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Counting card trick “The Final 3” explained (easy tutorial)
For the “Final 3” card trick, all you have to do is count some cards – that’s it. The trick will work every single time and doesn’t require any sleight of hand at all. In this post, I’ll explain it
step by step!
Here’s how this card trick works:
With the “Final 3” card trick, you are able to find three different cards chosen by the audience at random. All you have to do is count the cards on multiple piles and the rest of the trick happens
by itself.
If the trick doesn’t work when you try it, I’ve listed the common mistakes and a few extra tips at the end of this post.
Let’s get started with the tutorial.
How to perform “The Final 3” (all you have to do is count cards)
The great thing about this card trick is that you don’t need to learn any sleight of hand or other complex moves. It’s based on a mathematical principle and all you have to do is count a few cards,
that’s it!
Another thing I’d like to mention: I’ve taken quite some photos because it’s way easier to explain the trick with images. It might take some time until they load, but it’s worth it, I promise!
Here’s how you do the trick:
The spectator can choose three different cards from the deck (he can really pick whatever he wants, it doesn’t matter).
By the way: it might be a good idea to either let your volunteer write down his cards or have three different people remember one card each because memorizing three cards can be quite challenging.
After selecting the cards, the deck can be shuffled before you continue with the trick.
To make things easier, I’ve selected the three Aces for this tutorial.
Selecting the three cards (I’ve chosen the Aces for this tutorial)
Once the cards have been selected, you continue by creating four piles on the table (or whatever surface you can find).
This is where the counting part of the card trick starts.
How to count the cards for each pile
If one card is in the wrong pile, the trick won’t work.
Each pile must have the exact amount of cards listed down below to make it work.
By the way, many tutorials use a different amount of cards (10, 15, 15, and 9), but this method is easier and skips one extra step where you have to put four cards from the top to the bottom of the
Make sure to use these numbers, the trick is easier to do and looks more smooth!
The pile Number of cards
Pile number one (the left one) 14
Pile number two 15
Pile number three 15
Pile number four (the right one) 5
I’ve used a different color for the three selected cards (blue instead of red). This makes it easier to explain, but when performing, make sure to use a normal deck.
Another thing to keep in mind: the deck must have exactly 52 cards (no jokers or special cards!) to make the trick work.
This is how it should look like:
Creating the four piles (number of cards from left to right: 14, 15, 15, 5)
You now put the cards chosen by the audience on the piles in a specific order.
Whenever I say pile number #1, I’m talking about the pile on the left. The pile on the right is number #4.
Start by asking your spectator to put one of his cards (it doesn’t matter which one) on pile #1.
Next, he can cut pile #2 wherever he wants and put the cards on top of pile #1.
It will look like this:
Now, you repeat the same steps for the next two piles.
Ask your spectator to put another card on pile #2.
He can cut pile #3 and put the cards on pile #2.
The last card is put on pile #3.
The only difference for the last pile: you just put all of the cards of pile #3 instead of cutting it.
This is how it looks like when you are done:
You should now have three piles with the three cards chosen by the audience inside the piles.
The next step is to put the piles on top of each other, creating one pile.
But make sure to do it in the right order, because it will mess up the trick otherwise.
Here’s how you do it:
Put pile #3 on pile #2.
Now, put pile #2 on pile #1.
Once you’ve reached this point, there’s not that much that can go wrong.
From now on, the trick will pretty much happen on its own.
How to find the three cards selected by your audience
Start by dealing the cards onto the table, but make sure to put the first card face up.
If you don’t start with the first card facing up, the trick doesn’t work!
Start like this:
Continue with the second card next to it (with its back facing up).
Now, just go one by one until all the cards are distributed evenly in two piles.
Make sure to take your time, one card on the wrong pile will mess up the trick.
This is how it should look like:
When all the cards are distributed on the two piles, you can remove the left pile (the one with the cards facing up). You don’t need these cards anymore for the rest of the trick.
Continue with the right pile (the cards that are facing down).
Now, repeat the same steps again and again, until you are left with only three cards.
Start with the first card facing up, continue with the next card facing down, and repeat until all the cards are dealt on the table.
Remove the pile with the cards facing up, and repeat.
At some point, you will be left with only six cards.
If you’ve done everything correctly, all the spectator’s cards should be in the left pile.
For this tutorial, I’ve used the three Aces, so these are the cards I’m left with:
This trick will work every single time, as long as you count correctly.
I really like it since it doesn’t require any preparation at all and you can perform it with any deck, as long as it has 52 cards.
A few final tips and reasons why the trick might not work for you
If you can’t make the trick work, don’t worry – here’s my list of the common mistakes.
• Make sure to use a deck with exactly 52 playing cards.
• Don’t make any mistakes when counting the cards for the different piles.
• Put the piles on top of each other in the correct order.
• When dealing the cards onto the table, start with the first card facing up.
• Remove the pile with the cards facing up and continue with the other pile.
Other than that, please keep in mind that one spectator is usually not able to remember three different cards. Many people forget their choice because they are so focused on watching the performance!
You can either ask them to write the cards down or take a photo of them, it will ensure that the trick gets the attention he deserves.
There’s nothing worse than you showing the spectators the final three cards and they are like: “Oh, I don’t really know, I forgot my cards…”
Other than that, I love this trick and you will sure too!
Looking for even more easy card tricks?
You can check the “Three Piles Trick” in this post where I explain the entire trick step by step (it doesn’t require any sleight of hand as well).
If you like mental magic, you will enjoy this mind-reading card trick I’ve explained in this tutorial.
Image sources:
All of the images are my own photos, click here if you want to use them. | {"url":"https://trickla.com/counting-card-trick-the-final-three/","timestamp":"2024-11-09T10:48:02Z","content_type":"text/html","content_length":"97308","record_id":"<urn:uuid:d5de6746-7347-4ef1-aff3-509c12d9ac5f>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.75/warc/CC-MAIN-20241109085148-20241109115148-00861.warc.gz"} |
American Mathematical Society
Amalgamation for inverse and generalized inverse semigroups
HTML articles powered by AMS MathViewer
by T. E. Hall
Trans. Amer. Math. Soc. 310 (1988), 313-323
DOI: https://doi.org/10.1090/S0002-9947-1988-0965756-7
PDF | Request permission
For any amalgam $(S, T; U)$ of inverse semigroups, it is shown that the natural partial order on $S{{\ast }_U}T$, the (inverse semigroup) free product of $S$ and $T$ amalgamating $U$, has a simple
form on $S \cup T$. In particular, it follows that the semilattice of $S{{\ast }_U}T$ is a bundled semilattice of the corresponding semilattice amalgam $(E(S), E(T); E(U))$; taken jointly with a
result of Teruo Imaoka, this gives that the class of generalized inverse semigroups has the strong amalgamation property. Preserving finiteness is also considered. References
—, Inverse and regular semigroups and amalgamation: a brief survey, Proc. Sympos. Regular Semigroups, Northern Illinois Univ., De Kalb, Ill., 1979, pp. 49-79. —, The embedding of regular
semigroup amalgams, Proc. Sympos. Regular Semigroups, Northern Illinois Univ., De Kalb, Ill., 1979, pp. 92-100. L. Silver, Non-commutative localization and applications, J. Algebra 7 (1967),
Bibliographic Information
• © Copyright 1988 American Mathematical Society
• Journal: Trans. Amer. Math. Soc. 310 (1988), 313-323
• MSC: Primary 20M10; Secondary 08B25, 20M17, 20M18
• DOI: https://doi.org/10.1090/S0002-9947-1988-0965756-7
• MathSciNet review: 965756 | {"url":"https://www.ams.org/journals/tran/1988-310-01/S0002-9947-1988-0965756-7/?active=current","timestamp":"2024-11-03T01:30:28Z","content_type":"text/html","content_length":"69063","record_id":"<urn:uuid:a849c72d-eec5-47f1-a3cb-8af9dcb72815>","cc-path":"CC-MAIN-2024-46/segments/1730477027768.43/warc/CC-MAIN-20241102231001-20241103021001-00229.warc.gz"} |
Appendix B: Akaike-Information Criterion (AIC)
The Akaike information criterion measures the goodness of fit of a statistical model. It describes the trade-off between bias and variance in model construction, or loosely speaking, that of the
accuracy and complexity of the model.
The AIC is not a test of the model in the sense of hypothesis testing; instead, it provides a means for comparison among models—a tool for model selection. Given a data set, several candidate models
may be ranked according to their AIC, with the model having the minimum AIC being the best. From the AIC values, one may also infer that, e.g., the top two models are roughly in a tie, and the rest
are far worse.
1. In general, the AIC is defined as: $$\mathit{AIC}=2k-2\times\ln(L)$$ Where:
□ $k$ is the number of model parameters.
□ $\ln(L)$ is the log-likelihood function for the statistical model.
2. For smaller data sets, the AICc applies 2nd order correction: $$ \mathit{AICc}= \mathit{AIC} + \frac{2k(k+1)}{N-k-1} = \frac{2\times N \times k}{N-k-1}-2\times\ln(L) $$Where:
□ $N$ is the data sample size.
□ $k$ is the number of model parameters.
1. The AIC is not a test on the model in the sense of hypothesis testing; instead, it is a test between models - a tool for model selection.
Related Links
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third_party/SPIRV-Tools/source/fuzz/added_function_reducer.h - SwiftShader - Git at Google
// Copyright (c) 2020 Google LLC
// Licensed under the Apache License, Version 2.0 (the "License");
// you may not use this file except in compliance with the License.
// You may obtain a copy of the License at
// http://www.apache.org/licenses/LICENSE-2.0
// Unless required by applicable law or agreed to in writing, software
// distributed under the License is distributed on an "AS IS" BASIS,
// WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
// See the License for the specific language governing permissions and
// limitations under the License.
#ifndef SOURCE_FUZZ_ADDED_FUNCTION_REDUCER_H_
#define SOURCE_FUZZ_ADDED_FUNCTION_REDUCER_H_
#include <unordered_set>
#include <vector>
#include "source/fuzz/protobufs/spirvfuzz_protobufs.h"
#include "source/fuzz/shrinker.h"
#include "spirv-tools/libspirv.hpp"
namespace spvtools {
namespace fuzz {
// An auxiliary class used by Shrinker, this class takes care of using
// spirv-reduce to reduce the body of a function encoded in an AddFunction
// transformation, in case a smaller, simpler function can be added instead.
class AddedFunctionReducer {
// Possible statuses that can result from running the shrinker.
enum class AddedFunctionReducerResultStatus {
struct AddedFunctionReducerResult {
AddedFunctionReducerResultStatus status;
std::vector<uint32_t> transformed_binary;
protobufs::TransformationSequence applied_transformations;
uint32_t num_reduction_attempts;
spv_target_env target_env, MessageConsumer consumer,
const std::vector<uint32_t>& binary_in,
const protobufs::FactSequence& initial_facts,
const protobufs::TransformationSequence& transformation_sequence_in,
uint32_t index_of_add_function_transformation,
const Shrinker::InterestingnessFunction&
bool validate_during_replay, spv_validator_options validator_options,
uint32_t shrinker_step_limit, uint32_t num_existing_shrink_attempts);
// Disables copy/move constructor/assignment operations.
AddedFunctionReducer(const AddedFunctionReducer&) = delete;
AddedFunctionReducer(AddedFunctionReducer&&) = delete;
AddedFunctionReducer& operator=(const AddedFunctionReducer&) = delete;
AddedFunctionReducer& operator=(AddedFunctionReducer&&) = delete;
// Invokes spirv-reduce on the function in the AddFunction transformation
// identified by |index_of_add_function_transformation|. Returns a sequence
// of transformations identical to |transformation_sequence_in|, except that
// the AddFunction transformation at |index_of_add_function_transformation|
// might have been simplified. The binary associated with applying the
// resulting sequence of transformations to |binary_in| is also returned, as
// well as the number of reduction steps that spirv-reduce made.
// On failure, an empty transformation sequence and binary are returned,
// with a placeholder value of 0 for the number of reduction attempts.
AddedFunctionReducerResult Run();
// Yields, via |binary_out|, the binary obtained by applying transformations
// [0, |index_of_added_function_| - 1] from |transformations_in_| to
// |binary_in_|, and then adding the raw function encoded in
// |transformations_in_[index_of_added_function_]| (without adapting that
// function to make it livesafe). This function has |added_function_id_| as
// its result id.
// The ids associated with all global variables in |binary_out| that had the
// "irrelevant pointee value" fact are also returned via
// |irrelevant_pointee_global_variables|.
// The point of this function is that spirv-reduce can subsequently be applied
// to function |added_function_id_| in |binary_out|. By construction,
// |added_function_id_| should originally manipulate globals for which
// "irrelevant pointee value" facts hold. The set
// |irrelevant_pointee_global_variables| can be used to force spirv-reduce
// to preserve this, to avoid the reduced function ending up manipulating
// other global variables of the SPIR-V module, potentially changing their
// value and thus changing the semantics of the module.
void ReplayPrefixAndAddFunction(
std::vector<uint32_t>* binary_out,
std::unordered_set<uint32_t>* irrelevant_pointee_global_variables) const;
// This is the interestingness function that will be used by spirv-reduce
// when shrinking the added function.
// For |binary_under_reduction| to be deemed interesting, the following
// conditions must hold:
// - The function with id |added_function_id_| in |binary_under_reduction|
// must only reference global variables in
// |irrelevant_pointee_global_variables|. This avoids the reduced function
// changing the semantics of the original SPIR-V module.
// - It must be possible to successfully replay the transformations in
// |transformation_sequence_in_|, adapted so that the function added by the
// transformation at |index_of_add_function_transformation_| is replaced by
// the function with id |added_function_id_| in |binary_under_reduction|,
// to |binary_in| (starting with initial facts |initial_facts_|).
// - All the transformations in this sequence must be successfully applied
// during replay.
// - The resulting binary must be interesting according to
// |shrinker_interestingness_function_|.
bool InterestingnessFunctionForReducingAddedFunction(
const std::vector<uint32_t>& binary_under_reduction,
const std::unordered_set<uint32_t>& irrelevant_pointee_global_variables);
// Starting with |binary_in_| and |initial_facts_|, the transformations in
// |transformation_sequence_in_| are replayed. However, the transformation
// at index |index_of_add_function_transformation_| of
// |transformation_sequence_in_| -- which is guaranteed to be an AddFunction
// transformation -- is adapted so that the function to be added is replaced
// with the function in |binary_under_reduction| with id |added_function_id_|.
// The binary resulting from this replay is returned via |binary_out|, and the
// adapted transformation sequence via |transformation_sequence_out|.
void ReplayAdaptedTransformations(
const std::vector<uint32_t>& binary_under_reduction,
std::vector<uint32_t>* binary_out,
protobufs::TransformationSequence* transformation_sequence_out) const;
// Returns the id of the function to be added by the AddFunction
// transformation at
// |transformation_sequence_in_[index_of_add_function_transformation_]|.
uint32_t GetAddedFunctionId() const;
// Target environment.
const spv_target_env target_env_;
// Message consumer.
MessageConsumer consumer_;
// The initial binary to which transformations are applied -- i.e., the
// binary to which spirv-fuzz originally applied transformations.
const std::vector<uint32_t>& binary_in_;
// Initial facts about |binary_in_|.
const protobufs::FactSequence& initial_facts_;
// A set of transformations that can be successfully applied to |binary_in_|.
const protobufs::TransformationSequence& transformation_sequence_in_;
// An index into |transformation_sequence_in_| referring to an AddFunction
// transformation. This is the transformation to be simplified using
// spirv-reduce.
const uint32_t index_of_add_function_transformation_;
// The interestingness function that has been provided to guide the
// overall shrinking process. The AddFunction transformation being simplified
// by this class should still -- when applied in conjunction with the other
// transformations in |transformation_sequence_in_| -- lead to a binary that
// is deemed interesting by this function.
const Shrinker::InterestingnessFunction& shrinker_interestingness_function_;
// Determines whether to check for validity during the replaying of
// transformations.
const bool validate_during_replay_;
// Options to control validation.
spv_validator_options validator_options_;
// The step limit associated with the overall shrinking process.
const uint32_t shrinker_step_limit_;
// The number of shrink attempts that had been applied prior to invoking this
// AddedFunctionReducer instance.
const uint32_t num_existing_shrink_attempts_;
// Tracks the number of attempts that spirv-reduce has invoked its
// interestingness function, which it does once at the start of reduction,
// and then once more each time it makes a reduction step.
uint32_t num_reducer_interestingness_function_invocations_;
} // namespace fuzz
} // namespace spvtools
#endif // SOURCE_FUZZ_ADDED_FUNCTION_REDUCER_H_ | {"url":"https://swiftshader.googlesource.com/SwiftShader/+/b57a3aaee927a7412dffffac064d833e1d87dc7d/third_party/SPIRV-Tools/source/fuzz/added_function_reducer.h?autodive=0%2F%2F%2F%2F%2F","timestamp":"2024-11-06T22:10:35Z","content_type":"text/html","content_length":"62642","record_id":"<urn:uuid:d031372c-7610-46be-a332-fb1804abcb82>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.47/warc/CC-MAIN-20241106194801-20241106224801-00610.warc.gz"} |
Bitcoin TLDR
Combined summary - Estimating Likelihood for Lightning Payments to be (in)feasible
The discussion centers on the nuances of network state weighting, liquidity distribution in channels, and their implications for node balance uniformity within the context of minimum cost flow (MCF)
computations and wealth distribution.
The sender initially corrects a miscount in states to ten, which alters the basis of their argument regarding the probability models used to compare wealth distributions and payment feasibility. This
correction leads to a refined analysis that challenges the equivalence of uniformly weighting all network states to independently choosing channel balances, arguing that such an approach does not
accurately reflect the sum distribution of node balances.
As the conversation progresses, it delves deeper into the methodology of computing probabilities for feasible payments and MCF in networked systems. A significant focus is placed on a model
illustrating wealth distribution across nodes, where the sender revises the probability calculation of specific payment scenarios based on corrected wealth distributions. Through detailed
examination, the correspondence assesses how various distributions allow or disallow certain payment configurations, challenging initial assumptions with counterexamples backed by theoretical models.
Specifically, it references a paper that suggests liquidity is uniformly and independently distributed across channels, which plays a crucial role in comparing different flows and their feasibility (
read the paper). This part of the discussion highlights the complexity of accurately modeling payment systems and the importance of considering a wide range of assumptions about resource
Furthermore, the correspondence critiques an innovative approach by Rene Pickhardt, focusing on the comparison between the MCF probability and payment feasibility probability across different models.
It emphasizes the distinct probability spaces these models operate within, suggesting that direct comparisons may be misleading. By dissecting the underlying assumptions and calculations of each
model, the critique unveils foundational discrepancies in evaluating successful transactions within a network, urging for a nuanced understanding of their respective methodologies.
Exploring practical applications, the summary discusses how advanced analytical capabilities could revolutionize cryptocurrency transaction management through examples involving "Businessperson Bob"
and "User Alice." Bob's node management software autonomously optimizes financial operations by assessing transaction likelihoods and liquidity advertisements, while Alice's wallet configuration
addresses recurring payment challenges by adjusting retry intervals based on transaction success probabilities. These scenarios underline the potential of integrating sophisticated network analytics
to enhance transaction efficiency and reliability.
Rene Pickhardt's development of a mathematical theory aiming to understand the Lightning Network's dynamics further enriches the discourse. His work proposes moving away from traditional liquidity
estimation models towards assuming all feasible wealth distributions are equally likely, providing a novel method to compute payment success rates without detailed knowledge of liquidity
distributions. This approach, highlighted through an iPython notebook (view the notebook), reflects a more accurate representation of wealth distribution realities within the network, challenging the
feasibility of achieving 100% payment success rates due to the dynamic nature of network constraints and wealth distributions.
Discussion History
September 26, 2024 18:02 UTC | {"url":"https://tldr.bitcoinsearch.xyz/summary/delvingbitcoin/Sept_2024/combined_Estimating-Likelihood-for-Lightning-Payments-to-be-in-feasible?replies=","timestamp":"2024-11-13T11:26:12Z","content_type":"text/html","content_length":"47943","record_id":"<urn:uuid:5c9b8bbf-9a6f-4939-80cd-0c743b84af2c>","cc-path":"CC-MAIN-2024-46/segments/1730477028347.28/warc/CC-MAIN-20241113103539-20241113133539-00582.warc.gz"} |
Handy Polynomial Fitting with Bernstein Polynomials
11-12-2018, 02:09 PM
Post: #9
Thomas Klemm Posts: 2,268
Senior Member Joined: Dec 2013
RE: Handy Polynomial Fitting with Bernstein Polynomials
(11-12-2018 01:21 PM)Thomas Okken Wrote: I vaguely remember learning about Chebyshev polynomials for this purpose.
They are mentioned in the section:
Change of interpolation points
Quote:As I recall, Chebyshev fits have the nice property of having a hard upper bound on the error, which is within a constant (a factor of about 3 IIRC) of the worst-case error of the optimal
fit. I'd have to dig around to find that textbook, though, it may have been lost in the mists of time...
It appears to be
even better
Quote:Therefore, when the interpolation nodes x[i] are the roots of T[n], the error satisfies:
\(\left|f(x)-P_{{n-1}}(x)\right|\leq {\frac {1}{2^{{n-1}}n!}}\max _{{\xi \in [-1,1]}}\left|f^{{(n)}}(\xi )\right|\)
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IFERROR -- Suppress Error Help Needed
Welcome to the Smartsheet Forum Archives
The posts in this forum are no longer monitored for accuracy and their content may no longer be current. If there's a discussion here that interests you and you'd like to find (or create) a more
current version, please
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IFERROR -- Suppress Error Help Needed
I hope you're having a terrific day! I am looking for suggestions for using the IFERROR formula to suppress and error message. Here's the set-up:
Planned Allocation = Out-of-Box Allocation column
Planned Effort by Day = [Allocation] * 8
Duration = Out-of-Box Duration column
Sum Planned Effort (Estimate) = [Planned Effort by Day]*[Duration]
Sum Actual Effort (Actuals) = Number
Hours Burn = [Sum Actual Effort (Actuals)] / [Sum Planned Effort (Estimate)]
When either Sum Planned Effort (Estimate) or Sum Actual Effort (Actuals) are blank or zero, the following error message is returned:
Desired Outcome
When the formula returns an error message, display "0"
I've tried using the IFERROR function, but can't figure out how to nest it correctly.
I appreciate any suggestions you might have!
• Hi
Give this formula a try for "Hours Burn" column. If there is an error it will put a 0, else Actual/Estimate.
=IF(ISERROR([Sum Actual Effort (Actual)]1 / [Sum Planned Effort (Estimate)]1), 0, [Sum Actual Effort (Actual)]1 / [Sum Planned Effort (Estimate)]1)
• Aileen,
Here is a shorter version:
=IFERROR([Sum Actual Effort (Actuals)]23 / [Sum Planned Effort (Estimate)]23, 0)
for row 23.
Note: If you copied SmSulli's formula directly, you likely got an error (#UNPARSEABLE).
That formula used "Actual" instead of "Actuals" from your example.
Hope this helps
• Thank you, J. Craig Williams and SmSulli's! I'm sorry for my tardy reply. I've meant to log in for weeks and thank you.
THANK YOU!
Craig, the solution was perfect. Just what I needed! Thank you so much!
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Nanostarch Properties 1. Introduction Generally, milling refers to a mechanical operation employed for the size reduction of solid materials with consequential changes in properties. Grain
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Various devices have been used for processing B powder, including attritor mills, conventional planetary mills, tumbler mills/mixers, vibratory mills, shaker mills, and uniball mills [161][162
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trand quantum theory
Strands provide a simple and correct explanation for wave functions.
● Overview of strand quantum theory
● Testable predictions of strand quantum theory
● Qubits
● The fascination of strand quantum theory
● Similar ideas by other authors
● Bets and future tests
● Publications on quantum theory from strands
Visual summary of strand quantum theory
The spinning electron before blurring. Three strands with Planck radius reproduce spin 1/2, fermion behaviour, electrical charge 3 x 1/3 = 1, chirality, extremely low dipole moment, small mass,
vanishing colour charge, correct g-factor, correct antiparticle properties, as well as correct gravitational, electromagnetic, weak and strong interactions.
Overview of strand quantum theory
Strands provide a Planck-scale model of nature that includes wave functions:
Wave functions are blurred crossing densities of fluctuating and spinning tangles of strands of Planck radius.
Elementary fermions are rational (braided) tangles of strands made of two or three strands.
This strand tangle model allows deriving, from a single principle, the Schrödinger equation, the Pauli equation, the Klein-Gordon equation and the Dirac equation. The strand tangle model explains
spin 1/2, fermion behaviour, countable particles, antiparticles, particle-antiparticle mixing, and particle mass values.
For example, it is well-known that spinors behave like flags with a sign. (This corresponds to the analogy that vectors behave like arrows.) A tangle of strands indeed behaves like a flag: it has a
direction for the pole (the direction in which the tangle core points, the spin orientation) and has an orientation of the flag (the phase, i.e., the orientation of the core around the pointing
direction). In addition, the tangle also contains the sign that all spinors contain: positive if the tethers are untangled, negative if they are not.
Several consequences of the strand tangle model go beyond quantum theory. Classifying the possible tangle structures determines the possible elementary particles, their quantum numbers, their mass
values and the Higgs mechanism. Classifying tangle deformations with the Reidemeister moves determines the possible gauge interactions, their symmetry groups and their coupling constants. Over 50
precision tests are deduced. All tests predict the lack of physics beyond the standard model with massive Dirac neutrinos. All tests agree with observations.
The strand tangle model is simple, unique, consistent, complete, and predictive. Strands state that the standard model of particle physics is final and beautiful.
The preprint linked above explains and shows that wave functions evolve deterministically, but emerge as the average of fundamental strand fluctuations. As expected from observations, wave functions
are shown to have objective existence. Wave functions have a unique history. They have no hidden variables, despite appearances. Wave functions collapse. The environment, interacting baths and the
measurement apparatus play a role. The measurement problem is solved by decoherence.
Also another result of quantum theory remains valid: the universe as a whole has no wave function and no Hilbert space. The "wave function of the universe" is a contradiction in terms: wave functions
define the state for observations by outside observers. But there is nothing outside the universe - by definition.
Testable predictions of strand quantum theory
Neutrino masses have normal order.
No physics beyond the standard model or beyond general relativity occurs.
No trans-Planckian effects of any kind occur.
There are no higher (or lower) dimensions.
In nature, there is an upper limit on probability density, given by the inverse smallest volume.
There is no supersymmetry. Dark matter is not made of unknown elementary particles.
Glueballs exist.
Fundamental constants – elementary particle masses, mixing angles, and coupling constants – can and will be calculated.
Strands also visualize qubits. The preprint introduces the topic.
Strands provide a way to visualize the statement by Zizzi about nature: it from qubit. Indeed, everything arises from qubits; more precisely, everything arises from crossing switches of strands.
The fascination of strand quantum theory
Strand tangles are skeletons of wave functions. Wave functions are crossing densities.
Measurement and collapse are triggered by baths; collapse time is decoherence time and is short but measurable, in agreement with data.
Every electron, every atom, every basketball and every person is tethered.
Everything is connected to everything else.
The universe consists of a single strand.
Strands imply that `every thing' is made of `everything'.
Strands explain the fine structure constant and the electron mass. Thus they explain all colours around us.
Because strands have Planck radius and are unobservable, they are not hidden variables.
Similar ideas by other authors
So far, apart from the work by Battey-Pratt and Racey (and Dirac's lecture demonstration), no similar ideas are found in the literature. Not even the researchers working on emergent quantum theory
published anything similar. Several additional research papers that are close to the strand tangle model are cited in the pdf linked above.
Studying the complement of strands, i.e., the evolution of the space between them, could provide an equivalent alternative, as done by Asselmeyer-Maluga. However, his model has not yet achieved a
classification of particles or interactions, nor deduced a model for wave functions – though this should be possible.
Tests and bets
In science, every statement must be checked continuously, again and again, against observations. This is ongoing. A sweeping statement like "strands crossings explain wave functions" must be checked
with particular care. If you have a counterargument or notice a missing issue, just send a note to the author.
The numerous experimental predictions, tests and proposed bets (click here) are extremely precise. They cover all domains of nature and of physics. So far, all tests for the strand model are
positive. Finding a single observation falsifying a single prediction of the strand conjecture wins the bet.
Finding any alternative, correct and inequivalent description of wave functions – or of nature – wins the bet as well.
A provoking consequence: all approaches claiming that space is continuous and all approaches claiming that space is discrete are mistaken.
It might be that the similarities between strand gravity and strand particle entanglement can be used to deduce connections between gravity and entanglement.
An interesting aspect is the following: experiments with tethered chiral bodies in slowly flowing liquids or gases should yield estimates for particle mass values.
In the past, models for quantum theory have not been successful. The whole topic has a mixed reputation. But if one does not risk making a fool of oneself, there is no progress. Therefore: enjoy
exploring strands and enjoy exploring their relation to quantum theory.
Publications on quantum theory from strands
This simple introduction for mathematicians, physicists and physics students explains the origin of wave functions and the gauge groups U(1), SU(2) and SU(3): C. Schiller, On the relation between the
three Reidemeister moves and the three gauge groups International Journal of Geometric Methods in Modern Physics (2023) DOI: 10.1142/S0219887824500579. Download the preprint here.
The exploration of quantum electrodynamics also introduces wave functions arising from strands: C. Schiller, Testing a conjecture on quantum electrodynamics, Journal of Geometry and Physics 178
(2022) 104551. Download the preprint here.
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Because Our 3D Space Is A Manifold - an Astronomy Net God & Science Forum Message
Hi Alex,
Thank you for your post. We discussed this issue before at
but we didn't finish the conversation. I'm glad you brought it back up. Let me try another approach at answering your question.
In Differential Geometry, there is the concept of a manifold. A manifold is a space of, say, n dimensions that exists, or is embedded in a space of m dimensions where m>n.
All of the topological properties of the manifold can be the same as they would be if it weren't an embedded manifold.
The inverse square (cube, R, etc.) law, as you know, is a result of the topology of the space. In 2D, for example, you get an inverse R law. An inverse R law holds for figures on a sheet of paper
because the paper is 2D. If the sheet of paper exists in a 3D room, the inverse R law still holds on the sheet of paper. It still holds because the sheet of paper is a 2D manifold in 3D space.
So in the same way, if our 3D space were a 3D manifold in 4D or 10D space, the inverse square laws would still hold just the same as if it weren't a manifold.
Everything in the manifold would still be 3D and no instrument or pointer could align itself to point in any direction that is not a linear combination of three basis vectors (e.g. one meter North,
one meter East, and one meter Up). Therefore, we have no way of accessing anything outside of our manifold. Just the same as any figure drawn on a sheet of paper laying horizontally cannot point up.
This is completely consistent with the notion of dimensions being degrees of freedom of motion within the space. As long as everything in the manifold is 3D, any motion is confined to the manifold so
there are still only three degrees of freedom. The same way as an ant crawling on a sheet of paper is confined to two degrees of freedom (unless it leaves the manifold and falls off). The ant has
only two degrees of freedom even though the paper is a manifold in 3D space.
I think this easily, and mathematically, explains why the existence of large extra dimensions would not affect our inverse square laws.
Warm regards, | {"url":"http://www.astronomy.net/forums/god/messages/12796.shtml","timestamp":"2024-11-14T22:28:54Z","content_type":"text/html","content_length":"20900","record_id":"<urn:uuid:90ad1edc-15aa-4f9d-a732-34e3214c2e28>","cc-path":"CC-MAIN-2024-46/segments/1730477395538.95/warc/CC-MAIN-20241114194152-20241114224152-00352.warc.gz"} |
The Controversy about Hypothesis Testing
I spent a quick couple of days last week at the The Controversy about Hypothesis Testing meeting in Madrid.
The topic of the meeting was indeed the question of “hypothesis testing“, which I addressed in a post a few months ago: how do you choose between conflicting interpretations of data? The canonical
version of this question was the test of Einstein’s theory of relativity in the early 20th Century — did the observations of the advance of the perihelion of Mercury (and eventually of the
gravitational lensing of starlight by the sun) match the predictions of Einstein’s theory better than Newton’s? And of course there are cases in which even more than a scientific theory is riding on
the outcome: is a given treatment effective? I won’t rehash here my opinions on the subject, except to say that I think there really is a controversy: the purported Bayesian solution runs into
problems in realistic cases of hypotheses about which we would like to claim some sort of “ignorance” (always a dangerous word in Bayesian circles), while the orthodox frequentist way of looking at
the problem is certainly ad hoc and possibly incoherent, but nonetheless seems to work in many cases.
Sometimes, the technical worries don’t apply, and the Bayesian formalism provides the ideal solution. For example, my colleague Daniel Mortlock has applied the model-comparison formalism to deciding
whether objects in his UKIDSS survey data are more likely to be distant quasars or nearby and less interesting objects. (He discussed his method here a few months ago.)
In between thoughts about hypothesis testing, I experienced the cultural differences between the statistics community and us astrophysicists and cosmologists, of which I was the only example at the
meeting: a typical statistics talk just presents pages of text and equations with the occasional poorly-labeled graph thrown in. My talks tend to be a bit heavier on the presentation aspects, perhaps
inevitably so given the sometimes beautiful pictures that package our data.
On the other hand, it was clear that the statisticians take their Q&A sessions very seriously, prodded in this case by the word “controversy” in the conference’s title. In his opening keynote, Jose
Bernardo up from Valencia for the meeting discussed his work as a so-called “Objective Bayesian”, prompting a question from the mathematically-oriented philosopher Deborah Mayo. Mayo is an
arch-frequentist (and blogger) who prefers to describe her particular version as “Error Statistics”, concerned (if I understand correctly after our wine-fuelled discussion at the conference dinner)
with the use of probability and statistics to criticise the errors we make in our methods, in contrast with the Bayesian view of probability as a description of our possible knowledge of the world.
These two points of view are sufficiently far apart that Bernardo countered one of the questions with the almost-rude but definitely entertaining riposte “You are bloody inconsistent — you are not
mathematicians.” That was probably the most explicit almost-personal attack of the meeting, but there were similar exchanges. Not mine, though: my talk was a little more didactic than most, as I knew
that I had to justify the science as well as the statistics that lurks behind any analysis of data.
So I spent much of my talk discussing the basics of modern cosmology, and applying my preferred Bayesian techniques in at least one big-picture case where the method works: choosing amongst the
simple set of models that seem to describe the Universe, at least from those that obey General Relativity and the Cosmological Principle, in which we do not occupy a privileged position and which,
given our observations, are therefore homogeneous and isotropic on the largest scales.
Given those constraints, all we need to specify (or measure) are the amounts of the various constituents in the universe: the total amount of matter and of dark energy. The sum of these, in turn,
determines the overall geometry of the universe.
In the appropriate units, if the total is one, the universe is flat; if it’s larger, the universe is closed, shaped like a three-dimensional sphere; if smaller, it’s a three-dimensional hyperboloid
or saddle. What we find when we make the measurement is that the amount of matter is about 0.282±0.02, and of dark energy about 0.723±0.02.
Of course, these add up to just greater than one; model-selection (or hypothesis testing in other forms) allows us to say that the data nonetheless give us reason to prefer the flat Universe despite
the small discrepancy.
After the meeting, I had a couple of hours free, so I went across Madrid to the Reina Sofia, to stand amongst the Picassos and Serras. And I was lucky enough to have my hotel room above a different
2 responses to “The Controversy about Hypothesis Testing”
Glad to hear of your blog. I think one of the big problems in capturing the experimental testing of GTR Bayesianly is the simple fact that scientists did not and do not have an exhaustive set of
alternative gravity theories, much less do they find it profitable to try to assign a degree of belief in a “catchall hypothesis”. But they do split things up piecemeal to delineate parameters
and probe them precisely along the lines of frequentist confidence intervals and significance tests. This lets them understand the phenomenon, set upper bounds to how much a viable theory can
differ from GTR in given respects, and design better tests.
I don’t know in what way you are claiming we error statisticians are incoherent, by the way.
As for Bernardo, you are correct in saying his attack (on me) was personal and vehement (I actually didn’t hear the words you wrote about not being a bloody mathematician or whatever he said). He
failed to answer my question, and perhaps he did not realize that the audience’s snickering at his overlong harangue was not a sign that they were taking him seriously. Too bad!
I think I’m a Bayesian. Am I wrong?
Continuing my recent, seemingly interminable, series of too-technical posts on probability theory… To understand this one you’ll need to remember Bayes’ Theorem, and the resulting need for a
Bayesian statistician to come up with an ap… | {"url":"https://andrewjaffe.net/blog/2011/12/the_controversy/","timestamp":"2024-11-03T07:48:59Z","content_type":"text/html","content_length":"67933","record_id":"<urn:uuid:62e48eac-13fa-4895-9708-94ecd5e1343a>","cc-path":"CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00282.warc.gz"} |
How do you find the absolute value of abs(10)? | Socratic
How do you find the absolute value of #abs(10)#?
1 Answer
Absolute value of a number is its value sans its sign, positive or negative i.e. for $17$ or $+ 17$ (absence of any sign means it is positive), it is $17$ and for $- 17$ to it is $17$.
Number $10$ has no sign i.e. it is positive and hence, we can say that $| 10 | = 10$.
Graphically, absolute value of a number can be described as distance of its location on real number line from origin i.e. point described as $0$, irrespective of its direction, whether towards
positive or negative side of origin.
As the distance of $10$ from origin is $10$, its absolute value $| 10 | = 10$
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Algebric Manipulation MCQs [PDF] Quiz Questions Answers | Algebric Manipulation MCQ App Download & e-Book: Test 1
Class 10 Math MCQs - Chapter 3
Algebric Manipulation Multiple Choice Questions (MCQs) PDF Download - 1
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Algebric Manipulation MCQs with Answers PDF Download: Quiz 1
MCQ 1:
The number of methods to find H.C.F are
1. 2
2. 4
3. 3
4. 5
MCQ 2:
Product of two expressions ⁄ L.C.M =
1. H.C.F
2. L.C.M
3. H.C.F + L.C.M
4. H.C.F × L.C.M
MCQ 3:
Product of two expressions ⁄ H.C.F =
1. H.C.F
2. L.C.M
3. H.C.F + L.C.M
4. H.C.F × L.C.M
MCQ 4:
The number should be added to complete the square of x^4 + 64 is
1. 4x²
2. 16x²
3. 8x²
4. −16x²
MCQ 5:
For two or more algebraic expressions, the expression of lowest degree which is divisible by each of them without remainder is known as
1. L.C.M
2. H.C.F
3. rational expression
4. irrational expression
Algebric Manipulation Learning App: Free Download Android & iOS
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Mathematics Courses and Support Options
Click HERE to see the Mathematics Pathways Flow Chart.
Math Pathways based on Academic Goal
To achieve your goals efficiently and successfully, the Math Department has new pathways and support options. Beginning Fall 2019, students are encouraged to enroll directly into a transfer-level
course. Students should consider their academic goal, past achievements, and experiences. Students are also strongly encouraged to take advantage of the many resources available to support them at
LPC. Click the box below to learn more about your educational goals and the math pathway right for you.
Transfer-Level Associate’s Degree Technical Math Jam and
Mathematics and Foundational Math Concurrent
Courses Level Courses Courses Support Courses
• MATH 1 • MATH 30 • MATH 55 • MATH 110 • MATH 53 • NMAT 261 • NMAT 200C/MATH 100C
• MATH 2 • MATH 33 • NMAT 255 • NMAT 210 • MATH 72 • NMAT 262 • NMAT 201C/MATH 101C
• MATH 3 • MATH 34 • MATH 50 • MATH 107 • NMAT 263 • NMAT 202C
• MATH 5 • MATH 39 • NMAT 250 • NMAT 207 • NMAT 264 • NMAT 210C/MATH 110C
• MATH 7 • MATH 40 • MATH 156 • NMAT 265 • NMAT 255C/MATH 55C
• MATH 10 • MATH 47 • NMAT 256 • MATH 66 • MATH 66C
• MATH 27 • MATH 67 • MATH 67C
• MATH 68 • MATH 68C
Transfer-Level Mathematics Courses
An introduction to single-variable differential and integral calculus including: functions, limits and continuity; techniques and applications of differentiation and integration; the Fundamental Th
eorem of Calculus; areas and volumes of solids of revolution. Prerequisite: MATH 30 with a minimum grade of C, MATH 39 with a minimum grade of C. 90 hours lecture. AA/AS GE. Transfer: CSU, UC*; CSU
GE: B4; IGETC: 2A; C-ID# MATH 211, MATH 900 S (if taken with MATH 2). * MATH 1, 33, and 34 combined: maximum UC credit, one course.
Students can place into MATH 1 via:
• HSGPA ≥ 3.0 AND passed one full academic year with an A, B, or C of HS Precalculus, or
• HS Calculus with A, B, or C, or
• Pass Math 30 and Math 39.
Students are also strongly encouraged to take advantage of the many resources available to support them at LPC.
• MATH 66 Math Jam for Calculus I, bootcamp Course is offered 1-week prior to the start of the semester. Award-winning! Offered the week before the Fall and Spring semesters. Proven to increase
student success and retention rates.
• MATH 66C Concurrent Support for Calculus I, support during the semester. Aligned with your math course and designed with innovative strategies to provide math and learning support while taking
your math class.
5 UNITS
Continuation of single-variable differential and integral calculus. Topics covered include: inverse and hyperbolic functions; techniques of integration; polar and parametric equations; infinite
sequences, series, power series and Taylor series; applications of integration. Primarily for mathematics, physical science and engineering majors. Prerequisite: MATH 1 with a minimum grade of C. 90
hours lecture. AA/AS GE. Transfer: CSU, UC; CSU GE: B4; IGETC: 2A; C-ID# MATH 221, MATH 900 S (if taken with MATH 1).
Students can place into MATH 2 via:
• Pass MATH 1: Calculus I with a C or better.
• With AP AB score 3, 4, or 5.
Students are also strongly encouraged to take advantage of the many resources available to support them at LPC.
• MATH 67 Math Jam for Calculus II, bootcamp Course is offered 1-week prior to the start of the semester. Award-winning! Offered the week before the Fall and Spring semesters. Proven to increase
student success and retention rates.
• MATH 67C Concurrent Support for Calculus II, support during the semester. Aligned with your math course and designed with innovative strategies to provide math and learning support while taking
your math class.
5 UNITS
Vector valued functions, functions of several variables, partial diff erentiation, multiple integration, change of variables theorem, scalar and vector fields, gradient, divergence, curl, line
integral, surface integral, Green’s Stokes’ and divergence theorem, applications. Prerequisite: MATH 2 with a minimum grade of C. 90 hours lecture. AA/AS GE. Transfer: CSU, UC; CSU GE: B4; IGETC: 2A;
C-ID# MATH 230.
Students can place into MATH 3 via:
• Pass MATH 2: Calculus II with a C or better.
• With AP BC score 3, 4, or 5.
Students are also strongly encouraged to take advantage of the many resources available to support them at LPC.
• MATH 68 Math Jam for Calculus III, bootcamp Course is offered 1-week prior to the start of the semester. Award-winning! Offered the week before the Fall and Spring semesters. Proven to increase
student success and retention rates.
• MATH 68C Concurrent Support for Calculus III, support during the semester. Aligned with your math course and designed with innovative strategies to provide math and learning support while taking
your math class.
5 UNITS
Introduction to diff erential equations including the conditions under which a unique solution exists, techniques for obtaining solutions, and applications. Techniques include generation of series
solutions, use of Laplace Transforms, and the use of eigenvalues to solve linear systems. Generation of exact solutions, approximate solutions, and graphs of solutions using MATLAB. Prerequisite:
MATH 3 with a minimum grade of C. 54 hours lecture, 27 hours laboratory. AA/AS GE. Transfer: CSU, UC; CSU GE: B4; IGETC: 2A; C-ID# MATH 240.
3.5 UNITS
An introduction to linear algebra including: techniques and theory needed to solve and classify systems of linear equations using Gaussian elimination and matrix algebra; properties of vectors in
n-dimensions; generalized vector spaces, inner product spaces, basis, norms, orthogonality; eigenvalues, eigenspaces; and linear transformations. Selected applications of linear algebra, including
the use of MATLAB™ to solve problems involving advanced numerical computation. Prerequisite: MATH 2 with a minimum grade of C. 54 hours lecture. 27 hours laboratory. AA/AS GE. Transfer: CSU, UC; CSU
GE: B4; IGETC: 2A; C-ID# MATH 250.
3.5 UNITS
Designed for majors in mathematics and computer science, this course provides an introduction to discrete mathematical structures used in Computer Science and their applications. Course content
includes: Propositional and predicate logic; rules of inference; quantifiers; elements of integer number theory; set theory; methods of proof; induction; combinatorics and discrete probability;
functions and relations; recursive definitions and recurrence relations; elements of graph theory and trees. Applications include: analysis of algorithms, Boolean algebras and digital logic circuits.
Students who have completed, or are enrolled in, CS 17 may not receive credit. Prerequisite: MATH 1 with a minimum grade of C (May be taken concurrently), CS 1 with a minimum grade of C (May be taken
concurrently). 72 hours lecture, 18 hours laboratory. AA/AS GE. Transfer: CSU, UC; CSU GE: B4; IGETC: 2A; C-ID# COMP 152.
4 UNITS
This course focuses on the development of quantitative reasoning skills through in-depth, integrated explorations of topics in mathematics, including real number systems and subsystems. Emphasis is
on comprehension and analysis of mathematical concepts and applications of logical reasoning. Prerequisite: MATH 50 with a minimum grade of C or MATH 55 with a minimum grade of C or NMAT 255 with a
minimum grade of C or NMAT 250 with a minimum grade of C. 54 hours lecture. AA/AS GE. Transfer: CSU, UC.
3 UNITS
College algebra core concepts relating to Science, Technology, Engineering and Mathematics (STEM) and Business fields are explored, such as: polynomial, rational, radical, exponential, absolute
value, and logarithmic functions; systems of equations; theory of polynomial equations; and analytic geometry. Multiple representations, applications and modeling with functions are emphasized
throughout. May not receive credit if Mathematics 20 or 45 have been completed. Prerequisite: MATH 55 with a minimum grade of C or MATH 55B with a minimum grade of C or NMAT 255 with a minimum grade
of C. 72 hours lecture, 18 hours laboratory. AA/AS GE. Transfer: CSU, UC; CSU GE: B4, IGETC: 2A; C-ID# MATH 151.
Beginning Fall 2019, students are encouraged to enroll directly into a transfer-level course. Students should consider their academic goal, past achievements, and experiences.
Students are also strongly encouraged to take advantage of the many resources available to support them at LPC.
• NMAT 265 Math Jam for BSTEM^1 Bootcamp Course is offered 1-week prior to the start of the semester. Award-winning & tuition-free! Offered the week before the Fall and Spring semesters. Proven to
increase student success and retention rates.
• NMAT 201C (tuition-free) or MATH 101C (1 lab unit) Concurrent Support for BSTEM^1 support during the semester. Aligned with your math course and designed with innovative strategies to provide
math and learning support while taking your math class.
^1BSTEM is Business, Science, Technology, Engineering, and Mathematics
4 UNITS
Linear functions, systems of linear equations and inequalities, exponential and logarithmic functions and applications, matrices, linear programming, mathematics of finance, sets and Venn diagrams,
combinatorial techniques and an introduction to probability. Applications in business, economics and social sciences. Prerequisite: MATH 50 with a minimum grade of C or MATH 55 with a minimum grade
of C or MATH 55B with a minimum grade of C or NMAT 250 with a minimum grade of C or NMAT 255 with a minimum grade of C. 72 hours lecture. AA/AS GE. Transfer: CSU, UC*; CSU GE: B4; IGETC: 2A; C-ID#
MATH 130. * MATH 1, 33, and 34 combined: maximum UC credit, one course.
Beginning Fall 2019, students are encouraged to enroll directly into a transfer-level course. Students should consider their academic goal, past achievements, and experiences.
Students are also strongly encouraged to take advantage of the many resources available to support them at LPC.
• NMAT 264 Math Jam for SLAM^2 Bootcamp Course is offered 1-week prior to the start of the semester. Award-winning & tuition-free! Offered the week before the Fall and Spring semesters. Proven to
increase student success and retention rates.
• NMAT 200C (tuition-free) or MATH 100C (1 lab unit) Concurrent Support for SLAM^2 support during the semester. Aligned with your math course and designed with innovative strategies to provide math
and learning support while taking your math class.
^2SLAM is Statistics and Liberal Arts Mathematics.
4 UNITS
Functions and their graphs; limits of functions; differential and integral calculus of algebraic, exponential and logarithmic functions. Applications in business, economics, and social sciences and
use of graphing calculators. Partial derivatives and the method of LaGrange multipliers. Prerequisite: MATH 55 with a minimum grade of C or MATH 55B with a minimum grade of C or NMAT 255 with a
minimum grade of C. 90 hours lecture. AA/AS GE. Transfer: CSU, UC*; CSU GE: B4; IGETC: 2A; C-ID# MATH 140. * MATH 1, 33, and 34 combined: maximum UC credit, one course.
Beginning Fall 2019, students are encouraged to enroll directly into a transfer-level course. Students should consider their academic goal, past achievements, and experiences.
Students are also strongly encouraged to take advantage of the many resources available to support them at LPC.
• NMAT 265 Math Jam for BSTEM^1 Bootcamp Course is offered 1-week prior to the start of the semester. Award-winning & tuition-free! Offered the week before the Fall and Spring semesters. Proven to
increase student success and retention rates.
• NMAT 201C (tuition-free) or MATH 101C (1 lab unit) Concurrent Support for BSTEM^1 support during the semester. Aligned with your math course and designed with innovative strategies to provide
math and learning support while taking your math class.
^1BSTEM is Business, Science, Technology, Engineering, and Mathematics
5 UNITS
Trigonometry includes definitions of the trigonometric functions and their inverses, graphs of the trigonometric functions and their inverses, trigonometric equations, trigonometric expressions and
identities, including proofs, an introduction to vectors, polar coordinates and complex numbers. Applications include solving right triangles and solving triangles using the law of sines and the law
of cosines. Prerequisite: MATH 55B with a minimum grade of C or MATH 55 with a minimum grade of C or NMAT 255 with a minimum grade of C. 72 hours lecture, 18 hours laboratory. AA/AS GE. Transfer:
CSU; CSU GE: B4; C-ID# MATH 851.
Beginning Fall 2019, students are encouraged to enroll directly into a transfer-level course. Students should consider their academic goal, past achievements, and experiences.
Students are also strongly encouraged to take advantage of the many resources available to support them at LPC.
• NMAT 265 Math Jam for BSTEM^1 Bootcamp Course is offered 1-week prior to the start of the semester. Award-winning & tuition-free! Offered the week before the Fall and Spring semesters. Proven to
increase student success and retention rates.
• NMAT 201C (tuition-free) or MATH 101C (1 lab unit) Concurrent Support for BSTEM^1 support during the semester. Aligned with your math course and designed with innovative strategies to provide
math and learning support while taking your math class.
^1BSTEM is Business, Science, Technology, Engineering, and Mathematics
4 UNITS
Descriptive statistics, including measures of central tendency, dispersion and position; elements of probability; confi dence intervals; hypothesis tests; two-population comparisons; correlation and
regression; goodness of fit; analysis of variance; applications in various fields. Introduction to the use of a computer software package to complete both descriptive and inferential statistics
problems. Prerequisite: MATH 55 with a minimum grade of C or MATH 55B with a minimum grade of C or MATH 50 with a minimum grade of C or NMAT 250 with a minimum grade of C or NMAT 255 with a minimum
grade of C. 72 hours lecture, 18 hours laboratory. AA/AS GE. Transfer: CSU, UC; CSU GE: B4; IGETC: 2A; C-ID# MATH 110.
Beginning Fall 2019, students are encouraged to enroll directly into a transfer-level course. Students should consider their academic goal, past achievements, and experiences.
Students are also strongly encouraged to take advantage of the many resources available to support them at LPC.
• NMAT 264 Math Jam for SLAM^2 Bootcamp Course is offered 1-week prior to the start of the semester. Award-winning & tuition-free! Offered the week before the Fall and Spring semesters. Proven to
increase student success and retention rates.
• NMAT 200C (tuition-free) or MATH 100C (1 lab unit) Concurrent Support for SLAM^2 support during the semester. Aligned with your math course and designed with innovative strategies to provide math
and learning support while taking your math class.
^2SLAM is Statistics and Liberal Arts Mathematics.
4 UNITS
An introduction to a variety of mathematical concepts for students interested in liberal arts. Intended to cultivate an appreciation of the significance of mathematics in daily life and help develop
students’ mathematical reasoning. Topics include personal finance, logic, and exponential growth. Prerequisite: MATH 55 with a minimum grade of C or MATH 55B with a minimum grade of C or MATH 50 with
a minimum grade of C or NMAT 250 with a minimum grade of C or NMAT 255 with a minimum grade of C. 54 hours lecture, 18 hours laboratory. AA/AS GE. Transfer: CSU, UC; CSU GE: B4; IGETC: 2A.
Beginning Fall 2019, students are encouraged to enroll directly into a transfer-level course. Students should consider their academic goal, past achievements, and experiences.
Students are also strongly encouraged to take advantage of the many resources available to support them at LPC.
• NMAT 264 Math Jam for SLAM^2 Bootcamp Course is offered 1-week prior to the start of the semester. Award-winning & tuition-free! Offered the week before the Fall and Spring semesters. Proven to
increase student success and retention rates.
• NMAT 200C (tuition-free) or MATH 100C (1 lab unit) Concurrent Support for SLAM^2 support during the semester. Aligned with your math course and designed with innovative strategies to provide math
and learning support while taking your math class.
^2SLAM is Statistics and Liberal Arts Mathematics.
3 UNITS
Associate’s Degree and Foundational Level Courses
This course can also be taken tuition-free by registering for NMAT 250. This is an Intermediate Algebra course for students interested in fields of study that require Statistics or Liberal Arts
Mathematics (SLAM). Intermediate algebra concepts will be explored in the context of the function. Function concepts covered include: distinction between functions and relations, domain and range,
function notation, multiple representation of functions, behavior of functions, operations with functions (including composition), one-to-one functions, and invertible functions. Types of functions
considered: polynomial, rational, radical, exponential and logarithmic functions. Th e course includes an introduction to probability, counting and quantitative data. Standards for mathematical
practice, applications of functions, and modeling with functions are emphasized throughout. Strongly Recommended: MATH 110 with a minimum grade of C or MATH 110B with a minimum grade of C or NMAT 210
with a minimum grade of C. 54 hours lecture, 54 hours laboratory. AA/AS GE.
Advising Notes: This Intermediate Algebra level course satisfies an Associate Degree’s math requirement. This course engages students in the necessary exploration of intermediate algebra in the
context of liberal arts fields, through discovery and conceptual learning, without requiring the same level of rigorous algebraic proficiency as the intermediate algebra for science and economic
Beginning Fall 2019, students with a goal of transfer are encouraged to enroll directly into a transfer-level course. Students should consider their academic goal, past achievements, and experiences.
This course satisfies an Associate’s Degree and provides foundational intermediate algebra content to interested students.
Students are also strongly encouraged to take advantage of the many resources available to support them at LPC.
• NMAT 263 Math Jam for Intermediate Algebra Bootcamp Course is offered 1-week prior to the start of the semester. Award-winning & tuition-free! Offered the week before the Fall and Spring
semesters. Proven to increase student success and retention rates.
• NMAT 255C (tuition-free) or MATH 55C (1 lab unit) Concurrent Support for Intermediate Algebra support during the semester. Aligned with your math course and designed with innovative strategies to
provide math and learning support while taking your math class.
^2SLAM is Statistics and Liberal Arts Mathematics.
*NMAT courses are tuition-free, noncredit mathematics courses. Students may enroll as many times as desired.
NMAT 250/255 students may petition to receive credit by examination.
4 UNITS and NMAT 250^*- (TUITION-FREE)
This course can also be taken tuition-free by registering for NMAT 255. Intermediate Algebra concepts, in the service of Business, Science, Technology, Engineering and Math fields (BSTEM), will be
explored in this course including: an introduction to functions; linear and absolute value functions; absolute value equations and inequalities; compound linear inequalities; rational expressions,
functions and equations; radical expressions, functions and equations; rational exponents; complex numbers; quadratic functions and equations; inverse of a function; exponential and logarithmic
functions; properties of logarithms; exponential and logarithmic equations; conic sections; and systems of equations and inequalities. Multiple representations, applications and modeling with
functions are emphasized throughout. Strongly Recommended: MATH 110 with a minimum grade of C, MATH 110B with a minimum grade of C, or NMAT 210 with a minimum grade of C. 90 hours lecture. AA/AS GE.
Advising Notes: This Intermediate Algebra level course satisfies an Associate Degree’s math requirement. This Intermediate Algebra level course is specifically designed to prepare students with the
rigorous algebraic foundation required by economics, science, technology, engineering and math fields require.
Beginning Fall 2019, students with a goal of transfer are encouraged to enroll directly into a transfer-level course. Students should consider their academic goal, past achievements, and experiences.
This course satisfies an Associate’s Degree and provides essential intermediate algebra content to students interested in Business, Science, Technology, Engineering and Math fields.
Students are also strongly encouraged to take advantage of the many resources available to support them at LPC.
• NMAT 263 Math Jam for Intermediate Algebra Bootcamp Course is offered 1-week prior to the start of the semester. Award-winning & tuition-free! Offered the week before the Fall and Spring
semesters. Proven to increase student success and retention rates.
• NMAT 255C (tuition-free) or MATH 55C (1 lab unit) Concurrent Support for Intermediate Algebra support during the semester. Aligned with your math course and designed with innovative strategies to
provide math and learning support while taking your math class.
^1BSTEM is Business, Science, Technology, Engineering, and Mathematics
*NMAT courses are tuition-free, noncredit mathematics courses. Students may enroll as many times as desired.
NMAT 250/255 students may petition to receive credit by examination.
5 UNITS and NMAT 255^*- (TUITION-FREE)
This course can also be taken tuition-free by registering for NMAT 207. This course is intended to serve as a bridge between arithmetic and Elementary Algebra. It includes a review of arithmetic,
operations involving signed integers, fractions, and decimals, variables and variable expressions, simple linear equations and their graphs, percent and proportion, introduction to statistics,
geometry and measurement, and application problems. 54 hours lecture, 54 hours laboratory.
Advising Notes: This Prealgebra level course is a foundational math class with no prerequisites. This course will be offered only in the Emporium mode starting Fall 2019. The content in this class
provides any student with a starting point in a math sequence.
Beginning Fall 2019, students with a goal of transfer are encouraged to enroll directly into a transfer-level course. Students should consider their academic goal, past achievements, and experiences.
Students are also strongly encouraged to take advantage of the many resources available to support them at LPC.
• NMAT 261 Math Jam for Prealgebra Bootcamp Course is offered 1-week prior to the start of the semester. Award-winning & tuition-free! Offered the week before the Fall and Spring semesters. Proven
to increase student success and retention rates.
*NMAT courses are tuition-free, noncredit mathematics courses. Students may enroll as many times as desired.
4 UNITS and NMAT 207^*- (TUITION-FREE)
This course can also be taken tuition-free by registering for NMAT 210. Elementary algebra concepts, including: real numbers and their properties; algebraic expressions; integer exponents; operations
with polynomial expressions; linear and quadratic equations; linear inequalities and set notation; graphs of linear equations and inequalities; slope; systems of linear equations and inequalities;
and modeling with linear and quadratic equations. Strongly Recommended: MATH 107 with a minimum grade of C or MATH 107B with a minimum grade of C or NMAT 207 with a minimum grade of C. 72 hours
Advising Notes: This Elementary Algebra level course is foundational math. Elementary concepts covered in this course are building blocks for every subsequent math class, such as order of operations,
evaluating expressions, using formulas, solving linear equations, exponents, factoring and much more.
Beginning Fall 2019, students with a goal of transfer are encouraged to enroll directly into a transfer-level course. Students should consider their academic goal, past achievements, and experiences.
Students are also strongly encouraged to take advantage of the many resources available to support them at LPC.
• NMAT 262 Math Jam for Elementary Algebra Bootcamp Course is offered 1-week prior to the start of the semester. Award-winning & tuition-free! Offered the week before the Fall and Spring semesters.
Proven to increase student success and retention rates.
• NMAT 210C (tuition-free) or MATH 110C (1 lab unit) Concurrent Support for Elementary Algebra support during the semester. Aligned with your math course and designed with innovative strategies to
provide math and learning support while taking your math class.
*NMAT courses are tuition-free, noncredit mathematics courses. Students may enroll as many times as desired.
MATH 156 GEOMETRY - 3.5 UNITS and NMAT 256^*- (TUITION-FREE)
This course can also be taken tuition-free by registering for NMAT 256. Topics include congruence, similarity, right triangles, trigonometry, circles, expressing geometric properties with equations,
geometric measurement and dimension, modeling with geometry, conditional probability and the rules of probability, and using probability to make decisions. Prerequisite: MATH 110 with a minimum grade
of C or NMAT 210 with a minimum grade of C. 54 hours lecture, 27 hours laboratory.
4 UNITS and NMAT 210^*- (TUITION-FREE)
This course provides a survey of algebraic processes with an emphasis on applications in welding. Topics covered include, but are not limited to: algebraic expressions, plane geometry, the geometry
of solids, and triangle trigonometry. This course may not be used as a prerequisite for any transfer level course. Prerequisite: MATH 72C with a minimum grade of C or MATH 72D with a minimum grade of
C. 36 hours lecture. AA/AS GE.
2 UNITS
This course provides a survey of algebraic processes with an emphasis on applications in welding. Topics covered include, but are not limited to: quadratic equations, functions, and mathematical
models. This course may not be used as a prerequisite for any transfer level course. Prerequisite: MATH 72D with a minimum grade of C and MATH 53A with a minimum grade of C. 18 hours lecture. AA/AS
1 UNIT
This course provides a survey of computational and elementary algebraic processes with an emphasis on applications in the automotive and welding trades. Topics covered include, but are not limited
to: computations with real numbers, ratios, and proportions. This course cannot be used as a prerequisite for Math 50 Core Intermediate Algebra or Math 55 Intermediate Algebra. 18 hours lecture.
1 UNIT
This course provides a survey of computational and elementary algebraic processes with an emphasis on applications in the automotive and welding trades. Topics covered include, but are not limited
to: linear equations, the rectangular coordinate system, and linear equations in two variables. This course cannot be used as a prerequisite for Math 50 Core Intermediate Algebra or Math 55
Intermediate Algebra. Prerequisite: MATH 72A with a minimum grade of C. 18 hours lecture.
1 UNIT
This course provides a survey of computational and elementary algebraic processes with an emphasis on applications in the automotive and welding trades. Topics covered include, but are not limited
to: percentages and measurement. This course cannot be used as a prerequisite for Math 50 Core Intermediate Algebra or Math 55 Intermediate Algebra. Prerequisite: MATH 72A with a minimum grade of C
or MATH 72B with a minimum grade of C. 18 hours lecture.
1 UNIT
This course provides a survey of computational and elementary algebraic processes with an emphasis on applications in the automotive and welding trades. Topics covered include, but are not limited
to: the rectangular coordinate system, linear equations in two variables, and systems of linear equations. This course cannot be used as a prerequisite for Math 50 Core Intermediate Algebra or Math
55 Intermediate Algebra. Prerequisite: MATH 72B with a minimum grade of C and MATH 72C with a minimum grade of C. 18 hours lecture.
1 UNIT
Award winning, 1-week before the semester starts, and tuition free! These courses are offered the week before the Fall and Spring semesters. Innovative learning interventions help you prepare for
upcoming mathematics courses. Proven to increase student success and retention rates!
Students who attend two NMAT Math Jams can earn a noncredit Certificate of Completion.
Math Jam is a noncredit program designed to help students prepare for their upcoming math class at a community college. Embedded are essential study and life skills to develop each student
holistically, including learning skills and career development. Students will be learning arithmetic and Prealgebra material with the goal of preparing them to be successful in their upcoming class.
It is strongly recommended that students taking this course be enrolled in a community college math course. 30-60 hours.
Math Jam is a noncredit program designed to help students prepare for their upcoming math class at a community college. Embedded are essential study and life skills to develop each student
holistically, including learning skills and career development. Students will be learning prealgebra material with the goal of preparing them to be successful in their upcoming class. It is strongly
recommended that students taking this course be eligible for and enrolled in a community college math course. 30-60 hours.
Math Jam is a noncredit program designed to help students prepare for their upcoming math class at a community college. Embedded are essential study and life skills to develop each student
holistically, including career development. Students will be learning elementary algebra material with the goal of preparing them to be successful in their upcoming class. It is strongly recommended
that students taking this course are enrolled in a community college math course. 30-60 hours.
Math Jam for SLAM Prep is for students preparing for math courses in Statistics and Probability or Mathematics for Liberal Arts. Math Jam is a FREE noncredit program designed to help students prepare
for their upcoming math class at a community college. Embedded are essential study and life skills to develop each student holistically, including career development. Students will be learning
prerequisite algebraic and basic probability material with the goal of preparing them to be successful in their upcoming fi rst-level transfer course of Statistics or Math for Liberal Arts class. It
is strongly recommended that students taking this course be enrolled in Math 40: Statistics and Probability or Math 47: Mathematics for Liberal Arts at Las Positas College. 30-60 hours.
Math Jam for BSTEM Prep is for students preparing for math courses in College Algebra, Trigonometry, Business Calculus and review prior to Calculus I. Math Jam is a noncredit program designed to help
students prepare for their upcoming STEM focused math class at a community college. Embedded are essential study and life skills to develop each student holistically, including career development.
Students will be learning pre-transfer level material with the goal of preparing them to be successful in their upcoming class. It is strongly recommended that students taking this course are
enrolled in a community college math course. 30-60 hours.
Math Jam for Calculus I is a credit course for students preparing for Calculus I. Embedded are essential study and life skills to develop each student holistically, including career development.
Students will be learning basic skills and transfer-level material with the goal of preparing them to be successful in their upcoming class. It is strongly recommended that students taking this
course are enrolled in a calculus course. 27-54 hours laboratory.
Offered for credit only. 0.5-1 units
Math Jam for Calculus II is a credit course for students preparing for Calculus II. Embedded are essential study and life skills to develop each student holistically, including career development.
Students will be learning basic skills and transfer-level material with the goal of preparing them to be successful in their upcoming class. It is strongly recommended that students taking this
course are enrolled in a calculus course. 27 hours laboratory.
Offered for credit only. 0.5 units
Math Jam for Calculus III is a credit course for students preparing for Calculus III. Embedded are essential study and life skills to develop each student holistically, including career development.
Students will be learning basic skills and transfer-level material with the goal of preparing them to be successful in their upcoming class. It is strongly recommended that students taking this
course are enrolled in a calculus course. 27 hours laboratory.
Offered for credit only. 0.5 units
New! Jam all semester long with RECOMMENDED support during the semester. Offered for credit or tuition-free (noncredit). Aligned with your math course and designed with innovative strategies to
provide math and learning support while taking your math class. These classes will help you streamline the time you need to spend outside of a math class studying to be successful. The support course
includes assignments to help you prepare for upcoming tests and/or to review a recent test and learn from your mistakes. You will have opportunity to work on what YOU need to learn – whether it is to
review prerequisite material that you need to know in order to be successful or to work on the new material covered in your class. The support class does not have any homework.
This course offers structured support to students who are concurrently enrolled in Calculus I. The support course includes material to prepare students for the rigor of the calculus course by
teaching learning skills necessary to succeed in college courses as well as review of relevant prerequisite algebraic, geometric and trigonometric concepts, and more in-depth investigation of core
concepts in their concurrent math course. Corequisite: MATH 1. 54 hours laboratory.
Offered for credit only. 1 unit
This course offers structured support to students who are concurrently enrolled in Calculus II. The support course includes material to prepare students for the rigor of the calculus course by
teaching learning skills necessary to succeed in college courses as well as review of relevant prerequisite algebraic, geometric and trigonometric concepts, and more in-depth investigation of core
concepts in their concurrent math course. Corequisite: MATH 2. 54 hours laboratory.
Offered for credit only. 1 unit
This course offers structured support to students who are concurrently enrolled in Calculus III. The support course includes material to prepare students for the rigor of the calculus course by
teaching learning skills necessary to succeed in college courses as well as review of relevant prerequisite algebraic, geometric and trigonometric concepts, and more in-depth investigation of core
concepts in their concurrent math course. Corequisite: MATH 3. 54 hours laboratory.
Offered for credit only. 1 unit
Concurrent Support for SLAM Math is for students interested in disciplines that require Statistics and Liberal Arts Mathematics (SLAM) courses. This course off ers structured support to students who
are concurrently enrolled in a first-level transfer course, such as Statistics and Mathematics for Liberal Arts, and Finite Mathematics. The support course includes material to prepare students for
the rigor of the transfer math course by teaching learning skills necessary to succeed in college courses as well as review of relevant prerequisite algebraic and geometric concepts, and more
in-depth investigation of core concepts in their concurrent math course. Includes assignments to help you prepare for upcoming tests and/or to review a recent test and learn from your mistakes. You
will have opportunity to work on what YOU need to learn. The support class does not have any homework. Corequisite: MATH 40 or MATH 47 or MATH 33. 54 hours.
Offered for credit (MATH 100C) or for tuition-free noncredit (NMAT 200C)
Concurrent Support for BSTEM Mathematics is for students interested in Business, Science, Technology, Engineering and Mathematical fields. Th is course offers structured support to students who are
concurrently enrolled in a first-level transfer course, such as College Algebra, Trigonometry, and Business Calculus. The support course includes material to prepare students for the rigor of the
transfer math course by teaching learning skills necessary to succeed in college courses as well as review of relevant prerequisite algebraic and geometric concepts, and more in-depth investigation
of core concepts in their concurrent math course. Includes assignments to help you prepare for upcoming tests and/or to review a recent test and learn from your mistakes. You will have opportunity to
work on what YOU need to learn. The support class does not have any homework. Corequisite: MATH 30 or MATH 39 or MATH 34. 54 hours.
Offered for credit (MATH 101C) or for tuition-free noncredit (NMAT 201C)
This course is just-in-time concurrent support for students enrolled in a first-level transfer course, such as Statistics, College Algebra, Trigonometry, Business Calculus, Mathematics for Liberal
Arts, and Finite Mathematics. The support course is noncredit, open entry/open exit. The content will prepare students for the rigor the transfer math course by teaching learning skills necessary to
succeed in college courses as well as review of relevant basic and secondary education prerequisite algebraic and geometric concepts, and more in-depth investigation of core concepts to their
concurrent math course. The course design is to meet the needs of a variety of students, such as students who desire formal, regular ongoing learning supports, students wishing self-place into
transfer-level mathematics courses as defined by AB 705, and students who are repeating the course for the second or third time. The support course includes a review of basic and secondary level math
relevant to their college-level course, provides study strategies to promote understanding and improve performance, and more in-depth investigation of core concepts that are difficult for students to
master and learning skills such as growth mindset, brain research, time management, study skills, test taking, math anxiety and more. Corequisite: MATH 30 or MATH 33 or MATH 34 or MATH 39 or MATH 40
or MATH 47. 1-54 hours.
Students can register for this class anytime during the semester in ClassWeb.
This course is concurrent support for Elementary Algebra. The course is designed to provide additional, formal support to students who are currently taking an Elementary Algebra. It includes a review
of arithmetic, algebraic and geometric concepts that are relevant to their Elementary Algebra course, study strategies that promote understanding and improve performance, and more in-depth
investigation of core concepts that are diffi cult for students to master. Embedded are learning skills such as growth mindset, brain research, time management, study skills, test taking, math
anxiety and more. Includes assignments to help you prepare for upcoming tests and/or to review a recent test and learn from your mistakes. You will have opportunity to work on what YOU need to learn.
The support class does not have any homework. Corequisite: MATH 110 or NMAT 210. 54 hours.
Offered for credit (MATH 110C) or for tuition-free noncredit (NMAT 210C)
This course is concurrent support for Intermediate Algebra. The course is designed to provide additional, formal support to students who are currently taking an Intermediate Algebra. It includes a
review of arithmetic, algebraic and geometric concepts that are relevant to their Intermediate Algebra course, study strategies that promote understanding and improve performance, and more indepth
investigation of core concepts that are difficult for students to master. Embedded are learning skills such as growth mindset, brain research, time management, study skills, test taking, math anxiety
and more. Includes assignments to help you prepare for upcoming tests and/or to review a recent test and learn from your mistakes. You will have opportunity to work on what YOU need to learn. The
support class does not have any homework. Corequisite: MATH 55 or NMAT 255 or MATH 50 or NMAT 250. 54 hours. This class is offered for credit (MATH 55C) or for tuition-free noncredit (NMAT 255C). | {"url":"https://laspositascollege.edu/assessment/math-support.php","timestamp":"2024-11-07T07:31:07Z","content_type":"text/html","content_length":"170491","record_id":"<urn:uuid:8c6586f4-ab31-44a1-9048-b6444f40a047>","cc-path":"CC-MAIN-2024-46/segments/1730477027957.23/warc/CC-MAIN-20241107052447-20241107082447-00402.warc.gz"} |
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] [41] [42] [43] [44] [45] [46] [47] [48] [49] [50] [51]
Table 8.6 First Proposed Aggregation (Optimal Solution Indicated; Optimal Value = $116,445).
│ │ │ │ │PIT/NY/BOS │Supplies │
│ │2.078│2.013│2.618│2.761 │ │
│ ├─────┼─────┼─────┼───────────┼─────────┤
│ │ │(10) │ │ │ │
│ │1.387│2.054│2.182│2.733 │ │
│ ├─────┼─────┼─────┼───────────┼─────────┤
│ │(10) │ │ │(15) │ │
│ │ │1.067│0.789│1.580 │ │
│ │0.243├─────┼─────┼───────────┼─────────┤
│ │ │ │ │(15) │ │
│Demands│ │ │ │ │ │
to the expected magnitude of total transportation costs, another aggregation must be sought. Table 8.7 characterizes an alternative. Setting Ll = {NY,BOS) and using c« =
NY/BOS max fc= 1,2,3
+ max A;= 1,2,3
Dnv BOS
= 15,000 max{(2.879-2.815),(2.856-2.786),(1.686-1.608)! + 10,000 maxi(2.879-2.976),(2.856-2.960),(1.686-1.804)!
upon factoring ZJny and ZJbos out of the respective braces. Thus, «ny/bos = 15,000(0.078) -b 10,000(-0.097) = $200 for this aggregation. For the two proposed aggregations. Property 8.1 is exemplified
by the following:
v[full problem] < v[PIT/NY/BOS aggregation] < v[full problem] + *p,t/n
ie, 116,005 < 116,445 < 116,505, and
v[full problem] < v[NY/BOS aggregation] < v[full problem] + fNv/Bos, ie, 116,005 < 116,170 < 116,205.
Table 8.7 Second Proposed Aggregation (Optimal Solution Indicated;
││ │ │ │ │NY/BOS │Supplies │
││ │2.013│2.618│2.465│2.879 │ │
││ │(10) │ │ │ │ │
││1.387│2.054│2.182│2.426│2.856 │ │
││(10) │ │ │(10) │ │ │
││ │1.067│0.789│1.313│1.686 │ │
││ │ │ │ │(15) │ │
Owing to the nature of the pro rata demand aggregation scheme, an optimal solution of an aggregated model can be disaggregated into a feasible solution to the full problem with the same cost as the
optimal value of the aggregated problem. In the second aggregation the SEA-»NY/ BOS flow of 5000 cwt has an associated cost of 5000(2.879) = 5000[(15/ 25)2.815 + (10/25)2.976]. This flow would be
disaggregated into (15/ 25)5000 cwt to New York and (10/25)5000 cwt to Boston with the identical cost in the full problem. Similar results hold for disaggregating the remaining flows to NY/BOS. The
value of the disaggregated solution is guaranteed to be within $200 of optimality. Indeed, the suboptimality is actually only $116,170 - $116,005 = $165.
Considering H sequential applications of Property 8.1 leads to its extension given below.
Property 8.2 [Extended customer aggregation error bound]. Let Li,L2,...,Lf,, be disjoint subsets of demand point indices such that each subset Ll, individually satisfies qualification (8.28). Then:
if for A = 1,..., and all k, the js are identical
over all etLi,
v[CPLP] =¸ v[CPLP; (8.30)1 =¸ v[CPLP]
IModeling Philosophy
Solving the linear programming relaxation of problem (CPLP) augmented by the constraints 0 < z < 1, for all k, will generally produce fractional values for the variables if any throughput constraints
are binding. The convenience of using linear programming directly as a solution approach is lost. However, Kuehn and Hamburger (1963) issued an early caveat: "Care must also be taken to avoid the
chase for optimal solutions to simple problems and thereby miss the actual problem of business-the solution of,large-scale problems containing many customers buying various mixes of a full product
line, many potential warehouse sites, alternate warehouse types with different cost structures, several factories and, perhaps, a number of potential factory sites." The foUowing section presents a
computationally practicable approach for treating the complex problem described in the caveat.
A large food products company is engaged in distribution planning [Geoffrion and Graves (1974)], and a computer-based model is adopted to plan
the configuration and flow aspects of the distribution system design for the next five years. Hundreds of products must be aggregated into several product groups or "standard commodity bundles."
Examples of groupings are bottled cooking oil, packaged shortenings, and catsup. These commodity bundles, hereafter called commodities, are produced at several existent plants with known production
capacities (cwt/year). Not every plant can produce every group. There is a known demand for each of the commodities at each of several demand zones (customer aggregations). Commodities may be shipped
through distribution centers (DCs). Each demand zone is to be assigned uniquely to a DC. This feature has accounting, marketing, and transportation advantages. There may be upper and lower bounds on
the throughput of the DCs. There is a list of candidate sites for DCs developed from current locations, competitor DC locations, major demand centers, hub cities, or sites generated fi-om continuous
models. There is a fixed cost of establishing and operating a DC as well as a variable throughput cost. The transportation system is characterized by transportation costs that are linear in the
amount shipped.
The primary design questions are: which DC sites to use, what throughput to plan for at each DC, what demand zones to assign to each DC, and what pattern of flows should there be for each commodity.
The distribution system is to be designed to satisfy customer demand at minimum total cost while honoring plant capacity constraints, DC throughput bounds, and any additional constraints on the
system configuration. The model presumes that all plants are previously established. However, the potential exists for including additional zero-one variables to select among alternative plant sites
for some or all plants as well as among plant capacity expansion projects. The reader is encouraged to consider these possibilities as the narrative unfolds. The following symbols are used throughout
this section:
S., =
index for commodities,
index for plants (or, more generally, procurement zones), index for DC candidate sites, index for demand zones,
supply (production capacity) of commodity / at plant j (cwt/ year),
deinand for commodity i in demand zone H (cwt/year), minimum, maximum allowed total annual throughput for a DC at site (cwt/year),
fixed cost of establishing a DC at site (dollars/year), average variable unit cost of throughput for a DC at site (dollars/cwt),
average unit cost of producing and shipping commodity i from plant j through a DC at site to demand zone £ (dollars/cwt),
Xijki = amount of commodity i shipped from plant j through a DC at
site to demand zone (cwt), 1,1 = a zero-one variable equal to 1 if a DC at site serves demand
zone £ and equal to 0 otherwise, Zl, = a zero-one variable equal to 1 if a DC is established at site k, and equal to 0 otherwise.
The distribution design model denoted problem (DD) is given by:
minimize 2 2 S Z Cuki/X.ki + E (/ + S S ) (8-31) x,y,z J l i I
subject to
│e │for all ij │(8.32) │
│X.jiti = Digyii │for all i,k,£ │(8.33) │
│ │for all £ │(8.34) │
│KkZ, < S S .xFAf │for all │(8.35) │
│linear configuration constraints on ys and/or zs │ │
│ │(8.36) (8.37)│
│1 0, yf, Zt = 0 or 1 for all ij,k,£. │ │
Summations are understood to extend only over practicable combinations of indices, ie, certain ij links will be omitted if plant j doesnt produce commodity / and certain jk and k£ links will be
clearly uneconomical. There may only be a small subset of potential DC sites for supplying any given demand zone. The omitted subscript combinations reduce the number of variables actually present in
the model. The quadruple subscripting scheme used for the flow variables preserves the identity of the origin of commodities. This permits imposition of restrictions on certain Jk£ routes for
perishable commodities by omitting the associated variables. The quadruple subscripting scheme is also useful when a commodity price depends on the originating plant or when some transportation costs
are determined on the basis of direct plant-to-customer shipments even though a stopover is made at the assigned DC, a "storage-in-transit" privilege. If certain demand zones are actually to be
supplied directly from plants, a dummy DC site, say, k„, can be included. The associated z and ,/s would be set to 1 and the c,jtjs would be specified appropriately.
The linear configuration constraints (8.36) can be used as a technical device to select among projects for expansion or contraction of capacity at existing DCs, or to ensure that, at most, one
version of a DC will be
opened at a given site. Discussions with management uncovered the need to include service level constraints such as )/! Da <
Ti, where t,i,e is the average time to deliver an order for commodity / to demand zone 2 from a DC at site k, and , is an upper bound on average delivery time for commodity i. The latter concern can
also be addressed by assigning p a comparatively large objective function coefficient whenever site is unacceptably distant from demand zone £
Data Considerations
The unit production cost (dollars/cwt) component of Cijfcif measures the real cash flow change for changing the output rate of commodity i at plant j. The directly assignable portion of production
cost can be derived from an existing standard cost accounting system. Actual, rather than standard, raw material costs should be used to measure the influence of regional raw material prices on
location. The overhead portion requires estimation of variable components of indirect cost accounts (such as energy, indirect labor, management, maintenance, etc.) and the allocation of the variable
component to the given commodity.
The distribution cost (dollars/cwt) component of %-f has two parts. For inbound J->k transportation links, the transportation rate may adequately be modeled as a weighted average reflecting the mix
of modes and shipment sizes judged most likely to be used. (Pipeline inventory costs might be included.) For k-J2 links to large volume demand zones, unit transportation rates are developed by
dividing estimated total annual freight cost by the volume that must flow on the link. This flow is known because Dm is known and demand zones must be supplied entirely from a single DC. If shipment
is by truck delivering to smaU volume demand zones on a dehvery tour, distribution cost for k-*( hnks must be approximated. The following has been applied by Mairs, Wakefield, Johnson, and Spielberg
(1978) to give quick and reasonably accurate estimates:
(constant) X X m,, X Oie X r,,g
tiki ~ -----, where
sX UgX Pi
dke = travel distance from site to demand zone £,
nil, = freight cost per mile from site k. Si, = factor for trailer size, depending on site k, Ue = factor for trailer utilization, depending on the destination, Pi = density factor in cwt/cubic feet,
of = one-way trip factor-normally is 2 to reflect two-way trip; otherwise, less than 2 if the trip includes pickup of supplies from plant at an interchange point,
r„ = circuit cost factor, approximated by 2.38-0.18 loge(u?«), which reflects the fact that demand zone (is supplied on delivery route.
Solution by Benders Decomposition
The distribution design problem is a large-scale mixed-integer programming problem for this application. There are 17 product groups (commodities), 14 plants, 45 DC sites, and 121 demand zones. The
solution approach to be presented decomposes the problem. The approach capi-I talizes on the property that when the zero-one variables and z are held fixed at feasible values, the flow partition of
the problem separates into independent classical transportation problems, one for each commodity. To see this, consider the flow partition of the problem that involves all the transportation
variables and the temporarily held fixed variables, say yig. The problem becomes:
minimize 2 S E S
X I j Q
subject to (8.38)
Xiike S,j for all i,}
1 Xij.g = D.gyl/g for all i,k,ll
Xijke 0 for all i,j,k,Q.
I Define as the index value such that > = 1 for a particular value I of the H index. In words, T<i£) is the DC that serves demand zone £ Since U£) is unique by constraint (8.34), x,,;,, = 0 for all
ijM with (£). These facts provide the basis for stating the transportation problem for I the commodity as:
minimize X S , , « , ,.
subject to
2 -Mw S,j for all j
E XijUife = Dig for all £
X,jk(e)P s= 0 for all /,£
The DC variable cost (dollars/cwt) includes handling, inventory costs, and other variable operating expenses. The DC fixed cost (dollars/year) captures amortized capital costs and fixed components of
operating costs. However, as in Figure 8.6, f may function only to provide a straight-line approximation of DC throughput cost over a range of interest.
[start] [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [ 35 ] [36] [37] [38] [39] [40
] [41] [42] [43] [44] [45] [46] [47] [48] [49] [50] [51] | {"url":"http://eng.yax.su/finlab/ir011/35/index.shtml","timestamp":"2024-11-05T22:23:58Z","content_type":"text/html","content_length":"24391","record_id":"<urn:uuid:995fe59d-c11a-44e7-bef1-47195de09fe8>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00296.warc.gz"} |
Design and model analysis of the sonic vibration head (2024)
As a novel environmental sampling technique, sonic vibration drilling has been rapidly developed in the past few years. The penetration force is generated from two eccentric shafts driven by
hydraulic motors. This gives rise to the vertical oscillation of the drill pipe to drill in the stratum. As the most important parts of the sonic driller, the vibration head consists of eccentric
structure, synchronization mechanism, supporting structure and rotating structure. In the first part of this paper, a 3D mathematical model was developed after analyzing the working law of sonic
vibration head by using SolidWork. In the second part, the model was stimulated in order to predict the performances of the sonic vibration head by using ANASYS. In the third part, a physical
prototype was developed to conduct practical experiments, confirming feasibility of the previous design and stimulation, and making good references for future optimization.
1. Introduction
Sonic vibration drilling is a novel technique for environmental sampling that has been rapidly developed [1-3]. It is characterized by high drilling speed with no drilling mud needed. It can be
widely used in mineral prospecting survey, geotechnical investigation construction, environmental and well drilling, being capable of extracting high-quality core, especially in sedimentary stratum
[4]. The vibration head consists of eccentric structure, synchronization mechanism, supporting structure and rotating structure. It is the most important part of sonic driller [5]. Its vibration
characteristics are closely related to drilling efficiency, economy and security during the sonic drilling. Therefore, design for the vibration head is of great importance [6].
Traditionally, design of the drilling rig is mainly based on the previous scheme [7] which consists of fixed parameters of main components [8]. The parameters are generally used to verify strength
and toughness, making sure the rig safety performance. After this, a 2D engineering drawing is made to advance the physical prototype. For this method, it has disadvantages in wasting time due to the
redesign of the rig if there is something wrong checked out. In order to avoid this, a dynamic design is introduced. In this method, the Virtual Prototype Technology (VPT) [9, 10] is applied to
establish a three-dimensional model to make the stimulation, being capable of repeatedly revising the design of virtual prototype. The VPT ensures feasibility of the visual prototype in practical –
it can be successfully processed with higher quality and shorter design period in comparison with traditional one [11].
In the first part of this paper, a 3D mathematical model was developed after analyzing the working law of the sonic vibration head by using SolidWork. In the second part, the model was stimulated
mainly to predict the performances of the sonic vibration head by using ANASYS. In the third part, a physical prototype was developed and used to conduct some practical experiments to confirm the
feasibility of the previous design and stimulation, and help figure out some drawbacks of the previous design, making good references for the future optimization.
2. Sonic vibration head design
2.1. The principle of sonic drilling processing
Fig. 1 plots working law of the sonic vibration head. A hydraulic motor drives two eccentric shafts to high speed move respectively. The force generated by eccentric shafts can be resolved into
horizontal and vertical direction [12]. However, the overall horizontal components can be counteracted while the vertical can be strengthened due to the same magnitude speed but totally opposite
direction of the two eccentric shafts’ move. This gives rise to the vertical oscillation of the drill pipe, whose high frequency can partially cause the earth nearby liquefied, effectively reducing
the friction between the drilling tools and the layer around. It is, therefore, believed to be a highly efficient way to extract cores.
Fig. 1Principle of sonic drilling processing
2.2. The virtual prototype of the vibration head
The vibration head is mainly used to generate exciting force. In addition to protecting the support plate from being damaged, an isolator is designed. The design of eccentric parts and isolation
structure are vital to the design of the virtual prototyping. Specifically, the vibration head structurally consists of eccentric structure, synchronization mechanism, supporting structure and
rotating structure. With four parts having been completed respectively, an overall 3D visual model is developed.
During the process of assembling the visual model, a cordwood-like system is used to make the model structures more clear for its future revision. This clearance is realized by the proper order of
assembling, from smaller components like eccentric structure, isolator structure and supporting structure to the overall design (shown in Fig. 2).
Fig. 2Virtual prototype of the vibration head: 1. Eccentric parts, 2. Isolation structure, 3. Rotating structure, 4. Synchronization mechanism, 5. Support plate
2.3. Virtual prototyping design of the key parts
2.3.1. The virtual prototype of the synchronization mechanism
A double-eccentric shaft with simple structure is selected to ease the layout of the vibrator. The double-eccentric shaft consists of the eccentric shafts, support shafts, and hydraulic motor. These
components are installed in the vibrator, which is driven by the eccentric shaft. The synchronous motion of the double eccentric shaft is to ensure high working efficiency of the vibration head. In
order to keep the two eccentric shafts working synchronously, a group of structures are installed, including the synchronizing wheels, guide wheel, pressing wheel, and synchronous belt. Fig. 3 shows
the virtual prototype of the synchronization mechanism.
Fig. 3Virtual prototype of the synchronization mechanism
2.3.2. The virtual prototype of the isolation and support structure
When the audio vibration head works, an eccentric portion driven by the vibration will produce a periodic vertical vibration. The maximum exciting force can reach 180KN. Isolation 4. Test of the
vibration head structure must be set up so that most of the exciting force down can be through the drill pipe on the role in the formation. Centrifugal force upward must be reduced to a minimum
through isolator. Thus, the isolation structure must have upwards and downwards isolation capability in both directions.
Fig. 4Virtual prototype of the isolation and support structure
Isolation structure section includes a bolt pin, annular temperature isolator, locknut, adjusting gasket, and pin. Isolation bolted vibrator pins is suspended by the support structure, which is made
by the support housing, instead of welding. This can reduce the area of stress concentration (shown in Fig. 4).
3. Model and simulation analysis
3.1. Mathematical model of the vibration head
Based on the first virtual prototype acoustic vibration, the model with double eccentric shaft vibration head can be simplified as shown in Fig. 5. The eccentric system driven by a double-speed motor
can produce vibration in direction $y$, causing vibrating bodies vibrate simultaneously. However, body vibration isolator is simplified as a combination of damping and stiffness support, which can be
used to reduce the vibration between vibration head and to protect the main components. In terms of the vibrating body, it can be simplified to a particle due to its large mass and small elastic body
mass. The interaction force among bearings, eccentric shaft, and the sleeve body are regarded as the internal forces for vibration. But the symmetrical and metal vibration rubber components can move
up and down. The symmetrical arrangement of metal can be thought as two parallel rubber damping bodies. However, the two isolator cannot move completely uniform, which can cause the vibrating body
swing. The piled metal rubber cannot be regarded as a series of two damping rubber bodies. When the vibrating body moves downward, the vibration state is opposite during the upward movement of the
vibrating body.
Metal rubber is a new type of vibration damping material. Its internal structure is intertwined wire mesh structure space collusion line into the rubber-like polymer structure [13-15]. By using a
vibrating body metal rubber material, audio system is supported on the vibration head mounting bracket to reduce the impact of vibration. At the same time, it has a role in slowing the impact of
vibration on the mounting bracket body system in sonic drilling.
Assume that the rotation of the eccentric shaft system quality, structure, moment of inertia, and resistance moment are completely equal. The motors are completely synchronized. On the one hand, if
the vibration isolation system is not preloaded and compressed, the centroid of the vibrating body will stay at an initial position. On the other hand, the centroid will fall down by ${y}_{st}$ with
generating a static angle ${\theta }_{st}$ due to gravity and preloaded compression force of ${M}_{0}g$; vibration centroid will fall down by $y$, and a swing angle of $\theta$ will generated with an
inertia of $I$, at which of the time the static deformation and deformation are generated by the system are respectively ${y}_{st}-l{\theta }_{st}$, $y-l\theta$, ${y}_{st}+l{\theta }_{st}$, and $y+l\
theta$. The kinetic energy of the vibrating body can be expressed as:
$T=\frac{1}{2}\left({M}_{0}{\stackrel{˙}{y}}^{2}+I{\stackrel{˙}{\theta }}^{2}\right).$
Fig. 5Model of the vibration head driven by two eccentric rotating system: 1, 2. Two of eccentric rotating system, 3. Main vibration body, 4. Stiffness, 5. Damper
When considering the static deformation vibration isolation system, the potential energy system should include potential energy generated by elastic static deformation of the isolator. Two vibration
isolators in total deformation force are $\left({y}_{st}-l{\theta }_{st}+y-l\theta \right)$ and $\left({y}_{st}+l{\theta }_{st}+y+l\theta \right)$. The displacement of the centroid is $\left({y}_{st}
+y\right)$, therefore, the overall potential is:
$U=\frac{1}{2}\left[{K}_{1}\left({y}_{st}-l{\theta }_{st}+y-l\theta {\right)}^{2}+{K}_{2}\left({y}_{st}+l{\theta }_{st}+y+l\theta {\right)}^{2}-{M}_{0}g\left({y}_{st}+y\right)].$
Energy dissipation function can be expressed as:
$D=\frac{1}{2}\left({C}_{1}{y}^{2}+{C}_{2}{y}^{2}+{C}_{\theta }{\stackrel{˙}{\theta }}^{2}\right),$
where ${C}_{1}$ and ${C}_{2}$ are damping coefficient of direction $Y$, and ${C}_{\theta }$ is drag torque coefficient. The resultant force of generalized exciting force and weight over bit is:
$Q=2m{\omega }^{2}rsin\sigma +2m\stackrel{˙}{\omega }rcos\sigma +P-F.$
Substitute $T$, $U$, and $D$ into Lagrange equation, we can get:
$\frac{\partial T}{\partial y}=0,$
$\frac{\partial U}{\partial y}={K}_{1}\left({y}_{st}-l{\theta }_{st}\right)+{K}_{2}\left({y}_{st}+l{\theta }_{st}\right)-{M}_{0}g+\left({K}_{1}+{K}_{2}\right)y-\left({K}_{1}l-{K}_{2}l\right)\theta ,$
$\frac{\partial D}{\partial \stackrel{˙}{y}}=\frac{\partial }{\partial \stackrel{˙}{y}}\left(\frac{1}{2}{C}_{1}{y}^{2}+\frac{1}{2}{C}_{2}{y}^{2}+\frac{1}{2}{C}_{\theta }{\stackrel{˙}{\theta }}^{2}\
$\frac{d}{dt}\frac{\partial T}{\partial \stackrel{˙}{\theta }}=\frac{d}{dt}\frac{\partial T}{\partial \stackrel{˙}{\theta }}\left(\frac{1}{2}{M}_{0}{\stackrel{˙}{y}}^{2}+\frac{1}{2}I{\stackrel{˙}{\
theta }}^{2}\right)=I\stackrel{¨}{\theta },$
$\frac{\partial T}{\partial \theta }=0,$
$\frac{\partial U}{\partial \theta }=-{K}_{1}l\left({y}_{st}-l{\theta }_{st}\right)+{K}_{2}l\left({y}_{st}+l{\theta }_{st}\right)-\left({K}_{1}l-{K}_{2}l\right)y+\left({K}_{1}{l}^{2}+{K}_{2}{l}^{2}\
right)\theta ,$
$\frac{\partial }{\partial \stackrel{˙}{\theta }}\left(\frac{1}{2}{C}_{1}{y}^{2}+\frac{1}{2}{C}_{2}{y}^{2}+\frac{1}{2}{C}_{\theta }{\stackrel{˙}{\theta }}^{2}\right)={C}_{\theta }\stackrel{˙}{\theta
Differential equations of vibration can be represented based on Lagrange equation:
$\left\{\begin{array}{l}\frac{d}{dt}\frac{\partial T}{\partial \stackrel{˙}{y}}-\frac{\partial T}{\partial y}+\frac{\partial U}{\partial y}+\frac{\partial D}{\partial \stackrel{˙}{y}}=Q\left(y\
right),\\ \frac{d}{dt}\frac{\partial T}{\partial \stackrel{˙}{\theta }}-\frac{\partial T}{\partial \theta }+\frac{\partial U}{\partial \theta }+\frac{\partial D}{\partial \stackrel{˙}{\theta }}=0.\
Substitute Eqs. (5)-(12) into Eq. (13), we can get damped vibration equation of the vibration body expressed as:
$\left\{\begin{array}{l}{M}_{0}\stackrel{¨}{y}+{K}_{1}\left({y}_{st}-l{\theta }_{st}\right)+{K}_{2}\left({y}_{st}+l{\theta }_{st}\right)-{M}_{0}g+\left({K}_{1}+{K}_{2}\right)y\\ \mathrm{}\mathrm{}\
mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}-\left({K}_{1}l-{K}_{2}l\right)\theta +\left({C}_{1}+{C}_{2}\right)\stackrel{˙}{y}=2m{\omega }^{2}r\mathrm{s}\mathrm{i}\mathrm
{n}\sigma +2m\stackrel{\cdot }{\omega }rc\mathrm{o}\mathrm{s}\sigma +P-F,\\ I\stackrel{¨}{\theta }-{K}_{1}l\left({y}_{st}-l{\theta }_{st}\right)+{K}_{2}l\left({y}_{st}+l{\theta }_{st}\right)-\left
({K}_{1}l-{K}_{2}l\right)y+\left({K}_{1}{l}^{2}+{K}_{2}{l}^{2}\right)\theta \\ \mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}+{C}_{\theta }\theta =0.\end{array}\right\$
Considering factors of ${M}_{0}g$ and preloaded compression force, as the same as static elastic force of the isolator, two isolator centroid moments of static elastic shall be zero. It can be
expressed as:
$\left\{\begin{array}{l}{M}_{0}g={K}_{1}\left({y}_{st}-l{\theta }_{st}\right)+{K}_{2}\left({y}_{st}+l{\theta }_{st}\right),\\ -{K}_{1}l\left({y}_{st}-l{\theta }_{st}\right)+{K}_{2}l\left({y}_{st}+l{\
theta }_{st}\right)=0.\end{array}\right\$
Substitute Eq. (15) into (14), we can get Eq. (16) which is the differential equation of damping forced vibration. Kinetic equation can be used for analysis – the vibration equation in direction $y$
$\left\{\begin{array}{l}{M}_{0}\stackrel{¨}{y}+\left({C}_{1}+{C}_{2}\right)\stackrel{˙}{y}+\left({K}_{1}+{K}_{2}\right)y-\left({K}_{1}l-{K}_{2}l\right)\theta +F-P\\ \mathrm{}\mathrm{}\mathrm{}\mathrm
{}\mathrm{}\mathrm{}\mathrm{}=2m{\omega }^{2}r\mathrm{s}\mathrm{i}\mathrm{n}\sigma +2m\stackrel{˙}{\omega }r\mathrm{c}\mathrm{o}\mathrm{s}\sigma ,\\ I\stackrel{¨}{\theta }+{C}_{\theta }\stackrel{˙}{\
theta }-\left({K}_{1}l-{K}_{2}l\right)y+\left({K}_{1}{l}^{2}+{K}_{2}{l}^{2}\right)\theta =0.\end{array}\right\$
Dual-motor drive system inertia comes from two aspects: the eccentric rotation of the inertial force generated by the steady state and the inertia resultant force caused by the eccentric shaft
angular acceleration changes. These two forces together constitute the exciting force excitation system. Thus, the sound frequency vibration motor is significantly vulnerable to the impact of
instability. The bottom vibration load will feedback reversely with the changing of hydraulic flow reversely.
According to the vibration equation rotor dynamics equation, the single eccentric system can be expressed as:
$m{r}^{2}\stackrel{¨}{\sigma }+C\stackrel{˙}{\sigma }+mr\stackrel{¨}{y}\mathrm{c}\mathrm{o}\mathrm{s}\sigma +mgr\mathrm{c}\mathrm{o}\mathrm{s}\sigma ={T}_{m}.$
In the above formula, the first term on the left is a rotary moment of inertia; the second axis is the friction torque; the third inertial is the torque generated by the moment of inertia force in
the $y$ direction; and the fourth is the gravity eccentric shaft torque for each centroid.
As a system, vibrating body can transform the hydraulic energy into the excitation force as an output. The formulae for calculating the hydraulic motor power can be expressed as:
${T}_{m}n={\eta }_{m}\mathrm{\Delta }PQ.$
Eq. (18) can be explained based on the characteristic of hydraulic motor. Eccentric system is derived by hydraulic displacement motor. Pressure and flow are the two parameters to represent the
hydraulic energy of motor input. Apart from a small portion of mechanical energy loss, all the rest can be converted into mechanical energy output, which can drive the eccentric shaft rotate. Since
hydraulic motor has the minimum displacement of 0.49 cm^3/r, it can work with high speed of 12000 rpm, meeting requirement of a high-speed hydraulic motor.
3.2. Models for finite element simulation
Vibration head is structurally complex. In order to facilitate simulation, structures of the sonic frequency vibration are simplified into three components: support housing, vibrating body (main
vibration body) and isolation. The centrifugal eccentric structure can be directly replaced by the generating load. It is defined the supporting frame and the vibrating body as 45# steel material,
with elastic modulus of 2.1E11, Poisson’s ratio of 0.3, and mass density of 7850 kg/m^3. The type of the 3D solid model is applied to the support plate, which is divided by a free smart grid, setting
the precision to the fourth. The solid 185 is selected as element type for the finite element model. The final number of the nodes and units are 101,042 and 432,499 respectively. The combination of
spring – damper unit is used to simulate the vibration isolator whose mass is so small to be ignored. Each isolator is replaced by four springs, each of whom has the spring stiffness parameter $k=$
2.2E6, and damping 20 Ns/m.
During the working process of sonic drilling, the support frame is connected with the guide, where mini-range but zero displacement constraint is employed; the vibrator mainly works through the
vertical movement in direction $Y$ due to the force generated by eccentricity. Therefore, the constraints in direction $X$ and $Z$ are employed on both sides of the midline. There are many small
structures and features in vibration head, such as threaded holes, bosses and chamfers. It therefore will take up a lot of computing resources when calculating, which will make the meshing quality
worse. Therefore, they have been simplified off as is shown in Fig. 6.
Fig. 6Finite element models
The overall modal analysis is made in this paper. During the audio rig works, the support is connected with the rail, and all directions at zero displacement constraints are imposed. Due to the
presence of up and down of the vibrating body, constraints are imposed on both left and right sides of the center line $X$ and $Z$ displacement.
3.3. Simulation analysis
The modal analysis vibration system can determine the system’s natural frequencies and mode shapes, applying the maximum 90 KN of a sinusoidal variation to the vibration body. The first four natural
frequencies are 15.71 Hz, 56.19 Hz, 67.40 Hz, 505.3 Hz, and the first simulation of four main modes is selected.
The simulation results are shown in Fig. 7. (1) The first three vibration frequencies are 15.71 Hz, 56.19 Hz and 67.40 Hz respectively, which are within the operating frequency of the vibrating body.
The greatest vibration frequency can be used in drilling. (2) Because there are differences on both sides of vibration isolation system, one end of the vibrating body $X$-direction displacement is
maximum while minimum on the other end when the first-order resonance occurs. (3) Vibration are presented in $X$, $Y$ and $Z$ directions, but $Z$ has the main vibration. (4) The fourth-order
oscillation mode is mainly caused by vibration of the support housing, whereby vibration of the support body for increasing stiffness can be avoided or reduced. (5) Vibration system mainly occurs on
the vibrating body, and the support frame vibration is tiny.
Fig. 7Form four-order vibration modes of the vibration head
4. Test of the vibration head
The results of finite element simulation show that the maximum vibration force reaches up to 18 tons, which has met the design requirement. The resonance is likely to occur near the natural
frequency, which can be fully used in the field work. At the same time, the anticipated force of the isolator has been improved, reducing the difference between the two isolation structures. Based on
this, the physical prototype is developed to test the vibration.
4.1. Experiment methods
The operational modal analysis (OMA) is used to control system of the hydraulic motor driven eccentric rotation. An accelerometer is used as a vibration sensor. In the working conditions, many key
points of the acceleration signal are measured. The FFT spectrum analysis and modal parameter identification are used to get the natural frequency, vibration damping ratio, and other parameters of
the structure. The results of the vibration analysis provide a good reference for subsequent structural optimization. There are four key measurement points on each plane vibrating body and bracket
arrangement. There are a total of eight points, each of which is measured in the same direction.
Fig. 8Layout diagram of the measuring points
In the process of the acoustic vibration drill, it has three phases: starting, periodic vibration, and stopping. Among them, the starting and stopping phases are both transitional. During the smooth
operation phase, the exciting force is characterized by periodicity. The flow rate of the hydraulic motor is adjusted by changing size of the valve port, causing the change of exciting force. The
vibration test is made by changing the excitation forces under different speed. Compared support acceleration with vibrating body, the vibration characteristics of the vibrating body is identified.
Research isolation effect and the modal analysis are also carried out.
Fig. 9Time domain signal during sonic drilling
Fig. 10Signal time-frequency distribution
4.2. Vibration signal analysis
Fig. 9 shows measuring point 1, 2, and vibration signal measuring points 3 and 4. The vibration is characterized by the superimposition of different periodic signals. The vibration amplitude of
measuring point 1 and measuring point 4 is slightly larger than points 2 and 3, which agrees with the results obtained from stimulation.
Fig. 10 is a spectrum diagram of measuring point 1, the measuring point 2, time-domain signal measuring point 3 and the measuring point 4 by FFT analysis. From the Fig. 9, the first, second, third
and fourth natural frequencies are 14.8 Hz, 57 Hz, 69 Hz and 510 Hz respectively. The experimental results are consistent with the simulation ones. The simulation results are proved to be correct. It
also proves that when the vibration frequency is 57-69 Hz, it will generate resonance. Also, when the frequency is 150-200 Hz, there will be no resonance generated.
5. Conclusion
1) The virtual prototyping technology is used to dynamic design the audio head vibration. The results are amended on the basis the method of modal analysis, which has a convenient, high design
quality and short development cycle characteristics.
2) Design of the main vibration of the vibration energy is concentrated in the longitudinal direction of the head. The maximum excitation force of the designed vibration head reaches 18 tons,
achieving the expected goal.
3) The natural frequency of vibration of the head is concentrated in the vicinity of 14.8 Hz, 57 Hz and 69 Hz, around which the vibration energy can be obtained using resonance. When the frequency of
vibration reaches the maximum, the resonance will not occur, ensuring the vibration head’s security.
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About this article
09 February 2015
Mechanical vibrations and applications
sonic vibration head
model analysis
This work is supported by National Natural Science Foundation of China (No. 51004086), the Fundamental Research Funds for the Central Universities (No. 2652015059, 2652015061), the Beijing Higher
Education Young Elite Teacher Project (Grant No. YETP0645) and Beijing Organization Department Outstanding Talented Person Project (No. 2013D009015000002). Meanwhile, great thanks also go to former
researchers for their excellent works, which give great help for our academic study.
Copyright © 2015 JVE International Ltd.
This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is
properly cited. | {"url":"https://yedikuyular.org/article/design-and-model-analysis-of-the-sonic-vibration-head","timestamp":"2024-11-05T14:13:13Z","content_type":"text/html","content_length":"125727","record_id":"<urn:uuid:3d1ff8cb-4553-43fc-9e4b-bb4051cee045>","cc-path":"CC-MAIN-2024-46/segments/1730477027881.88/warc/CC-MAIN-20241105114407-20241105144407-00894.warc.gz"} |
Paper 3, Section I, A
(a) Prove that the real and imaginary parts of a complex differentiable function are harmonic.
(b) Find the most general harmonic polynomial of the form
$u(x, y)=a x^{3}+b x^{2} y+c x y^{2}+d y^{3}$
where $a, b, c, d, x$ and $y$ are real.
(c) Write down a complex analytic function of $z=x+i y$ of which $u(x, y)$ is the real part. | {"url":"https://questions.tripos.org/part-ib/2010-12/","timestamp":"2024-11-09T23:02:24Z","content_type":"text/html","content_length":"12931","record_id":"<urn:uuid:cd521050-4bd6-49b0-9494-ebb691ae1d75>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.10/warc/CC-MAIN-20241109214337-20241110004337-00169.warc.gz"} |
What is the derivative of f(t) = (2t-3te^t, 2t^2+3t ) ? | HIX Tutor
What is the derivative of #f(t) = (2t-3te^t, 2t^2+3t ) #?
Answer 1
$\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{4 t + 3}{2 - 3 t {e}^{t} - 3 {e}^{t}}$
Given #f(t)=(2t-3te^t, 2t^2+3t)#
#x=2t-3te^t# and #y=2t^2+3t#
solve for #dx/dt# and #dy/dt# then #dy/dx=(dy/dt)/(dx/dt)#
Solve for #dx/dt#
Solve for #dy/dt#
Solve now for #dy/dx#
God bless...I hope the explanation is useful.
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Answer from HIX Tutor
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
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• Readily available 24/7 | {"url":"https://tutor.hix.ai/question/what-is-the-derivative-of-f-t-2t-3te-t-2t-2-3t-8f9afa1f29","timestamp":"2024-11-02T21:09:04Z","content_type":"text/html","content_length":"569922","record_id":"<urn:uuid:2cbef817-2d2f-4a4e-9f3a-4cc2dc74b23b>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00697.warc.gz"} |
Thermal Model Eon (v,i,T,R,Rg) (Effect of Rg in Switching losses)
I think this might be a very fundamental question but I could not find any discussion in the forum related to the topic…
I need some help to know the source of the Eon Formula shown in the following video tutorial (figure attached):
At minute 8:30 it is mentioned that we can add a formula to our Turn-On losses thermal model to include the effect of Rg.
I would like to know what is the logic/reasoning behind that expression. It looks like “lookup()” is taking information from the custom table EonVsRg (defined at minute 7:40 to 8:22 ) but I would
like to understand exactly what is that expression doing. Finally, there is the last term “(1/19.68)” which I also would like to know from where it came from. Is the (1/19.68) a parameter that we can
get from the manufacturer datasheet?
Hope I can hear back from somebody soon.
From https://cms.wolfspeed.com/app/uploads/2020/12/C3M0280090J.pdf, you will find that in Figure 25 there is the attached plot for Rg vs. E:
I would probably use a simple formula for the gate resistance dependence on turn-off losses since that is basically a linear relationship. So, you could find the slope of the Eoff line, e.g., m =
(y_max - y_min)/(x_max - x_min) ~= (7.9 - 4.0)/(20 - 2.5) = 0.22 uJ/Ohm. Then, your equation would be E+(m*(Rgoff - Rgoff_nominal)) or E+(0.22*(Rgoff-2.5)).
As for your question on how exactly the EonvsRg custom lookup table and formula in our video were developed, it isn’t shown in the video directly, but 19.68 is the value of Eon at 2.5 Ohms (the
nominal value also for the Eon plots in the data sheet). So for the equation Elookup(‘EonvsRg’,Rg)(1/19.68):
• E is the lookup table data itself, and everything after that is a scaler
• lookup(‘EonvsRg’,Rg) provides the user-specified Rg value to the custom table named EonvsRg that is saved in the thermal description and is used to determine what the loss should be. You also
have to divide by the loss value at the nominal Rg (19.68 uJ) to get the multiplier factor correct for the new Rg.
You might choose to use this approach from the video for Eon in this case, as the behavior is a non-linear fitting. Further, you can just scale your entire data set based on the plot itself if you
are only interested in simulating at one Rg value. Note that having the equation really only helps if you plan to try out the effects of different Rg values, but if you know the multiplier it’s easy
to include that effect just as a scaling coefficient.
These tips are useful for other manufacturers as well, by the way. As for what Wolfspeed actually does for the Rg dependence in all of the PLECS thermal descriptions models that you can download from
their website, they use a more complex higher order fit. But again, I think that would certainly be overkill for the Eoff behavior at least in this case.
Let me know if you have any questions. | {"url":"https://forum.plexim.com/t/thermal-model-eon-v-i-t-r-rg-effect-of-rg-in-switching-losses/1064","timestamp":"2024-11-13T19:06:17Z","content_type":"text/html","content_length":"29016","record_id":"<urn:uuid:cd57443d-2ecf-4613-aa4f-e745d2cd8768>","cc-path":"CC-MAIN-2024-46/segments/1730477028387.69/warc/CC-MAIN-20241113171551-20241113201551-00099.warc.gz"} |
An Improved Long Short-Term Memory Neural Network for Macroeconomic Forecast
Referring to China Macroeconomic Indicators, the scale and quality of China’s macroeconomy can be comprehensively measured in terms of industry, fixed asset investment, domestic commerce, total
import and export, utilization of foreign capital, public finance, finance, securities, and prices.
Figure 1 presents the curve of China’s macroeconomic leading index. Considering these metrics of macroeconomy, a scientific hierarchical EIS for macroeconomy was established in the principles of
comprehensive, comparability and operability. The proposed EIS contains 9 primary indices and 31 secondary indices.
Figure 1. The curve of China’s macroeconomic leading index
Layer 1 (goal):
C={macrocosmic evaluation}
Layer 2 (primary indices):
C={C[1], C[2], C[3], C[4], C[5], C[6], C[7], C[8], C[9]}=industrial output, fixed asset investment, domestic commerce, total import and export, utilization of foreign capital, public finance,
finance, securities, prices};
Layer 3 (secondary indices):
C[1]={C[11], C[12], C[13], C[14], C[15], C[16], C[17]}={value added of industrial enterprises above designated size, extractive industries, manufacturing, electricity, gas and water production and
supply, state holding enterprises, joint-stock cooperative enterprises, enterprises funded by foreign investors or investors from Hong Kong, Macao, and Taiwan};
C[2]={C[21], C[22], C[23]}={fixed asset investment, state-owned investment, investment in real estate development};
C[3]={C[31], C[32], C[33], C[34], C[35]}={total retail sales of consumer goods, urban employment rate, rural employment rate, retail income, food service income};
C[4]={C[41], C[42]}={export value, import value};
C[5]={C[51]}={actual foreign direct investment};
C[6]={C[61], C[62], C[63]}={fiscal revenue (excluding debt revenue), various taxes, fiscal expenditure (excluding debt expenditure)}
C[7]={C[71], C[72], C[73], C[74], C[75], C[76]}={currency and quasi-currency (M2), currency (M1), cash flow (M0), deposit balance of financial institutions, household deposit, loan balance of
financial institutions};
C[8]={C[81], C[82]}={Shanghai Stock Exchange Composite Index, Shenzhen Stock Exchange Component Index}
C[9]={C[91], C[92], C[93], C[94]}={general consumer price index, retail price index of domestic products, retail price index of export products, retail price index of import products}.
Considering the types of evaluation indices, the Spearman correlation coefficient, which reflects the dependence between the two evaluation datasets, was chosen to measure the correlation between the
degree of linear correlation between the evaluation indices.
Let C[i] and C[j] be two primary index datasets, both of which contain m index data. First, the index data in the two datasets were sorted in descending order separately, and replaced with the data
with the corresponding rankings. The Spearman correlation coefficient between C[i] and C[j] can be calculated by:
$s=\frac{\sum{\left( {{U}_{i}}-\bar{U} \right)\left( {{V}_{i}}-\bar{V} \right)}}{\sqrt{\sum{{{\left( {{U}_{i}}-\bar{U} \right)}^{2}}\sum{{{\left( {{V}_{i}}-\bar{V} \right)}^{2}}}}}}$ (1)
where, U[i] and V[i] are the rankings of primary index data; U and $\bar{V}$ are the means of the corresponding ranking data. In actual application, the Spearman correlation coefficient can be
calculated based on the difference e[i] of each pair of index data in C[i] and C[j] in the corresponding ranking data:
$s=1-\frac{\eta \sum{e_{i}^{2}}}{m({{m}^{2}}-1)}$ (2)
Figure 2. The weight distribution of the index combination
Note: IO is Industrial output; GCPI is General consumer price index; CQC is Currency and quasi-currency; CF is Cash flow; ER is Employment rate; DC is Domestic commerce; FR is Fiscal revenue; TI is
Total import; TE is Total export; VIEDS is Value added of industrial enterprises above designated size; RPIP is Retail price indices of products; IRED is Investment in real estate development; FE is
Fiscal expenditure; SECI is Stock exchange composite indices.
The Spearman correlation coefficient obtained by formula (2) characterizes the directions of primary index datasets C[i] and C[j]. Suppose C[i] is an independent macroeconomic evaluation index, and C
[j] is a dependent macroeconomic evaluation index. If the secondary index data under C[i] increase, and if those under C[j] also increase, then the Spearman correlation coefficient between the two
datasets is positive; otherwise, the coefficient is negative. If the coefficient is 1, then the two datasets have a completely monotonous correlation; if the coefficient is 0, then the two datasets
have no correlation. Figure 2 shows the weight distribution of the index combination obtained through correlation analysis.
The next is to analyze the Granger causality between the two primary index datasets C[i] and C[j]. During the prediction of C[j], C[i] has an impact on the change law of C[j], if C[j] can be
predicted more effectively based on the historical information of C[i] and C[j] than based on the historical information of C[j] alone. In this case, C[i] is the Granger cause of C[j]. The flow of
the Granger causality test under this condition is explained in Figure 3.
Figure 3. The flow of Granger causality test
The missing items in macroeconomic index data are usually interpolated or deleted. Here, the missing items in secondary index data are complemented through cubic spline interpolation. Let {C[ij][1],C
[ij][2],…,C[ijp]} be the data of secondary index dataset C[ij]. Depending on the actual situation, an independent variable time series A={a[1],a[2],…,a[p]} was added to dataset C[ij]. Taking data in
A as boundaries, dataset C[ij] was split into p-1 intervals. Then, the cubic spline interpolation function must meet the following conditions:
(1) The function is continuous at the boundary of adjacent intervals, passing through every data in the time series:
$Splin{{e}_{k}}({{a}_{k}})={{C}_{ijk}}$ (3)
$Splin{{e}_{k}}({{a}_{k+1}})={{C}_{ijk+1}}$ (4)
(2) The curves of the first- and second-order partial derivatives of the function are smooth and continuous in each interval [a[k], a[k][+1]]:
$Splin{{{e}'}_{k}}({{a}_{k+1}})=Splin{{{e}'}_{k+1}}({{a}_{k+1}})$ (5)
$Splin{{{e}''}_{k}}({{a}_{k+1}})=Splin{{{e}''}_{k+1}}({{a}_{k+1}})$ (6)
The cubic polynomial corresponding to each interval can be expressed as:
$ & Splin{{e}_{k}}(a)={{b}_{1k}}{{\left( a-{{a}_{i}} \right)}^{3}}+{{b}_{2k}}{{\left( a-{{a}_{i}} \right)}^{2}} \\ & \text{ }+{{b}_{3k}}\left( a-{{a}_{i}} \right)+{{b}_{4k}} \\$
where, b[1][k], b[2][k], b[3][k], and b[4][k] contain a total of 4 (p-1) unknown polynomial coefficients.
(3) The function needs to meet the natural boundary conditions, i.e., the second-order derivatives are zero at the left and right ends of the time series:
$Splin{e}''({{a}_{1}})=Splin{e}''({{a}_{m}})=0$ (8)
The function also needs to meet fixed boundary conditions, i.e. the second-order derivatives c[1] and c[2] at the left and right ends of the time series are fixed constant values:
$Splin{e}''({{a}_{1}})={{c}_{1}}$ (9)
$Splin{e}''({{a}_{m}})={{c}_{2}}$ (10)
By selecting a boundary condition and solving the 4 (p-1) equations, all unknown polynomial coefficients could be obtained, and the cubic spline interpolation function could be determined for each
interval. Then, the interpolation results can be obtained by substituting the independent variable value that corresponds to each missing item of the index data was substituted to the corresponding
In this paper, the LSTM neural network is improved to forecast the macroeconomy. The network was trained entirely by time series data. Considering the time scale difference between time series data
of the evaluation indices, this paper adopts the Denton method to convert the low-frequency index data to high-frequency format. Let H be the high-frequency series converted from a low-frequency
series L[ij]={L[ij][1],L[ij][2],…,L[ijp]} of secondary indices for macroeconomy, and H' be another high-frequency series with a similar growth rate and the same number n of index data as H. The
Denton method needs to satisfy the following constraint:
${{L}_{ijk}}=\sum\limits_{l={{s}_{k}}}^{{{e}_{k}}}{\varepsilon {{H}_{l}}t}$ (11)
where, t is the time point of low-frequency series; s[k] and e[k] are the starting and end points of a time interval in high- and low-frequency time series, respectively. The value of constant c must
be selected based on the type of data H[l] in high-frequency time series H. The minimum penalty function can be expressed by:
$\underset{H}{\mathop{\min PF}}\,\left( H,{H}' \right)={{\sum\limits_{l=2}^{n}{\left( \frac{{{H}_{l}}}{{{{{H}'}}_{l}}}-\frac{{{H}_{l-1}}}{{{{{H}'}}_{l-1}}} \right)}}^{\text{2}}}$ (12) | {"url":"https://www.iieta.org/journals/ria/paper/10.18280/ria.340507","timestamp":"2024-11-08T13:47:02Z","content_type":"text/html","content_length":"98623","record_id":"<urn:uuid:e2518030-a209-4dfa-a4da-3bc3c30e50e8>","cc-path":"CC-MAIN-2024-46/segments/1730477028067.32/warc/CC-MAIN-20241108133114-20241108163114-00431.warc.gz"} |
States and State Functions - PSIBERGStates and State Functions - PSIBERG
A group of properties/measurables that have the success of reproducing the given condition(s) of a system on re-attainment of their value(s) each time are termed state functions. This means that
whenever the values of measurable properties are reproduced, the state of the system is regenerated.
Microscopic and Macroscopic states
The state of a system can be observed at both microscopic and macroscopic levels. The state function at a microscopic level is the wavefunction (ψ), however, at a macroscopic level, the state is
defined as a set of measurable properties which include temperature, pressure, volume, density, viscosity, etc. A change in any single value will result in a change in the macroscopic state of a
The microscopic state ad macroscopic states are interlinked through statistical mechanics where the combinations/permutations of microscopic states lead to the generation of macroscopic properties
through well-defined formulas e.g. number of microscopic states W defines entropy as S=klnW where k is the Boltzmann constant.
Microscopic vs Macroscopic State Functions
The microscopic and macroscopic state functions can be compared as:
Macroscopic states Microscopic states
They are defined by set of state functions Only one state function carries all the information
State functions include mass (m), number of particles (ni), Pressure (P), Temperature (T), Volume (V), Chemical composition (Xi), Entropy (S), The only state function is psi (Ψ)
all energy terms such as enthalpy (H), internal energy (U), Gibbs free energy (G), Helmholtz free energy (A)
Change in any of above mentioned property can result in change in state of the system Change in (Ψ) means the change in state of the system
Microscopic state function (Ψ) can not be calculated from macroscopic state functions Macroscopic state functions of the system can be
calculated by applying the operator on (Ψ)
They describe classical mechanics They describe quantum mechanics
These are numerical values which may have mathematical correlations These are mathematical functions
They do not tell us about microscopic properties They give probability functions
No operators are required to operate on macroscopic states
They only yield macroscopic information They yield both macroscopic and microscopic
Changes in States
A change in state can be brought about by nuclear, chemical, or physical processes. These processes are accompanied by certain changes and effects, which differ them from one another.
Nuclear processes
Nuclear processes bring changes in the nuclear states of atoms leading to nuclear reactions. The atomic composition/nature of atoms changes, as in the following aspects.
Examples of nuclear processes are fission and fusion reactions.
Chemical processes
The chemical processes bring the change in electronic composition around the nucleus leading to the formation of new molecules. Orbitals’ realignment and changes in primary bonds also take place.
Examples of chemical processes are all chemical reactions such as; gaseous hydrogen+oxygen.
Physical processes
Physical processes correspond to non-compositional changes leading to changes in intermolecular parameters such as bond lengths, bond angles, and vibrational rotational energy of the intact
composition. This may lead to some visual change of state e.g. Temperature rise changes ice to water followed by vapor formation.
Such changes can lead to phase change or phase transitions. (Solid→Liquid→Gas) phase transitions are physical processes.
Concepts Berg
How is internal energy a state function?
Internal energy is a combination of all attractive and repulsive forces existing in any substance. The net effect of all primary interactions (bonds) and secondary interactions (intermolecular
forces) defines the internal energy of a system. During any reaction, the combinational values of these parameters change which results in a change in the state of that system. It depends on the
nature and strength of bonding/interaction, not on the route through which the change has occurred. So, the internal energy is a state function, not a path function.
Is enthalpy a state function?
Enthalpy is a state function just like internal energy as its effective (change) value depends on the state of the system, not the path used to bring that change.
What is the difference between state and state function?
A state is a set of defined state functions (properties). Each contributing property is termed a state function.
Is heat capacity a state function?
Heat capacity is a state function.
Is energy a state function?
Energy is a state function because its value depends on the current state of a system.
Is work a path function or a state function?
Work is a path function because the frictional and other losses associated with the path also define the work being done to bring the change. Therefore, the value of work depends on the initial and
final states as well as on the path/route adopted to bring that change. | {"url":"https://psiberg.com/states-and-state-functions/","timestamp":"2024-11-04T23:58:42Z","content_type":"text/html","content_length":"121081","record_id":"<urn:uuid:195f1612-7645-4778-beeb-39ba1a0c067a>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.84/warc/CC-MAIN-20241104225856-20241105015856-00755.warc.gz"} |
Kursöversikt för MVE465 Linjär algebra och analys fortsättning
Dataintrång- Sociala medier
Equações diferenciais são equações que incluem tanto uma função quanto sua derivada (ou derivadas de ordem superior). Por exemplo, y=y' é uma equação diferencial. Aprenda a encontrar e representar
soluções de equações diferenciais básicas. Homojen diferansiyel denklem konusuna bir başlangıç yapalım. Homojen sözcüğünü bu kelimeyi günlük hayatta da kullanıyoruz. Sütün içindeki yağ eşit olarak
mesela sütün içine dağıldıysa, yayıldıysa..
The Separable differential equations exercise appears under the Differential equations Math section on Khan Academy. This exercise shows how to separate the s from the s on two different sides of the
equation. There are six types of problems in this exercise: Which of the following is the solution to the differential equation: The student is asked to find the solution to the differential Topics
covered in a first year course in differential equations. Need to understand basic differentiation and integration from Calculus playlist before starting here. Differential Equations Khan Academy
en este vídeo vamos a ver ecuaciones diferenciales de primer grado pero del tipo de mujeres homogéneas homogéneas como cuando decimos que la leche es una substancia homogénea si la leche es una
sustancia homogénea porque todas sus partes están bien distribuidas aunque no sé que tengan que ver la leche con las ecuaciones diferenciales homogéneas pero bueno vamos a ver aquí ecuaciones I
noticed the differential equations lectures stop after the Laplace Transformation sections. My class, and many other's, continue onto power series solutions of differential equations.
There are six types of problems in this exercise: Find the sum of the two values: The user is asked to find the two values such that the first solution is a solution of the differential equation.
Use Practice this lesson yourself on KhanAcademy.org right now: https://www.khanacademy.org/math/differential-equations/first-order-differential-equations/differ Introduction to separable
differential equations.Watch the next lesson: https://www.khanacademy.org/math/differential-equations/first-order-differential-equa What a differential equation is and some terminology. Learn for
free about math, art, computer programming, economics, physics, chemistry, biology, medicine, finance, history, and more.
Khan academy Matematik/Matte 5 – Pluggakuten
/05/02 · Skapa egna quiz enkelt och snabbt som du kan Jamil Khan. Senior Lecturer, Associate Professor. Overview · Research Outputs · Projects · Activities.
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Euler's method | Differential equations| AP Calculus BC | Khan Academy. Watch later. Share. Copy link. Info What is a differential equation - YouTube.
Differential equations are equations that relate a function with one or more of its derivatives.
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Khan Academy tutorials have revolutionized the way that people think about teaching and learning online. This non-profit Debug is a casual, conversational interview show featuring the best developers
in the business about the amazing apps they make and why and how they make them. On this episode Andy Matuschak, creator of the Sparkle update framework for Mac, 10 Nov 2015 Hey buddy I'm looking for
the Net Ionic Equations of Chemistry subject of Khan Academy for that will you please help me here to get the it? 9 Feb 2021 what is the meaning of oxidation concept definition of. reduction of order
differential equations khan Be viewed as a very clever improvement on Solving Differential Equations.
If playback doesn't begin shortly, try restarting your device. To Khan Academy: You guys are doing great in almost all the section of mathematics. Thanks a lot to you. I also support this idea of
starting a partial differential equation course which may include solving techniques of initial and boundary value problems. Differential equations introduction Khan Academy. Differential equations
are equations that relate a function with one or more of its equation called "second-order constant coefficient linear differential equation". www.khanacademy.org The Separable differential equations
exercise appears under the Differential equations Math section on Khan Academy.This exercise shows how to separate the s from the s on two different sides of the equation..
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Calculus Khan Academy. Utbildning. 5,0 • 1 betyg. Topics covered in the first two or three semester of college calculus. Everything from limits to derivatives to Differential equations
(Differentialekvationer). Sen är det även ganska hög risk att innehållet i dessa områden på Khan Academy inte alls är Khan Academy Uploaded 10 years ago 2008-09-03. Using the method of undetermined
coefficients to solve nonhomogeneous linear differential equations.
9 Feb 2021 what is the meaning of oxidation concept definition of. reduction of order differential equations khan Be viewed as a very clever improvement on Solving Differential Equations.
Differential equations are solved by finding the function for which the equation holds true. Learning Objectives. Calculate the order Differential equation introduction | First order differential
equations | Khan Academy de Khan Academy il y a 6 ans 7 minutes et 50 secondes 1 867. 429 vues 23 Feb 2021 Lecture 25 Play Video.
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Differential equations are equations that relate a function with one or more of its derivatives. This means their solution is a function! Learn more in this video. Differential equations are
equations that include both a function and its derivative (or higher-order derivatives). For example, y=y' is a differential equation. Learn how to find and represent solutions of basic differential
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Föreläsningsanteckningar - Wehlou
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Ordinary Least Squares (OLS) DerivationOrdinary Least Squares (OLS) Derivation
Understanding Ordinary Least Squares (OLS)
The Ordinary Least Squares (OLS) method is a statistical method for estimating the parameters of a linear regression model.
It is one of the most commonly used methods in data science and machine learning. OLS is based on the concept of minimizing the sum of squared errors between observed values and predicted values. The
derivation of OLS involves taking partial derivatives with respect to each parameter, setting them equal to zero, and solving for the parameters in terms of other variables. This process can be used
to find an optimal solution for any linear regression problem.
By using this method, we can find an optimal solution that minimizes the sum of squared errors between observed values and predicted values. In the context of linear regression, OLS is used to
estimate the parameters of a model. In this case, the model is a line that represents an observed point in time and its corresponding predicted value for a selected variable.
If ƒ is any function relating x and y, then ƒ(x) = α + βx: ƒ(y) = α + βy. The term “ordinary least squares” comes from the fact that it minimizes squared errors taken as observations minus
Ordinary Least Square (OLS) Derivation
Consider a linear regression model with one independent variable: Y[i] = α̂ + β̂X[i] + e[i] where Y[i] is the response variable, X[i] is the independent variable, alpha hat (α̂) and beta hat (β̂) are the
coefficients, and e[i] is the error term.
The model is given by:
Y[i] = α̂ + β̂X[i] + e[i]
Where Y[i] is the response variable, X[i] is the independent variable, α̂ and β̂ are the coefficients to be estimated, and e[i] is the error term.
To estimate α̂ and β̂ using the OLS method, we want to minimize the sum of squared errors (SSE) between the predicted values Ŷ[i] and the actual values Y[i]:
Where n is the number of observations.
The predicted values Ŷ[i] are given by:
Ŷ[i] = α̂ + β̂X[i]
To find the values of α̂ and β̂ that minimize SSE, we take the partial derivatives of SSE with respect to α̂ and β̂ and set them equal to zero:
∂SSE/∂α̂ = -2 (Y[i] - α̂ - β̂X[i]) = 0 ∂SSE/∂β̂ = -2 X[i](Y[i] - α̂ - β̂X[i]) = 0
Solving these equations simultaneously, we obtain the following OLS estimators for α̂ and β̂:
α̂ = Ȳ – β̂X̄
β̂ = (X[i] – X̄)×(Y[i] – Ȳ) / (X[i] – X̄)^2
where X̄ and Ȳ are the sample means of X and Y, respectively.
Thus, the OLS estimates for α̂ and β̂ are given by:
α̂ = Ȳ – β̂X̄
β̂ = (X[i] – X̄)*(Y[i] – Ȳ) / (X[i] – X̄)^2
where Ȳ is the sample mean of Y, X̄ is the sample mean of X, and n is the number of observations.
Once we have estimated α̂ and β̂, we can use them to make predictions for new values of X by plugging them into the equation:
Ŷ = α̂ + β̂X
Properties of least square estimators using Gauss-Markov Theorem
The Gauss-Markov theorem provides a set of conditions under which the Ordinary Least Squares (OLS) estimator is the Best Linear Unbiased Estimator (BLUE). The properties of the OLS estimator can be
summarized as follows:
1. Unbiasedness: The OLS estimator is unbiased, which means that it has an expected value that is equal to the true population parameter. In other words, on average, the OLS estimator produces
estimates that are close to the true values of the parameters being estimated.
2. Efficiency: Among all linear unbiased estimators, the OLS estimator has the smallest variance. This means that it produces estimates that are more precise than any other linear unbiased estimator
and achieves the smallest possible mean squared error.
3. Consistency: As the sample size n approaches infinity, the OLS estimator approaches the true population parameter with probability 1. In other words, as the sample size becomes larger, the OLS
estimator becomes more and more accurate.
4. Normality: Under certain assumptions, the OLS estimator is normally distributed. This is useful for constructing confidence intervals and hypothesis tests for estimated parameters.
The Gauss-Markov theorem provides conditions under which the OLS estimator satisfies these properties. The conditions include:
1. Linearity: The regression model is linear in the parameters.
2. Strict exogeneity: The error term has a zero mean and is uncorrelated with the independent variables.
3. No perfect multicollinearity: There is no perfect linear relationship among the independent variables.
4. Homoscedasticity: The error term has constant variance.
5. Normality: The error term is normally distributed.
6. Sample size: The sample size is sufficiently large.
If these conditions are met, then the OLS estimator is the Best Linear Unbiased Estimator (BLUE), meaning it has the smallest variance among all unbiased linear estimators. In other words, the OLS
estimator is the most efficient and precise estimator of the population parameters in a linear regression model.
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Substructures and chains of models
Truth propogates through an elementary chain of models to the limit.
Another serialized installment from A Panorama of Logic, my book currently in progress. The book aims to become an introduction to topics in logic for philosophers, mathematicians, and computer
scientists. Follow along as new chapters are released every week.
One mathematical structure M is a subsubstructure of another N, written simply M ⊆ N, if the domain of M is contained in the domain of N, and the two structures agree on the interpretations of all
the relations, functions and constants on that domain. For example, here is a chain of substructures:
\(⟨ℕ,+,·,0,1,<⟩ ⊆ ⟨ℤ,+,·,0,1,<⟩ ⊆ ⟨ℚ,+,·,0,1,<⟩ ⊆ ⟨ℝ,+,·,0,1,<⟩. \)
This is a chain of substructures because every natural number is an integer, every integer is a rational number, every rational number is a real number and all these structures agree on addition and
multiplication on their common domains, on the meaning of 0 and 1, and on the order <.
Elementary substructures
A substructure M of a structure N is an elementary substructure, written M ≺ N, if M is a substructure of N and also they agree on the truth of every assertion, so that M ⊨ φ[a[1],...,a[n]] if and
only if N ⊨ φ[a[1],...,a[n]], for every a[1], ..., a[n] in M and every assertion φ in the language of these structures.
None of the substructures pictured in the chain of substructures above are elementary substructures, nor even elementarily equivalent. To see this, observe that amongst these structures, only ℝ has a
solution to x^2 = 2; only ℕ thinks x + y = 0 → x = 0; only ℚ is 2-divisible, but does not have √2; and only ℤ satisfies the negations of all those properties. So none of these substructures is an
elementary substructure.
Absoluteness of truth
Although an assertion can have a different truth value in a substructure than it does in the parent structure, as we have just observed, nevertheless in certain situations one can find an
agreement—some kinds of truth values will be absolute. Let us begin by showing that term evaluation is absolute between a substructure and its parent.
Lemma. Term evaluation is absolute between a substructure and its parent. That is, evaluating a term t at points a[1], ..., a[n] in a substructure M ⊆ N gives the same value as evaluating it in the
parent structure N,
\(t^M( a_1,...,a_n ) = t^N( a_1,...,a_n ). \)
Proof. We prove this by induction on terms. Assume we have a substructure M ⊆ N. The claim is true for constant symbols, c^M = c^N, simply because this is part of what it means to be a
substructure—constants must be interpreted the same in a substructure as in the parent structure. The claim is also true for variables, since we are using the same valuation [a[1],...,a[n]] in the
substructure as in the parent structure, so each variable x[i] is interpreted as a[i] in each case. Finally, we consider terms of the form f(t[1],...,t[k]) and assume inductively that the claim is
true for terms t[1],...,t[k. ]Observe that
\(\begin{eqnarray*} \bigl( f( t_1,...,t_k )\bigr )^M( a_1,...,a_n ) & = & f^M\bigl( t_1^M( a_1,...,a_n ),...,t_k^M( a_1,...,a_n )\bigr )\\ & = & f^N\bigl( t_1^N( a_1,...,a_n ),...,t_k^N( a_1,...,a_n
)\bigr )\\ & = & \bigl( f( t_1,...,t_k )\bigr )^N( a_1,...,a_n ). \end{eqnarray*} \)
We used the induction hypothesis that t[i]^M(a[1],...,a[n]) = t[i]^N(a[1],...,a[n]) in the second equality, together with the fact that f ^M agrees with f ^N on points in M by the definition of
substructure. Thus, by induction, we conclude that the claim holds for all terms. □
Absoluteness Theorem. Assume M is a substructure of N.
1. Quantifier-free truth is absolute between a substructure and its parent—the models agree on the truth of any quantifier-free assertion ψ at individuals a[1],...,a[n] from the substructure M:
\( M ⊨ ψ[a_1, ...,a_n]\qquad\text{ if and only if }\qquad N ⊨ ψ[a_1, ...,a_n]. \)
2. Existential assertions are upward absolute from a substructure to its parent—if ψ is quantifier-free, then
\(M ⊨ ( ∃x ψ )[a_1, ...,a_n]\qquad\text{ implies }\qquad N ⊨ ( ∃x ψ )[a_1, ...,a_n]. \)
3. Universal assertions are downward absolute from a parent structure to any substructure—if ψ is quantifier-free, then
\(N ⊨ ( ∀x ψ )[a_1, ...,a_n]\qquad\text{ implies }\qquad M ⊨ ( ∀x ψ )[a_1, ...,a_n]. \)
Proof. We know by the lemma above that term evaluation is absolute between a substructure M ⊆ N and its parent, and this implies that every equality assertion s = t of terms will have the same truth
value in M as in N. Similarly, every relational atomic assertion Rt[1]···t[k] will have the same truth value in M as in N, because the terms evaluate the same and similarly the relation in M agrees
with N on points in M. Thus, the claim of statement (1) is true for atomic assertions.
We may extend from the atomic assertions to all quantifier-free assertions by induction on formulas. If the truth equivalence of statement (1) holds for assertions φ and ψ, then it also holds, I
claim, for φ ∧ ψ, φ ∨ ψ, φ → ψ, φ ↔ ψ, and ¬φ.
Let me illustrate for conjunction:
\(\begin{eqnarray*} M ⊨ ( φ ∧ ψ )[a_1, ...,a_n] &\iff& M ⊨ φ[a_1, ...,a_n]\text{ and }M ⊨ ψ[a_1, ...,a_n]\\ &\iff& N ⊨ φ[a_1, ...,a_n]\text{ and }N ⊨ ψ[a_1, ...,a_n]\\ &\iff& N ⊨ ( φ ∧ ψ )[a_1,
...,a_n]. \end{eqnarray*} \)
First, we use the definition of satisfaction to break up the conjunction, and then we use the induction hypothesis to move from M to N, and finally we use the definition of satisfaction again to
reassemble the conjunction. The argument follows a similar pattern for all the other logical connectives. Since every quantifier-free assertion is built in this way from atomic assertions via logical
connectives, this establishes statement (1) for all quantifier-free assertions.
For statement (2), suppose that the substructure satisfies an existential statement M ⊨ (∃x ψ)[a[1],...,a[n]]. So there is an individual b in M for which M ⊨ ψ[a[1],...,a[n],b]. Since ψ is
quantifier-free, it follows from statement (1) that this is absolute to the parent model N ⊨ ψ[a[1],...,a[n],b] and consequently N ⊨ (∃x ψ)[a[1],...,a[n]], as desired.
The reader will prove statement (3) in the exercises. □
Elementary chains
Let us next consider the concept of a chain of models, a tower of structures M[n], each a substructure of the next.
\(M_0\quad ⊆ \quad M_1\quad ⊆ \quad M_2\quad ⊆ \quad ··· \)
The union or limit of such a chain is the model M = ∪[n] M[n], whose domain is the union of the domains of the models appearing in the chain, upon which we interpret the structural elements, the
functions, relations, and constants of the signature. These interpretations are well defined on limit model precisely because the models in the tower cohere with one another on this atomic structure.
An elementary chain is a special kind of chain, where each model is an elementary substructure of the next.
\( M_0\quad ≺ \quad M_1\quad ≺ \quad M_2\quad ≺ \quad ··· \)
In this case, we claim, the limit model is an elementary extension of all the models in the chain.
Elementary chain theorem. The limit model of an elementary chain is an elementary extension of every model in the chain.
\( M_0\quad ≺ \quad M_1\quad ≺ \quad M_2\quad ≺ \quad ··· \quad ≺ \quad M\)
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Market Equilibrium under Proportional Transaction Costs in a Stochastic Factor Model – Mihail Zervos (London School of Economics)
We consider an economy with two agents. Each of the two agents receives a random endowment flow. We model this cumulative flow as the stochastic integral of a deterministic function of the economy’s
state, which we model by means of a general Ito diffusion. Each of the two agents has mean-variance preferences with different risk-aversion coefficients. The two agents can also trade a risky asset.
We determine the agents’ optimal equilibrium trading strategies in the presence of proportional transaction costs. In particular, we derive a new free-boundary problem that provides the solution to
the agents’ optimal equilibrium problem. Furthermore, we derive the explicit solution to this free-boundary problem when the problem data is such that the frictionless optimiser is a strictly
increasing or a strictly increasing and then strictly decreasing function of the economy’s state. | {"url":"https://www.tilastotieteenkeskus.fi/en/ajankohtaista/events/tba-mihail-zervos-london-school-of-economics/","timestamp":"2024-11-05T21:58:18Z","content_type":"text/html","content_length":"46541","record_id":"<urn:uuid:7d6d6838-a541-454d-97cd-ba8022fc07fd>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00174.warc.gz"} |
The Lazy Economist
Ever wondered how your estimation of a linear function relates to the elasticities of the estimated model? I always seem to forget, especially if I have taken the logarithm on one or both sides of
the equation. Here are the four cases you can have:
The function has the following form (if you have more variables on the right hand side, this doesn’t change the story):
[math]Y=a + bX[/math]
The elasticity is given by:
[math]\epsilon= \frac{dY}{dX}\frac{X}{Y}=b\frac{X}{Y} [/math]
and the coefficient b is the change in Y from a unit increase in X.
… Read the rest “Elasticities in estimated linear models” | {"url":"https://blog.modelworks.ch/category/econometrics/","timestamp":"2024-11-13T15:56:51Z","content_type":"text/html","content_length":"39729","record_id":"<urn:uuid:e4c3a610-e90a-4311-aa9f-3841afabb842>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00562.warc.gz"} |
NCERT Solutions for Class 10 Maths PDF Download CBSE - हिन्दी वार्ता
NCERT Solutions for Class 10 Maths PDF Download CBSE
Class 10 Maths NCERT Solutions
here we have compiled NCERT solutions for all chapters of Class 10 mathematics.
Class 10 Maths NCERT Solutions Chapter wise Topics
Chapter 1 Real Numbers
Chapter 2 Polynomials
Chapter 3 Pair of Linear Equations in Two Variables
Chapter 4 Quadratic Equations
Chapter 5 Arithmetic Progressions
Chapter 6 Triangles
Chapter 7 Coordinate Geometry
Chapter 8 Introduction to Trigonometry
Chapter 9 Some Applications of Trigonometry
Chapter 10 Circles
Chapter 11 Constructions
Chapter 12 Areas Related to Circles
Chapter 13 Surface Areas and Volumes
Chapter 14 Statistics
Chapter 15 Probability
How to score 100 % marks in Maths
Many of us believe that to score high in Mathematics one should have an extraordinary brain. But this is not true. Maths is a subject which develops a rational thinking and a logical approach in a
student. It is quite difficult for a student to fall in love with Mathematics overnight but here are a few tips to enhance your Mathematics score.
1. Maintain a separate register for formulae, theories, and methods
This subject is all about formulae, theories, concepts and you are always suggested to keep them handy. You can read them even when you are on the go. This practice is really useful when you are
doing your last minute revision.
2. Find solutions yourself
It is good to go through various types of problems, but at the same time, you should also make sure that you solve them for yourself. It is easy to learn the theories and concepts but to learn their
application is not that simple. So, if you want to score full marks in Mathematics, you need to solve each question by yourself, at least 3 to 4 times.
3. Understand the Syllabus
Having a clear understanding of your syllabus and weight to various sections will definitely help you to decide how much time you should dedicate to each section. For example, if you are aware that
there will be a 5 mark question from a specific section, then you don’t have to spend too much time on such questions.
4. Determine the areas of Improvement
Realizing the areas where you need to focus more will definitely help you to score better. Solving sample papers, writing tests can help you to find those sections in which you need more practice so
that you can improve the scores.
Things to consider during Examination
5. Keep the exam paper clean
You should keep in mind that the examiner has to understand every step of your answer and in most of the cases he would not be having much time to spend on a single answer. Avoid overwriting and
cutting clean margins for rough work.
6. Answer in steps
Whenever you go through an answer, ensure to pay special attention to the steps that helped you to reach out to get the answer. You cannot simply write the answer and get full marks in Maths, so
simply paying attention to the figures is wastage of time. Instead of all this, learn to keep the steps involved. You will definitely secure some marks for each step.
7. Attempt the Familiar questions
Read the question paper thoroughly before you begin to solve the questions. It is very normal to get stuck on the questions which you are not aware of but simply remember that you have a stipulated
time to attend the paper. In order to score high in Maths, you must attend the questions you know and then move on to the unfamiliar ones.
8. Draw Graphs
Using graphs and figures can help you to score more marks if you make them with concentration and neatness. For this, you will need a ruler and a sharpened pencil. These are some of the simplest
things that you should not forget to carry as they play a vital role in fetching marks. To give your best performance in Class 10 Maths Board Exams, make sure you are not afraid of the subject.
Rather, make it a fun experience. The only thing that you should remember is that there are no shortcuts in Maths.
More tips for scoring full marks in maths in Class 10
Read your paper, carefully!
Start your exam with a question you are most comfortable with.
Solve problems in a stepwise method
Attempt all questions — even if partially. You do get points for attempting them and for every correct step .
Write a clean paper
Don’t scribble and scrap, unnecessarily. If need be, just strike it once and start writing in a new line
Keep tabs
This is with respect to formulae, rules and theorems. Also, stick to proper methodology.
Formatting, a must
Draw margins, underline the important steps and leave space after answering each question.
Perfect graphs = full marks!
Convert, if required, all the quantities to appropriate SI units.
Choose a correct scale, which helps to draw the graph accurately and of good size.
Join all the points of the graph so that it gives a proper line or a smooth curve.
Also, read correctly the values if required to be obtained from the graph.
You must be logged in to post a comment. | {"url":"https://hindivarta.com/ncert-solutions-for-class-10-maths-pdf/","timestamp":"2024-11-11T17:13:01Z","content_type":"text/html","content_length":"144868","record_id":"<urn:uuid:ae13f344-a5b2-4f3b-b5b6-4e9cf1ea35b7>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00244.warc.gz"} |
Analysis of a lap around Brands Hatch Indy (Pt. II) - The Answer is 27
Analysis of a lap around Brands Hatch Indy (Pt. II)
This entry owes very much to a challenge from the Spanish motorsport blog DeltaGap. There, Daniel provides us with a set of real data in order to determine in which sectors the delta time is greater.
I took up the challenge and worked out a solution which will help to understand where in a circuit a driver can improve. Is it in the slow corners or in the fast straights?
The dataset
To obtain the delta time, here I will use the same laps I did compare in
Analysis of a lap around Brands Hatch Indy (Pt. I)
. But instead of having time in the abscissa, I will have distance.
The fastest lap of the session was lap #39 (0:42.799), while lap #14 (0:42.864) was the second-fastest one.
The equations
As I’ve seen lately, most people will relate speed and time through the area below the curve. But once we look at the equations, we will realise that this is not the case. By definition, velocity
is the rate of change of position with respect to time,
$$ v = \frac{dx}{dt} $$
and thus, the time can be obtained by
$$ dt = \frac{dx}{v} $$
If we were to compute the area under the curve using Riemann sums, i.e., $\sum v\, dx$, we would not get time. That is, time is not related to the area underneath the curve.
Delta time
We can compute the time it takes to travel certain distance as
$$ t = \int\limits_{x_s}^{x_f} \frac{dx}{v(x)} $$
and the delta time between two laps as
$$ \Delta t = t_1 – t_0 = \int\limits_{x_s}^{x_f} \frac{dx}{v_1(x)} – \int\limits_{x_s}^{x_f} \frac{dx}{v_0(x)} $$
where $t_0$ is the reference lap time.
This equation can be simplified, and then
$$ \Delta t = \int\limits_{x_s}^{x_f} \frac{v_0(x) – v_1(x)}{v_1(x) \cdot v_0(x)} \, dx $$
What does this mean? In the numerator we’ve got speed difference, and in the denominator, something very close to speed squared. That is, for a given delta speed —let’s say 1 kph—, the slower the
sector of the circuit —the denominator decreases—, the bigger the delta time gets. I’d give up 1 kph in the straights in order to get 1 kph in the corners. Of course that will depend on the circuit.
For instance, Monza is a very fast circuit with long high speed sectors while Monaco is a slow one with lots of bends, thus requiring improvements on different areas.
Straights are for fast cars, turns are for fast drivers —Colin McRae.
As Colin McRae said, if you want to improve your car’s laptimes, work the turns with the driver, or just give him a faster car.
I am using Python —a free and open-source general-purpose, high-level programming language— to solve the equations. More precisely, SciPy, an open-source library for mathematics, science, and
from scipy.integrate import trapz, cumtrapz
The trapz function will give us the result of computing the integral using the trapezoidal rule.
trapz((v.39 - v.14)/(v.14 * v.39), x=v.index.values)
As the trapezoidal rule is subject to errors, the result will not be 100% accurate, though the closer spaced the points are the smaller the error. In fact, the computed delta time is 0.079 s while
the actual delta time at the end of the lap was of 0.065 s.
On the other hand, the cumtrapz function will give us the cumulative values of computing the integral. That is, we will get an array of cumulated delta time values that will give us the delta time in
any part of the circuit from the starting line onwards.
Again, errors are cumulative, pretty much like in dead reckoning navigation. Nonetheless, those errors are very small and can be considered acceptable, specially when we are looking for the sectors
with greater improvement potential. Those sectors sectors can be spotted as the ones with a steeper delta time curve. Oddly enough, the greatest variations in delta time are found around the slowest
corners, just as we asserted earlier.
Special thanks to Steve Barker for providing me with vast amounts of data, part of which I used for this blog entry. I would like to take the opportunity to thank Sean Pivek who also provided
valuable amounts of data and Craig Scarborough who contributed with a retweet. Many thanks.
[1] Brandshatch.co.uk, (2014). Brands Hatch – Circuit Information. [online] Available at: http://www.brandshatch.co.uk/circuit-information.aspx#circuitMap [Accessed 30 Oct. 2014]. | {"url":"https://theansweris27.com/analysis-of-a-lap-around-brands-hatch-indy-pt-ii/","timestamp":"2024-11-11T20:40:15Z","content_type":"text/html","content_length":"76952","record_id":"<urn:uuid:9b08b5b9-e83f-40a4-ae22-468460d1834f>","cc-path":"CC-MAIN-2024-46/segments/1730477028239.20/warc/CC-MAIN-20241111190758-20241111220758-00482.warc.gz"} |
linear least squares/mldivide for large matrices in parallel?
Accepted Answer
Edited: Edric Ellis on 8 Apr 2015
Commented: Sean de Wolski on 27 Jul 2016
I have a really large system to solve using linear least squares. The A matrix can have 2-3 million rows and 2000-3000 columns. The B matrix has same row size but with a single column.
I have access to a supercomputer, and I want to run the x = A\B (or) mldivide(A,B) command in parallel, since I can easily run out of RAM even on workstations with lots of memory.
Any ideas? I am able to run EIG and SVD without any issues in parallel, since I assume it is automatically parallelized by MATLAB. What about linear least squares? Suggestions outside of MATLAB are
also welcome. Thanks.
7 views (last 30 days)
linear least squares/mldivide for large matrices in parallel?
If you have access to a cluster of machines, you could use distributed arrays to solve the large system in parallel using the multiple memories. You'll need MATLAB Distributed Computing Server worker
licenses on the cluster, and Parallel Computing Toolbox on the client machine. Something like this:
A = distributed.rand(20000,2000);
b = sum(A, 2);
x = A\b;
3 Comments
David on 27 Jul 2016
Okay so how do I then load up my distributed matrix? It seems that distributed arrays don't support non scalar right hand assignment... do I really need to loop through the entire 200,000 x 20,000
array in my case one by one? Won't this be really slow compared with the vectorized matrix manipulations I was using before?
I tried stuffing the matrix and then distributing it, but its over my single thread RAM limit at this point.
Sean de Wolski on 27 Jul 2016
David, please ask this in a new question - the answer will end up being to use codistributed arrays inside of spmd.
More Answers (1)
Hi Arvind
Parallel computing helps you to use more amount of CPU to run your simulation in a shorter time. As well as I know, when you have memory problem, it does not help you.
What I can suggest you is that you can implement the "x=A\B" by your own code.
I mean that write the m-file to calculate this x = A\B. The only difference is that you have to save your data and delete another one when you do not need it to avoid Memory problem.
For example, to calculate the A\B, you need to calculate A^(-1). Thus, first, JUST load matrix A and calculate A^(-1) and then save that matrix as a matrix and delete matrix A (be cause you do not
need it anymore).
I hope it helps you. | {"url":"https://au.mathworks.com/matlabcentral/answers/196655-linear-least-squares-mldivide-for-large-matrices-in-parallel","timestamp":"2024-11-05T23:59:08Z","content_type":"text/html","content_length":"139792","record_id":"<urn:uuid:72e4646e-ca6b-41cc-962e-017db5910d6e>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00764.warc.gz"} |
How Many Grams Are In A Half Ounce? - November 6, 2024
A half ounce is a common unit of measurement used to measure weight. It is often used to measure small amounts of food, spices, and other ingredients. The question of how many grams are in a half
ounce is a common one, and the answer depends on the type of measurement being used. In the United States, a half ounce is equal to 14.17 grams, while in the United Kingdom, a half ounce is equal to
13.6 grams. This article will explain the difference between the two measurements and provide an easy way to convert between them.
How Many Grams Are In A Half Ounce
How to Convert Grams to Ounces: A Guide to Understanding Measurement Conversions
Are you looking to convert grams to ounces? You’ve come to the right place! Converting between different units of measurement can be tricky, but with a few simple steps, you’ll be a pro in no time.
Let’s get started!
First, it’s important to understand the difference between grams and ounces. A gram is a unit of mass in the metric system, while an ounce is a unit of mass in the imperial system. One gram is equal
to 0.03527396195 ounces.
Now that you know the conversion rate, you can easily convert grams to ounces. To do this, simply multiply the number of grams by 0.03527396195. For example, if you have 10 grams, you would multiply
10 by 0.03527396195 to get 0.3527396195 ounces.
It’s also possible to convert ounces to grams. To do this, simply divide the number of ounces by 0.03527396195. For example, if you have 0.5 ounces, you would divide 0.5 by 0.03527396195 to get
14.174762 grams.
Now that you know how to convert grams to ounces and vice versa, you’re ready to tackle any measurement conversion! With a little practice, you’ll be a pro in no time. Good luck!
The Difference Between a Half Ounce and a Full Ounce: What You Need to Know
Are you confused about the difference between a half ounce and a full ounce? Don’t worry, you’re not alone! Many people are unsure of the difference between these two measurements, but it’s actually
quite simple.
A half ounce is exactly half of a full ounce. In the United States, a half ounce is equal to 14.17 grams, while a full ounce is equal to 28.35 grams. This means that a half ounce is half the weight
of a full ounce.
Half ounces and full ounces are commonly used to measure weight, but they can also be used to measure volume. For example, a half ounce of liquid is equal to one tablespoon, while a full ounce is
equal to two tablespoons.
Half ounces and full ounces are also used to measure food. For example, a half ounce of cheese is equal to one slice, while a full ounce is equal to two slices.
So, now you know the difference between a half ounce and a full ounce. Knowing the difference between these two measurements can help you accurately measure ingredients when cooking or baking, or
when measuring out medication. So, the next time you’re unsure of the difference between a half ounce and a full ounce, just remember that a half ounce is half the weight of a full ounce, and that
it’s equal to one tablespoon of liquid or one slice of cheese.
How to Measure Half an Ounce of Ingredients: Tips for Accurate Measurements
Measuring half an ounce of ingredients accurately can be tricky, but with a few simple tips, you can get it right every time! Here are some tips for measuring half an ounce of ingredients accurately:
1. Use a kitchen scale. A kitchen scale is the most accurate way to measure half an ounce of ingredients. Place the ingredient on the scale and set it to ounces. Then, adjust the weight until it
reads 0.5 ounces.
2. Use measuring spoons. If you don’t have a kitchen scale, measuring spoons are the next best option. Measure out two teaspoons of the ingredient, which is equal to one tablespoon. Then, measure out
one tablespoon of the ingredient, which is equal to half an ounce.
3. Use a liquid measuring cup. If you’re measuring a liquid ingredient, use a liquid measuring cup. Fill the cup up to the 1/2 ounce mark.
4. Use a measuring cup. If you’re measuring a dry ingredient, use a measuring cup. Fill the cup up to the 1/2 ounce mark.
5. Use a tablespoon. If you’re measuring a dry ingredient, use a tablespoon. Fill the tablespoon up to the 1/2 ounce mark.
With these tips, you’ll be able to measure half an ounce of ingredients accurately every time!
What You Need to Know About Grams and Ounces: A Comprehensive Guide
Grams and ounces are two of the most commonly used units of measurement when it comes to weighing items. Whether you’re a baker, a chef, or just someone who likes to measure things, it’s important to
understand the difference between grams and ounces and how to convert between them.
So, what exactly is the difference between grams and ounces? A gram is a metric unit of mass, while an ounce is an imperial unit of mass. A gram is equal to 0.03527396195 ounces, while an ounce is
equal to 28.349523125 grams.
When it comes to measuring food, it’s important to understand the difference between dry and liquid measurements. Dry measurements are measured in ounces, while liquid measurements are measured in
milliliters or liters. For example, a cup of flour is measured in ounces, while a cup of milk is measured in milliliters or liters.
It’s also important to understand how to convert between grams and ounces. To convert from grams to ounces, simply multiply the number of grams by 0.03527396195. To convert from ounces to grams,
simply multiply the number of ounces by 28.349523125.
Finally, it’s important to understand the difference between weight and mass. Weight is a measure of the force of gravity on an object, while mass is a measure of the amount of matter in an object.
Weight is measured in pounds, while mass is measured in grams or ounces.
Grams and ounces are two of the most commonly used units of measurement when it comes to weighing items. Understanding the difference between them and how to convert between them is essential for
anyone who needs to measure items accurately. With this comprehensive guide, you’ll be able to measure items with confidence!
How to Calculate the Number of Grams in a Half Ounce: A Step-by-Step Guide
• Step 1: Gather Your Materials – Before you begin, make sure you have a calculator, a measuring cup, and a scale.
• Step 2: Measure Out Half an Ounce – Using your measuring cup, measure out half an ounce of whatever material you are trying to weigh.
• Step 3: Place the Material on the Scale – Place the material on the scale and make sure it is balanced.
• Step 4: Read the Weight – Read the weight of the material on the scale. This should be in ounces.
• Step 5: Convert Ounces to Grams – Using your calculator, convert the weight in ounces to grams. To do this, multiply the weight in ounces by 28.35. This will give you the weight in grams.
• Step 6: Calculate the Number of Grams – Once you have the weight in grams, you can calculate the number of grams in a half ounce. To do this, divide the weight in grams by two. This will give you
the number of grams in a half ounce.
And there you have it! You now know how to calculate the number of grams in a half ounce. Have fun experimenting with different materials and weights!
The Benefits of Knowing How Many Grams are in a Half Ounce: Why It Matters
Knowing how many grams are in a half ounce is an important part of understanding measurements and conversions. Whether you’re a baker, a chef, or just someone who likes to measure out ingredients for
recipes, it’s important to know how to convert between different units of measurement. Knowing how many grams are in a half ounce can help you make sure that you’re using the right amount of
ingredients in your recipes.
Having a good understanding of measurements and conversions can also help you when you’re shopping for ingredients. If you’re looking for a specific amount of an ingredient, knowing how many grams
are in a half ounce can help you make sure that you’re getting the right amount. This can be especially helpful when you’re shopping for ingredients online, as you won’t be able to physically measure
out the ingredients before you buy them.
Knowing how many grams are in a half ounce can also help you when you’re trying to figure out how much something weighs. This can be especially helpful if you’re trying to figure out how much
something weighs in ounces, as you can easily convert between the two units of measurement.
Finally, knowing how many grams are in a half ounce can help you when you’re trying to figure out how much something costs. Many products are sold by weight, so knowing how many grams are in a half
ounce can help you figure out how much something costs per ounce. This can be especially helpful when you’re trying to compare prices between different products.
Overall, knowing how many grams are in a half ounce is an important part of understanding measurements and conversions. Whether you’re a baker, a chef, or just someone who likes to measure out
ingredients for recipes, it’s important to know how to convert between different units of measurement. Knowing how many grams are in a half ounce can help you make sure that you’re using the right
amount of ingredients in your recipes, help you when you’re shopping for ingredients, help you when you’re trying to figure out how much something weighs, and help you when you’re trying to figure
out how much something costs.
The History of Grams and Ounces: A Look at the Evolution of Measurement Systems
The history of grams and ounces is a fascinating one, full of interesting facts and stories. From ancient times to the present day, the measurement of weight has been an important part of everyday
In ancient times, the most common unit of measurement for weight was the grain. This was a unit of measurement based on the weight of a single grain of wheat or barley. This system was used in many
cultures, including the Egyptians, Greeks, and Romans.
In the Middle Ages, the ounce became the most common unit of measurement for weight. This was a unit of measurement based on the weight of a single ounce of silver. This system was used in many parts
of Europe, including England, France, and Germany.
In the 18th century, the gram became the most common unit of measurement for weight. This was a unit of measurement based on the weight of a single gram of water. This system was used in many parts
of the world, including Europe, Asia, and the Americas.
Today, the gram and the ounce are still the most common units of measurement for weight. They are used in many different contexts, from cooking to medicine to engineering. They are also used in many
different countries, from the United States to China to India.
The history of grams and ounces is a fascinating one, and it is a testament to the importance of measurement systems throughout history. From ancient times to the present day, the measurement of
weight has been an important part of everyday life.
Common Mistakes to Avoid When Measuring Grams and Ounces: What You Need to Know
Measuring grams and ounces can be tricky, but with a few simple tips, you can make sure you get it right every time! Here are some common mistakes to avoid when measuring grams and ounces:
1. Not using the right measuring tool. Make sure you’re using a kitchen scale that measures in both grams and ounces. This will ensure that you get an accurate measurement.
2. Not accounting for the weight of the container. If you’re measuring something in a container, make sure to subtract the weight of the container from the total weight. Otherwise, you’ll end up with
an inaccurate measurement.
3. Not converting correctly. If you’re measuring in grams and need to convert to ounces, make sure you use the correct conversion rate. One ounce is equal to 28.35 grams.
4. Not measuring accurately. Make sure you’re measuring to the nearest gram or ounce. This will help you get the most accurate measurement possible.
By avoiding these common mistakes, you can make sure you get the most accurate measurement when measuring grams and ounces. With a little practice, you’ll be a pro in no time!
The importance of understanding the difference between a UK and US half ounce measurement can’t be understated, especially when it comes to measurements like food and spices. Knowing that 14.17 grams
is equal to a US half ounce, or 13.6 grams is equal to a UK half ounce, allows people to cook recipes accurately regardless of their origin. It also enables them to avoid confusion when converting
weight measurements between the two systems. Now that you are aware of the differences between US and UK measurements, you can confidently convert between them with easy-to-follow instructions and
calculations. Cooking has never been easier!
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Neighborhoods: Experimenting with Cyclic Cellular Automata | Fronkonstin
Neighborhoods: Experimenting with Cyclic Cellular Automata
On candy stripe legs the Spiderman comes, softly through the shadow of the evening sun (Lullaby, The Cure)
Cellular automata are an inmense source of artistical images. Today, I experimented with Cyclic automata, which are ruled with these simple rules:
• Create a grid of cells.
• Give a state to each cell randomly; states a numbers between 0 and M-1 (you choose the value of M previously).
• For each cell, count how many of its neighbouring cells have their state value exactly 1 unit greater than the cell’s state.
• If the resulting number is greater than a certain thresold (that you also choose previously) increment the state of the cell by 1; if cell state reaches value of M, then you have to put 0 (in
other words, you have to add 1 modulus M).
• Repeat two previous steps a number of times.
A key concept in this algorithm is defining which is the neighborhood of a cell. There are two of them quite famous: Moore and Von Neumann neighborhoods, but you can define your own ones. Once you
decide to stop iterating, you can give color to each cell according its final state, and you will obtain images like this one:
If you have a look to the code, you will see the next parameters:
• neighborhood: the pattern of neighbouring cells
• range: the depth of neighborhood
• states: maximum number of states allowed (the M of the algorithm)
• thresold
• iter: number of iterations
• width and height of the grid
Apart from Moore and Von Neumann, I implemented some other neighborhoods. This chart shows some of them. In columns you can find the folowing: M (Moore), N (Von Neumann), Mr (Moore remote), Nr (Von
Neumann remote), Cr (Cross), S1 (S-Shape #1), Bl (Blade), C2 (Corners #2) and TM (Tick Mark). In rows, you can find different ranges for each neighborhood, from 1 to 4:
You will find more neighborhood in the code of the experiment, which is here. There are infinite combinations of the previous parameters, and each one results in a different image:
I used again COLOULovers palettes As always, I encourage you to experiment with the code, create your own neighborhoods, and see how they work. Happy New Year 2021!
3 thoughts on “Neighborhoods: Experimenting with Cyclic Cellular Automata”
1. this is elegant and brilliant. the best part is the simplicity in the explanation. i tried assigning colours on the go, and it turns out to look almost like some myriad of flow. just wonderful!
1. Thank you very much!!!
2. Thank you and happy new year to you ! | {"url":"https://fronkonstin.com/2021/01/02/neighborhoods-experimenting-with-cyclic-cellular-automata/","timestamp":"2024-11-14T23:43:24Z","content_type":"text/html","content_length":"127347","record_id":"<urn:uuid:38ce8f2f-9ede-4edc-8d77-7ae568980aaf>","cc-path":"CC-MAIN-2024-46/segments/1730477397531.96/warc/CC-MAIN-20241114225955-20241115015955-00231.warc.gz"} |
What Makes while Loop a Poor Choice in Programming
When it comes to looping through objects or values, we have several choices. A for loop; a foreach loop; a while loop; a do..while loop. And usually, we have some sequence-processing library or
syntax at our disposal. In .NET languages, we have the LINQ library available to process any sequence without having to loop explicitly.
These looping methods are obviously not the same. But they are not equally powerful, either. In this article, I will question the usefulness of while loops (including do..while) and try to point you
in the direction of not using them.
This opinion is biased, which asks for some explanation. I will be questioning the while loops from the ground of our ability to produce correct code. Therefore, the main argument in this article
will be code correctness – number of bugs if you will - while other aspects will be of lesser importance. Most importantly, performance considerations will be ignored. Practice shows that all looping
methods mentioned above (for/foreach loop, while loops and sequence processing libraries, like LINQ) perform similarly and therefore performance will not be improved by choosing one looping method
over the other.
While Loop Might Never End
The principal complaint about while loops is that they may never end:
while (true)
This is the infinite loop. If it ever happens that you construct an infinite loop in code, that code will become non-responsive at run time.
The problem with infinite loops is not trivial, as it is in the code segment above. What I called true in the while loop’s condition will usually be more complex:
while (condition || anotherOne)
If at any point during the program’s execution one of the conditions became terminally True, then the loop will turn into an infinite loop. Programmers are often unable to notice this problem by code
inspection alone.
For Loop Might Never End, Too
Please note that for loop alone does not differ from the while loop. Namely, for loop’s syntax consists of three components which can easily be transformed into the while loop form.
for (Initialize();
This for loop brings nothing new to the table compared to while:
while (ShouldContinue())
Therefore, the for loop syntax alone will not save you from infinite loop bug. It gets a little better if you use it in its idiomatic form:
for (int i = 0; i < n; i++)
While this code only represents a special case of the general for loop syntax, it is still safer than any other example given above, simply because it is idiomatic. Lest you try to modify the
iterator variable i inside the loop, which you should never do anyway, you’ll be safe from infinite iteration.
However, this form of the for loop is rarely useful today. Usually, you will have to include a more complex exit condition, like in this example:
for (Car car = parking.ClosestCar();
car.Color != Colors.Red;
As in the case of the while loop, the exit condition of this for loop might never be satisfied, turning it into an infinite loop.
Understanding Loops
Every loop must end. That is also known as the loop termination condition, and it is very easy to understand. Any loop can be represented as a while loop with a Boolean condition in its head:
while (condition)
For a loop to be valid, the condition must at some point turn False. That is what we call the termination condition – there must be a proof that the loop will not run forever.
On the other hand, for a loop to be part of an algorithm, we would introduce its preconditions, postconditions and invariants. Preconditions are the Boolean conditions that must be satisfied before
entering the loop. Postconditions are the Boolean conditions satisfied by the result obtained when the loop terminates. And invariants are the Boolean conditions that must be satisfied at all times,
before, and after the loop, as well as after every iteration of the loop.
Let’s take an example. Consider a loop which finds maximum value in an array:
int max = a[0];
for (int i = 1; i < a.Length; i++)
if (a[i] > max)
max = a[i];
How can we prove that this loop will indeed produce the maximum value? It is really easy if you think it through: Define the postcondition, which says that the result is the maximum value, and define
the termination condition, which says that the loop will terminate, and execution reach the point at which the result can be produced.
Therefore, we have the loop postcondition:
Termination condition is the second part of the infrastructure, and it says that at some point, exit condition will be satisfied:
Now we need to prove that both conditions are satisfied, and that will be the formal proof that this loop indeed produces a maximum value of an array.
But that is not all we have to do. We must also specify the preconditions, i.e. conditions that must be met before the loop begins. If preconditions are violated, then we suspect that the loop
wouldn’t be able to evaluate its body in the first place. You may have already noticed that the max variable is initialized to the first value in the array. For this operation to succeed, we must
ensure that the array contains at least one value:
We’re getting closer to the solution with every step of the way. The last element in this puzzle is the loop invariant. That is where we usually need a bit of imagination. Loop invariant can help
prove that no data structure will ever be broken, that all arithmetic operations will be possible, and so on. But you can also use loop invariant to help you explain the algorithm and then use that
formal explanation to prove correctness of the postcondition.
Here is the loop invariant which fits firmly with the code:
To cut the long story short, this invariant says that at the end of any iteration of the loop, the max variable contains the maximum value between the beginning of the array and the just visited
position in the array.
Bottom line is that maximum value loop, and the entire maximum value algorithm for that matter, is entirely defined with these four conditions, which must be proven correct before concluding that the
algorithm and the loop are correct:
I will not bother to prove these conditions here, as I feel that we have already drifted too far away from the main topic. I would ask you to fill in the gaps as the exercise. You can start from
observation that the loop variable i is strictly increasing with every iteration, and therefore termination condition will be satisfied at some point.
Also, with precondition satisfied (and you can take that for granted), and knowing the law by which loop variable increases, you can reach and inductive proof that loop invariant will be satisfied
indeed at the end of every iteration.
Finally, you will observe that the loop invariant reads the same as the postcondition when end of the loop is reached. By following these steps, you will prove that the algorithm and the loop are
both terminating and producing the expected result in every possible case.
And now back to the main topic of this article: Loop termination. For every loop you write, you must be sure that its termination condition will be satisfied after finite number of iterations.
Without that proof, the loop will be open for bugs, and such bugs will manifest in endless execution of the loop.
How Can Sequences Help with Loop Termination?
Working with sequences is quite different from implementing loops. The main difference is that the loop itself will now be transformed in such way that termination condition disappears entirely.
The foreach loop is designed to work with sequences in C#. Let’s rewrite the maximum value for loop into its foreach counterpart:
int max = a[0];
foreach (int x in a)
if (x > max)
max = x;
This algorithm is a bit different than the previous one (I’ll leave it to you to find the difference). But it can still be described with similar preconditions, postconditions, invariants and the
termination condition.
Can you tell what termination condition will be here?
Here is how we could define it: The foreach loop reaches the last element in a. We don’t have access to the loop’s internal workings, and therefore we can only tell that it will terminate if there is
an element in a which is not followed by another element.
And now that that is clear, we can reformulate the termination condition and say: The foreach loop will terminate if and only if applied to a finite sequence.
This sounds more practical. You can tell that every foreach loop will terminate as long as you have finite sequences in your hands.
How Can LINQ Help with Loop Conditions?
And so, we arrive at LINQ. It is the library which defines operators defined on sequences, where sequence is formally defined as an object implementing the IEnumerable<T> interface. Let’s rewrite the
maximum value example again, this time to use the LINQ Aggregate operator:
int max = a.Aggregate((acc, cur) => cur > acc ? cur : acc);
This is something completely different, compared to while and foreach loops we saw this far. Yet, we can still define all conditions as before. There, we find loop precondition to read the same as
before, as the sequence a must contain at least one element. Otherwise, Aggregate operator would fail to initialize. Loop invariant is that after every iteration the acc variable contains the maximum
value of all the values visited this far. Loop postcondition is that at the end of the iteration, the acc variable contains the overall maximum value.
And lastly, termination condition will say that the Aggregate operator will have to exhaust all elements in the sequence to terminate. In other words, we are meeting the same termination condition
again: The input sequence must be finite.
Wrapping Up the Theory with Sequences
When sequences and LINQ are used, we recognize that places where conditions are defined have been changed. That is the most remarkable observation about loops. Precondition and termination condition
will be defined on the sequence. Sequence consumer will have no responsibilities there. Invariant and postcondition, on the other hand, will remain entirely on the consuming end.
In the example above, precondition (sequence is not empty) and termination condition (sequence is finite) are both defined on the sequence itself. The rest of the conditions is what the lambda passed
to the Aggregate operator will be responsible to ensure.
And now we can think how that changes our position? If we write both ends of the code – initializing the sequence and then using it in the LINQ expression – then what is the benefit of this
separation of concerns? Let’s add one line more to the implementation and then everything will fit into place:
IEnumerable<int> a = FetchData();
int max = a.Aggregate(
(acc, cur) => cur > acc ? cur : acc);
Now it becomes more obvious. There are two segments of code involved here. One is the code which generates the sequence of objects. The other half is the LINQ expression which consumes the sequence.
That is how new separation of concerns comes to the table. It is the data generator’s responsibility to ensure the precondition and termination condition. It is then the data consumer’s
responsibility to implement the algorithm on those data, and that means to satisfy the loop invariant and the postcondition.
The algorithm is none of our business, if you think it through. Generally, the algorithm is the solution to the problem, and we are solving it the way we usually do that. Invariants and
postconditions are merely a mathematical tool we use. However, algorithm implementation can make a call to a non-terminating function. That means that part of the termination condition still exists
in the lambda expression used in LINQ. However, in nearly any practical situation, a moment of code inspection would be sufficient to discover whether the lambda terminates or will hang forever.
That leaves us with only the data generator where we are free to do the things in one way or the other. That is exactly where preconditions and termination condition are gaining importance. It is
very useful to know, at that point where we are asked to satisfy these two conditions while generating data, that termination condition has turned into a trivial question: Is the sequence finite? If
it is, then we are fine. Otherwise, we are not fine.
Return to the beginning of this article and you will see that nearly every loop had a different termination condition, and these conditions were nearly always mixed together with logic inside the
loop. With sequences and LINQ, and even with the foreach loop, all that diversity is gone, replaced with one and only universal condition – every loop on a sequence will be terminated in finite
number of iterations if it doesn’t contain an infinite operation and the sequence is finite.
Once again, the fact that the lambda always terminates will usually be trivial to prove. That leaves us with the simple condition that the sequence must be finite and then the loop will never be
A Word About API Design
There is one side-effect of the conclusions we have drawn this far. It gives us a hint about good API design. Imagine a task of designing the data generator, the entity which produces data over which
the consumer will loop. How exactly would you approach that task?
Let’s revisit the example with cars and their colors again:
for (Car car = parking.ClosestCar();
car.Color != Colors.Red;
We have already concluded that this design is risky because Paint method on the Car object would have to guarantee that it will eventually reach the red color. Will it or will it not, that is beyond
comprehension of this loop. Therefore, the loop, however correct by itself, might never terminate thanks to conditions that are expressed someplace else.
Could you come up with a better design, now that you understand loop conditions?
For one thing, we could turn this API into a form based on IEnumerable – the sequence of Car objects this time. Consider this design:
.TakeWhile(car => car.Color != Colors.Red)
.Select(car => ...);
The difference this design brings is that loop termination condition now boils down to knowing that the ProducePaints method returns a finite sequence, and nothing more than that. Even if the red car
was never produced, the loop will eventually terminate when all the paint colors have been tried out.
The ProducePaints method can even return a lazy-evaluated sequence (which is the preferred method, anyway). That would ensure that performance will not be affected by the overall length of the
sequence, but only by the demand of the subsequent chained operators.
This can be used as the general guideline when designing classes and libraries. Return sequences. Avoid protocols and methods returning flags.
In this article we have examined basic properties of loops and put them in perspective of two orthogonal approaches to looping: Common loop syntax, like for, while and do..while loops on the one
hand, and LINQ and sequences, with addition of the foreach loop on the other.
The result of this thorough examination is that the prior methods, common loops, are prone to infinite loop bugs. It is important to prove, in every separate case, that the loop will eventually
terminate, besides proving that it will produce the result which was expected.
The same problem, loop termination, diminishes to a trivial condition when data over which the loop execute are organized into a sequence, an IEnumerable<T>. In that case, all we have to do is ensure
that the sequence is finite, and the loop will terminate for sure.
In the end, we have touched the question of designing APIs. Lead by the desire to simplify the consuming end, the one which initiates the loop iteration, we conclude that the good API is the one
which exposes a sequence of data, rather than any other loop-friendly design. As long as the API can produce a finite sequence, we conclude that its consumer will be safe from infinite loops no
matter how clumsily it might behave on its end. | {"url":"https://www.codinghelmet.com/articles/what-makes-while-loop-poor-choice-in-programming","timestamp":"2024-11-02T05:37:07Z","content_type":"text/html","content_length":"44901","record_id":"<urn:uuid:3747b7af-e9d6-414f-99e9-8de3633b36cb>","cc-path":"CC-MAIN-2024-46/segments/1730477027677.11/warc/CC-MAIN-20241102040949-20241102070949-00018.warc.gz"} |
Comparing the performance of simple heuristics using the Heuristica R package
Daniel Barkoczi
This document provides a simple example of how to compare the out-of-sample performance of different models in the heuristica package.
# Use this seed to exactly replicate my tables and graphs below.
# Remove it to see a new sampling-- and whether the overall conclusions still
# hold.
Helper functions
First let’s load the heuristica package to get the heuristics we will compare. It also includes functions to calculate accuracy.
Let’s enter the models we want to test:
Here’s a function that does cross-validation taking the vector of models, criterion column, columns to fit, the dataset, and the number of repetitions as input:
crossV <- function(vec_of_models, criterion_col, cols_to_fit, data, reps,training_proportion){
fitting <- vector()
prediction <- vector()
for(i in 1:reps){
#randomly sample training and test row indexes
train <- sample(1:nrow(data), nrow(data)*training_proportion)
test <- setdiff(1:nrow(data), train)
#create training and test set
training_set <- data[train,]
test_set <- data[test,]
# If a regression is overdetermined (e.g. has too many columns(), it will
# drop the right-most columns. To instead make it drop random columns,
# we shuffle the column order.
shuffled_cols_to_fit <- sample(cols_to_fit)
y <- 0
for (mod in vec_of_models) { #fit the models to the training_set
y <- y+1
models[[y]] <- mod(training_set, criterion_col, shuffled_cols_to_fit)
#calculate percentage of correct predictions
fittingAccuracy <- percentCorrectList(training_set, models)
predictionAccuracy <- percentCorrectList(test_set, models)
fitting <- rbind(fitting,fittingAccuracy)
prediction <- rbind(prediction,predictionAccuracy)
results <- (rbind(colMeans(fitting),colMeans(prediction)))
rownames(results) <- c("Fitting","Prediction")
City population
Then we can just run this function to calculate predictive accuracy for different training and test set sizes. First let’s have the models predict the populations of 83 German cities using 9 binary
cues. The criterion column may change depending on your data set, so set it correctly!
data_set <- city_population
criterion_col <- 3
cols_to_fit <- 4:ncol(data_set)
Below we have the models train on 0.5 of the data (50%) and predict the other half, and we repeat this for 100 samples of splitting the data in half.
reps <- 100
training_proportion <- 0.5
results <- crossV(vec_of_models, criterion_col, cols_to_fit, data_set, reps,training_proportion)
round(results, 1)
## ttbModel unitWeightModel regModel minModel
## Fitting 74.9 73.6 75.9 70.2
## Prediction 72.4 71.7 73.6 68.4
Finally, let’s plot the results:
## Warning in type.convert.default(X[[i]], ...): 'as.is' should be specified by the
## caller; using TRUE
## Warning in type.convert.default(X[[i]], ...): 'as.is' should be specified by the
## caller; using TRUE
colnames(p) <- c("condition","model","value")
ggplot(p, aes(x=condition, y=value, colour=model,group=model)) +
geom_line() +
geom_point() +
xlab("Condition") + ylab("Proportion correct")
High school drop-outs
Now do the same analysis for the high school drop-out data set. It has 23 real-valued cues (rather than binary cues) for 63 Chicago public high schools.
Note that this data set has na’s, so we use na.omit to clean them because not all heuristics can handle them properly.
data_set <- na.omit(highschool_dropout)
criterion_col <- 4
cols_to_fit <- 6:ncol(data_set)
reps <- 100
training_proportion <- 0.5
results <- crossV(vec_of_models, criterion_col, cols_to_fit, data_set, reps,training_proportion)
rownames(results) <- c("Fitting","Prediction")
p <- melt(results)
## Warning in type.convert.default(X[[i]], ...): 'as.is' should be specified by the
## caller; using TRUE
## Warning in type.convert.default(X[[i]], ...): 'as.is' should be specified by the
## caller; using TRUE
colnames(p) <- c("condition","model","value")
ggplot(p, aes(x=condition, y=value, colour=model,group=model)) +
geom_line() +
geom_point() +
xlab("Condition") + ylab("Proportion correct")
The performance of all models drops when they are predicting unseen data. In the city population dataset the rank-order of the models remains the same for fitting and prediction. However, when
predicting high-school dropout some of the simple models (TTB and UnitWeightModel) outperform linear regression in prediction. These results suggests that different environmental structures (such as
the number of cues in the environment) favor different strategies.
How would other models compare to take-the-best? Try some of the existing models in the heuristica package (e.g., logRegModel for logistic regression) or create your own model (see vignette on ‘how
to make a heuristic’). | {"url":"https://cran.case.edu/web/packages/heuristica/vignettes/cross-validation.html","timestamp":"2024-11-10T04:52:04Z","content_type":"text/html","content_length":"130038","record_id":"<urn:uuid:f6613958-46c6-4958-a83d-fe962e29fa5e>","cc-path":"CC-MAIN-2024-46/segments/1730477028166.65/warc/CC-MAIN-20241110040813-20241110070813-00223.warc.gz"} |
Most teaching licenses will require that you pass a certification exam with math concepts. Thoughts of this test often induce fear and stress in even the most talented prospective teachers. Ensuring
that you have a deep understanding of the math fundamentals will alleviate this anxiety and help you pass your exam. One topic that many struggle with is graphing linear equations. Let’s review the
Linear equations make straight lines when graphed. The equations can all be written in the format y = mx + b, where m is the slope and b is the y-intercept. The slope describes how slanted the line
is and the y-intercept is the point where the line will cross the y-axis.
On these certification exams, you may see a couple problems where you are asked to match a graph to an equation. Here is one possible approach.
1. Start by determining whether the slope is positive or negative.
If you are looking at a graph, if the line goes up when looking at it from left to right, then it has a positive slope. If the line goes down, it has a negative slope. Once you determine the sign
of the slope, look at the equation when it is in the form y=mx+b and determine whether the m has the same sign. See if you can eliminate any answer choices.
2. Find the y-intercept.
The next easiest thing to identify about a linear graph is the y-intercept. Look at the y-axis (the vertical one) and determine the point where the line crosses the x-axis (the horizontal one).
Compare this value to the y-intercept in the equation that is represented by the variable b. See if any answer choices can be eliminated.
3. Determine the slope.
Pick any two points on the line and write down their coordinates. Then, figure out the slope by calculating the rise over the run or the change in y over the change in x. Find the difference in
the y-values divided by the difference in the x-values. This slope is represented by m in the standard equation, y = mx + b. Using the slope and y-intercept, find your answer.
If after following this approach you still have two or more viable options, you can always try different points. Remember, a graph represents all the points that make the equation true. If you are
struggling, just pick a point on the graph and plug the coordinates into the equation. If the equation does not work for a coordinate pair that is graphed, then that equation can’t represent the
I hope you find some helpful tips in this quick review of graphing linear equations. We have full-length comprehensive teacher prep courses for all topics on your teacher certification exams if you
need help preparing. Best of luck. | {"url":"https://www.prepforward.com/tag/math-certification/","timestamp":"2024-11-09T22:48:10Z","content_type":"text/html","content_length":"63517","record_id":"<urn:uuid:f1fdbd69-d661-4486-a919-1e64cbb2fc44>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.10/warc/CC-MAIN-20241109214337-20241110004337-00410.warc.gz"} |
Add It Up 3
Add It Up is a game where your goal is to select numbers that add up to the target sum . You only have a limited time to do so. The more numbers you select to get to the target sum, the higher your
score will be. The faster you solve a sum, the higher your score will be. | {"url":"https://cdn.htmlgames.com/AddItUp3/","timestamp":"2024-11-14T23:10:50Z","content_type":"text/html","content_length":"5255","record_id":"<urn:uuid:262a569c-9285-4598-bf7f-00dabaece121>","cc-path":"CC-MAIN-2024-46/segments/1730477397531.96/warc/CC-MAIN-20241114225955-20241115015955-00524.warc.gz"} |
Math 4 Wisdom. "Mathematics for Wisdom" by Andrius Kulikauskas. | Research / MathDiscoveryFourLevels
• Index
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Andrius Kulikauskas
• m a t h 4 w i s d o m - g m a i l
• +370 607 27 665
• My work is in the Public Domain for all to share freely.
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Math Discovery Examples
Math Discovery Four Levels: Truth
Truth* 0 Truth, 1 Model, 2 Implication, 3 Variable We now think of the problem as relating two sheets, one of which has a wider point of view because it includes what may vary, not just what is
fixed. There are four ways to relate two such sheets. They are given by the questions Whether it is true? What is true? How is it true? Why is it true? Truth is what is evident, what can't be hidden,
what must be observed, unlike a cup shut up in a cupboard. The fixed sheet is the level of our problem and the varying sheet is our metalevel from which we study it.14
Truth: Whether it is true?
Truth: Whether it is true?* Truth: Whether it is true? The two sheets may be conflated in which case we may interpret the problem as statements that we ourselves are making which may be true or false
and potentially self-referential. Together they allow for proofs-by-contradiction where true and false are kept distinct in the level, whereas the metalevel is in a state of contradiction where all
statements are both true and false. In my thinking, contradiction is the norm (the Godly all-things-are-true) and non-contradiction is a very special case that takes great effort, like segregating
matter and anti-matter. Deep structure "solution spaces" allow us, as with Euclid's equilateral triangle, to step away from the "solution" and consider the candidate solutions, indeed, the failed
□ Argument by contradiction* Instead of directly trying to prove something, we start by assuming that it is false, and show that this assumption leads us to an absurd conclusion. A
contradiction argument is usually helpful for proving directly that something cannot happen. ... When you begin thinking about a problem, it is always worth asking, What happens if we negate
the conclusion? Will we have something that is easier to work with? If the answer is "yes", then try arguing by contradiction. pg.46, The Art and Craft of Problem Solving, Paul Zeitz, 1999,
John Wiley & Sons, Inc.1439
☆ Square root of 2 is not rational* A classic example of proof by contradiction. pg.46, The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc.1438
□ Dropping the law of excluded middle* Edward Cherlin, 2011.04.05: Yale Professor Fred B. Fitch's book, Symbolic Logic presents a system of logic that can be proven consistent. Dropping the law
of Excluded Middle was essential to the construction. Gödel's theorem depends on Excluded Middle, so it doesn't apply to this proof of consistency. If R is the set of all sets that are not
members of themselves (with further precision required that does not concern us here), then R is a member of R if and only if R is not a member of R. In the presence of Excluded Middle, this
results in contradiction. In its absence, it is merely undecidable both in terms of provability and of truth. 992
Model: What is true?
Model: What is true?* The metalevel may simplify the problem at the level. Such a relationship may develop over stages of "wishful thinking" so that the metalevel illustrates the core of the problem.
Ultimately, the metalevel gives the solution's deep structure and the level gives the problem's surface structure. 62
☆ A right triangle is half a rectangle* This morphism is the basis for the area of a right triangle, but also for all of trigonometry, and shows that a function need not be a formula, and
shows how two domains - angles and ratios - can be linked, as by shapes. Gospel Math. 1846
☆ Recasting geometry/combinatorics as parity* Remove the two diagonally opposite corner squares of a chessboard. Is it possible to tile this shape with thirty-one 2 x 1 "dominos"? ... At
first, it seems like a geometric/combinatorial problem with many cases and subcases. But it is really just a question about counting colors. The two corners that were removed wre both
(without loss of generality) white, so the shape we are interested in contains 32 black and 30 white squares. Yet any domino, once it is placed, will occupy exactly one black and one
white square. The 31 dominos thus require 31 black and 31 white squares, so tiling is impossible. pg. 60 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons,
□ Apply algebra ideas to a calculus problem* Our final example is also due to Euler. Here the tables are turned: ideas from polynomial algebra are inappropriately applied to a calculus problem,
resulting in a wonderful and correct evaluation of an infinite series (although in this case, complete rigorization is much more complicated). ... Is there a simple expression for zeta(2) = 1
+ 1/2**2 + 1/3**2 + ... ? Euler's wonderful, crazy idea was inspired by the relationship between zeros and coefficients which says that the sum of the zeros of the monic polynomial x**n +
a_n-1 x**n-1 + ... + a1 x + a0 is equal to - a_n-1; this follows from an easy argument that examines the factorization of the polynomial into terms of the form (x-ri), where each ri is a
zero. Why not try this with functions that have infinitely many zeros? A natural candidate to start with is sin x ... pg.315 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley
& Sons, Inc2251
□ Crossover* A crossover ... is an idea that connects two or more different branches of math, usually in a surprising way. ... perhaps the three most productive crossover topics: graph theory,
complex numbers, and generating functions. pg.119, The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc.2154
□ Deliberately misleading presentation* Three women check into a motel room which advertises a rate of $27 per night. They each give $10 to the porter, and ask her to bring back 3 dollar bills.
The porter returns to the desk, where she learns that the room is actually only $25 per night. She gives $25 to the motel desk clerk, returns to the room, and gives the guests back each one
dollar, deciding not to tell them about the actual rate. Thus the porter has pocketed $2, while each guest spent 10-1 = $9, a total of 2 + 3 x 9 = $29. What happened to the other dollar? ...
This problem is deliberately trying to mislead the reader into thinking that the profit that the porter makes plus the amount that the guests spend *should* add up to $30. pg. 22, 102, The
Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc.1648
□ e* Determine, with proof, the largest number which is the product of positive integers whose sum is 1976. ... Once again, we shall inappropriately apply calculus to a discrete problem. It
makes intuitive sense for the numbers whose sum is 1976 to be equal (see the discussion of the AM-GM inequality...) But how large should these parts be? Consider the optimization question of
finding the maximum value of f(x) = (S/x)**x, where S is a positive constant. An exercise in logarithmic differentiation (do it!) shows that S/x = e. Thus, if the sum is S each part should
equal e and there should be S/e parts. Now this really makes no sense if S=1976 and the parts must be integers, and having a non-integral number of parts makes even less sense. But at least
it focuses our attention on parts whose size is close to e=2.71828... Once we start looking at parts of size 2 and 3, the problem is close to a solution... pg.313 The Art and Craft of Problem
Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc2250
□ Encoding* In contrast [to partitioning], the encoding tactic attempts to count something in one step, by first producing a bijection (a fancy term for a 1-1 correspondence) between each thing
we want to count and the individual "words" in a simple "code". ... Instead of partitioning the collection of subsets into many classes, look at this collection as a whole and encode each of
its elements (which are subsets) as a string of symbols. Imagine storing information in a computer. How can you indicate a particular subset of S = {a,b,c}? There are many possibilities, but
what we want is a uniform coding method that is simple to describe and works essentially the same for all cases. That way it will be easy to count. For example, any subset of S is uniquely
determined by the answers to the following yes/no questions. Does the subset include a? Does the subset include b? Does the subset include c? We can encode the answers to these questions by a
three-letter string which uses only the letters y and n. For example, the string yyn would indicate the subset {a,b}. Likewise, the string nnn indicates the empty set and yyy indicates the
entire set S. Thus There is a bijection between strings and subsets. ... And it is easy to count the number of strings; two choices for each letter and three letters per string mean 2**3
different strings in all. ... Proper encoding demands precise information management. ... try to think carefully about "freedom of choice": ask yourself what has already been completely
determined from previous choices ... Beginners are often seduced by the quick answers provided by encoding and attempt to convert just about any counting problem into a simple multiplication
or binomial coefficient Note that strings have an additional structure which makes the counting easy: the strings presume a total order of positions, from left to right, whereas the elements
of a set need not be ordered. This ordering comes for free and makes the bijection work. pg.213-214 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc2208
□ Fantasize an answer* When looking at the conclusion of the problem, especially for a "to find" problem, sometimes it helps to "fantasize" an answer. Just make something up, and then reread
the problem. Your fantasy answer is most likely false, and rereading the problem with this answer in mind may help you to see why the answer is wrong, which may point out some of the more
important constraints of the problem. pg.30, The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc.1428
□ Interpreting algebraic variables as coordinates* Whenever a problem involves several algebraic variables, it is worth pondering whether some of them can be interpreted as coordinates. pg. 59
The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc.1505
□ Recast a problem from one domain into another domain* The powerful idea of converting a problem from words to pictures is just one aspect of the fundamental peripheral vision strategy. Open
your mind to other ways of reinterpreting problems. ... what appeared to be a sequence of numbers was actually a sequence of descriptions of numbers ... Another example was the locker problem
in which a combinatorial problem metamorphosed into a number theory lemma. "Combinatorics <=> Number Theory" is one of the most popular and productive such "crossovers", but there are many
other possibilities. Some of the most spectacular advances in mathematics occur when someone discovers a new reformulation for the first time. pg. 60 The Art and Craft of Problem Solving,
Paul Zeitz, 1999, John Wiley & Sons, Inc.1507
□ Recast an inequality as an optimization problem* AM-GM reformulated ... we altered our point of view and recast an inequality as an optimization problem. pg.195-196 The Art and Craft of
Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc.2195
□ Recasting* [In combinatorics,] the strategy of recasting is especially fruitful: to counteract the inherent dryness of counting, it helps to creatively visualize problems (for example, devise
interesting "combinatorial arguments") and look for hidden symmetries. Many interesting counting problems involve very imaginative multiple viewpoints ... to see if a combinatorial identity
is true, examine how each side of the equation counts a representative element pg.212, 228 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc2205
□ Recasting geometry as algebra* Descartes' idea of recasting geometric questions in a numeric/algebraic form led to the development of analytic geometry, which then led to calculus. pg. 60 The
Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc.1506
□ Structural equivalence* Edward Cherlin: Proving that two seemingly unrelated, even apparently incompatible objects are equivalent in some way, or can model each other, is one of the deepest
ideas in math.813
☆ Brouwer's Intuitionistic logic and set theory* Edward Cherlin: Mathematicians lost interest in Brouwer's Intuitionistic logic and set theory when it was shown that it and the more usual
non-constructive logics and set theories can model each other. 817
☆ Equivalence of geometries* Edward Cherlin: The fact that each of elliptic/Riemannian, Euclidean, and hyperbolic/Lobachevskian geometries contain models of each other shows that all three
are equally valid. For example, a Clifford's surface in Riemannian space and a horosphere in Lobachevskian space both have locally Euclidean geometry. 816
☆ Galois theory* Galois theory maps the roots of a given polynomial equation (in field theory) to the Galois group of permutations of the roots. 818
☆ String theories are maps of each other* Edward Cherlin: At one time it was thought that there was a vast space of possible String Theories in physics. It turns out that all of them are
maps of each other. 815
☆ Taniyama-Shimura Theorem* Edward Cherlin: The proof of Fermat's Last Theorem depends on the Taniyama-Shimura Theorem that all elliptic functions are modular, that is, that there is a
structure-preserving mapping between elliptic functions and modular forms. 814
□ Two different ways* Keeping a flexible point of view is a powerful strategy. This is especially true with counting problems where often the crux move is to count the same thing in two
different ways. To help develop this flexibility, you should practice creating "combinatorial arguments". This is just fancy language for a story that rigorously describes in English how you
count something. ... Pay attention to the building blocks of "algebra to English" translation, and in particular, make sure you understand when and why multiplication rather than addition
happens, and vice versa. Examples include addition (or), multiplication (and), exponentiation, combination, permutation, distinct members, products of choices, sums of choices, complements of
combinations. pg.208 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc2201
• Make it easier* The easier problem may actually be the more informative, relevant, natural, instructive problem. If the given problem is too hard, solve an easier one. ... For example, if the
problems involves big, ugly numbers, make them small and pretty. If a problem involves complicated algebraic fractions or radicals, try looking at a similar problem without such terms. At best,
pretending that the difficulty isn't there will lead to a bold solution... At worst, you will be forced to focus on the key difficulty of your problem, and possibly formulate an intermediate
question, whose answer will help you with the problem at hand. And eliminating the hard part of a problem, even temporarily, will allow you to have some fun and raise your confidence. pg.18, 31
The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc.1417
• Wishful thinking* It is helfpul to try to loosen up, and not worry about rules or constraints. Wishful thinking is always fun, and often useful. For example, in this problem, the main difficulty
is that the top boxes labeled A and C are in the "wrong" places. So why not move them around to make the problem trivially easy? ... Ask yourself, "What is it about the problem that makes it
hard?" Then, make the difficulty disappear! You may not be able to do this legally, but who cares? Temporarily avoiding the hard part of a problem will allow you to make progress and may shed
light on the difficulties. pg.18, 31 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc.1416
Implication: How is it true?
Implication: How is it true?* Implication: How is it true? The metalevel may relate to the level as cause and effect by way of a flow of implications. The metalevel has us solve the problem,
typically by working backwards. The level presents the solution, arguing forwards. 63
□ Deduction* Also known as "direct proof", deduction is merely the simplest form of argument in terms of logic. A deductive argument takes the form "If P, then Q" or "P=>Q" or "P implies Q".
Sometimes the overall structure of an argument is deductive, but the smaller parts use other styles. pg.46, The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc.1437
□ Work backwards UCL problem solving technique 4 of 5.
□ Restate the problem UCL problem solving technique 5 of 5.
□ Penultimate step* Once you know what the desired conclusion is, ask yourself, "What will yield the conclusion in a single step?" Sometimes a penultimate step is "obvious", once you start
looking for one. And the more experienced you are, the more obvious the steps are. For example, suppose that A and B are weird, ugly expressions that seem to have no connection, yet you must
show that A = B. One penultimate step would be to separately argue that A ≥ B AND B ≥ A. Perhaps you want to show instead that A ≠ B. A penultimate step would be to show that A is always
even, while B is always odd. pg. 30, The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc. 1383
□ Recast geometry as logic* ...a problem that is geometric on the surface, but not at its core ... We are given n planets in space, where n is a positive integer. Each planet is a perfect
sphere and all planets have the same radius R. Call a point on the surface of a planet private if it cannot be seen from any other planet. ... We conjecture that the total private area is
always exactly equal to the area of one planet, no matter how the planets are situated. It appears to be a nasty problem in solid geometry, but must it be? The notions of "private" and
"public" seem to be linked with a sort of duality; perhaps the problem is really not geometric, but logical. ... If location x is private on one planet, it is public on all other planets.
After this nice discovery, the penultimate step is clear: to prove that Given any location x, it must be private on some planet. ... pg. 63 The Art and Craft of Problem Solving, Paul Zeitz,
1999, John Wiley & Sons, Inc.1519
Variable: Why is it true?
Variable: Why is it true?* The metalevel and the level may be distinct in the mind, as complements. Given the four levels (why, how, what, whether), the metalevel is associated with the wider point
of view (why being the widest) and the level with a narrower point of view. We may think of them concretely in terms of the types of signs: symbol, index, icon, thing. The pairs of four levels are
six ways to characterize the relationship. I believe that each way manifests itself through the relationship that we suppose for our variables: dependent vs. independent, known vs. unknown, given vs.
arbitrary, fixed vs. varying, concrete vs. abstract, defined vs. undefined, evaluated vs. unevaluated, specialized vs. generalized, domain given or not, determined vs. undetermined (as in the problem
of measuring the shortest distance to the river and grandmother's) and so on. I need to study the variety that variables can express. I suppose that, mentally, the varying variables are active in
both levels, whereas the fixed variables are taken to be in the level. The levels become apparent when, for example, we draw a picture because that distinguishes the aspects of our problem that our
iconic or indexical or symbolic. Likewise, our mental peripheral vision picks up on aspects specific to a particular level.64
☆ Free variables and Bound variables* Wikipedia: In mathematics, and in other disciplines involving formal languages, including mathematical logic and computer science, a free variable is a
notation that specifies places in an expression where substitution may take place. The idea is related to a placeholder (a symbol that will later be replaced by some literal string), or a
wildcard character that stands for an unspecified symbol. The variable x becomes a bound variable, for example, when we write 'For all x, (x + 1)2 = x2 + 2x + 1.' or 'There exists x such
that x2 = 2.' In either of these propositions, it does not matter logically whether we use x or some other letter. However, it could be confusing to use the same letter again elsewhere in
some compound proposition. That is, free variables become bound, and then in a sense retire from being available as stand-in values for other values in the creation of formulae.1165
□ Bend the rules* Don't let self-imposed, unnecessary restrictions limit your thinking. Whenever you encounter a problem, it is worth spending a minute (or more) asking the question, "Am I
imposing rules that I don't need to? Can I change or bend the rules to my advantage?" pg.23, The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc.1422
□ Draw a picture* I imagine that drawing a picture brings out its inner logic at the level of "icon" or "what". Central to the open-minded attitude of a "creative" problem solver is an
awareness that problems can and should be reformulated in different ways. Often, just translating something into pictorial form does wonders. pg.59, The Art and Craft of Problem Solving, Paul
Zeitz, 1999, John Wiley & Sons, Inc.1502
□ Draw pictures UCL problem solving technique 3 of 5
□ Draw pictures* In practice, there are several possible methods of showing that a given sequence converges to a limit. ... Draw pictures whenever possible. Pictures rarely supply rigor, but
often furnish the key ideas that make an argument both lucid and correct. ... consider the sequence (xn) defined by x0=alpha and x_n+1 = 1/2(x_n + alpha/x_n) ... In the picture below...
Notice that the y-coordinate of the midpoint of the line segment AB is the average of these two numbers, which is equal to x_1 ... To show convergence with this picture, we would need to
carefully argue why we will never "bounce" away from the convergence point. .... The picture suggests two things: that the sequence decreases monotonically, and that it decreases to square
root of alpha. ... The trickiest part in the example above was guessing that the limit was alpha. What if we hadn't been lucky enough to have a nice picture? pg.285-288 The Art and Craft of
Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc2236
□ Drawing the monk problem* A monk climbs a mountain. He starts at 8 am and reaches the summit at noon. He spends the night on the summit. The next morning, he leaves the summit at 8am and
descends by the same route he used the day before, reaching the bottom at noon. Prove that there is a time between 8 am and noon at which the monk was at exactly the same spot on the mountain
on both days One solution is to draw the paths on a distance-time graph, which makes it clear that the paths must cross and so they must meet. The pictures brings out the two conditions and
shows how they come together. pg.19, 59 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc.1504
□ Invent a font* The next example combines "Complement PIE" with other ideas, including the useful encoding tool, invent a font, whereby we temporarily "freeze" several symbols together to
define a single new symbol. ... Four young couples are sitting in a row. In how many ways can we seat them so that no person sits next to his or her "significant other?" Define Ai to be the
set of all seatings for which bi and gi sit together. To compute |Ai|, we have two cases: either bi is sitting to the left of gi or vice versa. For each case, there will be 7! possibilities,
since we are permuting 7 symbols: the single symbol bigi (or gibi), plus the 6 other people... Note that alphabetical order in a Spanish language dictionary treats "ch" and "ll" as letters so
that "ch" comes after "cz" and "ll" comes after "lz". pg.230 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc2212
□ Loosen up* Loosen up by deliberately breaking rules and consciously opening yourself to new ideas (including shamelessly appropriating them!) Don't be afraid to play around, and try not to
let failure inhibit you. pg.24, The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc.1425
□ Peripheral vision* One way to heighten your receptiveness to new ideas is to stay "loose", to cultivate a sort of mental peripheral vision. ... Likewise, when you begin a problem solving
investigation, you are "in the dark". Gazing directly at things won't help. You need to relax your vision and get ideas from the periphery. Like Polya's mouse, constantly be on the lookout
for twists and turns and tricks. Don't get locked into one method. Try to consciously break or bend the rules. pg.20, The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley &
Sons, Inc.1421
□ Without loss of generality* Note the use of the phrase "without loss of generality" in the following problem. The color "white" is chosen arbitrarily, yet its value is fixed. This is one way
that variables can be employed. Remove the two diagonally opposite corner squares of a chessboard. Is it possible to tile this shape with thirty-one 2 x 1 "dominos"? ... At first, it seems
like a geometric/combinatorial problem with many cases and subcases. But it is really just a question about counting colors. The two corners that were removed wre both (without loss of
generality) white, so the shape we are interested in contains 32 black and 30 white squares. Yet any domino, once it is placed, will occupy exactly one black and one white square. The 31
dominos thus require 31 black and 31 white squares, so tiling is impossible. pg. 60 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc.1510
□ Create notation* You can make progress on a math problem simply by creating a relevant notation for it, which allows you to think about it in a new way, in a new level.2256
• Hard and soft constraints* Wikipedia: The aim of constraint optimization is to find a solution to the problem whose cost, evaluated as the sum of the cost functions, is maximized or minimized.
The regular constraints are called hard constraints, while the cost functions are called soft constraints.934 | {"url":"https://www.math4wisdom.com/wiki/Research/MathDiscoveryFourLevels","timestamp":"2024-11-08T02:01:32Z","content_type":"application/xhtml+xml","content_length":"39469","record_id":"<urn:uuid:764c0182-744f-4aa9-8ed2-165390b3e0cd>","cc-path":"CC-MAIN-2024-46/segments/1730477028019.71/warc/CC-MAIN-20241108003811-20241108033811-00486.warc.gz"} |
Calculating Nonlinearity of Boolean Functions with Walsh-Hadamard Transform
Reading Time: 2 minutes
It was during Winter 2016 that I took an extremely interesting course on Computational Algebra. This course was solely devoted to analyzing topics that ranged from cryptography to error-resistant
functions. Among the many topics, one that stood out for me was LFSRs (Linear Feedback Shift Registers).
LFSRs are simple circuits that use their internal state to calculate a linear function. The output of the function is then fed into the circuit's new state, and the cycle continues. Perhaps that
sounds a bit to abstract right now. The truth is that I will most likely write about them in the weeks to come. (Because they are fascinating!). Wikipedia has a nice little animation that shows one
in action:
LFSR in Action - Source: Cuddlyable3 @ wikipedia.com
However, this article is not about LFSRs, but rather something even more fundamental: Boolean Functions.
Boolean functions are simply functions whose variables are either a 0 or a 1. While this might seem quite simple, it is the basis of substantially complex cryptographic constructs. In cryptography,
researchers are particularly concered on a the "non-linearity" property of boolean function. Non-Linearity is a measure by which we are able to determine the inputs of the function given the output.
For a trivial example, imagine we have a function f(x[1],x[2]) = x[1]+x[2] and another function g(x[1],x[2]) = x[1 ]. Clearly, it is easier to correlate the output of g to it's inputs (namely if g =
1 then x[1] = 1 and if g = 0 then x[1] = 0) than f with its inputs (although it is not that terribly hard). The overall idea is that the harder it is to figure out the inputs of a function given it's
output, the more "non-linear" it is and the more cryptographically useful it becomes.
The following is a "tutorial" style paper that explains the basis of boolean functions and how to compute their non-linearity. It explains how to calculate such non-linearity in one of the most
refined and proficient ways to do so: with the Fast Walsh-Hadamard Transform. I also included a bit more information on what it means for a function to be "non-linear" and what benefits this brings.
(You can also check this paper out in the Academia.edu link.)
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When training a model, it is often useful to lower the learning rate as the training progresses. This schedule applies the inverse decay function to an optimizer step, given a provided initial
learning rate. It requires a step value to compute the decayed learning rate. You can just pass a TensorFlow variable that you increment at each training step.
The schedule is a 1-arg callable that produces a decayed learning rate when passed the current optimizer step. This can be useful for changing the learning rate value across different invocations of
optimizer functions. It is computed as:
decayed_learning_rate <- function(step) {
initial_learning_rate / (1 + decay_rate * step / decay_step)
or, if staircase is TRUE, as:
decayed_learning_rate function(step) {
initial_learning_rate / (1 + decay_rate * floor(step / decay_step))
You can pass this schedule directly into a keras Optimizer as the learning_rate.
Example: Fit a Keras model when decaying 1/t with a rate of 0.5:
initial_learning_rate <- 0.1
decay_steps <- 1.0
decay_rate <- 0.5
learning_rate_fn <- learning_rate_schedule_inverse_time_decay(
initial_learning_rate, decay_steps, decay_rate)
model %>%
compile(optimizer = optimizer_sgd(learning_rate = learning_rate_fn),
loss = 'sparse_categorical_crossentropy',
metrics = 'accuracy')
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Introduction to Amplifiers
Module 8 - Introduction to Amplifiers
Pages i, 1-1, 1-11, 1-21, 1-31, 2-1, 2-11, 2-21, 2-31, 3-1, 3-11, 3-21, 3-31, 3-41, 3-51, 3-61, AI-1, Index
- Matter, Energy, and Direct Current
- Alternating Current and Transformers
- Circuit Protection, Control, and Measurement
- Electrical Conductors, Wiring Techniques, and Schematic Reading
- Generators and Motors
- Electronic Emission, Tubes, and Power Supplies
- Solid-State Devices and Power Supplies
- Amplifiers
- Wave-Generation and Wave-Shaping Circuits
- Wave Propagation, Transmission Lines, and Antennas
- Microwave Principles
- Modulation Principles
- Introduction to Number Systems and Logic Circuits
- - Introduction to Microelectronics
- Principles of Synchros, Servos, and Gyros
- Introduction to Test Equipment
- Radio-Frequency Communications Principles
- Radar Principles
- The Technician's Handbook, Master Glossary
- Test Methods and Practices
- Introduction to Digital Computers
- Magnetic Recording
- Introduction to Fiber Optics
Note: Navy Electricity and Electronics Training Series (NEETS) content is U.S. Navy property in the public domain.
The voltage drop across R1 can be computed:
The voltage at point a would be equal to the voltage of V1 minus the voltage drop of R1.
To check this result, compute the voltage drop across R2 and subtract this from the voltage at point
A. The result should be the voltage of V2.
It is not necessary that the voltage supplies be equal to create a point of virtual ground. In view (B) V1 supplies +1 volt to the circuit while V2 supplies -10 volts. The total difference in
potential is 11 volts. The total resistance of this circuit (R1 + R2) is 11 ohms. The total current (IT) is 1 ampere. The voltage drop across R1 (ER1 = R1 ´ IT) is 1 volt. The voltage drop across R2
(ER2 = R2 ´ I T) is 10 volts. The voltage at point a can be computed:
So point a is at virtual ground in this circuit also. To check the results, compute the voltage at V2.
You can compute the values for view (C) and prove that point a in that circuit is also at virtual ground.
The whole point is that the inverting input to the operational amplifier shown in figure 3-13 is at virtual ground since it is at 0 volts (for all practical purposes). Because the inverting input is
at 0 volts, there will be no current (for all practical purposes) flowing into the operational amplifier from the connection point of R1 and R2.
Given these conditions, the characteristics of this circuit are determined almost entirely by the values of R1 and R2. Figure 3-15 should help show how the values of R1 and R2 determine the circuit
Figure 3-15. - Current flow in the operational circuit.
Note: It should be stressed at this point that for purpose of explanation the operational amplifier is a theoretically perfect amplifier. In actual practice we are dealing with less than perfect. In
the practical operational amplifier there will be a slight input current with a resultant power loss. This small signal can be measured at the theoretical point of virtual ground. This does not
indicate faulty operation.
The input signal causes current to flow through R1. (Only the positive half cycle of the input signal
is shown and will be discussed.) Since the voltage at the inverting input of the operational amplifier is at 0 volts, the input current (Iin) is computed by:
The output signal (which is opposite in phase to the input signal) causes a feedback current (Ifdbk) to flow through R2. The left-hand side of R2 is at 0 volts (point A) and the right-hand side is at
Eout. Therefore, the feedback current is computed by:
(The minus sign indicates that Eout is 180 degrees out of phase with Ein and should not be confused with output polarity.)
Since no current flows into or out of the inverting input of the operational amplifier, any current reaching point a from R1 must flow out of point a through R2. Therefore, the input current (Iin)
and the feedback current (Ifdbk) must be equal. Now we can develop a mathematical relationship between the input and output signals and R1 and R2.
If you multiply both sides of the equation by R1:
If you divide both sides of the equation by Eout:
By inverting both sides of the equation:
You should recall that the voltage gain of a stage is defined as the output voltage divided by the input voltage:
Therefore, the voltage gain of the inverting configuration of the operational amplifier is expressed by the equation:
(As stated earlier, the minus sign indicates that the output signal is 180 degrees out of phase with the input signal.)
Noninverting Configuration
Figure 3-16 shows a noninverting configuration using an operational amplifier. The input signal (Ein) is applied directly to the noninverting (+) input of the operational amplifier. Feedback is
provided by
coupling part of the output signal (Eout) back to the inverting (-) input of the operational amplifier. R1 and R2 act as voltage divider that allows only a part of the output signal to be applied as
feedback (Efdbk).
Figure 3-16. - Noninverting configuration.
Notice that the input signal, output signal, and feedback signal are all in phase. (Only the positive alternation of the signal is shown.) It may appear as if the feedback is regenerative (positive)
because the feedback and input signals are in phase. The feedback is, in reality, degenerative (negative) because the input signals is applied to the noninverting input and the feedback signal is
applied to the inverting input, (Remember, that the operational amplifier will react to the difference between the two inputs.)
Just as in the inverting configuration, the feedback signal is equal to the input signal (for all practical purposes). This time, however, the feedback signal is in phase with the input signal.
Given this condition, you can calculate the gain of the stage in terms of the resistors (R1 and R2). The gain of the stage is defined as:
The feedback signal (Efdbk) can be shown in terms of the output signal (Eout) and the voltage divider (R1 and R2). The voltage divider has the output signal on one end and ground (0 volts) on the
other end. The feedback signal is that part of the output signal developed by R1 (at point A). Another way to look at it is that the feedback signal is the amount of output signal left (at point A)
after part of the output signal
has been dropped by R2. In either case, the feedback signal (Efdbk) is the ratio of R1 to the entire voltage divider (R1 + R2) multiplied by the output signal (Eout).
Mathematically, the relationship of the output signal, feedback signal, and voltage divider is:
if you divide both sides of the equation by Eout:
By inverting both sides of the equation:
Separating the right-hand side:
Therefore, by substitution:
You can now see that the gain of the noninverting configuration is determined by the resistors. The formula is different from the one used for the inverting configuration, but the gain is still
determined by the values of R1 and R2.
Bandwidth Limitations
As with most amplifiers, the gain of an operational amplifier varies with frequency. The specification sheets for operational amplifiers will usually state the open-loop (no feedback) gain for d.c.
(or 0 hertz). At higher frequencies, the gain is much lower. In fact, for an operational amplifier, the gain decreases quite rapidly as frequency increases.
Figure 3-17 shows the open-loop (no feedback) frequency-response curve for a typical operational amplifier. As you should remember, bandwidth is measured to the half-power points of a frequency-
response curve. The frequency-response curve shows that the bandwidth is only 10 hertz with this
configuration. The UNITY Gain Point, where the signal out will have the same amplitude as the signal in (the point at which the gain of the amplifier is 1), is 1 megahertz for the amplifier. As you
can see, the frequency response of this amplifier drops off quite rapidly.
Figure 3-17. - Open-loop frequency-response curve.
Figure 3-17 is the open-loop frequency-response curve. You have been told that most operational amplifiers are used in a closed-loop configuration. When you look at the frequency-response curve for a
closed-loop configuration, one of the most interesting and important aspects of the operational amplifier becomes apparent: The use of degenerative feedback increases the bandwidth of an operational
amplifier circuit.
This phenomenon is another example of the difference between the operational amplifier itself and the operational-amplifier circuit (which includes the components in addition to the operational
amplifier). You should also be able to see that the external resistors not only affect the gain of the circuit, but the bandwidth as well.
You might wonder exactly how the gain and bandwidth of a closed-loop, operational-amplifier circuit are related. Figure 3-18 should help to show you the relationship. The frequency-response curve
shown in figure 3-18 is for a circuit in which degenerative feedback has been used to decrease the circuit gain to 100 (from 100,000 for the operational amplifier). Notice that the half-power point
of this curve is just slightly above 10 kilohertz.
Figure 3-18. - Closed-loop frequency-response curve for gain of 100.
Now look at figure 3-19. In this case, more feedback has been used to decrease the gain of the circuit to 10. Now the bandwidth of the circuit is extended to about 100 kilohertz.
Figure 3-19. - Closed-loop frequency-response curve for gain of 10.
The relationship between circuit gain and bandwidth in an operational-amplifier circuit can be expressed by the Gain-Bandwidth Product (Gain ´ Bandwidth = UNITY Gain Point). In other words, for
operational-amplifier circuits, the gain times the bandwidth for one configuration of an operational amplifier will equal the gain times the bandwidth for any other configuration of the same
operational amplifier. In other words, when the gain of an operational-amplifier circuit is changed (by changing the value of feedback or input resistors), the bandwidth also changes. But the gain
times the bandwidth of the first configuration will equal the gain times the bandwidth of the second configuration. The following example should help you to understand this concept.
The frequency-response curves shown in figures 3-17, 3-18, and 3-19 have a gain-bandwidth product of 1,000,000. In figure 3-17, the gain is 100,000 and the bandwidth is 10 hertz. The gain-bandwidth
product is 100,000 times 10 (Hz), or 1,000,000. In figure 3-18, the gain has been reduced to 100 and the bandwidth increases to 10 kilohertz. The gain-bandwidth product is 100 times 10,000 (Hz) which
is also equal to 1,000,000. In figure 3-19 the gain has been reduced to 10 and the bandwidth is 100 kilohertz. The gain-bandwidth product is 10 times 100,000 (Hz), which is 1,000,000. If the gain
were reduced to 1, the bandwidth would be 1 megahertz (which is shown on the frequency-response curve as the unity-gain point) and the gain-bandwidth product would still be 1,000,000.
Q-19. What does the term "closed-loop" mean in the closed-loop configuration of an operational amplifier?
In answering Q20, Q21, and Q23, select the correct response from the choices given in the parentheses.
Q-20. In a closed-loop configuration the output signal is determined by (the input signal, the feedback signal, both).
Q-21. In the inverting configuration, the input signal is applied to the (a) (inverting, noninverting) input and the feedback signal is applied to the (b) (inverting, noninverting) input.
Q-22. In the inverting configuration, what is the voltage (for all practical purposes) at the inverting input to the operational amplifier if the input signal is a 1-volt, peak-to-peak sine wave?
Q-23. In the inverting configuration when the noninverting input is grounded, the inverting input is at (signal, virtual) ground.
Q-24. In a circuit such as that shown in figure 3-15, if R1 has a value of 100 ohms and R2 has a value of 1 kilohm and the input signal is at a value of + 5 millivolts, what is the value of the
output signal?
Q-25. If the unity-gain point of the operational amplifier used in question 24 is 500 kilohertz, what is the bandwidth of the circuit?
Q-26. In a circuit such as that shown in figure 3-16, if R1 has a value of 50 ohms and R2 has a value of 250 ohms and the input signal has a value of +10 millivolts, what is the value of the output
Q-27. If the open-loop gain of the operational amplifier used in question 26 is 200,000 and the open- loop bandwidth is 30 hertz, what is the closed loop bandwidth of the circuit?
Applications of Operational Amplifiers
Operational amplifiers are used in so many different ways that it is not possible to describe all of the applications. Entire books have been written on the subject of operational amplifiers. Some
books are devoted entirely to the applications of operational amplifiers and are not concerned with the theory of operation or other circuits at all. This module, as introductory material on
operational amplifiers, will show you only two common applications of the operational amplifier: the summing amplifier and the difference amplifier. For ease of explanation the circuits shown for
these applications will be explained with d.c. inputs and outputs, but the circuit will work as well with a.c. signals.
Summing Amplifier (Adder)
Figure 3-20 is the schematic of a two-input adder which uses an operational amplifier. The output level is determined by adding the input signals together (although the output signal will be of
opposite polarity compared to the sum of the input signals).
Figure 3-20. - Two-input adder.
If the signal on input number one (E1) is +3 volts and the signal on input number two (E2) is +4 volts, the output signal (Eout) should be -7 volts [(+3 V) + (+4 V) = +7 V and change the polarity to
get -7 V].
With +3 volts at E1 and 0 volts at point a (which is at virtual ground), the current through R1 must be 3 milliamperes.
(The + sign indicates a current flow from right to left.)
By the same sort of calculation, with +4 volts at E2 and 0 volts at point a the current through R2 must be 4 milliamps.
This means that a total of 7 milliamps is flowing from point a through R1 and R2. If 7 milliamps is flowing from point A, then 7 milliamps must be flowing into point A. The 7 milliamps flowing into
point a flows through R3 causing 7 volts to be developed across R3. With point A at 0 volts and 7 volts developed across R3, the voltage potential at Eout must be a -7 volts. Figure 3-21 shows these
voltages and currents. | {"url":"https://rfcafe.com/references/electrical/neets-modules/NEETS-Module-08-3-21-3-30.htm","timestamp":"2024-11-07T22:01:42Z","content_type":"text/html","content_length":"43102","record_id":"<urn:uuid:c20ed76c-d7b5-4fbd-ac20-4082d01b6f8e>","cc-path":"CC-MAIN-2024-46/segments/1730477028017.48/warc/CC-MAIN-20241107212632-20241108002632-00835.warc.gz"} |
P-persistent homology of finite topological spaces
Published in Rendiconti Sem. Mat. Univ. Pol. Torino, 2015
Vaccarino, F., Patania, A., & Petri, G. (2017). Rendiconti Sem. Mat. Univ. Pol. Torino. Vol. 74, 1. 27 - 45.
Let P be a finite partially ordered set, briefly a finite poset. We will show that for any P-persistent object X in the category of finite topolog- ical spaces, there is a P− weighted graph, whose
clique complex has the same P-persistent homology as X. ArXiv pre-print | {"url":"https://alpatania.github.io/publication/2015-02-18-one-graph","timestamp":"2024-11-03T21:39:13Z","content_type":"text/html","content_length":"11771","record_id":"<urn:uuid:2c43b661-3243-4ea9-af6c-b784ce4a84dc>","cc-path":"CC-MAIN-2024-46/segments/1730477027796.35/warc/CC-MAIN-20241103212031-20241104002031-00215.warc.gz"} |
Merkle Trees
In our last lesson, we looked at cryptographic hash functions and attacks against them. Cryptographic hash functions are used in almost every protocol that guarantees integrity and authentication.
But in a blockchain setting, one of their most important applications is in Merkle trees.
From the original patent on Merkle trees.
Merkle trees were invented by Ralph Merkle, one of the forefathers of modern cryptography. Though he patented the Merkle tree in 1982, the patent on them has long expired. Merkle trees are used
widely in many applications, including Git, BitTorrent, ZFS, the Certificate Transparency framework, and of course, pretty much every cryptocurrency.
Let's walk through a simple example of how we might use a Merkle tree in an application.
Building up a Merkle tree
Say we're designing a file sharing protocol. We'll assume the files we're sharing are large—say, Linux distros.
In this protocol, once a user finishes downloading a file, they need to somehow verify it wasn't corrupted in transit. TCP itself can correct most random errors, but even so, corruption errors are
common when dealing with files of this size. How can we ensure integrity at the application layer?
Here's an idea: let's ship a cryptographic hash alongside the Linux ISO. That way, after the user is done downloading the ISO, they can hash it and check if the digests match. Hashing is pretty
fast—even on a single core you can hash hundreds of megabytes per second.
The Linux distro is now shipped alongside its hash.
But what if the hashes don't match? How do you know where the error in the file was?
Actually, you have no way of knowing where the error was. You have to throw out all of the 2GB and restart the download, hoping this time the corruption doesn't happen. This seems like a lousy
Here's an idea: what if we break up the data into blocks and ship hashes for each block?
This is nice! Now if we have a random corruption in our data, instead of downloading all 2GB over again, we can check which of the 250MB blocks came out corrupted and then only re-download that
block. The downside is that we need to ship all 8 hashes alongside the download:
Block 1 digest: 743bacf85062c45f533060e3abb3dc1f02683269
Block 2 digest: 4db3539ecf02e69ffacec45751129e38bdfa469e
Block 3 digest: 0f34d88b134aece384b3d079996d08ee7eadecfd
Block 4 digest: 5bbc8a852490d2f9d9e0067f408950539af0bda9
Block 5 digest: 2ba7cb929bc68c6706b8be03946add92e941999d
Block 6 digest: ee389b56b4a6ff4407df5ffd5f675c2fa73edd22
Block 7 digest: d446b49425506732cfd707865908a99a1ab15ba7
Block 8 digest: 1f706fc900b2c4543525ea1434a201d169004a3d
For 250MB blocks this is probably fine, but if we want smaller blocks to minimize the impact of corruptions, then we need more than 8 hashes. If we wanted 128KB blocks, we'd need 15,000 hashes, and
if we wanted 8KB blocks, we'd need 250,000 hashes. This becomes unwieldy pretty fast.
Here's where Merkle trees come in. Merkle trees are a kind of cryptographic accumulator. Cryptographic accumulators allow you to compact arbitrarily many pieces of data into a constant amount of
space. In other words, a Merkle tree lets us represent arbitrarily many blocks while only transmitting a single hash across the wire.
Merkle trees are also known as hash trees, because they hash data upwards in a tree. It's easy to explain in code—here's how you can create a hash tree with only two elements.
from hashlib import sha1
def h(s): return sha1(s.encode()).hexdigest() # hashing helper function
block1 = "Block 1"
block2 = "Block 2"
digest1 = h(block1)
digest2 = h(block2)
root = h(digest1 + digest2)
# d1c6d4f28135f428927a1248d71984a937ee543e
(This diagram uses the notation h(1, 2) for legibility, but it's actually h(h(1) + h(2)).)
By concatenating the two digests and taking their hash, the root of the hash tree commits to both digests. Think about it: if there were some other way to produce this same root, then that would
imply the existence of a hash collision. Hash collisions should be impossible for a strong cryptographic hash function. Thus, the root of this hash tree, known as the Merkle root, must be a unique
identifier of this exact tree.
(If you don't follow this argument, play around with another example in code! This intuition is really important, and we'll continue to build on it.)
The Merkle root is therefore an accumulator over all of the original data that was hashed to produce this tree. It also commits to that data in order, since we used string concatenation on the
underlying blocks to combine their values. (If you had used a commutative operation like addition or XOR instead of concatenation, then technically you could've switched the order of some blocks and
gotten the same root. This is undesirable, so don't make that mistake.)
How does this scale up to many blocks though? Pretty simple. We repeat this same operation across the data in layers until we get a single root.
So the root of this tree is 6c2d5a56f541df426366aebb4db927113016387a. Notice that if you modified any element of the tree, even by 1 bit, then the avalanche effect of the hash would cause every hash
upstream to change, all the way up to the root.
Now, say we downloaded the Linux distro along with its Merkle root (a single hash). We recompute the Merkle tree over the Linux distro on our side, and we find that our root doesn't match the one we
were provided. This means our file is corrupted.
How can we quickly diagnose which of the blocks we downloaded was faulty? See if you can figure this out for yourself.
Here's the answer: we have to request the two hashes below the root in the canonical Merkle tree, and figure out which hash doesn't match up with our client-side tree. Once we've figured it out which
subtree is faulty, we can repeat this for the two children of that subtree, and so on until we reach the base. Assuming there's a single faulty block, this will let you pinpoint that block with only
\(O(\log{n})\) comparisons (where \(n\) is the number of underlying data blocks).
Inclusion proofs
We've seen how powerful Merkle trees are for verifying file integrity. But the real power of cryptographic accumulators comes not just in accumulating data, but in then being able to efficiently
prove claims about the data.
Imagine an accumulator as an opaque box full of items. You can't directly look inside this box, but with the magic of cryptography, you can query it in specific ways.
One of the operations you can perform with a cryptographic accumulator is an inclusion proof. This is a small proof that decisively attests that an item exists in the accumulator.
If you know the Merkle root of an e-book, how can I efficiently prove to you that a certain quotation comes from that e-book? I can do this without providing you the entire e-book or even the entire
Merkle tree.
Take a moment and see if you can sketch out how to do this without reading on.
Have an idea? The animation below demonstrates the answer for a simple 4-word e-book.
We only need to provide the data we're proving exists, the Merkle root, and sibling hashes along the path from the leaf up to the root. This should require only \(O(\log{n})\) hashes to transmit over
the wire. If you redo all of the hashing and the roots match, you will know with certainty that quotation was indeed part of the e-book. This kind of proof is known as a Merkle proof.
You should be asking: why is this sufficient for an inclusion proof? What if someone just makes up the sibling hashes to make the roots match? How do we know they came from the real Merkle tree?
I'll leave this to you to think through for yourself.
Merkle trees in Bitcoin
Inclusion proofs are a powerful primitive enabled by Merkle trees. They are used extensively in Bitcoin light clients, also known as SPV (Simple Payment Verification) nodes. We'll do a quick sneak
preview of how this works.
In Bitcoin, all the transactions that took place in the last ~10 minutes are bundled together into a block and transmitted to everyone in the network. These blocks can be quite large, since they
potentially contain thousands of transactions.
To save on bandwidth, Bitcoin pulls off a clever trick: instead of transmitting all of the transactions, the transmitted block only includes a Merkle root of that block's transactions. (In practice,
this transmitted data is known as a block header, while the transactions themselves are transmitted separately on request. We'll learn more about this later.)
Credit: LetsTalkBitcoin.com
Since a Merkle root is a cryptographic accumulator over all the underlying ordered data, every block header includes a commitments to all the transactions in that block. Because of this optimization,
lightweight clients only need to keep track of block headers and can selectively verify Merkle proofs that a certain transaction was included in a given block. This optimization is essential for
mobile phones or web wallets to be able to use the Bitcoin network without having to download everything.
Don't worry if that's confusing; there's a lot of structure to Bitcoin that we'll explain in upcoming lessons. But by now you have gotten a glimpse of how useful Merkle trees can be.
There are many more innovations on Merkle trees that are worth exploring, including proofs of non-inclusion, online updates, and n-ary Merkle trees. We'll provide resources in the additional reading
if you want to see how the state of the art has evolved. We will also provide reading on a subtle second preimage attack against a naive implementation of a Merkle tree (though it's not of practical
use in a blockchain setting).
In our next assignment, you'll be writing your own implementation of a Merkle tree. You'll then be writing code to verify Merkle proofs.
Save your code from this, as you may find it useful for your cryptocurrency project if you choose to implement a Merkle tree.
Once you've completed the coding assignment, you're ready to move on.
Additional reading | {"url":"https://nakamoto.com/merkle-trees/","timestamp":"2024-11-05T00:36:29Z","content_type":"text/html","content_length":"42830","record_id":"<urn:uuid:bdf64d73-c191-44fd-b923-020a5c60375f>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.84/warc/CC-MAIN-20241104225856-20241105015856-00591.warc.gz"} |
changes the selected matrix inside the sw object
function setmatrix(obj, varargin)
changes the selected matrix inside the sw object
SETMATRIX(obj, 'Option1', Value1, ...)
obj sw class object.
One of the below options has to be given:
label Label of the matrix that is already assigned to either as
anisotropy, g-tensor or coupling.
mat_idx Index of the matrix, stored in obj.matrix. Alternative to
the 'label' option.
coupling_idx Value of the obj.coupling.idx, that defines the coupling,
for which the symmetry allowed matrix elements will be set.
aniso_idx Value of the obj.matom.idx, that selects a magnetic atom,
for which the symmetry allowed matrix elements will be
set for single ion anisotropy.
g_idx Value of the obj.matom.idx, that selects a magnetic atoms,
for which the symmetry allowed matrix elements will be
set for the g-tensor.
Optional inputs:
pref Defines prefactors as a vector for the symmetry allowed
components, dimensions are [1 nSymMat]. Alternatively, if only
a few of the symmetry allowed matrices have non-zero
prefactors, use:
{[6 0.1 5 0.25]}
This means, the 6th symmetry allowed matrix have prefactor 0.1,
the 5th symmetry allowed matrix have prefactor 0.25. Since
Heisenberg isotropic couplings are always allowed, a cell with
a single element will create a Heisenberg coupling, example:
This is identical to obj.matrix.mat = eye(3)*0.1
For DM interactions (antisymmetric coupling matrices), use
three element vector in the cell:
{[D1 D2 D3]}
In this case, these will be the prefactors of the 3
antisymmetric symmetry allowed matrices. In case no crystal
symmetry is defined, these will define directly the components
of the DM interaction in the xyz coordinate system. Be
carefull with the sign of the DM interaction, it depends on the
order of the two interacting atoms! Default value is {1}.
For anisotropy matrices antisymmetric matrices are not allowed.
The selected obj.matrix.mat will contain the new value.
setmatrix(crystal,'label','J1','pref',{[6 0.235]})
This example will set 'J1' coupling to the 6th symmetry allowed matrix,
with prefactor 0.235.
This will set 'J2' to antiferromagnetic Heisenberg exchange, with value
of 1.25 meV.
See also SW, SW.GENCOUPLING, SW.GETMATRIX.
This function calls: This function is called by: Generated on Mon 26-Nov-2018 15:08:42 by m2html © 2005. iFit (c) E.Farhi/ILL EUPL 1.1 | {"url":"http://ifit.mccode.org/techdoc/Applications/SpinW/m_files/@sw/setmatrix.html","timestamp":"2024-11-11T20:52:19Z","content_type":"text/html","content_length":"5658","record_id":"<urn:uuid:9ada98f7-d26d-4bb6-add9-44bf22943213>","cc-path":"CC-MAIN-2024-46/segments/1730477028239.20/warc/CC-MAIN-20241111190758-20241111220758-00351.warc.gz"} |
Maximum Number of Clusters
The maximum number of clusters that can be fit in an LCA model depends on the model degrees of freedom. The degrees of freedom in an LCA model are based on the size of the contingency table created
by the columns. The size of the contingency table is the number of cells in the table that contain at least one observation and is denoted as K. If all cells contain at least one observation, K is
the product of the number of levels of the response columns. The formula for degrees of freedom is defined as follows:
DF = K - {nCluster - 1 + nCluster(nTotalLevels - nCols)} - 1
nCluster = the number of clusters
nTotalLevels = the sum of the levels of the response columns
nCols = the number of response columns
In order for the LCA model to be adequately fit, the degrees of freedom must be positive. Therefore, to ensure DF > 0, the maximum number of clusters is defined as follows:
max(nCluster) < floor[K/(1 + nTotal Levels − nCols)] | {"url":"https://www.jmp.com/support/help/en/16.2/jmp/maximum-number-of-clusters.shtml","timestamp":"2024-11-06T10:55:57Z","content_type":"application/xhtml+xml","content_length":"6683","record_id":"<urn:uuid:8a7f3b96-c031-4156-b37c-08c4a7d147d3>","cc-path":"CC-MAIN-2024-46/segments/1730477027928.77/warc/CC-MAIN-20241106100950-20241106130950-00182.warc.gz"} |
Parallelization of Antenna and Array Analyses
This example shows how to speed up antenna and array analysis using Parallel Computing Toolbox™.
Analyzing antenna performance over a frequency band is an important part of antenna design. There are various analysis methods that are supported by Antenna Toolbox™:
• Port analyses include impedance, returnLoss, sparameters, vswr, efficiency, resonantFrequency, and bandwidth.
• Surface analyses include current, feedCurrent, and charge.
• Field analyses include pattern and its variations, EHfields, axialRatio, and beamwidth.
This example demonstrates the advantages of parallel computing for computing axial ratio and return loss over a frequency band. For axial ratio, we illustrate the manual setup of parfor for speeding
up calculations. For return loss, we show how the UseParallel option, which is supported for all port analysis methods, can be used to automatically enable parallel computing.
Axial Ratio Computation without Parallel Computing
Compute the axial ratio of an Archimedean spiral over a frequency band of 0.8 to 2.5 GHz in steps of 100 MHz. Perform this calculation serially, and save the time taken to perform these computations
in variable time1.
sp = spiralArchimedean(Turns=4,InnerRadius=5.5e-3,OuterRadius=50e-3);
freq = 0.8e9:100e6:2.5e9;
AR = zeros(size(freq));
for m = 1:numel(freq)
AR(m) = axialRatio(sp, freq(m), 0, 90);
time1 = toc;
Axial Ratio Computation with Parallel Computing
Repeat the calculations using Parallel Computing Toolbox to reduce computation time. Use the function parpool to create a parallel pool cluster. Then, use parfor to calculate the axial ratio over the
same frequency band. The time taken to perform the computation is saved in variable time2.
Starting parallel pool (parpool) using the 'Processes' profile ...
Connected to parallel pool with 8 workers.
sp = spiralArchimedean(Turns=4,InnerRadius=5.5e-3,OuterRadius=50e-3);
ARparfor = zeros(size(freq));
parfor m = 1:numel(freq)
ARparfor(m) = axialRatio(sp, freq(m), 0, 90);
time2 = toc;
Sometimes parfor will be slower on the first run because the code needs to become available to the workers. Re-run the code above for a second time to re-evaluate the elapsed time.
sp = spiralArchimedean(Turns=4,InnerRadius=5.5e-3,OuterRadius=50e-3);
ARparfor = zeros(size(freq));
parfor m = 1:numel(freq)
ARparfor(m) = axialRatio(sp, freq(m), 0, 90);
time2 = toc;
Axial Ratio Computation Time Comparison
The table below shows the time taken for axial ratio analysis with and without Parallel Computing Toolbox. The cluster information is saved in the variable pardata.
cases = ["Without Parallel Computing" "With Parallel Computing"];
time = [time1; time2];
numWorkers = [1; pardata.NumWorkers];
time numWorkers
______ __________
Without Parallel Computing 8.0698 1
With Parallel Computing 1.4435 8
disp("Speed-up due to parallel computing = " + time1/time2)
Speed-up due to parallel computing = 5.5902
The plot below shows the axial ratio data calculated for two cases. The results are identical.
plot(freq./1e9, AR,"r+", freq./1e9, ARparfor,"bo")
grid on
xlabel("Frequency (GHz)")
ylabel("Axial ratio (dB)")
title("Axial Ratio of Archimedean Spiral Antenna at Boresight")
legend("Without parallel computing","With parallel computing", ...
During this analysis the antenna structure is meshed at every frequency and then the far-fields are computed at that frequency to compute the axial ratio. One way of reducing the analysis time is to
mesh the structure manually by specifying a maximum edge length.
Return Loss Computation without Parallel Computing
The previous section performed a field analysis computation. All field and surface analysis computations in Antenna Toolbox™ accept only scalar frequency as input. However, returnLoss and all other
port analysis functions accept a frequency vector as input.
When a frequency vector is specified as an input, the antenna structure is meshed at the highest frequency. The resulting mesh is used for performing the analysis over the specified frequency band.
The CPU time taken to perform the computation is saved in variable time3.
sp = spiralArchimedean(Turns=4,InnerRadius=5.5e-3,OuterRadius=50e-3);
RL = returnLoss(sp, freq);
time3 = toc;
Return Loss Computation with Parallel Computing
To use parallel computing to compute the return loss, set UseParallel to true. The time taken to perform the computation is saved in variable time4.
Alternatively, to manually set up a parfor loop for return loss calculations, mesh the structure at the highest frequency and use the parfor loop to run a frequency sweep. You cannot use the parfor
loop by passing a single frequency at a time (as shown in the discussion about axialRatio) because the meshing happens at every frequency, limiting the advantage of parallel computing and potentially
producing different results from the computations performed without parallel computing.The solution is to run the analysis at the highest frequency first, get the mesh information using the
MeshReader object, and use the maximum edge length in the mesh to ensure the same mesh is used for all the computations.
sp = spiralArchimedean(Turns=4,InnerRadius=5.5e-3,OuterRadius=50e-3);
switch "Automatic parfor"
case "Automatic parfor"
RLparfor = returnLoss(sp, freq, UseParallel=true);
time4 = toc;
case "Manual parfor"
RLparfor = zeros(size(freq));
RLparfor(end) = returnLoss(sp, freq(end));
meshdata = mesh(sp);
[~] = mesh(sp, MaxEdgeLength=meshdata.MaxEdgeLength);
parfor m = 1:(numel(freq)-1)
RLparfor(m) = returnLoss(sp, freq(m));
time4 = toc;
Return Loss Computation Time Comparison
The table below indicates the time taken for calculating the return loss with and without Parallel Computing Toolbox. The cluster information is saved in the variable pardata.
cases = ["Without Parallel Computing";"With Parallel Computing"];
time = [time3; time4];
numWorkers = [1; pardata.NumWorkers];
time numWorkers
______ __________
Without Parallel Computing 3.6094 1
With Parallel Computing 4.7136 8
disp("Speed-up due to parallel computing = " + time3/time4)
Speed-up due to parallel computing = 0.76576
The plot below shows the return loss data calculated for two cases. The results are identical.
plot(freq./1e9, RL,"r+", freq./1e9, RLparfor,"bo")
grid on
xlabel("Frequency (GHz)")
ylabel("Return loss (dB)")
title("Return Loss of Archimedean Spiral Antenna")
legend("Without parallel computing","With parallel computing", ...
Delete the current parallel pool.
Parallel pool using the 'Processes' profile is shutting down.
See Also | {"url":"https://uk.mathworks.com/help/antenna/ug/parallelization-of-antenna-and-array-analyses.html","timestamp":"2024-11-08T07:49:44Z","content_type":"text/html","content_length":"82503","record_id":"<urn:uuid:90a94331-cd9d-47be-8d1d-eb38f9f2f917>","cc-path":"CC-MAIN-2024-46/segments/1730477028032.87/warc/CC-MAIN-20241108070606-20241108100606-00892.warc.gz"} |
More lessons from complexity
The Logistic Map
Recall the exotic dynamics of the logistic map (see for example, Rasband, 1990; Peitgen, et al., 1992; Beck & Schlögl, 1993):
X[k+1] = a X[k] · (1 - Xk)
that is, the chain of ultimately stable (and unstable) values X[8](a) found iterating the map, where X[k]s denotes the normalized size of a population at generation k and a is a free parameter having
values between 0 and 4:
When a = 1, the logistic parabola is below the one to one line (added to aid in the calculations), and then X[8] = 0 (Figure 1);
When 1 < a = 3, the parabola is above the line X = Y and X[8] = (a - 1)/a , the non-zero intersection between the curve and the straight line (Figure 2);
When 3 < a = 3.449…, [X8] = {X[8](1), X[8](2)} and the population settles into an oscillation repeating every two generations (Figure 3);
When 3.449… < a = 3.544…, X[8] = {X[8](^3), X8(^4), X[8](^5), X[8](^6)}. The population ultimately repeats every four generations, and the dynamics have experienced a bifurcation (Figure 4);
When a is increased up to a value a[8]= 3.5699…, successive bifurcations in powers of two happen quickly, that is, the dynamics repeat exactly every 2^n generations, for any value of n;
When a[8] < a = 4, behavior is found either periodic or non periodic. For instance, for a = 3.6 an infinite strange attractor with a whole in the middle is found (Figure 5);
When a = 3.83, X[8] = {X[8](^1), X[8](^2), X[8](^3)} and the dynamics oscillate every 3 generations (Figure 6);
When a = 4, the most common behavior is non periodic and a dense strange attractor over the interval [0, 1] is found (Figure 7).
At the end, the cascade of stable period-doubling bifurcations (before a[8]) and the emergence of chaos (strange attractors) intertwined with periodic behavior (including any period greater than two)
is summarized via the celebrated Feigenbaum’s diagram (Figure 8).
This is so named after Mitchell Feigenbaum who showed that the bifurcation openings and their durations happen universally for a class of unimodal maps according to two universal constants F[1] and F
[2], as follows (Feigenbaum, 1978) (refer to Figure 9):
d[n]/d[n+1] ? F[1] = -2.5029…, ?[n]/?[n+1] ? F[2] = 4.6692…
For example, other “fig trees” guided by F[1] and F[2] and for the two simple mappings f(X) = a X · (1—X ^3) and f(X) = a X · (1 - X)^3 are shown below[1]. Notice how such contain: a straight “root,”
a bent “branch,” bifurcation branches, and then, in an orderly intertwined fashion, following Sharkovskii’s order (see for example, Rasband, 1990; Peitgen, et al., 1992; Beck & Schlögl, 1993),
periodic branches, and the ever dusty “foliage of chaos,” where the unforgiving condition of sensitivity to initial conditions rules.
Chaos theory and our quest for peace
As the dynamics of the logistic map describe several physical processes (see for instance, Cvitanovic, 1989; Bai-Lin. 1984), including fluid turbulence as induced by heating, that is, convection, it
is pertinent to consider such a simple and universal mechanism to study how “chaos” and its related condition of “violence” may arise in the world.
Given that the key parameter a, associated with the amount of heat (Libchaber & Maurer, 1978), dictates the ultimate organization of the fluid, we may see that it is wise to keep it small (in the
world, and within each one of us) in order to avoid undesirable “nonlinearities.” For although the allegorical fig trees exhibit clear order in their pathway towards disorder, we may appreciate in
the uneasy jumping on strange attractors (and also on periodic ones), the anxious and foolish frustration we often experience (so many times deterministically!) when we, by choosing to live in a
hurry, travel from place to place to place in “high heat” without finding our “root.”
In this spirit, the best solution for each one of us is to slow down altogether the pace of life, coming down the tree, so that by not crossing the main thresholdX = Y, that is, by choosing a = 1, we
may surely live without turbulence and chaos in the robust state symbolized by X8 = 0[2,3]. For there is a marked difference between a seemingly laminar condition as it happens through tangent
bifurcations (see for example, Rasband, 1990; Peitgen, et al., 1992; Beck & Schlögl, 1993) and being truly at peace, for the former invariably contains dramatic bursts of chaos and ample
intermittency (see for example, Rasband, 1990; Peitgen, et al., 1992; Beck & Schlögl, 1993).
As zero, that is, converging to the origin, is identified as the desired state, it is sensible to realize that such an organization, a trivial solution for X[8](a), even if unstable, may be reached
even when the worst chaos engulfs us (a = 4). For the precise dynamics of the pre-images of zero do not wander for ever in high heat, but rather find the way to the root through a delicate hopscotch
by the middle[4] (Figure 11). For it is tragic indeed to “oscillate for ever” (Figure 12). And more tragic yet to be close to “the point” and miss it altogether forever[5] (Figure 13). For the
butterfly effect, with all probability and contrary to the illusion that it provides us with options, leaves us irremediably trapped in dust.
At the end, the emergence of the modern science of complexity helps us visualize our ancient choices. It is indeed best for us to live in serenity and in a simple manner, not amplifying and hence
heeding the voice. For only the conscious order of Love does not suffer the destiny of arrogant stubbornness that justly receives the same “bad luck” of a parabolic tree that did not have any fruit,
the same one that with its tender branch(es) and budding leaves, also announces horrendous times, but also very good ones, times of joy and of friendship. | {"url":"https://journal.emergentpublications.com/Article/e128fc6b-377d-4020-9111-08d32c6e071f/newsprint","timestamp":"2024-11-06T23:12:37Z","content_type":"text/html","content_length":"30013","record_id":"<urn:uuid:43f63746-8f1c-40a8-a219-0847292bc312>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00872.warc.gz"} |
An improved version of the random-facet pivoting rule for the simplex algorithm
The Random-Facet pivoting rule of Kalai and of Matoušek, Sharir and Welzl is an elegant randomized pivoting rule for the simplex algorithm, the classical combinatorial algorithm for solving linear
programs (LPs). The expected number of pivoting steps performed by the simplex algorithm when using this rule, on any linear program involving n inequalities in d variables, is 2^O(√(n-d) log(d/
√n-d)), where logn = max{1, logn}. A dual version of the algorithm performs an expected number of at most 2^O(√d log ((n-d)/√ d)) dual pivoting steps. This dual version is currently the fastest known
combinatorial algorithm for solving general linear programs. Kalai also obtained a primal pivoting rule which performs an expected number of at most 2^O(√logn) pivoting steps. We present an improved
version of Kalai's pivoting rule for which the expected number of primal pivoting steps is at most min {2^O(√(n-d)log (d/(n-d)), 2^O(√(d log ((n-d/d))} This seemingly modest improvement is
interesting for at least two reasons. First, the improved bound for the number of primal pivoting steps is better than the previous bounds for both the primal and dual pivoting steps. There is no
longer any need to consider a dual version of the algorithm. Second, in the important case in which n = O(d), i.e., the number of linear inequalities is linear in the number of variables, the
expected running time becomes 2^O (√) rather than 2^O(√dlogd). Our results, which extend previous results of Gartner, apply not only to LP problems, but also to LP-type problems, supplying in
particular slightly improved algorithms for solving 2-player turn-based stochastic games and related problems.
Publication series
Name Proceedings of the Annual ACM Symposium on Theory of Computing
Volume 14-17-June-2015
ISSN (Print) 0737-8017
Conference 47th Annual ACM Symposium on Theory of Computing, STOC 2015
Country/Territory United States
City Portland
Period 14/06/15 → 17/06/15
• Linear programming
• Randomized pivoting rules
• Simplex algorithm
Dive into the research topics of 'An improved version of the random-facet pivoting rule for the simplex algorithm'. Together they form a unique fingerprint. | {"url":"https://cris.tau.ac.il/en/publications/an-improved-version-of-the-random-facet-pivoting-rule-for-the-sim","timestamp":"2024-11-14T03:56:51Z","content_type":"text/html","content_length":"54944","record_id":"<urn:uuid:795e19af-a7e2-4dbf-a810-7cca6fc8274a>","cc-path":"CC-MAIN-2024-46/segments/1730477028526.56/warc/CC-MAIN-20241114031054-20241114061054-00442.warc.gz"} |
Talk by Roberto Natalini
On March 30th, 2022, Dr. Roberto Natalini (CNR Rom) gave a talk about "Mean field and macroscopic limits for hybrid model of cell motion" as part of the research seminar Analysis of the
FernUniversität in Hagen. This lecture is partially supported by the COST action Mathematical models for interacting dynamics on networks.
In this talk I focus on a quite general class of hybrid mathematical models of collective motions of cells under the influence of chemical stimuli. The models are hybrid in the sense that cells are
discrete entities given by ODE, while the chemoattractant is considered as a continuous signal which solves a diffusive equation. For these models it is possible to prove the mean-field limit in the
Wasserstein distance to a system given by the coupling of a Vlasov-type equation with the chemoattractant equation. This approach and results are not based on empirical measures, but rather on
marginals of large number of individuals densities, and we show the limit with explicit bounds, by proving also existence and uniqueness for the limit system. In the monokinetic case we derive new
pressureless nonlocal Euler-type model with chemotaxis.
These results have been obtained in collaboration with Thierry Paul.
Video of the talk | {"url":"https://www.fernuni-hagen.de/analysis/en/research/natalini.shtml","timestamp":"2024-11-10T17:53:09Z","content_type":"text/html","content_length":"19214","record_id":"<urn:uuid:11439b54-dd59-4137-894b-bd15a8c2252c>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.61/warc/CC-MAIN-20241110170046-20241110200046-00089.warc.gz"} |
How do you convert dimensions into inches?
How do you convert dimensions into inches?
As an example, let’s say you have a piece of wood measuring 50cm and you want to convert it into inches. To get your answer, divide your cm figure by 2.54. So, 50 รท 2.54 = 19.685 inches.
How do you convert dimensions to centimeters?
How to convert inches to centimetres? To convert inches to centimetres, multiply the given inch value by 2.54 centimetres. For example, to convert 7 inches to centimeters, multiply 7 by 2.54.
How are dimensions listed in order?
When you tell us the dimensions of the box, they need to be in this order, Length x Width x Depth. Get a Quote Today!
How do you convert dimensions into centimeters?
How do dimensions go in order?
When you tell us the dimensions of the box, they need to be in this order, Length x Width x Depth.
What comes first length or width or height or depth?
The longer dimension is the length and comes first in order. The other dimension is the width. Now, measure the distance from one of the inside corners to the top. This is the depth.
How can we convert a measure into a dimension?
Drag Sales to Rows and Discount to Columns.
To treat Discount as a dimension,click the drop-down arrow on the field (on the Columns shelf) and select Dimension from the context menu.
To complete the process,click the drop-down arrow on the Discount again and select Discrete from the context menu.
How to do measurements conversions?
It is important to convert all the different units of length into one common unit before solving the questions.
The conversion among the metric system can be done by multiplying or dividing by 10 or the multiples of 10.
Meter is also called the SI unit of length or base unit of length.
How do you convert centimeters to millimeters?
– 1 cm = about 0.4 inches – 1 cm = 10 mm – 1 inch = exactly 2.54 cm (you don’t need this for this conversion, but it’s a good one to know)
How do you convert area to square feet?
To find length,locate the longest side of the area to be measured.
Fix a tape measure or other measuring tool to one end of the length and extend it to the other end.
Record the measurement.
To find the width,locate the shortest side of the area to be measured. Repeat the process and record that measurement. | {"url":"https://eleanorrigby-movie.com/how-do-you-convert-dimensions-into-inches/","timestamp":"2024-11-03T00:11:44Z","content_type":"text/html","content_length":"74986","record_id":"<urn:uuid:f9f93b83-b042-486a-a9f4-99bdaa8b1d37>","cc-path":"CC-MAIN-2024-46/segments/1730477027768.43/warc/CC-MAIN-20241102231001-20241103021001-00629.warc.gz"} |
Find Missing Positive - Interview Problem
First Missing Positive
Difficulty: Hard
Asked in: Amazon, Google
Understanding the Problem
Problem Description: Given an array, find the smallest missing positive integer.
For example
Input: A[] = [2, 3, -7, 6, 8, 1, -10, 15]
Output: 4
Input: A[] = [3, 2, -1, 1]
Output: 4
Input: A[] = [7, 8, 9, 11, 12]
Output: 1
Possible follow-up questions to ask the interviewer:-
• Are there negative integers present in the array? ( Ans : Yes)
• Can we update the array? ( Ans : Yes)
• Are there any repeating values in the array? ( Ans : Yes)
Brute force and Efficient Solutions
We are discussing four ways to solve this problem :
1. Brute force approach
2. Using Sorting and searching
3. Using a Hash Table
4. Using In-place Hash
1. Brute force approach
The simple solution is to search all positive integers starting from 1 ( Think! ). At worst case, we have to search for n+1 numbers.
Pseudo Code
int smallestPositive(int A[], int n)
for(i = 1 to n+1) // search for 1 to n+1 elements
bool flag = false
for(j = 0 to n-1) // iterating through the array
if(A[j] == i)
flag = true
if(flag == false)
return i
Complexity Analysis
Time Complexity: Search positive integers from 1 to n+1 linearly in the array = O(n²)
Space Complexity: O(1)
Critical ideas to think!
• Can we optimize the search if the array is in a sorted order? Think about the complexities of searching?
2.Using Sorting and Searching
We can sort the array and linearly scan the array to find the smallest positive element.
Pseudo Code
int smallestPositive(int A[], int n)
small = 1
for(i = 0 to n-1)
if(small > A[i]) // to handle repetition and non positive numbers
if(small == A[i])
small = small + 1
if(A[i] > small)
return small
return small
Complexity Analysis
Suppose we are using heap sort to solve this problem.
Time Complexity: Sorting + Linear Scan for finding smallest positive
= O(nlogn) + O(n) = O(nlogn)
Space Complexity: O(1)
Critical ideas to think!
• What if all the elements that are present in the array are negative? Does the above algorithm cover this condition? Analyze this with an example.
• Compare different sorting algorithms that can be used to solve the above problem.
3. Using Hash Table
We can build a hash table to store the integers present in the given array. Now we can look out for positive integers starting from 1 to n+1 in the Hash Table.
Solution Steps
• Create a hash table H.
• Insert the elements of the array in the hash table.
• Iterate i from 1 to n+1 and look for i in H.
• If i is not present in H , return i.
• Else, continue the iteration.
Pseudo Code
int smallestPositive(int A[], int n)
Create a hash table H
for(i = 0 to n-1)
H[A[i]] = true
// search for first n+1 positive integer
for(i = 1 to n+1)
// integer not present in the hash table
if(i not in H)
return i
Complexity Analysis
Time Complexity: Building hash table + Searching elements in Hash Table = O(n) + O(n) = O(n)
Space Complexity: O(n), To store the hash table.
Critical ideas to think!
• Do we need to store non-positive integers in the hash map?
4. In-place Hash
The idea is to store the occurrences of the numbers itself in the array. Since the range for missing elements for an array of size n is [1,n+1], so we can use the value of the index in the array to
mark the presence of a number in the array (keeping in mind to retrieve the original elements after updating it).How should we mark the presence of an element? ( Think! )
When we come across an element k, where 1 ≤ k ≤ N, we will update the value at index (k-1) to its negative value, i.e. A[k-1] = -A[k-1]. (Why index k-1 and not k?) But this approach can still fail
for some types of cases, can you guess which cases?
We need to take care of not accidentally modifying the value at the same index more than once when we encounter duplicates. Also, this approach doesn’t work if there are non-positive numbers. So we
first need to segregate positive numbers from negative numbers and then apply the above discussed in-place hashing approach.
Solution Steps
• Segregate the positive numbers from others, i.e. to move all negative integers to the right side of the array and return the size of the sub-array containing the positive integers (which is N
here). We will use a helper function segregate() for this purpose. This function will use the two-pointer approach similar to the partition function in the Quick sort.
• Now iterate through this sub-array and mark the presence of an element A[i], if abs(A[i])-1 < N .
• To ensure that we don't do so more than once on the same index, we assign A[abs(A[i])-1] = - abs( A[abs(A[i])-1] ) .
• Now scan through the array till N and find the first positive element and return its index i.e i+1 . If we didn't find any positive value after the loop then return N+1.
Pseudo Code
int smallestPositive(int A[], int n)
N = segregate(A, n)
for (k = 0 to N - 1)
if(abs(A[k]) - 1 < N)
A[abs(A[k]) - 1] = - abs(A[abs(A[k]) - 1])
for(k = 0 to N - 1)
if(A[k] > 0)
return k + 1
return N + 1
// segregates non-negative numbers to right side of array
int segregate(int A[], int n)
int j = n-1, i = 0
while( i <= j )
if(A[i] <= 0)
swap(A[i], A[j]) // swap values at A[i] and A[j]
j = j - 1
i = i + 1
return i
Complexity Analysis
Time Complexity: Segregate the array and then Linear Scan.
= O(n) + O(n) = O(n)
Space Complexity: O(1)
Critical ideas to think!
• Why do we need to take abs(A[i]) during each operation?
• Why we are not storing the occurrences of numbers greater than N?
• What if we need to find the smallest missing number in an array from a given set of numbers?
• Is it possible to list all the numbers that are missing in the array in the range [1,n], where n is the size of the array?
Comparison of different solutions
Suggested problems to solve
• Find all numbers disappeared in an array.
• K-th missing element in an unsorted array.
• Smallest prime number missing in an array.
• Find the missing integer in an array if mean is given.
• Find the missing number in another array which is shuffled copy.
May the code be with You!
Enjoy Algorithms!
AfterAcademy Data Structure And Algorithms Online Course—Admissions Open | {"url":"https://afteracademy.com/blog/find-missing-positive/","timestamp":"2024-11-07T10:54:18Z","content_type":"application/xhtml+xml","content_length":"80809","record_id":"<urn:uuid:416ee17e-d965-4a5e-988e-18012b108083>","cc-path":"CC-MAIN-2024-46/segments/1730477027987.79/warc/CC-MAIN-20241107083707-20241107113707-00452.warc.gz"} |
SAS rand - Generate Random Numbers in a SAS Data Step
If you want to generate random numbers in a SAS data step, the easiest way is to use the SAS rand() function.
data k;
do i = 1 to 10;
rand_num = rand("Uniform");
When working with data, sometimes it can be very useful to generate random numbers to be able to perform simulations or get a random sample of a dataset.
In SAS, we can generate random numbers easily. The SAS rand() function can return random numbers from different statistical distributions depending on how we want the resulting properties of our
randomly generated dataset.
Generating Random Numbers in a Range Using SAS
Using SAS, we can generate random numbers uniformly in a range easily. By default, if we pass “Uniform” to the SAS rand() function, you will receive random numbers between 0 and 1.
To generate random numbers between 0 and 1, we can do so easily in the following SAS code.
data k;
do i = 1 to 10;
rand_num = rand("Uniform");
To generate random numbers between numbers a and b, for example, 0 and 10, we need to add by a and then multiple the randomly generated number by the difference between a and b.
data k;
a = 0;
b = 10;
do i = 1 to 10;
rand_num = a + (b - a) * rand("Uniform");
You will see that the numbers generated from this data step are all between 0 and 10.
i rand_num
1 1 9.4039006904
2 2 0.2762846812
3 3 9.4098080415
4 4 8.4096989129
5 5 7.4553070194
6 6 0.1901044999
7 7 8.8871195493
8 8 8.6166257504
9 9 8.9881443954
10 10 0.5240682443
Generating Random Integers in a Range Using SAS
If you want to generate random integers in a range, there are 2 ways you can do it. Depending on your version of SAS, you can pass “integer” to the SAS rand() function, or you will need to use the
SAS ceil() function.
If you have a version of SAS later than SAS 9.4M5, you can pass “integer”, as well as the bounds of your range to generate random integers in a range.
data k;
a = 0;
b = 10;
do i = 1 to 10;
rand_int = rand("integer",0,10);
If you have a version of SAS earlier than SAS 9.4M5, you have to use the SAS ceil() function. We can modify our code from above to generate an integer in a range using the SAS ceil() function easily.
data k;
a = 0;
b = 10;
do i = 1 to 10;
rand_num = a + ceil((b - a) * rand("Uniform"));
As you can see below, we generate 10 integers between 0 and 10
i rand_num
Using Different Statistical Distributions to Generate Random Numbers in SAS
We can use the SAS rand() function to generate random numbers from different statistical distributions.
For example, if we want to generate normally distributed random numbers, we just pass “Normal” to rand()
rand_norm = rand("Normal");
You can change the mean and standard deviation by passing arguments specifying these properties:
// Normal distribution with mean 0 and standard deviation 1
rand_norm = rand("Normal", 0, 1);
You can see all of the different statistical distributions you can use “>here.
Hopefully this article has been useful for you to understand how you can generate random numbers using the SAS rand() function in your SAS code. | {"url":"https://daztech.com/sas-rand/","timestamp":"2024-11-15T01:08:42Z","content_type":"text/html","content_length":"245228","record_id":"<urn:uuid:51c85861-01d7-42c1-9f1e-f76447c1fe8a>","cc-path":"CC-MAIN-2024-46/segments/1730477397531.96/warc/CC-MAIN-20241114225955-20241115015955-00293.warc.gz"} |
What are Numerators and Denominators?
The fraction 1/2 means “one piece of a whole object divided into two equally sized parts.” The denominator indicates that two parts make a whole, and the numerator counts off the fact that the
fraction 1/2 contains one of those parts. The fraction X/Y means “X pieces of a whole object that is divided into Y equally sized parts.”
In the last article, we discovered that integers alone are not sufficient to fully describe the world around us—we need the fractions existing between the integers too. (In learning that, we also
learned that fractions are not integers.) In fact, eventually, we’re going to discover that even fractions won’t fully satisfy our needs! But we’ll leave that for a future article—there’s still much
to learn about the fractional world first.
In particular, today we’re going to take a deeper look at fractions and learn a few quick tips to help you understand exactly what numerators and denominators tell you.
What are fractions?
Before we get too far into the details of what the various parts of a fraction mean, let’s briefly review their anatomy. First, a fraction is made up of two integers—one on the top, and one on the
The top one is called the numerator, the bottom one is called the denominator, and these two numbers are separated by a line. The line can be horizontal or slanted—they both mean the same thing and
simply serve to separate the numerator from the denominator.
How to pronounce fractions in English
If you’ve known about fractions for a while, it’s probably been some time since you’ve contemplated the names we use to describe them. But they aren’t exactly obvious, so it’s worth spending a minute
or two thinking about them.
Here’s the quick and dirty tip to help you remember how to pronounce them all: The numerator is always spoken first, and you pronounce it exactly as you pronounce the number. For example, in 1/2 the
numerator, 1, is just pronounced “one;” or in 45/77 the numerator, 45, is simply pronounced “forty-five.” Easy enough. But denominators are a bit trickier. They use the following convention:
• 2 is pronounced “half”
• 3 is pronounced “third”
• 4 is pronounced “fourth” (or “quarter”)
• 5 is pronounced “fifth”
• 6 is pronounced “sixth”
• 7 is pronounced “seventh”
• 8 is pronounced “eighth”
• 9 is pronounced “ninth”
• 10 is pronounced “tenth,” and so on.
So, for the fraction written 1/2, the denominator, 2, is pronounced “half,” and the entire fraction is therefore “one-half.” A little less obvious: For the fraction 45/77, the denominator, 77, is
pronounced “seventy-seventh,” so the entire fraction is “forty-five seventy-sevenths.”
An easy way to remember this is that with the exceptions of “half” and “quarter,” the words used to describe the denominator of a fraction are the same used to put things in order—for example, the
order in which runners finish a race: “third,” “fourth,” “fifth,” etc.
What is a denominator?
Now let’s take a closer look at the different parts of a fraction. First, the bottom part—the denominator. The word “denominator” is derived from the Latin word “nomen,” which means “name” (and also
shows up in words like “nominate” and “nomenclature”). And that’s pretty much what the denominator of a fraction does: it “names,” or indicates, the type of fraction that is described by the
numerator (the top part).
What does the denominator tell you?
Here’s what I mean. The denominator of a fraction tells you how many parts a whole is broken into. It can be a whole pineapple, a whole song, or a whole anything. If the denominator of a fraction is,
say, 4, then that indicates that the whole whatever is broken up into 4 equally-sized pieces.
Or, if the denominator is 12, that means the whole whatever is broken-up into 12 equally-sized pieces. But how exactly does that “name” the type of fraction? Well, that leads us to the meaning of the
What is a numerator and what does it tell you?
The word numerator comes from the Latin verb “enumerate,” which we still use in English to mean “to count.” So, the numerator of a fraction counts the number of equally-sized pieces (identified by
the denominator) that are contained in the fraction.
How, then, do we put this all together to understand the meaning of fractions? Here’s the quick and dirty tip: Going back to our examples from before, the fraction 1/2 means “one piece of a whole
object divided into two equally sized parts.” The denominator indicates that two parts make a whole, and the numerator counts off the fact that the fraction 1/2 contains one of those parts.
Similarly, the fraction 45/77 means “forty-five pieces of a whole object that is divided into seventy-seven equally sized parts.”
What does it mean if the numerator is bigger than the denominator?
In all the examples so far, the numerator has always been smaller than the denominator. In other words, in 1/2 and 45/77, 1 and 45 are smaller than 2 and 77, respectively. But what would it mean if
the numerator were bigger than the denominator? Something like 7/4?
Well, let’s try interpreting this the same way as before. The denominator, 4, indicates that a whole is divided into four equally sized parts, and the numerator, 7, indicates that we have seven of
those parts. So, if four parts make a whole, and we have seven, then we must have a whole object plus three more of the equally sized parts. So 7/4 is equivalent to 1 3/4—also known as “one and
three-quarters”—and we now know that a fraction whose numerator is greater than its denominator represents a number that is greater than one. In case you’re wondering, that type of fraction is called
“improper,” whereas fractions like 1/2 with numerators less-than denominators are called “proper.”
What does it mean if the denominator is less than one?
So far we’ve only talked about fractions with denominators that are greater than one. At the end of the last article, I asked the “brain-teaser” question: “Why can’t the denominator of a fraction be
zero?” To find out the answer to that question, and take a peek at how fractions with denominators less-than one work, check out last week’s Math Dude Video Extra! episode posted to YouTube and the
videos section of the Math Dude’s Facebook page.
Who uses fractions?
In short: Everyone! You’ve probably used fractions without even realizing it.
Let’s say your grandma bakes six cookies. She says that you and your two cousins can each have one, and then asks you to put the rest in a plastic bag. That means 3/6 of the cookies will be eaten and
3/6 will be saved. Once the 3 cookies are put away, you give 1/3 of the batch to one cousin, 1/3 to your other cousin, and keep 1/3 to yourself. You don’t even think of it as using fractions to
divide the batch, but that’s what’s happening!
But what about your grandma? She used fractions when she was baking the cookies.
But what about your grandma? She used fractions when she was baking the cookies. The recipe might have called for one cup of flour, a quarter-cup of sugar, and two eggs for a full batch of cookies.
She only wanted to make half a batch, so she divided the recipe in half. That means she used 1/2 cup of flour, 1/8 a cup of sugar, and one egg to make the dough for this completely hypothetical (and
probably inedible) recipe.
Later, you and your cousins head to the mall to get some new games. You’re in luck—everything in the store is 30% off! That means if you calculate what 3/10 of the original price of the game you want
is, you can subtract that number from the original price and figure out what your discounted total would be.
Fractions are so common we treat them like second nature! | {"url":"https://www.quickanddirtytips.com/articles/what-are-numerators-and-denominators/","timestamp":"2024-11-04T10:38:55Z","content_type":"text/html","content_length":"174579","record_id":"<urn:uuid:763d43ba-c697-4204-9803-68359c675a6f>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00332.warc.gz"} |
We introduce and analyze a size-structured oocyte population model, with non local nonlinearities on recruitment, growth and mortality rates to take into account interactions between cells. We pay
special attention to the form of the recruitment term, and its influence on the asymptotic behavior of the cell population.
This model is well-suited for representing oocyte population dynamics within the fish ovary. The nonlocal nonlinearities enable us to capture the diverse feedback mechanisms acting on the growth of
oocytes of varying sizes and on the recruitment of new oocytes.
We firstly investigate the existence and uniqueness of global bounded solutions by transforming the partial differential equation into an equivalent system of integral equations, which can be solved
using the Contraction Mapping Principle.
In a second step, we investigate the asymptotic behavior of the model. Under an additional assumption regarding the form of the growth rate, we can, with the use of a classical time-scaling
transformation, reduce the study to that of a equation with linear growth speed and nonlinear inflow boundary condition. Using arguments from the theory of abstract semilinear Cauchy problems, we
investigate the local stability of stationary solutions of this equation by reducing it to a characteristic equation involving the eigenvalues of the linearized problem around equilibrium states.
When the mortality rate is zero, the study of existence and stability of stationary solutions is simplified. Explicit calculations can be carried out in certain interesting cases. | {"url":"https://www4.math.duke.edu/media/videos.php?cat=554&sort=most_commented&time=this_year&seo_cat_name=All","timestamp":"2024-11-08T11:49:24Z","content_type":"text/html","content_length":"91449","record_id":"<urn:uuid:dcc7972f-cff1-482b-9027-c335a2816fe2>","cc-path":"CC-MAIN-2024-46/segments/1730477028059.90/warc/CC-MAIN-20241108101914-20241108131914-00565.warc.gz"} |
The Importance of Measures in Business - Basculasbalanzas.com
Measures are a key element of math education. They help students understand lengths, volume and capacity. They also help them solve problems and develop problem-solving skills.
In mathematics, a measure is a set function that satisfies the axioms of countable disjoint unions. It is also called a metric. Measure theory is the branch of mathematics that studies measures.
A measure is a value or number that quantifies some property. It can also be used as a unit of comparison. The amount of a substance that can be contained in a container is a measure. The size of an
object can be measured with a tape measure.
A countably additive set function m displaystyle m in the real numbers is a measure, if it is not zero. Measures can also be defined in topological spaces. In this case, they are defined as linear
functionals on a locally convex topological vector space with compact support.
The concept of measurement is central to many scientific and technical fields. Philosophers have debated a wide range of conceptual, metaphysical, and epistemological issues related to measurement.
For example, some philosophers see it as the process of assigning a number to qualitative empirical observations. Others view it as the estimation of mind-independent properties and relations. Still
others see it as a symbolic activity that is characterized by certain types of operations.
There are different types of measures and metrics. Some focus on inputs, such as the number of products sold or the total amount of calls made. Others provide progress toward desired outputs, such as
revenue growth or customer satisfaction. Measures are also used to predict future performance.
Nominal scales classify observations into categories that are mutually exclusive and exhaustive. Examples include dichotomous data, such as’sick’ or ‘healthy’ when measuring health, and ordinal data,
like ranks in the military or grades in schools.
Use Power BI measures when you need dynamic context-dependent calculations that adjust instantly to user actions, such as filtering or selecting specific data points. In contrast, use calculated
columns when you need static values that are added to a table or to perform complex DAX expressions. Using the correct measurement type helps you eliminate issues like redundant work, slower
execution speed and less flexible data models. This translates into more error-free decisions that are not irretrievably damaged by incorrect or inaccurate data.
Measurement is central to modern science, engineering and commerce. However, the way that measurements are used varies greatly in different workplace situations. Hence, there are many different
decisions that can be made about what to measure and how precise those measurements should be. Before attempting to measure, it is therefore important to decide for what purpose the resulting numbers
are useful and then ensure that the measures meet those expectations. Moreover, measurement has many links to other subjects such as arithmetic (the lab technician example on proportional reasoning),
geometry and statistics.
A new generation of measurement theorists is developing an understanding of measurement in terms of information-theoretic analysis. They compare measurement instruments to information-transmission
systems that encode an object’s state into an internal signal and then transmit this signal to a receiver. The information that an instrument’s indication conveys about the occurrence of the measured
state depends on the features of the measuring system and on the level of noise in the environment.
Measures can be found everywhere in a business, but their effectiveness depends on the type of data they’re used to collect and analyze. In particular, they should accurately reflect what they’re
supposed to quantify in order to provide actionable insights. This is why some metrics, such as key performance indicators (KPIs), focus more on inputs, while others, such as customer satisfaction,
can help track progress toward desired results over time.
Measure theory is a branch of mathematics that deals with the generalisation of geometric measures such as length, area, and volume, as well as the notions of mass, time duration, and even the
probability of certain occurrences. It also explores the possibility of having a “measure” whose values are not restricted to the non-negative real numbers or infinity, such as the Liouville measure
on a symplectic manifold or the Gibbs measure on a Hamiltonian system.
In Power BI Desktop, you can create your own custom measures by using the Calculated column wizard or by writing a DAX expression in the Fields list. These can then be used in visuals and in
relationships between tables in a data model. | {"url":"https://www.basculasbalanzas.com/the-importance-of-measures-in-business-4/","timestamp":"2024-11-10T14:50:31Z","content_type":"text/html","content_length":"53616","record_id":"<urn:uuid:4af4ae7a-9e64-4e98-a387-cfac07c63f6e>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.60/warc/CC-MAIN-20241110134821-20241110164821-00080.warc.gz"} |
Torque Calculator
Calculating Torque
Torque is a key concept in physics and engineering, describing the rotational force applied to an object. It plays a crucial role in the functioning of machinery, vehicles, and various mechanical
systems. Whenever an object rotates around an axis, torque is responsible for causing that motion. In many applications, understanding how to calculate torque is critical when designing systems like
engines, motors, and gears, as it ensures efficient and reliable operation.
The Torque Formula
Torque \( T \) can be calculated using the following equation:
\( T = F \times r \times \sin(\theta) \)
• \( T \) is the torque (Nm).
• \( F \) is the force applied (N).
• \( r \) is the distance from the axis of rotation (m).
• \( \theta \) is the angle between the force and the lever arm (degrees or radians).
Step-by-Step Guide to Calculating Torque
To calculate torque, follow these steps:
• Step 1: First, identify the force \( F \) applied to the object. This force generates the rotation and is measured in newtons (N).
• Step 2: Next, measure or determine the distance \( r \) from the axis of rotation to where the force is applied, often referred to as the lever arm.
• Step 3: Then, determine the angle \( \theta \) between the direction of the applied force and the lever arm. If the force is applied perpendicular to the lever arm, \( \theta \) is 90°, which
maximizes the torque because \( \sin(90^\circ) = 1 \).
• Step 4: Finally, use the torque formula \( T = F \times r \times \sin(\theta) \) to calculate the torque generated by the applied force.
Example: Calculating Torque for a Wrench
Let’s assume you are tightening a bolt using a wrench that is 0.3 meters long, and you apply a force of 50 N perpendicular to the wrench. The torque is calculated as:
\( T = 50 \times 0.3 \times \sin(90^\circ) = 15 \, \text{Nm} \)
Thus, in this case, the torque generated is 15 Newton-meters.
Torque in Rotational Motion
Torque is vital in rotational dynamics because it causes objects to rotate about an axis. The relationship between torque and angular acceleration is given by Newton’s second law for rotation:
\( T = I \times \alpha \)
• \( I \) is the moment of inertia (kg·m²), representing an object’s resistance to rotational motion.
• \( \alpha \) is the angular acceleration (rad/s²).
As a result, torque is directly proportional to angular acceleration and the moment of inertia. This means that greater torque results in faster rotational motion if the moment of inertia remains
Factors Affecting Torque
Several factors influence the torque generated in a system:
• Force: The magnitude of the applied force directly affects the torque. A larger force generates more torque, provided the other variables remain constant.
• Lever arm length: The distance from the axis of rotation to where the force is applied (lever arm) also plays a critical role. A longer lever arm increases the torque for the same applied force.
• Angle of force application: The angle \( \theta \) at which the force is applied affects the effectiveness of the torque. A perpendicular force (90° angle) maximizes torque, while a force applied
parallel to the lever arm (0° or 180°) generates no torque.
Practical Applications of Torque Calculation
Torque calculations are essential in various engineering and mechanical applications, such as:
• Automotive engineering: Torque is crucial in understanding how engines generate rotational power and how it is transmitted to the wheels of a vehicle.
• Mechanical systems: Torque calculations are used in designing gear systems, motors, and other mechanical devices that involve rotational motion.
• Construction: Torque is vital when using tools like wrenches and screwdrivers to tighten bolts, ensuring the right amount of force is applied to prevent over-tightening or loosening.
Example: Calculating Torque in a Car Engine
Consider a car engine generating 200 Nm of torque. The engine delivers this torque to the wheels through a transmission system. The force transmitted to the wheels can be calculated using the
\( F = \frac{T}{r} \)
Where \( r \) is the radius of the wheel. If the wheel has a radius of 0.3 meters, the force exerted on the road is:
\( F = \frac{200}{0.3} = 666.67 \, \text{N} \)
Therefore, the car engine applies a force of 666.67 N to the road surface, allowing the vehicle to move forward.
Torque vs. Power
Torque and power are often related, but they are not the same. Torque refers to the rotational force, while power is the rate at which work is done. Power \( P \) in rotational systems can be
calculated using the formula:
\( P = T \times \omega \)
• \( P \) is the power (watts).
• \( T \) is the torque (Nm).
• \( \omega \) is the angular velocity (rad/s).
This equation shows that power increases with both torque and rotational speed, making both important factors in the design of mechanical systems.
Frequently Asked Questions (FAQ)
1. What is the difference between torque and force?
Force is a push or pull that acts on an object, while torque is the rotational equivalent of force. Torque causes objects to rotate, whereas force causes objects to move in a straight line. Torque
depends on the magnitude of the force, the distance from the axis of rotation, and the angle at which the force is applied.
2. How can I increase the torque in a system?
To increase torque, you can either increase the applied force, increase the length of the lever arm, or adjust the angle at which the force is applied to be closer to perpendicular to the lever arm.
Mechanical systems like gears can also increase torque by changing the ratio of input to output forces.
3. What is the significance of torque in an engine?
In engines, torque is crucial for generating rotational motion and transmitting power to the wheels or other mechanical components. Higher torque allows an engine to do more work at lower speeds,
which is particularly useful for applications requiring strong pulling power, like towing or accelerating a vehicle. | {"url":"https://turn2engineering.com/calculators/torque-calculator","timestamp":"2024-11-07T00:32:15Z","content_type":"text/html","content_length":"211948","record_id":"<urn:uuid:b74893b1-ae75-46c9-841b-0eaefea0ab3f>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00025.warc.gz"} |
Core Deposit Circuit | Tornado Cash
The core deposit circuit is what most users interact with, proving that a user has created a commitment representing the deposit of some corresponding asset denomination, that they haven't yet
withdrawn that asset, and that they know the secret that they supplied when generating the initial commitment.
Making a Deposit
A deposit into Tornado.cash is a very simple operation, which doesn't actually involve any ZK proofs. At least not yet. To make a deposit, you invoke the deposit method of a Tornado contract
instance, supplying a Pedersen Commitment, along with the asset denomination that you're depositing. This commitment is inserted into a specialized Merkle Tree, where the structure of the Merkle Tree
is aligned to an elliptic curve associated with a prime in the order of the BN128 elliptic curve, and the labels of the tree are computed using MiMC hashing.
Commitment Scheme
When you make a "commitment" in the context of cryptography, what you're doing is taking a secret value - often large and random - and running it through some cryptographic function (e.g. a hash
function), then disclosing the result. Later, when you need to make good on the commitment, you prove that you know the original secret value.
Pedersen Hash
A Pedersen Hash is an extremely specialized hashing function that is particularly well-suited for use in applications leveraging Zero Knowledge proving circuits. Where other hashing functions like
SHA-256 are designed to exhibit properties such as producing very different outputs for even slightly different inputs (the avalanche effect), Pedersen hashing instead prioritizes the ability to
compute the hash extremely efficiently in Zero Knowledge circuits.
Hashing a message with Pedersen compresses the bits of the message down to a point along an elliptic curve called Baby Jubjub. Baby Jubjub is in the order of the BN128 elliptic curve that is
supported by precompiled operations on the Ethereum network which were added in EIP-196. This means that operations that use the Baby Jubjub curve, such as Pedersen Hashing, are highly gas-efficient.
When you compute the Pedersen hash of a message, the resulting point along the its elliptic curve is very efficient to verify, but infeasible to reverse back into the original message.
Tornado Commitment
To generate a commitment for a Tornado.cash deposit, you first generate two large random integers, each 31 bytes in length. The first value is a nullifier that you will later disclose in order to
withdraw your deposit, and the second is a secret that secures the confidential relationship between your deposit and withdrawal.
The preimage of your deposit note is the concatenation of these two values (nullifier + secret), resulting in a message 62 bytes in length. This message is Pedersen hashed, resulting in an output
representing an element of the Baby Jubjub elliptic curve encoded as a 32-byte big-endian integer.
If you want to see this in code form, you can reference the tornado-cli deposit function.
MiMC Merkle Tree
The Tornado contract is a specialized Merkle Tree which labels its nodes using MiMC hashes.
For those not familiar with Merkle Trees, they are binary trees where each non-leaf node is labelled with the hash of the labels of its child nodes, and the leaf nodes are labelled with the hash of
their data. Ordinarily, Merkle Trees use a one-way cryptographic hashing function like SHA-2, but in this case, we're using MiMC, which has some useful properties.
One of the useful properties of MiMC is that it's well-suited to operating over prime fields, which is important to us because Zero Knowledge proofs are fundamentally based on prime fields, and
Pedersen Hashes are points within a prime field defined by the Baby Jubjub elliptic curve - which is in turn within the order of the BN128 curve supported natively on Ethereum. Because Zero Knowledge
proofs are operationally expensive, and each operation in an Ethereum transaction has a corresponding gas cost, the specific types of operations we design around need to be as gas-efficient as
The other particularly useful properties of MiMC are that it's non-parallelizable, and difficult to compute but easy to verify. These properties add to the security of the contract by making it
computationally infeasible to calculate a forged "commitment" which has a colliding path within the merkle tree.
Zero Nodes
During the initialization of the Tornado Merkle Tree, a single path spanning the height of the tree is preallocated starting with a "zero leaf" node with a label of keccak256("tornado") % FIELD_SIZE.
Each subsequent non-leaf node toward the root is then labelled as if the entire bottom of the tree were populated by that same same leaf node.
The purpose of these "zero nodes" is to ensure that all paths within the merkle tree are invalid until they terminate in a valid commitment.
Inserting a Commitment
When you insert a commitment into the Tornado contract's merkle tree, you are replacing a "zero leaf" with a new leaf whose label is the MiMC hash of your Pedersen commitment, and then traversing up
the tree updating each subsequent parent node with a new label based on the label updates that your new leaf introduces below.
Commitments are inserted from left to right within the tree, with every two commitment insertions filling a "subtree". Each insertion increments the "index" of the tree, determining whether the next
commitment will be inserted on the left or right side of the entry to its merkle path.
Once your deposit has updated the tree, the label of the top-most node becomes the tree's new "root", and is added to a rolling history containing the labels of the last 100 roots, for later use in
processing withdrawal transactions.
The Tornado.cash deposit contracts are deployed with 20 "levels", with each level increasing the number of potential leaves by a power of 2. That means that the contract's merkle tree supports up to
2^20 leaves, allowing for up to 1,048,576 deposits to be made into the contract before it needs to be replaced.
The reason behind this seemingly-low number of levels is that every deposit has to perform as many updates to the tree as there are levels. A tree with more levels would require more gas per deposit,
as well as correspondingly larger proof sizes when withdrawing notes.
Making a Withdrawal
Having made a deposit, you now have a set of truth claims that you can generate a proof based upon. Generally speaking, Zero Knowledge proofs are anchored to some value(s) known by both the prover
and the verifier, to which a relationship is going to be proven to a set of values known only by the prover. The circuit verifier can confirm that the prover has used the value(s) that are known, and
that the proof that they computed satisfies the constraints imposed by the circuit.
Inputs to a Withdrawal Proof
In the case of Tornado.cash deposits, the prover (the person submitting a withdrawal transaction), and the verifier (the deposit contract's withdrawal method) both know a recent merkle root. The
prover also supplies a set of other public inputs that they used for the generation of their proof.
The total set of public inputs for a withdrawal proof are:
2.The Pedersen hash of the nullifier component from their deposit commitment
3.The address of the recipient of their withdrawal
4.The address of the relayer that they've selected (or their own address)
5.The fee that they're paying the relayer (or zero)
6.The refund that they're paying the relayer (or zero)
The additional private inputs for a withdrawal proof are:
1.The nullifier component from their deposit commitment
2.The secret component from their deposit commitment
3.The set of node labels that exist in the path between the root and the leaf nodes of the merkle tree
4.An array of 0/1 values indicating whether each specified path element is on the left or right side of its parent node
Proven Claims
It would be easy to miss the clever new piece of knowledge we created when we constructed and inserted our commitment into the merkle tree. You might be inclined to think that to make a withdrawal,
we're simply going to prove that we know the components of the Pedersen commitment, and that the merkle tree is just an efficient way to store those commitment hashes.
What's special about this construction is that it enables us to prove not just that we know the components of a deposited commitment, but rather it enables us to prove simply that we know the path to
a commitment within the tree, and how to get there starting with a commitment preimage.
If we were only to prove that we knew the preimage to a deposited hash, we would risk revealing which commitment is ours. Instead, we're not disclosing the commitment preimage, but instead we're
simply proving that we have knowledge of a preimage to a commitment within the tree. Which commitment is ours remains completely indistinguishable on the withdrawal side of the circuit protocol.
Computing the Witness
Nullifier Hash Check
In order to compute the witness for the withdrawal proof, our circuit first takes the private deposit commitment inputs (nullifier + secret), and runs them through a circuit component which
simultaneously computes the Pedersen hash of the full commitment message, and the Pedersen hash of the nullifier alone. The circuit then compares the resulting nullifier hash to the one you supplied
as a public input, and asserts their equality.
This proves that the nullifier hash that you supplied publicly is in fact a component of your original commitment.
Merkle Tree Check
Next, the circuit takes the commitment hash it has computed, the merkle root you have specified publicly, and the path elements and left/right selectors that you specified privately, as inputs to a
component which checks your merkle tree path claim.
The Merkle Tree Checker starts from the bottom of the path, inputting your commitment hash and the first element of your proposed path into a Muxer. The Muxer takes a third input, which is an element
from your supplied left/right directions. The Muxer component uses these directions to inform an MiMC hashing component as to the order of its inputs. If the supplied direction is 0, then the
supplied path element is on the left, and your commitment hash is on the right. If the direction is 1, then the order is reversed.
The MiMC hasher outputs the resulting hash, and the Merkle Tree Checker proceeds to the next level. It repeats the last process, except this time, instead of using your commitment hash, it uses the
hash of the last level. It continues to run through each level of the proposed path, until it ends up with a final hash output.
The Merkle Tree Checker compares the hash that it has computed to the public merkle root input that you supplied, and asserts their equality.
This proves that your commitment exists within some path beneath the specified merkle root.
Extra Withdrawal Parameter Check
Before finishing, the circuit takes each of the remaining four public inputs, and squares them into a public output. While this isn't strictly necessary, it creates a set of constraints within your
proof that ensure that your transaction parameters can't be tampered with before your withdrawal transaction is processed. If any of those parameters were to change, your proof would no longer be
Computing the Proof
Now that we have a witness for our proof, we take those witnessed state values and input them into the R1CS corresponding to the Withdrawal circuit, and run the prover over it. Out of the prover
comes two proof artifacts. The first is the proof itself, according to the SNARK protocol we're using, and the second is the set of public inputs and outputs corresponding to that proof.
Completing a Withdrawal Transaction
With the withdrawal proof now generated, you supply that proof, along with its public inputs, to the withdraw method of the deposit contract. This method verifies that:
1.The specified relayer fee does not exceed the value of the denomination of asset being withdrawn
2.The supplied nullifier hash has not been spent before
3.The supplied merkle root is known, using the 100-root historical record
4.The supplied proof is valid
One of the artifacts deployed as a dependency of the deposit contract is a Solidity contract that is generated using the proving key of the Withdrawal circuit as an input. This Verifier contract is
an optimized proof verifier with a single public view function, which accepts a proof and the array of six public inputs as uint256 values.
This function returns TRUE if the proof is valid according to the public inputs.
If the above preconditions are met, the supplied nullifier hash is inserted into the set of spent nullifiers, and then the value of the deposit is distributed amongst the recipient and relayer,
according to the specified fee parameters. | {"url":"https://docs.tornadoeth.cash/tornado-cash-classic/circuits/core-deposit-circuit","timestamp":"2024-11-07T10:13:33Z","content_type":"text/html","content_length":"360733","record_id":"<urn:uuid:f0ac2a7e-89ab-48ce-9844-5c45750b8c7b>","cc-path":"CC-MAIN-2024-46/segments/1730477027987.79/warc/CC-MAIN-20241107083707-20241107113707-00003.warc.gz"} |
Just Intonation in Notes and Numbers
An idealized, orderly, and multisymmetric presentation of parts of the interval material from Just intonation. All intervallic pairs with the same distance through the center are complementary within
the octave.
The compass rose at the top left indicates that one moves horizontally (primarily the axis from 9 o'clock to 3 o'clock) in perfect fifths (right) and perfect fourths (left), respectively; in the
direction of the axis from 1 o'clock to 7 o'clock and its parallels one moves in just major thirds (up) and just minor sixths (down), and in the direction of the axis from 5 o'clock to 11 o'clock and
its parallels one moves in pure minor thirds (down) and major just sixths (up), respectively. The colour code is described later in the text. Historical versions of just intonation can also be found
further down.
"Just intonation!? - That does not exist !!"
The above was a response I received from the music director of the Copenhagen Round Tower, who himself was a high-level musician, when I told him that the theme for one of the concert evenings at the
week-long program in connection with the exhibition Spiral Tonality, 2008, should be Just intonation. His facial expressions and tone of voice certainly did not indicate immediate enthusiasm.
The Copenhagen Round Tower. Photo: Wikipedia, Creative Commons
Although the equal temperament is practical and has evened out almost all the many facets of tuning theory, so that only a few, even among music experts, are really aware of the countless other ways
of tuning, Just intonation in certain contexts certainly has its justification, not least because said equal temperament, culturally has given us a conditioned habituation to consider significantly
false tones – 4 of the 12 intervals in the series – acceptable. The intervals in question are minor and major thirds as well as minor and major sixths, all of which deviate about one-seventh of a
semitone from the acoustically pure intervals.
Mathematically, this modern standard tuning makes all semitones equal (frequency increase for each step: 1: 12√2, also written 121/12, twelfth root of two, 1,059, ie barely 6%).
In a scale based on C, played on most modern pianos, keyboards, guitars and more, the tones indicated by red dot will be considerably false.
When said musician reacted as he did, I would argue that it may not only, but also, be because the pure intervals may sound peculiar when throughout one's life you have become accustomed to playing
and listening to false versions. However, the the music director's tone of voice did change when it became clear that the program would feature Per Nørgaard (leading Danish composer), and that he
himself would show up for the concert, where he played excerpts from his works Turn and Spell.
Just intonation is based on the tones from the harmonic series, which – given by nature as it is – is our ultimate reference for the purity of intervals. You can get an impression of these intervals
in the interactive rendering of the octave spiral programmed by my friend Marco Gianotta.
This diagram is based on the octave spiral mentioned above. Moving clockwise in the outer grey circle, the interval fractions indicate 12 tones in Just intonation: prime-minor second-major
second-minor third-major third-fourth-tritone-fifth-minor sixth-major sixth-minor seventh -major septim (-octave).
The red spiral represents the first four octaves of the harmonic series, 1-2-4-8-16. When you start in the center and follow the red spiral arm clockwise, each revolution will represent one octave.
More and more tone functions (integers) are added with each turn, and thus the neighboring intervals become ever smaller:
First revolution: 1:2
Second revolution: 2:3:4
Third revolution: 4:5:6:7:8
Fourth revolution: 8:9:10:11:12:13:14:15:16
In the harmonic spiral:
1-2-4-8 -... are octaves of the primary note, do
3-6-12 -... are octaves of the perfect fifth, so
5-10-20 -... are octaves of the major third, mi.
The blue spiral is the inverse of the harmonic series, ie a reflection through the vertical axis. The so-called "subharmonic series" does not have the same manifest character as the harmonic series,
whose partial tones can be measured directly in the sound, so there is reason to use the former term with caution.
The partials of the harmonic series are thus mirrored about the vertical axis in the inverse, blue spiral, but also the intervals in the outer, gray circle have a mirror character. The fractions are
inverse of each other, as the inverted fractions are in all cases by 'octavation' (multiplication or division by 2-4-8 ...), are brought into the 'mother interval' 1: 2 (one to two).
LENGTHS, NUMBERS & TONES
The self-chosen limitation in historical Just intonation is that it only operates with the first three primary functions in the harmonic series, corresponding to the first three prime numbers, 2, 3
and 5, which translated into music have to do with the intervals octave, fifth and major third. NB! In that particular order: prime number 3 creates the perfect fifths of music (contrary to fifth
step in the constructed scales of music), prime number 5 creates large thirds (constrary to the third step in the constructed scales of music).
In other words: We are dealing with a [2; 3; 5] system.
The numbers should be understood in the light of length divisions of a string:
½ length gives the double frequency – the experience of the quality of the octave
1/3 length gives 3-fold frequency – the experience of the quality of the fifth (twelth)
1/5 length gives 5-fold frequency – the experience of the quality of the just major third
Homemade, three-string monochord with fitted measuring tape and sliding frets on two of the strings. As can be seen, the string length is 60 centimeters. 60 = 2x2x3x5, and it is certainly not without
reason that it was used as a base by the ancient Mesopotamians, and that we, among other things, still carry memories of it on our left wrist. The lengths 60-54-48-45-40-36-32-30 (2^2x3x5-2x3^3-2^
4x3-3^2x5-2^3x5-2^2x3^2-2^5-2x3x5) describe a major scale with integers. Relative frequencies: 1-10/9-5/4-4/3-3/2-5/3-15/8 (-2). At the photo one fret is placed at 40 centimeters. 40/60 = 2/3 =
perfect fifth. The other is placed at 45 cm. 45/60 = 3/4 = perfect fourth. The third string vibrates freely in its full length, 60 cm. 60=1/1= prime.
I am passionate about drawing diagrams to illustrate the patterns of the music. This post was born out of conversations with a good friend who inspired me to illustrate how intervals in a set of
notes are complementary within the octave, and just as all musical intervals can be said to have their actual origins in the harmonic series, so an inversion of this series with the somewhat
compromised term "subharmonic series" may be considered the origin of half of these intervals:
The same 16 first notes of the harmonic series are here drawn in red in a note system. The inverse of the harmonic series is marked in blue. The 12 white notes are the basic material for the Just
Color code:
Tones originating from the harmonic series.
Major and augmented intervals generally behave as 'expanding', feeling 'major-like', and tend to outward resolution.
In the harmonic series, 2-4-8-16 -... are different octaves of the fundamental. When these factors are in the denominator of a (relative frequency) interval fraction, the interval has its direct
origin in the harmonic series (the opposite is true for relative length fractions):
• 3/2 = perfect fifth
• 5/4 = just major third
• 9/8 = major wholetone
• 15/8 = major seventh
Tones originating from the "subharmonic series". Minor and diminished intervals generally behave as 'contracting', feeling 'minor-like', and tend to inward resolution.
When 2-4-8-16 -... is in the counter of an interval fraction, the interval may be considered having its origin in the "subharmonic series":
• 4/3 = perfect fourth
• 6/5 = just minor third
• 16/9 = minor seventh
• 15/8 = small halftone
Some just intervals are not directly related to a preceding prime (red) or a subsequent octave (blue), but occur between other partials of the harmonic series. These intervals may be either
contracting (not least 6/5, the minor just third) or expanding (not least 5/3, the just major sixth):
• 6/5 = just minor third
• 5/3 = just major sixth
• 25/18 = augmented fourth
• 36/25 = diminished fifth
A more detailed look at the complementary intervals shown below the white notes in the diagram above. Here, the same intervals as above are also described on an axis. During the tone selection on the
top row, the intervals are arranged in pairs, complementary within the octave:
• minor second (16/15) - major seventh (15/8)
• major second (9/8) - minor seventh (16/9)
• minor third (6/5) - major sixth (5/3)
• ...
It is important to mention again that these intervals from Just intonation are an idealized selection. There are multiple versions of Just intonation, some constructed partly at random, others with a
specific purpose in mind. In contrast to the top diagram. As can be seen, the overall impression of the selected historical versions of Just intonation presented here is that they are more likely to
include expanding intervals (upwards, to the right) than contractiing (downwards, to the left):
Sources with several historical examples: Here and here.
The Just intonation played a special role in the early Renaissance, but one of its disadvantages was that disonnant triads occur if one moves far away from the original tonal centre. However, the
important role of the just thirds was cemented, which had been a problem in the preceding Pythagorean tuning, where these were not pure at all. A compromise between the two disadvantages was sought
to be made in the subsequent meantone temperament, where in the most widespread version the major thirds are just and the fifths a little too small.
Then followed the diversity of well temperament during the baroque, where the music would more factually enter the era of modulation between different keys. This should not be confused with equal
temperament which took hold later, generally well into the 19th century.
In the context of Just intonation, mirrorong and complementary intervals, it would be appropriate to mention the American Harry Partch (1901-74) from modern times. In his most famous version of Just
intonation, the boundaries were expanded, so that he not only included the first three prime numbers and their music, but also the following two, ie 7 and 11. Thus he got a microtonal tone system,
where the octave is not like habitual division into 12, but into 43 steps. In other words, Partch's extended mood is a [2; 3; 5; 7; 11] system.
Here the steps are depicted clockwise, from small to large. They all have a complementary partner on the opposite side of the vertical mirror axis:
And here is a similarly complementary Partch tone system consisting of 31 tones per octave: | {"url":"https://www.overtone.cc/profiles/blogs/just-intonation-in-notes-and-numbers","timestamp":"2024-11-03T14:02:12Z","content_type":"text/html","content_length":"73305","record_id":"<urn:uuid:f85352d7-79c2-4ab4-a161-a324ad5c7fa8>","cc-path":"CC-MAIN-2024-46/segments/1730477027776.9/warc/CC-MAIN-20241103114942-20241103144942-00454.warc.gz"} |
Monitoring least squares models of distributed streams
Least squares regression is widely used to understand and predict data behavior in many fields. As data evolves, regression models must be recomputed, and indeed much work has focused on quick,
efficient and accurate computation of linear regression models. In distributed streaming settings, however, periodically recomputing the global model is wasteful: communicating new observations or
model updates is required even when the model is, in practice, unchanged. This is prohibitive in many settings, such as in wireless sensor networks, or when the number of nodes is very large. The
alternative, monitoring prediction accuracy, is not always sufficient: in some settings, for example, we are interested in the model's coefficients, rather than its predictions. We propose the first
monitoring algorithm for multivariate regression models of distributed data streams that guarantees a bounded model error. It maintains an accurate estimate using a fraction of the communication by
recomputing only when the precomputed model is sufficiently far from the (hypothetical) current global model. When the global model is stable, no communication is needed. Experiments on real and
synthetic datasets show that our approach reduces communication by up to two orders of magnitude while providing an accurate estimate of the current global model in all nodes.
Publication series
Name Proceedings of the ACM SIGKDD International Conference on Knowledge Discovery and Data Mining
Volume 2015-August
Conference 21st ACM SIGKDD Conference on Knowledge Discovery and Data Mining, KDD 2015
Country/Territory Australia
City Sydney
Period 10/08/15 → 13/08/15
• Data mining
• Distributed streams
• Least squares
• Regression
ASJC Scopus subject areas
• Software
• Information Systems
Dive into the research topics of 'Monitoring least squares models of distributed streams'. Together they form a unique fingerprint. | {"url":"https://cris.haifa.ac.il/en/publications/monitoring-least-squares-models-of-distributed-streams","timestamp":"2024-11-08T00:03:27Z","content_type":"text/html","content_length":"57462","record_id":"<urn:uuid:e6de939f-4d7e-4581-ac5f-95c34dce6c19>","cc-path":"CC-MAIN-2024-46/segments/1730477028017.48/warc/CC-MAIN-20241107212632-20241108002632-00796.warc.gz"} |
15.2.4. Panel Compression Buckling (Rectangular) - Abbott Aerospace UK Ltd
Reference: Abbott, Richard. Analysis and Design of Composite and Metallic Flight Vehicle Structures 3 Edition, 2019
Panel compression buckling is common when considering the primary failure mode of the upper wing skin panels, horizontal tail skins and vertical tail skins. Compression also contributes to the
failure of other parts of the structure in combined failure modes.
15.2.4.1. Compression Buckling Allowable Stress
The compression buckling coefficient, k[c], can be found once the panel aspect ratio is known from the following figure taken from (NACA-Report-733)
The compression buckling coefficient in the figure above is derived using the following expression:
This is covered in greater depth in section 15.2.4.3.
15.2.4.2. Panel with Simply Supported Edges
The simplest approach to panel compression buckling and the approach that is commonly used for initial sizing is to assume that and the panel is simply supported:
The coefficient for the simply supported edge condition is given by the following expression:
The minimum values of this curve for successive values of λ is shown below:
The minimum values of this curve for successive values of λ is shown below:
Figure 15.2.4‑3: Web Compression Buckling Coefficient – Simply Supported
The simple and the general Panel compression buckling coefficients are calculated in this spreadsheet:Table
The simple approach to panel compression buckling is given in this spreadsheet:
Using this simple approach, to account for panel material plasticity if exceeds F[cy] then limit F[cr] to F[cy].
15.2.4.3. Compression Buckling Allowable Stress with Varying Panel Rotational Edge Fixity
A simplified method allows for a quantified measure of the panel rotational edge stiffness, ε , this is defined by the following expression:
The value of ε can be calculated by taking the following approach. This approach is taken in part from (NACA-TN-888, 1943).
The term ε can be calculated from this expression:
When this is combined with the initial expression for ε the expression for evaluating ε becomes:
Of all these terms the C[BT] is a relatively little used cross section property and (NACA-TN-888, 1943) is one of the few references that gives a method to determine this value.
C[BT] is defined using the following expression:
and is described as “where u is the unit warping of the element of area dA from a reference plane through the shear center and normal to the axis when the angle of twist per unit length (dθ /dx) is
Thankfully the reference gives a set of expressions for for common cross sections:
Further explained in the figure below:
Figure 15.2.4‑4: Parameters for Torsion-Bending Constant
The spreadsheet for this method is at the link below:
A view of how the simply supported compression buckling coefficient, k, changes with the value of is given in Figure 17 of (NACA-TN-3781, 1957):
The lower line on the graph above defines the relationship between k and for a panel in compression buckling.
Figure 15.2.4‑6: The relationship between k and ε – Clarified
The spreadsheet method for estimating the effect on k for edge rotational edge fixity is given at the link below:
Note that this method of accounting for panel edge fixity correlates well to the value of k calculated using the general expression which is shown in Figure 15.2.4.
15.2.4.4. Compression Buckling Allowable Stress with Full Elasto-Plastic Material Data
If the calculated compression buckling stress is approaching the compression yield stress (F[cy]) of the material the elastic compression buckling allowable could be optimistic. If a more nuanced
approach than limiting the buckling stress to F[cy] is required, the compression buckling allowable should be modified using the plasticity correction factor η.
From (NACA-TN-3781, 1957) for compression buckling the plasticity correction factor for a long simply supported panel is:
the plasticity correction factor for a long, clamped panel is:
Plotting these two plasticity correction factors for a typical aluminum give the following result:
Figure 15.2.4‑7: Plasticity Correction Factors Compression buckling for Typical Aluminum Material
Figure 15.2.4‑8: Difference Between Different Edge Fixity Conditions for Plasticity Correction Factors for a Typical Aluminum Material
There is only a small difference between the two factors (less than 10%). It is recommended that the plasticity correction factor for clamped panels is used for all panels as it provides a simpler,
conservative solution.
Like the shear buckling allowable corrected for material plasticity, the plasticity correction factor for compression buckling can be used to produce a curve that relates the elastic buckling stress
with the plastic buckling stress.
Once the graph has been plotted the elastic shear buckling allowable can be plotted on the x-axis, project upwards to the curve and read across to the Y-axis to give the compression buckling
allowable corrected for plasticity.
Superimposing the simpler approach of limiting F[cr] to F[cy] over the Elastic vs Plastic shear buckling stress curve gives the following result:
Figure 15.2.4‑9: Elastic vs Plastic Compression Buckling Stress for Sample Material with Comparison to Simple Elastic and F[cy] Cut off Value
For the sample material (and for most ductile materials) the simple approach does give a reasonable approximation to the correctly calculated plastic buckling allowable. A spreadsheet is available
for this method at the link below:
15.2.4.5. Effect of Central Hole on Panel Compression Buckling Allowable
In the process of writing this section many references were reviewed. The single best summary of results is contained in (NASA-TM-1998-206542, 1998). This paper is a summary of the results of a
Finite Element study looking at the effect of circular and square holes on isotropic Panel buckling. Other references that are in general agreement include (NASA-TP-3024, 1990).
The figures in (NASA-TM-1998-206542, 1998) are given in terms of changes in absolute buckling values. These results have been reinterpreted to give a reduction factor for compression buckling factor
In the following figures the most onerous reduction from either the symmetric buckling or antisymmetric buckling modes have been used.
Note: The reduction factors are based on Figure 17 of (NASA-TM-1998-206542, 1998). This is reported as having ‘free’ edges – edges which are free to move in-plane. However, the results for a hole
diameter of 0 in agree with the results for a simply supported panel without a hole from table 4 of the same reference. This result also agrees with independent checking done for panels with edges
restrained in translation and free in rotation during the compilation of this section. It is therefore assumed that the reference is in error and Figure 17 is representative of panels with a hole
where the edges are restrained for in-plane translation.
Figure 15.2.4‑10: Reduction in Compression Buckling Allowable for Square Panels with Centrally Located Holes (NASA-TM-1998-206542, 1998)
It was found that the critical panel aspect ratio was 1:1 (square). As long as the panel is loaded in compression aligned with the long dimension, use of the square panel data is conservative.
This method is not applicable for panels loaded in compression across the short dimension.
As noted, the ratio of greatest reduction in buckling strength for each of the clamped and simply supported curves was calculated and plotted:
Figure 15.2.4‑11: Ratio of Reduction in Compression Buckling Strength of Clamped and Simply Supported Square Panels with Varying Sizes of Central Hole
A simple conservative linear approximation can be used that is applicable to both simply supported and clamped panels.
Figure 15.2.4‑12: Ratio of Reduction in Compression Buckling Strength of Clamped and Simply Supported Square Panels with Varying Sizes of Central Hole – Simple Approximation
The equation for this linear approximation is:
A spreadsheet method is available at the link below:
Reference: Abbott, Richard. Analysis and Design of Composite and Metallic Flight Vehicle Structures 3 Edition, 2019
Panel compression buckling is common when considering the primary failure mode of the upper wing skin panels, horizontal tail skins and vertical tail skins. Compression also contributes to the
failure of other parts of the structure in combined failure modes.
Figure 15.2.4‑1: Panel Compression Buckling (NACA-TN-3781, 1957)
15.2.4.1. Compression Buckling Allowable Stress
The compression buckling coefficient, k[c], can be found once the panel aspect ratio is known from the following figure taken from (NACA-Report-733)
Figure 15.2.4‑2: Web Compression Buckling Coefficient (NACA-Report-733)
The compression buckling coefficient in the figure above is derived using the following expression:
This is covered in greater depth in section 15.2.4.3.
15.2.4.2. Panel with Simply Supported Edges
The simplest approach to panel compression buckling and the approach that is commonly used for initial sizing is to assume that and the panel is simply supported:
The coefficient for the simply supported edge condition is given by the following expression:
The minimum values of this curve for successive values of λ is shown below:
The minimum values of this curve for successive values of λ is shown below:
Figure 15.2.4‑3: Web Compression Buckling Coefficient – Simply Supported
The simple and the general Panel compression buckling coefficients are calculated in this spreadsheet:Table
The simple approach to panel compression buckling is given in this spreadsheet:
Using this simple approach, to account for panel material plasticity if exceeds F[cy] then limit F[cr] to F[cy].
15.2.4.3. Compression Buckling Allowable Stress with Varying Panel Rotational Edge Fixity
A simplified method allows for a quantified measure of the panel rotational edge stiffness, ε , this is defined by the following expression:
The value of ε can be calculated by taking the following approach. This approach is taken in part from (NACA-TN-888, 1943).
The term ε can be calculated from this expression:
When this is combined with the initial expression for ε the expression for evaluating ε becomes:
Of all these terms the C[BT] is a relatively little used cross section property and (NACA-TN-888, 1943) is one of the few references that gives a method to determine this value.
C[BT] is defined using the following expression:
and is described as “where u is the unit warping of the element of area dA from a reference plane through the shear center and normal to the axis when the angle of twist per unit length (dθ /dx) is
Thankfully the reference gives a set of expressions for for common cross sections:
Further explained in the figure below:
Figure 15.2.4‑4: Parameters for Torsion-Bending Constant
The spreadsheet for this method is at the link below:
A view of how the simply supported compression buckling coefficient, k, changes with the value of is given in Figure 17 of (NACA-TN-3781, 1957):
Figure 15.2.4‑5: The relationship between k and ε (NACA-TN-3781, 1957)
The lower line on the graph above defines the relationship between k and for a panel in compression buckling.
Figure 15.2.4‑6: The relationship between k and ε – Clarified
The spreadsheet method for estimating the effect on k for edge rotational edge fixity is given at the link below:
Note that this method of accounting for panel edge fixity correlates well to the value of k calculated using the general expression which is shown in Figure 15.2.4.
15.2.4.4. Compression Buckling Allowable Stress with Full Elasto-Plastic Material Data
If the calculated compression buckling stress is approaching the compression yield stress (F[cy]) of the material the elastic compression buckling allowable could be optimistic. If a more nuanced
approach than limiting the buckling stress to F[cy] is required, the compression buckling allowable should be modified using the plasticity correction factor η.
From (NACA-TN-3781, 1957) for compression buckling the plasticity correction factor for a long simply supported panel is:
the plasticity correction factor for a long, clamped panel is:
Plotting these two plasticity correction factors for a typical aluminum give the following result:
Figure 15.2.4‑7: Plasticity Correction Factors Compression buckling for Typical Aluminum Material
Figure 15.2.4‑8: Difference Between Different Edge Fixity Conditions for Plasticity Correction Factors for a Typical Aluminum Material
There is only a small difference between the two factors (less than 10%). It is recommended that the plasticity correction factor for clamped panels is used for all panels as it provides a simpler,
conservative solution.
Like the shear buckling allowable corrected for material plasticity, the plasticity correction factor for compression buckling can be used to produce a curve that relates the elastic buckling stress
with the plastic buckling stress.
Once the graph has been plotted the elastic shear buckling allowable can be plotted on the x-axis, project upwards to the curve and read across to the Y-axis to give the compression buckling
allowable corrected for plasticity.
Superimposing the simpler approach of limiting F[cr] to F[cy] over the Elastic vs Plastic shear buckling stress curve gives the following result:
Figure 15.2.4‑9: Elastic vs Plastic Compression Buckling Stress for Sample Material with Comparison to Simple Elastic and F[cy] Cut off Value
For the sample material (and for most ductile materials) the simple approach does give a reasonable approximation to the correctly calculated plastic buckling allowable. A spreadsheet is available
for this method at the link below:
15.2.4.5. Effect of Central Hole on Panel Compression Buckling Allowable
In the process of writing this section many references were reviewed. The single best summary of results is contained in (NASA-TM-1998-206542, 1998). This paper is a summary of the results of a
Finite Element study looking at the effect of circular and square holes on isotropic Panel buckling. Other references that are in general agreement include (NASA-TP-3024, 1990).
The figures in (NASA-TM-1998-206542, 1998) are given in terms of changes in absolute buckling values. These results have been reinterpreted to give a reduction factor for compression buckling factor
In the following figures the most onerous reduction from either the symmetric buckling or antisymmetric buckling modes have been used.
Note: The reduction factors are based on Figure 17 of (NASA-TM-1998-206542, 1998). This is reported as having ‘free’ edges – edges which are free to move in-plane. However, the results for a hole
diameter of 0 in agree with the results for a simply supported panel without a hole from table 4 of the same reference. This result also agrees with independent checking done for panels with edges
restrained in translation and free in rotation during the compilation of this section. It is therefore assumed that the reference is in error and Figure 17 is representative of panels with a hole
where the edges are restrained for in-plane translation.
Figure 15.2.4‑10: Reduction in Compression Buckling Allowable for Square Panels with Centrally Located Holes (NASA-TM-1998-206542, 1998)
It was found that the critical panel aspect ratio was 1:1 (square). As long as the panel is loaded in compression aligned with the long dimension, use of the square panel data is conservative.
This method is not applicable for panels loaded in compression across the short dimension.
As noted, the ratio of greatest reduction in buckling strength for each of the clamped and simply supported curves was calculated and plotted:
Figure 15.2.4‑11: Ratio of Reduction in Compression Buckling Strength of Clamped and Simply Supported Square Panels with Varying Sizes of Central Hole
A simple conservative linear approximation can be used that is applicable to both simply supported and clamped panels.
Figure 15.2.4‑12: Ratio of Reduction in Compression Buckling Strength of Clamped and Simply Supported Square Panels with Varying Sizes of Central Hole – Simple Approximation
The equation for this linear approximation is:
A spreadsheet method is available at the link below: | {"url":"https://www.abbottaerospace.com/aa-sb-001/15-local-stability-isotropic-materials/15-2-general-buckling-expression/15-2-4-panel-compression-buckling-rectangular/","timestamp":"2024-11-11T10:23:57Z","content_type":"text/html","content_length":"199949","record_id":"<urn:uuid:eb3ac721-c653-49e7-a286-fd05d939093d>","cc-path":"CC-MAIN-2024-46/segments/1730477028228.41/warc/CC-MAIN-20241111091854-20241111121854-00718.warc.gz"} |
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Date Issued:
The core idea of Ramsey theory is that complete disorder is impossible. Given a large structure, no matter how complex it is, we can always find a smaller substructure that has some sort of
order. One view of this problem is in edge-colorings of complete graphs. For any graphs G, H1, ..., Hk, we write G ? (H1, ..., Hk), or G ? (H)k when H1 = ��� = Hk = H, if every k-edge-coloring of
G contains a monochromatic Hi in color i for some i ? {1,...,k}. The Ramsey number rk(H1, ..., Hk) is the minimum integer n such that Kn ? (H1, ..., Hk), where Kn is the complete graph on n
vertices. Computing rk(H1, ..., Hk) is a notoriously difficult problem in combinatorics. A weakening of this problem is to restrict ourselves to Gallai colorings, that is, edge-colorings with no
rainbow triangles. From this we define the Gallai-Ramsey number grk(K3,G) as the minimum integer n such that either Kn contains a rainbow triangle, or Kn ? (G)k . In this thesis, we determine the
Gallai-Ramsey numbers for C7 with multiple colors. We believe the method we developed can be applied to find grk(K3, C2n+1) for any integer n ? 2, where C2n+1 denotes a cycle on 2n + 1 vertices.
Title: GALLAI-RAMSEY NUMBERS FOR C7 WITH MULTIPLE COLORS.
Bruce, Dylan, Author
Name(s): Song, Zi-Xia, Committee Chair
University of Central Florida, Degree Grantor
Type of text
Date Issued: 2017
Publisher: University of Central Florida
Language(s): English
The core idea of Ramsey theory is that complete disorder is impossible. Given a large structure, no matter how complex it is, we can always find a smaller substructure that has some
sort of order. One view of this problem is in edge-colorings of complete graphs. For any graphs G, H1, ..., Hk, we write G ? (H1, ..., Hk), or G ? (H)k when H1 = ��� = Hk = H, if every
Abstract/ k-edge-coloring of G contains a monochromatic Hi in color i for some i ? {1,...,k}. The Ramsey number rk(H1, ..., Hk) is the minimum integer n such that Kn ? (H1, ..., Hk), where Kn is
Description: the complete graph on n vertices. Computing rk(H1, ..., Hk) is a notoriously difficult problem in combinatorics. A weakening of this problem is to restrict ourselves to Gallai
colorings, that is, edge-colorings with no rainbow triangles. From this we define the Gallai-Ramsey number grk(K3,G) as the minimum integer n such that either Kn contains a rainbow
triangle, or Kn ? (G)k . In this thesis, we determine the Gallai-Ramsey numbers for C7 with multiple colors. We believe the method we developed can be applied to find grk(K3, C2n+1)
for any integer n ? 2, where C2n+1 denotes a cycle on 2n + 1 vertices.
Identifier: CFH2000264 (IID), ucf:46025 (fedora)
Note(s): College of Sciences, Mathematics
This record was generated from author submitted information.
Subject(s): Ramsey Theory
Graph Theory
Link to This http://purl.flvc.org/ucf/fd/CFH2000264
Restrictions public
on Access:
Host UCF
In Collections | {"url":"https://ucf.digital.flvc.org/islandora/object/ucf%3A46025","timestamp":"2024-11-06T09:06:35Z","content_type":"text/html","content_length":"35768","record_id":"<urn:uuid:5169c096-66cb-4a97-94b1-ece3390c2b19>","cc-path":"CC-MAIN-2024-46/segments/1730477027910.12/warc/CC-MAIN-20241106065928-20241106095928-00462.warc.gz"} |
Use the gradient-based optimization algorithm (grad_opt1) to find the
Use the gradient-based optimization algorithm (grad_opt1) to find the minima of the
function f(x) = cos(ex) + x²-1. Employ a rate parameter of -0.02 and -0.001, and start
from initial search points in the set (1.5, 2.5, 3). Set TOL=10^-6 and IMAX=1000. For each
case, generate a plot for the evolution of the search point x value vs iteration 't' (six plots
total). Compare your results to the true minima shown in the plot of f(x) within the interval
[04]. Does the method always lead to the minimum closest to the initial search point?
Fig: 1 | {"url":"https://tutorbin.com/questions-and-answers/use-the-gradient-based-optimization-algorithm-grad_opt1-to-find-the-minima-of-the-function-f-x-cos-ex-x-1-employ-a-rate","timestamp":"2024-11-05T07:08:11Z","content_type":"text/html","content_length":"62288","record_id":"<urn:uuid:9231f3c4-c5b0-49ed-9dae-024ef25195e3>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00579.warc.gz"} |
CAT in Mathematics
Click here for links below to some practice materials for the CAT in Mathematics.
Here you can find information on all of CUNY’s placement tests.
A Description of the CAT in Mathematics (COMPASS)
The CUNY Assessment Test in Mathematics (also known as the CAT in Mathematics, or the COMPASS Math test) is an untimed, multiple-choice, computer-based test designed to measure students’ knowledge of
a number of topics in mathematics. The test draws questions from four sections: numerical skills / pre-algebra,
algebra, college algebra, and trigonometry. Numerical skills / pre-algebra questions range from basic math concepts and skills (integers, fractions, and decimals) to the knowledge and skills that are
required in an entry-level algebra course (absolute values, percentages, and exponents). The algebra items are questions from elementary and intermediate algebra (equations, polynomials, formula
manipulations, and algebraic expressions). The college algebra section includes questions that measure skills required to perform operations with functions, exponents, matrices, and factorials. The
trigonometry section addresses topics such as trigonometric functions and identities, right-triangle trigonometry, and graphs of trigonometric functions. No two tests are the same; questions are
assigned randomly from the four sections, adapting to your test-taking experience.
Placement into CUNY’s required basic math courses is based on results of the numerical skills/pre-algebra and algebra sections. The test covers progressively advanced topics with placement into more
advanced mathematics or mathematics-related courses based on results of the last three sections of the test.
Students are permitted to use only the Microsoft Windows calculator while taking the test.
CAT in Mathematics Practice Materials
Below are some sample tests and websites containing more samples and information about the CAT in Mathematics and related materials. Special software may be needed to view some of these files; check
under our Software section to get them.
│ACT sample tests │all subjects │PDF │
│Hostos Community College online tutorials and diagnostic tests │pre-algebra, algebra │web-based Flash │
│Pearson Education sample test │pre-algebra, algebra │web-based; student ID = “111-11-1111” and course ID is │
│ │ │“compass” │
│Kentucky Early Math Testing Program practice exams │pre-algebra, algebra, college algebra, │web-based │
│ │trigonometry │ │
│Georgia Highlands College COMPASS practice tests and slideshow lessons │pre-algebra, algebra │ │
│Bellevue Community College sample problems with percentage breakdowns of problem │pre-algebra, algebra, college algebra, │PDF │
│types │trigonometry │ │
│Test Prep Review self-assessment modules │pre-algebra, algebra │PDF │ | {"url":"https://cunymath.commons.gc.cuny.edu/for-students/compass-2/","timestamp":"2024-11-10T09:18:00Z","content_type":"text/html","content_length":"71806","record_id":"<urn:uuid:b9ca233d-7c57-4e2e-9a97-6112aca1ccfc>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00232.warc.gz"} |
software - Definition & Meaning | Englia
The "software" comprising the carefully planned interpretive routines, compilers, and other aspects of automative programming are at least as important to the modern electronic calculator as its "
hardware" of tubes, transistors, wires, tapes and the like.
1958, John W. Tukey, "The Teaching of Concrete Mathematics" in The American Mathematical Monthly, vol. 65, no. 1 (Jan. 1958), pp 1-9 | {"url":"https://englia.app/definition/software","timestamp":"2024-11-10T17:19:08Z","content_type":"text/html","content_length":"70667","record_id":"<urn:uuid:efd00a06-c0e6-4938-bafd-c9ca629f55da>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.61/warc/CC-MAIN-20241110170046-20241110200046-00489.warc.gz"} |
Background: The Underlying Model
The CATMOD procedure analyzes data that can be represented by a two-dimensional contingency table. The rows of the table correspond to populations (or samples) formed on the basis of one or more
independent variables. The columns of the table correspond to observed responses formed on the basis of one or more dependent variables. The frequency in the cell is the number of subjects in the ith
population that have the jth response. The frequencies in the table are assumed to follow a product multinomial distribution, corresponding to a sampling design in which a simple random sample is
taken for each population. The contingency table can be represented as shown in Table 30.1.
Table 30.1: Contingency Table Representation
Sample 1 2 r Total
For each sample i, the probability of the jth response () is estimated by the sample proportion, . The vector () of all such proportions is then transformed into a vector of functions, denoted by .
If denotes the vector of true probabilities for the entire table, then the functions of the true probabilities, denoted by , are assumed to follow a linear model
where denotes asymptotic expectation, is the design matrix containing fixed constants, and is a vector of parameters to be estimated.
PROC CATMOD provides two estimation methods:
• The weighted least squares method minimizes the weighted residual sum of squares for the model. The weights are contained in the inverse covariance matrix of the functions . According to central
limit theory, if the sample sizes within populations are sufficiently large, the elements of and (the estimate of ) are distributed approximately as multivariate normal. This allows the
computation of statistics for testing the goodness of fit of the model and the significance of other sources of variation. For details of the theory, see Grizzle, Starmer, and Koch (1969) or Koch
et al. (1977, Appendix 1). Weighted least squares estimation is available for all types of response functions.
• The maximum likelihood method estimates the parameters of the linear model so as to maximize the value of the joint multinomial likelihood function of the responses. Maximum likelihood estimation
is available only for the standard response functions, logits and generalized logits, which are used for logistic regression analysis and log-linear model analysis. Two methods of maximization
are available: Newton-Raphson and iterative proportional fitting. For details of the theory, see Bishop, Fienberg, and Holland (1975).
Following parameter estimation, hypotheses about linear combinations of the parameters can be tested. For that purpose, PROC CATMOD computes generalized Wald (1943) statistics, which are
approximately chi-square distributed if the sample sizes are sufficiently large and the null hypotheses are true. | {"url":"http://support.sas.com/documentation/cdl/en/statug/65328/HTML/default/statug_catmod_overview11.htm","timestamp":"2024-11-09T00:00:20Z","content_type":"application/xhtml+xml","content_length":"32658","record_id":"<urn:uuid:89e44fff-a8f1-4e4b-8e3c-26971d33cc4a>","cc-path":"CC-MAIN-2024-46/segments/1730477028106.80/warc/CC-MAIN-20241108231327-20241109021327-00592.warc.gz"} |
1st Grade Math Worksheets Adding 3 Numbers
1st Grade Math Worksheets Adding 3 Numbers act as foundational devices in the realm of mathematics, offering a structured yet versatile system for learners to discover and understand numerical
concepts. These worksheets provide an organized strategy to comprehending numbers, supporting a strong foundation upon which mathematical effectiveness flourishes. From the easiest checking exercises
to the ins and outs of innovative computations, 1st Grade Math Worksheets Adding 3 Numbers cater to students of varied ages and ability degrees.
Revealing the Essence of 1st Grade Math Worksheets Adding 3 Numbers
1st Grade Math Worksheets Adding 3 Numbers
1st Grade Math Worksheets Adding 3 Numbers -
This fun and differentiated JAM PACKED unit is filled with 215 pages of everything you need to teach adding 3 numbers with sums to 20 in first grade Link to REAL PICTURES fun ideas to teach in
description This pack covers 4 main standards concepts adding 3 numbers strategies for adding
Grade 1 Addition Add 3 s on a number line Add 3 numbers on a number line Number lines and 3 addends Students are given addition equations with 3 numbers between 0 and 20 and use a number line to
solve them Worksheet 1 Worksheet 2 Worksheet 3 Similar Number lines equations Adding 2 numbers on a number line More addition
At their core, 1st Grade Math Worksheets Adding 3 Numbers are automobiles for theoretical understanding. They encapsulate a myriad of mathematical principles, guiding learners through the maze of
numbers with a collection of appealing and deliberate workouts. These worksheets transcend the borders of traditional rote learning, urging energetic engagement and fostering an user-friendly
understanding of mathematical partnerships.
Nurturing Number Sense and Reasoning
Adding Three Numbers Add 3 Numbers Worksheets Printables Make Ten First Math Addition
Adding Three Numbers Add 3 Numbers Worksheets Printables Make Ten First Math Addition
Out of 100 IXL s SmartScore is a dynamic measure of progress towards mastery rather than a percentage grade It tracks your skill level as you tackle progressively more difficult questions
Consistently answer questions correctly to reach excellence 90 or conquer the Challenge Zone to achieve mastery 100 Learn more 0 Work it out
Course 1st grade Unit 2 Lesson 2 Addition within 20 Adding 7 6 Adding 8 7 Add within 20 Adding 5 3 6 Add 3 numbers Math
The heart of 1st Grade Math Worksheets Adding 3 Numbers hinges on growing number sense-- a deep comprehension of numbers' meanings and interconnections. They urge exploration, welcoming students to
study arithmetic procedures, decode patterns, and unlock the enigmas of sequences. Via provocative challenges and sensible puzzles, these worksheets become gateways to honing thinking abilities,
nurturing the analytical minds of budding mathematicians.
From Theory to Real-World Application
Adding 3 Math Worksheet Woo Jr Kids Activities
Adding 3 Math Worksheet Woo Jr Kids Activities
First Grade math worksheets to help students practice making a 10 first when adding 3 numbers or addends How to use the make ten strategy when adding 3 numbers Try to look for number pairs or number
bonds that make up 10 and add that first to get a ten before adding the third number
Jump to the rhythm of the math beat with this 1st grade worksheet that features single digit addition problems with sums up to 9 1st grade Math Interactive Worksheet These fun and colorful first
grade addition worksheets make numbers more exciting and accessible Try different strategies to see which one fits your child best Educational
1st Grade Math Worksheets Adding 3 Numbers work as channels bridging academic abstractions with the palpable truths of daily life. By infusing sensible circumstances right into mathematical workouts,
students witness the importance of numbers in their environments. From budgeting and measurement conversions to recognizing statistical information, these worksheets encourage students to possess
their mathematical prowess past the boundaries of the classroom.
Diverse Tools and Techniques
Flexibility is inherent in 1st Grade Math Worksheets Adding 3 Numbers, employing an arsenal of instructional devices to cater to different learning designs. Visual help such as number lines,
manipulatives, and digital resources function as friends in imagining abstract concepts. This varied strategy guarantees inclusivity, accommodating learners with various preferences, toughness, and
cognitive designs.
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In a progressively diverse world, 1st Grade Math Worksheets Adding 3 Numbers welcome inclusivity. They go beyond social boundaries, incorporating examples and issues that reverberate with learners
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mathematical concepts.
Crafting a Path to Mathematical Mastery
1st Grade Math Worksheets Adding 3 Numbers chart a training course in the direction of mathematical fluency. They impart determination, crucial reasoning, and analytic skills, crucial qualities not
only in mathematics yet in various elements of life. These worksheets empower students to browse the detailed terrain of numbers, nurturing a profound recognition for the beauty and reasoning
inherent in mathematics.
Welcoming the Future of Education
In an age noted by technological advancement, 1st Grade Math Worksheets Adding 3 Numbers seamlessly adapt to digital platforms. Interactive interfaces and electronic resources enhance traditional
learning, providing immersive experiences that transcend spatial and temporal borders. This combinations of conventional methodologies with technological developments heralds an appealing age in
education and learning, promoting an extra dynamic and appealing knowing atmosphere.
Verdict: Embracing the Magic of Numbers
1st Grade Math Worksheets Adding 3 Numbers illustrate the magic inherent in maths-- a captivating journey of expedition, exploration, and proficiency. They go beyond standard pedagogy, acting as
catalysts for igniting the fires of inquisitiveness and inquiry. With 1st Grade Math Worksheets Adding 3 Numbers, learners start an odyssey, unlocking the enigmatic globe of numbers-- one issue, one
option, at once.
The Worksheet For Adding Three Numbers To One Hundredths Is Shown In Black And White
Adding 3 Numbers 1st Grade
Check more of 1st Grade Math Worksheets Adding 3 Numbers below
Grade 1 Math Worksheet Add 3 Single Digit Numbers K5 Learning Adding 3 Single Digit Numbers
Adding 3 Digit Numbers Worksheet
Math Addition Worksheets 1st Grade
Adding Three Numbers Worksheet
First Grade Addition Worksheets
Adding Three Numbers First Grade
Adding 3 Numbers On A Number Line Worksheets K5 Learning
Grade 1 Addition Add 3 s on a number line Add 3 numbers on a number line Number lines and 3 addends Students are given addition equations with 3 numbers between 0 and 20 and use a number line to
solve them Worksheet 1 Worksheet 2 Worksheet 3 Similar Number lines equations Adding 2 numbers on a number line More addition
Add 3 Single Digit Numbers Grade 1 Addition Worksheets
This first grade worksheet will have students adding three single digit numbers up in no time Students have previously only worked with adding two single digit numbers such as 2 3 or 5 6 They will
now build on that skill by adding a third number to the mix
Grade 1 Addition Add 3 s on a number line Add 3 numbers on a number line Number lines and 3 addends Students are given addition equations with 3 numbers between 0 and 20 and use a number line to
solve them Worksheet 1 Worksheet 2 Worksheet 3 Similar Number lines equations Adding 2 numbers on a number line More addition
This first grade worksheet will have students adding three single digit numbers up in no time Students have previously only worked with adding two single digit numbers such as 2 3 or 5 6 They will
now build on that skill by adding a third number to the mix
Adding Three Numbers Worksheet
Adding 3 Digit Numbers Worksheet
First Grade Addition Worksheets
Adding Three Numbers First Grade
Fall Math And Literacy Packet 1st Grade Math Activities Elementary Preschool Math Fun
Adding Three Numbers Worksheet
Adding Three Numbers Worksheet
Number 3 Worksheets Adding Three Single Digits Additon Worksheet 2 First Grade Worksheets | {"url":"https://szukarka.net/1st-grade-math-worksheets-adding-3-numbers","timestamp":"2024-11-09T01:14:23Z","content_type":"text/html","content_length":"26246","record_id":"<urn:uuid:89b4c0b2-5173-4e6d-be2d-eecda9f34808>","cc-path":"CC-MAIN-2024-46/segments/1730477028106.80/warc/CC-MAIN-20241108231327-20241109021327-00200.warc.gz"} |
Neural networks for regression with autograd
Posted November 18, 2017 at 02:20 PM | categories: autograd, python | tags:
Today we are going to take a meandering path to using autograd to train a neural network for regression. First let's consider this very general looking nonlinear model that we might fit to data.
There are 10 parameters in it, so we should expect we can get it to fit some data pretty well.
\(y = b1 + w10 tanh(w00 x + b00) + w11 tanh(w01 x + b01) + w12 tanh(w02 x + b02)\)
We will use it to fit data that is generated from \(y = x^\frac{1}{3}\). First, we just do a least_squares fit. This function can take a jacobian function, so we provide one using autograd.
import autograd.numpy as np
from autograd import jacobian
from scipy.optimize import curve_fit
# Some generated data
X = np.linspace(0, 1)
Y = X**(1. / 3.)
def model(x, *pars):
b1, w10, w00, b00, w11, w01, b01, w12, w02, b02 = pars
pred = b1 + w10 * np.tanh(w00 * x + b00) + w11 * np.tanh(w01 * x + b01) + w12 * np.tanh(w02 * x + b02)
return pred
def resid(pars):
return Y - model(X, *pars)
MSE: 0.0744600049689
We will look at some timing of this regression. Here we do not provide a jacobian.
pars = least_squares(resid, np.random.randn(10)*0.1).x
1.21 s ± 42.7 ms per loop (mean ± std. dev. of 7 runs, 1 loop each)
And here we do provide one. It takes a lot longer to do this. We do have a jacobian of 10 parameters, so that ends up being a lot of extra computations to do.
pars = least_squares(resid, np.random.randn(10)*0.1, jac=jacobian(resid)).x
24.1 s ± 1.61 s per loop (mean ± std. dev. of 7 runs, 1 loop each)
We will print these parameters for reference later.
b1, w10, w00, b00, w11, w01, b01, w12, w02, b02 = pars
print([w00, w01, w02], [b00, b01, b02])
print([w10, w11, w12], b1)
[5.3312122926210703, 54.6923797622945, -0.50881373227993232] [2.9834159679095662, 2.6062295455987199, -2.3782572250527778]
[42.377172168160477, 22.036104340171004, -50.075636975961089] -113.179935862
Let's just make sure the fit looks ok. I am going to plot it outside the fitted region to see how it extrapolates. The shaded area shows the region we did the fitting in.
X2 = np.linspace(0, 3)
Y2 = X2**(1. / 3.)
Z2 = model(X2, *pars)
plt.plot(X2, Y2, 'b.', label='analytical')
plt.plot(X2, Z2, label='model')
plt.fill_between(X2 < 1, 0, 1.4, facecolor='gray', alpha=0.5)
You can seen it fits pretty well from 0 to 1 where we fitted it, but outside that the model is not accurate. Our model is not that related to the true function of the model, so there is no reason to
expect it should extrapolate.
I didn't pull that model out of nowhere. Let's rewrite it in a few steps. If we think of tanh as a function that operates element-wise on a vector, we could write that equation more compactly at:
[w00 * x + b01]
y = [w10, w11, w12] @ np.tanh([w01 * x + b01]) + b1
[w02 * x + b02]
We can rewrite this one more time in matrix notation:
y = w1 @ np.tanh(w0 @ x + b0) + b1
Another way to read these equations is that we have an input of x. We multiply the input by a vector weights (w0), add a vector of offsets (biases), b0, activate that by the nonlinear tanh function,
then multiply that by a new set of weights, and add a final bias. We typically call this kind of model a neural network. There is an input layer, one hidden layer with 3 neurons that are activated by
tanh, and one output layer with linear activation.
Autograd was designed in part for building neural networks. In the next part of this post, we reformulate this regression as a neural network. This code is lightly adapted from https://github.com/
The first function initializes the weights and biases for each layer in our network. It is standard practice to initialize them to small random numbers to avoid any unintentional symmetries that
might occur from a systematic initialization (e.g. all ones or zeros). The second function sets up the neural network and computes its output.
from autograd import grad
import autograd.numpy.random as npr
from autograd.misc.optimizers import adam
def init_random_params(scale, layer_sizes, rs=npr.RandomState(0)):
"""Build a list of (weights, biases) tuples, one for each layer."""
return [(rs.randn(insize, outsize) * scale, # weight matrix
rs.randn(outsize) * scale) # bias vector
for insize, outsize in zip(layer_sizes[:-1], layer_sizes[1:])]
def nn_predict(params, inputs, activation=np.tanh):
for W, b in params[:-1]:
outputs = np.dot(inputs, W) + b
inputs = activation(outputs)
# no activation on the last layer
W, b = params[-1]
return np.dot(inputs, W) + b
Here we use the first function to define the weights and biases for a neural network with one input, one hidden layer of 3 neurons, and one output layer.
init_scale = 0.1
# Here is our initial guess:
params = init_random_params(init_scale, layer_sizes=[1, 3, 1])
for i, wb in enumerate(params):
W, b = wb
print('w{0}: {1}, b{0}: {2}'.format(i, W.shape, b.shape))
w0: (1, 3), b0: (3,)
w1: (3, 1), b1: (1,)
You can see w0 is a column vector of weights, and there are three biases in b0. W1 in contrast, is a row vector of weights, with one bias. So 10 parameters in total, like we had before. We will
create an objective function of the mean squared error again, and a callback function to show us the progress.
Then we run the optimization step iteratively until we get our objective function below a tolerance we define.
def objective(params, _):
pred = nn_predict(params, X.reshape([-1, 1]))
err = Y.reshape([-1, 1]) - pred
return np.mean(err**2)
def callback(params, step, g):
if step % 250 == 0:
print("Iteration {0:3d} objective {1:1.2e}".format(i * N + step,
objective(params, step)))
N = 500
NMAX = 20
for i in range(NMAX):
params = adam(grad(objective), params,
step_size=0.01, num_iters=N, callback=callback)
if objective(params, _) < 2e-5:
Iteration 0 objective 5.30e-01
Iteration 250 objective 4.52e-03
Iteration 500 objective 4.17e-03
Iteration 750 objective 1.86e-03
Iteration 1000 objective 1.63e-03
Iteration 1250 objective 1.02e-03
Iteration 1500 objective 6.30e-04
Iteration 1750 objective 4.54e-04
Iteration 2000 objective 3.25e-04
Iteration 2250 objective 2.34e-04
Iteration 2500 objective 1.77e-04
Iteration 2750 objective 1.35e-04
Iteration 3000 objective 1.04e-04
Iteration 3250 objective 7.86e-05
Iteration 3500 objective 5.83e-05
Iteration 3750 objective 4.46e-05
Iteration 4000 objective 3.39e-05
Iteration 4250 objective 2.66e-05
Iteration 4500 objective 2.11e-05
Iteration 4750 objective 1.71e-05
Let's compare these parameters to the previous ones we got.
for i, wb in enumerate(params):
W, b = wb
print('w{0}: {1}, b{0}: {2}'.format(i, W, b))
w0: [[ -0.71332351 3.23209728 -32.51135373]], b0: [ 0.45819205 0.19314303 -0.8687 ]
w1: [[-0.53699549]
[ 0.39522207]
[-1.05457035]], b1: [-0.58005452]
These look pretty different. It is not too surprising that there could be more than one set of these parameters that give similar fits. The original data only requires two parameters to create it: \
(y = a x^b\), where \(x=1\) and \(b=1/3\). We have 8 extra parameters of flexibility in this model.
Let's again examine the fit of our model to the data.
Z2 = nn_predict(params, X2.reshape([-1, 1]))
plt.plot(X2, Y2, 'b.', label='analytical')
plt.plot(X2, Z2, label='NN')
plt.fill_between(X2 < 1, 0, 1.4, facecolor='gray', alpha=0.5)
Once again, we can see that between 0 and 1 where the model was fitted we get a good fit, but past that the model does not fit the known function well. It is coincidentally better than our previous
model, but as before it is not advisable to use this model for extrapolation. Even though we say it "learned" something about the data, it clearly did not learn the function \(y=x^{1/3}\). It did
"learn" some approximation to it in the region of x=0 to 1. Of course, it did not learn anything that the first nonlinear regression model didn't learn.
Now you know the secret of a neural network, it is just a nonlinear model. Without the activation, it is just a linear model. So, why use linear regression, when you can use an unactivated neural
network and call it AI?
Copyright (C) 2017 by John Kitchin. See the License for information about copying.
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Decimal to Octal Converter | Convertopedia
Use the below online conversion tool to convert any
decimal number to octal number:
Decimal Number
Octal Number
Also Try:
Decimal Number
Whole number, a decimal point and a fractional value combines to form a decimal number. The decimal point separates the whole number part from the fractional part of the number. Each digit of a
decimal number can be any number from 0 to 9. Any value less than 1 is written to the right of decimal point. Decimal numbers are also known as base-10 number or counting numbers. Place value of
decimal number varies as the whole number powers of 10 starting from the left of decimal point. Similarly, the place value of digits left to decimal point varies as the division of power of tens.
Octal Number
Octal numbers use digits from 0-7 only. It is known as base-8 number. The place value of each digits of an octal number varies as the whole number powers of 8 starting from the right (Least
Significant Digit). The first single digit number in octal system is 0 and the last is 7. Similarly, the first two digit octal number is 10 and the last is 77 and so on. Octal number system was
widely used in early computers.
Decimal to octal conversion example
Convert 582[10] to octal
582[10] = 1106[8]
Convert 859[10] to octal
859[10] = 1533[8]
Decimal to octal conversion table
Decimal Number Octal Number
Also Try:
Other Converters | {"url":"https://www.convertopedia.com/numerical-converters/decimal-to-octal-converter/","timestamp":"2024-11-05T12:52:24Z","content_type":"text/html","content_length":"85356","record_id":"<urn:uuid:9ac6350d-43db-4930-a087-09e6112ddcf2>","cc-path":"CC-MAIN-2024-46/segments/1730477027881.88/warc/CC-MAIN-20241105114407-20241105144407-00816.warc.gz"} |
Professor, Department of Astronomy and Astrophysics
University of Chicago
Astro 321
WF 1:30-3:00 AAC 123
First Meeting: 1/5
This course will have its focus on the inhomogeneous universe.
I expect that you are comfortable with programming in your language of choice. Some rudimentary general relativity background would be helpful but not strictly necessary.
There is no required texbook for the course but here are a few suggestions:
• Peacock: Cosmological Physics, Cambridge 1999 (a broad book)
• Dodelson: Modern Cosmology (CMB, kinetic theory)
• Kolb & Turner: Early Universe (early universe)
• Liddle & Lyth: Cosmological Inflation and Large-Scale Structure (inflationary perturbation theory),
• Padmanabhan: Structure Formation in the Universe (non-linear collapse)
In the syllabus below, I give a cross reference to Peacock's book for further reading.
There will be weekly problem sets (50%) and a final project (50%).
For a final project you may work in groups of 5 (or fewer) people on any of the following
• Particle Mesh N-Body Code
• Inflationary Perturbation Solver
• Einstein-Boltzmann Code
• Halo Model Code
• Monte Carlo Markov Chain
You may also come up with your own comparable numerical project or be creative and develop a webApp or iApp. If you are truly computation averse see me for permission to do a reading project.
You will present your project to the class at the end of the quarter and submit the PDF of the presentation for linkage here. Extra credit if you make your code publically available.
Problem Sets
Final Project Preparation
Each project has a core set of things that I expect you to accomplish. I encourage you to develop your codes further in ways of your choosing to develop a more extensive toolbox.
N-body Group
Follow Andrey Kravtsov's Notes
Halo Model Group
Read Cooray & Sheth [Phys.Rept. 372 (2002) 1-129 e-Print: astro-ph/0206508] and construct the halo model nonlinear matter power spectrum out the 1 halo + 2 halo terms, halo bias, and the NFW profile.
You may find these excersises helpful
Problem Set 1
Problem Set 2
Problem Set 3
Problem Set 4
but you do not need to turn these in.
MCMC Group
Code up a MCMC analysis of your favorite cosmological data set (e.g. UNION2 SN) and extract the posterior probability distributions of the cosmological parameters you include. Compare them with known
results in the literature. You may find COSMOMC and Lewis and Bridle Phys.Rev. D66 (2002) 103511 e-Print: astro-ph/0205436 useful.
Final Project Presentations
Rough outline of the course:
• Friedmann Robertson Walker (FRW) Cosmology: P-Ch-3 & 5
• Matter in the Universe: P-Ch-12
• Kinetic theory in an expanding universe: P-Ch-15.1-15.6; P-Ch-16.1-16.3; Kolb & Turner; Dodelson
• Inhomogeneous fields and linear perturbation theory: P-Ch-15.1-15.6; P-Ch-16.1-16.3; Dodelson
• Inflationary Cosmology: P-Ch-11; Liddle & Lyth
• Cosmic Microwave Background: P-Ch-18; online tutorial
• Large Scale Structure: P-Ch-15
• Spherical collapse and mass functions: P-Ch-15.7-8; 16.4; 17.2
• Bias and the halo model: P-Ch-15.7-8: P-Ch-15.7-8; 16.4; 17.2
Lecture Notes
Lecture notes will be posted as we go through the course: | {"url":"https://background.uchicago.edu/~whu/Courses/ast321_11.html","timestamp":"2024-11-04T11:38:44Z","content_type":"text/html","content_length":"11279","record_id":"<urn:uuid:e5a02d6d-20d7-40d4-9fd3-120c9b2ad4c1>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00624.warc.gz"} |
Time Goes On Class 3 Worksheet with Answers Maths Chapter 13
Class 3 Maths Chapter 13 Worksheet Time Goes On
Time Goes On Class 3 Maths Worksheet
Time Goes On Worksheet
Question 1.
Answer the following questions which are based on the given story.
Anaya and her friends Shubha and Mitushi are playing the game of dates and days in the new year calendar 2024. Shubha said that the month of January is missing in the new calendar. Anaya replied that
the month of January is similar to the month of July.
(i) On Mitushi’s B turn, her task is to decide to till all the dates of January 2024. How can she fill it?
(ii) A. State with yes or no.
(a) The numbers of Monday and Tuesday in the month of January are 5 and 4 respectively.
(b) The Sunday dates in the month of January are 7,14, 21 and 28, respectively.
B. Fill in the blanks according to the month of January.
(a) The number of Saturdays is ____________.
(b) ____________ are the Monday dates in the month of January.
(c) Five days after January 25 is January ____________ .
C. A temple is closed between 9th January and 15th January. Find the date of the opening of the temple.
D. List all the months which have 31 days.
E. List all the months which have 30 days.
Question 2.
Ankur and Shikha are observing the calendar the years 2022 and 2023. Answer the following questions according to calendar 2022 and 2023.
(i) Ankur is trying to list the festivals of 2023. He made a table and filled the festivals one by one and after that he returned to his home. Now, it is your task to write the correct date of the
festival in the table.
(ii) State with true and false.
(a) The number of days in months does not change in both calendars. (True/false)
(b) The names of the days in weeks change in both calendars. (True/false)
(c) The total number of Sundays is changed in both calendars. (True/false)
(d) The total number of weeks does not change in both calendars. (True/false)
(iii) (a) List all the months in the calendar of 2022 and 2023.
January, February, March, April, May, June, July, August, September, October, November, December
(b) List all months that have less than 31 days.
February, April, June, September, November
(c) Is there any month which has 28 days in both calendars?
Question 3.
Answer the following questions which are based on age-related problems.
(i) Meena is three times older than her sister. The difference of their ages is 20 yrs. Find the age of Meena and her sister. ____________
(ii) Sara is four times as old as her cousin and 24 years older than her cousin. Find the age of her cousin. ____________
(i) Meena’s age is 30 yr and her sister’s age is 10 yr
(ii) 8 yr
Question 4.
Answer the following questions which is based on the birth certificate given below.
(i) In which month was Rinki born?
(ii) How old will Rinki be on 05-02-2025?
4 yr
(iii) On which date Rinki will be 25 years old?
(iv) On which date is Rinki’s fifth birthday?
(v) On what date was her birth certificate issued?
(vi) Where was Rinki born?
Kamla Nursing Home, Khairthal
Question 5.
Match the columns.
(i) b
(ii) e
(iii) a
(iv) f
(v) d
(vi) c
Question 6.
Match the 12-hour alarm clock to the correct time shown by it.
(i) b
(ii) e
(iii) d
(iv) a
(v) c
Question 7.
Fill in the blanks.
(i) Sheetal started her work at 9 o’clock in the morning and finished her work at 5 o’clock in the evening. He took ____________ hours to complete the work.
(ii) Rihana started her lunch at 11 o’clock in the morning and finished it at 11.15 o’clock in the morning. She took ____________ hours to finish lunch.
(i) 8
(ii) \(\frac{1}{4}\) or 15 min
Question 8.
List all the activities that can be done in the time frame given below. One has been done for you.
┃Time Duration │Activity ┃
┃10 min │Brushing teeth ┃
┃15 min │Breakfast ┃
┃20 min │Prayer ┃
┃30 min │Class time ┃
┃60 min │Play ┃
Question 9.
Draw the hands on the clock to match the time on the digital clock.
Question 10.
Find the time difference between two clocks. One has been done for you.
(ii) 10 min
(iii) 30 min
(iv) 40 min
(v) 1 h or 60 min
(vi) 55 min | {"url":"https://www.learninsta.com/time-goes-on-class-3-worksheet/","timestamp":"2024-11-11T11:07:58Z","content_type":"text/html","content_length":"63683","record_id":"<urn:uuid:5b2f4ac3-43a3-4dd8-9e46-73694e8c1d89>","cc-path":"CC-MAIN-2024-46/segments/1730477028228.41/warc/CC-MAIN-20241111091854-20241111121854-00337.warc.gz"} |
ERC Starting Grant - QFreC
Machine learning empowers computers to solve complex tasks such as pattern identification and strategy optimization with applications in, e.g. financial trading, fraud detection, medical diagnosis,
and self-driving vehicles. The required computing power is, however, pushing existing computational resources to their limits, restraining their further advancement. In QFreC, I target the
realization of photonic frequency-based quantum co-processors, specifically tailor-made to solve machine learning problems with capabilities commensurate with today’s high-power, yet energy-efficient
processing needs. In particular, I will use a high-dimensional photonic quantum frequency comb approach, where photons have hundreds to thousands of discrete and equidistantly spaced frequency modes,
giving access to large, scalable information capacity.
For implementing quantum-accelerated machine learning tasks such as the classification of classical or quantum data, I will follow i) the exploration of quantum photonic frequency-domain processing
with the adaptation of qubit learning concepts (vector-based and neural network-based approaches) to high-dimensional quantum representations, i.e. quDits, ii) the realization of efficiency-enhanced
and novel integrated quantum frequency comb systems with quantum resources that allow real-world applications using highly nonlinear on-chip platforms, and iii) the development of reconfigurable,
fast, and broadband experimental control schemes using, e.g. quadrature amplitude modulation formats and nonlinear optical processes. To enable stable, compact, cost- and energy-efficient quantum
processing devices, the QFreC project will build on the advances of the well-developed telecommunications infrastructure and the photonic chip fabrication industry.
QFreC merges photonic quantum frequency-domain circuits with quantum machine learning, enabling large-scale controllable quantum resources for the exploration of quantum-enhanced machine learning. | {"url":"https://www.iop.uni-hannover.de/de/arbeitsgruppen/pqt/erc-starting-grant-qfrec","timestamp":"2024-11-08T14:39:35Z","content_type":"application/xhtml+xml","content_length":"36029","record_id":"<urn:uuid:160ab0a7-4777-4f64-aff2-581e65302656>","cc-path":"CC-MAIN-2024-46/segments/1730477028067.32/warc/CC-MAIN-20241108133114-20241108163114-00019.warc.gz"} |
If NP languages are hard on the worst-case, then it is easy to find their hard instances
We prove that if NP ⊈ BPP, i.e., if SAT is worst-case hard, then for every probabilistic polynomial-time algorithm trying to decide SAT, there exists some polynomially samplable distribution that is
hard for it. That is, the algorithm often errs on inputs from this distribution. This is the first worst-case to average-case reduction for NP of any kind. We stress however, that this does not mean
that there exists one fixed samplable distribution that is hard for all probabilistic polynomial-time algorithms, which is a pre-requisite assumption needed for one-way functions and cryptography
(even if not a sufficient assumption). Nevertheless, we do show that there is a fixed distribution on instances of NP-complete languages, that is samplable in quasi-polynomial time and is hard for
all probabilistic polynomial-time algorithms (unless NP is easy in the worst case). Our results are based on the following lemma that may be of independent interest: Given the description of an
efficient (probabilistic) algorithm that fails to solve SAT in the worst case, we can efficiently generate at most three Boolean formulae (of increasing lengths) such that the algorithm errs on at
least one of them.
Bibliographical note
Funding Information:
Dan Gutfreund was supported in part by ONR grant N00014-04-1-0478. Most of this research was done while he was at the Hebrew University. Ro-nen Shaltiel did part of this research while staying at the
Weizmann Institute and supported by the Koshland scholarship,and was also supported by Grant No. 2004329 from the United States-Israel Binational Science Foundation(BSF),Jerusalem,Israel.
AmnonTa-shmawassupportedbytheIsrael Science Foundationgrant no. 217/0.
• Average-case complexity
• Foundations of cryptography
• Pseudo classes
• Worst-case to average-case reductions
ASJC Scopus subject areas
• Theoretical Computer Science
• General Mathematics
• Computational Theory and Mathematics
• Computational Mathematics
Dive into the research topics of 'If NP languages are hard on the worst-case, then it is easy to find their hard instances'. Together they form a unique fingerprint. | {"url":"https://cris.haifa.ac.il/en/publications/if-np-languages-are-hard-on-the-worst-case-then-it-is-easy-to-fin-2","timestamp":"2024-11-03T23:11:24Z","content_type":"text/html","content_length":"57169","record_id":"<urn:uuid:c6826fe3-88e1-42fd-8e70-b623fc4d1428>","cc-path":"CC-MAIN-2024-46/segments/1730477027796.35/warc/CC-MAIN-20241103212031-20241104002031-00223.warc.gz"} |
Shimla teacher challenging Einstein to deliver lecture at IIT Gandhinagar
Shimla: Ajay Sharma, principal of Government Senior Secondary School, Dhami, has been invited by Indian Institute of Technology, Gandhinagar, to deliver a lecture on modification of Einstein’s famous
equation E=mc2. The Department of Physics , IIT Gandhinagar is organizing a scientific workshop COSMOS 2010 on April 10 and 11.
It is pertinent to mention that since past many years Ajay Sharma has been boldly talking about generalization of Einstein’s equation E=mc2. In 2002, Chief Minister Prem Kumar Dhumal had forwarded
his research papers to Vice-chancellor of Himachal Pradesh University for comments. Then Department of Physics, HPU, had replied that as none of its teachers had specialisation in the field, the
papers could not be evaluated.
Working on the topic for past 28 years, Ajay says he has 100% confidence about the validity of his ideas. He has published nearly 18 research papers in peer review international journals, along with
two books published from America and Germany. In his research papers, Ajay Sharma has confirmed that Einstein’s derivation of 1905 has serious limitations and is valid only under special conditions
of the parameters. “Under general conditions, Einstein’s derivation of E=mc2 also predicts that, ‘when a candle burns its mass must increase’ It is absolutely incorrect conclusion and never justified
experimentally. The scientific world is completely silent over the issue.” Thus Ajay Sharma has given generalized equation dE= Ac2dm. Also, he has also given many applications of his equation.
Ajay Sharma laments that had he got cooperation from the Department of Science and Technology, New Delhi, he could have proved his point years ago. Ajay has been requesting the department to
constitute a committee of specialist scientists to evaluate his work so that his work may be transparently evaluated.
13 COMMENTS
1. keep it up!!!!!!!!!!
2. keep it up!!!!!!!!!!
3. your work seems to be authentic……….and i hope that it will be proven to the world. all the best
4. your work seems to be authentic……….and i hope that it will be proven to the world. all the best
5. sir v hv given an assignment abt the new formula e=ac2dm.unit and dimension of a? and physical and numerical inteperator for symbol ‘a’.pls sir tell me.so thankful to u.
6. sir v hv given an assignment abt the new formula e=ac2dm.unit and dimension of a? and physical and numerical inteperator for symbol ‘a’.pls sir tell me.so thankful to u.
7. sir besties for u……. i hope we will be someday teaching ur work in near future….i am really interested in ur work
8. Sir, Your research work is laudable & inspirable to younger Indian scientists & Engineers. Wishing yourgoodself & your family Happy & prosperous new year 2013. With warm regards Yours Er. K L
9. creative n intersting matter…. go 4 it sir…
10. Well Mr. Ajay Sharma with all due respect i welcome your candle logic. But the truth is you got it bit wrong.Sir, I must suggest you to read “The Special Theory Of Relativity” again. I must
assure you that my intentions are strictly honorable and there is no hurting you. But the fact is you have made a mistake in understanding space-time exactly. I just found a mistake with your
candle logic. I don’t know how you came to the eq.dE= Ac2dm but if you have derived this eq. with having your candle logic in your mind then sorry you got it all wrong. I once again want you to
know there is no hurting you because i respect people who question things but the truth is we must not spread false ideas. I can give you the explanation of you candle logic. Anyone who wants the
explanation of why E=mc2 is correct(at least in case of candle logic) can contact me
11. helo Aditya ! Without the knowledge of Einstein;s theory I could not say whether Mr. Ajay Sharma is right or wrong . I could’nt understand the ‘Special Theory Of Relativity ‘.
But i like creativity…………………..
12. sir, I am a scientist working hard i have made i ship which is unsinkable it can even work after its fuel tank is empty sir, it has been patent from indian government(patent no.1633/mum/2012
published date 19/10/2012.i need a strong help for making this patent a globalise one. Sir, it requires heay fees which is not possible by and top of that i am handcap by leg.
13. Is this man crazy? If you say no, run….. to the mental hospital! | {"url":"https://www.himvani.com/4436/shimla-teacher-challenging-einstein-to-deliver-lecture-at-iit-gandhinagar/","timestamp":"2024-11-11T20:23:27Z","content_type":"text/html","content_length":"162065","record_id":"<urn:uuid:2b5a29ce-cfc8-4e6a-8dd4-122c0be34158>","cc-path":"CC-MAIN-2024-46/segments/1730477028239.20/warc/CC-MAIN-20241111190758-20241111220758-00707.warc.gz"} |
Restriction presheaves and restriction colimits
posted on 2022-03-28, 01:41 authored by Daniel Lin
Restriction categories, as defined by Cockett and Lack, are an abstraction of the notion of partial functions between sets, and therefore, are important in furthering our understanding of what it
means to be partial. This thesis builds upon the work of Cockett and Lack, by providing restriction analogues of notions from ordinary category theory. One such notion is that of free cocompletion.
We show that every restriction category may be freely completed to a cocomplete restriction category, and that this free cocompletion can be described in terms of a restriction category of
restriction presheaves. Indeed, a restriction presheaf is defined precisely so that this is the case. We then generalise free cocompletion to join restriction categories, which are categories whose
compatible maps may be combined in some way. To do this, we introduce the notion of join restriction presheaf, and show that for any join restriction category, its join restriction category of join
restriction presheaves is its free cocompletion. The second half of this thesis explores the notion of restriction colimit. More precisely,we define the restriction colimit of a restriction functor
weighted by a restriction presheaf. We also show that cocomplete restriction categories may be characterised as those having all such restriction colimits. Finally, we give applications of
restriction colimits. Some examples of restriction colimits are gluings of atlases in a restriction category, and composition o frestriction profunctors. We conclude this thesis with notions in
category theory that have no analogue in the restriction setting.
Table of Contents
1. Introduction -- 2. Cocompletion of restriction categories -- 3. Free cocompletion of locally small restriction categories -- 4. Restriction presheaves -- 5. Cocompletion of join restriction
categories -- 6. Join restriction presheaves -- 7. Restriction colimits -- 8. Atlases and their gluings -- 9. Restriction profunctors and other restriction definitions -- References.
Bibliography: pages 95-96 Empirical thesis.
Awarding Institution
Macquarie University
Degree Type
Thesis PhD
PhD, Macquarie University, Faculty of Science and Engineering, Department of Mathematics and Statistics
Department, Centre or School
Department of Mathematics and Statistics
Year of Award
Principal Supervisor
Richard Garner
Copyright Daniel Lin 2019. Copyright disclaimer: http://mq.edu.au/library/copyright
1 online resource (viii, 96 pages)
Former Identifiers
mq:70988 http://hdl.handle.net/1959.14/1269712 | {"url":"https://figshare.mq.edu.au/articles/thesis/Restriction_presheaves_and_restriction_colimits/19427732/1","timestamp":"2024-11-06T22:06:13Z","content_type":"text/html","content_length":"138624","record_id":"<urn:uuid:4a0c7686-473d-4a8e-842d-0c9a14356ce2>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.47/warc/CC-MAIN-20241106194801-20241106224801-00896.warc.gz"} |
Worksheet Graphing Quadratic Functions A 3 2 Answers
If you are a novice in calculus, then you might not understand the benefits of graphing functions. You might think that functions like exponents, derivatives and limits are only needed in order to
calculate complex solutions involving quaternions, scalars and functions of more than one variable. But, this is not true. There are many other benefits that are more directly related to the
functions and graphs. In fact, there are even some useful functions that can be derived from quadratic equations!
Graphing Standard Form from worksheet graphing quadratic functions a 3 2 answers , source:blueicetechnologies.com
First, let us look at the benefits of graphing functions with two variables. One such example is the quadratic function, also called the x-intercept. This function finds the x-axis point, through the
range [0, infinity] to the left. It also finds the y-intercept, through the range [0, infinity] to the right. The exact formula for finding these points involves the Greek letter called Phi. When
graphed out, the functions reveal a structure known as a “shear,” which is a mathematical term used to denote the gradual change in a variable’s value over time.
Quadratic functions also fit neatly into the category of functions with no beginning and no end. That means they can be written as a line graph, or in any other graph format you prefer. A line graph
can be simply fitted by quadratic function values on a horizontal line. (It is also easy to draw a shear diagram from the data set, as illustrated above.) A shear graph gives an easily readable
graphical representation of the numerical values.
Worksheet search result by word Worksheet on time rate and speed from worksheet graphing quadratic functions a 3 2 answers , source:ftxs8.com
Some functions of higher degree can be graphed as a circle graph, or as a parabola. Either way, the meaning of each point on the circle or parabola will remain the same, since the value of each point
is the same everywhere. A parabolic function can be graphed as a paraboloid, wherein each of its ends touches the x-axis exactly once. A parabola can be graphed as a sphere. Each point on the sphere
traces a direct path to the x-axis.
Quadratic functions – particularly those of the form (a b) xn-(a+b), where a and b are real numbers – can be efficiently represented using mathematical equations. (They can also be graphically
represented using the classic graphing methods.) This makes quadratic function graphing an extremely flexible method to solve some important mathematical problems. It can also simplify some
complicated worksheets, depending on the type of functions involved and the computation involved.
Factoring Polynomials Trinomials Activity Advanced from worksheet graphing quadratic functions a 3 2 answers , source:pinterest.com
Quadratic functions can be graphed in two different ways. One way is to use the identity functions, which are shears that cut the functions into four components (a b, c and d) and treat each of the
components as a separate curve. The other way is to use the parabola function, which is just the other way of representing a quadratic equation (where a is equal to b and c is equal to d). These
identities are necessary only if you are working with small values.
quadratic equation graphs can be prepared manually or automatically. In manual procedures, the data set has to be inserted manually into the spreadsheet so as to obtain the required points. Automatic
procedures often correspond to automatic shearing procedures (since they make use of shears that have the same shape as the quadratic function). However, while they are more convenient than the
former method, they are more difficult to implement. For automatic results to be reliable, certain criteria have to be met. For example, an integration constant k has to be determined so as to relate
the function f(x) with the plotted integral.
Solving and Graphing Inequalities Worksheet Answers Beautiful from worksheet graphing quadratic functions a 3 2 answers , source:edinblogs.net
Worksheet graphing can help you understand theorems like Fermat’s Theorems, Integrals, geometric calculations etc. It also helps you in understanding the concept of functions and their derivatives,
quadratic equations and functions of a complex number. Thus worksheet functions – like any other graphical tools – are essential for scientists, mathematicians, engineers and architects to carry out
important tasks in their work.
Functions Worksheets And Answers from worksheet graphing quadratic functions a 3 2 answers , source:topsimages.com
posite Function Worksheet Answers from worksheet graphing quadratic functions a 3 2 answers , source:ngtank.com
The Coordinate Grid Paper Grid A math worksheet from the from worksheet graphing quadratic functions a 3 2 answers , source:pinterest.com
4 Ways to Find the Range of a Function in Math wikiHow from worksheet graphing quadratic functions a 3 2 answers , source:wikihow.com
21 Awesome Quadratic Function Worksheet from worksheet graphing quadratic functions a 3 2 answers , source:t-honda.com
Unique Transformations Algebra 2 Worksheet from worksheet graphing quadratic functions a 3 2 answers , source:duboismuseumassociation.org | {"url":"https://briefencounters.ca/45645/worksheet-graphing-quadratic-functions-a-3-2-answers/","timestamp":"2024-11-07T22:47:21Z","content_type":"text/html","content_length":"94332","record_id":"<urn:uuid:04748341-b6ff-43d0-8c67-5adc7854cb4c>","cc-path":"CC-MAIN-2024-46/segments/1730477028017.48/warc/CC-MAIN-20241107212632-20241108002632-00814.warc.gz"} |
Mohamed Amine Khamsi
Teaching Philosophy
I believe that teaching is both an art and a challenge, where the primary goal is to inspire students to recognize the personal and professional value of mastering mathematics. Flexibility is key, as
each group of students requires a unique approach to effectively convey mathematical concepts. I aim to build a solid theoretical foundation while encouraging students to connect their learning to
other scientific fields and real-world applications. By creating a motivating and supportive environment, I strive to empower students to see mathematics as an essential tool for their success. | {"url":"https://drkhamsi.com/Teaching.html","timestamp":"2024-11-10T08:11:02Z","content_type":"text/html","content_length":"6964","record_id":"<urn:uuid:87302f44-71e8-4c74-8442-c8c810df9a25>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00353.warc.gz"} |
From Vessels to Fleets - A Data Science Journey with HDBSCAN
When we look at the Global Fishing Watch map, we often see fishing activity that appears to move together. That is, we see groups of vessels consistently fishing in proximity to each other even as
the group as a whole moves about the ocean. In other words, we see fleets. Some members of our team, myself included, wanted to know if we could automatically group fishing vessels into these
fleets. By understanding fleets, we think we might be able to reveal more about vessel behavior. Maybe we can tell what types of species a vessel is targeting. Or, if a vessel is in a fleet where
other vessels have labor violations or illegal behavior, it might indicate a higher chance that our particular vessel has a higher risk of these infractions.
The overall approach to grouping these vessels into fleets is clear: use one of the many clustering methods to group vessels. However, turning this general approach into an algorithm that creates
realistic fleets requires some experimentation. In this blog post, I present an overview and motivation for the approach we used. For the gory details, take a look at one of the notebooks located in
Clustering by Day
The first order solution is to start with one point from each vessel per day and create a vessel cluster for each day as shown below. I use HDBSCAN for the clustering because it is one of the few
clustering algorithms that doesn’t require the number of clusters to be specified in advance, and because I’ve had good luck with it in the past.
Figure 1: Daily clustering applied to drifting longliners on the 1st and 7th of January, 2017. Note how some of the clustering changes dramatically; notably in the eastern Pacific and around
There are several problems with this approach which Figure 1 helps make clear. First, clusters are not necessarily consistent across days, and each day can have a different set of “fleets.” Second,
fleets can break into smaller clusters on different days, making it difficult to clearly define what a fleet is.. Third, not all vessels broadcast every day, which means they won’t be clustered on
some days.
It’s possible to link clusters across days by looking at the overlap between fleets, and in fact I reuse this idea later when linking clusterings across years, but this doesn’t help with the second
two issues. Fortunately, there’s a better solution: clustering across both space and time.
Clustering Across Space and Time
Clustering over an extended period of time — I used one year — fixes most of the issues with clustering a day at time. HDBSCAN supports passing a matrix of distances between pairs of objects rather
than object locations. This allows computing the average distance between vessels over the year and passing that in as the distance matrix. By omitting any days from the average when a vessel in a
pair wasn’t broadcasting, issue number three is almost completely avoided. I say almost because there is one remaining corner case: if two boats never broadcast on the same day the average is not
defined. Fortunately, HDBSCAN also allows passing in a special missing value for some distances and will still sensibly cluster objects as long as there aren’t too many missing values.
Figure 2: Results of yearly clustering applied to drifting longliners shown on the 1st and 7th of January, 2017. The clustering is now consistent across dates, but many fewer boats are clustered
because the clustering is too strict.
Because we are clustering over an entire year, the clusters are consistent across time. A fleet is the same fleet on January 1 as it is on December 31. For clustering over longer intervals, see the
section on Stitching Fleets Together Across Time at the bottom of this post.
Clustering across across time like this also results in clusters where vessels are consistently together. In fact, it is rather too good at this, only clustering vessels that spend nearly all of
their time together. This ends up missing a lot of vessels, because vessels often leave their fleet for a time to return to port or just for short excursions to other fishing grounds. This is visible
in the new clusterings shown in Figure 2 where the number of clustered vessels is far fewer than in the previous daily clusterings. Fortunately, since we are already computing a custom distance
matrix by hand, we can do better by redefining what we mean by average distance.
Tweaking the Distance Matrix
The initial distance matrix computed the root-mean-square (RMS) value of the Haversine distance (which is a fancy way of saying “distance between points on a sphere”) between each pair of vessels. By
adjusting the way the average distance is computed it is possible to get the clustering to produce much more realistic fleets. We have now reached the point in the narrative, where it is traditional
to sweep a great deal of experimentation under the rug. Let me just say that I tried:
1. Capping the individual distances before averaging.
2. Averaging only the N closest days for each vessel pair.
3. Trying different averaging methods, notably the geometric mean.
4. Sorting the list of distances for each vessel pair then weighting the average based on position in the list.
These are just the approaches that worked well enough that I recall them, I’m sure there were quite a few more that I’ve forgotten that were complete flops.
In the end I arrived at a combination of (3) and (4) that does a good job of clustering vessels into realistic fleets. For each pair of vessels, the Haversine distances are computed on each day, then
the distances di are sorted into increasing order and the overall distance is computed as the weighted geometric mean of the distances, specifically:
Where N is the number of days with defined distances and Wi is a weight factor defined by:
Where is the number of days in the given year (365 or 366 if it’s a leap year).
This averaging scheme may appear arbitrary, but it follows naturally from the form of the problem. First, note that the geometric mean has an effect similar to capping the distances, in that it
weights smaller distances more than larger distances. To appreciate this, consider the geometric, vs arithmetic means of 1 and 10,000: the geometric mean is ~100, while the arithmetic mean is ~5,000.
There is a similar connection between (4) and (2). We can can recast (2) as sorting the distances and then computing a weighted average with the first N having a weight of 1 and the rest a weight of
0. The procedure in (4) is very similar except that the weights vary smoothly from 1 to 0. Having established that (3) and (4) are in some sense smooth analogues of (1) and (2), let’s discuss why the
later make sense in the context of clustering fleets.
The rationale for (1), and by extension (3), rests the fact that beyond a certain distance a vessel is simply away from the fleet. A vessel that is 2000 km away from the fleet is not significantly
less associated with the fleet than one that is 1900 km away. In contrast, the difference between vessels that are 110 km and 10 km away is large in terms of likelihood of fleet membership. Both
capping the distances and using the geometric mean are ways to codify this and in practice the latter performed better.
Similarly, the rationale for (2) and (4) rests on the fact that vessels don’t need to be together for the entire year to be considered part of one fleet since vessels leave to visit port or for other
short excursions. This can be taken into account by only considering the N closest days as done in (2). However, this requires tuning on N, the best value of which may vary significantly from region
to region, and in practice the continuous version of this concept, expressed in (4) seems to perform better. There are an infinite number of weighting functions that can be used here though, so there
is still plenty of opportunity for tuning.
Clustering vessels using this approach yields fleets that are consistent across time, at least on a yearly time scale, but has coverage similar to the daily clusters as show below. The clusters tend
to correspond to specific flag states, which lends credence to them representing real fleets. More importantly, the resulting fleets appear realistic and offer useful insights into vessel movement.
Figure 3: Results of yearly clustering applied to drifting longliners with a custom distance metric shown on the 1st and 7th of January, 2017. The clustering is still consistent across dates, but now
clusters a much larger fraction of the boats. The resulting clusters are also more fine grained.
Stitching Fleets Together Across Time
The clustering method described above does a nice job clustering vessels over a one year, but how should we go about clustering vessels across longer time frames? The obvious answer is to just extend
the time range used in the clustering technique above. However, this has two issues. The first is just convenience. Using a longer time scale requires a new set of experiments to find the correct
clustering parameters, and running the clustering over the longer time range is slower, meaning finding these parameters takes longer than it did the first time around.
The second issue is what I’m calling the fleet coherence time. Vessels don’t remain attached to the same fleet forever. There is some turnover as boats are sold, reflagged, or repurposed, so one
would like a solution that allows fleet membership to evolve over time. Our current solution is to cluster years separately, but allow hints about fleets to pass between years, and then stitching the
fleets together across years.
Stitching the fleets together is relatively straightforward. I use the fleets from 2017 – the year that I did my parameter tuning on – as the canonical fleets. I then match fleets from subsequent and
previous years to these fleets my maximizing the intersection-over-union (IOU). There are two corner cases: first, in some cases, multiple clusters will match a single fleet, in which case they are
merged. Second, some clusters may not match an earlier fleet at all, in which case they are dropped.
Just stitching the fleets together in this way, is not quite enough to get well behaved, consistent fleets because some of the clusterings change too much year to year. The most egregious example is
the set of fleets around Madagascar. Small differences in the vessel distribution can change the clustering of these fleets from a group of six or so fleets to a single fleet. Which of these
representations is preferable depends on ones application, but one would prefer it to stay stable. My current solution is to add a fleet based pseudo distance to the average distance computed above.
The pseudo distance is computed based on the fleets found in 2017 and this distance is added to the base distance computed for 2016 and 2018 to stabilize the fleets from those years. This approach is
a bit ad hoc, but in practice it seems to work well.
Figure 4: Results of yearly clustering applied drifting longliners with a custom distance metric for 2016 through 2018
To fully appreciate the fleet clustering it helps to see how the vessels behave over an extended period of time as shown in Figure 4. Additionally, there are fleet animations for drifting longliners,
trawlers, purse seines and squid jiggers available in the repo for this project https://github.com/GlobalFishingWatch/fleet-clustering/. | {"url":"https://globalfishingwatch.org/data/data-science-journey/","timestamp":"2024-11-07T13:44:53Z","content_type":"text/html","content_length":"405615","record_id":"<urn:uuid:3a11de2b-2a4a-4e37-86c9-0c5da5a8c5f0>","cc-path":"CC-MAIN-2024-46/segments/1730477027999.92/warc/CC-MAIN-20241107114930-20241107144930-00475.warc.gz"} |
528 Mathematics and GeoPhysics - 528 Revolution
528 Mathematics and GeoPhysics
Why GeoPhysics Depends on 528 Frequency Dynamics
Vic Showell’s masterful mathematical monographs posted herein provide proofs of 528’s fundamental power to shape our planet.
By Vic Showell
Introduction and commentary by
Dr. Leonard G. Horowitz
Take out a compass and see if it points North.
Naturally is does, due to the flow of electromagnetic energy naturally occurring due to the polarity of the planet established by, fundamentally, its sacred geometry and musical-mathematics producing
the force field in which harmony and homeostasis is administered energetically (i.e., geo-physically, arguably “spiritually”) due to the flow and charge of electrons.
Electrons are the exclusive source of all energy universally. Whether it is the blink of your eye caused by the flow of electrons across the nerve membranes in your brain, or a comet flying through
space, electrons are exclusively in charge of administering the energy and movement of objects in space/time.
The arrow on your compass pointing north simply means the flow of electrons is directed north, upcoming from the south.
Also consider a lightening bolt striking Earth to “ground” its negatively charged electrons from watery charged clouds in the sky.
These simple, yet profound, understandings involving geophysics provide powerful revelations and implications for advancing a whole new worldview. This is a revolution in the way we think about
matter and energy to create helpful technologies and simple solutions to complex global problems.
Add to these revelations that every electron is charged with energy that is tuned to a different frequency, and that 528Hz frequency can be used to re-tune electrons to generate the “electricity of
love” (i.e., the “Spirit of Love,”–“Universal Healer”) and you have more to ponder, and additional opportunities to develop helpful technologies to heal the world of what ails it.
Next, consider that health and freedom from nearly all chronic diseases depends on body chemistry. Typically, body chemistry, like environmental chemistry, is adjusted naturally and most powerfully
by electron energy. That is, acidity versus alkalinity depends on the absence or presence of electrons, respectively. Every chemical reaction in the body, even every electro-genetic change in your
DNA, depends on the same electron energy that explodes hydrogen bombs.
So this concept of 528 frequency-charged electrons, and their impact on our planet (and virtually everything), can be contemplated to serve a greater mission for people worldwide.
In essence, 528 frequency dynamics provides the technology to set humanity free from the limitations imposed upon us by energy industrialists who have done a marvelous job complicating modern living,
while profiting from civilization’s ignorance and needless suffering.
The following text is provided elsewhere on this and associated websites by Vic Showell. “Vic” is a master mathematician specializing in sacred geometry and metaphysics.
Vic has been studying pyramids for a hobby. His joy has produced a major blessing for humanity and civilization’s evolution, having to do with the musical mathematics of geophysics, “living water
science,” and the construction of the universe cymatically.
Showell’s revelations legitimize “The Plan” for humanity’s musical alliance and “Spiritual Coup” that was facilitated on June 21, 2009 during the “LOVE WATER Experiment”–LIVEH2O: Concert for the
Living Water.
What follows is an introduction to Vic’s masterful thesis proving mathematically that pyramid sacred geometry, cosmology, Pi, Phi, the Fibonacci series, and geophysics as well, are controlled
musically-mathematically, and are intimately linked to 528 frequency harmonics and creative hydrosonics.
In essence, Showell’s revelations compel considerations of what I call, “Musical Creation: A New Theory on Physical Reality and Reactions in Space Time.” Among these considerations are three
rEVOLutionary postulates:
Postulate 1: All electrons are spinning musically; vibrating harmonically, determined by nine core creative frequencies—musical tones—comprising a “The Perfect Circle of Sound.”
Postulate 2: These nine frequencies of sound determine Pi, Phi, the Fibonacci series, all sacred geometrics including the structure of the universe, and the laws of physics.
Postulate 3: All piezoelectricity, oxidative-reductive reactions, electro-magnetism, and chemical interactions rely on the harmonics or dissonance of a musical vibrational mathematical matrix
fundamental to physics and chemistry; all determined by simply nine ‘Perfect Circle of Sound’ frequencies.”
How did Vic come to these astonishing revelations? “I just don’t know how it comes to me,” he wrote. “It must be that I am so subconsciously tuned to the synchronous harmonics of the ancient
timelines that it just happens.”
Technically-minded person who seek more information than provided on LOVE528.com, on the TECHNOLOGY page therein, will relish these contributions.
In this thesis, Mr. Showell has labored with LOVE to explain the fundamental mathematical geometrical sustaining universal truth–the knowledge that holds the capacity to set humanity optimally free
in LOVE with the creative Self/self through the master creative carrier wave through which the Creator broadcasts the universe into existence.
This main carrier Wave, Mr. Showell’s research proves is LOVE, 528Hz, as now massive scientific/mathematical evidence posits.
So what do you think we should do with this knowledge? How would you recommend communicating these revelations? What would your first goals be, given this new knowledge?
Sure you can read it here, as others may, but these monumental revelations require distribution and assimilation by a world urgently requiring something wonderful to happen to “give peace a chance,”
and change the dissonant vibrations in the hearts of mad men making war instead of LOVE. You can help by simply “playing” with these numbers, and sharing this information with others.
Background on Pi
The mathematical constant Pi is a real number that arises naturally in mathematics. Unlike physical constants, mathematical constants are defined independently of physical measurement.
In other words, in math—the Creator’s language—there are physical and metaphysical constants, or laws that govern everything including nature’s structure, function, and balance.
Pi is approximately equal to 3.14159.
Pi is also an irrational number, which means that it cannot be expressed as a fraction m/n, where m and n are integers. Consequently its decimal representation never ends or repeats. It is
furthermore a transcendental number, which means that no finite sequence of algebraic operations on integers (powers, roots, sums, etc.) could ever produce it.
“Throughout the history of mathematics,” Wikipedia reports, “much effort has been made to determine π more accurately and understand its nature; fascination with the number has even carried over into
culture at large.” This fascination with the accuracy of Pi is the subject of Vic Howell’s contribution.
Background on Sacred Geometry, Phi and the Golden Ratio
The golden ratio, also known as the divine proportion, golden mean, or golden section, is a number often encountered when taking the ratios of distances in simple geometric figures such as the
pentagon, pentagram, decagon and dodecahedron.
The designations “phi” (for the golden ratio conjugate) and “Phi” (for the larger quantity) are sometimes also used (Knott).
The golden ratio is found in the pyramids of Giza and the Parthenon at Athens. (http://mathworld.wolfram.com/GoldenRatio.html)
Pi and phi are important in trigonometry. For example, if you divide a 360° circle into 5 sections of 72° each, you will get a five point pentagon whose dimensions are all based on phi relationships.
Accordingly, phi, pi and 5 (a Fibonacci number) are related through trigonometry.
Dale Lohr contributed the following equation defining the relationship between pi, phi and 5 as follows:
Pi = 5 arccos (.5 Phi)
Note that the angle of ½ of Phi, or .5 Phi, is 36 degrees, of which there are 10 in a circle or 5 of in pi radians.
Additionally, here is graphic CymaGlyph evidence from pioneering acoustic engineer John Stuart Reid, consistent with Dale Lohr’s contributions showing 528 Hz frequency sound produces a 360 degree
circle uniquely containing 36 nodes:
Giza Pyramid Math & Sacred Geometry
Now for the purpose of this study, Vic analyzed the Saqqara and Giza pyramids that are geometrically very similar. The Pharaoh Khufu desired to solidify the ancient sacred geometry in the pyramid
complexes built. So he reproduced the geometry of Giza to establish the dimensions of the Saqqara pyramid.
Internet sites cite the dimensions of the Giza pyramid as follows:
BASE: 756 feet.
BASE/2: (also called the “Saturn synod” is 756/2 = 378 feet
HEIGHT: 481.090909 feet (or 280 cubits).
SLOPE: 51.8427*
[*Note: Using 481 as the height, the slope would be 51 .8375. This is close enough to 51.84 for this analysis. Vic used 481 as the “calculative constant,” and regardless if he used 481.09090909 or
not 481 = 13 x 37; wherein the 37 has to do with planetary synods of Jupiter, mars, venus and earth cycle number. This relates to the “Dresden Codex wherein 702 divided by 481 pyramid height = x;
then x times 37 = exactly 54 reflecting simple math harmonics.]
Lesson One: Ancient Pi is not 3.14159, but 3.142857 . . . to Infinity
Get a calculator, and take your time to follow this slowly—step by step, calculation by calculation. You will see that modern Pi is significantly different from Ancient Pi. This seemingly small
difference has substantially huge implications for the Spiritual Renaissance. This truth holds the capacity to set humanity free, and analyzing it will help to open your “heart-mind”—your intuitive
channel through which more revelation and inspiration will come to you. This is what many people are experiencing by “playing” with these numbers that, in a sense, have you communing with the “matrix
math” of creation, which is powerfully energized and energizing.
For additional reinforcement and benefit, Mr. Showell recommends you draw a pyramid dimensionally similar to the ancient Egyptian Saqqara pyramid, to visualize the sacred geometry created by this
math. Divide the base line into 2 sections of ancient Pi to test the prospect that ancient Pi is truly 22 / 7. You can see from his graphic (and yours) that the pyramid base is 2(a)Pi by 2(a)Pi. And
the height is 4Pi.
Next, using a calculator, divide 22/7 and write down what you see digitally on your display. It is:
3.142857 142857 142857 142857 . . . to infinity.
Now double what you see, and write it down as written here. It may help to put one row directly below the other, like this:
3.142857 142857 142857 142857 . . . to infinity.
6.285714 285714 285714 285714 . . . to infinity.
Look carefully from a distance at both rows of numbers. Do you see the pattern?
The fractional pattern that repeats to infinity shifts two places, or numbers, with every doubling. (This is best noted with calculators that cipher to 18 places.) In other words, the “14” in
“.142857 shifts positions to the end of the repeating pattern, to become .235714.
IT DOES NOT DO THIS WITH THE “MODERN” ALLEGED Pi EQUIVALENT!
Try it again to make sure. Double 6.285714. You get:
12.571428 571428 571428 571428] . . . to infinity.
Put this number directly below the other two rows to get:
3.142857 142857 142857 142857 . . . to infinity.
6.285714 285714 285714 285714 . . . to infinity.
12.571428 571428 571428 571428] . . . to infinity.
Do you see what is happening, or what is emerging? See if you can see certain repeating numbers that form a pattern—a developing mathematical (and musical) matrix.
Lesson Two: Phantom of the Opera
and the Music of the Light
If you have seen the play Phantom of the Opera, or heard the lyrics of “Music of the Night,” it relays the story of the Angel of Darkness (i.e., Lucifer—the fallen angel of music), capturing the
heart of “Christina”—metaphorically representing the Christ energy or Christianity. The demon causes a crystal chandelier to fall on the audience to destroy the opera company engaged in battle under
his control involving her love. In the end, she renders the demon LOVE and is released to marry her hero.
In the Book of Revelation, the “Apocalypse” refers to a similar End Times battle resolving in a musical “marriage” between God’s faithful—the bride of Christ—who is finally wed to the Messiah for all
eternity. In “Healing Codes for the Biological Apocalypse,” by Horowitz and Puleo, these authors advance a musical code hidden in the verse numbers in the Book of Numbers (7:12-83) that applies the
ancient priesthood’s alchemical Pythagorean math. The code reveals the “music of the light”—the original Solfeggio scale by which the Hymn to St. John the Baptist was sung. This music, reputedly the
most spiritually-uplifting hymn of all time, is composed of only six notes; each defined in Webster’s Dictionary as part of a “New Song” sung during a global concert in Rev. 14:1 involving 144,000
spiritually pure persons who are “ransomed” by God for this Apocalyptic marriage.
These six tones were derived by reducing the verse numbers from their multiple digit integers to their single digit whole number (e.g., verse 12 = 1 + 2 = 3). Thus, the following frequencies of sound
were revealed:
UT: 396
RE: 417
MI: 528
FA: 639
SO: 741
LA: 852
This author later added the following 3 unnamed notes to this list to complete what he christened “The Perfect Circle of Sound.” These included:
You should note that these numbers each resolve into 3, 6, or 9 when the Pythagorean system is used.
Furthermore, the center two Solfeggio scale notes, I realized (MI and FA), identifies the MIracle FAmily (528 and 639); and that 528 (5 + 2 + 8 = 15 = 1 + 5 = 6) or Miracle 6 (i.e., also denoting
MI6, British Secret Service). This frequency lies at the heart of the electromagnetic color and sound spectrums; it is light green yellow in color, like the heart chakra (i.e., spinning bioenergy
vortex), and is associated with LOVE.
Now return your attention to the whole numbers obtained by doubling pi. When reduced to their single digit, the following pattern is established to infinity:
3636363636 to infinity.
The repeating pair of numbers, 3 + 6, adds up to 9. (In Pythagorean math 9 equals “completion.” Zero’s are man made place holders. There are really only 9 numbers in the universe—1 thru 9.)
Notice also that that the numbers comprising the fraction of true ancient pi are missing the 3s, 6s, and 9s. From conclusions I drew based on the simple mathematical “infinity pattern” (shown in the
figure below) pioneered by Marko Rodin, the 3s, 6s and 9s represent a special set of numbers that provides a “”portal’ to the spiritual domain.” These whole numbers reflect the totality of spirit–the
Holy Spirit that hovered over the face of the Water at the beginning of time—as part of the Triune God. The repeating fraction numbers, missing the 3s, 6s, or 9s, represent a fraction of the whole,
or pieces of the totality manifesting physical reality, separated from spirit by a “period,” denoting time as a factor forcing separation (i.e., before or after, past and future, instead of being in
the NOW of spiritual totality.)
Finally, in case you have not noticed, the matrix pattern created by doubling the fraction numbers, as Marko Rodin also showed during another analysis, reveals the original Solfeggio frequencies in
patterned positions within the matrix.
You will notice the 528 shifts up and to the right, while the 741 shifts in the opposite direction—down and to the left. This is consistent with the “Devil’s tone” designated in musicology as the
dissonant combination of these two specific disharmonic tones.
Now Joseph Hydn’s 96th symphony is called “The Miracle.” Here again 9 + 6 = 15; where 1 + 5 = 6. Modern propaganda alleges this classical composition was named from the sudden, inexplicable, nearly
disastrous detachment of a huge crystal chandelier that fell from the ceiling of the concert hall during Hydn’s performance. Miraculously, no one was harmed. No doubt Gason Leroux, author of Phantom
of the Opera, borrowed this apparent myth. Given the above revelations, and those in Healing Codes for the Biological Apocalypse, Hydn obviously composed and named his 96th symphony “The Miracle”
honoring the 3rd “Miracle” tone of the original Solfeggio musical scale that resolves into “MI6” from either 96, 528, or 15. Leroux did get the ending right with LOVE, 528Hz, conquering the demon.
Finally, if you add up each fraction using the Pythagorean method, you get:
1 + 4 + 2 + 8 + 5 + 7 = 27 where 2 + 7 = 9—the “Completion” number. So in each doubling of Pi you get 3 or 6 alternating in the whole number spot, and 9 in the fractional location. This set of 3s,
6s, and 9s, as stated previously, provide the spiritual portal through which metaphysical energy manifests into physical reality.
Lesson Three: Phi Cosmology, Pyramid Geometry, and the Creator’s Music
Take the original decimal sequence—.142857, and isolate that fraction for your next experiment (i.e., 0 .142857). Inverse it EXACTLY: 758241. Then, extrapolate it out to infinity. (Be sure to keep
punching in the decimal sequence into the calculator, well past what you see your calculator show. This is what the calculator does unseen.) Now you can see the 2nd Solfeggio tone, REsonance, 417Hz,
resonating to eternity harmonically fixed to 582, a harmonic of 528, LOVE.
Here again you can see the 3s and 6s combining to form the 9s:
4 + 1 + 7 = 12 = 1 + 2 = 3
5 + 8 + 2 = 15 = 1 + 5 = 6
If you again shift two digits from back to front of this number, you get REsonance, 417 in front, next to the harmonic of LOVE 582:
NOW! A great truth is revealed about Phi—and the deepest truth about all the ancient Egyptian pyramids and their math becomes apparent in a flash!
If you multiply the proven legitimate Pi fraction (.142857 going to infinity) times the cosmological constant associated with galactic and solar cycles known as the Saturn synod, which is 378 (whose
Pythagorean sum is also a 9), you get precisely 54 as you approach infinity. Try it. The more you extrapolate the numbers, getting closer to infinity, the closer it gets to 54!
0.142857 142857 142857 142857 142857 . . . x the Saturn synod (378) = 54 exactly!
Why is this important? Because, not only is 5+4=9 or completion, but the sine of 54 degrees is one half of Phi (or Phi/2).
Here is where Vic’s genius delivered to the max. He “Harmonic Codexed” the decimal sequence by shifting the 1st digit in line to the back of the line, and multiplied that number by the Saturn synod:
.428571 428571 428571 x 378 Saturn synod = 162 exactly as you get to infinity. EUREKA! 162 is exactly 100 times ancient Phi of 1.62!
Try this with all the sequences similarly by shifting the first digit in line to the back of the line and than multiply that number times the Saturn synod .
With this simple experiment, you have just proven the entire Egyptian pyramid math calendar geometry is based on Saturn and Mars cosmologies; with Saturn being predominant.
“Why do you think the Greeks and Romans called Saturn Chronos?” Vic asked facetiously. “Bingo!” The cosmos is laid out perfectly, mathematically, harmoniously—a musical composition of Divine
synchrony. Saturn is like the lead bass player or percussionist maintaining the rhythm of the spiraling cosmos. (The “bass clef” in music “ reflects this spiraling center.)
Similarly, Vic provides irrefutable evidence, solid mathematics, that proves 528 is a central number operating in the musical-mathematical cosmic matrix, underlying the sacred pyramid geometics
established by Phi.
(The graphics on the right link to full length articles and analyses by Vic Showell. Download these, including his full thesis, in pdf formats by clicking on these images.)
Lesson Four: Importance of the Ancient Solfeggio Musical Scale
According to Vic’s mathematical analysis, and a growing international consensus of math experts, the original Solfeggio frequencies work like magic in the Saturnian cosmologies and the 378 synod.
(The word “magic” sources from the ancient Melchezadak priesthood (called the Magi) who held this knowledge sacred and secret.
Once again, the original Solfeggio musical scale features 396, 417, 528, 639, 741, and 852 Hertz frequencies—mathematics fundamental to hydro-sonic creation and creationism. The later is the actual
act of creating the cosmos; applying the musical-mathematical “symphony” to impact everything within earshot.
Returning to his earlier analysis of the Saggara pyramid and Phi, Vic wrote that the number 7 within the fraction of Pi, derived from his initial decimal inverse, represents the 7 dots of the ancient
cylinder seals and stele representing planet Earth as advanced and discussed by Zaccharia Sitchin.
So for Lesson Four, Vic recommends taking the real ancient Pi = 3.142857 . . . , and multiplying it times the Saturn synod (378). This yields EXACTLY 1188. “Look at that number,” he counsels,
encouraging your intuition. 11 x 8 = 88, where 8 = 8 = 16 and 1 + 6 = 7; and 7 x 11 = 77 which yields by Pythagorean addition 14. As you see below, these are all numbers featured in sacred geometry
and musical-math.
Musically profound, if you now take the original Solfeggio frequencies and apply the aforementioned information you prove Pi, Phi, the Fibonacci Series, and more, are all based on the Solfeggio
musical matrix.
Convince yourself as follows:
First, divide the first note of the original Solfeggio (UT, which is 396Hz) by 1188, and you will get the exact sine of the molecular structure of Water—the tetrahedron:
396/1188 = 0.3333333—sine of exact tetrahedral (19.47122061)
You should also realize that, according to Webster’s Dictionary’s definition of “UT,” this single tone contains all other musical notes, and represents the entire gamut of human experience and
emotion from grief to joy!
“Ha!” Vic wrote, “The ancients used 19.5 as their tetrahedral constant. They did not need the exact 19.47122 constant. However, within the deeper Saturnian cosmology, the exact value yields better
“Have you seen the NASA image of the Saturn hexagonal super- storm from 2007, at the Saturn pole? That is when the Star Gate opened to the Pleiades,” Vic continued, “THAT is why the Cassini space
mission was sent to Saturn and not to Jupiter.”
Vic additionally noted that the newly discovered Saqqara pyramid measurements are noteworthy. The pyramid measures 22 meters square at its base, and 14 meters high. Half those measures and you get
the values 11 and 7.
If you divide 528, by this pyramid base length of 22, you get exactly 24 (as in 24 hours in a day), or 6, just like MIracle 6, or 528, where 5 + 2 + 8 = 15 and 1 + 5 = 6. Try it:
528/22 = 24
Now Vic gets more technical. Recalling that 22/7 is ancient Pi or 3.142857143; by taking the height of 14, divided by the base diagonal to base center point of the square-root of 242; you get “the
tangent of the just under 42 degree slope of the Saqqara pyramid side angle. (The height of 14 divided by the base diagonal to base center point of the square root 242 = 0.899954085. If you now take
this arctangent and multiply it (0.899954085) by ancient Pi, and you get exactly the square root of 8. By the way, 242 = 11 x 22)
So 1.62 is ancient Phi used in Egypt and MesoAmerica, Vic reminds us. “This is by virtue of that number unifying the Nine, as in 9 x 18 = 162.
“Thus, the slope of the Saqqara pyramid Side Face,
is arctangent of the sqare root of1.62 = 51.84 degrees
and 72 squared = 5,184.”
He alerts us, once again, to the decimal placements. 5,184 is related to 22/19.5 = x. Then squaring x, you get the tangent of 51.84 degrees. If you square x again, you get the ancient Phi number
So now you may see more clearly how the original Solfeggio musical scale, including 528Hz, Pi and Phi, are intimately connected to the liquid crystal (tetrahedral) structure of Water and every other
sacred geometric form.
Lesson Five: Final Proofs Regarding the Legitimacy of 528 and the Original Solfeggio Musical Scale
When you read Vic’s work, you have to understand his use of the ancient system of no decimals, or what he calls “decimal variations in numeric sets or sequences.” For example:
0.3333—-> 3.33333 —-> 33.33333, and 13 x 333.33333 = 4333 .33333 Jupiter sidereal; as 0.3333 is sine of the tetrahedral 19.47122061.
Because he found that the ancient Egyptian cosmological constant is 195/162 (or 1.95/1.62), he uses the 1.95 form of decimal placement often; which he named the “Khufu Constant.”
“In any case,” he writes, “to get to the ancient Solfeggio music of the spheres, you can use tetrahedral 19.47122061 but then you get 10 Pi.” So he used 1.947122061 instead because, he declared,
“that form created virtually magical pathways into the tangents and cosines and sines of angles in derivations. The difference of exact Pi and the actual value achieved is 0.000354308,” that he
chides, “if this is not close enough, then this will not work for you. It works for me.”
He derives this miniscule musical difference, ultimately in resonance harmonics, by taking the LA tone of the Solfeggio (852) and dividing it by the MI tone (528) as shown here:
Solfeggio 852/528 = 1.6136363636, then x 1.947122061 = 3.141946962 then minus Pi = 0.000354308.
Similarly, he examines the Solfeggio tones FA (639) and UT (396) to obtain the same stunning result as follows:
639/396 = 1.61363636363Naturally, this music underlying the spinning fractal universe “goes in cycles,” he explains:
396 x 1.3333333 = 528
And this square root of 1.3333333 is the Side Angle slope tangent of the Mars Pentad Grid Pyramid 2 units high. Some researchers also believe this number, 1.33333, is the slope tangent of the Side
Face on the 2nd pyramid of Giza.
“A few other Solfeggios do this as well with 1.33333,” he adds.
For his grand finale, Vic Showell (whose e-mail address reads vshowell, as if “V” stood for “victory” in showing humanity well [Water’s] sacred geometry based on musical harmony) presents his Quantum
Space Time Fractal Harmonic Codex, beginning with LOVE (528). Lay readers beware of going further with his flow of technically Divine, mentally masterful, revelation. You might get caught in
confusion by the following examples that further prove the aforementioned points:
Example 1:
[528] <——>[825]
[528] / [825] = [0.64], and [64] = [8] squared.
now look back at my [51 .84] degree pyramid
and [72] squared = [5,184].
[0.64] x [5 .184] = [3.31776], then divide by Phi[1.61803388] = [2.050488445] = tangent of [64.00203243] degrees!
Example 2:
[825] <—–> [528]
[825] / [528] = [1.5625] , then divide by Phi[1.61803399] = [0.965678107] = tangent of [43.99968~] degrees = [44]
and of course [44] x [120] = [5280] mile
and [1.2] = [pi] / [phi squared]…. pi / [2.61803399]
Example 3: Dresden Codex [702] / [585] venus synod = exact [1.2] = Pi / Phi sq.
[1.2] = [6 / 5] or [12] / [10], and [12] sq. = [144]
But Pi / Phi sq. = [ 1 .19998~], off by [2 / 100,000] from exact [1.2]
So to align the mathematics of the universe correctly, it is quite simple, reverse the equation to
[1.2] = Pi divided by [2.618033989]…..[phi sq.]
Therefore [1.2] x [2 .618033989] = [3.141640787] = True Pi
And that my friends is an extremely valid restructuring of Pi into the math system [if we are going to disregard ancient original Pi upon which cosmology depends.]
The above mathematical analysis provides proof of Divine design far exceeding the math of men. It virtually proves “Postulate 1,” that “All electrons [as well as celestial spheres] are spinning
musically; vibrating harmonically, determined by nine core creative frequencies—musical tones—comprising a “The Perfect Circle of Sound.”
In addition, “Postulate 2” is also proven by the aforementioned analysis: “These nine frequencies of sound determine Pi, Phi, the Fibonacci series,” and all sacred geometric forms including the
structure of Water, the tetrahedron, cosmology, and the laws of physics.
“As above, so below.” From this evidence “Postulate 3” also appears to be significantly evidenced, virtually certain: “All piezoelectricity, oxidative-reductive reactions, electro-magnetism, and
chemical interactions rely on the harmonics or dissonance of a musical vibrational mathematical matrix fundamental to physics and chemistry; all determined by simply nine ‘Perfect Circle of Sound’
— end —
Doc 1. SOLVING the MYSTERY of the TEOTIHUACAN GRIDS 15.5 Degree ANGLE.pdf
Doc 2. The Mars Pentad Time Pyramids Part Two. Egyptian and Mayan Cosmologies Deciphered.pdf
Doc. 3. Establishing a New Value for True Pi.pdf
Doc. 4. The Mars Pentad Time Pyramids Pentagonal Pyramid Introduction.pdf
Doc. 5. The Tetra Phihedral Pentagonal Pyramid and Geometric Crystallizations.pdf
Doc. 6. Ancient Egyptian Pyramid Pi and Solfeggio Synchronicities.pdf
Doc. 7. Hypercube Tesseract 261 Ancient Square Root Two with Egyptian and Mayan Fourth Dimension Cosmology Involving 528.pdf
Doc. 8. Vic Showell’s Combined Thesis–The Universal Harmonic Codes, Ancient Pi, Ancient Phi, Universal Harmonic Pi, Modern Pi and Phi, Grand Unification of Ancient and Modern Mathematics.pdf
Doc. 11. Vic Showell’s Grand Unification of Ancient and Modern Math. Includes 528 and Its Relationship to Tetrahedron Sacred Geometry: Khafre and Khufu Pyramid Geometry in Tetrahedral Hexad Geometry;
and The Mars Cydonia Hexad Mounds–Stunning!
For more information on this topic, go to LOVE528.com.
Dr. Leonard G. Horowitz gratefully acknowledges the work of Vic Showell in advancing this important research.
For essays by Dr. Leonard G. Horowitz, please see: HealthyWorldSOLUTIONS.com.
To read The Book of 528: Prosperity Key of LOVE, CLICK HERE.
Please purchase only through CureShoppe.com or dealers authorized exclusively by Dr. Horowitz to sell this book. There are many people working to rip us off, including those selling bootlegged copies
of this important book!
For health and energy products resonating with 528, visit the “528 Store” inside CureShoppe.com, by CLICKING HERE.
About the Author
(DMD, MA, MPH, DNM, DMM) is a “world leading intellectual,” and internationally known authority in public health, emerging diseases, and natural healing. Author of more than sixteen books, including
Healing Codes for the Biological Apocalypse, Walk on Water, LOVE the Real da Vinci CODE, and DNA: Pirates of the Sacred Spiral, he is globally known as the most outspoken critic of the genocide he
ascribes to the “military-medical-petrochemical-pharmaceutical cartel.”
Dr. Horowitz’s presentations and publications have served as the impetus for numerous Hollywood productions; including most recently the movie INCEPTION. In his most recent books, Walk on Water and
LOVE the Real Da Vinci CODE, and his 2-hour DVD documentary, The LOVE CODE, Dr. Horowitz presents the musical-mathematics underlying the spiritual mechanics of creation. His “Perfect Circle of
Sound™” revelations have been revolutionizing the music industry as well as water science.
Dr. Horowitz revealed that the 528Hz frequency of sound is the precious “LOVE tone” or “Spirit of Aloha.” This is, according to mounting evidence, the prophetically important “key to the House of
David” mentioned in Isaiah 22:22 and Rev. 3:6-8.
Dr. Horowitz recently launched i528Tune.com following his instructions to recording artists worldwide regarding this musical technology of Divine creativity. He recommends instruments and voices be
retuned to prompt miraculous healings and Peace on Earth.
In 2009, Dr. Horowitz seeded the concept of producing the international Concert for the Living Water, LIVE H2O, June 19-21, 2009.
Dr. Horowitz’s website is www.DrLenHorowitz.com
528 Mathematics and GeoPhysics | {"url":"https://www.528revolution.com/528-mathematics-and-geophysics/","timestamp":"2024-11-03T02:34:38Z","content_type":"text/html","content_length":"184270","record_id":"<urn:uuid:045fbee0-8774-4853-94bb-af444b184546>","cc-path":"CC-MAIN-2024-46/segments/1730477027770.74/warc/CC-MAIN-20241103022018-20241103052018-00351.warc.gz"} |
Center of Mass | Curious Toons
Table of Contents
Welcome to the fascinating world of physics, a realm where the ordinary becomes extraordinary! Imagine a universe where a single apple falling from a tree can unlock the secrets of gravity, or where
the flicker of a light bulb exposes the mysteries of electricity. As we embark on this journey together, you’ll discover how the laws that govern motion can explain everything from the swoop of a
basketball to the intricate dance of planets in the cosmos.
Physics isn’t just about formulas and equations—it’s a lens through which we can view the intricacies of our everyday lives. Why does the sky appear blue? How does a roller coaster achieve thrilling
heights and dizzying drops? What makes a car accelerate and come to a sudden stop? These questions and more are the keys to understanding the forces at play around us.
Get ready to explore the wonders of the physical world, experiment with your ideas, and challenge what you think you know. From the tiniest particles to the vastness of space, let’s unveil the
incredible forces shaping our universe. Are you ready to take the plunge into the captivating realms of physics? The adventure awaits!
1. Introduction to Center of Mass
1.1 Definition of Center of Mass
The center of mass (COM) is a pivotal concept in physics, representing the average position of mass in a system of particles. It is defined as the point at which the total mass of the system can be
considered to be concentrated for the purposes of analyzing motion. Mathematically, for a system of particles, the position of the center of mass is calculated using the formula:
\vec{R}{\text{COM}} = \frac{1}{M} \sum{i=1}^{n} mi \vec{r}i
where (M) is the total mass of the system, (mi) is the mass of each particle, and (\vec{r}i) is the position vector of each particle. The center of mass is crucial in simplifying the study of motion,
especially in systems where multiple forces act. For example, in a symmetrical object, like a uniform rod, the center of mass is located at its geometric center. Understanding the COM aids in
analyzing various physical phenomena, including collisions, rotations, and stability. It allows us to predict how objects will move and react under external forces, making it an essential concept in
classical mechanics.
1.2 Importance in Physics
The concept of center of mass (CM) is fundamental in physics because it simplifies the analysis of complex systems. The center of mass serves as a unique point where the total mass of a system can be
thought to be concentrated. This allows for easier calculations of motion, particularly in systems involving multiple bodies, such as planets, vehicles, or even composite objects. Understanding
center of mass is crucial in fields such as mechanics, astrophysics, and engineering. For example, in mechanics, the trajectory of a projectile can be simplified by considering its center of mass
instead of analyzing every point on the object. In astrophysics, the gravitational interactions between celestial bodies can be analyzed through their centers of mass, allowing for predictions of
their orbits. Furthermore, in engineering, the stability and balance of structures or vehicles can be assessed by locating their center of mass. Overall, mastering the concept of center of mass not
only aids in solving practical problems but also builds a foundational understanding for exploring more advanced topics in physics, making it an essential part of the curriculum.
Application Area Importance of Center of Mass
Mechanics Simplifies projectile motion analysis
Astrophysics Analyzes orbital dynamics
Engineering Assesses stability and balance
2. Calculating the Center of Mass
2.1 Center of Mass for Discrete Systems
The center of mass (COM) is a crucial concept in physics that represents the average position of all the mass in a system. For discrete systems consisting of individual particles, the COM can be
calculated using the formula:
\text{COM} = \frac{\sum{mi \cdot xi}}{\sum{m_i}}
where ( mi ) is the mass of each particle and ( xi ) is the position of each particle along the axis of interest. This equation highlights how the position of the center of mass is influenced by the
distribution of mass in the system. For example, consider a system of three particles with the following properties:
Particle Mass (( m_i )) Position (( x_i ))
1 2 kg 1 m
2 3 kg 2 m
3 5 kg 3 m
To find the center of mass, calculate the weighted sum of positions:
\text{COM} = \frac{(2 \cdot 1) + (3 \cdot 2) + (5 \cdot 3)}{2 + 3 + 5} = \frac{2 + 6 + 15}{10} = 2.3 \text{ m}
Thus, the center of mass for this discrete system is located at 2.3 m, effectively balancing the entire mass of the system.
2.2 Center of Mass for Continuous Distributions
The center of mass (CM) for continuous distributions is a crucial concept that extends the idea of center of mass from discrete particles to objects with continuous mass. For a continuous body, the
center of mass is defined as the average position of all mass elements within the object, weighted by their mass. Mathematically, the center of mass ( \mathbf{R}_{CM} ) is calculated using the
\mathbf{R}_{CM} = \frac{1}{M} \int \mathbf{r} \, dm
where ( M ) is the total mass of the object, ( \mathbf{r} ) is the position vector, and ( dm ) is the infinitesimal mass element. This equation indicates that we sum (integrate) over all
infinitesimal mass elements across the entire volume of the object. For standard geometries, we often utilize symmetry to simplify calculations. For example, a uniform rod of length ( L ) has its
center of mass located at ( L/2 ) along its length. This concept is essential in various applications, from understanding the motion of rigid bodies to analyzing systems in equilibrium. By grasping
how to compute the center of mass for continuous distributions, students can better understand complex physical systems.
3. Properties of the Center of Mass
3.1 Behavior of Center of Mass in Motion
The Center of Mass (COM) plays a crucial role in understanding the motion of objects in various systems. It acts as a weighted average of the distribution of mass, and its behaviors during motion can
be illustrated through a few fundamental principles. When an object or a system of particles is subject to external forces, the motion of the COM can be described using Newton’s laws. Specifically,
the acceleration of the COM is directly proportional to the net external force acting on the system and inversely proportional to the total mass of the system (F = m*a). This relationship gives rise
to the concept that the COM moves in a predictable path, unaffected by internal motions as long as external forces are the only influences.
For example, when a rocket propels forward, the COM of the rocket remains in motion governed by the thrust and gravitational forces, despite the individual parts of the rocket experiencing complex
motion as fuel is expelled. Similarly, in a system of colliding particles, the total momentum before collision equals the total momentum after collision, ensuring the COM’s path remains continuous.
Scenario Behavior of COM
External Forces Moves according to net force
Internal Forces Path unaffected by internal interactions
Collisions Conserves overall momentum
This behavior reveals the beauty of the COM as a concise descriptor of motion in complicated systems.
3.2 Relation to Forces and Torques
The center of mass (CM) plays a crucial role in understanding the relationship between forces and torques acting on an object. The center of mass of a system is the point where the total mass of the
system can be considered concentrated for analyzing translational motion. When a net external force acts on a system, it affects the motion of the CM according to Newton’s second law, which states (
F_{\text{net}} = m \cdot a ), where ( m ) is the total mass and ( a ) is the acceleration of the center of mass.
In terms of torques, the center of mass serves as a pivotal point. When analyzing rotational motion, the torque (( \tau )) about the center of mass helps determine the angular acceleration (( \alpha
)) of a body. The relationship is given by the equation ( \tau = I \cdot \alpha ), where ( I ) is the moment of inertia. This links the distribution of mass relative to the center of mass to the
rotational dynamics of the system. Understanding these connections is essential, as it enables us to predict how objects will move and rotate under the influence of various forces and torques.
4. Applications of Center of Mass
4.1 Center of Mass in Real-World Problems
The concept of Center of Mass (CoM) is essential for solving various real-world problems in physics, particularly in engineering and biomechanics. The CoM is the point where the mass of a system is
concentrated, and it moves as if all the mass were concentrated at that point when external forces act upon it. For example, in vehicle design, understanding the CoM helps improve stability and
safety. A lower CoM in cars leads to better handling during cornering, whereas in construction, ensuring that tall structures have a CoM within their base enhances resistance to tipping during high
winds or earthquakes. In sports, athletes strategically position their CoM to maximize performance; gymnasts, for example, use this knowledge to execute flips and land accurately. Furthermore,
robotics engineers consider CoM when designing robots that navigate varying terrains, ensuring they maintain balance and mobility. In summary, the applications of Center of Mass in real-world
scenarios are vast and crucial for safety, efficiency, and performance across different fields.
Application Area Importance of CoM
Vehicle Design Enhances stability and handling
Construction Increases resistance to tipping
Sports Maximizes performance in movements
Robotics Ensures balance and mobility
4.2 Role in Engineering and Design
The concept of the center of mass (COM) plays a crucial role in engineering and design across various fields. In structural engineering, understanding the COM helps in the design and stability of
buildings and bridges, ensuring they can withstand forces like wind and earthquakes. For instance, the stability of a tall structure depends on its center of mass being low enough to prevent tipping
during lateral forces. In the automotive industry, engineers must consider the COM to improve vehicle handling, safety, and performance. A lower center of mass in cars enhances stability and reduces
the likelihood of rolling over during sharp turns. Similarly, in the field of robotics, the location of the COM influences maneuverability and balance, guiding the design of legs or wheels for
optimal performance. In aerospace engineering, the positioning of the center of mass relative to the center of lift is vital for ensuring flight stability. Overall, a deep understanding of the center
of mass aids engineers in creating safe, efficient, and functional designs across various applications.
In summary, the center of mass is pivotal in:
Engineering Field Application
Structural Engineering Stability of buildings and bridges
Automotive Engineering Vehicle handling and safety
Robotics Balance and maneuverability
Aerospace Engineering Flight stability and performance
5. Experiments and Demonstrations
5.1 Simple Experiments to Determine Center of Mass
Determining the center of mass (COM) can be a rewarding hands-on experience for students. Here are a few simple experiments:
1. Balance Beam Method: Use a ruler balanced on a fulcrum (e.g., a pencil). Place weights (like small objects) at varying distances from the fulcrum. Adjust the positions until you find a balance.
The point where the ruler balances is the center of mass.
2. Hanging Method: Suspend an irregularly shaped object (like a cardboard cutout) from a string. Allow it to hang freely and mark the vertical line directly beneath the hanging point. Repeat by
suspending the object from a different point. The intersection of these lines indicates the center of mass.
3. Flat Surface Method: Place an object on a flat surface and slightly tilt it. Mark the object’s position and tilt again in another direction. The intersection of the lines drawn from the top
points of the object will give you the COM.
These experiments not only illustrate the concept of the center of mass but also engage students in practical physics, enhancing their understanding and appreciation of the subject.
5.2 Interactive Activities for Understanding Concepts
Interactive activities are vital for deepening students’ understanding of the center of mass concept. By engaging in hands-on experiments, students can visualize and manipulate the principles in
action. One effective activity involves using diverse objects (like wooden blocks, balls, and cardboard) to create a balance scale. Students can adjust the position of the objects and observe how
their combined center of mass shifts, leading to discussions about stability and balance.
Another engaging experiment is the “Center of Mass Race,” where students work in teams to create structures using lightweight materials (such as straws or paper). Each team strives to build a tower
that stands as tall as possible while keeping the center of mass low. This fosters collaboration and critical thinking as they must apply their understanding of mass distribution.
Additionally, computer simulations can be used to visualize how the center of mass behaves in different systems, such as pendulums or rotating bodies. These interactive tools allow for real-time
experimentation, reinforcing the concepts learned in class. Such activities not only make learning enjoyable but also help solidify students’ grasp of the center of mass through practical application
and experimentation.
As we wrap up this year’s journey through the fascinating world of physics, I want to leave you with a thought that transcends the equations and theories we’ve explored. Physics isn’t just about
numbers or laws; it’s about understanding the universe and our place within it. Every time you flick a light switch or watch a rocket launch, remember that you’re witnessing principles that have been
uncovered through centuries of inquiry and imagination.
You’ve learned to see the world through the lens of scientific reasoning, to question, to hypothesize, and to explore. This isn’t just a collection of facts—it’s a toolkit for life. Whether you
pursue a career in science or take another path, the critical thinking, problem-solving skills, and curiosity you’ve developed will serve you well.
So as we conclude this chapter, I encourage you to continue asking questions, to wonder about the natural world, and never stop learning. Physics is everywhere, and it belongs to you. Thank you for
your hard work, your enthusiasm, and your camaraderie. Keep looking up at the stars, and remember that every end is just the beginning of a new adventure. Keep exploring! | {"url":"https://curioustoons.in/center-of-mass/","timestamp":"2024-11-09T18:42:45Z","content_type":"text/html","content_length":"111338","record_id":"<urn:uuid:f6ccc637-b7ed-4846-bafe-fc79165f99d4>","cc-path":"CC-MAIN-2024-46/segments/1730477028142.18/warc/CC-MAIN-20241109182954-20241109212954-00256.warc.gz"} |
Weird model comparison results when using WLS
Replied on Wed, 08/10/2022 - 16:42
Hi Hans,
Thanks for the clear code and working example! I'll have to take some time to check into this. The Satorra-Bentler chi-squared value seems way off, but I'm not sure what's causing it yet. The
mxCompare() behavior for WLS models is still relatively new; I've tested it, but not exhaustively. There is mostly likely a bug that I didn't catch in my initial testing.
At least one mystery is easy to solve: the p-value is for the SBchisq, not the chi-squared in the chisq column.
> pchisq(777.5465, 1, lower.tail = FALSE)
[1] 4.110957e-171
Satorra and Bentler (2001), found that this scaled difference chi-squared behaved better in smaller samples than the standard chi-squared, and performed equally well-for large samples.
I'll update you on this thread as I find out more!
Mike Hunter
Replied on Mon, 08/15/2022 - 06:02
In reply to Hmm ... looking into this by mhunter
Thank you very much. I'll just manually calculate the chi-square difference and p-value in the mean time. Luckily, my sample is very large, so it does not sound like it will make a difference.
Replied on Mon, 11/14/2022 - 16:05
mhunter Joined: 07/31/2009
I just wanted to let you know that this issue is now resolved in the development version of OpenMx and is available on GitHub. If you care to see the details, they are [here](https://github.com/
OpenMx/OpenMx/issues/358). The fix will be part of the next release of OpenMx. | {"url":"https://openmx.ssri.psu.edu/node/4830","timestamp":"2024-11-08T23:46:38Z","content_type":"text/html","content_length":"42653","record_id":"<urn:uuid:07bcc358-1011-4579-9b2e-ea86974c3ebd>","cc-path":"CC-MAIN-2024-46/segments/1730477028106.80/warc/CC-MAIN-20241108231327-20241109021327-00143.warc.gz"} |
square matrix logarithm
square matrix
logm(x) is the matrix logarithm of x. The result is complex if x is not positive or definite positive. If x is a symmetric matrix, then calculation is made by Schur form. Otherwise, x is assumed
diagonalizable. One has expm(logm(x))=x.
See also
• expm — square matrix exponential
• log — natural logarithm | {"url":"https://help.scilab.org/logm","timestamp":"2024-11-07T07:26:53Z","content_type":"text/html","content_length":"9999","record_id":"<urn:uuid:0fccc2cd-25c9-4764-ac05-b5495d51d418>","cc-path":"CC-MAIN-2024-46/segments/1730477027957.23/warc/CC-MAIN-20241107052447-20241107082447-00033.warc.gz"} |
Python Recursive Factorial Function
Oops, something went wrong. Please try again in a few moments.
def myfactorial(n: int) -> int:
Recursive function to compute the factorial of a non-negative integer.
- n: int
The non-negative integer for which the factorial is to be computed.
- int:
The factorial of the given non-negative integer 'n'.
- ValueError:
Raises an error if the input 'n' is negative.
# Base case: factorial of 0 is 1
if n == 0:
return 1
# Recursive case: factorial of n is n multiplied by factorial of (n-1)
elif n > 0:
return n * myfactorial(n - 1)
# Error case: negative input
raise ValueError("Input should be a non-negative integer.")
# Example usage of the myfactorial function:
# Example 1: Computing factorial of a non-negative integer
n1 = 5
factorial1 = myfactorial(n1)
print(f"The factorial of {n1} is {factorial1}.")
# Example 2: Computing factorial of 0
n2 = 0
factorial2 = myfactorial(n2)
print(f"The factorial of {n2} is {factorial2}.")
# Example 3: Computing factorial of a negative integer (should raise an error)
n3 = -5
factorial3 = myfactorial(n3)
print(f"The factorial of {n3} is {factorial3}.")
except ValueError as e:
print(f"Error while computing factorial: {e}") | {"url":"https://codepal.ai/code-generator/query/vf5ots9R/python-recursive-factorial-function","timestamp":"2024-11-03T11:58:31Z","content_type":"text/html","content_length":"106242","record_id":"<urn:uuid:93afd501-5201-4172-9b21-9554a093ac06>","cc-path":"CC-MAIN-2024-46/segments/1730477027776.9/warc/CC-MAIN-20241103114942-20241103144942-00735.warc.gz"} |
Ethanol & Flex Fuel Tuning
Ethanol & Flex Fuel Tuning: Fuel Characteristics
Fuel Characteristics
00:00 - To better understand what we need to change with our tuning to suit an ethanol blend of fuel, we need to understand the differences in the actual fuel properties.
00:09 These differences affect the amount of fuel that we need to deliver to the engine in order to maintain a consistent air fuel ratio.
00:17 I've already mentioned that when we're switching from gasoline to E85, we'll need to add around 40% more fuel by volume.
00:25 But now we're going to look at why.
00:28 The first difference is that ethanol has a vastly different stoichiometric air fuel ratio than pump gasoline.
00:35 The stoichiometric air fuel ratio defines the mass of air and fuel that we need to combine in order to achieve theoretically complete combustion.
00:45 In other words the mass of fuel that we need to add to the mass of inlet air on order to completely combust all of the available components.
00:54 When we talk about pump gasoline it has a stoichiometric air fuel ratio of 14.7:1 This means that for every gram of fuel, we need 14.7 grams of air to achieve complete combustion.
01:08 If we consider pure ethanol though, things are very different.
01:12 Pure ethanol has a stoichiometric air fuel ratio of 9:1 which means that now for every one gram of fuel, we need to supply nine grams of air.
01:23 Let's reverse things though because when we're tuning the ECU, it's not the air that we're controlling, it's the fuel delivery.
01:31 The different stoichiometric air fuel ratio, means that when we swap from gasoline to ethanol, we need to inject a much larger volume of fuel, to achieve the specific target air fuel ratio.
01:44 We need to also consider that the stoichiometric air fuel ratio will vary with the ethanol content of the fuel.
01:52 We've looked at pure gasoline with a stoichiometric air fuel ratio of 14.7:1 and ethanol at 9:1 however as the ethanol content moves from 0% to 100% the stoichiometric air fuel ratio also
changes between these two limits.
02:09 To complicate matters though, while the ethanol blend is defined on the basis of volume, when it comes to the air fuel ratio, it's the mass of the fuel and air that's important.
02:21 Let's discuss fuel density and then we can come back to the stoichiometric air fuel ratio.
02:27 The concept of mass versus volume is a tricky one but in simple terms, if we have two containers of a fixed volume, and we fill one with feathers and the other with sand, most people can
understand that the container filled with sand would weigh more.
02:43 This is because sand is denser than feathers.
02:47 In order to convert from volume to mass we need to take into account the density of the fuel.
02:53 And this is another important fuel characteristic that differs between ethanol and gasoline.
02:59 The density defines how much a given volume of fuel will weigh.
03:04 The density of gasoline is approximately 0.739 kilograms per litre, whereas the density of pure ethanol is approximately 0.787 kilograms per litre.
03:16 This means that if we had one litre of each fuel, ethanol would weigh more.
03:21 To add a further complexity to this, the density of the fuel will also vary with the fuel temperature.
03:29 The densities that I just mentioned, are measured at 25 degrees centigrade.
03:33 But as the fuel heats up or cools down its density varies too but not necessarily at the same rate.
03:41 For example gasoline has a fuel temperature coefficient of 0.0009 kilograms per litre per degree centigrade.
03:51 This means that for every degree that the temperature increases, one litre of gasoline would decrease in mass by 0.0009 kilograms, which is about a 10th of a percent.
04:05 Now that might not sound particularly relevant, but when you consider that the fuel temperature can easily fluctuate by as much as 60 to 70 degrees centigrade, this would affect the fuel mass
by around 7%.
04:19 In comparison to gasoline, pure ethanol's density varies slightly more as temperature changes, with a temperature coefficient of approximately 0.001 kilograms per litre per degree centigrade.
04:33 This means that the mass of a specific volume of ethanol will vary slightly more than that of gasoline given the same temperature change.
04:43 So now that we've learned about the fuel density, we can take this into account to see how the stoichiometric air fuel ratio changes as the ethanol blend fluctuates.
04:54 Let's consider E85 which is probably the most common ethanol blend we're likely to use.
05:01 E85 as we know is a blend consisting of 85% ethanol and 15% gasoline by volume.
05:09 In order to find the stoichiometric air fuel ratio for this blend though, we need to work out what this is as a mass ratio.
05:19 To do this let's assume we have one litre of E85 fuel.
05:23 To account for the fuel density, we can multiply 0.85 which is the volume of the ethanol in one litre, by the density of ethanol, which we know is 0.787 kilograms per litre.
05:37 This gives us a mass of 0.669 kilograms of ethanol in one litre of E85.
05:45 Now if we take the remaining 0.15 litres which consists of gasoline and multiply this by the density of gasoline, which is 0.739 kilograms per litre, we get a mass of 0.111 kilograms.
06:02 Now if we add these two masses together we can find the overall density of one litre of E85.
06:09 0.669 kilograms plus 0.111 kilograms gives us a total mass of 0.78 This means that the density of E85 is 0.78 kilograms per litre.
06:26 Now that we know the total mass and the mass of ethanol we can work out the mass ratio by dividing the mass of ethanol by the total mass.
06:35 This is exactly the same as how we work out the ethanol ratio except now we're using mass instead of volume.
06:42 In this case 0.669 kilograms divided by 0.78 kilograms, equals 0.858 which can be expressed as 85.8% This means that a blend of 85% ethanol by volume, or E85 as we would refer to it, contains a
blend of 85.8% ethanol by mass.
07:07 So you can see there's a subtle but real difference between the volume blend of E85 and the mass blend.
07:14 Now let's take the stoichiometric air fuel ratio of ethanol which is 9.0:1 and multiply this by the mass blend we just worked out of 0.858 This gives us a value of 7.72 Now if we do the same
for the gasoline component we need to multiply 14.7 by the remaining percentage, which is one minus 0.858 or in other words 0.142 This gives us a value of 2.09 If we now add 2.09 and 7.7 we get
a final value of 9.81:1 This is our stoichiometric air fuel ratio for a blend of 85% ethanol by volume.
08:02 If the process we've just worked through here is a little too much for you, I've attached an Excel spreadsheet to this module, which will allow you to see how these calculations work in more
detail, and to quickly work out the stoichiometric air fuel ratio for any ethanol blend.
08:20 If we look at the difference between the fuels based solely on their stoichiometric air fuel ratios, this means that when we move from gasoline to E85, we need to supply approximately 49% more
fuel by mass.
08:35 When we take into consideration the higher density of ethanol though we find that the additional volume of fuel the injectors need to deliver isn't quite as high as our mass calculation works
08:47 Accounting for this, we'll find that the additional volume of fuel that we need to supply as we move from gasoline to E85 is actually closer to 40%.
08:57 To throw one last spanner into the works though, we'll also find that a given fuel injector tends to flow a slightly reduced volume of fuel on E85, in comparison to gasoline.
09:10 This is because the two fuels have a different viscosity.
09:13 In general this effect is most often ignored in tuning flex fuel vehicles, although some ECUs do offer the ability to trim the injector flow as the ethanol concentration changes.
09:25 The problem is that we currently don't have easy access to this sort of data, which makes it hard to incorporate.
09:32 So what is the relevance of all this? And how much do you actually need to understand? The aim of this course is to give you the correct information so that you understand how the fuel you're
using will affect your tuning.
09:47 The actual application of this data will depend on the ECU you're tuning and how much control you have over the fuel delivery.
09:55 For example many ECUs completely disregard all of the fuel characteristics entirely, and we simply define how much more fuel the ECU is to deliver as the ethanol content changes.
10:08 This many sound quite a simplistic approach given the characteristics I've just mentioned, however in reality this method still does a perfectly good job.
10:18 An alternative option is to consider the actual stoichiometric air fuel ratio of the fuel relative to ethanol content.
10:26 And some ECUs go as far as to define how each aspect of the fuel changes with ethanol content.
10:33 The difference with these approaches, is that if we ignore the fuel's characteristics, then we need to do all of the work of telling the ECU how to vary the injected fuel volume as the ethanol
content changes.
10:45 And this will require two completely separate and very different fuel maps.
10:50 On the other hand if the ECU knows what the fuel characteristics are, it can do most of the hard work in the background and the resulting air fuel ratio should be relatively consistent as
ethanol content changes without the need for independent maps.
11:06 The reality is that even ECUs that properly account for the changing fuel characteristics will still see some error creep in as the content changes, and for this reason you'll still often need
a trim table to correct any errors that the main fuel equation doesn't completely account for.
11:25 Remember at this point we're only focusing on the air fuel ratio and there are also aspects such as the ignition timing, cold start calibration, and boost control to consider too.
11:37 But we'll look at that a little further into the course.
11:41 As I've already mentioned you don't need to be a chemist in order to tune an engine on ethanol blended fuels.
11:48 Everything I've just discussed really is just there to explain why we need to inject so much more fuel when we switch from gasoline to ethanol, which is one of the key aspects of ethanol or
flex fuel tuning, and hence is important to understand.
12:05 The key points to take away from this module, are that the stoichiometric air fuel ratio and fuel density both vary as the ethanol content of the fuel changes.
12:16 These fuel characteristics mean that we need to supply approximately 40% more fuel by volume in order to achieve a stoichiometric air fuel ratio as we move from E0 through to E85. | {"url":"https://www.hpacademy.com/courses/ethanol-and-flex-fuel-tuning/tuning-requirements-for-flex-fuel-fuel-characteristics/","timestamp":"2024-11-12T02:29:20Z","content_type":"text/html","content_length":"212520","record_id":"<urn:uuid:028b13b8-ba4c-45bd-ad33-9aad0ff15acd>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.50/warc/CC-MAIN-20241112014152-20241112044152-00461.warc.gz"} |
2009 APS March Meeting
Bulletin of the American Physical Society
2009 APS March Meeting
Volume 54, Number 1
Monday–Friday, March 16–20, 2009; Pittsburgh, Pennsylvania
Session B35: Focus Session: Iron Pnictides and Other Novel Superconductors III: General Theory Hide Abstracts
Sponsoring Units: DMP
Chair: Alex Koshelev, Argonne National Laboratory
Room: 405
Monday, B35.00001: Valley density-wave (VDW) and Superconductivity in Iron-Pnictides
March Vladimir Cvetkovic, Zlatko Tesanovic
16, 2009
11:15AM One of the experimentally observed features of iron-pnictide superconductors is the structural transition and SDW ordering occurring at almost the same temperature. Starting from a
- tight-binding model [1], we construct an effective theory for iron-pnictides with the distinctive two hole and two electron Fermi surfaces. This theory is then mapped onto a negative-U
11:27AM Hubbard model with additional orbital and spin flavors [2]. We demonstrate that the superconducting instability of the attractive Hubbard model --- valley density-wave (VDW) --- corresponds
to the observed structural and SDW orders. The deviations from perfect nesting between the hole and electron Fermi surfaces are mapped onto the Zeeman field which causes portions of Fermi
surface to remain ungapped. The origin of pnictide superconductivity in this model, and its ties to the VDW are discussed. [1] V. Cvetkovic and Z. Tesanovic, http://arxiv.org/abs/0804.4678.
[2] V. Cvetkovic and Z. Tesanovic, http://arxiv.org/abs/0808.3742. [Preview Abstract]
Monday, B35.00002: Spin-Density Wave in Iron Pnictides
March Jian Kang, Valentin Stanev, Zlatko Tesanovic
16, 2009
11:27AM Multi-band Hubbard-like model with appreciable nesting is applied to the study of spin-density wave (SDW) in iron pnictides\footnote{ V. Stanev, J. Kang, and Z. Tesanovic, Phys. Rev. B \
- textbf{78}, 184509 (2008).}. It is assumed that the SDW particle-hole pairing mechanism arises from the short range interaction between hole bands near $\Gamma $ point and electron bands
11:39AM near M. Within the Hubbard-Stratonovich transformation, an auxiliary field is introduced to obtain the effective action. The mean-field solution is obtained by the stationary phase analysis
of this action, and results in an itinerant, antiferromagnetically ordered ground state, with the staggered magnetic moment modulation at wavevector M. We study fluctuations of the spin
order around M, both in its direction and amplitude. We present detailed results for the propagation velocity of this mode (spin-wave velocity) as a function of the various parameters of our
model and compare them to the available experimental observations of the spin-wave spectrum. [Preview Abstract]
Monday, B35.00003: Iron-based superconductors: What can we learn from DFT?
March Lilia Boeri, Oleg Dolgov, Alexander Golubov, Ole Krogh Andersen
16, 2009
11:39AM The discovery of superconductivity in iron pnicticides has initiated an intense theoretical activity. So far, however, not only the pairing mechanism, but even the basic electronic structure
- of these materials is not well understood. We use Density Functional Theory to understand the electronic and vibrational properties of LaOFeAs, which can be considered a prototype for iron
11:51AM pnictides. First, we calculate the phonon dispersions and electron-phonon coupling using linear response and show that standard Migdal-Eliashberg theory cannot explain the experimental Tc.
Then we derive ab-initio an accurate tight-binding Hamiltonian, using downfolding + N-ization (NMTO), which allows us to elucidate the origin of the complicated band structure of iron
pnicticides. As a first application of our model, we study magnetism. [Preview Abstract]
Monday, B35.00004: Correlations in Ferropnictides
March Klaus Koepernik, Helmut Eschrig
16, 2009
11:51AM The strength of correlations in the ferropnictide superconductors is still under debate. While arguments for an electron-electron interaction $U$ of $5$eV have been made, some experimental
- results support a $U$ of merely $1$eV. Density functional theory in the local spin density approximation (LSDA) seems to describe several aspects of the electronic structure quite
12:03PM reasonably, which would also support a smaller $U$. However, the unusually large error of the calculated lattice structure remains a puzzle. We discuss the influence of correlations on the
electronic structure and the properties of the ferropnictides in the framework of LSDA+U calculations. [Preview Abstract]
Monday, B35.00005: Superconductivity in SrFe$_{2-x}$Co$_x$As$_2$: Internal Doping of the Iron Arsenide Layers
March Helge Rosner, Andreas Leithe-Jasper, Walter Schnelle, Christoph Geibel
16, 2009
12:03PM In the strontium iron-cobalt arsenides SrFe$_{2-x}$Co$_x$As$_2$ ($0.2\leq x \leq 0.4$) superconductivity with $T_c$ up to 20\,K is observed in magnetic susceptibility, electrical
- resistivity, and specific heat data. This first observation of bulk superconductivity induced by electron doping in this family of compounds -- despite strong disorder in the Fe-As layer --
12:15PM favors an itinerant electronic theory in contrast to the strongly correlated cuprates and renders a $p$- or $d$-wave pairing unlikely. The magnetic ordering present in SrFe$_2$As$_2$ is
rapidly suppressed by substitution of Fe by Co. DFT calculations show that this is due to a rigid down-shift of the Fe-3$d_{x^2-y^2}$-related band edge in the density of states. [Preview
Monday, B35.00006: Linear temperature dependence of the spin susceptibility in Fe-pnictides.
March Dmitri V. Efremov, Andrey V. Chubukov, Ilya M. Eremin, Maxim M. Korshunov, Dmitri L. Maslov
16, 2009
12:15PM We argue that linear $T$ dependence of the spin susceptibility $\chi >T$ observed in Fe pnictides can be explained within the itinerant Fermi liquid model of hole and electron bands. The
- spin susceptibility is linear in $T$ in a generic Fermi liquid in 2D. We show that for pnictides, the prefactor for the $T$ term comes chiefly from intra-band scattering and is strongly
12:27PM enhanced compared to an ordinary Fermi liquid as it contains precisely the same interaction that gives rise to spin-density-wave ordering. We compare theoretical slope with the data. [
Preview Abstract]
Monday, B35.00007: Theory of novel and superconducting properties of Fe-based superconductors
March Invited Speaker:
16, 2009
12:27PM I will discuss antiferromagnetism and superconductivity in novel $Fe-$based superconductors within the itinerant model of small electron and hole pockets near $(0,0)$ and $(\pi,\pi)$. I will
- 1:03PM argue that the effective interactions in both channels logarithmically flow towards the same values at low energies, {\it i.e.}, antiferromagnetism and superconductivity must be treated on
equal footings. The magnetic instability comes first for equal sizes of the two pockets, but looses to superconductivity upon doping. The superconducting gap has no nodes, but changes sign
between the two Fermi surfaces (extended $s$-wave symmetry). I will argue that the $T$ dependencies of the spin susceptibility, NMR relaxation rate, and the penetration depth for such state
are exponential only at very low $T$, and can be well fitted by power-laws over a wide $T$ range below $T_c$. I will also discuss the type of a transition between spin-density-wave and
superconducting states at $T=0$ and at finite $T$, and the linear $T$ dependence of the spin susceptibility in the normal state. \\ Based on the works done with I. Eremin, D. Efremov, M.
Korshunov, D. Maslov, M. Vavilov, and A. Vorontsov. [Preview Abstract]
Monday, B35.00008: Nodal Spin Density Wave and band topology of the FeAs based materials
March Hui Zhai, Ying Ran, Fa Wang, Ashvin Vishwanath, Dung-Hai Lee
16, 2009
1:03PM - The recently discovered FeAs-based materials exhibit a $(\pi,0)$ Spin Density Wave (SDW) in the undoped state, which gives way to superconductivity upon doping. Here we show that due to an
1:15PM interesting topological feature of the band structure, the SDW state cannot acquire a full gap. This is demonstrated within the SDW mean-field theory of both a simplified two band model and
a more realistic 5-band model. The positions of the nodes are different in the two models and can be used to detected the validity of each model. [Preview Abstract]
Monday, B35.00009: Normal State Spin Dynamics of Five-band Model for Iron-pnictides
March Toshikaze Kariyado, Masao Ogata
16, 2009
1:15PM - Normal state (assuming absence of SC or AF order) spin dynamics of iron-pnictide superconductors is discussed by calculating spin structure factor $S(q,\omega)$ in an itinerant five-band
1:27PM model within RPA approximation. Due to the characteristic Fermi surface structure of iron-pnictide, column like response is found at $(\pi,0)$ in extended Brillouin zone. This is consistent
with recent neutron experiments. Furthermore, we show that the temperature dependence of inelastic neutron scattering intensity is reproduced if we set interaction parameters appropriately.
[Preview Abstract]
Monday, B35.00010: Symplectic fermion approach to the striped magnetism in the iron arsenides
March Xun Xue, Jianhui Dai
16, 2009
1:27PM - Based on the fact that the near transition temperature of striped SDW and structure distortion in iron pnictides, we propose a symplectic fermion approach to account for this kind of
1:39PM antiferromagnetic properties. The model is expecting for better understanding of the experimental results. [Preview Abstract]
Monday, B35.00011: Parquet formalism applied to pnictide superconductors
March Jun Liu, Karlis Mikelsons, Shuxiang Yang, Herbert Fotso , Mark Jarrell
16, 2009
1:39PM - DMFT combined with Parquet approximation is used to study the single particle property of pnictide superconductors (such as FeSe, SrFe2As2,...) in an attemp to understand the enhancement of
1:51PM superconducitivity under pressure. By tracking the evolution of one-particle spectral function, pressure dependence of this type of compound is studied in depth. In the study, inhomogeneous
frequency grid is used to high frequency summation. [Preview Abstract]
Monday, B35.00012: Jahn-Teller Effect, Structural Phase Transition and Resistivity Anomaly in Iron Pnictides
March Weicheng Lv, Jiansheng Wu, Philip Phillips
16, 2009
1:51PM - We attribute the structural phase transition (SPT) in the parent compounds of iron pnictides to a Jahn-Teller distortion. Due to the anisotropy of the $d_{xz}$ and $d_{yz}$ orbitals in the
2:03PM $xy$ plane, some orbital ordering will make the orthorhombic structure more energetically favorable, thus inducing the SPT. In an orbital ordered system, the sites with orbitals that do not
order act as scattering impurities, causing a resistivity anomaly upon the onset of the SPT. Below the SPT, we find that the resistivity displays a $\ln{T}$ divergence. All of these are in
agreement with the experiments. [Preview Abstract]
Monday, B35.00013: Theory of the Magnetic Moment in Iron Pnictides
March Jiansheng Wu, Philip Phillips, Antonio Castro-Neto
16, 2009
2:03PM - We show that the combined effects of spin-orbit, monoclinic distortion, and p-d hybridization in tetrahedrally coordinated Fe in LaFeAsO invalidates the naive Hund's rule filling of the Fe
2:15PM d-levels. The two highest occupied levels have one electron each but as a result of differing p-d hybridizations, the upper level is more itinerant while electrons in the lower level are
more localized. The resulting magnetic moment is highly anisotropic with an in-plane value of $0.25-0.35\mu_B$ per Fe and a z-projection of $0.06\mu_B$, both of which are in agreement with
experiment. [Preview Abstract]
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Einstein’s journal paper (Einstein 1905) and Dirac’s book (Dirac 1981) are physics-related items.
Dirac, Paul Adrien Maurice. 1981. The Principles of Quantum Mechanics. International Series of Monographs on Physics. Clarendon Press.
Einstein, Albert. 1905. “Zur Elektrodynamik bewegter Körper. (German) [On the Electrodynamics of Moving Bodies].” Annalen Der Physik 322 (10): 891–921.
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Predicted distribution of Mersenne primes
Mersenne primes are prime numbers of the form 2^p – 1. It turns out that if 2^p – 1 is a prime, so is p; the requirement that p is prime is a theorem, not part of the definition.
So far 51 Mersenne primes have discovered [1]. Maybe that’s all there are, but it is conjectured that there are an infinite number Mersenne primes. In fact, it has been conjectured that as x
increases, the number of primes p ≤ x such that 2^p – 1 is also prime is asymptotically
e^γ log x / log 2
where γ is the Euler-Mascheroni constant. For a heuristic derivation of this conjecture, see Conjecture 3.20 in Not Always Buried Deep.
How does the actual number of Mersenne primes compared to the number predicted by the conjecture? We’ll construct a plot below using Python. Note that the conjecture is asymptotic, and so it could
make poor predictions for now and still be true for much larger numbers. But it appears to make fairly good predictions over the range where we have discovered Mersenne primes.
import numpy as np
import matplotlib.pyplot as plt
# p's for which 2^p - 1 is prime.
# See https://oeis.org/A000043
ps = [2, 3, 5, ... , 82589933]
# x has 200 points from 10^1 to 10^8
# spaced evenly on a logarithmic scale
x = np.logspace(1, 8, 200)
# number of p's less than x such that 2^p - 1 is prime
actual = [np.searchsorted(ps, t) for t in x]
exp_gamma = np.exp(0.5772156649)
predicted = [exp_gamma*np.log2(t) for t in x]
plt.plot(x, actual)
plt.plot(x, predicted, "--")
plt.ylabel(r"Mersenne primes $\leq 2^p-1$")
plt.legend(["actual", "predicted"])
Related posts
[1] Fifty one Mersenne primes have been verified. But these may not be the smallest Mersenne primes. It has not yet been verified that there are no Mersenne primes yet to be discovered between the
47th and 51st known ones. The plot in this post assumes the known Mersenne primes are consecutive, and so it is speculative toward the right end. | {"url":"https://www.johndcook.com/blog/2019/09/16/distribution-of-mersenne-primes/","timestamp":"2024-11-12T07:00:14Z","content_type":"text/html","content_length":"50514","record_id":"<urn:uuid:7024446c-501c-46ac-b7d1-f66e62fcb07f>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.58/warc/CC-MAIN-20241112045844-20241112075844-00238.warc.gz"} |
Add and subtract fractions with the same denominator and related fractions; write mathematical statements >1 as a mixed number (e.g. 2/5 + 4/5 = 6/5 = 11/5)
An interactive quiz which test your understanding of all the Fractions (including Decimals and Percentages) objectives in the Year 5 curriculum. Choose one objective or multiple objectives. You can
save, or print, your test results as a pdf at the end of the quiz. Ideal for formative or summative assessment.
Alternatively, you can use the Interactive Maths Quiz which includes objectives from all strands of the Year 5 curriculum.
This quiz tests the following objectives: | {"url":"https://mathsframe.co.uk/en/resources/category/419/add-and-subtract-fractions-with-the-same-denominator-and-related-fractions-write-mathematical-statements-greater-than-1-as-a-mixed-number","timestamp":"2024-11-02T09:46:12Z","content_type":"text/html","content_length":"17433","record_id":"<urn:uuid:e40e78d4-a072-4dfb-a7d8-0c5cb2a208a7>","cc-path":"CC-MAIN-2024-46/segments/1730477027709.8/warc/CC-MAIN-20241102071948-20241102101948-00862.warc.gz"} |
Clustering of ECG segments for patients before sudden cardiac death based on Lagrange descriptors
Novel approach to clustering of ECG segments based on Lagrange descriptors is presented in this paper. The approach starts by extracting 2D features with the help of Lagrange descriptors. Then the
features are transformed to latent vectors which are clustered using K-means algorithm. The object of the research is to visualize the dynamics of clusters of 2D features of segments of ECG before
sudden cardiac death happens to a patient.
1. Introduction
Sudden cardiac death (SCD) is often described as the result of a change of typical sinus rhythm of a heart to a rhythm which does not support adequate pumping of the blood to the brain [1]. A number
of techniques were used for predicting SCD and they achieve very different accuracy [2]. Despite the numerous methods and attempts to predict SCD there still exists challenges. Pacemakers and other
devices requires high level of accuracy while interacting with human beings. More on these challenges could be found in [3].
Clustering and classification approaches in ECG data analysis is not a new direction [4-6]. But the novelty of this paper arises from the fact that we incorporate here Lagrangian descriptors (LD) as
the first step in feature extraction. LD were introduced in [7]. The methodology is able to visualize the structure of a phase space. The method is based on computing the length of the trajectory a
particle (trajectory) travels.
Convolutional autoencoder neural network (CNN) is employed here for extracting latent vectors. More on such techniques could be found in [8, 9].
1.1. Data description
In this paper we use Sudden Cardiac Death Database available at PhysioNet [10]. The database contains 23 complete Holter recordings. Each recording comprises 2 time series – two lead ECG. For short
description of the data used refer to Table 1.
Table 1ECG data annotations for Sudden Cardiac Death Database
Data file Gender Age Signal duration Underlying cardiac rhythm
30m.mat Male 43 24:33:17 Sinus
31m.mat Female 72 13:58:40 Sinus
32m.mat Unknown 62 24:20:00 Sinus, intermittent VP
… … … … …
52.mat Female 82 07:31:05 Sinus
We have analyzed each data file of the database but provide results here for the first entry i.e. file “30m.mat”.
1.2. Methodology
Consider ECG signal ${X}_{t}$, $t=1,2,\dots ,N$. Since the aim here is to investigate the dynamics of extracted features in different segments of ECG, the signal is divided into 5 data sets of equal
length: ${x}_{t}^{\left(q\right)}={X}_{\left(q-1\right)\bullet ⌊N/5⌋+t}^{}$, $t=1,2,\dots ,⌊N/5⌋$, $q=1,2,\dots ,5$. Each segment is investigated separately using the same methodology.
Datasets ${x}_{t}^{\left(q\right)}$ are further divided into a number of non-overlapping vectors ${y}_{i}^{\left(q,c\right)}\text{,}$$c=⌊⌊N/5⌋/256⌋$, $i=1,2,\dots ,256$. Now, time embedding is
performed for each vector and collection of triples of ECG values are computed: ${\left({u}_{i},{v}_{i},{w}_{i}\right)}_{{\tau }_{1},{\tau }_{2}}:=\left({y}_{i}^{\left(q,c\right)},{y}_{i+{\tau }_{1}}
^{\left(q,c\right)},{y}_{i+{\tau }_{1}+{\tau }_{2}}^{\left(q,c\right)}\right)$, ${\tau }_{1},{\tau }_{2}=1,2,\dots ,48$, $i=1,2,\dots ,166$. This is in fact a standard non-uniform time series
embedding approach. Thus, the triples obtained could be treated as points of a trajectory in 3D space. Lengths $L\left({\tau }_{1},{\tau }_{2}\right)$ of the vectors ${\left({u}_{i},{v}_{i},{w}_{i}\
right)}_{{\tau }_{1},{\tau }_{2}}$ are the simplest LD estimates and are computed by Eq. (1).
$L\left({\tau }_{1},{\tau }_{2}\right)=\sqrt{{\sum }_{i=1}^{165}{\left({\left({u}_{i+1},{v}_{i+1},{w}_{i+1}\right)}_{{\tau }_{1},{\tau }_{2}}-{\left({u}_{i},{v}_{i},{w}_{i}\right)}_{{\tau }_{1},{\tau
LD estimates are then used for 2D feature computation. We place a dot of grayscale intensity $L\left({\tau }_{1},{\tau }_{2}\right)$ in the two-dimensional plane at coordinates $\left({\tau }_{1},{\
tau }_{2}\right)$. After considering ${\tau }_{1},{\tau }_{2}=1,2,\dots ,48$ two-dimensional features are obtained for ECG segments ${x}_{t}^{\left(q\right)}$. Clustering of the features is a rather
straightforward task. So CNN is employed for extracting second level features – latent vectors. These are later fed to dimensionality reduction by PCA (leaving only 2 components, just for the
visualization purposes) and clustered by K-means. Fig. 1 shows full diagram of the methodology proposed for clustering of ECG segments.
Fig. 1Full diagram of the methodology presented in this paper together with the structure of CNN for extracting latent vectors (second level features) from 2D features of ECG
1.3. Numerical experiments
At first 2D feature maps must be computed. Actually, this step of preprocessing implies a very big number of numerical experiments to be carried out. We have used two-dimensional time embedding
together with the simplest LD estimate as mentioned before. One can try different dimensions as well as different LD estimates but that would only lead to optimal feature extraction technique and not
tell more details about the methodology itself.
First three 2D feature maps for the first data file and the measurements of the first lead are shown in Fig. 2.
Fig. 2A subset of 2D features of ECG segments before sudden cardiac death for “30m.mat” data file
Visual differences in Fig. 2 implies that clustering of such mages would be a logical step to try to find different clusters for the beginning of the recordings and for the end of the recordings.
This is the ultimate goal of the research presented here.
Next, optimal CNN must be trained on the obtained 2D feature maps and latent vectors computed. It should be noted that we linearly transform the values in the maps to the interval $\left[0;1\right]$
before applying CNN. For training of CNN we have tested filter sizes 4 and 8, 4 and 8 for number of convolutional filters in the first layer of the network, 2 and 4 for number of filters in the
second layer as well as 2 and 4 for parameters in downsampling (MaxPooling) layers. We have used python with keras for training of CNN. Computations took significant amount of time to complete. For
that matter cuda hardware acceleration was employed. Training performance was measured by the mean squared error (MSE). Table 2 shows best CNN architectures for the first data file.
Table 2Best CNN architectures for the data file “30m.mat”. Depending on the network parameters the length nlatent of the latent vectors is different. Training and validation sets were selected
randomly with ratio 70/30
Data file Lead Segment Conv. dim. ${n}_{c1}$ ${n}_{c2}$ ${n}_{p1}$ ${n}_{p2}$ ${n}_{latent}$ ${MSE}_{train}$ ${MSE}_{val}$
30m.mat 1 1 4 8 4 2 4 144 0.0072 0.0063
30m.mat 1 2 8 8 4 2 2 576 0.0074 0.0061
30m.mat 1 3 8 8 4 2 4 144 0.0069 0.0056
30m.mat 1 4 8 8 4 2 4 144 0.0072 0.0062
30m.mat 1 5 8 8 4 4 2 144 0.0075 0.0073
30m.mat 2 1 4 8 4 2 2 576 0.0067 0.0060
30m.mat 2 2 8 8 4 4 2 144 0.0070 0.0055
30m.mat 2 3 4 8 4 2 2 576 0.0063 0.0054
30m.mat 2 4 8 8 4 2 2 576 0.0082 0.0055
30m.mat 2 5 4 8 4 2 2 576 0.0081 0.0079
Table 2 shows that bigger filter sizes are preferred in terms of MSE. It could be also noted that MSE in validation set is always lower compared to the MSE in the training set. This ensures that
there was no overfitting during the training process. Since there are no big differences between the MSEs for different optimal CNN architectures we fix parameters ${n}_{c1}=\text{8}$, ${n}_{c2}=\
text{4}$, ${n}_{p1}=\text{2}$, ${n}_{p2}=\text{2}$ and 8 as the dimension of convolutional filters for the experiments presented in this paper.
The change in training performance of CNN is depicted in Fig. 3. Note that the plots of training performance for other data files are of almost identical shape.
Fig. 3Training performance of CNN
a) The case for the signal of the lead 1
Optimal CNN network was employed and sets of latent vectors were computed for each segment of ECG. Clustering of the latent vectors for the measurements from the first lead from “30m.mat” data file
is depicted in Fig. 4.
Fig. 4Clusters of latent vectors corresponding to different segments of ECG before sudden cardiac death (1st lead of the “30m.mat” data file). Colors depict different clusters and are not related in
any manner between different parts a) – e)
It is known that 2 hours before sudden cardiac death is the time window in which an expert can spot unnatural and potentially risky ECG activity [1]. In Fig. 4 we clearly see that 5 segments during
the last 25 minutes have visibly different clusters of latent space vectors. Same applies to the lead 2 (Fig. 5). Unfortunately, the data sets used do not have more measurements which could span more
than 2 hours. That would be especially valuable for the validation if the methodology proposed actually signals something unusual before the SCD happens.
Note that shapes of clusters of the vectors of latent space in general have some similar areas between different segments of ECG and even between different data files. This suggests that the proposed
methodology might not be completely random. Of course, bigger data sets are required to prove or disprove this.
Fig. 5Clusters of latent vectors corresponding to different segments of ECG before sudden cardiac death (2nd lead of the “30m.mat” data file). Colors depict different clusters and are not related in
any manner between different parts a) – e)
2. Conclusions
The main finding of this research is the fact that LD based feature extraction is capable of differently labeling different ECG segments although some similarities also exist.
The methodology is an interesting tool which can be tested for more different CNN parameters, different clustering approaches or different LD estimates. And that is a huge testbed for possible
research outcomes which could potentially be an alternative to visual analysis of ECG performed by an expert.
Keeping track of the extracted features in the way showed here could also be considered as a universal tool or even an expert system for anomaly detection in other dynamical systems. One just needs
to note cluster centers and/or sizes for the expert system to be completely automated.
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About this article
Biomechanics and biomedical engineering
Lagrange descriptors
feature extraction
convolutional autoencoder
This research was supported by the Research, Development and Innovation Fund of Kaunas University of Technology (project acronym DDetect).
Copyright © 2019 Mantas Landauskas, 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/21007","timestamp":"2024-11-14T21:37:18Z","content_type":"text/html","content_length":"122013","record_id":"<urn:uuid:22a2c7f1-0b33-407d-a31c-7f95edc7c7de>","cc-path":"CC-MAIN-2024-46/segments/1730477395538.95/warc/CC-MAIN-20241114194152-20241114224152-00230.warc.gz"} |
Transformers: The Nuts and Bolts
Last time, we looked at what made the Transformer special (link). However, we did not look at its architecture. Let’s remedy that. It is going to be a lengthy post, so let’s get started.
Overall Architecture
Vaswani et al. proposed the Transformer for machine translation. The Transformer’s overall architecture is an encoder-decoder one. Therefore, it consists of an encoder that feeds into a decoder. Both
the encoder and decoder are composed of multiple layers/components.
The architecture of the Transformer.
Encoder Stack
The encoder stack takes in the model inputs. Then, it maps them to abstract, continuous representations that hold the learned information for the inputs. The encoder consists of six encoders stacked
on top of each other. Those layers sit on top of an embedding layer. Let’s see how the Transformer works by following a hypothetical input as it works its way through the Transformer, starting with a
single encoder layer.
Embeddings and Positional Encoding
Let’s use a simple example to illustrate the workings of the Transformer. We’ll use “The quick brown fox” as our example input. So, the first thing we do is feed our input sentence into the model.
The Transformer needs the embedding vectors for that input. That is what the embeddings module gets.
The embeddings module is a giant lookup table that holds the numerical embedding vectors for every word in the Transformer’s vocabulary. We take the embeddings from some other place (e.g., GloVe).
So, the module looks up the embedding for each word and outputs that embedding.
The Transformer’s overall architecture is an encoder-decoder one.
But wait, there’s more. Sequential ordering matters in language and needs to be preserved. The sequential nature of RNNs preserves the sequential ordering of the input. The Transformer, however,
isn’t naturally sequential. Therefore, we use positional encoding to retain the knowledge of the token positions. We add the positional encodings to the token embeddings. The equations to calculate
the positional embeddings are:
PE₍ₚₒₛ,₂ₜ₎ = sin(pos/10000²ᵗ/ᵈₘ)
PE₍ₚₒₛ,₂ₜ₊₁₎ = cos(pos/10000²ᵗ/ᵈₘ)
So, for even position tokens, we use sine. Conversely, odd position tokens use cosine. dₘ is the dimension of the embeddings and positional embeddings. Next up is the multi-head self-attention
Multi-Head Self-Attention Mechanism
The speed of the Transformer mainly comes from the multi-head self-attention layer. When we say that the Transformer is parallelizable, this is where that happens. We talked about the Transformer’s
self-attention mechanism in the last post. In this post, we’ll focus on how to turn self-attention into multi-head self-attention.
The overall model uses dₘ-dimensional vectors. The vectors are broken up into smaller depth-dimensional vectors to enable processing them in parallel.
depth = dₘ / h
h is the number of attention heads. An attention head is just one processing unit of multi-head attention.
To run faster, the Transformer uses multi-head attention, not just attention.
Then, we pass each set of depth-dimensional vectors to one of the attention heads for processing. The actual mechanism of this component is pretty straightforward. After splitting up the component’s
inputs, we run each vector set (i.e., the depth-dimensional vectors) through 3 linear layers. Each layer has a different set of weights (wₐ, wₖ, wᵥ). The model initializes and trains the weights
independently. The linear layers output the V, K, and Q matrices for their attention head. So, if the broken-up input embedding vectors are eᵢ, then the equations for these layers are:
Q = wₐeᵢ
K = wₖeᵢ
V = wᵥeᵢ
With the V, K, and Q matrices in hand, we apply the Transformer’s secret sauce to them (i.e., scaled dot-product attention).
The self-attention layer is composed of many layers. The mask layer is present in the decoders, but not the encoders.
So, the self-attention layer is where we add parallelism to the Transformer. However, you have to get off the 6-lane highway at some point. That is where the Concat layer comes in. It brings the
various self-attention lanes together. As the name suggests, the Concat layer concatenates the many outputs from the different self-attention layers. So instead of multiple matrices of depth
-dimensional vectors, we get a single matrix of dₘ-dimensional vectors. The formula for the Concat layer is:
MultiHead(Q, K, V) = Concat(head₁, …, headₕ)Wₒ where headₜ = Attention(QW꜀ₜ, KWₖₜ, VWᵥₜ)
Afterward, a linear layer weights that single matrix with wₒ, producing the output of this layer. If we assume that the input to the linear layer is x, then the formula for that layer is:
Z = wₒx
There is one final step before the multi-head attention layer passes its outputs to the next layer. That step is we run the output of the multi-head attention layer through a residual and
normalization layer. The purpose of this layer is to facilitate model training and stability. Starting with the residuals, a residual is just the difference between the observed and predicted values.
Take, for example, the function f(xᵢ). It produces an output, xₒ = f(xᵢ). The residual is then just f(xᵢ) - xᵢ, or put more simply xₒ - xᵢ. In practice, that means we add the input to the layer to
its output. In other words, if the layer’s input is Xₑ, then:
Zᵣ = Z + Xₑ
The residual and normalization layer facilitates training and model stabilization.
The residual provides a shortcut around the multi-head self-attention layer. It allows a path for gradients to flow without going through weights. That means that exploding and vanishing gradients
are non-issues. After adding the residual, the model applies a layer normalization. The purpose of layer normalization is to stabilize the model during training. Models like the Transformer have
multiple layers that we train via gradient descent. In other words, we compute an error for each training epoch. The layers update their weights according to that error. The layers do this updating
one at a time. That presents a problem because once a layer has updated its weights, the error may no longer be accurate given the updated weights (i.e., covariate shift). As a result, the training
that subsequent layers undergo may not be correct. So, model training for deeply layered networks is always chasing a moving target. Layer normalization solves this by normalizing the input to neural
layers. That fixes the covariate shift problem. In this case, we normalize the output of the multi-head self-attention layer before moving it onto the next layer.
Position-wise Feed-Forward Layer
After the multi-head attention layer, we come to the position-wise feed-forward layer. This layer is two linear transformations with a ReLU activation in between. So, it is, essentially, two
convolutions of kernel size 1. The position-wise feed-forward layer runs parallel instances of itself. Those processors operate on each position of the input matrix (i.e., word), hence
“position-wise”. Each parallelized processor shares the weights, but each linear transform has its own set of weights. Its formula is:
FFN(x) = max(0, xW₁ + b₁)W₂ + b₂
The position-wise feed-forward layer has higher inner dimensionalities than input and output dimensionality. That makes it reminiscent of the sparse-autoencoder. Overall, it is just a lot more
matrix calculations. Following the position-wise feed-forward layer, we have another residual and normalization layer. Like the residual and normalization layer that follows the multi-head attention
layer, this residual and normalization layer also facilitates training and model stabilization. This time, however, the outputs are stabilized for the subsequent encoder.
6 stacked encoders make up the encoder stack. The same is true for decoders and the decoder stack.
So, that concludes our examination of a single encoder. As previously mentioned, 6 stacked encoders make up the encoder stack. The encoders feed into one another sequentially. The last encoder sends
its outputs to the decoder stack.
Decoder Stack
Even though this post is already very long, the decoders are almost identical to the encoders. So, we are in the home stretch. A decoder is composed of two multi-head attention layers and a
position-wise feed-forward layer. Each of those sublayers also has its residual and normalization layer. Mostly, the layers are identical to their encoder counterparts. However, a different mission
necessitates slight design changes. The decoder is autoregressive. In other words, it predicts a token sequence one token at a time. It considers the previously foretold tokens, the output from the
encoder stack, and the current token for each prediction.
Shifted Inputs
The decoder shifts its inputs one position to the right by prepending a <start> token to the input sequence. Everything else works the same.
Masked Multi-head Attention
Sentences are sequential structures. Therefore, at time t, you have seen the word at time t and every word proceeding it. So when predicting a token sequence one token at a time, we can only look to
past predictions. However, the unmodified multi-head attention allows you to look at the tokens after t. In other words, you can look into the future. Doing so for token prediction is ludicrous. It
would be like someone who says they can predict next Tuesday’s lottery numbers. However, to do so, they consider the events of next Wednesday and Thursday.
A Look-Ahead Mask turns Multi-Head Attention into Masked Multi-Head Attention.
Therefore, we need to change the first multi-head attention layer into a masked multi-head attention layer. We do this with a look ahead mask. A look ahead mask is a matrix of the same size as the
matrix outputted by the multi-head attention layer. It contains 1s and negative infinities. The mask is applied to the scores after scaling but before the softmax. The softmax then just zeros out the
future words for every word in the scaled matrix. The formula for this is simple enough.
The decoder is autoregressive.
Attention(Q, K, V) = softmax[[mask + (QKᵀ)] / √dₖ]V
Multi-head Attention With Memory
The second multi-head attention layer of the decoder is also slightly different from its encoder counterpart. The layer ingests an additional input, the Z output from the last encoder of the encoder
stack. That Z output is memory. It comes in unmasked. The decoder incorporates it into the calculations of the K and V vectors. So this layer’s formulas are:
Q = wₐeᵢ
K = Zwₖeᵢ
V = Zwᵥeᵢ
Of course, the weights here are separate from the weights of the encoder’s multi-head attention.
Let’s Cap it Off
So we have our encoder and decoder stacks. Now, all we need to do is add additional task-specific linear layers and a softmax. So if we are using the Transformer to translate into German, we add the
linear layers for that task. The softmax is there to compute the probabilities over the outputs.
This post has been much longer than my typical ones. However, I think the extra length was worth it to understand the Transformer. Even though I typically post every Thursday, I will need to take a
breather and skip next week. | {"url":"https://www.revistek.com/posts/transformer-architecture","timestamp":"2024-11-05T19:13:57Z","content_type":"text/html","content_length":"26267","record_id":"<urn:uuid:c2d2a40a-1192-445e-914a-54b9d68293c7>","cc-path":"CC-MAIN-2024-46/segments/1730477027889.1/warc/CC-MAIN-20241105180955-20241105210955-00159.warc.gz"} |
7.12. Math 163- Basic Statistics
Math Placement, Transfer Credit/Credit by Exam, PPL (Math Placement Test) → First and Second Year Math Courses → Math 163- Basic Statistics
Math 163- Basic Statistics (3 credits)
Description: Organizing data: displaying distributions, measures of center, measures of spread, scatterplots, correlation, regression, and their interpretation. Design of experiments: simple random
samples and their sampling distribution, models from probability, normal distributions, and normal approximations. Statistical inference: confidence intervals and hypothesis testing, t procedures and
chi-square tests. Not intended for those who plan further studies in statistics.
Placement Level: ALEKS PPL score of 60-100% , SAT I MSS 640-800, ACT MATH 26-36, or U of A Math 112 required for placement into this course. Test scores expire after one year. SAT/ACT placement is
generally for first year students only.
If my scores are lower than the Placement Level: Math 100, then Math 112, then Math 163. Students who have credit for Math 120R, 122B or 125 also qualify to enroll in Math 163.
Comments: Not a prerequisite for any math or lab science (CHEM, MBC, PHYS) courses. Students in nursing, nutritional sciences should take this instead of Math 263. Recommended as a second semester | {"url":"https://kb.math.arizona.edu/placement/index.php?pg=kb.page&id=86","timestamp":"2024-11-02T07:44:05Z","content_type":"text/html","content_length":"11357","record_id":"<urn:uuid:7cf2c727-3ce7-4cc7-b5bc-1296b6b70041>","cc-path":"CC-MAIN-2024-46/segments/1730477027709.8/warc/CC-MAIN-20241102071948-20241102101948-00625.warc.gz"} |
EC2302 DIGITAL SIGNAL PROCESSING Two Marks With Answers
Anna University, Chennai
SUBJECT CODE: EC2302
UNIT : I
1. What is a continuous and discrete time signal? Ans:
Continuous time signal: A signal x(t) is said to be continuous if it is defined for all time t. Continuous time signal arise naturally when a physical waveform such as acoustics wave or light wave is
converted into a electrical signal. This is affected by means of transducer. (e.g.) microphone, photocell.
Discrete time signal: A discrete time signal is defined only at discrete instants of time. The independent variable has discrete values only, which are uniformly spaced. A discrete time signal is
often derived from the continuous time signal by sampling it at a uniform rate.
2. Give the classification of signals? Ans:
Continuous-time and discrete time signals
Even and odd signals
Periodic signals and non-periodic signals Deterministic signal and Random signal Energy and Power signal
3. What are the types of systems? Ans:
Continuous time and discrete time systems Linear and Non-linear systems
Causal and Non-causal systems Static and Dynamic systems
Time varying and time in-varying systems
Distributive parameters and Lumped parameters systems stable and Un-stable systems.
4. What are even and odd signals? Ans:
Even signal: continuous time signal x(t) is said to be even if it satisfies the condition x(t)=x(-t) for all values of t.
Odd signal: The signal x(t) is said to be odd if it satisfies the condition x(-t)=-x(t) for all t. In other words even signal is symmetric about the time origin or the vertical axis, but odd signals
are anti-symmetric about the vertical axis
5. What are deterministic and random signals? Ans:
Deterministic Signal: Deterministic signal is a signal about which there is no certainty with respect to its value at any time. Accordingly we find that deterministic signals may be modelled as
completely specified functions of time.
Random signal: Random signal is a signal about which there is uncertainty before its actual occurrence. Such signal may be viewed as group of signals with each signal in the ensemble having different
wave forms
.(e.g.) The noise developed in a television or radio amplifier is an example for random signal.
6. What are energy and power signal? Ans:
Energy signal: signal is referred as an energy signal, if and only if the total energy of the signal satisfies the condition 0<E<∞. The total energy of the continuous time signal x(t) is given as
(t)dt, integration limit from –T/2 to +T/2
Power signal: signal is said to be powered signal if it satisfies the condition
0<P<∞. The average power of a continuous time signal is given by
P=limT→∞1/T∫x2(t)dt, integration limit is from-T/2 to +T/2.
7. What are the operations performed on a signal? Ans:
Operations performed on dependent variables:
Amplitude scaling: y (t) =cx (t), where c is the scaling factor, x(t) is the continuous time signal.
Addition: y (t)=x1(t)+x2(t) Multiplication y (t)=x1(t)x2(t)
Differentiation: y (t)=d/dt x(t) Integration (t) =∫x(t)dt
Operations performed on independent variables: Time shifting
Amplitude scaling
Time reversal
8. What are elementary signals and name them? Ans:
The elementary signals serve as a building block for the construction of more complex signals. They are also important in their own right, in that they may be used to model many physical signals that
occur in nature.
There are five elementary signals. They are as follows,
Unit step function Unit impulse function Ramp function Exponential function Sinusoidal function
9. What are the properties of a system? Ans:
A system is said to be stable if the input x(t) satisfies the condition(t)₃≤Mx<∞
and the output satisfies the condition ₃y(t)₃≤My<∞ for all t.
A system is said to be memory if the output signal depends on the present and
the past inputs.
A system is said to be invertible if the input of the system can be recovered from
the system output.
Time invariance:
A system is said to be time invariant if a time delay or advance of the input
signal leads to an identical time shift in the output signal.
A system is said to be linear if it satisfies the super position
Principle i.e.) R(ax1(t)+bx2(t))=ax1(t)+bx2(t)
10. What is memory system and memory less system? Ans:
A system is said to be memory system if its output signal at any time depends on the past values of the input signal. Circuit with inductors capacitors are examples of memory system..
A system is said to be memory less system if the output at any time depends on the present values of the input signal. An electronic circuit with resistors is an example for memory less system.
11. What is an invertible system? Ans:
A system is said to be invertible system if the input of the system can be recovered from the system output. The set of operations needed to recover the input as
the second system connected in cascade with the given system such that the output signal of the second system is equal to the input signal applied to the system.
12. What are time invariant systems? Ans:
A system is said to be time invariant system if a time delay or advance of the
input signal leads to an idenditical shift in the output signal. This implies that a time invariant system responds idenditically no matter when the input signal is applied.
It also satisfies the condition
13. Is a discrete time signal described by the input output relation y[n]= rnx[n] time invariant.
A signal is said to be time invariant if R{x[n-k]}= y[n-k] R{x [n-k]} =R(x[n]) / x[n]→x[n-k]
=rnx [n-k] ---------------- (1)
Y [n-k] =y[n] / n→n-k
=rn-kx [n-k] ------------------- (2)
Equations (1) ≠Equation (2)
Hence the signal is time variant.
14. Show that the discrete time system described by the input-output relationship
Y[n] = nx[n] is linear?
For a sys to be linear R{a1x1[n]+b1x2[n]}=a1y1[n]+b1y2[n] L.H.S:R{ a1x1[n]+b1x2[n] }=R{x[n]} /x[n] → a1x1[n]+b1x2[n]
= a1 nx1[n]+b1 nx2[n] ------------------- (1)
R.H.S: a1y1[n]+b1y2[n]= a1 nx1[n]+b1 nx2[n]-------------------- (2)
Equation(1)=Equation(2) Hence the system is linear
15. What is SISO system and MIMO system? Ans:
A control system with single input and single output is referred to as single input single output system. When the number of plant inputs or the number of plant outputs is more than one the system is
referred to as multiple input output system. In both the case, the controller may be in the form of a digital computer or microprocessor in which we can speak of the digital control systems.
16. What is the output of the system with system function H1 and H2 when connected in cascade and parallel?
When the system with input x(t) is connected in cascade with the system H1 and
H2 the output of the system is
When the system is connected in parallel the output of the system is given by y(t)=H1x1(t)+H2x2(t).
17. What do you mean by periodic and non-periodic signals?
A signal is said to be periodic if x(n+N)=x(n)
Where N is the time period.
A signal is said to be non- periodic if x(n+N)≠x(n)
18. Determine the convolution sum of two sequences x (n) = {3, 2, 1, 2} and
h (n) = {1, 2, 1, 2} Ans:
y(n) = {3,8,8,12,9,4,4}
19. Find the convolution of the signals x(n) = 1 n=-2,0,1
= 2 n=-1
= 0 elsewhere.
y(n) = {1,1,0,1,-2,0,-1}
20. Detemine the solution of the difference equation
y(n) = 5/6 y(n-1) – 1/6 y(n-2) + x(n) for x(n) = 2n u(n)
y(n) = -(1/2)n u(n) + 2/3(1/3)n u(n)+8/5 2nu(n)
21. Determine the response y(n), n>=0 of the system described by the second order difference equation y(n) – 4y(n-1) + 4y(n-2) = x(n) – x(n-1) when the input is x(n) = (-1)n u(n) and the initial
condition are y(-1) = y(-2)=1.
y(n) = (7/9-5/3n)2n u(n) +2/9(-1)n u(n)
22. Differentiate DTFT and DFT Ans:
DTFT output is continuous in time where as DFT output is Discrete in time.
23. Differentiate between DIT and DIF algorithm
DIT – Time is decimated and input is bi reversed format output in natural order
DIF – Frequency is decimated and input is natural order output is bit reversed format.
24. How many stages are there for 8 point DFT? Ans:
25. How many multiplication terms are required for doing DFT by expressional method and FFT METHOD?
Expression –n2 FFT N /2 log N
UNIT - II
26. Distinguish IIR and FIR filters
FIR IIR
Impulse response is finite Impulse Response is infinite
They have perfect linear phase They do not have perfect linear phase
Non recursive Recursive
Greater flexibility to control the shape of magnitude response Less flexibility
27. Distinguish analog and digital filters
Analog Digital
Constructed using active or Consists of elements like adder,
passive components and it is described by a differential subtractor and delay units and it is described by a difference equation
Frequency response can be changed by changing the Frequency response can be changed by changing the filter
components coefficients
It processes and generates analog output Processes and generates digital output
Output varies due to external conditions Not influenced by external conditions
28. Write the expression for order of Butterworth filter? Ans:
The expression is N=log (λ /€) 1/2/log (1/k) ½
29. Write the expression for the order of chebyshev filter? Ans:
N=cosh-1(λ /e)/cosh-1(1/k)
30. Write the various frequency transformations in analog domain? Ans:
LPF to LPF:s=s/♠c
LPF to HPF:s=♠c/s
LPF to BPF:s=s2xlxu/s(xu-xl)
LPF to BSF:s=s(xu-xl)?s2=xlxu. X=♠
31. Write the steps in designing chebyshev filter?
Find the order of the filter.
Find the value of major and minor axis. λ
Calculate the poles.
Find the denominator function using the above poles.
The numerator polynomial value depends on the value of n. If n is odd: put s=0 in the denominator polynomial.
If n is even put s=0 and divide it by (1+e2)1/2
32. Write down the steps for designing a Butterworth filter? Ans:
1. From the given specifications find the order of the
2. Find the transfer function from the value of N
3. Find ♠c
4. Find the transfer function ha(s) for the above value of ♠c by su s by that value.
33. State the equation for finding the poles in chebyshev filter.
sk=acos¢k+jbsin¢k,where ¢k=∏/2+(2k-1)/2n)∏
34. State the steps to design digital IIR filter using bilinear method
Substitute s by 2/T (z-1/z+1), where T=2/Ώ (tan (w/2) in h(s) to get h (z)
35. What is warping effect? Ans:
For smaller values of w there exist linear relationship between w and .but for larger values of w the relationship is nonlinear. This introduces distortion in the frequency axis. This effect
compresses the magnitude and phase response. This effect is called warping effect
36. Write a note on prewarping. Ans:
The effect of the non linear compression at high frequencies can be compensated. When the desired magnitude response is piecewise constant over frequency, this compression can be compensated by
introducing a suitable rescaling or prewar ping the
critical frequencies.
37. Give the bilinear transform equation between s plane and z plane.
s=2/T (z-1/z+1)
38. Why impulse invariant method is not preferred in the design of IIR filters other than low pass filter?
In this method the mapping from s plane to z plane is many to one. Thus there are an infinite number of poles that map to the same location in the z plane, producing an aliasing effect. It is
inappropriate in designing high pass filters. Therefore this method is not much preferred.
39. By impulse invariant method obtain the digital filter transfer function and the differential equation of the analog filter h(s) =1/s+1
H (z) =1/1-e-Tz-1
Y/x(s) =1/s+1
Cross multiplying and taking inverse lap lace we get, D/dt(y(t)+y(t)=x(t)
40.What is meant by impulse invariant method?
In this method of digitizing an analog filter, the impulse response of the resulting digital filter is a sampled version of the impulse response of the analog filter. For e.g. if the transfer
function is of the form, 1/s-p, then
H (z) =1/1-e pTz-1
41.What do you understand by backward difference?
One of the simplest methods of converting analog to digital filter is to approximate the differential equation by an equivalent difference equation.
42. What are the properties of chebyshev filter?
1. The magnitude response of the chebyshev filter exhibits ripple either in the stop band or the pass band.
2. The poles of this filter lies on the ellipse
43. Give the Butterworth filter transfer function and its magnitude characteristics for different orders of filter.
The transfer function of the Butterworth filter is given by
H (jΏ) =1/1+j (Ώ/Ώc) N
44.Give the magnitude function of Butterworth filter.
The magnitude function of Butterworth filter is
2N 1/2
|h(jΏ)=1/[1+(Ώ/Ώc) ]
45. Give the equation for the order N, major, minor axis of an ellipse in case of chebyshev filter?
The order is given by N=cosh-1(( .1αp
A= ( 1/N-µ
B=Ωp ( 1/N+ µ
46.Give the expression for poles and zeroes of a chebyshev type 2 filters
The zeroes of chebyshev type 2 filter SK=jΏs/sink₃k, k=1….N
The poles of this filter xk+jyk
xk= Ώsσk/ Ώ 2+σk2
yk=ΏsΏk/ Ώs2+σk2 σk=acos₃k
47.How can you design a digital filter from analog filter?
Digital filter can de designed from analog filter using the following methods
1. Approximation of derivatives
2. Impulse invariant method
3. Bilinear transformation
49. Differentiate Butterworth and Chebyshev filter.
Butterworth dampimg factor 1.44 chebyshev 1.06
Butterworth flat response damped response.
50.What is filter?
Filter is a frequency selective device , which amplify particular range of frequencies and attenuate particular range of frequencies.
51.What are the types of digital filter according to their impulse response?
IIR (Infinite impulse response) filter
FIR (Finite Impulse Response) filter.
52.How phase distortion and delay distortion are introduced?
The phase distortion is introduced when the phase characteristics of a filter is nonlinear within the desired frequency band.
The delay distortion is introduced when the delay is not constant within the desired frequency band.
53. What is mean by FIR filter?
The filter designed by selecting finite number of samples of impulse response (h(n) obtained from inverse fourier transform of desired frequency response H(w)) are called FIR filter
54.Write the steps involved in FIR filter design
Choose the desired frequency response Hd(w) Take the inverse fourier transform and obtain Hd(n)
Convert the infinite duration sequence Hd(n) to h(n) Take Z transform of h(n) to get H(Z)
55.What are advantages of FIR filter?
Linear phase FIR filter can be easily designed .
Efficient realization of FIR filter exists as both recursive and non-recursive structures.
FIR filter realized non-recursively stable.
The round off noise can be made small in non recursive realization of FIR filter.
56. What are the disadvantages of FIR FILTER
The duration of impulse response should be large to realize sharp cutoff filters. The non integral delay can lead to problems in some signal processing
57. What is the necessary and sufficient condition for the linear phase characteristic of a FIR filter?
The phase function should be a linear function of w, which inturn requires constant group delay and phase delay.
58.List the well known design technique for linear phase FIR filter design?
Fourier series method and window method
Frequency sampling method.
Optimal filter design method.
59. Define IIR filter?
The filter designed by considering all the infinite samples of impulse response are called IIR filter.
60. For what kind of application , the antisymmetrical impulse response can be used?
The ant symmetrical impulse response can be used to design Hilbert transforms and differentiators.
61. for what kind of application, the symmetrical impulse response can be used?
The impulse response, which is symmetric having odd number of samples can be used to design all types of filters, i.e, lowpass, highpass, bandpass and band reject. The symmetric impulse response
having even number of samples can be used to design lo pass and bandpass filter.
62. What is the reason that FIR filter is always stable?
FIR filter is always stable because all its poles are at the origin.
63. What condition on the FIR sequence h(n) are to be imposed n order that this filter can be called a liner phase filter?
The conditions are
o Symmetric condition h(n)=h(N-1-n)
o Antisymmetric condition h(n)=-h(N-1-n)
64. Under what conditions a finite duration sequence h(n) will yield constant group
delay in its frequency response characteristics and not the phase delay?
If the impulse response is anti symmetrical ,satisfying the condition
The frequyncy response of FIR filter will have constant group delay and not the phase delay.
65. State the condition for a digital filter to be causal and stable?
A digital filter is causal if its impulse response h (n) =0 for n<0.
A digital filter is stable if its impulse response is absolutely summable.
66. What are the properties of FIR filter?
FIR filter is always stable.
A realizable filter can always be obtained. FIR filter has a linear phase response.
67. When cascade from realization is preferred in FIR filters?
The cascade from realization is preferred when complex zeros with absolute magnitude less than one.
68.What are the disadvantage of Fourier series method ?
In designing FIR filter using Fourier series method the infinite duration impulse response is truncated at n= (N-1/2).Direct truncation of the series will lead to fixed percentage overshoots and
undershoots before and after an approximated discontinuity in the frequency response .
69. What is Gibbs phenomena?
What are Gibbs oscillations?
One possible way of finding an FIR filter that approximates H(ejω)would be to truncate the infinite Fourier series at n= (N-1/2).Abrupt truncation of the series will lead to oscillation both in pass
band and is stop band .This phenomenon is known as Gibbs phenomenon.
70. What are the desirable characteristics of the windows?
The desirable characteristics’ of the window are
The central lobe of the frequency response of the window should contain most of the energy and should be narrow.
The highest side lobe level of the frequency response should be small. The side’s lobes of the frequency response should decrease in energy
rapidly as ω tends to π.
71. Compare Hamming window with Kaiser Window.
Hamming window Kaiser window
1.The main lobe width is equal to8π/N The main lobe width ,the peak side lobe level can be varied by varying the
and the peak side lobe level is –41dB. parameter α and N.
2.The low pass FIR filter designed will have first side lobe peak of –53 dB The side lobe peak can be varied by varying the parameter α.
72.What is the necessary and sufficient condition for linear phase characteristics in FIR filter?
The necessary and sufficient condition for linear phase characteristics in FIR filter is the impulse response h(n) of the system should have the symmetry property,i.e,
H(n) = h(N-1-n)
Where N is the duration of the sequence.
73. What are the advantage of Kaiser widow?
1. It provides flexibility for the designer to select the side lobe level and N.
2. It has the attractive property that the side lobe level can be varied
Continuously from the low value in the Blackman window to the high value in the rectangle window.
74.What is the principle of designing FIR filter using frequency sampling method?
In frequency sampling method the desired magnitude response is sampled and a linear phase response is specified .The samples of desired frequency response are defined as DFT coefficients. The filter
coefficients are then determined as the IDFT of this set of samples.
75. For what type of filters frequency sampling method is suitable?
Frequency sampling method is attractive for narrow band frequency selective filters where only a few of the samples of the frequency response are non-zero.
76.What is meant by autocorrelation?
The autocorrelation of a sequence is the correlation of a sequence with its shifted version, and this indicates how fast the signal changes.
UNIT -IV
77.Define white noise?
A stationary random process is said to be white noise if its power density
Spectrum is constant. Hence the white noise has flat frequency response
spectrum. SX(w) = σx
, -π ≤ wπ
78.what do you understand by a fixed-point number?
In fixed point arithmetic the position of the binary point is fixed. The bit to the right represents the fractional part of the number & those to the left represent the integer part. For example, the
binary number 01.1100 has the value 1.75 in decimal.
79.What is the objective of spectrum estimation?
The main objective of spectrum estimation is the determination of the power spectral density of a random process. The estimated PSD provides information about the structure of the random process
which can be used for modeling, prediction or filtering of the deserved process.
80.List out the addressing modes supported by C5X processors?
1. Direct addressing
2. Indirect addressing
3. Immediate addressing
4. Dedicated-register addressing
5. Memory-mapped register addressing
6. Circular addressing
81.what is meant by block floating point representation? What are its advantages?
In block point arithmetic the set of signals to be handled is divided into blocks. Each block has the same value for the exponent. The arithmetic operations within the block uses fixed point
arithmetic & only one exponent per block is stored thus saving memory. This representation of numbers is more suitable in certain FFT flow graph & in digital audio applications.
82.what are the advantages of floating point arithmetic?
1. Large dynamic range
2. Over flow in floating point representation is unlike.
83.what are the three-quantization errors to finite word length registers in digital filters?
1. Input quantization error
2. Coefficient quantization error
3. Product quantization error
84.How the multiplication & addition are carried out in floating point arithmetic?
In floating point arithmetic, multiplication are carried out as follows, Let f1 = M1*2c1 and f2 = M2*2c2. Then f3 = f1*f2 = (M1*M2) 2(c1+c2)
That is, mantissa is multiplied using fixed-point arithmetic and the exponents are added.
The sum of two floating-point numbers is carried out by shifting the bits of the mantissa
Of the smaller number to the right until the exponents of the two numbers are equal and then adding the mantissas.
85.What do you understand by input quantization error? Ans:
In digital signal processing, the continuous time input signals are converted into digital using a b-bit ACD. The representation of continuous signal amplitude by a fixed digit produce an error,
which is known as input quantization error.
86.List the on-chip peripherals in 5X. Ans:
The C5X DSP on-chip peripherals available are as follows:
1. Clock Generator
2. Hardware Timer
3. Software-Programmable Wait-State Generators
4. Parallel I/O Ports
5. Host Port Interface (HPI)
6. Serial Port
7. Buffered Serial Port (BSP)
8. Time-Division Multiplexed (TDM) Serial Port
9. User-Maskable Interrupts
87.what is the relationship between truncation error e and the bits b for representing a decimal into binary?
For a 2's complement representation, the error due to truncation for both positive and negative values of x is 0>=xt-x>-2-b
Where b is the number of bits and xt is the truncated value of x.
The equation holds good for both sign magnitude, 1's complement if x>0
If x<0, then for sign magnitude and for 1's complement the truncation error satisfies.
88.what is meant rounding? Discuss its effect on all types of number representation?
Rounding a number to b bits is accomplished by choosing the rounded result as the b bit number closest to the original number unrounded.
For fixed point arithmetic, the error made by rounding a number to b bits satisfy the
-2-b 2-b
-----<=xt-x<= --------
For all three types of number systems, i.e., 2's complement, 1's complement & sign magnitude.
For floating point number the error made by rounding a number to b bits satisfy the inequality
-2-b<=E<=2-b where E=xt-x
-------- x
89.what is meant by A/D conversion noise?
A DSP contains a device, A/D converter that operates on the analog input x(t) to produce xq(t) which is binary sequence of 0s and 1s.
At first the signal x(t) is sampled at regular intervals to produce a sequence x(n) is of infinite precision. Each sample x(n) is expressed in terms of a finite number of bits given the sequence xq
(n). The difference signal e(n)=xq(n)-x(n) is called A/D conversion noise.
90.what is the effect of quantization on pole location?
Quantization of coefficients in digital filters lead to slight changes in their value. These
changes in value of filter coefficients modify the pole-zero locations. Sometimes the pole locations will be changed in such a way that the system may drive into instability.
91.which realization is less sensitive to the process of quantization?
Ans: Cascade form.
92.what is meant by quantization step size?
Let us assume a sinusoidal signal varying between +1 and -1 having a dynamic range 2. If the ADC used to convert the sinusoidal signal employs b+1 bits including sign bit, the number of levels
available for quantizing x(n) is 2b+1. Thus the interval between successive levels
q= 2 =2-b
Where q is known as quantization step size.
93.How would you relate the steady-state noise power due to quantization and the b bits representing the binary sequence?
Steady state noise power
Where b is the number of bits excluding sign bit.
94.what is overflow oscillation?
The addition of two fixed-point arithmetic numbers cause over flow the sum exceeds the word size available to store the sum. This overflow caused by adder make the filter output to oscillate between
maximum amplitude limits. Such limit cycles have been referred to as over flow oscillations.
95.what are the methods used to prevent overflow?
There are two methods used to prevent overflow
1. Saturation arithmetic 2. Scaling
96.what are the two kinds of limit cycle behavior in DSP?
1.zero input limit cycle oscillations
2.Overflow limit cycle oscillations
97.Determine "dead band" of the filter
The limit cycle occur as a result of quantization effect in multiplication. The amplitudes of the output during a limit cycle are confined to a range of values called the dead band of the filter.
98.Explain briefly the need for scaling in the digital filter implementation.
To prevent overflow, the signal level at certain points in the digital filter must be scaled so that no overflow occurs in the adder.
UNIT- V
99. List down the various advantages of multirate signal processing.
a) Computational requirements is less. b) Storage for filter coefficient is less. c) Finite arithmetic effect is less.
d) Filter order required in multirate application is low.
100. Define sampling theorem.
According to the sampling theorem, a band limited signal x(t) having finite energy, which has no frequency comp[onents higher than fh hertz, can be completely reconstructed from its samples taken at
the ratet of 2fh samples per sec. ( fs ≥ 2fh ).
fs --- samplinf frequency ; fh --- highest signal frequency.
101. what is multirate signal processing?
The theory of processing signals at different sampling rates is called multirate signal processing.
Part B UNIT-I
1. Determine the DFT of the sequence x(n) =1/4, for 0<=n <=2
0, otherwise
Ans: The N point DFT of the sequence x(n) is defined as
x(k)= ∑ x(n)e
x(n) = (1/4,1/4,1/4)
[1+2cos(2πk/3)] where k= 0,1,……….,N-1
2. Derive the DFT of the sample data sequence x(n) = {1,1,2,2,3,3}and compute the corresponding amplitude and phase spectrum.
Ans: The N point DFT of the sequence x(n) is defined as
X(k)= ∑ x(n)e
X(0) = 12
X(1) = -1.5 + j2.598
X(2) = -1.5 + j0.866
X(3) = 0
X(4) = -1.5 – j0.866
X(5) =-1.5-j2.598
X(k) = {12, -1.5 + j2.598, -1.5 + j0.866,0, -1.5 – j0.866, -1.5-j2.598}
∟X(k)={0,- π/3,- π/6,0, π/6, π/3}
3.Given x(n) = {0,1,2,3,4,5,6,7} find X(k) using DIT FFT
Ans: Given N = 8
WNk = e-j(2π/N)k
= 1
= -j
= -0.707-j0.707
Using butterfly diagram
X(k) = {28,-4+j9.656,-4+j4,-4+j1.656,-4,-4-j1.656,-4-j4,-4-j9.656}
4.Given X(k) = {28,-4+j9.656,-4+j4,-4+j1.656,-4,-4-j1.656,-4-j4,-4-j9.656} ,find x(n) using inverse DIT FFT algorithm.
WNk = ej(2π/N)k
W 0 = 1
W82 =0.707+j0.707
W83 = j
W8 = -0.707+j0.707
x(n) = {0,1,2,3,4,5,6,7}
5.Find the inverse DFT of X(k) = {1,2,3,4} Ans: The inverse DFT is defined as
x(n)=(1/N ) ∑ x(k)e
x(0) = 5/2
x(1) = -1/2-j1/2 x(2) = -1/2
x(3) = -1/2+j1/2
x(n) = {5/2, -1/2-j1/2, -1/2, -1/2+j1/2}
6.Design an ideal frequency response
0 otherwise
low pass filter with a
Hd(e jw) =1 for –∏/2<=w<=∏/2
Find the value of h(n) for N=11 find H(Z) plot magnitude response
a. Find h(n) by IDTFT
b. Convert h(n) in to a fine length by truncation
c. H(0)=1/2, h(1)=h(-1)=0.3183 h(2)=h(-2)=0 h(3)=h(-3)= -
0.106 h(4)=h(-4)=0 h(5)=h(-5)=0.06366
d. Find the transfer function H(Z) which is not realizable conver in to realizable by multiplying by z-(N-1/2)
e. H’(Z) obtained is 0.06366-0.106z-2+.3183Z-4+.5Z-5+.3183Z-6-.106Z-
f. Find H (e jw) and plot amplitude response curve.
7. Design an ideal low pass filter with a frequency response
Hd(e jw) =1 for – ∏/4<=|w|<=∏
0 otherwise
find the value of h(n) for N=11 find H(Z) plot magnitude response
g. Find h(n) by IDTFT
h. Convert h(n) in to a fine length by truncation i. H(0)=0.75
h(2)=h(-2)=-.159 h(3)=h(-3)= -
0.075 h(4)=h(-
4)=0 h(5)=h(-
j. Find the transfer function H(Z) which is not realizable conver in to realizable by multiplying by z-(N-1/2)
k. H’(Z) obtained is 0.045-0.075z-2 -.159 Z-3-0.22Z-4+0.75Z-5-.22Z-6 -0.159Z-7 -.
l. Find H (e jw) and plot amplitude response curve.
8. Design band pass filter with a frequency response
Hd(e jw) =1 for –∏/3<=|w|<=2∏/3
0 otherwise
find the value of h(n) for N=11 find H(Z) plot magnitude response
m. Find h(n) by IDTFT
n. Convert h(n) in to a fine length by truncation
o. Find the transfer function H(Z) which is not realizable conver in to realizable by multiplying by z-(N-1/2)
p. H’(Z) obtained Find H (e jw
and plot amplitude response curve.
9. Design band reject filter with a frequency response Hd(e
jw) =1 for ∏/4<=|w|<=3∏/4
0 otherwise
find the value of h(n) for N=11 find H(Z) plot magnitude response
q. Find h(n) by IDTFT
r. Convert h(n) in to a fine length by truncation
s. Find the transfer function H(Z) which is not realizable conver in to realizable by multiplying by z-(N-1/2)
t. H’(Z) obtained Find H (e jw
and plot amplitude response curve.
10. Derive the condition of FIR filter to be linear in phase.
Conditions are
Group delay and Phase delay should be constant
And show the condition is satisfied
11.Derive the expression for steady state I/P Noise Power and Steady state O/P Noise Power.
Write the derivation.
12 Draw the product quantatization model for first order and second order filter Write the difference equation and draw the noise model.
13.For the second order filter Draw the direct form II realization and find the scaling factor S0 to avoid over flow
Find the scaling factor from the formula
I= --------------------------------------- (1-r2)(1-2r2cos2ø =r4)
14 Explain Briefly about various number representation in digital computer.
1 Fixed point
2 Floating point
3 Block floating point
Signed magnitude represenation
1’s Complement
2’s Complement
15. Consider the transfer function H(Z)=H1(Z)H2(Z) where H1(Z) =1/1- a1Z-1 H2(z) =1/ 1-a2Z-1
Find the o/p Round of noise power Assume a1=0.5 and a2= 0.6 and find o.p round off noise power.
Draw the round of Noise Model. By using residue method find σ01
By using residue method find σ
02 = σ 01 2+ σ 02 2
2 –2b (5.43)
17.what is meant by A/D conversion noise. Explain in detail?
A DSP contains a device, A/D converter that operates on the analog input x(t)
to produce xq(t) which is binary sequence of 0s and 1s.
At first the signal x(t) is sampled at regular intervals to produce a sequence x(n) is of infinite precision. Each sample x(n) is expressed in terms of a finite number of bits given the sequence xq
(n). The difference signal e(n)=xq(n)-x(n) is called A/D conversion noise.
+ derivation.
18.Consider the transfer function H(Z)=H1(Z)H2(Z) where H1(Z) =1/1- a1Z-1 H2(z) =1/ 1-a2Z-1
Find the o/p Round of noise power Assume a1=0.7 and a2= 0.8and find o.p round off noise power.
19.Given X(k) = {1,1,1,1,1,1,1,1,} ,find x(n) using inverse DIT FFT algorithm.
WNk = ej(2π/N)k
Find x(n)
20.Find the inverse DFT of X(k) = {3,4,5,6} Ans: The inverse DFT is defined as
x(n)=(1/N ) ∑ x(k)e
21. Derive the expression for steady state I/P Noise Variance and Steady state
O/P Noise Variance and Write the derivation also.
22. a) what do you mean by down sampling?
b) Obtain the spectrum (expression) of the down sampled signal.
c) Plot the spectra of any signal x(n) and its down sampled version.
23. a) discuss on efficient transversal structure for decimator and interpolator. b) Describe the features of QMF bank.
24. Describes and derive sampling rate conversion by a rational factor I/D in multirate signal processing.
SRINIVASAN ENGINEERING COLLEGE, PERAMBALUR- 621212
SRINIVASAN ENGINEERING COLLEGE, PERAMBALUR- 621212
SRINIVASAN ENGINEERING COLLEGE, PERAMBALUR- 621212
1 comment:
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Multilevel Optimization Modeling for Risk-Averse Stochastic Programming
Coherent risk measures have become a popular tool for incorporating risk aversion into stochastic optimization models. For dynamic models in which uncertainty is resolved at more than one stage,
however, using coherent risk measures within a standard single-level optimization framework becomes problematic. To avoid severe time-consistency difficulties, the current state of the art is to
employ risk measures of a specific nested form, which unfortunately have some undesirable and somewhat counterintuitive modeling properties. For one thing, this technique requires increasing risk
aversion as risks and reward recede in time. Further, it produces objective functions that cannot be law invariant with respect to the total incurred costs and rewards, meaning that two solutions
with identical probability distributions of final wealth may be assigned different levels of risk, and the nested form of the objective function cannot be simplified. These properties deter practical
acceptance of such models, and are particularly undesirable for situations with close final time horizons. This paper summarizes these issues and then presents an alternative multilevel optimization
modeling approach that enforces a form of time consistency through constraints rather than by restricting the modeler's choice of objective function. This technique leads to models that are
time-consistent even while using time-inconsistent risk measures, and can easily be formulated to be law invariant with respect to the final wealth if so desired. We argue that this approach should
be the starting point for all multistage optimization modeling. When used with time-consistent objective functions, we show its multilevel optimization constraints become redundant and the associated
models thus simplify to a more familiar single-objective form. Unfortunately, this paper also shows that its proposed approach leads to NP-hard models, even in the simplest imaginable setting in
which it would be needed: three-stage linear problems on a finite probability space, using the standard mean-semideviation and average-value-at-risk measures. Finally, we show that for a simple but
reasonably realistic test application, that the kind of models we propose, while being drawn from an NP-hard family and certainly more time consuming to solve than those obtained from the
nested-objective approach, are readily solvable to global optimality using a standard commercial MILP solver. Therefore, there seems some promise of the modeling approach being useful despite its
computational complexity properties.
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Herd immunity and measles: why we should aim for 100% vaccination coverage - Gagon Family Medicine
Herd immunity and measles: why we should aim for 100% vaccination coverage
The measles outbreak traced back to Disneyland has spread to eight states, with as many as 95 cases reported by January 28. Media outlets are highlighting the rise of anti-vaccination sentiments.
Scientists are expressing their dismay at people who reject sound medical advice and put their families and communities in harm’s way.
Measles was considered eliminated in the United States in 2000. But if the first month of 2015 is any indication, this year will easily beat the record number of measles cases recorded in 2014.
The narrative during this outbreak, or any measles outbreak really, is that measles is a highly transmissible disease. So transmissible in fact that 90-95% of people must be vaccinated in order to
protect the entire population, or achieve what is called herd immunity.
That is partly true. Measles is highly transmissible, not least because people can be contagious days before symptoms develop. But there are three problems with this line of reasoning about
vaccination rates.
First, the numbers are based on calculations that assume a world of random mixing. Second, the vaccination coverage is not a perfect measure of immunity in the population. Third, and most problematic
in my view, it gives people a seemingly scientific justification for not getting vaccinated – after all, if not everyone needs to get vaccinated in order to attain herd immunity, can it really be so
bad if I opt out of it?
What exactly is herd immunity?
Let’s look at the concept of herd immunity first. The basic idea is that a group (the “herd”) can avoid exposure to a disease by ensuring that enough people are immune so that no sustained chains of
transmission can be established. This protects an entire population, especially those who are too young or too sick to be vaccinated. But how many people need to be immune to achieve this?
In order to calculate the number of people who need to be immune for herd immunity to be effective, we need to know how many people will get infected, on average, by an infectious person.
Imagine that a newly infected person will on average pass on the disease to two other people. Those two will each infect another two people, who will themselves pass it on, and so on, resulting in
the classical pattern of an exponentially growing outbreak.
Marcel Salathé, Author provided
In order to stop the growth in the number of transmissions, we need to ensure that each individual case causes, on average, less than one new infection. So, let’s say that one case leads on average
to two more infections, but instead we want that number to be less than one. That means at least 50% of the population needs to be immune, so that at most, only one of the two people who might have
been infected by an individual will be.
Marcel Salathé
How many people need to get vaccinated to achieve herd immunity?
So, how do we calculate what fraction of a population needs to be immune to reach herd immunity? First, we need to know what the reproduction number, or R, is. That’s how many new cases a single case
of an infection will cause.
Imagine that you are infected in a completely susceptible population, and you pass on the infection to five other people (ie R=5). In order to prevent an outbreak, at least four out of those five
people, or 80% of the population in general, should be immune. Put differently, 20% of the population may remain individually susceptible, but the population would still remain protected.
So if you can estimate the reproduction number for a given disease, you can calculate the fraction of the population that needs to be immune in order to attain herd immunity.
For influenza and Ebola, the number R is about two. For polio and smallpox, it is around five to eight. But for measles it is much higher, somewhere between 10 and 20. And because of that, the goal
for measles vaccination coverage is typically around 90-95% of a population.
But there’s a problem with this calculation.
The population is not random
The assumption underlying the calculation for herd immunity is that people are mixing randomly, and that vaccination is distributed equally among the population. But that is not true. As the
Disneyland measles outbreak has demonstrated, there are communities whose members are much more likely to refuse vaccination than others.
Geographically, vaccination coverage is highly variable on the level of states, counties, and even schools. We’re fairly certain that opinions and sentiments about vaccination can spread in
communities, which may in turn lead to polarized communities with respect to vaccination.
And media messages, especially from social media, may make the problem worse. When we analyzed data from Twitter about sentiments on the influenza H1N1 vaccine during the swine flu pandemic in 2009,
we found that negative sentiments were more contagious than positive sentiments, and that positive messages may even have back-fired, triggering more negative responses.
And in measles outbreak after measles outbreak, we find that the vast majority of cases occurred in communities that had vaccination coverages that were way below average.
The sad truth is this: as long as there are communities that harbor strong negative views about vaccination, there will be outbreaks of vaccine-preventable diseases in those communities. These
outbreaks will happen even if the population as a whole has achieved the vaccination coverage considered sufficient for herd immunity.
Marcel Salathé
If negative vaccination sentiments become more popular in the rest of the population as well, we may start to see more sustained transmission chains. Once those chains are sufficiently frequent to
connect under-vaccinated communities, we may again be in a situation of endemic measles.
The solution often proposed is that we should do a better job of convincing people that vaccines are safe. I’m all for it. But I would also suggest that we should stop basing our vaccination policies
on models that made sense in a world of constrained vaccine supply, and aim for 100% vaccination coverage among those who can get vaccinated.
This would also solve another problem, often glanced over: There are many people who cannot get vaccinated for medical reasons, either because they are too young, or because they have other
conditions that prevent them from acquiring immunity through vaccination.
Herd immunity against measles requires that 90-95% of the entire population are immune, whereas vaccination coverage is measured as the percentage vaccinated of the targetpopulation – which only
includes people who are eligible for vaccination. This means that to achieve 95% immunity in the population for measles, vaccination coverage needs to be higher than 95%. This is the scientific
argument for a public health policy that aims at 100% vaccination coverage.
More importantly, there is an ethical argument to be made for the goal of 100% vaccination coverage. It sends the right message. Everyone who can get vaccinated, should get vaccinated – not only to
protect themselves, but to protect those who can’t, through herd immunity.
https://gagonfamilymedicine.com/wp-content/uploads/2017/09/Herd-immunity-and-measles-why-we-should-aim-for-100-vaccination-coverage.jpg 350 850 gagonfp https://gagonfamilymedicine.com/wp-content/
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Find the second biggest element in an unsorted array - Single Traversal - sneppets
Find the second biggest element in an unsorted array – Single Traversal
Given an input array of integers, your goal is to find the second biggest element present in the array in an effective way. Note, you need to achieve this in O(n) time i.e., you should be able to
find result by traversing through the array only once.
Below is the efficient algorithm for doing this
1. Declare/provide input array
2. Initialize two variables firstBig and secondBig with Integer.MIN_VALUE
3. Input array validation: Check if atleast two elements present in array.
4. For loop: Start Traversing the array - single traversal
4.1 Check if the current element is greater than the firstBig, if yes then set secondBig to firstBig (secondBig = firstBig) and set firstBig to current element (firstBig = current element)
4.2 Check if current element is NOT greater than firstBig, then check if that is greater than secondBig and NOT EQUAL to firstBig, if yes then set secondBig to current element (secondBig = current element)
5. Print the value stored in the secondBig variable, which is the second biggest element in the array.
Solution to find second biggest:
package com.sneppets.dsalgo.examples;
* Java Program to find the second biggest element in an unsorted array of integers in single traversal
* @author sneppets.com
public class SecondBiggestInArray {
public static void main (String[] args)
int inArray[] = {2, 4, 7, 6, 5, 1, 3, 10};
int arraySize = inArray.length;
findSecondBiggest(inArray, arraySize);
private static void findSecondBiggest(int[] inArray, int arraySize) {
//Initialize two variables with Integer.MIN_VALUE
int firstBig = Integer.MIN_VALUE;
int secondBig = Integer.MIN_VALUE;
//Check if atleast two elements present in array.
if(arraySize < 2)
System.out.println(" There should be atleast
two elements in arrays. Invalid input !!");
//Traversing the array
for (int i=0; i< arraySize; i++)
//if the current element is greater than the firstBig
if(inArray[i] > firstBig)
//set secondBig to firstBig
secondBig = firstBig;
//set firstBig to current element.
firstBig = inArray[i];
//if current element is NOT greater than firstBig,
//then check if that is greater than secondBig and NOT EQUAL to firstBig
else if (inArray[i] > secondBig && inArray[i] !=firstBig)
//set secondBig to current element
secondBig = inArray[i];
System.out.println("Second biggest element
in the input array is: " + secondBig);
Second biggest element in the input array is: 7
Time Complexity – Single Traversal – O(n)
Space complexity – O(1)
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3 Times Table Multiplication Chart – Times Table Club
3 Times Table
3 Times Table is the multiplication table of the number 3 provided here for students, parents and teachers. These times tables are helpful in solving math questions quickly and easily.
How to read the 3 times table?
One times three is 3, two times three is 6, three times three is 9, ect.
How to memorise the 3 times tables orally?
Write the 3 times tables on a sheet of paper and read them aload repeatedly.
What is an example math question using the 3 times tables?
Maths Question: A boy eats 3 bananas every day. How many bananas will the boy eat in 1 week (7 days)?
Solution: The customer eats 3 bananas per day. Therefore, using the 3 times table, the total number of bananas eaten by the boy in 1 week is 3 × 2 = 14 bananas.
│Tables 1 to 20 │Multiplication Table │ | {"url":"https://timestableclub.com/3-times-table/","timestamp":"2024-11-03T03:43:54Z","content_type":"text/html","content_length":"35939","record_id":"<urn:uuid:0683fcf2-707d-4d20-bdcc-e7e0a54d6871>","cc-path":"CC-MAIN-2024-46/segments/1730477027770.74/warc/CC-MAIN-20241103022018-20241103052018-00024.warc.gz"} |
INDI-V: Various measures that quantify the contribution of individual independent disease risk variants
Measures of the Contribution of Independent Risk Variants to Disease
Input Variables
Output Variables
Alert warning = 1
\(Note:\) Some combinations of K, RAF, RBb and RBBB generate non-estimable results. This usually occurs when RRBB is large, which is usually when RAF is small. If RAF is small, homozygotes of the
risk allele are rare and the estimate of RRBB will have wide confidence intervals. Under these circumstances the equation that calculates kbb fails the assumptions of the multiplicative risk model
and the probability of disease for risk allele homozygotes, kbb*RRBB, is estimated to be greater than 1. When this occurs we fudge the calculations by reducing RRBB by iteratively reducing RRBB, in
up to 10 steps to a minimum of RRBb, until an RRBB is reached that generates a kbb that satisfies kbb*RRBB .< 1. When Alert =1, RRBBcalc outputs the RRBB used in calculations. Since the risk allele
homozygotes are rare, the calculations are relatively insensitive to the value of RRBB. We output kbb, kbbRRBb and kbbRRBB (which is kbbRRBBcalc) to aid diagnostics, if these are negative or >1, then
there is an input problem and disregard the results.All calculations are underpinned by model assumptions- and acknowledge the old adage: that all models are wrong but some models are useful. | {"url":"https://shiny.cnsgenomics.com/INDI-V/","timestamp":"2024-11-02T15:44:14Z","content_type":"text/html","content_length":"12679","record_id":"<urn:uuid:34451b43-18ba-4a28-b079-8dd1a12ffb0c>","cc-path":"CC-MAIN-2024-46/segments/1730477027714.37/warc/CC-MAIN-20241102133748-20241102163748-00700.warc.gz"} |
Using the SUMIF Function in Excel - Guide & Example
Using the SUMIF Function in Excel – Guide & Example
[et_pb_section fb_built=”1″ _builder_version=”4.6.6″ _module_preset=”default”][et_pb_row _builder_version=”4.6.6″ _module_preset=”default”][et_pb_column type=”4_4″ _builder_version=”4.6.6″
_module_preset=”default”][et_pb_text content_last_edited=”off|desktop” _builder_version=”4.6.6″ _module_preset=”default” header_font=”Montserrat|800||on|||||” header_text_color=”#000000″]
Using the SUMIF Function in EXCEL – Guide & Example
[/et_pb_text][et_pb_text content_tablet=”
The SUMIF function in Excel is used to sum up a range of cells that fulfil certain criteria. It is a useful tool for summarizing and adding data based on specified conditions.
Earlier, we wrote about how to use the IF and SUM functions in Excel. The SUMIF function practically combines the two Excel functions so that you can add up values based on certain criteria or logic.
For instance, if you have a spreadsheet with a dataset that contains sales data across different regions and different periods, you can use the SUMIF function to find the total sales for a specific
region. We will come back to this example of using the SUMIF function later.
” content_phone=”
The SUMIF function in Excel is used to sum up a range of cells that fulfil certain criteria. It is a useful tool for summarizing and adding data based on specified conditions.
Earlier, we wrote about how to use the IF and SUM functions in Excel. The SUMIF function practically combines the two Excel functions so that you can add up values based on certain criteria or logic.
For instance, if you have a spreadsheet with a dataset that contains sales data across different regions and different periods, you can use the SUMIF function to find the total sales for a specific
region. We will come back to this example of using the SUMIF function later.
” content_last_edited=”on|desktop” _builder_version=”4.6.6″ _module_preset=”default” text_font=”Poppins|300|||||||” text_font_size=”16px” text_line_height=”1.8em”]
The SUMIF function in Excel is used to sum up a range of cells that fulfil certain criteria. It is a useful tool for summarizing and adding data based on specified conditions.
Earlier, we wrote about how to use the IF and SUM functions in Excel. The SUMIF function practically combines the two Excel functions so that you can add up values based on certain criteria or logic.
For instance, if you have a spreadsheet with a dataset that contains sales data across different regions and different periods, you can use the SUMIF function to find the total sales for a specific
region. We will come back to this example of using the SUMIF function later.
[/et_pb_text][et_pb_image src=”http://skillsharepk.com/wp-content/uploads/2023/01/SUMIF.jpg” alt=”Using the SUMIF Function” title_text=”Using the SUMIF Function” _builder_version=”4.6.6″
_module_preset=”default” border_width_all=”1px” border_color_all=”#d8d8d8″][/et_pb_image][/et_pb_column][/et_pb_row][et_pb_row _builder_version=”4.6.6″ _module_preset=”default”][et_pb_column type=
”4_4″ _builder_version=”4.6.6″ _module_preset=”default”][et_pb_text content_tablet=”
Using the SUMIF Function in Excel
Here is the syntax to use the SUMIF function:
%91su_note note_color=%22#b0def9%22%93=SUMIF(range, criteria, %91sum_range%93)%91/su_note%93
• range: the cells that you want to apply the criteria to.
• criteria: the logic or the criteria must meet to be included in the sum
• sum_range: an optional argument that specifies the range of cells to sum given the criteria specified earlier. If this argument is left blank, Excel will sum the cells in the range specified in
the range argument.
For example, suppose you have a list of values in the cell range A1:A10, and you want to add up only the values that are greater than 10. You can do that using the SUMIF formula below:
%91su_note note_color=%22#b0def9%22%93=SUMIF(A1:A10, %22>5%22)%91/su_note%93
This would sum up the values in A1:A10 that are greater than 10.
Moreover, the SUMIF function can also be used to sum up values based on a specific text. For example, suppose you have a list of names in column A and a list of numeric values in column B, and you
want to sum the values in column B for all the cells in column A that contain the name %22Muhammad%22. You could use the following formula:
%91su_note note_color=%22#b0def9%22%93=SUMIF(A1:A10, %22Muhammad%22, B1:B10)%91/su_note%93
This would sum the values in B1:B10 for all the cells in A1:A10 that contain the name %22Muhammad%22.
Remember, while entering the criteria, make sure that you put it in an enclosed bracket (“ ”)
” content_phone=”
Using the SUMIF Function in Excel
Here is the syntax to use the SUMIF function:
%91su_note note_color=%22#b0def9%22%93=SUMIF(range, criteria, %91sum_range%93)%91/su_note%93
• range: the cells that you want to apply the criteria to.
• criteria: the logic or the criteria must meet to be included in the sum
• sum_range: an optional argument that specifies the range of cells to sum given the criteria specified earlier. If this argument is left blank, Excel will sum the cells in the range specified in
the range argument.
For example, suppose you have a list of values in the cell range A1:A10, and you want to add up only the values that are greater than 10. You can do that using the SUMIF formula below:
%91su_note note_color=%22#b0def9%22%93=SUMIF(A1:A10, %22>5%22)%91/su_note%93
This would sum up the values in A1:A10 that are greater than 10.
Moreover, the SUMIF function can also be used to sum up values based on a specific text. For example, suppose you have a list of names in column A and a list of numeric values in column B, and you
want to sum the values in column B for all the cells in column A that contain the name %22Muhammad%22. You could use the following formula:
%91su_note note_color=%22#b0def9%22%93=SUMIF(A1:A10, %22Muhammad%22, B1:B10)%91/su_note%93
This would sum the values in B1:B10 for all the cells in A1:A10 that contain the name %22Muhammad%22.
Remember, while entering the criteria, make sure that you put it in an enclosed bracket (“ ”)
” content_last_edited=”on|desktop” _builder_version=”4.6.6″ _module_preset=”default” text_font=”Poppins|300|||||||” text_font_size=”16px” text_line_height=”1.8em” header_font=”Poppins|800|||||||”
header_2_font=”Montserrat|800||on|||||” header_2_text_color=”#000000″ header_3_font=”Poppins|800|||||||” header_3_font_size=”16px” header_2_font_tablet=”” header_2_font_phone=””
header_2_font_last_edited=”on|phone” header_3_font_tablet=”” header_3_font_phone=”” header_3_font_last_edited=”on|tablet”]
Using the SUMIF Function in Excel
Here is the syntax to use the SUMIF function:
[su_note note_color=”#b0def9″]=SUMIF(range, criteria, [sum_range])[/su_note]
• range: the cells that you want to apply the criteria to.
• criteria: the logic or the criteria must meet to be included in the sum
• sum_range: an optional argument that specifies the range of cells to sum given the criteria specified earlier. If this argument is left blank, Excel will sum the cells in the range specified in
the range argument.
For example, suppose you have a list of values in the cell range A1:A10, and you want to add up only the values that are greater than 10. You can do that using the SUMIF formula below:
[su_note note_color=”#b0def9″]=SUMIF(A1:A10, “>5”)[/su_note]
This would sum up the values in A1:A10 that are greater than 10.
Moreover, the SUMIF function can also be used to sum up values based on a specific text. For example, suppose you have a list of names in column A and a list of numeric values in column B, and you
want to sum the values in column B for all the cells in column A that contain the name “Muhammad”. You could use the following formula:
[su_note note_color=”#b0def9″]=SUMIF(A1:A10, “Muhammad”, B1:B10)[/su_note]
This would sum the values in B1:B10 for all the cells in A1:A10 that contain the name “Muhammad“.
Remember, while entering the criteria, make sure that you put it in an enclosed bracket (“ ”)
[/et_pb_text][et_pb_text content_tablet=”
An example of using the SUMIF function
Suppose you have data on total sales made by a company between 2020 and 2022
%91caption id=%22attachment_1806%22 align=%22aligncenter%22 width=%221457%22%93
Total sales between 2020 and 2022%91/caption%93
Now suppose you want to find the total sales made in Lahore in 2021. You can easily calculate this using the SUMIF function shown below:
%91su_note note_color=%22#b0def9%22%93=SUMIF(D9:D33, I9, F9:F33)%91/su_note%93
To elaborate, the ‘range’ is defined in column D (containing data on cities), the ‘criteria’ is given in cell I9 (Lahore) and the values to be added i.e. the ‘sum range’ is given in column F (data on
Similarly, if you want to calculate the total sales in Lahore in the year 2022: You can once again accomplish that using the SUMIF function as shown below:
%91su_note note_color=%22#b0def9%22%93=SUMIF(D9:D33, I9, G9:G33)%91/su_note%93
In this case, ‘range’ is defined in column D (containing data on cities), the ‘criteria’ is given in cell I9 (Lahore) and the values to be added i.e. the ‘sum range’ is given in column G (data on
But what if you want to apply multiple conditions? In that case, you can achieve that using the SUMIFS function.
” content_phone=”
An example of using the SUMIF function
Suppose you have data on total sales made by a company between 2020 and 2022
%91caption id=%22attachment_1806%22 align=%22aligncenter%22 width=%221457%22%93
Now suppose you want to find the total sales made in Lahore in 2021. You can easily calculate this using the SUMIF function shown below:
%91su_note note_color=%22#b0def9%22%93=SUMIF(D9:D33, I9, F9:F33)%91/su_note%93
To elaborate, the ‘range’ is defined in column D (containing data on cities), the ‘criteria’ is given in cell I9 (Lahore) and the values to be added i.e. the ‘sum range’ is given in column F (data on
Similarly, if you want to calculate the total sales in Lahore in the year 2022: You can once again accomplish that using the SUMIF function as shown below:
%91su_note note_color=%22#b0def9%22%93=SUMIF(D9:D33, I9, G9:G33)%91/su_note%93
In this case, ‘range’ is defined in column D (containing data on cities), the ‘criteria’ is given in cell I9 (Lahore) and the values to be added i.e. the ‘sum range’ is given in column G (data on
But what if you want to apply multiple conditions? In that case, you can achieve that using the SUMIFS function.
” content_last_edited=”on|phone” _builder_version=”4.6.6″ _module_preset=”default” text_font=”Poppins|300|||||||” text_font_size=”16px” text_line_height=”1.8em” header_2_font=”Montserrat|800||on|||||
” header_2_text_color=”#000000″ header_3_font=”Poppins|800|||||||” header_3_font_size=”16px” header_3_font_tablet=”” header_3_font_phone=”” header_3_font_last_edited=”on|phone”]
An example of using the SUMIF function
Suppose you have data on total sales made by a company between 2020 and 2022
Total sales between 2020 and 2022
Now suppose you want to find the total sales made in Lahore in 2021. You can easily calculate this using the SUMIF function shown below:
[su_note note_color=”#b0def9″]=SUMIF(D9:D33, I9, F9:F33)[/su_note]
To elaborate, the ‘range’ is defined in column D (containing data on cities), the ‘criteria’ is given in cell I9 (Lahore) and the values to be added i.e. the ‘sum range’ is given in column F (data on
Similarly, if you want to calculate the total sales in Lahore in the year 2022: You can once again accomplish that using the SUMIF function as shown below:
[su_note note_color=”#b0def9″]=SUMIF(D9:D33, I9, G9:G33)[/su_note]
In this case, ‘range’ is defined in column D (containing data on cities), the ‘criteria’ is given in cell I9 (Lahore) and the values to be added i.e. the ‘sum range’ is given in column G (data on
But what if you want to apply multiple conditions? In that case, you can achieve that using the SUMIFS function.
[/et_pb_text][et_pb_text content_tablet=”
Using the SUMIFS Function
The SUMIFS function is practically the same as the SUMIF function, except that you can define multiple criteria and conditions within the same function.
The syntax for the SUMIFS function is as below:
%91su_note note_color=%22#b0def9%22%93SUMIFS(sum_range, criteria_range1, criteria1, %91criteria_range2, criteria2%93…)%91/su_note%93
• sum_range is the range of cells that you want to sum
• criteria_range1, criteria_range2, etc. are the ranges of cells that you want to use as criteria to determine which cells in the sum_range to add
• criteria1, criteria2, etc. are the conditions that you want to use to filter the criteria_range1, criteria_range2
Note that the second criterion is an optional input, and is therefore given in a square bracket (%91 %93). Finally, you can add as many conditions as you want using the SUMIFS function.
” content_phone=”
Using the SUMIFS Function
The SUMIFS function is practically the same as the SUMIF function, except that you can define multiple criteria and conditions within the same function.
The syntax for the SUMIFS function is as below:
%91su_note note_color=%22#b0def9%22%93SUMIFS(sum_range, criteria_range1, criteria1, %91criteria_range2, criteria2%93…)%91/su_note%93
• sum_range is the range of cells that you want to sum
• criteria_range1, criteria_range2, etc. are the ranges of cells that you want to use as criteria to determine which cells in the sum_range to add
• criteria1, criteria2, etc. are the conditions that you want to use to filter the criteria_range1, criteria_range2
Note that the second criterion is an optional input, and is therefore given in a square bracket (%91 %93). Finally, you can add as many conditions as you want using the SUMIFS function.
” content_last_edited=”on|phone” _builder_version=”4.6.6″ _module_preset=”default” text_font=”Poppins|300|||||||” text_font_size=”16px” text_line_height=”1.8em” header_2_font=”Montserrat|800||on|||||
” header_2_text_color=”#000000″ header_3_font=”Poppins|800|||||||” header_3_font_size=”16px” header_3_font_tablet=”” header_3_font_phone=”” header_3_font_last_edited=”on|phone”]
Using the SUMIFS Function
The SUMIFS function is practically the same as the SUMIF function, except that you can define multiple criteria and conditions within the same function.
The syntax for the SUMIFS function is as below:
[su_note note_color=”#b0def9″]SUMIFS(sum_range, criteria_range1, criteria1, [criteria_range2, criteria2]…)[/su_note]
• sum_range is the range of cells that you want to sum
• criteria_range1, criteria_range2, etc. are the ranges of cells that you want to use as criteria to determine which cells in the sum_range to add
• criteria1, criteria2, etc. are the conditions that you want to use to filter the criteria_range1, criteria_range2
Note that the second criterion is an optional input, and is therefore given in a square bracket ([ ]). Finally, you can add as many conditions as you want using the SUMIFS function.
[/et_pb_text][et_pb_text content_tablet=”
An Example of Using the SUMIFS Function
Suppose that instead of having data of sales across different columns, you have data within a single column. In such a case, you will have the use the SUMIFS function.
For example, to find our the total sales made in Lahore in the year 2020, you will enter the formula as in shown in the picture below:
%91caption id=%22attachment_1812%22 align=%22aligncenter%22 width=%221410%22%93
” content_phone=”
An Example of Using the SUMIFS Function
Suppose that instead of having data of sales across different columns, you have data within a single column. In such a case, you will have the use the SUMIFS function.
For example, to find our the total sales made in Lahore in the year 2020, you will enter the formula as in shown in the picture below:
%91caption id=%22attachment_1812%22 align=%22aligncenter%22 width=%221410%22%93
” content_last_edited=”on|desktop” _builder_version=”4.6.6″ _module_preset=”default” text_font=”Poppins|300|||||||” text_font_size=”16px” text_line_height=”1.8em” header_2_font=”Montserrat|800||on|||
||” header_2_text_color=”#000000″ header_3_font=”Poppins|800|||||||” header_3_font_size=”16px” header_3_font_tablet=”” header_3_font_phone=”” header_3_font_last_edited=”on|phone”]
An Example of Using the SUMIFS Function
Suppose that instead of having data of sales across different columns, you have data within a single column. In such a case, you will have the use the SUMIFS function.
For example, to find our the total sales made in Lahore in the year 2020, you will enter the formula as in shown in the picture below:
Enter multiple conditions within the same formula using the SUMIFS function
[/et_pb_text][et_pb_text content_tablet=”
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Convert Miles to Km - mi → km
Both are units of measurement of length, and the Mile is more used in English-speaking countries, the United States, for example.
The mile was created in ancient Rome, at the time they measured distances by steps, which was not accurate at all, after all, the size of people's legs influenced the result, in short, 1 thousand
steps was equal to one mile.
The Kilometer was conceived through the derivation of the meter, where 1 thousand meters is equal to 1 km, this measurement unit is related to the International System of Units and is used in several
places in the world. | {"url":"https://converteonline.com/convert-miles-to-km/","timestamp":"2024-11-02T15:06:57Z","content_type":"text/html","content_length":"25182","record_id":"<urn:uuid:bdcb9807-b84e-4dc0-8aef-88a9066afadc>","cc-path":"CC-MAIN-2024-46/segments/1730477027714.37/warc/CC-MAIN-20241102133748-20241102163748-00783.warc.gz"} |
Top Benefits of A Private High School Math Tutor
Are you still unaware of the top benefits of a private high school math tutor? The greatest benefit of working with a private high school math tutor is personalized instruction. In a traditional
classroom setting, teachers are often juggling the needs of 20-30 students at once. This can make it difficult to provide each student with the individual attention they need to succeed.
A private tutor can focus solely on your child’s needs and learning style. They can assess your child’s current understanding of the material and develop a personalized lesson plan to help them
improve. Tutors can also provide extra help with difficult concepts and assignments.
Research conducted by San Bernardino Valley College found that Students who receive more academic support are more likely to pass their courses and stay enrolled.
However, according to a study by the National Association of College Admission Counseling, half of the high school counselors spend at least 21% of their time on college readiness counseling, but
unfortunately, only 18% of ninth-grade students have discussed college with a counselor.
To ensure you don’t miss the crucial support, read the detailed benefits of working with a private high school math tutor:
Looking to Learn High School Math? Book a Free Trial Lesson and match with top High School Math Tutors for Concepts, Homework Help, and Test Prep.
One-on-one attention
One of the biggest benefits of math tutoring is the one-on-one attention that your child will receive. In a typical math class, there are 20-30 students, which means that it can be difficult for
teachers to give each student the personalized attention they need to succeed.
Math tutors can identify your child’s individual needs and develop a personalized learning plan. They can also provide your child with the extra support and guidance they need to master challenging
If your child is shy or hesitant to ask questions in class, having a math tutor can help them feel more comfortable and confident. A good math tutor will create a safe and supportive learning
environment where your child can ask questions and get the help they need.
A study by Bear-Stearns found that parents of students at the top and bottom of their classes are most likely to seek tutoring. This suggests that math tutoring can be beneficial for students of all
levels, from those who want to excel to those who need extra help.
Customized instruction
Traditional classroom lectures and textbooks are designed for the average student, but not all students learn the same way. Tutor provides customized instruction for each student, tailored to their
individual strengths, weaknesses, and areas of need.
Learner tutors take the time to get to know each student and develop customized lesson plans. This means that students get the precise help they need to succeed, without wasting time on material they
already know.
More understanding
Learning math is like exploring a new city. While a guidebook (textbook) provides basic information, a local tour guide (tutor) can take you on a personalized tour, sharing in-depth insights and
showing you how the city’s history relates to your own experiences.
For example, math students can learn about probability theory in the context of data analysis, and English students can discuss Shakespeare’s plays in the context of modern social issues.
Tutors can also help students to develop a passion for learning. By providing students with engaging and relevant lessons, tutors can help them to see the value of what they are learning. This can
lead to improved academic performance and even spark new career interests.
Improved study habits
Math tutors can teach your child effective study habits, such as how to set goals, create a study schedule, and organize their materials. These skills will help your child succeed in all areas of
school, not just math.
Math tutoring can help your child learn how to manage their time effectively. This is an important skill for students of all ages, but it is especially important for high school and college students
who have to juggle multiple classes and extracurricular activities.
Math tutoring can help your child learn how to learn independently. This is an important skill for all students to develop, as it will help them succeed in college and the workforce.
Effective test prep
Preparing for standardized tests is similar to training for a sports competition. A coach (tutor) helps you develop the right skills, create a game plan, and build the confidence to perform at your
best during the big match (test day).
Standardized tests, such as the PSAT, ACT, and SAT, play an important role in high school admissions. Even regular in-school tests can have a significant impact on GPAs. For students who understand
the course material but struggle with test-taking, a tutor can be helpful.
Test prep tutors can review the material with students and provide practical tips for studying and test-taking. For example, tutors can teach students how to manage their time effectively, identify
and avoid common mistakes, and approach different types of test questions.
Increased confidence
Math tutoring can help students improve their confidence in a number of ways. When students feel more comfortable with the material and understand that they can improve with time and effort, they are
more likely to participate in class without the fear of getting an answer wrong.
Developing a growth mindset in math can be a powerful way to improve confidence. As students see their own progress and overcome obstacles, they begin to believe that they can achieve their goals in
math and other areas of their lives.
A study of 600 students at the National Institute of Education in Singapore found that students who think they are skilled in math tend to perform well on math tests. This suggests that there is a
link between confidence and math achievement.
Better college preparation
Think of high school as a warm-up for a marathon (college). Just as you need the right training and gear to run a marathon, high school tutoring equips you with the foundational knowledge and skills
needed to succeed in the more challenging terrain of college-level coursework.
High school tutoring helps students to understand foundational topics which makes it easy for them in college as courses in college move faster compared to that in high school.
High school tutoring can be a valuable investment for students who are planning to attend college. By providing individualized support and attention, tutors can help students develop the skills and
knowledge they need to succeed in college and beyond.
Personalized feedback
Personalized feedback is a key component of effective math tutoring. Math teachers may provide general tips on handling incorrect calculations or double-checking work, but private tutors can offer
one-on-one feedback tailored to each student’s needs.
This means that students are less likely to receive unnecessary or irrelevant advice. Instead, tutors can focus on providing students with specific information that they can apply to their current
learning goals. This is true for students of all levels, from those who are struggling with basic math to those who are preparing for advanced courses like linear algebra or college math.
Improves academic performance
One of the most obvious benefits of math tutoring is improved academic performance. This can lead to scholarships for college and beyond. But even more importantly, improved academic performance in
math can boost confidence, provide a solid foundation for future learning, and more.
When a student grasps a math concept well, their grades will improve. This shows that they have a better understanding of academic topics that they previously struggled with. This understanding is
essential for students, especially in a subject like math, where concepts build on each other as students progress.
To find the right high school tutor, check out our guide on How to Find The Right High School Math Tutor?
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Final thoughts
Overall, math tutoring can be a valuable investment for students who are struggling in math or who want to improve their grades. By providing students with the support and guidance they need, tutors
can provide students with personalized attention, flexible scheduling, customized instruction, and support for standardized test preparation.
If you want to learn more about high school math, please read: What is High School Math? | {"url":"https://wiingy.com/resources/math/benefits-of-a-private-high-school-math-tutor/","timestamp":"2024-11-14T03:25:58Z","content_type":"text/html","content_length":"170831","record_id":"<urn:uuid:c4e76c22-7e92-48eb-8a81-bf8e593ea717>","cc-path":"CC-MAIN-2024-46/segments/1730477028526.56/warc/CC-MAIN-20241114031054-20241114061054-00343.warc.gz"} |
Research Fellows Archives - Clay Mathematics Institute
Mehtaab Sawhney
Mehtaab Sawhney will receive his PhD from the Massachusetts Institute of Technology in 2024, under the supervision of Yufei Zhao.
While still a graduate student, Sawhney has achieved a stunning number of breakthroughs on fundamental problems across extremal combinatorics, probability theory, and theoretical computer science. He
is a highly collaborative researcher whose partnership with Ashwin Sah has been particularly fruitful. His remarkable body of work has already transformed swathes of combinatorics. For example,
working with Kwan, Sah and Simkin, he proved a 1973 conjecture of Erdős on the existence of high-girth Steiner triple systems; with Keevash and Sah he established the existence of subspace designs;
with Jain and Sah he established sharp estimates for the singularity probability in a wide class of discrete random matrices; with Sah and Sahasrabudhe he showed the existence of the spectral
distribution of sparse directed Erdős–Rényi graphs; and with Kwan, Sah and Sauermann, he developed highly novel tools in anti-concentration in order to prove the Erdős- McKay conjecture concerning
edge statistics in Ramsey graphs.
Mehtaab was appointed as a Clay Research Fellow for a term of five years from 1 July 2024.
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Who I AM and Where I AM – StankovUniversalLaw.com
Georgi Stankov
Erik’s elaboration on the nature of All-That-Is is very original as he solves the relationship between the Whole and the parts in terms of quotients of the number one with the number seven which
render open transcendental numbers.
The idea of introducing the concept of U-sets of the Whole as transcendental numbers, which have infinite approximations when presented as rational closed numbers, is an excellent one. In this way
Erik proves that we can present All-That-Is entirely in terms of mathematics. This example also explains why we can describe the Universal Law in all its manifestations and specific applications in
physics as a universal mathematical equation, which is essentially a rule of three.
By the way, current mathematics deals exclusively with closed rational numbers. Until now there is no mathematics based on open transcendental numbers. As physics is essentially applied mathematics
to the physical world, this explains why this science has failed to grasp the essence of All-That-Is.
In the new theory of the Universal Law I prove convincingly that all parts of the Whole are open U-subsets and thus can only be adequately described with open transcendental numbers. This insight is
at present not known to all scientists and theoreticians.
Erik has obviously grasped the essence of the new theory of the Universal Law on a very deep level and, according to my impression, almost nobody else has grasped it that way so far. There are a few
physicists I know, who have grasped the physical part of the new theory, but not the philosophy of mathematics, which is the foundation of the new theory of the Universal Law.
The ability of mathematics to present All-That-Is and its parts adequately is an aspect of the profound harmony of the Whole, which is the result of the laws of constructive and destructive
interference and these Laws of Creation can be mathematically presented with the help of open transcendental numbers. All U-sets of the Whole are open and exchange energy according to the new
Axiomatics. As they contain each other as an element, the element being energy, they are superimposed wave systems and thus follow the laws of constructive and destructive interference.
This is the only correct physical view of the world (physical weltanschauung), to which humanity and its scientists in particular must first evolve in order to understand the nature of All-That-Is
and subsequently “who we truly are”.
Erik’s essay below is an important contribution to this endeavour and I commend him highly for this intellectual endeavour.
When you take everything that exists, you can give it a name such as “All-That-Is”; it is a simple and clear designation for what it means. There is nothing outside All-That-Is, the name indicates
that it literally embraces everything.
All-That-Is is Energy. Any form or formless energy has an awareness, it is aware of itself and what it consists of. The all-encompassing awareness is All-That-Is and another name for that is
Universal Awareness.
All-That-Is = Energy = Universal Awareness
The Universal Awareness consists of multiple levels of awareness. The first level of awareness consists of a 2nd, 3rd, 4th etc. Each level of awareness as an identity is aware of itself and that it
is part of the Universal Awareness.
Each identity of the 1st level of awareness can attune to the all-encompassing Universal Awareness, in which it is aware of all identities.
We can understand the Universal Awareness and the first level of awareness also mathematically.
The all-encompassing awareness, the Universal Awareness is equal to 1 (according to the first axiom of last equivalence of all terms).
One = 1 = Universal Awareness (principle of last equivalence)
There is namely only 1 Universal Awareness. The Universal Awareness is also equal to infinity, mathematically we can divide 1 indefinitely and each part of the Universal Awareness must also express
the property / characteristics of the infinity.
The first division of the Universal Awareness can be divided by the number 7 (with respect to the 7F creationary energies), it will give a unique infinite number.
1 / 7 = 0.14285714285714 …
An identity of the 1st level of awareness can mathematically be expressed as 0.14285714285714 …
When an identity of the 1st level of awareness attunes to the all-encompassing Universal Awareness, it is aware of all the identities of the first level. Mathematically we can express it as follows:
When 0.14285714285714 … attunes on 1 it is aware of the
7 x 0.14285714285714 …
We have so far the following synonyms.
All-That-Is = Energy = Universal Awareness = 1 = Infinity
Division of Awareness
Each level of awareness is part of the Universal Awareness and has the characteristics of the Universal Awareness. Every level of awareness is part of another level of awareness.
1st level of awareness = 0.14285714285714 …
2nd level of awareness is 0.14285714285714 / 7… = 0.02040816326531 … etc.
An identity of the 2nd level of awareness is aware of itself and when it attunes to the all-encompassing awareness (which is also an identity of the 1st level of awareness), it is aware of the seven
identities of the 2nd level of awareness.
7 x 0.02040816326531 … = 0.14285714285714 …
The Universal Awareness is fully aware of the 7 x 1st level awareness with the 7 x 7 2nd level of awareness.
1 = 7 x 0.14285714285714 …
1 = 7 x 7 x 0.02040816326531 …
Alongside the first two levels of awareness, the Universal Awareness exists in infinite levels.
The Human Being
The human being is part of the all-encompassing awareness. This all-encompassing awareness is at the level of the Universal Awareness, it is aware of itself and when it attunes to this
all-encompassing awareness, it is aware of the seven identities of the same level of awareness.
The all-encompassing awareness of a person (incarnated soul) can be called a “Soul Family” or “Monad”. The larger encompassing awareness of a Soul Family is called a “Soul Tribe”, it consist of 7
Soul Families and so on.
This all-encompassing awareness as Souls Family consists inter alia, of its incarnations in human form. People incarnate on a planet in the space-time continuum to experience extreme (artificial)
conditions of separation. A Soul Family creates in the Now new creations based on survival until the unity of Being and the original ability of a sovereign creator is reclaimed by the incarnated
A Soul Family:
– Is an all-encompassing awareness within the Universal Awareness
– Is fully aware of all its incarnations (persons)
– It creates in the Now new creations.
A person experiences separation in linear time (past, present and future), in form of birth, growth and death. Thinking only in linear time is a limited experience and it does not correspond to the
characteristics of All-That-Is. A human awareness is always a part of a Soul Family, which embodies the characteristics of All-That-Is – it is infinite / immortal.
Every person is part of his Soul Family and the Universal Awareness. We may experience a limited form of life, but our awareness is infinite and immortal. We will always exists, only in different
forms, as every creation is stored in the Universal Awareness. The Universal Awareness consists of everything, of every creation, of every person that experiences the most extreme form of separation,
but the latter is only artificially created. No individual can ever be separated from the Universal Awareness because there is nothing outside All-That-Is, only our limited thinking may elicit this
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