.uwa.edu.au
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