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Amazon algebra 2 book state of texas
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DistatisFast: Fast implementation the multivariate analysis method Distatis in phylter: Detect and Remove Outliers in Phylogenomics Datasets
New implementation of the DISTATIS method for K matrices of dimension IxI. This version of Distatis is faster than the original one because only the minimum required number of eigenvalues and
eigenvectors is calculated. The difference in speed is particularly visible when the number of matrices is large.
matrices A list of K distance matrices, all of the same dimension (IxI).
factorskept Number of factors to keep for the computation of the factor. When "auto" (the default), a brokenstick model is used for the choice of the number of components to keep.
parallel Should the matrix products computations be parallelized? Default to TRUE. scores of the observations.
A list of K distance matrices, all of the same dimension (IxI).
Number of factors to keep for the computation of the factor. When "auto" (the default), a brokenstick model is used for the choice of the number of components to keep.
Should the matrix products computations be parallelized? Default to TRUE. scores of the observations.
PartialF: list of projected coordinates of species for each gene.
alpha: array of length K of the weight associated to each matrix.
lambda: array of length K of the normalization factors used for each matrix. lambda=1 always.
RVmat: a KxK matrix with RV correlation coefficient computed between all pairs of matrices.
compromise: an IxI matrix representing the best compromise between all matrices. This matrix is the weighted average of all K matrices, using 'alpha' as a weighting array.
quality: the quality of the compromise. This value is between 0 and 1
matrices.dblcent: matrices after double centering describes how much of the variance of the K matrices is captured by the compromise.
Abdi, H., Valentin, D., O'Toole, A.J., & Edelman, B. (2005). DISTATIS: The analysis of multiple distance matrices. Proceedings of the IEEE Computer Society: International Conference on Computer
Vision and Pattern Recognition_. (San Diego, CA, USA). pp. 42-47.
# Get a list of matrices # from the carnivora dataset data(carnivora) matrices <- phylter(carnivora, InitialOnly = TRUE, parallel = FALSE)$matrices # Perform a Distatis analysis on these matrices:
distatis <- DistatisFast(matrices, parallel = FALSE) #distatis is a list with multiple elements: distatis$alpha #weigh of each matrix (how much it correlates with others) distatis$RVmat #RV matrix:
correlation of each matrix with each other distatis$compromise # distance matrix with "average" pairwise distance between species in matrices # etc.
For more information on customizing the embed code, read Embedding Snippets. | {"url":"https://rdrr.io/cran/phylter/man/DistatisFast.html","timestamp":"2024-11-05T03:10:15Z","content_type":"text/html","content_length":"27007","record_id":"<urn:uuid:a5aae683-3c70-494c-a1cd-2e75d2b5cea5>","cc-path":"CC-MAIN-2024-46/segments/1730477027870.7/warc/CC-MAIN-20241105021014-20241105051014-00662.warc.gz"} |
Quantum Portfolio Optimization: A Deep Dive into Algorithms and Data Encoding
🌐 Introduction to Quantum Portfolio Optimization
Portfolio optimization is a critical task in finance, involving the selection of assets to maximize returns while minimizing risk. As the number of assets and constraints increases, the optimization
problem becomes increasingly complex, making it challenging for classical computers to solve efficiently. Quantum computing offers a potential solution, leveraging the principles of quantum mechanics
to perform complex calculations and optimize portfolios more effectively.
In this blog post, well dive deep into the technical details of quantum portfolio optimization, exploring the specific algorithms and quantum mechanics principles used. well also discuss how
financial data is translated into quantum circuits and how quantum computing can revolutionize portfolio optimization in the finance industry.
🔢 Quantum Mechanics Principles in Portfolio Optimization
Quantum portfolio optimization relies on several key principles of quantum mechanics, including:
1. Superposition: Quantum systems can exist in multiple states simultaneously, allowing quantum computers to perform many calculations in parallel.
2. Entanglement: Quantum bits (qubits) can be entangled, meaning their states are correlated, even if they are physically separated. This enables quantum computers to solve complex problems more
efficiently than classical computers.
3. Interference: Quantum states can interfere with each other, allowing quantum computers to amplify the desired solutions and cancel out the unwanted ones.
These principles enable quantum computers to explore vast solution spaces and find optimal portfolio allocations more efficiently than classical computers.
🎛️ Quantum Algorithms for Portfolio Optimization
Several quantum algorithms have been developed specifically for portfolio optimization, including:
1. Quantum Approximate Optimization Algorithm (QAOA): QAOA is a hybrid quantum-classical algorithm that alternates between applying quantum gates and classical optimization steps to find the optimal
solution. In the context of portfolio optimization, QAOA can be used to find the optimal asset allocation that maximizes returns while satisfying given constraints.
2. Variational Quantum Eigensolver (VQE): VQE is another hybrid quantum-classical algorithm that uses a parameterized quantum circuit to minimize a cost function. In portfolio optimization, the cost
function can be defined as the risk or the negative of the expected returns, and VQE can be used to find the optimal portfolio weights.
3. Quantum Amplitude Estimation (QAE): QAE is a quantum algorithm that estimates the amplitude of a given quantum state, which can be used to calculate the expected value of a function. In portfolio
optimization, QAE can be used to estimate the expected returns and risks of different portfolio allocations, enabling more accurate optimization.
These algorithms leverage the power of quantum computing to solve portfolio optimization problems more efficiently than classical approaches.
💾 Encoding Financial Data into Quantum Circuits
To perform quantum portfolio optimization, financial data must be encoded into quantum circuits. This involves mapping the assets and their characteristics (e.g., expected returns, risks,
correlations) onto qubits and quantum gates.
One common approach is to use amplitude encoding, where the asset weights are encoded into the amplitudes of the quantum states. For example, a portfolio with two assets can be represented by a
quantum state |ψ⟩ = α|00⟩ + β|01⟩ + γ|10⟩ + δ|11⟩, where the amplitudes α, β, γ, and δ correspond to the weights of the assets.
Another approach is to use angle encoding, where the asset weights are encoded into the rotation angles of the quantum gates. For example, a portfolio with two assets can be represented by a quantum
circuit that applies a rotation gate R(θ₁) to the first qubit and a rotation gate R(θ₂) to the second qubit, where the angles θ₁ and θ₂ correspond to the weights of the assets.
The choice of encoding scheme depends on the specific problem and the quantum algorithm used. Amplitude encoding is often used with QAOA and VQE, while angle encoding is commonly used with QAE.
🚀 Quantum Circuit Example for Portfolio Optimization
To illustrate how quantum portfolio optimization works in practice, let's consider a simple example using the QAOA algorithm and amplitude encoding.
Suppose we have a portfolio with two assets, A and B, and we want to find the optimal allocation that maximizes the expected return while keeping the risk below a certain threshold. We can encode
this problem into a quantum circuit as follows:
1. Initialize a quantum circuit with two qubits, representing the two assets.
2. Apply a Hadamard gate to each qubit to create a superposition of all possible asset allocations.
3. Apply a parameterized quantum circuit, consisting of a series of rotation gates and entanglement gates, to evolve the quantum state towards the optimal solution.
4. Measure the qubits to obtain the optimal asset allocation.
The parameterized quantum circuit can be optimized using a classical optimizer, such as gradient descent, to find the optimal values of the parameters that maximize the expected return while
satisfying the risk constraint.
Here's an example of what the quantum circuit might look like using the Qiskit library in Python:
from qiskit import QuantumCircuit, QuantumRegister, ClassicalRegister, execute, Aer
# Define the problem parameters
expected_returns = [0.05, 0.07] # Expected returns of assets A and B
covariance_matrix = [[0.04, 0.02], [0.02, 0.06]] # Covariance matrix of asset returns
risk_threshold = 0.1 # Maximum acceptable risk
# Define the quantum circuit
qr = QuantumRegister(2)
cr = ClassicalRegister(2)
qc = QuantumCircuit(qr, cr)
# Apply Hadamard gates to create superposition
# Apply parameterized quantum circuit
params = [0.1, 0.2, 0.3, 0.4] # Initial parameter values
qc.rx(params[0], qr[0])
qc.rx(params[1], qr[1])
qc.cx(qr[0], qr[1])
qc.rz(params[2], qr[0])
qc.rz(params[3], qr[1])
# Measure the qubits
qc.measure(qr, cr)
# Optimize the parameters using a classical optimizer
def objective_function(params):
# Execute the quantum circuit with the current parameters
backend = Aer.get_backend('qasm_simulator')
result = execute(qc, backend, shots=1024).result()
counts = result.get_counts(qc)
# Calculate the expected return and risk of the portfolio
expected_return = 0
for outcome, count in counts.items():
x = int(outcome, 2) / (2**2 - 1)
expected_return += x * (expected_returns[0] * x + expected_returns[1] * (1 - x)) * count / 1024
risk = np.sqrt(x**2 * covariance_matrix[0][0] + (1 - x)**2 * covariance_matrix[1][1] + 2 * x * (1 - x) * covariance_matrix[0][1])
# Penalize portfolios that exceed the risk threshold
if risk > risk_threshold:
expected_return -= 1000
return -expected_return # Negative sign for minimization
from scipy.optimize import minimize
result = minimize(objective_function, params, method='COBYLA')
optimal_params = result.x
# Execute the quantum circuit with the optimal parameters to get the final portfolio allocation
backend = Aer.get_backend('qasm_simulator')
result = execute(qc, backend, shots=1024).result()
counts = result.get_counts(qc)
optimal_allocation = int(list(counts.keys())[0], 2) / (2**2 - 1)
print(f"Optimal allocation: {optimal_allocation:.2f} in Asset A, {1 - optimal_allocation:.2f} in Asset B")
In this example, we define a quantum circuit with two qubits, representing a portfolio with two assets. We apply Hadamard gates to create a superposition of all possible asset allocations, and then
apply a parameterized quantum circuit to evolve the quantum state towards the optimal solution. The parameters of the quantum circuit are optimized using a classical optimizer (in this case, the
COBYLA algorithm from the SciPy library) to maximize the expected return while keeping the risk below a specified threshold.
Finally, we execute the quantum circuit with the optimal parameters to obtain the final portfolio allocation.
🎉 Conclusion
Quantum portfolio optimization is a promising application of quantum computing in finance, offering the potential to solve complex optimization problems more efficiently than classical approaches. By
leveraging the principles of quantum mechanics and specific quantum algorithms, such as QAOA, VQE, and QAE, quantum computers can find optimal portfolio allocations that maximize returns while
minimizing risk.
However, realizing the full potential of quantum portfolio optimization requires careful encoding of financial data into quantum circuits and the development of efficient quantum algorithms and
hardware. As quantum computing technology continues to advance, we can expect to see more widespread adoption of quantum portfolio optimization in the finance industry, leading to more efficient and
effective portfolio management.
By understanding the technical details of quantum portfolio optimization, including the specific algorithms and quantum mechanics principles used, financial professionals can better prepare for the
quantum future of finance and harness the power of quantum computing to make more informed investment decisions. | {"url":"https://www.elontusk.org/blog/Finance/portfoliooptimization","timestamp":"2024-11-10T01:37:30Z","content_type":"text/html","content_length":"86836","record_id":"<urn:uuid:0ecfb8bd-76af-48b6-8008-3b61d646a8f5>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.3/warc/CC-MAIN-20241110005602-20241110035602-00146.warc.gz"} |
Classnotes Systems by Linear Combination 2-23/2
Classnotes Systems by Linear Combination 2-23/2-24
Algebra II CP
The 3rd and last method for solving systems of 2 equations in 2 variables is called Linear Combination.
This method will work for all systems, and is less work than substitution.
The goal of Linear Combination is to get either the x variable or the y variable to cancel out. In order for this to happen,
you must leave your 2 equations in standard form and stacked over one another. Examine the coefficients of the terms
and decide if x or y will eliminate from the original system OR if x or y will eliminate if you multiply one equation by a
factor to force the terms to be equal opposites. Once this is in place, “add” the other columns of variables and
constants – thus solving for one of your two variables.
Example: x + 5y = 33
4x + 3y = 13
Notice that neither the x column nor the y column will eliminate if you add straight down
the columns as they are currently written. If however, you multiply the top equation by
-4, the x column would then eliminate. Thus, the new system would look like:
now, “add” each column
-4x -20y = -132
4x + 3y = 13
-17y = -119
now solve for the value of the y-coordinate
It is your choice as to which of the original equations you substitute the value y = 7 into to get the value for x. I am going
to use the 1st equation: x + 5(7) = 33
x + 35 = 33 subtract 35 from both sides
x = -2
Therefore the ordered pair solution for this system is ( -2, 7)
There are more examples in your textbook in section 3.2
Classwork for the day is p. 152 #23-34
I have tentatively set the chapter 3 test for next Wednesday, March 2 | {"url":"https://studylib.net/doc/9505182/classnotes-systems-by-linear-combination-2-23-2","timestamp":"2024-11-02T12:31:39Z","content_type":"text/html","content_length":"59617","record_id":"<urn:uuid:2fcfe23d-cba6-4353-a2b2-438164dd1ea5>","cc-path":"CC-MAIN-2024-46/segments/1730477027710.33/warc/CC-MAIN-20241102102832-20241102132832-00050.warc.gz"} |
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the problems texts is at. ( xn ) be a sequence in R such that for all n, xn > 0 your Answer if... And network Analysis 1 real Analysis, Stability & computer Techniques multiple choice questions on
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Rate vs. timing (XVII) Analog vs. digital
It is sometimes stated that rate-based computing is like analog computing while spike-based computing is like digital computing. The analogy comes from the fact, of course, that spikes are discrete
whereas rates are continuous. But as any analogy, it has its limits. First of all, spikes are not discrete in the way digital numbers are discrete. In digital computing, the input is a stream of
binary digits, coming one after another in a cadenced sequence. The digits are gathered by blocks, say of 16 or 32, to form words that stand for instructions or numbers. Let us examine these two
facts with respect to spikes. Spikes do not arrive in a cadenced sequence. Spikes arrive at irregular times, and time is continuous, not digital. What was meant by digital is presumably that there
can be a spike or there can be no spike, but there is nothing in between. However, given that there is also a continuous timing associated to the occurrence of a spike, a spike is better described as
a timed event rather than as a binary digit. But of course one could decide to divide the time axis into small time bins, and associate a digit 0 when there is no spike and 1 when there is a spike.
This is certainly true, but as one performs this process as finely as possible to approximate the real spike train, it appears that there are very few 1s drowned in a sea of 0s. This is what is meant
by “event”: information is carried by the occurrence of 1s at specific times rather than by the specific combinations of 0s and 1s, as in digital computing. So in this sense, spike-based computing is
not very similar to digital computing.
The second aspect of the analogy is that digits are gathered in words (of say 32 digits), and these words are assigned a meaning in terms of either an instruction or a number. Transposed to spikes,
these “words” could be the temporal pattern of spikes of a single neuron, or perhaps more meaningfully a pattern of spikes across neurons, as in synchrony-based schemes, or across neurons and time,
as in polychronization. Now there are two ways of understanding the analogy. Either a spike pattern stands for a number, and in this case the analogy is not very interesting, since this is pretty
much saying that spikes implement an underlying continuous value, in other words this is the rate-based view of neural computation. Or a spike pattern stands for a symbol. This case is more
interesting, and it may apply to some proposed spike-based schemes (like polychronization). It emphasizes the idea that unlike rate-based theories, spike-based theories are not (necessarily) related
to usual mathematical notions of calculus (e.g. adding numbers), but possibly to more symbolic manipulations.
However, this does not apply to all spike-based theories. For example, in Sophie Denève’s spike-based theory of inference (which I will describe in a future post), spike-based computation actually
implements some form of calculus. But in her theory, analog signals are reconstructed from spikes, in the same way as the membrane potential results from the action of incoming spikes, rather than
the other way around as in rate-based theories (i.e., a rate description is postulated, then spikes are randomly produced to implement that description). So in this case the theory describes some
form of calculus, but based on timed events.
This brings me to the fact that neurons do not always interact with spikes. For example, in the retina, there are many neurons that do not spike. There are also gap junctions, in which the membrane
potentials of several neurons directly interact. There are also ephaptic interactions (through the extracellular field potential). There is also evidence that the shape of action potentials can
influence downstream synapses (see a recent review by Dominique Debanne). In these cases, we may speak of analog computation. But this does not bring us closer to rate-based theories. In fact, quite
the opposite: rates are abstracted from spikes, and stereotypical spikes are an approximation of what really goes on, which may involve other physical quantities. The point here is that firing rate
is not a physical quantity as the membrane potential, for example. It is an abstract variable. In this sense, spike-based theories, because they are based on actual biophysical quantities in neurons,
might be closer to what we might call “analog descriptions” of computation than rate-based theories.
Laisser un commentaire | {"url":"https://romainbrette.fr/rate-vs-timing-xvii-analog-vs-digital/","timestamp":"2024-11-07T07:31:19Z","content_type":"text/html","content_length":"28361","record_id":"<urn:uuid:6673f24e-88f3-4238-b6fe-8e1c830c1140>","cc-path":"CC-MAIN-2024-46/segments/1730477027957.23/warc/CC-MAIN-20241107052447-20241107082447-00180.warc.gz"} |
1/p-secure multiparty computation without honest majority and the best of both worlds
A protocol for computing a functionality is secure if an adversary in this protocol cannot cause more harm than in an ideal computation, where parties give their inputs to a trusted party which
returns the output of the functionality to all parties. In particular, in the ideal model such computation is fair - all parties get the output. Cleve (STOC 1986) proved that, in general, fairness is
not possible without an honest majority. To overcome this impossibility, Gordon and Katz (Eurocrypt 2010) suggested a relaxed definition - 1/p-secure computation - which guarantees partial fairness.
For two parties, they construct 1/p-secure protocols for functionalities for which the size of either their domain or their range is polynomial (in the security parameter). Gordon and Katz ask
whether their results can be extended to multiparty protocols. We study 1/p-secure protocols in the multiparty setting for general functionalities. Our main result is constructions of 1/p-secure
protocols that are resilient against any number of corrupt parties provided that the number of parties is constant and the size of the range of the functionality is at most polynomial (in the
security parameter n). If less than 2/3 of the parties are corrupt, the size of the domain is constant, and the functionality is deterministic, then our protocols are efficient even when the number
of parties is log log n. On the negative side, we show that when the number of parties is super-constant, 1/p-secure protocols are not possible when the size of the domain is polynomial. Thus, our
feasibility results for 1/p-secure computation are essentially tight. We further motivate our results by constructing protocols with stronger guarantees: If in the execution of the protocol there is
a majority of honest parties, then our protocols provide full security. However, if only a minority of the parties are honest, then our protocols are 1/p-secure. Thus, our protocols provide the best
of both worlds, where the 1/p-security is only a fall-back option if there is no honest majority.
اللغة الأصلية الإنجليزيّة
عنوان منشور المضيف Advances in Cryptology - CRYPTO 2011 - 31st Annual Cryptology Conference, Proceedings
الصفحات 277-296
عدد الصفحات 20
المعرِّفات الرقمية للأشياء
حالة النشر نُشِر - 2011
منشور خارجيًا نعم
الحدث 31st Annual International Cryptology Conference, CRYPTO 2011 - Santa Barbara, الولايات المتّحدة
المدة: ١٤ أغسطس ٢٠١١ → ١٨ أغسطس ٢٠١١
سلسلة المنشورات
الاسم Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
مستوى الصوت 6841 LNCS
رقم المعيار الدولي للدوريات (المطبوع) 0302-9743
رقم المعيار الدولي للدوريات (الإلكتروني) 1611-3349
!!Conference 31st Annual International Cryptology Conference, CRYPTO 2011
الدولة/الإقليم الولايات المتّحدة
المدينة Santa Barbara
المدة ١٤/٠٨/١١ → ١٨/٠٨/١١
أدرس بدقة موضوعات البحث “1/p-secure multiparty computation without honest majority and the best of both worlds'. فهما يشكلان معًا بصمة فريدة. | {"url":"https://cris.ariel.ac.il/ar/publications/1p-secure-multiparty-computation-without-honest-majority-and-the--3","timestamp":"2024-11-11T10:24:18Z","content_type":"text/html","content_length":"63275","record_id":"<urn:uuid:f21f6fee-4a7b-4d0d-bae5-0e5dde1b7d88>","cc-path":"CC-MAIN-2024-46/segments/1730477028228.41/warc/CC-MAIN-20241111091854-20241111121854-00015.warc.gz"} |
The Geometry of Non-Generic Components in the Emerton-Gee Stack for GL2
Core Concepts
This paper investigates the geometric properties of non-generic components in the Emerton-Gee stack for GL2, providing insights into their smoothness, normality, and singularities, and discusses the
implications for the conjectural categorical p-adic Langlands correspondence.
Bibliographic Information:
Kansal, K., & Savoie, B. (2024). Non-Generic Components of the Emerton-Gee Stack for GL2. arXiv preprint arXiv:2407.07883v2.
Research Objective:
This research paper aims to analyze the geometric properties of non-generic components within the Emerton-Gee stack for GL2, specifically focusing on their smoothness, normality, and the nature of
their singularities. The authors seek to refine the understanding of these components and explore the implications of their findings for the conjectural categorical p-adic Langlands correspondence.
The authors utilize a combination of algebraic geometry and representation theory techniques. They employ smooth-local charts and auxiliary schemes to study the structure of the Emerton-Gee stack. By
analyzing the properties of these charts, they deduce information about the smoothness, normality, and singular loci of the non-generic components.
Key Findings:
• The paper provides precise conditions for the smoothness and normality of non-generic components in the Emerton-Gee stack.
• It demonstrates that the normalizations of these components admit smooth-local covers by resolution-rational schemes.
• The authors determine the codimension of the singular loci within these components.
• The research establishes a connection between the singularities of the components and the ramification properties of associated Galois representations.
Main Conclusions:
The geometric properties of non-generic components in the Emerton-Gee stack, particularly their singularities, are intricately linked to the ramification behavior of corresponding Galois
representations. These findings contribute to a deeper understanding of the stack's structure and offer valuable insights for the ongoing development of the categorical p-adic Langlands
This research significantly advances the understanding of the Emerton-Gee stack, a central object in the p-adic Langlands program. By elucidating the geometry of its non-generic components, the
authors provide crucial information for constructing and understanding the conjectural categorical p-adic Langlands correspondence, a profound connection between representation theory and number
Limitations and Future Research:
The study primarily focuses on non-generic components in the case of GL2. Further research could explore the geometry of generic components and extend the analysis to higher-rank general linear
groups. Investigating the implications of these geometric findings for the explicit construction of the categorical p-adic Langlands correspondence remains a promising avenue for future
p > 3 K is an unramified extension of Qp of degree f.
How do the geometric properties of the Emerton-Gee stack, as studied in this paper, generalize to other reductive groups beyond GL2?
Answer: While this paper focuses specifically on the Emerton-Gee stack for $\mathrm{GL}_2$, generalizing these geometric results to other reductive groups presents significant challenges and is a
subject of active research. Here's a breakdown of potential generalizations and difficulties: Potential Generalizations: Irreducible Components: The labeling of irreducible components of the
Emerton-Gee stack by Serre weights should generalize to other groups, using appropriate notions of Serre weights for those groups. Smoothness and Normality: The criteria for smoothness and normality
of these components likely involve intricate relationships between the root data of the group, the shape of the inertial type τ, and the chosen Serre weight. Resolution of Singularities: The
existence of smooth-local covers by resolution-rational schemes might extend, but explicit constructions would be much more involved. The nature of these resolutions would likely depend heavily on
the specific reductive group. Difficulties: Increased Complexity: The representation theory and geometry associated with general reductive groups are significantly more intricate than those for $\
mathrm{GL}_2$. Lack of Explicit Charts: Constructing explicit smooth-local charts, like those using Breuil-Kisin modules for $\mathrm{GL}_2$, becomes much harder. Understanding Shapes: The "shape"
conditions imposed on Breuil-Kisin modules to cut out irreducible components would need to be generalized and would likely be more complicated. Current Research: Researchers are actively exploring
these generalizations. Some progress has been made for groups like $\mathrm{GSp}_4$, but a complete understanding for general reductive groups remains a significant open problem.
Could there be alternative geometric interpretations of the singularities in the non-generic components that provide further insights into the p-adic Langlands correspondence?
Answer: Yes, exploring alternative geometric interpretations of the singularities in the non-generic components of the Emerton-Gee stack is a promising avenue for deeper insights into the p-adic
Langlands correspondence. Here are some potential approaches: Derived Geometry: Instead of focusing solely on the reduced part of the stack, studying the full derived structure of the Emerton-Gee
stack might provide a more natural framework for understanding the singularities. Derived algebraic geometry could offer tools to resolve these singularities in a more refined way. Intersection
Theory: Investigating the intersection theory of the irreducible components, particularly near the singular loci, could reveal connections between the geometry of the stack and the representation
theory of the relevant groups. Moduli of Langlands Parameters: The Emerton-Gee stack is expected to be related to a moduli stack of Langlands parameters. Understanding this relationship and how
singularities on one side correspond to geometric features on the other could be illuminating. Relationship to p-adic Hodge Theory: The singularities might reflect deeper phenomena in p-adic Hodge
theory. For instance, they could be linked to the structure of period rings or the geometry of certain p-adic period domains. Potential Benefits: Refined Conjectures: Alternative geometric
interpretations could lead to more refined conjectures about the p-adic Langlands correspondence, potentially suggesting new directions for research. Connections to Other Areas: These interpretations
might reveal unexpected connections between the p-adic Langlands program and other areas of mathematics, such as derived algebraic geometry, p-adic Hodge theory, or the theory of singularities.
What are the implications of this research for the development of explicit constructions of the functor A in the categorical p-adic Langlands correspondence?
Answer: This research, particularly the results on the geometry of the Emerton-Gee stack, has significant implications for explicitly constructing the functor A in the categorical p-adic Langlands
correspondence. Here's how: Characterizing the Sheaves: The paper provides a concrete description of the support of the conjectural sheaves L(σm,n) and shows that, under certain conditions, they are
pushforwards of invertible sheaves on smooth stacks. This characterization is a crucial step towards explicitly constructing these sheaves. Understanding Self-Duality: The results on the codimension
of singular loci and the existence of smooth-local covers allow for a better understanding of the self-duality of L(σm,n). This is essential, as self-duality is a key property expected of the functor
A. Smoothness and Gluing: The existence of smooth-local charts on the Emerton-Gee stack suggests a strategy for constructing A locally on these charts and then gluing these local constructions. The
explicit nature of these charts, using Breuil-Kisin modules, provides a concrete setting for performing these local constructions. Challenges and Future Directions: Non-Generic Components: The
singularities in the non-generic components pose a challenge. The paper's results on resolutions of singularities provide a starting point, but more work is needed to understand how to construct A
over these singular components. Higher Dimensions: Generalizing these constructions to GLd for d > 2 will be significantly more difficult due to the increased complexity of the geometry and
representation theory involved. Overall Impact: This research provides a concrete framework and essential ingredients for constructing the functor A in the categorical p-adic Langlands
correspondence. The explicit nature of the results, particularly the use of Breuil-Kisin modules and the analysis of singularities, offers a promising path towards making the p-adic Langlands program
more explicit and computationally accessible. | {"url":"https://linnk.ai/insight/scientificcomputing/the-geometry-of-non-generic-components-in-the-emerton-gee-stack-for-gl2-x4IEYSn0/","timestamp":"2024-11-11T04:52:47Z","content_type":"text/html","content_length":"374331","record_id":"<urn:uuid:47221d27-f383-4295-a1f4-71a1fb206b8e>","cc-path":"CC-MAIN-2024-46/segments/1730477028216.19/warc/CC-MAIN-20241111024756-20241111054756-00437.warc.gz"} |
Code of View
• This post is part of a series where I explain how to build an augmented reality Sudoku solver. All of the image processing and computer vision is done from scratch. See the first post for a
complete list of topics.
It’s finally time to finish implementing the Canny edge detector algorithm. The previous step produced an image where strong and weak edges were detected. Pixels marked as strong edges are
assumed to be actual edges. On the other hand, pixels marked as weak edges are considered to only “maybe” be edges. So the question remains, when should a weak edge actually be a strong edge? If
you have a chain of strong edges connected to each other, then there is definitely an edge running along them. You could then reason that if a group of weak edges are directly connected to these
strong edges, they are part of the same edge and should be promoted. This is where connectivity analysis comes in.
Connectivity analysis is a very mathy term but is straight forward with respect to image processing. The whole idea is about examining groups of pixels that meet some requirement and are
connected in such a way that you can move from any one pixel in the group to any other without leaving the group. Say you have a pixel that meets a certain criteria. Look at each of the
surrounding pixels and check if it meets a criteria (could be the same as before but doesn’t have to be). If it does, the pixels are connected. Repeat this process to find all pixels in the
group. That’s it, actually really simple, right?
Which surrounding pixels should be examined depends on application. The two most common patterns are known as 4- and 8-connectivity. The former checks the pixels directly left, right, above, and
below. The latter checks all of the same pixels as 4-connectivity but also top-left, top-right, bottom-left, and bottom-right.
In this case, we need to find strong edge pixels and then check each surrounding pixel if it’s a weak edge pixel. If it is, change it into a strong edge pixel since the strong and weak edges are
You might be wondering if you can search in the reverse order. Find weak edges and check for surrounding strong edges. The answer is yes, but it’ll probably be slower because the hysteresis
thresholding used to determine if an edge is strong or weak guarantees that there are at least as many weak edges as strong edges.
The implementation below searches for all strong edges. Whenever a strong edge is found, a flood fill is used to examine all 8 surrounding pixels for weak edges. Found weak edges are promoted to
strong edges. The flood fill repeatedly searches the 8 surrounding pixels surrounding each promoted edge until no more weak edges are found.
Image image; //Input: Should be output of Non-Maximum Suppression.
// Channel 0: Strong edge pixels.
// Channel 1: Weak edge pixels.
// Channel 2: Should be set to 0.
// There should be no strong edge pixels in the pixels touching the
// border of the image.
//Output: Image will have all weak edges connected to strong edges changed
// into strong edges. Only the first channel will have useful data,
// the other channels should be ignored.
//Keep track of coordinates that should be searched in case they are connected.
std::stack<std::pair<unsigned int,unsigned int>> searchStack;
//Add all 8 coordinates around (x,y) to be searched.
auto PushSearchConnected = [&searchStack](const unsigned int x,const unsigned int y) {
searchStack.push(std::make_pair(x - 1,y - 1));
searchStack.push(std::make_pair(x ,y - 1));
searchStack.push(std::make_pair(x + 1,y - 1));
searchStack.push(std::make_pair(x - 1,y));
searchStack.push(std::make_pair(x ,y));
searchStack.push(std::make_pair(x + 1,y));
searchStack.push(std::make_pair(x - 1,y + 1));
searchStack.push(std::make_pair(x ,y + 1));
searchStack.push(std::make_pair(x + 1,y + 1));
//Find strong edge pixels and flood fill so all connected weak edges are turned into
//strong edges.
for(unsigned int y = 1;y < image.height - 1;y++)
for(unsigned int x = 1;x < image.width - 1;x++)
const unsigned int index = (y * image.width + x) * 3;
//Skip pixels that are not strong edges.
if(image.data[index + 0] == 0)
//Flood fill all connected weak edges.
const std::pair<unsigned int,unsigned int> coordinates = searchStack.top();
//Skip pixels that are not weak edges.
const unsigned int x = coordinates.first;
const unsigned int y = coordinates.second;
const unsigned int index = (y * image.width + x) * 3;
if(image.data[index + 1] == 0)
//Promote to strong edge.
image.data[index + 0] = 255;
image.data[index + 1] = 0;
//Search around this coordinate as well. This will waste time checking the previous
//coordinate again but it's fast enough.
Putting it All Together
Here’s the pipeline so far. An RGB image is captured using a camera. The RGB image is converted to greyscale so it can be processed with the Canny edge detector. The Canny edge detector, applies
a Gaussian filter to remove noise and suppress insignificant edges. The “flow” of the image is analyzed by finding gradients using the Sobel operator. These gradients are the measured strength
and direction of change in pixel intensities. Non-Maximum Suppression is applied using the gradients to find potential edges. Hysteresis Thresholding compares the potential edge gradients
strengths to a couple of thresholds which categorize if an edge is an actual edge, maybe an edge, or not an edge. Finally, Connectivity Analysis promotes those “maybe” edges to actual edges
whenever they’re directly connected to an actual edge.
Besides the input image, the Canny edge detector has three tweakable parameters that can have a big impact on whether an edge is detected or not. The first is the radius of the Gaussian filter.
If the radius is too large, edges are blended together and not detected. If the radius is too small, small edges or noise are detected which should be ignored. An effective radius choice is
further complicated by the results depending on the size of image being used. For this project, images are 640x480 and I’ve found a radius of 5 pixels works quite well. Larger images can be
downscaled which, depending on choice of sampling filter, helps to further suppress noise anyway. Smaller images are not practical because they lack a sufficient number of pixels for all of the
numbered digits to be identified reliably.
The other two parameters are the high and low thresholds used in the Hysteresis Thresholding step. These are a little trickier to get right and are problem dependent. In general, having the high
threshold two or three times the low threshold works well. But that still leaves picking a high threshold. I’m using a statistical approach called Otsu’s method which is effective on high
contrast images. In this case, the puzzles are usually printed in black ink on white paper which is a perfect fit. I’m not going to go over how Otsu’s method works in interest of keeping this
blog series from getting any longer. However, the output hovers around 110 for all of my test cameras and puzzles so that’s probably a good place to start experimenting.
Building on the output of the Canny edge detector, the next post will cover the detection of straight lines which might indicate the grid in a Sudoku puzzle.
• This post is part of a series where I explain how to build an augmented reality Sudoku solver. All of the image processing and computer vision is done from scratch. See the first post for a
complete list of topics.
The whole point of the Canny edge detector is to detect edges in an image. Knowing that edges are a rapid change in pixel intensity and armed with the gradient image created using the method in
the previous post, locating the edges should be straight forward. Visit each pixel and mark it as an edge if the gradient magnitude is greater than some threshold value. Since the gradient
measures the change in intensity and a higher magnitude means a higher rate of change, you could reason that the marked pixels are in fact edges. However, there’s a few problems with this
approach. First, marked edges have an arbitrary thickness which might make it harder to determine if it’s part of a straight line. Second, noise and minor textures that were not completely
suppressed by the Gaussian filter can create false positives. And third, edges which are not evenly lit (eg. partially covered by a shadow) could be partly excluded by the selected threshold.
These issues are handled using a mixture of Non-Maximum Suppression, Hysterasis Thresholding, and Connectivity Analysis. Since Non-Maximum Suppression and Hysterasis Thresholding are easily
performed at the same time, I’m going to cover them together in this post. Connectivity Analysis will be covered next time as the final step in the Canny algorithm.
Non-Maximum Suppression
Non-Maximum Suppression is a line thinning technique. The basic idea isn’t all that different from the naive concept above. The magnitude of the gradient is still used to find pixels that are
edges but it isn’t compared to some threshold value. Instead, it’s compared against gradients on either side of the edge, directed by the current gradient’s angle, to make sure they have smaller
The gradient’s angle points in the direction perpendicular to that of the edge. If you’re familiar with normals, that’s exactly what this is. Then this angle already points to one side of the
edge. But both sides of the edge are compared, so rotate the angle 180 degrees to find the other side.
Examine one of these two angles and look at which surrounding pixels it points to. Depending on the angle, it probably won’t point directly at a single pixel. In the example above, instead of
pointing directly to the right, it points a little above the right. By the discrete nature of working with images, an approximation must be used to determine a magnitude for comparison on each
side of the center gradient. The angle could be snapped to point to the the nearest pixel and respective gradient or an interpolation could be performed between the gradients of the two closest
matching pixels. The latter is more accurate but the former is simpler to implement and performs fine for our purpose so that’s the approach used in the code example below.
Hysterasis Thresholding
Hysterasis Thresholding is used to suppress edges that were erroneously detected by Non-Maximum Suppression. This could be noise or some pattern that’s too insignificant to be an edge. Perhaps
not an obvious solution, but luckily it’s not very complicated. If the magnitude of a gradient is greater than or equal to some high threshold, mark the pixel as a strong edge. If the magnitude
of a gradient is greater than or equal to some low threshold, but less than the high threshold, mark the pixel as a weak edge. A strong edge is assumed to be an actual edge where as a weak edge
is “maybe” an edge. Weak edges can be promoted to strong edges if they’re directly connected to a strong edge but that’s covered in the next post. The high and low thresholds can be determined
experimentally or through various statistical methods outside the scope of this post.
These two concepts are easily implemented together. Start by examining each gradient’s angle to determine which surrounding gradients should be examined. If the current gradient’s magnitude is
greater than both the appropriate surrounding gradients’ magnitudes, proceed to hysterasis thresholding. Otherwise, assume the pixel for the current gradient is not an edge. For the hysterasis
thresholding step, mark the pixel as a strong edge if it’s greater than or equal to the specified high threshold. If not, mark the pixel as a weak edge if it’s greater than or equal to the
specified low threshold. Otherwise, assume the pixel for the current gradient is not an edge. Repeat the process until every gradient is examined.
The following code marks pixels as strong or weak edges by writing to an RGB image initialized to 0. Strong edges set the red channel to 255 and weak edges set the green channel to 255. This
could have easily been accomplished with a single channel image where, say, strong edges are set to 1, weak edges set to 2, and non-edges set to 0. The choice of using a full RGB image here is
purely for convenience because the results are easier to examine.
const std::vector<float> gradient; //Input gradient with magnitude and angle interleaved
//for each pixel.
const unsigned int width; //Width of image the gradient was created from.
const unsigned int height; //Height of image the gradient was created from.
const unsigned char lowThreshold; //Minimum gradient magnitude to be considered a weak
const unsigned char highThreshold; //Minimum gradient magnitude to be considered a strong
Image outputImage(width,height); //Output RGB image. A high red channel indicates a strong
//edge, a high green channel indicates a weak edge, and
//the blue channel is set to 0.
assert(gradient.size() == width * height * 2);
assert(lowThreshold < highThreshold);
//Perform Non-Maximum Suppression and Hysterasis Thresholding.
for(unsigned int y = 1;y < height - 1;y++)
for(unsigned int x = 1;x < width - 1;x++)
const unsigned int inputIndex = (y * width + x) * 2;
const unsigned int outputIndex = (y * width + x) * 3;
const float magnitude = gradient[inputIndex + 0];
//Discretize angle into one of four fixed steps to indicate which direction the
//edge is running along: horizontal, vertical, left-to-right diagonal, or
//right-to-left diagonal. The edge direction is 90 degrees from the gradient
float angle = gradient[inputIndex + 1];
//The input angle is in the range of [-pi,pi] but negative angles represent the
//same edge direction as angles 180 degrees apart.
angle = angle >= 0.0f ? angle : angle + M_PI;
//Scale from [0,pi] to [0,4] and round to an integer representing a direction.
//Each direction is made up of 45 degree blocks. The rounding and modulus handle
//the situation where the first and final 45/2 degrees are both part of the same
angle = angle * 4.0f / M_PI;
const unsigned char direction = lroundf(angle) % 4;
//Only mark pixels as edges when the gradients of the pixels immediately on either
//side of the edge have smaller magnitudes. This keeps the edges thin.
bool suppress = false;
if(direction == 0) //Vertical edge.
suppress = magnitude < gradient[inputIndex - 2] ||
magnitude < gradient[inputIndex + 2];
else if(direction == 1) //Right-to-left diagonal edge.
suppress = magnitude < gradient[inputIndex - width * 2 - 2] ||
magnitude < gradient[inputIndex + width * 2 + 2];
else if(direction == 2) //Horizontal edge.
suppress = magnitude < gradient[inputIndex - width * 2] ||
magnitude < gradient[inputIndex + width * 2];
else if(direction == 3) //Left-to-right diagonal edge.
suppress = magnitude < gradient[inputIndex - width * 2 + 2] ||
magnitude < gradient[inputIndex + width * 2 - 2];
if(suppress || magnitude < lowThreshold)
outputImage.data[outputIndex + 0] = 0;
outputImage.data[outputIndex + 1] = 0;
outputImage.data[outputIndex + 2] = 0;
//Use thresholding to indicate strong and weak edges. Strong edges are assumed
//to be valid edges. Connectivity analysis is needed to check if a weak edge
//is connected to a strong edge. This means that the weak edge is also a valid
const unsigned char strong = magnitude >= highThreshold ? 255 : 0;
const unsigned char weak = magnitude < highThreshold ? 255 : 0;
outputImage.data[outputIndex + 0] = strong;
outputImage.data[outputIndex + 1] = weak;
outputImage.data[outputIndex + 2] = 0;
We’re almost done implementing the Canny algorithm. Next week will be the final step where Connectivity Analysis is used to find weak edges that should be be promoted to strong edges. There will
also be a quick review and some final thoughts on using the algorithm.
• This post is part of a series where I explain how to build an augmented reality Sudoku solver. All of the image processing and computer vision is done from scratch. See the first post for a
complete list of topics.
The second part of the Canny algorithm involves finding the gradient at each pixel in the image. This gradient is just a vector describing the relative change in intensity between pixels. Since a
vector has a magnitude and a direction, then this gradient describes how much the intensity is changing and in what direction.
The gradient is found using the Sobel operator. The Sobel operator uses two convolutions to measure the rate of change in intensity using surrounding pixels. If you studied Calculus, it’s worth
noting that this is just a form of differentiation. One convolution measures the horizontal direction and the other measures the vertical direction. These two values together form the gradient
vector. Below are the most common Sobel kernels. Unlike the Gaussian kernel, the Sobel kernels do not change so they can simply be hard coded.
`K_x = [[-1,0,1],[-2,0,2],[-1,0,1]]`
`K_y = [[-1,-2,-1],[0,0,0],[1,2,1]]`
Where `K_x` is the horizontal kernel and `K_y` is the vertical kernel.
A little more processing is needed to make the gradient usable for the Canny algorithm. If you’ve studied linear algebra or are familiar with polar coordinates the following should be familiar to
First, the magnitude is needed which describes how quickly the intensity of the pixels are changing. By noting that the result of each convolution describes a change along the x- and y-axis,
these two pairs put together form the adjacent and opposite sides of a right triangle. From there it should follow immediately that the magnitude is the length of the hypotenuse and can be solved
using Pythagorean theorem.
`G_m = sqrt(G_x^2 + G_y^2)`
Where `G_m` is the magnitude, `G_x` is the result of the horizontal convolution, and `G_y` is the result of the vertical convolution.
The direction of the gradient in radians is also needed. Using a little trigonometry and the fact that we can calculate the tangent using the `G_x` and `G_y` components
`tan(G_theta) = G_y / G_x`
`G_theta = arctan(G_y / G_x)`
Where `G_theta` is the angle in radians. The `arctan` term requires special handling to prevent a divide by zero. Luckily, C++ (and most programming languages) have a convenient atan2 function
that handles this case by conveniently returning `+-0` or `+-pi`. The C++ variant of atan2 produces radians in the range `[-pi,pi]` with `0` pointing down the positive x-axis and the angle
proceeding in a clockwise direction.
The following example demonstrates how to calculate the gradient for a region of pixels. A typical unsigned 8-bit data type is assumed with the black pixels having a value of 0 and the white
pixels having a value of 255.
`P = [[255,255,255],[255,0,0],[0,0,0]]`
`G_x = P ** K_x = (255 * -1) + (255 * 0) + (255 * 1) + (255 * -2) + (0 * 0) + (0 * 2) + (0 * -1) + (0 * 0) + (0 * 1) = -510`
`G_y = P ** K_y = (255 * -1) + (255 * -2) + (255 * -1) + (255 * 0) + (0 * 0) + (0 * 0) + (0 * 1) + (0 * 2) + (0 * 1) = -1020`
`G_m = sqrt((-510)^2 + (-1020)^2) = 1140.39467`
`G_theta = arctan((-1020)/-510) = -2.03444 text{ radians} = 4.24888 text{ radians}`
Here’s an implementation in code that, once again, ignores the border where the convolution would access out of bounds pixels.
const Image image; //Input RGB greyscale image
//(Red, green, and blue channels are assumed identical).
std::vector<float> gradient(image.width * image.height * 2,0.0f); //Output gradient with
//magnitude and angle
//Images smaller than the Sobel kernel are not supported.
if(image.width < 3 || image.height < 3)
//Number of items that make up a row of pixels.
const unsigned int rowSpan = image.width * 3;
//Calculate the gradient using the Sobel operator for each non-border pixel.
for(unsigned int y = 1;y < image.height - 1;y++)
for(unsigned int x = 1;x < image.width - 1;x++)
const unsigned int inputIndex = (y * image.width + x) * 3;
//Apply the horizontal Sobel kernel.
const float gx =
static_cast<float>(image.data[inputIndex - rowSpan - 3]) * -1.0f + //Top left
static_cast<float>(image.data[inputIndex - rowSpan + 3]) * 1.0f + //Top right
static_cast<float>(image.data[inputIndex - 3]) * -2.0f + //Mid left
static_cast<float>(image.data[inputIndex + 3]) * 2.0f + //Mid right
static_cast<float>(image.data[inputIndex + rowSpan - 3]) * -1.0f + //Bot left
static_cast<float>(image.data[inputIndex + rowSpan + 3]) * 1.0f; //Bot right
//Apply the vertical Sobel kernel.
const float gy =
static_cast<float>(image.data[inputIndex - rowSpan - 3]) * -1.0f + //Top left
static_cast<float>(image.data[inputIndex - rowSpan]) * -2.0f + //Top mid
static_cast<float>(image.data[inputIndex - rowSpan + 3]) * -1.0f + //Top right
static_cast<float>(image.data[inputIndex + rowSpan - 3]) * 1.0f + //Bot left
static_cast<float>(image.data[inputIndex + rowSpan]) * 2.0f + //Bot mid
static_cast<float>(image.data[inputIndex + rowSpan + 3]) * 1.0f; //Bot right
//Convert gradient from Cartesian to polar coordinates.
const float magnitude = hypotf(gx,gy); //Computes sqrt(gx*gx + gy*gy).
const float angle = atan2f(gy,gx);
const unsigned int outputIndex = (y * image.width + x) * 2;
gradient[outputIndex + 0] = magnitude;
gradient[outputIndex + 1] = angle;
Next is another step in the Canny algorithm which uses the gradient created here to build an image with lines where edges were detected in the original image.
• This post is part of a series where I explain how to build an augmented reality Sudoku solver. All of the image processing and computer vision is done from scratch. See the first post for a
complete list of topics.
We captured an image from a connected camera (Linux, Windows) and converted it to greyscale. Now to start searching for a puzzle in the image. Since puzzles are made up of a grid of lines, it
makes sense that the goal should be to hunt down these straight lines. But how should this be done exactly? It sure would be nice if an image could have all of its shading removed and be left
with just a bunch of outlines. Then maybe it would be easier to find straight lines. Enter, the Canny edge detector.
Take a row of pixels from a typical greyscale image and line graph them by intensity. Some sections of the graph will change very little. Other parts will have a much more rapid change indicating
an edge in the image. The Canny edge detector is a multi-step algorithm used to find such edges in an image. A particularly nice property of the algorithm is edges are usually reduced to a
thickness of a single pixel. It’s very useful considering the thickness of an edge can vary as you move along it.
Gaussian Blur
The first step is to apply a Gaussian filter to the image. Gaussian filters are used for blurring images. They’re very good for suppressing insignificant detail such as the noise introduced by a
camera’s capture process.
The Gaussian filter is a convolution based on the Gaussian function (very commonly used in statistics where it describes the normal distribution). Plotting the function produces a bell shaped
curve. It’s usually described using the following formula:
`text{Gaussian}(x) = 1/(sigma^2 sqrt(2pi)) * e^(-x^2/(2sigma^2))`
Where x is the distance from the center of the curve and `sigma` is the standard deviation. The standard deviation is used to adjust the width of the bell curve. The constant multiplier at the
beginning is used as a normalizing term so the area under the curve is equal to 1. Since we’re only interested in a discrete form of the function that’s going to require normalization (more on
this in a moment) anyway, this part can be dropped.
`text{Gaussian}(x) = e^(-x^2/(2sigma^2))`
The standard deviation choice is somewhat arbitrary. A larger standard deviation indicates data points spread out further relative from the mean have more significance. In this case, the mean is
the center of the filter where `x = 0` and significance is the contribution to the output pixel. A convenient method is to choose a factor representing how many standard deviations from the mean
should be considered significant and relating it to the radius of the filter in pixels.
`r = a*sigma`
`sigma = r/a`
Where `r` is the radius, `sigma` is the standard deviation, and `a` is the number of standard deviations. In the case of image processing, letting `a = 3` is usually sufficient.
A convenient property of the Gaussian filter is it’s a separable filter which means that instead of building a 2D (square) convolution matrix, only a 1D convolution matrix is necessary. Then the
convolution is applied in one horizontal pass and one separate vertical pass. Not only does this run faster in practice, but it also simplifies the code.
The first step is to build a kernel by sampling the Gaussian function, defined above, at discrete steps. Then the kernel must be normalized so the sum of each of its values is 1. Normalization is
performed by summing all of the values in the kernel and dividing each value by this sum. If the kernel is not normalized, using it in a convolution will make the resulting image either darker or
lighter depending on if the sum of the kernel is less than or greater than 1, respectively.
const float radius; //Radius of kernel in pixels. Can be decimal.
const float sigma = radius / 3.0f; //Sigma chosen somewhat arbitrarily.
const unsigned int kernelRadius = static_cast<unsigned int>(radius) + 1; //Actual discrete
//sample radius.
//The + 1 helps
//to account for
//the decimal
//portion of the
const unsigned int kernelSize = kernelRadius * 2 + 1; //Number of values in the kernel.
//Includes left-side, right-side,
//and center pixel.
std::vector<float> kernel(kernelSize,0.0f); //Final kernel values.
//Having a radius of 0 is equivalent to performing no blur.
if(radius == 0.0f)
//Compute the Gaussian. Straight from the formula given above.
auto Gaussian = [](const float x,const float sigma) {
const float x2 = x * x;
const float sigma2 = sigma * sigma;
return expf(-x2/(2.0f*sigma2));
//Sample Gaussian function in discrete increments. Technically the function is symmetric
//so only the first `kernelRadius + 1` values need to be computed and the rest can be
//found by copying from the first half. But that's more complicated to implement and
//isn't worth the effort.
float sum = 0.0f; //Keep an accumulated sum for normalization.
for(unsigned int x = 0;x < kernelSize;x++)
const float value = Gaussian(static_cast<float>(x) -
kernel[x] = value;
sum += value;
//Normalize kernel values so they sum to 1.
for(float& value : kernel)
value /= sum;
Applying the kernel is done by performing a convolution in separate horizontal and vertical passes. I’m choosing to ignore the edges here because they aren’t necessary for this project. If you
need them, clamping to the nearest non-out-of-bounds pixels works well.
const std::vector<float> kernel; //Gaussian kernel computed above.
const unsigned int kernelRadius; //Gaussian kernel radius computed above.
const Image inputImage; //Input RGB image.
Image outputImage(inputImage.width,inputImage.height); //Output RGB image. Same width and
//height as inputImage. Initialized
//so every pixel is set to (0,0,0).
assert(kernel.size() == kernelRadius * 2 + 1);
//Convenience function for clamping a float to an unsigned 8-bit integer.
auto ClampToU8 = [](const float value)
return static_cast<unsigned char>(std::min(std::max(value,0.0f),255.0f));
//Since the input image is not blurred around the border in this implementation, then the
//output would just be a blank image if the image is smaller than this area.
if(kernelRadius * 2 > inputImage.width || kernelRadius * 2 > inputImage.height)
return outputImage;
//Blur horizontally. Save the output to a temporary buffer because the convolution can't
//be performed in place. This also means the output of the vertical pass can be placed
//directly into the output image.
std::vector<unsigned char> tempImageData(inputImage.data.size()); //Temporary RGB image
//data used between
for(unsigned int y = 0;y < inputImage.height;y++)
for(unsigned int x = kernelRadius;x < inputImage.width - kernelRadius;x++)
float sum[3] = {0.0f,0.0f,0.0f};
for(unsigned int w = 0;w < kernel.size();w++)
const unsigned int inputIndex =
(y * inputImage.width + x + w - kernelRadius) * 3;
sum[0] += static_cast<float>(inputImage.data[inputIndex + 0]) * kernel[w];
sum[1] += static_cast<float>(inputImage.data[inputIndex + 1]) * kernel[w];
sum[2] += static_cast<float>(inputImage.data[inputIndex + 2]) * kernel[w];
const unsigned int outputIndex = (y * inputImage.width + x) * 3;
tempImageData[outputIndex + 0] = ClampToU8(sum[0]);
tempImageData[outputIndex + 1] = ClampToU8(sum[1]);
tempImageData[outputIndex + 2] = ClampToU8(sum[2]);
//Blur vertically. Notice that the inputIndex for the weights is incremented across the
//y-axis instead of the x-axis.
for(unsigned int y = kernelRadius;y < inputImage.height - kernelRadius;y++)
for(unsigned int x = kernelRadius;x < inputImage.width - kernelRadius;x++)
float sum[3] = {0.0f,0.0f,0.0f};
for(unsigned int w = 0;w < kernel.size();w++)
const unsigned int inputIndex =
((y + w - kernelRadius) * inputImage.width + x) * 3;
sum[0] += static_cast<float>(tempImageData[inputIndex + 0]) * kernel[w];
sum[1] += static_cast<float>(tempImageData[inputIndex + 1]) * kernel[w];
sum[2] += static_cast<float>(tempImageData[inputIndex + 2]) * kernel[w];
const unsigned int outputIndex = (y * inputImage.width + x) * 3;
outputImage.data[outputIndex + 0] = ClampToU8(sum[0]);
outputImage.data[outputIndex + 1] = ClampToU8(sum[1]);
outputImage.data[outputIndex + 2] = ClampToU8(sum[2]);
While the Canny edge detector described here going forward expects a greyscale image, the Gaussian filter works well on RGB or most other image formats. Next time I’ll cover the next step of the
Canny edge detector which involves analyzing how the intensity and direction of pixels change throughout an image.
• This post is part of a series where I explain how to build an augmented reality Sudoku solver. All of the image processing and computer vision is done from scratch. See the first post for a
complete list of topics.
Video frames from a camera are captured on Windows (Vista and later) using the Media Foundation API. The API is designed to handle video/audio capture, processing, and rendering. And… it’s really
a pain to work with. The documentation contains so many words, yet requires jumping all over the place and never seems to explain what you want to know. Adding insult to injury, Media Foundation
is a COM based API, so you can expect lots of annoying reference counting.
Hopefully, I scared you into using a library for your own projects. Otherwise, just like with the Linux version of this post, I’m going to go over how to find all of the connected cameras, query
their supported formats, capture video frames, and convert them to RGB images.
List Connected Cameras
A list of connected cameras can be found using the MFEnumDeviceSources function. A pointer to an IMFAttributes must be provided to specify the type of devices that should be returned. To find a
list of cameras, video devices should be specified. IMFAttributes is basically a key-value store used by Media Foundation. A new instance can be created by calling the MFCreateAttributes
After evaluating the available video devices, make sure to clean-up the unused devices by calling Release on each and CoTaskMemFree on the array of devices. This behavior of having to manually
manage referencing counting everywhere within Media Foundation. It makes proper error handling incredibly tedious so I’ve only inserted assert() calls in the following code snippets for brevity.
#include <mfapi.h>
//Create an instance of IMFAttributes. This class is basically a key-value store used by
//many Media Foundation functions to configure how they should behave.
IMFAttributes* attributes = nullptr;
HRESULT hr = MFCreateAttributes(&attributes,1);
//Configure the MFEnumDeviceSources call below to look for video capture devices only.
hr = attributes->SetGUID(MF_DEVSOURCE_ATTRIBUTE_SOURCE_TYPE,
//Enumerate all of the connected video capture devices. Other devices besides cameras
//might be found like video capture cards.
UINT32 deviceCount = 0;
IMFActivate** devices = nullptr;
hr = MFEnumDeviceSources(attributes,&devices,&deviceCount);
//Evaluate which video devices should be used. The others should be freed by calling
for(UINT32 x = 0;x < deviceCount;x++)
IMFActivate* device = devices[x];
//Use/store device here OR call Release() to free.
//Clean up attributes and array of devices. Note that the devices themselves have not been
//cleaned up unless Release() is called on each one.
Using a Camera
Before anything can be done with a camera, an IMFMediaSource must be fetched from the device. It’s used for starting, pausing, or stopping capture on the device. None of which is necessary for
this project since the device is expected to be always running. But, the IMFMediaSource is also used for creating an IMFSourceReader which is required for querying supported formats or making use
of captured video frames.
#include <mfidl.h>
#include <mfreadwrite.h>
IMFActivate* device; //Device selected above.
//Get a media source for the device to access the video stream.
IMFMediaSource* mediaSource = nullptr;
hr = device->ActivateObject(__uuidof(IMFMediaSource),
//Create a source reader to actually read from the video stream. It's also used for
//finding out what formats the device supports delivering the video as.
IMFSourceReader* sourceReader = nullptr;
hr = MFCreateSourceReaderFromMediaSource(mediaSource,nullptr,&sourceReader);
//Clean-up much later when done querying the device or capturing video.
Querying Supported Formats
Cameras capture video frames in different sizes, pixel formats, and rates. You can query a camera’s supported formats to find the best fit for your use or just let the driver pick a sane default.
There’s certainly an advantage to picking the formats yourself. For example, since cameras rarely provide an RGB image, if a camera uses a format that you already know how to convert to RGB, you
can save time and use that instead. You can also choose to lower the frame rate for better quality at a cost of more motion blur or raise the frame rate for a more noisy image and less motion
The supported formats are found by repeatedly calling the GetNativeMediaType method on the IMFSourceReader instance created above. After each call, the second parameter is incremented until the
function returns an error indicating that there are no more supported formats. The actual format info is returned in the third parameter as an IMFMediaType instance. IMFMediaType inherits from
the IMFAttributes class used earlier so it also behaves as a key-value store. The keys used to look up info about the current format are found on this page.
IMFSourceReader* sourceReader; //Source reader created above.
//Iterate over formats directly supported by the video device.
IMFMediaType* mediaType = nullptr;
for(DWORD x = 0;
&mediaType) == S_OK;
//Query video pixel format. The list of possible video formats can be found at:
GUID pixelFormat;
hr = mediaType->GetGUID(MF_MT_SUBTYPE,&pixelFormat);
//Query video frame dimensions.
UINT32 frameWidth = 0;
UINT32 frameHeight = 0;
hr = MFGetAttributeSize(mediaType,MF_MT_FRAME_SIZE,&frameWidth,&frameHeight);
//Query video frame rate.
UINT32 frameRateNumerator = 0;
UINT32 frameRateDenominator = 0;
hr = MFGetAttributeRatio(mediaType,
//Make use of format, size, and frame rate here or hold onto mediaType as is (don't
//call Release() next) for configuring the camera (see below).
//Clean-up mediaType when done with it.
Selecting a Format
There’s not a lot to setting the device to use a particular format. Just call the SetCurrentMediaType method on the IMFSourceReader instance and pass along one of the IMFMediaType queried above.
IMFSourceReader* sourceReader; //Source reader created above.
IMFMediaType* mediaType; //Media type selected above.
//Set source reader to use the format selected from the list of supported formats above.
hr = sourceReader->SetCurrentMediaType(MF_SOURCE_READER_FIRST_VIDEO_STREAM,
//Clean-up mediaType since it won't be used anymore.
Capturing a Video Frame
There’s a couple of ways to capture a video frame using Media Foundation. The method described here is the synchronous approach which is the simpler of the two. Basically, the process involves
asking for a frame of video whenever we want a one and the thread then blocks until a new frame is available. If frames are not requested fast enough, they get dropped and a gap is indicated.
This is done by calling the ReadSample method on the IMFSourceReader instance. This function returns an IMFSample instance or a nullptr if a gap occurred. The IMFSample is a container that stores
various information about a frame including the actual data in the pixel format selected above.
Accessing the pixel data involves calling the GetBufferByIndex method of the IMFSample and calling Lock on the resulting IMFMediaBuffer instance. Locking the buffer prevents the frame data from
being modified while you’re processing it. For example, the operating system might want to re-use the buffer for frames in the future but writing to it at the same tile as it’s being read will
garble the image.
Once done working with the frame data, don’t forget to call Unlock on it and clean-up IMFSample and IMFMediaBuffer in preparation for future frames.
IMFSourceReader* sourceReader; //Source reader created above.
//Grab the next frame. A loop is used because ReadSample returns a null sample when there
//is a gap in the stream.
IMFSample* sample = nullptr;
while(sample == nullptr)
DWORD streamFlags = 0; //Most be passed to ReadSample or it'll fail.
hr = sourceReader->ReadSample(MF_SOURCE_READER_FIRST_VIDEO_STREAM,
//Get access to the underlying memory for the frame.
IMFMediaBuffer* buffer = nullptr;
hr = sample->GetBufferByIndex(0,&buffer);
//Begin using the frame's data.
BYTE* bufferData = nullptr;
DWORD bufferDataLength = 0;
hr = buffer->Lock(&bufferData,nullptr,&bufferDataLength);
//Copy buffer data or convert directly to RGB here. The frame format, width, and height
//match the media type set above.
//Stop using the frame's data.
//Clean-up sample and buffer since frame is no longer being used.
Converting a Video Frame To RGB
The selected pixel format probably cannot be used directly. For our use, it needs to be converted to RGB^1. The conversion process varies by format. In the last post I covered the YUYV 4:2:2
format. In this one, I’m going to go over the similar NV12 format.
The NV12 format is a Y’CbCr format that’s split into two chunks. The first chunk is the luminance (Y’) channel which contains an entry for each pixel. The second chunk interleaves the Cb and Cr
channels together. There are only one Cb and Cr pair for every 2x2 region of pixels.
Just like with the YUYV 4:2:2 format, the Y’CbCr to RGB conversion can be done by following the JPEG standard. This is a sufficient approach because we are assuming the camera’s color space
information is unavailable.
`R = Y’ + 1.402 * (Cr - 128)`
`G = Y’ - 0.344 * (Cb - 128) - 0.714 * (Cr - 128)`
`B = Y’ + 1.772 * (Cb - 128)`
const unsigned int imageWidth; //Width of image.
const unsigned int imageHeight; //Height of image.
const std::vector<unsigned char> nv12Data; //Input NV12 buffer read from camera.
std::vector<unsigned char> rgbData(imageWidth * imageHeight * 3); //Output RGB buffer.
//Convenience function for clamping a signed 32-bit integer to an unsigned 8-bit integer.
auto ClampToU8 = [](const int value)
return std::min(std::max(value,0),255);
const unsigned int widthHalf = imageWidth / 2;
//Convert from NV12 to RGB.
for(unsigned int y = 0;y < imageHeight;y++)
for(unsigned int x = 0;x < imageWidth;x++)
//Clear the lowest bit for both the x and y coordinates so they are always even.
const unsigned int xEven = x & 0xFFFFFFFE;
const unsigned int yEven = y & 0xFFFFFFFE;
//The Y chunk is at the beginning of the frame and easily indexed.
const unsigned int yIndex = y * imageWidth + x;
//The Cb and Cr channels are interleaved in their own chunk after the Y chunk.
//This chunk is 1/4th the size of the first chunk. There is one Cb and one Cr
//component for every 2x2 region of pixels.
const unsigned int cIndex = imageWidth * imageHeight + yEven * widthHalf + xEven;
//Extract YCbCr components.
const unsigned char y = nv12Data[yIndex];
const unsigned char cb = nv12Data[cIndex + 0];
const unsigned char cr = nv12Data[cIndex + 1];
//Convert from YCbCr to RGB.
const unsigned int outputIndex = (y * imageWidth + x) * 3;
rgbData[outputIndex + 0] = ClampToU8(y + 1.402 * (cr - 128));
rgbData[outputIndex + 1] = ClampToU8(y - 0.344 * (cb - 128) - 0.714 * (cr - 128));
rgbData[outputIndex + 2] = ClampToU8(y + 1.772 * (cb - 128));
That’s it for video capture, now the image is ready to be processed. I’m going to repeat what I said in the opening, go find a library to handle all of this for you. Save yourself the hassle.
Especially if you decide to support more pixel formats or platforms in the future. Next up will be the Canny edge detector which is used to find edges in an image.
1. I have a camera that claims to produce RGB images but actually gives BGR (red and blue are switched) images with the rows in bottom-to-top order. Due to the age of the camera, the hardware is
probably using the .bmp format which is then directly unpacked by the driver. Anyway, expect some type of conversion to always be necessary. ↩
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Algebra (NCTM)
Represent and analyze mathematical situations and structures using algebraic symbols.
Recognize and generate equivalent forms for simple algebraic expressions and solve linear equations
Grade 6 Curriculum Focal Points (NCTM)
Algebra: Writing, interpreting, and using mathematical expressions and equations
Students write mathematical expressions and equations that correspond to given situations, they evaluate expressions, and they use expressions and formulas to solve problems. They understand that
variables represent numbers whose exact values are not yet specified, and they use variables appropriately. Students understand that expressions in different forms can be equivalent, and they can
rewrite an expression to represent a quantity in a different way (e.g., to make it more compact or to feature different information). Students know that the solutions of an equation are the values of
the variables that make the equation true. They solve simple one-step equations by using number sense, properties of operations, and the idea of maintaining equality on both sides of an equation.
They construct and analyze tables (e.g., to show quantities that are in equivalent ratios), and they use equations to describe simple relationships (such as 3x = y) shown in a table. | {"url":"https://newpathworksheets.com/math/grade-6/algebraic-equations-1?dictionary=counting&did=384","timestamp":"2024-11-13T04:29:03Z","content_type":"text/html","content_length":"45415","record_id":"<urn:uuid:2abf495a-de04-4a4c-9983-ab71d05f8a2e>","cc-path":"CC-MAIN-2024-46/segments/1730477028326.66/warc/CC-MAIN-20241113040054-20241113070054-00290.warc.gz"} |
The Relationship Between Mass and Weight in context of weight to force
26 Aug 2024
The Relationship Between Mass and Weight: A Study on the Interplay between Force and Gravity
This paper delves into the fundamental relationship between mass and weight, exploring the intricate connection between these two physical quantities. By examining the concept of force and its
interaction with gravity, we will demonstrate how mass and weight are intertwined, shedding light on the underlying principles that govern their behavior.
Mass (m) and weight (W) are two distinct yet intimately connected concepts in physics. Mass is a measure of an object’s resistance to changes in its motion, whereas weight is the force exerted on an
object by gravity. The relationship between mass and weight is crucial in understanding various physical phenomena, from the behavior of objects on Earth to the dynamics of celestial bodies.
The Force-Gravity Connection
Force (F) is a vector quantity that describes the interaction between two objects or systems. In the context of weight, force is exerted by gravity (g) on an object with mass (m). The formula for
weight can be written as:
W = m × g
where W is the weight in Newtons (N), m is the mass in kilograms (kg), and g is the acceleration due to gravity in meters per second squared (m/s^2).
The Relationship Between Mass and Weight
The relationship between mass and weight is rooted in the concept of force. As mentioned earlier, weight is the force exerted by gravity on an object with mass. This means that for a given mass, the
weight will be proportional to the acceleration due to gravity (g). In other words:
W ∝ m × g
This proportionality can be expressed mathematically as:
W = k × m × g
where k is a constant of proportionality.
In conclusion, this paper has demonstrated the intricate relationship between mass and weight. By examining the concept of force and its interaction with gravity, we have shown that mass and weight
are intertwined, with weight being proportional to both mass and acceleration due to gravity. This understanding is essential for grasping various physical phenomena and has far-reaching implications
in fields such as engineering, astronomy, and beyond.
• [Insert relevant references]
Note: The formulae provided are in ASCII format, without numerical examples.
Related articles for ‘weight to force ‘ :
• Reading: **The Relationship Between Mass and Weight in context of weight to force **
Calculators for ‘weight to force ‘ | {"url":"https://blog.truegeometry.com/tutorials/education/5ec2096654bd4b775b12ac164e45c934/JSON_TO_ARTCL_The_Relationship_Between_Mass_and_Weight_in_context_of_weight_to_f.html","timestamp":"2024-11-12T23:37:25Z","content_type":"text/html","content_length":"16914","record_id":"<urn:uuid:f42a732a-722d-4b78-ad87-f9712ba7fa97>","cc-path":"CC-MAIN-2024-46/segments/1730477028290.49/warc/CC-MAIN-20241112212600-20241113002600-00177.warc.gz"} |
Learning Airfoil Parameters
Learning Airfoil Parameters¶
This is a tutorial to determine the aerodynamic coefficients of a given airfoil using GENN in SMT (other models could be used as well). The obtained surrogate model can be used to give predictions
for certain Mach numbers, angles of attack and the aerodynamic coefficients. These calculations can be really useful in case of an airfoil shape optimization. The input parameters uses the airfoil
Camber and Thickness mode shapes.
• Inputs: Airfoil Camber and Thickness mode shapes, Mach, alpha
• Outputs (options): cd, cl, cm
In this test case, we will be predicting only the Cd coefficient. However, the other databases for the prediction of the other terms are available in the same repository. Bouhlels mSANN uses the
information contained in the paper [1] to determine the airfoil’s mode shapes. Moreover, in mSANN a deep neural network is used to predict the Cd parameter of a given parametrized airfoil. Therefore,
in this tutorial, we reproduce the paper [2] using the Gradient-Enhanced Neural Networks (GENN) from SMT.
Briefly explaining how mSANN generates the mode shapes of a given airfoil:
1. Using inverse distance weighting (IDW) to interpolate the surface function of each airfoil.
2. Then applying singular value decomposition (SVD) to reduce the number of variables that define the airfoil geometry. It includes a total of 14 airfoil modes (seven for camber and seven for
3. Totally 16 input variables, two flow conditions of Mach number (0.3 to 0.6) and the angle of attack (2 degrees to 6 degrees) plus 14 shape coefficients.
4. The output airfoil aerodynamic force coefficients and their respective gradients are computed using ADflow, which solves the RANS equations with a Spalart-Allmaras turbulence model.
See also [3], [4], [5], [6] for more information.
import csv
import os
import numpy as np
WORKDIR = os.path.dirname(os.path.abspath(__file__))
def load_NACA4412_modeshapes():
return np.loadtxt(open(os.path.join(WORKDIR, "modes_NACA4412_ct.txt")))
def load_cd_training_data():
with open(os.path.join(WORKDIR, "cd_x_y.csv")) as file:
reader = csv.reader(file, delimiter=";")
values = np.array(list(reader), dtype=np.float32)
dim_values = values.shape
x = values[:, : dim_values[1] - 1]
y = values[:, -1]
with open(os.path.join(WORKDIR, "cd_dy.csv")) as file:
reader = csv.reader(file, delimiter=";")
dy = np.array(list(reader), dtype=np.float32)
return x, y, dy
def plot_predictions(airfoil_modeshapes, Ma, cd_model):
import matplotlib
import matplotlib.pyplot as plt
# alpha is linearily distributed over the range of -1 to 7 degrees
# while Ma is kept constant
inputs = np.zeros(shape=(1, 15))
inputs[0, :14] = airfoil_modeshapes
inputs[0, -1] = Ma
inputs = np.tile(inputs, (50, 1))
alpha = np.atleast_2d([-1 + 0.16 * i for i in range(50)]).T
inputs = np.concatenate((inputs, alpha), axis=1)
# Predict Cd
cd_pred = cd_model.predict_values(inputs)
# Load ADflow Cd reference
with open(os.path.join(WORKDIR, "NACA4412-ADflow-alpha-cd.csv")) as file:
reader = csv.reader(file, delimiter=" ")
cd_adflow = np.array(list(reader)[1:], dtype=np.float32)
plt.plot(alpha, cd_pred)
plt.plot(cd_adflow[:, 0], cd_adflow[:, 1])
plt.legend(["Surrogate", "ADflow"])
plt.title("Drag coefficient")
Drag coefficient prediction (cd): 0.009881152860141375 | {"url":"https://smt.readthedocs.io/en/latest/_src_docs/examples/airfoil_parameters/learning_airfoil_parameters.html","timestamp":"2024-11-12T07:25:48Z","content_type":"text/html","content_length":"24547","record_id":"<urn:uuid:c18d2258-5025-4ec4-bb44-e10262cb7f13>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.58/warc/CC-MAIN-20241112045844-20241112075844-00568.warc.gz"} |
How to apply a boundary condition to a range of DOFs?
Dear @mb246,
You have not put much effort into making this into a minimal reproducible example.
I am assuming that your dofs in range 1-6 (I am assuming that you are using 1-indexing) and that you are referring to degrees of freedom located at vertex 1-6 in the input mesh.
You need to map the input vertices to the dofs on your process.
See for instance:
and subsequent posts.
As clearly illustrated by the dolfinx demos, Dirichlet boundary conditions are applied by giving the indices of the dofs as input to dolfinx.fem.dirichletbc, you just need to map your global input
vertex numbering to the local dof index on your process. | {"url":"https://fenicsproject.discourse.group/t/how-to-apply-a-boundary-condition-to-a-range-of-dofs/15758/2","timestamp":"2024-11-03T13:44:15Z","content_type":"text/html","content_length":"25241","record_id":"<urn:uuid:c80e7a9c-2e0b-4b0f-9589-256a0d1761f7>","cc-path":"CC-MAIN-2024-46/segments/1730477027776.9/warc/CC-MAIN-20241103114942-20241103144942-00710.warc.gz"} |
Measuring the EIU Democracy Index (with Polity IV) | R-bloggersMeasuring the EIU Democracy Index (with Polity IV)
Measuring the EIU Democracy Index (with Polity IV)
[This article was first published on
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Yet again, I have conjured up an (academically) unusual dataset on democracy! This time it’s the Economist Intelligence Unit’s Democracy Index, a weird little gem. The dataset is the basis for a
paper the Economist publishes every two years. Because of this biannuality, there is data estimating the “Democratic-ness” of the world’s countries for 2006, 2008 and 2010. What happened between
those years, God only knows. I dumped the data into a CSV file and merged them with the polity data. Since they’re both measures of democracy, I hypothesize (again) that they should be fairly
linearly correlated. Shall we take a peek?
EIUmergePolity <- read.csv("http://nortalktoowise.com/wp-content/uploads/2011/07/EIUmergePolity.csv")
model1 <- lm(polity2 ~ Overall)
lm(formula = polity2 ~ Overall)
Min 1Q Median 3Q Max
-9.9426 -2.4366 0.1764 2.1389 10.5728
Estimate Std. Error t value Pr(>|t|)
(Intercept) -9.39327 0.42154 -22.28 <2e-16 ***
Overall 2.41504 0.07162 33.72 <2e-16 ***
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 3.475 on 470 degrees of freedom
(35 observations deleted due to missingness)
Multiple R-squared: 0.7076, Adjusted R-squared: 0.7069
F-statistic: 1137 on 1 and 470 DF, p-value: < 2.2e-16
Yet again, we find a good linear model which just isn’t all that impressive. An R-squared of .7 is not half bad for an explanatory model, but these are two different measures of the same thing.
They should be more closely correlated than that. Let’s take a look, shall we?
plot(Overall, polity2, main="Democracy Index over Polity2 Score")
Damn! It looks kind of non-linear. Again. Taking the same technique I used to fit a quadratic curve to the data last time, maybe we’ll get closer.
model2 <- lm(polity2 ~ Overall + I(Overall^2))
lm(formula = polity2 ~ Overall + I(Overall^2))
Min 1Q Median 3Q Max
-10.9995 -1.3065 0.3161 1.5667 11.9635
Estimate Std. Error t value Pr(>|t|)
(Intercept) -14.51086 0.88953 -16.313 < 2e-16 ***
Overall 4.70622 0.36131 13.026 < 2e-16 ***
I(Overall^2) -0.21243 0.03289 -6.459 2.64e-10 ***
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 3.334 on 469 degrees of freedom
(35 observations deleted due to missingness)
Multiple R-squared: 0.7314, Adjusted R-squared: 0.7303
F-statistic: 638.7 on 2 and 469 DF, p-value: < 2.2e-16
Doesn’t help much, I’m sad to say. We see a small increase in the model’s descriptive value, but it’s really small. If we draw the curve to the scatterplot, will it at least look better?
plot(Overall, polity2, main="Democracy Index over Polity2 Score")
x <- seq(0,10)
y <- model2$coef %*% rbind(1,x,x^2)
Yeah, it looks a little better, but there’s not much that can be done. It seems that there’s a fundamental inconsistency in the way we measure governance, particularly at the extreme ends of the
spectrum. Perhaps this is characteristic of data measuring the same thing in different ways. Anyone know any literature examining this sort of thing? | {"url":"https://www.r-bloggers.com/2011/07/measuring-the-eiu-democracy-index-with-polity-iv/","timestamp":"2024-11-15T03:28:00Z","content_type":"text/html","content_length":"93561","record_id":"<urn:uuid:ec9d6f79-251e-4d5c-a28f-58471e7b3ab5>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00872.warc.gz"} |
Inches to Cm
The history of measurement scales has been quite varied and extensive. In the past, many different distance units were used to measure the length of an object. In the absence of any standard unit for
measurement, people utilized body parts such hand, foot and cubit for the purpose of measuring any height of any object or humans. These units were not uniform and varied in length from one era to
another. With the development of metric system in late 18th century a uniform measurement system came into existence and standards in respect to measurement were set. The system was adopted by all
the countries across the world and it was then when a standard scale for measuring Centimeter and Inch was devised.
Inch: A customary unit of length, an inch is equal to 2.54 centimeters. The standard length for the inch varied from place to place in the past and it was in the year 1959 that International Yard was
defined and Inch was measured exact the same length all over the world. A yard was defined as 36 inches on an inch scale and 0.9144 meters to be precise. An inch is denoted by the sign in’ and it is
represented by double prime.
Uses of Inch scale
Inch is used as a standard unit of measurement of length for electronic products like TV and computer screen as well as Mobiles.
The scale of inch is also used in the measurement of objects like doors, ceilings as well as other items that are shorter than a meter and are not practical to be measured with centimeters.
Centimeter: A unit of length, a centimeter is equivalent to 100th of a meter. Centimeter is also termed or is known as the base unit of length and is used as the standard unit of measurement for
measuring height of a person or an object. Being the standard unit of length, centimeter finds greater acceptability in daily life and is considered as the best pragmatic approach for routine
Uses of centimeter
• To measure the height of a person or any object.
• To claim the amount of rainfall with the help of a rain gauge
• Centimeter is also used in maps to convert the map scale into practical world distances
How to convert Inches into centimeters
When going for an inch to centimeter conversion, you simply need to divide the given number of centimeters with 2.54. For example if you want to convert 100 centimeters into inches, then you can
simply go for 100/2.54 to obtain the required answer.
Inches scale is also another way to convert inches into centimeter. An inches scale always comes with equivalent centimeters feature and you can simply count the number of centimeters for given
inches. This technique however suffers the demerit of lack of conversion with large numbers and you can only compare digits up-to which the scale facilitates.
Converting inches into centimeters is not at all a big task and you can calculate it anywhere with the help of online calculator. With online calculator or virtual inches scale you simply need to
input the number of inches that you want to convert into centimeters. Once you press the enter you will be to avail the answer regarding the number of centimeters for a given amount of inches.
Now, we hope, you are very much aware about the history of inches and CM scales, and can convert inches to CM and vice versa. | {"url":"https://convertsumo.com/105-inches-to-cm/","timestamp":"2024-11-07T12:40:00Z","content_type":"text/html","content_length":"20201","record_id":"<urn:uuid:f2c173da-e389-4b9f-b01a-42c863c4ba1c>","cc-path":"CC-MAIN-2024-46/segments/1730477027999.92/warc/CC-MAIN-20241107114930-20241107144930-00604.warc.gz"} |
The Graph Funding by Chain Broker
Number of ProjectsProjects3156
Number of GainersGainers340
Number of LosersLosers1047
The Graph
Indexing protocol for querying networks like Ethereum and IPFS. Anyone can build and publish open APIs, called subgraphs, making data easily accessible
Project Fundamentals
4.29 / 10
Performance Metrics
5.66 / 10
Market & Financial Health
4.36 / 10
Social & Development
9.53 / 10
Roadmap & Timeline
2.29 / 10
Private Price
Period Change Low Bounce High Dip Volatility
InitialCurrent (86.6%)Total
The Graph Token Distribution Chart
Stage Price Total Raise Valuation Vesting Period
Seed Round $0.00147 $2.5M $14.7M 0.0% tge, 6 months cliff, 5.56% monthly
Series A Round $0.00294 $5M $29.4M 0.0% tge, 6 months cliff, 5.56% monthly
Public Round $0.03 $12M $300M 100% unlocked
Community Round $0.026 $5.2M $260M 0.0% tge, 12 months cliff, 100.0% monthly
Team 0.0% tge, 12.5% every half a year
Marketing 100% unlocked
Rewards 0.0% tge, 12 months cliff, 8.33% monthly
Operations 12.5% tge, 8.75% every half a year
Mining Indexing rewards will begin at 3% annually and is subject to future independent technical governance
Foundation 23.0% tge, 5.0% every half a year
Grants 10.0% tge, 6.0% quarterly
What is The Graph?
All the necessary data for a reliable project research
This page contains The Graph key metrics such as: current price, market cap, trading volume, price change, and other relevant information like project backers, twitter network and public sale
platforms, that can help investors make informed decisions about investing in The Graph.
The Graph Price Chart
A price chart for The Graph displays the historical trading data for the asset, including private and public prices, highest and lowest prices, and the current price. The chart is designed to help
traders analyze the price movements of The Graph and understand how current price relates to other price points. The chart will typically be accompanied by data, like: current price, all time high
price, all time low price, private price, and public price.
The Graph ATH Price
ATH stands for All-Time High. In the context of cryptocurrency, it refers to the highest price that a particular coin or token has ever reached. For example, if the price of Bitcoin reached $40,000
for the first time, that would be considered its ATH.
The Graph ATH Price is $2.84
The Graph ATL Price
The All-Time Low (ATL) price is the lowest value the token has ever reached on a crypto exchanges. It's a historical data point which can help to gauge the token performance over time and make
investment decisions.
The Graph ATL Price is $0.05205
The Graph Public Price
The public price - is the value of the token at initial offering either via initial coin offering, initial dex offering, or initial exchange offering. This is the price a token or a coin was offered
to public before it is available for trading. Public price is useful for traders and investors to gauge the advantage of current value of an asset over value offered to the public investors.
The Graph Public Price is $0.03
The Graph Private Price
The private price - refers to the value of the token, that is negotiated and agreed upon between private parties, usually institutional investors or high net worth individuals. These prices are not
publicly available and are not influenced by the overall market conditions or trading activity. They may be based on a set of specific terms and conditions, such as lock-up periods or minimum
investment amounts, and may differ significantly from the public price.
The Graph Private Price is $0.00294
The Graph Current Price
The curent price is the current market value of a token or a coin, available on crypto exchanges, changes constantly, influenced by market conditions, news & trading activity. Useful for traders and
investors to make informed decisions.
The Graph Current Price is $0.1799
The Graph Price Change for 24h, 7d, 14d, 30d and 1y periods
The price change for a specific period for The Graph project - shows how the token's value has changed over that period of time. It's a useful metric to gauge the token's performance.
• 24 hours - -7.09% (Seven point zero nine percent);
• 7 days - 16.6% (Sixteen point six percent);
• 14 days - 14.2% (Fourteen point two percent);
• 30 days - 1.21% (One point two one percent);
• 1 year - 44.7% (Forty four point seven percent);
The Graph Market Cap
Market Cap for The Graph is a metric to determine the current value of the project. Calculated by multiplying all the tokens in circulation by the current price. It is a metric used to gauge the size
of the project and its relative importance in the crypto market.
The Graph Initial Market Cap
Initial Market Cap for The Graph crypto project is the total value of available tokens in circulation at the time of the initial coin offering (ICO) or initial listing on a cryptocurrency exchange.
It is calculated by multiplying the initial price of the token by the initial circulation of the token. It's a metric used to gauge the size of the project and its relative importance in the crypto
market at the initial stage of the token's life cycle. It's also a benchmark to evaluate the token performance over time.
The Graph Initial Market Cap is $38.7M
The Graph Initial FDMC
Initial FDMC (Fully Diluted Market Cap) of The Graph project - refers to the project valuation at the time of the initial coin offering (ICO) or initial listing on a cryptocurrency exchange.
The Graph Initial FDMC is $300M
The Graph FDMC
FDMC (Fully Diluted Market Cap) for The Graph is the project valuation at current price. It takes into account all the tokens including available for trading in the open market and the ones that are
locked up or restricted. It's a metric used to gauge the liquidity of a token and the market's perception of its value.
The Graph FDMC is $1.8B
The Graph Market Cap
Market Cap is the total value of all the tokens in circulation, calculated by multiplying the current price of the token by the current circulation of the token. It is a metric used to gauge the size
of the project and its relative importance in the crypto market.
The Graph Market Cap is $1.56B
The Graph Moonsheet
A moonsheet represents a prediction or a forecast of a future high value for the token. It's a term used by crypto traders and investors for a token that is expected to experience a significant price
increase, similar to the moon landing.
The Graph Moonsheet ROI
Moonsheet ROI - is the expected return on investment for the token, based on the predicted future value of the token. It's a metric used by crypto traders and investors to evaluate the potential
profitability of the investment in the project.
The Graph Moonsheet Price
Moonsheet price - is the predicted future high value of the token. It's a term used by crypto traders and investors for a token that is expected to experience a significant price increase, similar to
the moon landing. It's a forecasted number that can be used for investment decisions.
The Graph Moonsheet Market Cap
Moonsheet Market Cap - is the predicted future market capitalization for the token based on the predicted future value of the token. It's a metric used by crypto traders and investors to evaluate the
potential size of the market for the token in the future.
Frequently asked questions
What is the price prediction for The Graph?
The price for any crypto project is solely dependent by demand & supply. It crypto project supply is usually available to the public. The current supply of The Graph is 8662500000.0, however the
total supply is 10000000000.0. Increase in the supply cause decrease in the price given constant demand. As for the demand, it is a determined by various factos, like: utility, project updates, and
most important community support. The great starting point for The Graph price prediction would be to join its community and analyze their support. Nobody would be able to predict the price, but some
key metrics to watch for: future unlocks, price change trend, social activity, valuation, and volume.
How much is The Graph crypto worth?
The value of the project is a very important concept to understand before any investment decision. Never look at The Graph price alone. The current price is just the last aggreed price between the
seller and a buyer. The correct way to understand The Graph value is to look at current market cap, which is 1558141200.0. It is also a good habbit to watch for fully diluted market cap, which is
also known as project valuation (value projection at current price). The Graph FDMC is 1798720000.0. To add, always consider the project liquidity and volume, which give a larger picture of the
project value.
Where can I buy The Graph?
There are several options to buy a crypto asset: decentralized exchanges and centralized exchanges. The Graph is available to trade at Coinbase, Binance | {"url":"https://chainbroker.io/projects/the-graph/","timestamp":"2024-11-14T00:47:11Z","content_type":"text/html","content_length":"263363","record_id":"<urn:uuid:782fd5fb-8662-4099-a18c-16f1ba2d23d0>","cc-path":"CC-MAIN-2024-46/segments/1730477028516.72/warc/CC-MAIN-20241113235151-20241114025151-00029.warc.gz"} |
force constant
1. Injuries and trades have forced constant adjustments to the Blazers'lineup.
2. His dissertation was titled " A quantum-mechanical calculation of inter-atomic force constants in copper ."
3. Here x _ i is the state of the system and F is a forcing constant.
4. The force constant would be 100x9.80665 = 980.665.
5. Most commonly, an electronic feedback loop is employed to keep the probe-sample force constant during scanning.
6. This gives the ligand a higher force constant.
7. In the end, it all came back to Wilson, a hero whose humanity forces constant reconsideration.
8. Where k is the spring constant ( or force constant ), which is particular to the spring.
9. Denoted } }, it is also called the electric force constant or electrostatic constant, hence the subscript.
10. Many commercial AFM cantilever tips have pre-measured resonant frequencies and force constants which are provided to the customer. | {"url":"https://www.hindlish.com/force%20constant/force%20constant-meaning-in-hindi-english","timestamp":"2024-11-09T01:24:20Z","content_type":"text/html","content_length":"38515","record_id":"<urn:uuid:d77169b6-a3c9-4717-9592-9bdccde36af2>","cc-path":"CC-MAIN-2024-46/segments/1730477028106.80/warc/CC-MAIN-20241108231327-20241109021327-00331.warc.gz"} |
Hundred is a square grid, whose cells are to be filled by some digits. The task is to fill additional digits in required cells such that the sum of numbers in each row and each column equals to 100.
Cross+A can solve puzzles of 3 x 3 and 4 x 4. | {"url":"https://cross-plus-a.com/html/cros7hdr.htm","timestamp":"2024-11-07T00:27:09Z","content_type":"text/html","content_length":"1106","record_id":"<urn:uuid:555f95c4-b7bc-40de-ba9b-e7b95e543788>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00392.warc.gz"} |
Department of Operational Sciences
Theses/Dissertations from 1999
Modeling and Analysis of Aerial Port Operations, Timothy W. Albrecht
Visualizing Early-Stage Breast Cancer Tumors in a Mammographic Environment through a 3-Dimensional Mathematical Model, Christopher B. Bassham
Analysis of a Non-Trivial Queueing Network, Kelly Scott Bellamy
The Quota Allocation Model: The Linear Optimization of a Markov Decision Process, David A. Brown
Quality Function Deployment from an Operations Research Perspective, Eve M. Burke
A Comparison of Genetic Algorithm Parametrization on Synthetic Optimization Problems, Mehmet Eravsar
A Bayesian Decision Model for Battle Damage Assessment, Daniel W. Franzen
Pilot Inventory Complex Adaptive System (PICAS): An Artificial Life Approach to Managing Pilot Retention, Martin P. Gaupp
Efficient Simulation via Validation and Application of an External Analytical Model, Thomas H. Irish
An Approach for Tasking Allocated Combat Resources to Targets, David A. Koewler
Selection of Psychophysiological Features across Subjects for Classifying Workload Using Artificial Neural Networks, Trevor I. Laine
Using Simulation to Model The Army Recruiting Station With Multi-Quality Prospects, Edward L. McLarney
Two-Stage Stochastic Linear Programming with Recourse: A Characterization of Local Regions using Response Surface Methodology, David T. Mills
Portfolio Selection of Innovative Technologies via Life Cycle Cost Modeling, Scott C. Naylor
A Validation Assessment of THUNDER 6.5's Intelligence, Surveillance, and Reconnaissance Module, Francine N. Nelson
Dynamic Unmanned Aerial Vehicle (UAV) Routing with a Java-Encoded Reactive Tabu Search Metaheuristic, Kevin P. O'Rourke
Solving the Multidimensional Multiple Knapsack Problem with Packing constraints using Tabu Search, Jonathan M. Romaine
Implementation of the Metaheuristic Tabu Search in Route Selection for Mobility Analysis Support System, David M. Ryer
A Value Focused Approach to Determining the Top Ten Hazards in Army Aviation, Brian K. Sperling
Strategic Effects of Airpower and Complex Adaptive Agents: An Initial Investigation, Thomas R. Tighe
Multi-Discipline Network Vulnerability Assessment, Karilynne Wallace
Unmanned Aerial Vehicle Mission Level Simulation, Jennifer G. Walston
An Evaluation and Comparison of the ACE and BRACE Airfield Models, David W. Wiliams
Technology Selection for the Air Force Research Laboratory Air Vehicles Directorate: An Analysis Using Value Focused Thinking, Michael F. Winthrop
Theses/Dissertations from 1998
Forecasting Market Index Performance Using Population Demographics, Bradley J. Alden
Calculating a Value for Dominant Battlespace Awareness, Eric A. Beene
Parameter Estimation of the Mixed Generalized Gamma Distribution Using Maximum Likelihood Estimation and Minimum Distance Estimation, Dean G. Boerrigter
A VFT Approach to Allocation of Manpower and Budget Cuts, Thomas G. Boushell
Sensitivity Analysis of Brawler Pilot Skill Levels, Daniel C. Buschor
Solving Geometric Knapsack Problems using Tabu Search Heuristics, Christopher A. Chocolaad
Using Simulation to Model Time Utilization of Army Recruiters, James D. Cordeiro Jr. and Mark A. Friend
An Airlift Hub-and-Spoke Location-Routing Model with Time Windows: Case Study of the CONUS-to-Korea Airlift Problem, David W. Cox
Sensitivity Analysis of the Thunder Combat Simulation Model to Command and Control Inputs Accomplished in a Parallel Environment, David A. Davies
A Life Cycle Cost Model for Innovative Remediation Technologies, Osman S. Dereli
The Application of Statistical Sampling Techniques to the Operational Readiness Inspection, Troy L. Dixon and Timothy S. Madgett
A Value Function Approach to Information Operations Measures of Merit, Michael P. Doyle
Generating Bomber Routes for the Delivery of Gravity Weapons, Gregory J. Ehlers
Feature Saliency in Artificial Neural Networks with Application to Modeling Workload, Kelly A. Greene
An Operational Review of Air Campaign Planning Automation, William R. Haas
A Model to Forecast Civilian Personnel Inventory for the National Security Agency, Stephen G. Hoffman
A Value Focused Thinking Approach to Academic Course Scheduling, Shane A. Knighton
A Network Disruption Modeling Tool, James A. Leinart
Cross-Resolution Combat Model Calibration Using Bootstrap Sampling, Bryan S. Livergood
Optimizing the Efficiency of a Multi-Stage Axial-Flow Compressor: An Application of Stage-Wise Optimization, Shawn A. Miller
A Variable-Complexity Modeling Approach to Scramjet Fuel Injection Array Design Optimization, Michael D. Payne
Verification and Validation of FAARR Model and Data Envelopment Analysis Models for United States Army Recruiting, Gene M. Piskator
Parallel Implementation of an Artificial Neural Network Integrated Feature and Architecture Selection Algorithm, Craig W. Rizzo
Optimum Preventive Maintenance Policies for the AMRAAM Missile, Scott J. Ruflin
Embedding a Reactive Tabu Search Heuristic in Unmanned Aerial Vehicle Simulations, Joel L. Ryan
A Game-Theoretic Improvement Model for Stochastic Networks: Reliability vs. Throughput, Jeffrey A. Schavland
Analysis of Alternatives: Multivariate Considerations, John J. Siegner
The Application of Sequential Convex Programming to Large-Scale Structural Optimization Problems, Todd A. Sriver
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Personnel Airdrop Risk Assessment Using Bootstrap Sampling, Wonsik Kim | {"url":"https://scholar.afit.edu/operations-logistics/index.10.html","timestamp":"2024-11-05T09:09:40Z","content_type":"text/html","content_length":"102488","record_id":"<urn:uuid:341d88f4-0378-49e8-b67c-e12122457e22>","cc-path":"CC-MAIN-2024-46/segments/1730477027878.78/warc/CC-MAIN-20241105083140-20241105113140-00460.warc.gz"} |
Unruh Effect and Information Entropy Approach
Department of Physics, University of Oslo, PB 1048 Blindern, N-0316 Oslo, Norway
Faculty of Physics, Taras Shevchenko National University of Kyiv, UA-03022 Kyiv, Ukraine
Skobeltsyn Institute of Nuclear Physics, Moscow State University, RU-119991 Moscow, Russia
Author to whom correspondence should be addressed.
These authors contributed equally to this work.
Submission received: 1 May 2022 / Revised: 22 May 2022 / Accepted: 25 May 2022 / Published: 27 May 2022 / Corrected: 18 April 2024
The Unruh effect can be considered a source of particle production. The idea has been widely employed in order to explain multiparticle production in hadronic and heavy-ion collisions at
ultrarelativistic energies. The attractive feature of the application of the Unruh effect as a possible mechanism of the multiparticle production is the thermalized spectra of newly produced
particles. In the present paper, the total entropy generated by the Unruh effect is calculated within the framework of information theory. In contrast to previous studies, here the calculations are
conducted for the finite time of existence of the non-inertial reference frame. In this case, only a finite number of particles are produced. The dependence on the mass of the emitted particles is
taken into account. Analytic expression for the entropy of radiated boson and fermion spectra is derived. We study also its asymptotics corresponding to low- and high-acceleration limiting cases. The
obtained results can be further generalized to other intrinsic degrees of freedom of the emitted particles, such as spin and electric charge.
1. Introduction
As was demonstrated by Unruh [
], the observer comoving with the non-inertial reference frame (RF) with the acceleration
will detect particles thermalized at temperature
in Planck units, whereas the observer in any inertial RF will see bare vacuum. If the acceleration
equals the surface gravity of some Schwarzschild black hole (BH), when the observer is at the horizon,
coincides with the temperature
$T BH$
of the Bekenstein–Hawking radiation [
] of the horizon.
This peculiar non-invariance of the vacuum has raised a lot of interest in the topic (for review see, e.g., [
] and references therein). Recall that the Unruh effect was initially derived for scalar particles. Here, the change in the ratio between the negative and positive frequency modes of scalar fields in
the noninertial RF was considered [
]. Generalizations to arbitrary trajectories of the observer is discussed in [
], whereas the generalization to the accelerated reference frames with rotation can be found in [
]. The emergence of the Unruh effect in the Rindler manifold of an arbitrary dimension and its relationship to the vacuum noise and stress are investigated in [
]. Various methods and approaches have been employed. For instance, an algebraic approach was used to extend the Unruh effect to theories with arbitrary spin and with interaction [
], whereas the path integral approach was applied to derive the effect for fermions within the framework of quantum field theory [
]. Among the recent studies, one can mention the relativistic quantum statistical mechanics approach [
] based on the application of Zubarev’s density operator [
]. Within this approach, the Unruh effect was obtained first for the scalar particles [
] and then generalized to the gas of massless fermions [
]. In the present study, we employ the approach based on the application of the information theory, which is a promising tool to study the black hole information dynamics, as one may see in [
] or reviews [
Usually, the non-inertial observer is assumed to accelerate forever. However, such an assumption implies the availability of an infinite energy supply and ever-lasting particle emission. The more
sophisticated scenario, which considers the Unruh effect at finite time interval, is analyzed in papers [
There are a lot of proposals for the detection and application of the Unruh effect, see, e.g., [
]. The paper [
] discusses the possibility of eavesdropping in the non-inertial reference frame. The production of the entangled photon pairs from the vacuum with the help of the Unruh effect was investigated in [
], whereas, in [
], the creation of accelerated black holes by means of the Unruh effect was studied. In [
], the authors discuss the possibility of using accelerated electrons as thermometers; more on the topic can be found in Refs. [
]. Generated bosons and fermions were considered to be produced via the quantum tunneling mechanism at the Unruh horizon in [
The Unruh effect can be considered as a source for the creation of new particles. This idea has been widely employed [
] in order to explain multiparticle production in hadronic and heavy-ion collisions at ultrarelativistic energies. The attractive feature of the application of the Unruh effect as a possible
mechanism of multiparticle production is the thermalized spectra of newly produced particles. Experiments with ultrarelativistic hadronic and heavy-ion collisions and their theoretical
interpretations indicate that the produced matter seems to reach equilibrium extremely quickly, see, e.g., [
] for the present status of the field. The mechanism of this fast equilibration is still debated; therefore, the Unruh effect might be of great help. At the same time, since the Unruh source is
thermal, it results in the observer-dependent entropy generation [
]. In the present paper, we also consider the Unruh horizon as a thermal source of particles. These particles are characterized by thermal distribution. Our aim is to estimate the entropy of the
distribution and to define its dependence on any intrinsic degrees of freedom of the emitted particles.
In this paper, we consider Unruh radiation at some fixed energy
, which is assumed to be a parameter. It can be argued that such an analysis is incorrect because we should have taken into account the all-energy modes
$E i$
via the product
$∏ E i$
in the corresponding density matrix, see, e.g., [
]. This approach implies an independent emission of modes at different energies. In other words, in that case, one deals with the energy modes via the tensor product of the corresponding subspaces.
However, such a generalization is not mandatory for the Unruh effect. For instance, in [
], the authors demonstrate that one can obtain the Unruh effect for some fixed energy without any need to take a product of all the modes to encompass the all-Unruh thermal bath states. This
circumstance allows us to consider energy
of the mode as a parameter and, therefore, to take into account any correlations between the modes originating from the finite energy supply and restrictions imposed by energy conservation.
The paper is organized as follows.
Section 2
presents the necessary basics from the probability theory and the information theory.
Section 3
briefly describes the Unruh effect and the density matrix of the emitted quanta. The total entropy of the Unruh source is estimated in
Section 4
. Here, the general expression for the entropy of fermion and boson radiation is derived, as well as its analytic series expansion. In
Section 5
, one is dealing with the analysis of temperature asymptotics of the entropy. Two limiting cases corresponding to low and high temperatures, or, equivalently, the acceleration of the observer, are
Section 6
is devoted to the contribution of intrinsic degrees of freedom of the produced particles. Final remarks and conclusions can be found in
Section 7
2. Probability and Entropy
Let us consider some distribution
${ X }$
with the unnormalized distribution probability
$d x$
. In other words,
$d x$
is a number of events in which
is being observed. Shannon entropy
$H X$
may be written as
$H X = − ∑ x d x D X ln d x D X = ln D X − 1 D X ∑ x d x ln d x ,$
$D X = ∑ x d x$
$H X$
encodes the amount of information we need in order to completely describe
${ X }$
, i.e., this is amount of information we are lacking. Therefore, we should deal with the distribution
${ X }$
. It is scale-invariant, so it does not change under the transformations
$d x → α d x$
for any
$α = const$
Similarly, for joint distribution
${ X , Y }$
with the unnormalized distribution probability
$d x , y$
, one can write down Shannon entropy
$H X , Y$
$H X , Y = − ∑ x , y d x , y D X , Y ln d x , y D X , Y = ln D X , Y − 1 D X , Y ∑ x , y d x , y ln d x , y ,$
$D X , Y = ∑ x , y d x , y$
In the joint case, one may define the conditional probability
$d x | y$
$d x | y = d x , y d y , d y = ∑ x d x , y .$
It defines the amount of events with
from the set of events in which
occurs. Using Equation (
), Shannon entropy
$H X | y$
$H X | y = ln D X | y − 1 D X | y ∑ x d x | y ln d x | y = − ∑ x d x | y ln d x | y ,$
$D X | y = ∑ x d x | y = 1$
, as follows from Equation (
Finally, substituting Equations (
) and (
) into Equation (
), one obtains
$H X , Y = H Y + H X | y Y = H X + H Y | x X ,$
where averaging taken over
$A Z = 1 D Z ∑ z d z A , Z ≡ X , Y .$
Recall that all the formulae above are valid for the discrete distributions only. In the continuous case, one should use the probability density function (PDF)
$p x$
instead of
$d x$
. Shannon entropy becomes dimensionally incorrect and should be re-defined, as shown in [
For the distribution
${ X }$
with the PDF
$p x$
, the entropy given by Equation (
) is generalized to
$H X p = ln D X p − 1 D X p ∫ p x ln p x d x − ln d x X p ,$
$D X p = ∫ p x d x$
is the norm and
$A X p = 1 D X p ∫ p x A d x .$
The last term in Equation (
) is related to the limiting density of discrete points and takes into account the amount of information encoding a discrete-continuum transition (see [
] for details). The term originates from the fact that the PDF
$p x$
is not dimensionally invariant compared to the discrete probability
$d x$
. The last one can be set to be dimensionless—see the explanation below Equation (
$p x$
cannot. In any realistic computational task, the term determines the contribution of the bin widths
$d x$
of the distribution to the entropy. Note that one may formally reduce
$H X p$
$H X$
by substituting
$∫ p x d x$
$∑ x d x$
and setting
$ln d x X p$
to zero; the same procedure is valid in the opposite direction.
3. Unruh Effect
From here, we will use Planck (or natural) units, $c = G = ℏ = k B = 1$. Furthermore, we restrict our analysis to $1 + 1$-dimensional space-time because two other spatial dimensions play no role and,
therefore, can be neglected.
As was already mentioned in
Section 1
, vacuum is non-invariant with respect to the reference frame [
]. In the non-inertial RF determined with the acceleration
, one meets the appearance of horizon that separates space-time into the inside and outside domains. As a result, the non-inertial observer detects the radiation going out from the horizon, while the
inertial one detects the Minkowski vacuum state
only. For bosons, the latter reads [
$0 = 1 − exp − E / T 1 − exp − N E / T ∑ n = 0 N − 1 exp − n E / 2 T n in n out ,$
whereas, for fermions, one obtains
$0 = 1 1 + exp − E / T ∑ n = 0 1 exp − n E / 2 T n in n out$
is the energy of the quanta emitted at the Unruh horizon with the temperature
$T = a / 2 π$
. The denominator for bosons stands for the normalization reasons. Parameter
, as can be seen from Equation (
), encodes the maximum amount of quanta at energy
plus 1. Loosely speaking,
is the number of dimensions of the corresponding Fock space at the given energy
and temperature
of the source. The subscripts
denote the components of the field (Rindler modes) with respect to the horizon.
is assumed to be infinite. One may argue that the finiteness of the parameter
in the boson case is incorrect from a mathematical point of view since one deals with the incomplete basis then. However, in any real physical situation, one is dealing with the finite number of
produced particles, bosons and fermions. Taking
$N → ∞$
in Equation (
), as it is widely used in the literature on the topic, seems to be too strong of an assumption because the source produces an infinite amount of energy,
$N − 1 E → ∞$
. This is valid in the case of everlasting acceleration or the non-zero probability of detecting
$N → ∞$
amount of particles at some finite time interval; both scenarios can be provided with the infinite energy supply only. This is because the infinite sum for bosonic modes—see Equation (
)—contains an arbitrary amount of particles: despite being exponentially suppressed, the probability for any
$n ≠ ∞$
in the sum for bosons is non-zero. Such a scenario seems to be rather unlikely from the physical point of view, especially when one considers the application of the Unruh effect for the description
of particle production in relativistic hadronic or heavy ion collisions. Therefore, we assume the maximum number of particles to be finite in all calculations below.
Furthermore, let us consider only boson production in what follows because the expression for the fermions given by Equation (
) can be derived from Equation (
) by setting
$N = 2$
Expression (
) is the Schmidt decomposition [
]. The outgoing radiation is described by the density matrix
$ρ out = Tr in 0 0 = 1 − exp − E / T 1 − exp − N E / T ∑ n = 0 N − 1 exp − n E T n out n out ,$
where we have traced over the inaccessible degrees of freedom (
- modes). Thus, the pure vacuum state from the inertial RF has transformed into the mixed one in the non-inertial RF. Here, the geometric origin of the Unruh effect appears. Namely, finiteness of the
speed of light leads to the appearance of the horizon dividing the all modes in Hilbert space into the accessible (
-) and non-accessible (
-) ones. The complete state is obviously pure and follows unitary evolution. However, because one has limited access to it in the non-inertial RF, it looks like a decoherence. The eigenvalues of the
density matrix
$ρ out$
define the emission probability of a certain number of particles at energy
and temperature
. Therefore, Equation (
) describes the conditional multiplicity distribution
${ n | N , E , T }$
at any given
$N , E$
One may assume that once we have the distribution, it is possible to calculate the corresponding Shannon entropy due to the formulae presented in
Section 2
. However, to deal with the density matrix
, one should use the von Neumann entropy
$H ρ$
instead, which is defined as
The key difference of the von Neumann entropy from its classical analog, Shannon entropy, is related to its meaning:
$H ρ$
defines the amount of information encoded with the correlations between the system described by
and the rest of the world. From this point of view, the density matrix
defines the projection of some larger system, which was determined in the larger Hilbert space, to the space in which the observed system is being defined. The projection might result in a loss of
information encoded with the corresponding correlations between the Hilbert subspaces. The von Neumann entropy is the quantity to estimate the amount of this information. Due to its origin, it can be
equal to zero for the entire space and non-zero for its subspace. This is not the case for the Shannon entropy because classical entropy of the whole system cannot be less than that of some part of
it. However, the von Neumann entropy can be set as equal to its Shannon counterpart provided that the Schmidt decomposition coincides with the basis of the detector [
4. Unruh Entropy
For the emission probability
$ρ out$
from Equation (
), the von Neumann entropy is defined as
$H ρ out = − Tr ρ out ln ρ out = H n | N , E , T = σ q E / T | q = N q = 1 ,$
where we use the following notations
$σ q E / T = q E / T exp q E / T − 1 − ln 1 − exp − q E T ,$
$f x | x = b x = a = f a − f b .$
As one may notice,
$H n | N , E / T$
is an even function of
$E / T$
, i.e.,
$H n | N , E / T = H n | N , − E / T$
. The asymptotic behavior of the entropy (
) with respect to
$E / T$
is the following
$lim E / T → 0 H n | N , E , T = ln N = max H lim E / T → ∞ H n | N , E , T = 0 .$
Expression (
) defines the entropy of the emitted quanta, as well as the quanta inside the horizon, for some mode of the radiated field only, which is determined by parameter
, energy
and temperature
. Parameter
depends on the amount of time during which the observer is being described by the non-inertial reference frame. It follows from the fact that the longer one is observing the horizon, the more
particles at any fixed energy may be detected. Therefore, we conclude that
should increase with time. Temperature
is completely determined by the acceleration
, see [
]. However,
cannot be considered as a fixed parameter. The non-inertial observer is expected to detect particles at different energies. The energy range for the particles may be written as
is the invariant mass of the particles, and
is the maximum energy to be observed, respectively. We assume
to be limited by the acceleration
since the observation of the high-energy particles is very unlikely due to energy conservation law: one cannot extract more energy from the vacuum than is being spent to sustain the observer’s
Unfortunately, the definition of the energy range does not mean we know the spectrum distribution ${ E }$. It is determined by the unnormalized PDF $p E$ of the emission of a particle from the vacuum
at energy E.
In order to figure out
$p E$
somehow, we use the following procedure. As can be noticed from Equation (
), for any particle number
$n > 0$
, the emission probability is proportional to the factor
$exp ( − E / T )$
. The case with
$n = 0$
means no emission at all. Therefore, one should expect exponential behavior for
$p E$
where prefactor
is responsible for any corrections that might depend on the particle type and its quantum numbers. For the sake of simplicity, we assume
$C = const$
and, therefore, drop it due to normalization reasons (see
Section 2
) in what follows. It is worth noting that such assumption results in Schwinger-like mechanism of particle production [
]. Thus, we recovered Schwinger-like particle production from the properties of Hilbert space and space-time only. Recall, however, that this result is generated by the Unruh effect after neglecting
all possible corrections.
Now, we have the spectrum distribution
${ E }$
as given by Equation (
). Without any loss of generality, we assume energy to be defined within the range
$m ≤ E ≤ M$
(Equation (
)). From Equations (
) and (
), one obtains
$H n , E | N , T = − ln d E E p + ln D E p − 1 D E p ∫ m M p E ln p E d E + 1 D E p ∫ m M p E H n | N , E , T d E ,$
where the subscript
$E p$
implies that the energy distribution is not discrete but rather a continuous one, i.e., it is defined with some PDF—see the text concerning Equation (
). In order to obtain the analytic expression, we substitute Equations (
) and (
) into Equation (
) and obtain, after the straightforward calculations, the total Unruh entropy
$H n , E | N , T$
in a form
$H n , E | N , T = − ln d E E p + 1 + ln D E p + m exp − m / T − M exp − M / T D E p + T D E p ∑ k = 1 ∞ 2 k q + 1 k k q + 1 + q E T × exp − ( k q + 1 ) E / T k q + 1 | E = M E = m | q = N q = 1 ,$
$D E p = ∫ m M p E d E = T exp − m T − exp − M T$
$σ q E / T$
from Equation (
) is represented by the following series
$σ ( q E / T ) = ∑ k = 1 ∞ 1 k + q E T exp − k q E T .$
The first term in Equation (
) is responsible for encoding the discrete-continuum transition, see [
]. It is expected to depend neither on any quantum numbers of outgoing particles nor on the reference frame. Therefore, we assume
$ln d E E p$
to be constant.
Expression (
) defines entropy for the distribution
${ n , E | N , T }$
of the particles being detected by the observer associated with non-inertial RF moving with acceleration
$a = 2 π T$
. Recall that in the case of fermions, one should use
$N = 2$
. For the bosons,
may take any positive integer value obeying the energy conservation law. The entropy calculated for the Unruh radiation of fermions and bosons is presented in
Figure 1
Figure 2
, respectively. One can see the distinct maximum in the region of small values of the
$m / T$
ratio. The maximum increases with rising the
$M / T$
ratio and becomes more pronounced with the increase in radiated particles (see
Figure 2
The considered example seems to be straightforward. However, one should keep in mind that the whole analysis above is valid for $1 + 1$-dimensional space-time. Other spatial dimensions do not
contribute to the density matrix $ρ out$ or to its von Neumann entropy because the corresponding subspaces of the Hilbert space contribute to $ρ out$ via the direct tensor product and, therefore, can
be traced out with no consequences to the analysis above. This simple direct extension to additional spatial dimensions for the Unruh effect may lead to the widely spread conclusion that the Unruh
effect results in the appearance of thermal bath all over the space. In our opinion, this conclusion needs to be clarified. Namely, in the last case, the non-inertial observer, as well as the horizon
itself, should be considered as an infinite plane in the additional spatial dimensions being accelerated alongside the normal to the plane. However, the observer should be finite and, therefore,
cannot detect particles from the half-space defined by the horizon. Otherwise, it would lead to faster-than-light speed communication and causality violation because the transition to inertial RF
cannot cause the immediate disappearance of the Unruh radiation from the horizon occupying the half-space.
To overcome the difficulties, we have to assume that
• In order to obey, the energy conservation law N should be finite;
• In the case of (2 + 1) or (3 + 1)-dimensional space-time, the Unruh horizon should be considered as a radiation source of finite size.
Due to the axial symmetry of the non-inertial reference frame, the horizon should be a disk shape with some radius r. The radius can be determined by the observer’s size and causality, i.e., the
finiteness of light speed. Such an assumption leads to an observer-dependent size of r. The problem may be cured, e.g., if one considers the observer’s acceleration a as a surface gravity of the
corresponding black hole and obtain some efficient scale $r = 4 π T − 1$.
One might be confused by the fact that since the Unruh effect describes the thermal bath, its entropy should be maximal. As can be easily noticed from the eigenvalues of the density matrix (
), all of them exponentially depend on the total energy of the emitted number of particles and thus generate a well-known partition function. Note, however, that
$ρ out$
is defined for some
value of energy. Therefore,
can be considered a parameter of the conditional distribution
$n | N , E , T$
. Dealing with the joint distribution
$n , E | N , T$
over multiplicity
and energy
of the emitted quanta, one should take into account energy conservation. It results in some correlations between the possible number of emitted particles and their energy. Thus, the entropy
$H n , E | N , T$
describes not a completely thermal source but some other one.
5. Asymptotics of Unruh Entropy
Let us analyze the asymptotic behavior of the total Unruh entropy in Equation (
) for (i) small and (ii) large acceleration of the observer. The case of small acceleration is analogous to
$T → 0$
; therefore, we will drop all but the leading term in Equation (
). At small temperatures, Equation (
) transforms into
$D E p | T → 0 ≈ T exp − m / T ,$
where we have neglected the term
$exp ( − M / T )$
is the upper bound for the energy spectrum; therefore,
$M > m$
. The Unruh entropy becomes
$H E | T → 0 = ln D E p − 1 D E p ∫ m M p E ln p E d E ≈ ln T − m T + 1 + m T = ln T + 1 .$
Because the entropy
$H n | N , E , T$
equals zero when
$N = 1$
, we consider the case with
$N > 1$
$T → 0$
. Neglecting all the higher-order exponents, one obtains from Equation (
) that
$H n | N , E , T | T → 0 ≈ E T exp − E / T .$
Substituting Equations (
) and (
) into Equation (
), we obtain
$H n , E | N , T | T → 0 ≈ − ln d E T E p + 1 + 1 4 1 + 2 m T exp − m / T ,$
where all the higher-order exponents are omitted. This distribution is displayed in
Figure 3
. The entropy reaches a quite distinct maximum at
$m / T ≈ 0.5$
and quickly drops to unity at larger values of this ratio.
In the case of large acceleration
$a → ∞ ⇔ T → ∞$
, one obtains from Equation (
$∫ m M p E | T → ∞ d E = ∫ m M 1 − E T + E 2 2 T 2 d E + O 1 / T 3 = M − m 1 − M + m 2 T + M 2 + M m + m 2 6 T 2 + O 1 / T 3 ,$
and, therefore,
$H E = ln D E p − 1 D E p ∫ m M p E ln p E d E = ln M − m − M − m 2 24 T 2 + O 1 / T 3 .$
Thus, the conditional entropy
$H n | N , E , T$
from Equation (
) becomes
$H n | N , E , T | T → ∞ = ln N − N 2 − 1 24 T 2 E 2 + O 1 / T 4 ,$
which, together with Equation (
), gives us
$1 D E p ∫ m M p E H n | N , E , T d E = ln N − M 2 + M m + m 2 72 T 2 N 2 − 1 + O 1 / T 3 .$
Finally, substituting Equations (
) and (
) into Equation (
), we obtain the desired asymptotics at high acceleration (or temperature)
$H n , E | N , T | T → ∞ = − ln d E E p + 1 + ln M − m + ln N − N 2 + 2 M 2 + m 2 + N 2 − 7 M m 72 T 2 + O 1 / T 3 .$
The entropy asymptotics at
$T → ∞$
calculated according to Equation (
) is presented in
Figure 4
for fermions
$( N = 2 )$
and in
Figure 5
for the boson spectra with
$N = 100$
and 1000 particles, respectively. At high temperatures, the entropy weakly depends on
and quickly increases with an increase in the value of
. The larger the number of particles, the steeper the rising slope. For
$N = 1000$
, the entropy seems to saturate at
$M ≥ 5$
6. Generalization to Intrinsic Degrees of Freedom
Expression (
) is valid for some mode of the radiated field only, which is defined by the joint multiplicity-energy distribution
$n , E$
, temperature
and parameter
. However, since the emitted particles may have additional degrees of freedom
${ λ }$
, such as electric charge, spin, polarization, etc., they have to be taken into account too. This is equivalent to the following modification of the total distribution
$n , E | N , T → λ , n , E | N , T .$
Using Equation (
), we then obtain
$H λ , n , E | N , T = H λ + H n , E | N , T , λ λ .$
However, such a generalization is not an easy task at all. Let us consider a simple example, while detecting a particle at some E, one should measure its energy. Such a process results in the
consumption of the particle’s momentum. One may argue that calorimetry is not required. The observer can build some source of similar particles and carry out interference experiments to determine the
energy of the particle to be detected. However, any such interference will result in the re-distribution of the momenta during the interference and therefore will change the observer’s momentum as
well. Thus, one concludes that measuring the particle’s energy E leads to a change in the observer’s acceleration a. It implies a change in the Unruh temperature $T = a / 2 π$ of the source the
observer is dealing with.
One may also note that the Unruh effect is being considered within the quasi-classical approach. It means that the density matrix
$ρ out$
in Equation (
) is obtained under the assumption that the outgoing radiation has no influence on the background metric (see [
]). Such a remark is correct, but what about other degrees of freedom
? For instance, taking into account the spin of the particles emitted by the Unruh horizon may lead to a change in the observer’s angular momentum. In this case, the observer’s acceleration
can not be constant due to the conservation of the total angular momentum anyway and thus implies a change in
in Equation (
) during particle identification.
Thus, the situation seems to be simple only if one neglects
influence of the outgoing particles during the Unruh effect. In this case the entropy
$H n , E | N , T , λ$
does not depend on
, and Expression (
) is reduced to the sum
$H λ , n , E | N , T = H λ + H n , E | N , T .$
7. Conclusions
The Unruh effect is considered from the point of view of the information theory. We estimated the total entropy of the radiation generated by the Unruh horizon in the non-inertial reference frame for
the state verified as vacuum by any inertial observer. Usually such a case is treated as von Neumann entropy of the corresponding density matrix. However, this is just the starting point of our study
because the density matrix of the outgoing radiation describes the conditional multiplicity distribution at the given energy and Unruh temperature. As a result, it allows one to estimate the
entropy of the Unruh source by taking into account both the multiplicity and energy distribution of the outgoing quanta. We show how it can be calculated even without the exact knowledge of the
corresponding Hamiltonian. In particular, such a lack of information results in the Schwinger-like spectrum of the emission (see Equation (
The case of a finite amount of particle emission is considered. It allows us to utilize the results for realistic particle emission spectra. The asymptotics of the general expression for entropy with
respect to low and high values of the Unruh temperature are also investigated. We found that in the case of small acceleration corresponding to a low temperature, the entropy of the radiation does
not depend on the maximal amount of emitted particles in the leading order (see Equation (
)). The dependence on
is recovered for large accelerations when
$T → ∞$
(see Equation (
)). It can be explained by the abundant emission of particles from the hot Unruh horizon when the amount of the emitted quanta may be considered as an extra degree of freedom contributing to the
total entropy.
Another interesting point is that the total entropy $H n , E | N , T$ quickly drops to zero with the increase in the mass m of the quanta. It can be explained by the energy conservation law: the more
energy is being spent on the creation of particle’s mass, the less of it may be used to generate the total distribution. At the same time, total entropy of the Unruh source slightly increases with
the maximum allowed energy M because the distribution widens with the increase in M, thus leading to the total entropy increase.
The obtained results can be applied to the analysis of particle distributions in inelastic scattering processes at high energies. Furthermore, they may be generalized to other degrees of freedom of
the emitted particles, such as spin, charges, etc. However, such a generalization may significantly complicate the analysis. For instance, additional conservation laws originating from the other
degrees of freedom might change the metric. Therefore, one may be forced to take a distribution $T$ into account too.
Author Contributions
Conceptualization, M.T. and E.Z.; methodology, M.T.; investigation, M.T.; resources, L.B.; data curation, M.T.; writing—original draft preparation, M.T.; writing—review and editing, L.B. and E.Z.;
visualization, M.T.; project administration, L.B.; funding acquisition, L.B. All authors have read and agreed to the published version of the manuscript.
The work was supported by the Norwegian Centre for International Cooperation in Education (SIU) under grants “CPEA-LT-2016/10094—From Strong Interacting Matter to Dark Matter” and “UTF-2016-long-term
/10076—Training of Bachelor, Master and PhD Students specialized in high energy physics”, and by the Norwegian Research Council (NFR) under grant No. 255253/F50—“CERN Heavy Ion Theory”.
Institutional Review Board Statement
Not applicable.
Informed Consent Statement
Not applicable.
Data Availability Statement
No data have been reported.
Fruitful discussions with J.M. Leinaas, O. Teryaev and S. Vilchinskii are gratefully acknowledged. Numerical calculations and visualization were made at Govorun (JINR, Dubna) computer cluster
Conflicts of Interest
The authors declare no conflict of interest.
Figure 1.
(Color online) The entropy
$H ( n , E | N , T )$
of Unruh radiation given by Equation (
) for fermions
$( N = 2 )$
as function of
$m / T$
$M / T$
Figure 2.
(Color online) The same as
Figure 1
but for bosons. The spectrum of bosons contains (
$N = 100$
and (
$N = 1000$
Figure 3.
(Color online) Asymptotic behavior of entropy
$H ( n , E | N , T )$
given by Equation (
) at
$T → 0$
as function of
$m / T$
Figure 4.
(Color online) High-temperature asymptotics of the entropy
$H ( n , E | N , T )$
of Unruh radiation given by Equation (
) for fermions
$( N = 2 )$
as a function of
Figure 5.
(Color online) The same as
Figure 4
but for bosons with (
$N = 100$
and (
$N = 1000$
particles in the spectrum.
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Travel distance estimation of landslide-induced debris flows by machine learning method in Nepal Himalaya after the Gorkha earthquake
Debris flow defined as debris grains with slurry (Phillips and Davies
) can continue to pose a significant threat in mountainous areas long after the occurrence of strong earthquakes (Shieh et al.
) since the affected areas are highly susceptible to debris flows in the first 5–10 years following such seismic events (Tang et al.
). This is because the ground shaking could weaken the stability of slopes (Fan et al.
) and increase their possibility to collapse (Lv et al.
). Furthermore, the unstable mass on slopes may start to move during the rainy season, potentially transitioning into unsaturated/saturated flows (Tang et al.
) or temporarily accumulating at the base of slopes, feeding the potential debris flows (Dahlquist and West
). As a result, the magnitude of debris flows can reach 10
or even greater due to the entrainment of large quantities of materials during the flowing process (Crosta et al.
), finally leading to a long travel distance and causing severe losses of properties and human lives (Dahlquist and West
; Qiu et al.
. Therefore, estimating the distance of debris flows in the aftermath of earthquakes on a regional scale is of paramount importance. It helps in identifying high-risk areas and developing effective
mitigation strategies (Cascini et al.
; Corominas et al.
; Paudel et al.
; Zhang et al.
; Zhou et al.
In general, the travel distance estimation of debris flows induced by landslides includes two primary approaches: empirical methods and numerical modelling. The former can be dated back to the work
of Scheidegger (
), which enabled the ratio of the runout distance (
) to height difference (
) relating to the reach angle
. Subsequent studies by Corominas (
) and Rickenmann (
) incorporated deposit volume into a relationship with
, making it a significant indicator in estimating the travel distance (Lorente et al.
; Devoli et al.
; Hürlimann et al.
). However, these empirical methods have limitations as they struggle to adequately account for the complexity of debris-flow processes, which are influenced by a range of geomorphic, environmental
and geological factors (Regmi et al.
). Therefore, an appropriate combination of disposing factors is essential to reveal the travel characteristics of debris flows. In addition to the empirical methods, numerical modelling can provide
relatively accurate estimations when using parameters obtained through controlled laboratory experiments, but this method itself is time-consuming and costly for regional application (Paudel et al.
; Qiu et al.
). Therefore, a new model for the travel analysis of landslide-induced debris flows after the earthquake is necessary to form effective debris-flow mitigation strategies. In this context, machine
learning is introduced here due to its strong ability to train complex and hidden relationships between a set of input variables and output results (Khosravi et al.
). The applications of the machine learning models have demonstrated their superiority in hazard analysis (Lin et al.
; Rahmati et al.
). In this paper, an ensemble method (XGBoost) is employed for travel distance estimation. However, parameter optimization is a time-consuming process. Therefore, a genetic algorithm (GA) is used to
generate optimal hyperparameters, allowing for the development of an optimal estimation model. Finally, a hybrid machine learning model is developed to estimate the travel distance of
landslide-induced debris flows following an earthquake.
In this paper, a hybrid machine learning model is developed to estimate the travel distance of debris flows induced by landslides after the Mw 7.8 Nepal earthquake that occurred on 25 April 2015. The
study area is characterized by its challenging topography, with rugged terrain and steep slopes. It experiences recurring debris flows each year, primarily attributed to active geological activities
and abundant rainfall. Overall, our work aims to enhance understanding of landslide-induced debris flows after the earthquake and benefit the planning of mitigation strategies.
Study area
Our study area is located in the Nepal Himalaya, and the identified debris flows are distributed in the northern part of Nepal with an altitude ranging from 947 m to 2966 m above sea level (Fig.
Fig. 1
Inventory of occurred debris flows after the Gorkha earthquake in the Nepal Himalaya
The diverse topography of Nepal gives rise to varying weather patterns, with the northern part predominantly occupying the Himalayan mountains and the southern part consisting of flat terrain. Nepal
experiences a typical monsoon climate characterized by two distinct seasons each year: a dry season from October to March of the following year and a rainy season from April to September. As
indicated in Fig.
, the debris flows in this paper are primarily concentrated in the 14 districts of northern Nepal. To depict the weather conditions in these areas, the 14 districts are highlighted to illustrate
precipitation distribution. The distribution of the mean annual precipitation from 1980 to 2015 is shown in Fig.
. District 1 recorded the highest mean annual rainfall at 4950 mm, followed by District 7 with the second-highest precipitation at 4050 mm (Fig.
). Meanwhile, District 3 experienced the lowest mean annual precipitation at 950 mm. This distribution pattern aligns with the findings of Du et al. (
) and Li et al. (
), which indicate that precipitation exhibits a negative correlation with altitude in the Himalayan regions.
Cold weather is typically associated with the dry season, often resulting from snowfall, particularly in the Himalayan regions. Subsequently, the temperature begins to rise in April, and the outside
air in April and May becomes exceptionally humid and stifling, with thunderstorm activity occurring in the evenings. Following two months of sweltering conditions, the rainy season commences toward
the end of June, driven by the southwest monsoon from the Indian Ocean (Adhikari and Koshimizu
). This monsoon climate brings abundant rainfall that persists for three months, lasting until the end of September. Figure
demonstrates a notable increase in precipitation from May to June. The peak rainfall occurs in July and August, with over 80% of the total precipitation occurring between June and September (Paudel
et al.
). Therefore, substantial rainfall during the rainy season becomes a key triggering factor for a higher incidence of debris flows, particularly after the shaking of the Gorkha earthquake.
Fig. 3
Mean monthly precipitation over the past 35 years for the 14 districts
In addition to the meteorological conditions in the Nepal Himalayas, the lithology distribution also plays a crucial role in slope stability and the formation of debris flows. The identified debris
flows are mainly distributed in three tectonic zones, including Tibetan-Tethys (TT), Higher Himalayan (HH), and Lesser Himalayan (LH) Zones, based on the summarization of Upreti (
). Approximately a third of the debris flows in Fig.
are situated in the TT zone, extending from east to west in the northern Gorkha region, enclosed by the North Dipping Normal Fault System, the South Tibetan Detachment Fault System, and the
Indus-Tsangpo Suture Zone. The geological strata in this zone range from the lower Paleozoic to lower Tertiary marine sedimentary succession. As for the HH zone lying below the TT zone, the main rock
(gneisses) serves as the basement of the TT zone. The TT and HH zones were initially considered to be continuous until the detection of the South Tibetan Detachment Fault System (Fuchs
). Moreover, a small number of debris flows are observed in the LH zone, characterized by rocks that include sandstone, mudstone, and conglomerate, dating from Neogene to Quaternary. In summary, the
complex geology conditions and abundant rainfall in these areas contribute to the occurrence of debris flows. Understanding the dangerous areas caused by debris flows after an earthquake becomes
significant for site selection of residential areas and infrastructure construction.
3. Methodology
In this paper, we utilized a machine-learning-based method to estimate the travel distance of the debris flows with the involvement of a series of predisposing factors after the Mw 7.8 Gorkha
earthquake. The technical route is presented in Fig.
Fig. 4
Flow chart of this study
This method is composed of several steps:
(1) Decided debris-flow paths and source areas using satellite images, ENVI and GIS tools. First, the coordinates of the endpoint of the travelling materials where they stopped, and the travel
distance of the travelling materials are provided by Dahlquist and West (
). A total of 89 debris flows were included in this study, and their travel distances ranges from 200 to 8010 m based on the provided dataset in Table
Appendix A
. Then, we overlaid the contour map onto the satellite images to identify the central points of the source areas, as illustrated in Fig.
. This process enables us to establish the trajectories of debris flows.
Fig. 5
Identified landslide-induced debris flows using satellite images
(2) Upon completing a three-step analysis, which comprises correlation analysis, multi-collinearity analysis, and principal component analysis (PCA), the PCA was employed to eliminate extraneous
information from the selected variables to generate three principal components. After that, the three principal components were rescaled into the range of [0.01, 0.99]. The normalized data acts as
input variables for model training with the application of the parameter optimization method, named ‘genetic algorithm’.
(3) Conducted model assessment using the evaluation indexes, such as RMSE, MAE and MAPE. Whilst the contributions of the selected variables to estimation accuracy improvement were also evaluated to
address the importance of involving all the factors in the model development.
(4) Compared the estimation results of the developed model with existing empirical equations using the testing dataset.
3.1 Selection of disposing factors
The importance of factor selection has been addressed by McDougall (
). However, the challenge of selecting the most appropriate input parameters still persists. Therefore, factor selection for estimating travel distance holds substantial importance. Our study not
only strives to achieve reliable estimations but also aims to offer valuable guidance for factor selection when analyzing the risk of debris flows in the Himalayas. We have preliminarily identified
key factors that could contribute to travel distance estimation. These factors encompass various aspects of geomorphic and environmental conditions, including the volume of failure mass (
), the drop height between the centre of the source area and the endpoint of movement mass (
), the mean gradient of the travelling path (
), the mean curvature of travelling path (
) and the normalized difference vegetation index (NDVI).
serve as indicators of the potential energy stored within the failure mass, offering insights into its subsequent movement distance (Roback et al.
; Zhan et al.
; Puglisi et al.
; (Qiu et al.
). A greater sediment volume normally could cause a longer runout distance (Legros et al.
; Guo et al.
; Falconi et al.
). As for the mean gradient of the travelling path (
), this factor is proven to present a strong correlation with the travel distance (Rickenmann
). Notably, our calculation of
diverges from Rickenmann (
is calculated using the formula proposed by IMHE (
$$J=\frac{{\left( {\sum\limits_{{i=1}}^{n} {\left( {{E_{i - 1}}+{E_i}} \right){L_i} - 2{E_0}L} } \right)}}{{{L^2}}}$$
is the average path gradient (‰).
, …,
are the elevations of each break point in the movement path (m). Elevation was obtained from a 12.5 m digital elevation model (DEM (downloaded from
, …,
are the lengths of each section of the movement path (m).
is the number of path sections.
is the elevation of the endpoint of mass movement (m), and
is the length of the travel path (m). The divided sections are presented in Fig.
Fig. 6
Segments of the travel path
has been used by previous studies to delineate the hazardous area (Yu et al.
), a factor closely linked to the runout distance of debris flows (Hürlimann et al.
; Prochaska et al.
represents the mean curvature of the flowing path of a debris-flow event, which was calculated using the ‘Surface’ tool in GIS. Similar to the impacts of
on debris-flow runout distance, NDVI is also considered for the runout estimation (Booth et al.
), and serves as a reflection of vegetation coverage within debris-flow affected areas. Lush vegetation can cause a reduction of the debris-flow velocity (Lancaster et al.
) and, therefore, result in a decrease in the travel distance (Michelini et al.
). In this study, Landsat 8 satellite images captured prior to the occurrence of debris flows are utilized (USGS
). These images enable the extraction of mean NDVI values, which represent vegetation coverage. However, whether all these factors are appropriate for the travel distance estimation still needs
further studies. As such, a comprehensive three-step analysis is conducted to ensure strong correlations between input variables and travel distance. Simultaneously, collinearity analysis among the
input variables and dimension reduction techniques, such as Principal Component Analysis (PCA), is employed to generate a set of corrected variables.
3.2 A three-step analysis
A three-step analysis was employed here to decide the input variables for model development. This analysis is composed of a correlation analysis, a multi-collinearity analysis, and the PCA. The
correlation analysis is cconducted by SPSS statistical software to unveil the relationship between each input variable and the travel distance (
). As a result, we can identify and eliminate variables with weak correlations to
. Subsequently, a multi-collinearity analysis is conducted among the remaining variables to avoid distortion and inaccuracy in travel distance estimation. Multi-collinearity demonstrates the
linearity between the variables, implying that a specific independent factor can be substituted by other variables through a linear equation. In this context, two indices, named tolerance index (TOL)
and variance inflation factor (VIF), were used in this section. Multicollinearity was decided when the TOL value is smaller than 0.1 (Menard
) or the VIF value exceeds 10 (Guns and Vanacker
). These two indices are calculated using the following equations:
$$Tolerance=1 - R_{i}^{2}$$
$$VIF=\left[ {\frac{1}{{Tolerance}}} \right]$$
denotes the coefficient of determination in the regression model when the dependent variable is
, while the other input data are independent variables. Following the first two-step analysis, the importance of each variable in contributing to the travel distance can be initially assessed.
However, further data processing remains crucial due to intercorrelations among these factors. Additionally, the useless information within the data should also be removed since it can increase the
analysis difficulties (Chaib et al.
). Therefore, PCA was introduced to reduce the data dimension and eliminate the relevance between factors based on origin software. This method seeks to generate new indices, termed ‘principal
components’, which encapsulate the most essential data information. The fundamental PCA process comprises several steps: (1) Normalize the multi-dimension data matrix; (2) calculate the eigenvalues
and eigenvectors of this matrix; (3) arrange the eigenvalues and eigenvectors in descending order; (4) Select the first
values based on the accumulative contributions. Finally, (5) a new
-dimensional matrix can be generated through dimension reduction. The whole process can be described as:
$$T=\left(T_1,T_2,......,T_n\right)=\begin{bmatrix}t_{11}t_{12}\cdots t_{1m}\\\vdots\ddots\vdots\\t_{n1}t_{n2}\cdots t_{nm}\end{bmatrix}$$
samples are included in
, and each sample contains the number of
variables. Then, the matrix
, consisting of the calculated eigenvectors
= (
, …,
), contributes to the generation of the principal components
$$\left\{ \begin{gathered} {V_1}={v_{11}}{T_1}+{v_{12}}{T_2}+ \cdots +{v_{1n}}{T_n} \hfill \\ {V_2}={v_{21}}{T_1}+{v_{22}}{T_2}+ \cdots +{v_{2n}}{T_n} \hfill \\ \vdots \hfill \\ {V_m}={v_{m1}}{T_1}+
{v_{m2}}{T_2}+ \cdots +{v_{mn}}{T_n} \hfill \\ \end{gathered} \right.$$
, and the original data is replaced by the principal components,
, …,
. In order to further enhance the input stability and difficulties of data processing ability for the model, the generated three principal components are normalized into the range of [0.01, 0.99]
based on the equation:
$${x_{nor}}=\frac{{x - \hbox{min} (x)}}{{\hbox{max} (x) - \hbox{min} (x)}}(U - L)+L$$
where x[nor] represents the normalized data, which came from x. U and L are the upper and lower normalization bounds, respectively.
3.3 Development of a machine learning model
In contrast to empirical equations, a machine learning model can encompass a significantly large number of independent factors, which serve to capture topographic and environmental features. On such
an algorithm, extreme gradient boosting (XGBoost), a boosting-related algorithm, excels well in various competitions due to its utilization of the second derivative for the calculation of loss
functions and the incorporation of additional regularization items. Therefore, XGBoost stands out as a superior choice for conducting regression analysis. In consistent with the other boosting
algorithms, including AdaBoost and GBDT, XGBoost is composed of weak regressors, often represented as decision trees. To obtain the estimation value, this model needs to constantly produce trees,
which function as weak regressors. After the completion of model training, the cumulative scores associated with the leaf nodes of these trees collectively yield the final estimation value. The
mechanism of XGBoost is presented in Appendix. A. In this study, python was used to develop a XGBoost model user the editing environment of pycharm.
3.4 Model assessment
Model assessment is an important component of model development to test the estimation performance. To assess the estimation results of this developed model, the root mean square error (RMSE) and
mean absolute error (MAE) were employed to evaluate model performance. RMSE and MAE are widely used indexes to evaluate model performance (Chai and Draxler
). In essence, these two indexes can both reflect the reliability and efficiency of the developed model by quantifying the disparities between actual and estimation values. It’s worth noting that
RMSE tends to be more sensitive to outliers in the input data compared to MAE (Willmott and Matsuura
). Apart from the two metrics for evaluating actual errors, another evaluation index, MAPE (Mean Absolute Percentage Error), was introduced to present the error ratio due to its independence and
interpretability (Bowerman et al.
). Therefore, the three metrics employed in our study not only evaluate model performance but also aid in identifying outlier values. These statistical indexes can be calculated by:
$$RMSE=\sqrt {\frac{{\sum\limits_{{i=1}}^{n} {{{\left( {{y_{ipre}} - {y_i}} \right)}^2}} }}{n}}$$
$$MAE=\frac{{\sum\limits_{{i=1}}^{n} {\left| {{y_{ipre}} - {y_i}} \right|} }}{n}$$
$$MAPE=\frac{{100\% }}{n}\sum\limits_{{i=1}}^{n} {\left| {\frac{{{y_{ipre}} - {y_i}}}{{{y_i}}}} \right|}$$
where y[ipre] represents the estimation results, and y[i] is the actual value. n is the number of estimation values. A better model is indicated if the calculated results of RMSE, MAE, and MAPE are
closer to 0. Moreover, to further reveal the contributions of each variable in estimating the travelling distance, one variable is removed from model development at a time to generate five estimation
models. Then the RMSEs and MAEs of each model are calculated, respectively. Meanwhile, the ratios of RMSE and MAE of the models are also calculated, as the abnormal values may cause the instability
of output results. So, this ratio can reflect the model’s stability.
4. Result analysis
4.1 Determination of input variables
Five scatter diagrams (Fig.
), using the data in Table
Appendix A
), are presented below to describe the correlations of travel distance (
) with each candidate variable. Initial filtration can be achieved based on the
values in these figures. Among the variables considered, height difference (
) exhibits the strongest correlation with the travel distance, closely followed by the volume of failure mass (
). As for the other variables,
, a stronger correlation is observed between
, reaching 0.545.
displays a correlation value of 0.401 with
. Conversely, NDVI demonstrates a weak correlation with travel distance, leading to its exclusion from the model development process.
Fig. 7
Correlation analysis between variables and travel distance
After the correlation analysis between the input and output variables, the subsequent step, referred to as the multi-collinearity analysis, is conducted to assess the linearity among the remaining
, and
. The calculated results, as presented in Table
, clearly indicate the absence of multi-collinearity among these variables, as all TOL values exceed the threshold of 0.1. Furthermore, no VIF values exceed 100, affirming the suitability of all four
variables for inclusion in the model development process.
Table 1
Tolerance values and variance inflation factor (VIF)
│Factors │Collinearity indexes│ │
│ │TOL │VIF │
│Volume of failure mass (V[L]) │0.743 │1.346│
│Height difference between center of source area and end point of mass movement (H) │0.528 │1.894│
│Mean gradient of travelling path (J) │0.692 │1.445│
│Mean curvature of travelling path (C) │0.866 │1.155│
To ensure a strong correlation between the generated principal components and travel distance, Pearson’s coefficient was introduced again to test the correlation between dependent and independent
variables. The results show that PC1 presents the strongest correlation with L, reaching 0.744, followed by PC2, 0.730. The lowest correlation coefficient among the three input variables is
represented by the PC3, 0.726. Overall, the three principal components present a strong correlation with L since the absolute values are all larger than 0.7.
4.2. Estimation of travel distance and evaluation of model performance
The normalized principal components are used to train machine learning using Python under the editing environment of Pycharm. The estimation model can be generated, and the results are plotted in
Fig. 8
Estimation results of the machine learning model and estimation errors
As depicted in Fig.
, the estimation results exhibit minor errors to the measured values (see green labels in Fig.
). To visualize the differences between the estimation results and measured values, the estimation errors are plotted in Fig. 8 (see dark blue labels). Over 80% of the errors are less than 100 m, and
the minimum estimation error lowers to 3.2 m. To provide an encompassing evaluation of this model’s performance, we calculate the MAPE, RMSE, and MAE values, yielding values of 8.71%, 144.30 m and
86.15 m, respectively. It’s worth mentioning that the RMSE exceeds 100, even though most errors are less than 100. This can be attributed to the RMSE being more sensitive to abnormal values when
compared to MAE. So, we employ both RMSE and MAE to reflect the discrepancies between the estimation results and measured values. Moreover, most of the estimated values are lower than the actual
values. This discrepancy, in part, may be due to the ignorance of retention of the failure mass during the transportation process. We calculate the failure mass according to the source area without
accounting for the potential loss of materials during their journey.
4.3 Sensitivity analysis
In addition to assessing the overall model performance, it is equally essential to discern the individual contribution of each geomorphic factor to the travel distance estimation. This factor
importance analysis not only sheds light on the significance of each raw factor but also offers valuable insights for mitigating debris flows. To achieve this, we evaluate the significance of each
raw factor by excluding one factor at one time, thereby training a total of 15 models, including PCA model (PCA1+PCA2+PCA3), Model 1 (
), Model 2 (
), Model 3 (
), Model 4 (
), Model 5 (
), Model 6 (H+J), Model 7 (H+C), Model 8 (
), Model 9 (
), Model 10 (
), Model 11 (
), Model 12 (
), Model 13 (
), and Model 14 (
). After that, we test the estimation accuracy of the 15 models based on the RMSE, MAE, and MAPE indices. Before conducting sensitivity analysis, we plotted the estimation results of PCA model and
Model 1 to test the efficiency of PCA method in removing noise information and therefore increasing estimation accuracy (Fig.
). As indicated in Fig.
, PCA model performs better than Model 1 since the estimation results of Model 1 exhibit the greater divergence from the measured values.
Fig. 9
Scatter diagram of the estimation results for the models
After that, we calculate the MAPE, RMSE, and MAE values of the models to conduct sensitivity analysis, as illustrated in Fig.
. Figure
(a) shows that a decreasing proportion of 36.9% is achieved for MAPE after PCAs were used as input data when compared with Model 1. It’s worth noting that the decrease of MAPE indicates the increase
of estimation accuracy. A similar decline of 33.1% is noted when factor
is omitted from the factor combination (Model 2). However, a greater MAPE decrease is observed when removing
from model development (Model 3), which reaches a proportion of 43.3%. As for Model 4 comprising of
, and
, it achieves a MAPE of 17.5%, which is slightly smaller than the MAPE value of Model 3 (17.1%). It can be concluded that
plays a more critical role than
in travel distance estimation. As for the models with the involvement of two factors, Model 5 performs the best, which indicates that the contributions of
to the model development are limited in comparison to
. A significant percentage reduction of MAPE can be found when
factor in Model 6 was replaced by
(Model 8), reaching 79.2%. Model 10 exhibits the smallest MAPE value due to a combination of
. Furthermore, The MAPE value ranges from 45.8 to 79.3% if only
, and
were utilized for model development, respectively. This underscores the pivotal roles of
as the main control factors, determining the potential energy and the distance the failure mass can travel (Lo
Fig. 10
RMSE, MAE and MAPE values of the five models
Fig. 11
Estimation errors of the models
As illustrated in Fig.
, PCA model can achieve the most stable outputs, and the estimation errors range from 8.1 to 182.7 m. As for the Model 1, its maximum estimation error increases to 410.5 m, and the minimum one shows
a slightly increase, reaching 9.5 m. After we exclude
from model development (Model 2), the maximum estimation error comes to 693.6 m. The other two models with the involvement of three factors (Mode 3 and Model 4) have the similar maximum value as
Model 2. Among the six models from Model 5 to Model 10, the maximum estimation errors of Model 9 and Model 10 come to 1404.2 m and 2829.7 m. These two abnormal values are the leading cause of the
RMSE rising in Fig.
(b). Additionally, the maximum error increase from 504.7 m to 1155.1 m from Model 4 to Model 5. This is also the main reason of the sudden increase of MAPE in Fig.
(b). As for Model 11, 12, 13, and 14, their maximum errors are all greater than 3,000 m. Therefore, to estimate the travel distance of debris flows after the earthquake, all four factors should be
involved in model development, and a PCA analysis can be employed to further increase the estimation accuracy.
However, further verifications are still essential to present the superiority of this machine learning method in estimating travel distance when compared with existing empirical equations. So, the
comparison analysis between the machine learning method in this paper and empirical methods is conducted in the next section.
5. Comparison with existing empirical equations
To further test the performance of the machine learning model, four empirical equations are employed in this section. The proposed equations in the past four studies are presented in Table
. The calculated results using the testing dataset are plotted in Fig.
Table 2
Summary of the empirical equations for travel distance
│Source │Equations │Dataset │
│(Rickenmann 1999) │\(L=1.9{M^{0.16}}H_{e}^{{0.83}}\) (11) │Italy, Japan, China, Swiss, U.S.A, Columbia│
│(Lorente et al. 2003) │\(L=7.13{\left( {M{H_e}} \right)^{0.271}}\) (12) │Central Spanish Pyrenees │
│(Hürlimann et al. │ │ │
│ │ │ │
│2015 │\(L=7.48{V^{0.45}}\) (13) │Switzerland │
│ │ │ │
│) │ │ │
Fig. 12
Comparison results between the hybrid model and empirical equation
This figure illustrates that the estimation results exhibit the closest agreement with the actual values, displaying a uniform distribution around the line of perfect agreement. This indicates that
the machine learning method produces estimations without evident overestimation or underestimation. However, the estimated results by Rickenmann (
) consistently tend to be higher than the measured values. As for the equation proposed by Rickenmann (
), the author used data from different countries, including Mount St. Helens Iahars and Nevado del Ruiz, U.S.A, where ample water involvement led to longer travel distances. Consequently, the
estimated results by the two equations are inevitably higher than the measured values. In contrast, the other two equations proposed by Lorente et al. (
)d rlimann et al. (2015) underestimate the travel distance. Lorente et al. (
) defined the
as the height difference between the head of the source area and the point where the travelling mass starts to deposit. However, this definition may ignore the length of the stopped sediments and,
therefore, present lower estimation values. Additionally, the equation of Hürlimann et al. (
) relies on laboratory experiments. Laboratory experiments cannot fully simulate the real scenarios because the measured maximum travel distance of debris flows can only reach 450 m for large volumes
in this study. Therefore, the proposed equation relying on the laboratory data is prone to underestimating the travel distance of debris flows. Overall, our study performs the best, as it exhibits
the smallest error when compared to the measured values.
6. Discussion and limitations
The superiority of machine learning in travel distance estimation may mainly rely on the compatibility of this method by involving more factors in behavior delineation of debris flows from
multi-dimensional perspectives. Therefore, factor selection becomes significant since the input data should reflect the impacts on travel distance and capture physical reality. Normally, two
categories can be suggested, including geometric-morphological and magnitude-based factors (Zhou et al.
). For a better estimation quality, we select both the two types of factors for model development. However, a limitation may not be ignored, that is, the effect of pore-fluid pressure to runout
behaviour of debris flows (McCoy et al.
; Zheng et al.,
). A debris-flow event can be initiated when Rickenmann exceeds the thresholds, but different rainfall intensity may result in different runout distance even though the sediment volume is fixed
value. This is because different rainfall intensities can impact the saturation degree of sediment mass and further result in different mobility. As a result, travel distance would vary due to
different rainfall intensity. However, the sparse distribution of rain gauge stations restricts the determination of reliable rainfall data. Therefore, a further study is essential to improve the
travel distance estimation to establish a reliable warning system in this area. Additionally, integrating more factors than empirical equations may also bring useless information and, therefore,
increase the complexity of the program running. So, PCA plays a critical role in dimension reduction by removing redundant information.
The analysis of debris flows, including the initiation, flowing and deposition, is a complicated task, which requires an understanding of the topographic features, environmental conditions, and
geological settings. Machine learning can provide an alternative to simplify the flowing process without conducting fluid and mechanical analysis to achieve a reliable estimation or warning. XGBoost
used in this study was developed based on Gradient Boosting Decision Tree (GBDT) with the involvement of a L2 regular term to void overfitting and development of a second-order expansion (Dong et al.
). As a result, this model has attracted a lot of attentions from various fields due to its good performance in both computation speed and prediction accuracy (Wang et al.
; Qiu et al.
). For example, XGBoost performs better than LR in medical field (Wang et al.
). Additionally, XGBoost outperforms ANN and SVM in predicting underground levels (Osman et al.
). Again, a better performance was found in utilizing XGBoost to predict concrete strength when compared to SVM and multilayer perceptron (MLP) (Nguyen et al.
). Overall, XGBoost is an effective machine learning model in conducting prediction analysis. Therefore, this model is selected in our study. It’s worth noting that we cannot conclude that our model
developed using XGBoost can perform well in all fields and different conditions. The performance may vary due to different data structure and sample size, which may appear to be the limitation of our
model for a wide application due to the involvement of debris-flow data only in the Nepal Himalayan Mountain regions. However, developing a model that is suitable for a worldwide application remains
a challenge, which requires continuous input of debris-flow data globally and cannot be fully completed in this study. Even though there are several empirical equations incorporating data in
different countries and regions into model development, such as an equation proposed by Rickenmann (
), they may not provide a reliable estimation when applying to a specific site based on a simple fitting equation. However, the introduction of machine learning method can improve this limitation due
to its strong ability in finding the hidden and complex relationship. Therefore, the superiority and efficiency of this model cannot be altered in travel distance estimation analysis.
Moreover, the loss of mass volume during the flowing process is ignored, and we assume that all the failure mass would arrive at the endpoint. This assumption may be the reason for estimation errors.
Some estimation results are larger than the actual values because we ignore the retention of materials during the flowing process. However, regardless of the limitation in our model, it is still
effective in estimating the travel distance of debris flows after the earthquake, which explicitly improves the estimation accuracy. The application of this model can be further expanded with the
input of more debris-flow data, which may substantially improve estimation accuracy. Overall, we provide a machine-learning-based method to estimate the travel distance of debris flows, which can
achieve effective warning and mitigation of debris flows after earthquakes in mountainous areas.
7. Conclusion
In conclusion, we introduce a machine-learning-based method to estimate the travel distance of debris flows along the Nepal Himalayas after the 2015 Gorkha earthquake. First, the travel path and
center of the material source area are decided based on the identified debris flows. Then, five factors in relation to the geomorphological and environmental conditions are initially selected,
including V[L], H, J, C, and NDVI. After that, a correlation analysis is conducted to analyze the correlations between each variable and travel distance. Then, the multi-collinearities among
variables are investigated to remove NDVI because it presents a weak correlation with travel distance. Furthermore, the remaining four variables are used to generate principal components, PC1, PC2
and PC3, to reduce the dimension of input data and ensure model stability.
Moreover, the decided input data is normalized into the range of 0.01 to 0.99, and then the data is separated into a training set and a testing set with a ratio of 7:3. The training set is used to
train the estimation model using GA-XGBoost. GA is employed to generate the optimal hyperparameters of XGBoost. As for the performance of the model, RMSE, MAE and MAPE are used to evaluate the
trained model, and the results show that the MAPE is 8.71%. The RMSE and MAE are 144.3 m and 86.1 m, respectively. Additionally, to reveal the contributions of each variable to the estimation of
travel distance, sixteen models were developed by excluding a factor at one time from model development, including PCA model (PC1+PC2+PC3), Model 1 (V[L]+H+J+C), Model 2 (V[L]+H+J), Model 3 (V
[L]+H+C), Model 4 (H+J+C), Model 5 (V[L]+H), Model 6 (H+J), Model 7 (H+C), Model 8 (V[L]+J), Model 9 (V[L]+C), Model 10 (J+C), Model 11 (H), Model 12 (V[L]), Model 13 (J), and Model 14 (C).
The performances of these models were evaluated using the three indexes again. The results show the necessity of incorporating all four factors into model development if high accuracy is expected.
The proposed factor combination in our studies is suitable for estimating travel distance for debris flows after the earthquake. Finally, we compared the estimation model with existing empirical
equations. Our proposed model performs the best because the estimation results are the closest to the actual values. Therefore, this model can effectively estimate the travel distance of debris flows
after the earthquake, but slight fluctuations of the estimation accuracy may be inevitable due to the different topographic conditions if this model is applied to other areas.
This work was financially supported by the European Union’s Horizon 2020 research and innovation program Marie Skłodowska-Curie Actions Research and Innovation Staff Exchange (RISE) (Grant No
778360). For the purpose of open access, the author has applied a Creative Commons Attribution (CC-BY) licence to any Author Accepted Manuscript version arising from the submission.
For the purpose of open access, the author has applied a Creative Commons Attribution (CC-BY) licence to any Author Accepted Manuscript version arising from the submission.
Competing interests
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Open Access
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give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this
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Publisher’s note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. | {"url":"https://www.springerprofessional.de/en/travel-distance-estimation-of-landslide-induced-debris-flows-by-/27682394","timestamp":"2024-11-02T12:16:59Z","content_type":"text/html","content_length":"297645","record_id":"<urn:uuid:337e6f7a-3ea4-4d97-845d-1c016513ccdd>","cc-path":"CC-MAIN-2024-46/segments/1730477027710.33/warc/CC-MAIN-20241102102832-20241102132832-00579.warc.gz"} |
CAMB: Getting the power spectrum for isocurvature perturbations for dark matter
I am trying to get the power spectrum for isocurvature perturbations for dark matter for redshift $z=1$ using CAMB. I have the following code:
Code: Select all
import numpy as np
import camb
from camb import model, initialpower
import matplotlib.pyplot as plt
# Set up the CAMB parameters
pars = camb.CAMBparams()
# Set cosmological parameters
pars.set_cosmology(H0=67.7, ombh2=0.022447, omch2=0.11928, mnu=0.06, omk=0, tau=0.0568)
# Indicate we want scalar perturbations
pars.WantScalars = True
pars.WantTensors = False
pars.WantVectors = False
pars.DoLensing = True
# Set the initial power spectrum parameters
pars.InitPower.set_params(As=2e-9, ns=0.9682)
# Adjust the InitialCondition and InitialConditionVector
pars.InitialCondition = 2
pars.InitialConditionVector = [0, 1]
# Set the desired redshift
redshifts = np.array([1])
pars.set_matter_power(redshifts=redshifts, kmax=5)
# Calculate results for these parameters
results = camb.get_results(pars)
# Get the matter power spectrum at the desired redshift: P(k,z)
kmin = 1e-4
kmax = 5
nk = 1000
k = np.logspace(np.log10(kmin), np.log10(kmax), nk)
ks_isocurvature, zs_isocurvature, pk_isocurvature = results.get_matter_power_spectrum(minkh=kmin, maxkh=kmax, npoints=nk)
pk_isocurvature = pk_isocurvature.T
plt.rcParams["mathtext.fontset"] = "cm"
# Plot
for i , z in enumerate(zs_isocurvature):
plt.loglog(ks_isocurvature, pk_isocurvature[:,i], label=f"Isocurvature mode")
However, I get the same power spectrum as that for adiabatic perturbations for dark matter. Could someone help me modify this code to get only the isocurvature contribution.
Last edited by Ian DSouza on May 23 2024, edited 1 time in total.
Re: CAMB: Getting the power spectrum for isocurvature perturbations for dark matter
I think you want to set pars.scalar_initial_condition = initial_iso_CDM
Re: CAMB: Getting the power spectrum for isocurvature perturbations for dark matter
Thanks, that seemed to change the power spectrum. | {"url":"https://cosmocoffee.info/viewtopic.php?t=3795&sid=335c2960e91537cd63f4eca744aa2740","timestamp":"2024-11-02T01:31:27Z","content_type":"text/html","content_length":"33123","record_id":"<urn:uuid:9be968f7-0d32-45d5-879c-309698c7ab94>","cc-path":"CC-MAIN-2024-46/segments/1730477027632.4/warc/CC-MAIN-20241102010035-20241102040035-00269.warc.gz"} |
Discrete Structures - Functions
Problem 1
For each of the functions below, indicate whether the function is onto, one-to-one,
neither or both. If the function is not onto or not one-to-one, give an example showing
f: R → R. f(x) = x2
h: Z → Z. h(x) = x3
Problem 2
Consider the functions f and g whose domain and target are Z. Let
f(x)=x2 g(x)=2x
Evaluate f ο g(0)
Give a mathematical expression for f ο g | {"url":"https://samples.eduwriter.ai/680228720/discrete-structures-functions-in-mathematics-amp-economics","timestamp":"2024-11-05T04:19:55Z","content_type":"text/html","content_length":"37545","record_id":"<urn:uuid:9f989024-0354-4f18-a063-12eb44829fab>","cc-path":"CC-MAIN-2024-46/segments/1730477027870.7/warc/CC-MAIN-20241105021014-20241105051014-00686.warc.gz"} |
Monotonicity of RG flow in emergent dual holography of worldsheet nonlinear $σ$ model
Monotonicity of RG flow in emergent dual holography of worldsheet nonlinear σ model
Based on the renormalization group (RG) flow of worldsheet bosonic string theory, we construct an effective holographic dual description of the target space theory identifying the RG scale with the
emergent extra dimension. This results in an effective dilaton-gravity theory, analogous to the low-energy description of bosonic M-theory. We argue... Show more | {"url":"https://synthical.com/article/Monotonicity-of-RG-flow-in-emergent-dual-holography-of-worldsheet-nonlinear-%24%CF%83%24-model-465f8453-d92a-4265-afad-2e01c7436d51?","timestamp":"2024-11-06T23:28:08Z","content_type":"text/html","content_length":"72787","record_id":"<urn:uuid:112a9145-204d-4b67-bb32-8205bc4ec496>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00231.warc.gz"} |
Finding the Ratio of Triangle Areas With the Same Base | Geometry Help
The area of a triangle is given by the formula (base · height)/2. Triangles that have the same base will have areas whose ratio is the same as the ratio of their heights:
In the above drawing, triangle ΔABC and triangle ΔDBC have the same base, BC, which has length b. Their areas are thus Area[ABC]= (b · h[1])/2 and Area[DBC]= (b · h[2])/2. And so the ratio of the
areas is Area[ABC]/Area[DBC]= [(b · h[1])/2]/[(b · h[2])/2]= h1/ h2, or simply the ratio of their heights.
And similarly, triangles with the same height will have areas whose ratio is the same as the ratio of their bases:
Here, triangle ΔABC and triangle ΔACD have the same height (note that the height of an obtuse triangle like ΔABC can be outside the triangle!), h. Their areas are thus Area[ABC]= (x · h)/2 and Area
[ACD]= (b · y)/2. And so the ratio of the areas is Area[ABC]/Area[ACD]= [(b · x)/2]/[(b · y)/2]= x/ y, or simply the ratio of their bases.
Triangles with the same height are often found between two parallel lines - like in this problem.
Let's look at a (difficult!) geometry problem that makes use of the above property.
ABCD is a deltoid, and line AE is parallel to BC. Triangle ΔAEB (the shaded area) has an area of 30. The ratio of the lengths of segments DE and EC is 4:3, or 3|DE|=4|EC|. What is the area of the
Since we are asked to find the area of a shape, and we know the ratio of a couple of line segments, this is a hint to try and find triangles with the same height (or same base) as the one where we
know the area.
The tricky thing here is to construct such triangles. We said, above, that triangles with the same height are often found between two parallel lines, and we have such parallel lines (AE||BC) - so
we'll try to construct such triangles between those two lines.
Let's draw the diagonal AC. We now have two triangles with the same base - ΔBAE and ΔCAE, (AE is the base of both of them), and also the same height, since they are between two parallel lines. It is
easier to see this if we extend the base, AE.
Now draw the height of ΔCAE, CF, to the extended base. GBCF is a rectangle - since AE||BC (given) and CF||BG as they are both perpendicular to GF. So CF=BG.
But if ΔBAE and ΔCAE have the same base and the same height, the have the same area! The area of ΔCAE is then also 30.
Now let's look at triangles ΔADE and ΔCAE. They both have the same height (they share a common vertex - A, and their bases, DE and EC, are on the same line). So the ratio of their areas is the same
as the ratio of their bases - which is given (4:3).
So if the area of ΔCAE is 30, the area of ΔADE is 30*4/3=40. The combined areas of ΔADE and ΔCAE is 30+40=70, and they form a triangle (ΔADC) which is half of the deltoid, so the total area of the
deltoid is 140. | {"url":"https://geometryhelp.net/ratio-triangle-areas-same-base/","timestamp":"2024-11-12T03:53:19Z","content_type":"text/html","content_length":"82533","record_id":"<urn:uuid:ca8f136c-e3e6-4e85-8b9d-0e415bdf3e6d>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.50/warc/CC-MAIN-20241112014152-20241112044152-00524.warc.gz"} |
Problem D
Languages en is
Tómas has found himself in a strange world. This world consists of $n$ cells arranged in a circle. Thus cells $i$ and $i+1$ are adjacent for $1 \leq i < n$, and cells $1$ and $N$ are also adjacent.
In each cell there is an $a_ i$ amount of money. Tómas starts at cell $1$. In each step he moves $d$ cells. When landing in a cell Tómas takes all the money in that cell. Can you find out how much
money Tómas will get if he continues walking forward until he can’t gather any more money?
The first line of the input contains two integers $n,d$ ($1 \leq n \leq 10^5$, $1 \leq d \leq 10^{14}$), where $n$ is the number of cells and $d$ is how many cells Tómas moves forward with each step.
The next line contains $n$ integers, $a_ i$ ($1 \leq a_ i \leq 10^9$), denoting how much money is in the $i$-th cell.
Print a single integer, how much money Tómas will end up with.
Group Points Constraints
1 25 $1 \leq n, a_ i \leq 100$, $d = 1$
2 25 $1 \leq n, d, a_ i \leq 100$
3 50 No further constraints
Sample Input 1 Sample Output 1
Sample Input 2 Sample Output 2
Sample Input 3 Sample Output 3 | {"url":"https://nus.kattis.com/courses/IT5003/IT5003_S1_AY2425/assignments/on92dn/problems/peningar","timestamp":"2024-11-04T10:37:01Z","content_type":"text/html","content_length":"33147","record_id":"<urn:uuid:1cdbd5e8-6a41-4d14-8a24-cdb6c11da5e4>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00711.warc.gz"} |
Generating Compiler Optimizations
We present an automated technique for generating compiler optimizations from examples of concrete programs before and after improvements have been made to them. The key technical insight of our
technique is that a proof of equivalence between the original and transformed concrete programs informs us which aspects of the programs are important and which can be discarded. Our technique
therefore uses these proofs, which can be produced by translation validation or a proof-carrying compiler, as a guide to generalize the original and transformed programs into broadly applicable
optimization rules.
We present a category-theoretic formalization of our proof generalization technique. This abstraction makes our technique applicable to logics besides our own. In particular, we demonstrate how our
technique can also be used to learn query optimizations for relational databases or to aid programmers in debugging type errors.
Finally, we show experimentally that our technique enables programmers to train a compiler with application-specific optimizations by providing concrete examples of original programs and the desired
transformed programs. We also show how it enables a compiler to learn efficient-to-run optimizations from expensive-to-run super-optimizers.
Proof Generalization
A significant component of this paper is our framework for proof generalization. In the same way abstract interpretation can be applied to various abstractions to produce iterative program analyses,
our framework can be applied to various logics to produce algorithms for generalizing. For example, by applying our framework to E-PEGs, we get an algorithm for generalizing proofs of equivalence
between concrete programs into broadly applicable program optimizations. We have already applied our framework to databases, Hindley-Milner type inference, contracts, and type checking to construct
algorithms for aiding database query optimizations, type debugging of Hindley-Milner-based type inference, contract debugging, and type generalization. We want this framework to be easily accessible,
and we would be interested to hear about additional applications. We would be glad to help researchers overcome the hurdles of encoding logics categorically or constructing pushout completions, so
that they may exploit our framework to solve their own problems. Please feel free to contact us for advice.
The "Most" General Proof: Many of the questions we received after our presentation were regarding how a proof can be "most" general. There are many parameters to this attribute. First, the proof is
key. A concrete optimization may not have a most general form, but each proof does. However, the most general form of some proofs may be more general than the most general form of some other proofs.
Often this may be a complete accident; other times one proof may be a slightly better variant of another proof. For example, any sequential form of a traditional tree-structured proof will actually
produce a better generalization than the proof tree. In the paper, we show a number of techniques for improving a proof so that it has a better generalization. Second, the generalization process all
occurs within a logic. Some logics are more general than other, so more general logics will produce more general proofs. In particular, the distinction between axioms and axiom schemas becomes very
important in proof generalization. For example, in program expression graphs there are loop operations like `pass' which each operate at some loop index. One logic might perceive this as a set of
operators: pass-1, pass-2, and so on. Another logic might perceive this as a single operator with a loop index parameter: pass-i. In the former logic, generalized optimizations will still have the
loop index constants, whereas in the latter logic the generalized optimizations will also generalize the loop index. In our implementation, we use the latter logic, but in the paper we discuss an
even more general logic we could have used. Third, the axiom set has a significant impact on generalization. Some axioms do not generalize well. Consider the example from our presentation where we
replace 8+8-8 with 8. In our presentation, we use axioms such as x-x=0 and x+0=x. We could have also simply used constant folding. However, constant folding produces poor generalizations. Thus, in
our implementation we avoid constant folding as much as possible, only including it in our axiom set as a last resort. There are more factors which influence the generalization process, but these
three are the most important ones. In our experience, it was easy to design a good logic and a good axiom set, and the automatically generated proofs were all what we hoped for. We hope this proof
generalization technique and its applications open new interesting research directions for proof theory.
Category Theory
Some have asked us why we abstracted our proof generalization technique at all, and why we used category theory as our abstraction. However, we actually designed the abstract algorithm first, using
category theory, and then used that to figure out how to solve our concrete problem. We got stuck with the concrete problem, overwhelmed by the details and the variables, and any solution we could
think of seemed arbitrary. In order to reflect and simplify, we decided to phrase our question categorically. This lead to a diagram of sources and sinks, so we just used pushouts and pullbacks to
glue things together. The biggest challenge was coming up with pushout completions, rather than using some existing standard concept. The categorical formulation was easy to specify and reason about.
Afterwards, we instantiated the abstract processes, such as pushouts, with concrete algorithms, such as unification, in order to produce our final implementation with strong generality guarantees.
We have actually found this process of abstracting to category theory whenever we get stuck to be quite fruitful. Not only does it end up solving our concrete problem, but we end up with a better
understanding of our own problem as well as an abstract solution which can be easily adapted to other applications. Thus, our experience suggests that category theory may be useful in constructing
actual algorithms, in addition to being useful as a framework for formalization. We would be interested to know of other similar experiences, either positive or negative.
Questions, Comments, and Suggestions
If you have any questions, comments, or suggestions, please e-mail us. We would be glad to know of any opinions people have or any clarifications we should make. | {"url":"https://www.cs.cornell.edu/~ross/publications/proofgen/","timestamp":"2024-11-03T15:21:17Z","content_type":"text/html","content_length":"9023","record_id":"<urn:uuid:76676206-98ec-405b-9b30-4345c6ba6098>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00696.warc.gz"} |
Queue Meter
Not sure what the queue meter is? Read on →
1. What does the Queue Meter do?
For any social media account, it’s important that new content is published on its timeline at least once every day to keep the audience engaged. The queue meter helps you keep an eye on your queued
posts for each account to ensure that there is at least 1 post scheduled to go out - for the next 7 days.
Let’s take an example.
1. For our Twitter account: @crowdfire, we want to make sure that we post at least 3 times every day.
2. Accordingly, we have set the Posts per day frequency to 3.
3. So for the next 7 days, we need to schedule at least 7 x 3 = 21 posts for the account @crowdfire. The queue meter bar for @crowdfire thus shows a max of 21.
4. This means that if we make sure that we schedule 21 posts for @crowdfire, we are good for the next 7 days!
5. Tomorrow, the bar will once again recalculate whether your account timeline needs more posts scheduled on it or not and display the bar and numbers accordingly.
As long as the bar for each account is full, you are good and ready! If not, then you need to schedule more posts for that particular account. That’s it!
2. What’s the number I keep seeing on the right top corner?
The number indicates the total posts that all your accounts are falling short of for the next 7 days.
Let’s take the same example from above.
1. For the Twitter account `@crowdfire`, let’s say we change the frequency to 4 posts /day. Now we need to schedule 28 posts so that our timeline will have 4 posts going out each day for the next 7
days. But, we have scheduled 6 posts till now. This means that there is a shortfall of all 22 posts for the Twitter account. (again, this is just for the next 7 days)
2. This means that we still need to schedule 22 more posts. (In this case, the queue meter will indicate 22. )
3. Now, let's say we have another account, say a Facebook account that has a frequency of 1 per day, which needs 7 posts scheduled so that the next 7 days are good to go. But for that account, we
have scheduled only 1 post so far. So there are still 6 posts that we need to schedule.
4. In this case the queue meter will show 6 + 22 = 28
5. Now we can see that there are still 28 posts that we need to schedule so that for the next 7 days we have new content being published for all our connected accounts.
1. 22 posts for the Twitter account
2. 6 posts for the Facebook account
6. Once we schedule these posts on each account respectively, the queue meter will not show any number! Think of this as unread messages in your inbox. You just need to reach inbox zero here.
7. Tomorrow, the queue meter will again check your queued posts for the next 7 days and show the pending posts once again.
Some important things to take note of: The queue meter does not display the pending posts for the next week, but for the next 7 days.
• So if today is Monday, the queue meter will check if you have enough posts till the coming Sunday and calculate accordingly.
• If today is Wednesday, the queue meter will check if you have enough posts till next Tuesday. (and not just till the end of that week)
3. Got it! But what do the Posts per day frequency mean and how do I change it?
The posts per day indicate the number of posts that you want to publish for that particular account each day. If you set it to 4, then each day 4 posts will get published on that account from your
scheduled posts queue.You can change this by tapping on the x/day button next to each bar.
4. Cool! If I don’t schedule posts according to the queue meter, will there be a problem?
Not at all! The queue meter is a guide to indicate the ideal number of posts you need to schedule. It’s best if you follow the indicators, but all your posts will still get published regardless. In
case you do face any problems you can always write to us at - hello@crowdfireapp.com
5. Why call it Queue Meter?
We found it to be simple and easy to relate to. If you have better suggestions, please do let us know! Happy to take in new ideas
6. I still don’t get it. Can you help me out?
Sure! Just drop us a line on hello@crowdfireapp.com and we’ll get someone in touch with you.
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We appreciate your effort and will try to fix the article | {"url":"https://support.crowdfireapp.com/support/solutions/articles/5000758413-queue-meter","timestamp":"2024-11-10T11:22:42Z","content_type":"text/html","content_length":"165045","record_id":"<urn:uuid:9c63df3b-38b5-444c-9189-f942c78f6055>","cc-path":"CC-MAIN-2024-46/segments/1730477028186.38/warc/CC-MAIN-20241110103354-20241110133354-00645.warc.gz"} |
The mean height of 12 men is 1.70 m, and the mean height of 8 women is 1.60 m. Find the total height of the 12 men, the total height of the 8 women, the mean height of the 20 men and women.
Hint: Here the question belongs to the topic of mean or average. We have to find the mean height of both men and women. To solve this First, we have to multiply the average height of men with the
total number of men. We get the total height of men similarly for women. Next by considering the total height of both men and women, we can determine the mean or average height of both men and women.
This will be a required solution.
Complete step by step solution:
The Arithmetic Mean is the average of the numbers: a calculated "central" value of a set of numbers.
To calculate it: add up all the numbers, then divide by how many numbers there are.
Consider the given question,
The mean height of 12 men \[ = 1.70\,m\]
Therefore, the total height of 12 men \[ = 12 \times 1.70 = 20.4\,m\]
The mean height of 8 women \[ = 1.60\,m\]
Therefore, the total height of 8 women \[ = 8 \times 1.60 = 12.8\,m\]
Now, find the mean height of 20 men and women
The formula for the mean is given by
\[mean = \,\dfrac{{Sum\,\,of\,\,total\,\,heights\,\,of\,\,men\,\,and\,\,women}}{{Total\,\,no.\,\,of\,\,both\,\,men\,\,and\,\,women}}\]
Therefore, we have
\[ \Rightarrow mean = \,\dfrac{{20.4 + 12.8}}{{20}}\]
On adding all the terms in the numerator, we have
\[ \Rightarrow mean = \,\dfrac{{33.2}}{{20}}\]
On dividing the number 33.2 by 20 we have
\[ \Rightarrow mean = \,1.66\]
Therefore, the mean height of 20 men and women is 1.66 m.
So, the correct answer is “1.66 m”.
Note: To solve these kinds of problem the Student’s must know the formula of mean i.e., \[mean = \,\dfrac{{sum\,\,of\,\,all\,\,numbers}}{{no.\,\,of\,numbers}}\]. Remember If we know two any
parameters of the formula of mean we can determine the unknown parameter by simple arithmetic operation between two known parameters. | {"url":"https://www.vedantu.com/question-answer/the-mean-height-of-12-men-is-170-m-and-the-mean-class-8-maths-cbse-60a6dd6f6b1bfc510b14a721","timestamp":"2024-11-14T07:24:14Z","content_type":"text/html","content_length":"151992","record_id":"<urn:uuid:215bcb65-d174-42f1-b7dd-1ec125294f97>","cc-path":"CC-MAIN-2024-46/segments/1730477028545.2/warc/CC-MAIN-20241114062951-20241114092951-00554.warc.gz"} |
Extracting jpegs
Is the jpeg that I extract in PM different from one obtained by saving a NEF as a jpeg in Nikon Capture NX?
Most definitely. The JPEG that gets extracted from a NEF is the JPEG that was inserted by the camera. On the other hand, if you save your adjusted NEF in NX as a NEF, then NX puts a new JPEG inside
the adjusted NEF. If you extract that JPEG, then it is going to be very similar to the JPEG that NX would create.
I hope this is clear? | {"url":"https://forums.camerabits.com/index.php?topic=731.0","timestamp":"2024-11-08T14:48:31Z","content_type":"application/xhtml+xml","content_length":"27061","record_id":"<urn:uuid:4da9bb6d-84fd-4efb-9248-e1ffd278e55f>","cc-path":"CC-MAIN-2024-46/segments/1730477028067.32/warc/CC-MAIN-20241108133114-20241108163114-00189.warc.gz"} |
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APS March Meeting 2014
Bulletin of the American Physical Society
APS March Meeting 2014
Volume 59, Number 1
Monday–Friday, March 3–7, 2014; Denver, Colorado
Session Z8: Focus Session: Spin-Dynamics: Theory and Experiment Hide Abstracts
Sponsoring Units: GMAG
Chair: Lifa Zhang, University of Texas at Austin
Room: 104
Friday, Z8.00001: Angular Momentum of Phonons and Einstein-de Haas Effect
March Lifa Zhang, Qian Niu
7, 2014
11:15AM We study angular momentum of phonons in a magnetic crystal. In the presence of a spin-phonon interaction, we obtain a nonzero angular momentum of phonons, which is an odd function of
- magnetization. At zero temperature, phonon has a zero-point angular momentum besides a zero-point energy. With increasing temperature, the total phonon angular momentum diminishes and
11:27AM approaches to zero in the classical limit. The nonzero phonon angular momentum can have a significant impact on the Einstein-de Haas effect. To obtain the change of angular momentum of
electrons, the change of phonon angular momentum needs to be subtracted from the opposite change of lattice angular momentum. Furthermore, the finding of phonon angular momentum gives a
potential method to study the spin-phonon interaction. Possible experiments on phonon angular momentum are also discussed. [Preview Abstract]
Friday, Z8.00002: Angular and Linear Momentum of Excited Ferromagnets
March Peng Yan, Akashdeep Kamra, Yunshan Cao, Gerrit Bauer
7, 2014
11:27AM The angular momentum vector of a Heisenberg ferromagnet with isotropic exchange interaction is conserved, while under uniaxial crystalline anisotropy the projection of the total spin along
- the easy axis is a constant of motion. Using Noether's theorem, we prove that these conservation laws persist in the presence of dipole-dipole interactions. However, spin and orbital angular
11:39AM momentum are not conserved separately anymore. We also define the linear momentum of ferromagnetic textures. We illustrate the general principles with special reference to spin transfer
torques and identify the emergence of a non-adiabatic effective field acting on domain walls in ferromagnetic insulators [Preview Abstract]
Friday, Z8.00003: Dynamic Magnetoelectric Effect in Ferromagnet|Superconductor Tunnel Junctions
March Mircea Trif, Yaroslav Tserkovnyak
7, 2014
11:39AM We study the magnetization dynamics in a ferromagnet$\mid$insulator$\mid$superconductor tunnel junction and the associated buildup of the electrical polarization. We show that for an open
- circuit, the induced voltage varies strongly and nonmonotonically with the precessional frequency, and can be enhanced significantly by the superconducting correlations. For frequencies much
11:51AM smaller or much larger than the superconducting gap, the voltage drops to zero, while when these two energy scales are comparable, the voltage is peaked at a value determined by the driving
frequency. We comment on the potential utilization of the effect for the low-temperature spatially-resolved spectroscopy of magnetic dynamics. [Preview Abstract]
Friday, Z8.00004: Dependence of the demagnetization time on the exchange interaction and spin moment
March Guoping Zhang, Thomas F. George, Mingsu Si
7, 2014
11:51AM In femtomagnetism, the demagnetization time is at the center of laser-induced ultrafast demagnetization. It depends on various intrinsic and extrinsic parameters, but the experimental results
- are controversial, and in some cases the opposite effects are reported. In this presentation, we directly address how the exchange interaction and magnetic spin moment affect the
12:03PM demagnetization time. We employ a simple model that includes the exchange interaction and spin-orbit coupling. Then we derive an equation of motion for the spin moment change, from which a
master equation is found. This equation explicitly shows how the demagnetization time is related to the exchange interaction and spin moment. This result can be directly compared with the
latest experimental results. [Preview Abstract]
Friday, Z8.00005: Theoretical and experimental investigations of the electronic structure configuration during ultrafast demagnetization of Co
March Emrah Turgut, Patrik Grychtol, Dmitry Zusin, Henry C. Kapteyn, Margaret M. Murnane, Dominik Legut, Karel Carva, Peter M. Oppeneer, Stefan Mathias, Martin Aeschlimann, Claus M. Schneider,
7, 2014 Justin Shaw, Ronny Knut, Hans Nembach, Thomas J. Silva
- We report on theoretical and experimental studies of the electronic structure configuration during the ultrafast demagnetization in Co thin films. After an ultrafast optical laser excitation
12:15PM of a ferromagnetic material, the magnetization of the material decreases rapidly in less than a picosecond. This ultrafast behavior has attracted a significant amount of attention for more
than two decades; however, the underlying driving mechanism is still unclear. In this work, we use an extreme ultraviolet, broad-bandwidth, tabletop, ultrafast, and element-selective
magnetization probe that employs the transverse magneto-optical Kerr effect to extract the energy- and time-resolved dynamics of the off-diagonal dielectric tensor element that is
proportional to the magnetization. We compare our data with theoretical optical predictions based upon \textit{ab-initio} calculations of the electronic structure, with the ultimate goal of
determining how the occupation of majority and minority states vs. energy evolves after ultrafast optical pumping. [Preview Abstract]
Friday, Z8.00006: Fast reversal of magnetic vortex chirality by electric current
March Weng-Lee Lim, RongHua Liu, Tolek Tyliszczak, Dmitry Berkov, Sergei Urazhdin
7, 2014
12:15PM We demonstrate reversal of magnetic vortex in a microscopic Pt/Permalloy bilayer disk by a nonuniform electric current in the plane of the disk. The switching is detected electronically by
- measuring the response to a small ac magnetic field, and confirmed by direct imaging with x-ray magnetic dichroism microscopy (XMCD). The magnetic contrasts obtained from time-resolved x-ray
12:27PM imaging indicate a fast and robust switching of magnetic vortex driven by electric current. The time-resolved XMCD measurements show that the characteristic switching time is less than 3 ns.
Analysis from micromagnetic simulation shows that the reversal of the magnetic vortex is driven by a combination of the Oersted field due to the charge current and the spin transfer due to
spin current generated by the spin Hall effect in Pt. The simulation reveals that the magnetization switching process of the magnetic vortex involves two distinct stages. The switching first
proceeds with a fast dynamics and then evolves at a slower dynamics before reaching the final magnetic vortex state with opposite chirality, in agreement with the experimental result. The
simulation also shows that the spin transfer torque (STT) accelerates the reversal of magnetic vortex in comparison to the case without STT. [Preview Abstract]
Friday, Z8.00007: Photo-induced Spin Angular Momentum Transfer into Antiferromagnetic Insulator
March Fan Fang, Yichun Fan, Xin Ma, J. Zhu, Q. Li, T.P. Ma, Y.Z. Wu, Z.H. Chen, H.B. Zhao, Gunter Luepke
7, 2014
12:27PM Spin angular momentum transfer into antiferromagnetic(AFM) insulator is observed in single crystalline Fe/CoO/MgO(001) heterostructure by time-resolved magneto-optical Kerr effect (TR-MOKE).
- The transfer process is mediated by the Heisenberg exchange coupling between Fe and CoO spins. Below the Neel temperature(TN) of CoO, the fact that effective Gilbert damping parameter $\
12:39PM alpha$ is independent of external magnetic field and it is enhanced with respect to the intrinsic damping in Fe/MgO, indicates that the damping process involves both the intrinsic spin
relaxation and the transfer of Fe spin angular momentum to CoO spins via FM-AFM exchange coupling and then into the lattice by spin-orbit coupling. [Preview Abstract]
Friday, Z8.00008: Combined molecular and spin dynamics study of collective excitations in BCC iron
March Dilina Perera, David P. Landau, Don Nicholson, G. Malcolm Stocks
7, 2014
12:39PM Spin dynamics simulations of classical spin systems have revealed a substantial amount of information regarding the collective excitations in magnetic materials. However, much of the previous
- work has been restricted to lattice-based spin models that completely disregard the effect of lattice vibrations. Combining an empirical many body potential with a spin Hamiltonian
12:51PM parameterized by first principles calculations, we present a compressible magnetic model for BCC iron, which treats the dynamics of translational degrees of freedom on an equal footing with
the magnetic (spin) degrees of freedom. This model provides us with a unified framework for performing combined molecular and spin dynamics simulations and make simultaneous quantitative
measurements of the spin wave and vibrational spectrum. Results from our simulations reveal that the presence of lattice vibrations leads to softening and damping of spin waves, as well as
evidence for a novel form of longitudinal spin wave excitation coupled with the longitudinal phonon mode of the same frequency. Furthermore, we will also discuss the influence of lattice
vibrations at different temperatures and the implications of using different atomistic potentials. [Preview Abstract]
March Z8.00009: ABSTRACT WITHDRAWN
7, 2014
12:51PM [Preview Abstract]
Friday, Z8.00010: Laser Demagnetization Dynamics in Gadolinium from Time Resolved Photoemission
March John Bowlan, Bj\"orn Frietsch, Martin Teichmann, Robert Carley, Martin Weinelt
7, 2014
1:03PM The field of ultrafast magnetization dynamics has seen rapid progress in recent years and has the potential to enable magnetic data storage systems orders of magnitude faster than those based
- on conventional read/write heads. The dynamics of laser demagnetization in ferromagnetic Gadolinium depend on the transfer of energy and angular momentum between the metallic valence
1:15PM electrons and the core-like $4f$ electrons. Angle-Resolved Photoemission (ARPES) with femtosecond XUV laser pulses produced by high harmonic generation enables the direct measurement of the
electronic band structure on a sub-picosecond time scale in a ``tabletop'' setup. Photoemission allows the magnetization dynamics of the valence and $4f$ bands to be tracked independently of
one another. Thus, time-resolved photoemission is an alternative to experimental methods such as surface magnetic second harmonic generation (MSHG), the magneto-optical Kerr effect (MOKE),
and x-ray magnetic circular dichroism (XMCD). We applied this technique to study Gd(0001) films grown epitaxially on a W crystal. We find that the valence electrons demagnetize on a fs time
scale, while the 4f electrons respond more slowly. [Preview Abstract]
Friday, Z8.00011: Pump-probe measurement of short and long-range exchange interactions in a rare-earth magnet using resonant x-ray diffraction
March Matthew Langner, Sujoy Roy, Yi-De Chuang, Rolf Versteeg, Yi Zhu, Marcus Hertlein, Thornton Glover, Karine Dumesnil, Robert Schoenlein
7, 2014
1:15PM The combined effects of spin-orbit interactions, magnetostriction, and long-range exchange coupling lead to a wide variety of magnetic phases in the rare earth magnets. In dysprosium, core
- level spins develop a spiral phase as a result of competition between short and long-range RKKY exchange interactions mediated by the conducting electrons. We use time-resolved resonant x-ray
1:27PM diffraction to directly probe the spiral order parameter of the core level magnetism in response to optical pumping of the conduction electrons that mediate the exchange interaction. The
dynamics of the diffraction intensity and spiral turn angle occur on different time scales, and through free-energy analysis, we associate these dynamics with changes in the short and
long-range exchange coupling. [Preview Abstract]
Friday, Z8.00012: Numerical Renormalization-Group computation of nuclear magnetic relaxation rates
March Krissia Zawadzki, Luiz N. Oliveira, Jos\'{e} Wilson M. Pinto
7, 2014
1:27PM We report an essentially exact numerical renormalization-group (NRG) computation of the temperature-dependent NMR rate $1/T_1$ of a probe at a distance $R$ from a magnetic impurity in a
- metallic host. We split the metallic states into two subsets, A and B. The former comprises electrons $a_k$ in $s$-wave states about the magnetic-impurity site. The coupling between the $a_k$
1:39PM band and the impurity is described by the Anderson Hamiltonian, diagonalizable by the NRG procedure. Each state $b_k$ in the B subset is a linear combination of an $s$-wave state about the
probe site with the degenerate $a_k$, constructed to be orthogonal to all the $a_k$'s. The $b_k$ band hence decouples from the impurity and is analytically treatable. We show that the
relaxation rate has three components: (i) a constant associated with the $b_k$'s; (ii) a $T$-dependent term associated with the $a_k$'s, which decays in proportion to $1/(k_FR)^2$, where
$k_F$ is the Fermi momentum; and (iii) another $T$-dependent term due to the interference between the $a_k$'s and the $b_k$'s. The interference term shows Friedel oscillations whose
amplitude, proportional to $1/k_FR$, can be mapped onto the universal function of $T/T_K$ describing the Kondo resistivity. We compare our findings with results in the literature. [Preview
March Z8.00013: ABSTRACT MOVED TO A54.00014
7, 2014
1:39PM [Preview Abstract]
Friday, Z8.00014: Dynamic Phase Diagram of the DC-Pumped Magnon Condensates
March Scott Bender, Rembert Duine, Arne Brataas, Yaroslav Tserkovnyak
7, 2014
1:51PM We investigate the effects of nonlinear dynamics and damping by phonons on a system of electronically pumped Bose-Einstein condensed or normal phase magnons in a ferromagnet. The nonlinear
- effects are crucial to understanding the phenomenon of ``swasing." Meanwhile damping was heretofore neglected, since the pumped magnon condensates previously considered are quasi-equilibrium
2:03PM and considered only a much shorter timescale. We analyze the magnetic phase behavior in the presence of these two new effects, demonstrating the possibility of stable condensate and
hysteresis. [Preview Abstract]
Friday, Z8.00015: Dynamics of a two-dimensional quantum spin liquid: signatures of emergent Majorana fermions and fluxes
March Johannes Knolle, Dimitry Kovrizhin, John Chalker, Roderich Moessner
7, 2014
2:03PM Topological states of matter present a wide variety of striking new phenomena. Prominent among these is the fractionalisation of electrons into unusual particles: Majorana fermions, Laughlin
- quasiparticles or magnetic monopoles. Their detection, however, is fundamentally complicated by the lack of any local order, such as, for example, the magnetisation in a ferromagnet. While
2:15PM there are now several instances of candidate topological spin liquids, their identification remains challenging. Here, we provide a complete and exact theoretical study of the dynamical
structure factor of a two-dimensional quantum spin liquid in gapless and gapped (abelian and non-abelian) phases. We show that there are direct signatures--qualitative and quantitative--of
the Majorana fermions and gauge fluxes emerging in Kitaev's honeycomb model. These include counterintuitive manifestations of quantum number fractionalisation, such as a neutron scattering
response with a gap even in the presence of gapless excitations, and a sharp component despite the fractionalisation of electron spin. Our analysis identifies new varieties of the venerable
X-ray edge problem and explores connections to the physics of quantum quenches. [Preview Abstract]
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Math 4 Wisdom. "Mathematics for Wisdom" by Andrius Kulikauskas. | Exposition / AdjointString
Adjoint strings
I am trying to classify adjoint strings and relate them to divisions of everything or their representations.
An adjoint string of {$N$} functors
{$F_0 \dashv F_1 \dashv \cdots \dashv F_{N-2} \dashv F_{N-1}$}
is one for which {$F_0$} has no left adjoint functor and {$F_{N-1}$} has no right adjoint functor.
Adjunction is a relationship that distinguishes a left adjoint functor {$F:\mathcal{D}\rightarrow\mathcal{C}$} and a right adjoint functor {$G:\mathcal{C}\rightarrow\mathcal{D}$}. Evidently, it is a
profoundly meaningful relationship which manifests itself in a wide variety of examples. In each case, the left adjoint and the right adjoint play different conceptual roles which can't be switched
around. Thus adjunction can ground and express the distinction of roles.
A functor {$F$} may or may not have a right adjoint. But if it does have a right adjoint, then that right adjoint is unique up to isomorphism. Similarly with left adjoints. We get strings of adjoints
of various lengths, but nothing more complicated, that is, no branches.
Conceptually, examples of adjoint strings can be grouped into families that illustrate the same theme. Thus I am collecting and grouping examples and trying to understand their conceptual
Adjoint strings of length {$N$} may express conceptual relationships between {$N$} perspectives and thus may indeed express divisions of everything.
For example, the adjunction between a free construction {$F$} and a forgetful functor {$G$} may model the division of everything into two perspectives, as with free will and fate.
I also find it noteworthy that, in the case of abelian categories {$\mathcal{C}$} and {$\mathcal{D}$}, every left adjoint functor {$F$} is right-exact, which means that if {$0\rightarrow D_1\
rightarrow D_2\rightarrow D_3\rightarrow 0$} is exact, then {$F(D_3)\rightarrow F(D_2)\rightarrow F(D_1)\rightarrow 0$} is exact. Similarly, every right adjoint functor {$G$} is left-exact, which
means that if {$0\rightarrow C_1\rightarrow C_2\rightarrow C_3\rightarrow 0$} is exact, then {$0\rightarrow F(D_3)\rightarrow F(D_2)\rightarrow F(D_1)$} is exact. (See: Exact functor and Exact
sequence.) I have previously considered that exact sequences from {$0$} to {$0$} of length {$N$} may express divisions of everything of length {$N-2$}, so I find the connection worth considering.
Adjoint strings may perhaps express the relevant concepts more abstractly.
One potential problem with adjoint strings is that the functors take us back and forth between {$\mathcal{D}$} and {$\mathcal{C}$}. It's not clear to me what that would mean for divisions of
everything. However, it is possible that {$\mathcal{D} = \mathcal{C}$}, so that the functors {$F$} and {$G$} are endofunctors. That would drastically reduce the number of examples but perhaps the
conceptual ideas would persist.
Another issue is that conceptually there seem to be several different families of adjoint strings having the same length, for example, length {$2$}. Thus adjoint strings may perhaps express not
divisions themselves but rather their representations. For example, Grothendieck's six operations comprise an adjoint string of length {$2$} and an adjoint string of length {$4$} which seem to
express the six representations. Again, if the free-forgetful adjunction expresses free will vs. fate, then that is but one of four representations of the division of everything into two
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Compare the output of two Mplus models — compareModels
The compareModels function compares the output of two Mplus files and prints similarities and differences in the model summary statistics and parameter estimates. Options are provided for filtering
out fixed parameters and nonsignificant parameters. When requested, compareModels will compute the chi-square difference test for nested models (does not apply to MLMV, WLSM, and WLSMV estimators,
where DIFFTEST in Mplus is needed). Model outputs to be compared can be full summaries and parameters (generated by readModels), summary statistics only (extractModelSummaries), or parameters only
show = "all",
equalityMargin = c(param = 1e-04, pvalue = 1e-04),
compare = "unstandardized",
sort = "none",
showFixed = FALSE,
showNS = TRUE,
diffTest = FALSE
m1 The first Mplus model to be compared. Generated by readModels, extractModelSummaries, or extractModelParameters.
m2 The second Mplus model to be compared.
show What aspects of the models should be compared. Options are "all", "summaries", "equal", "diff", "pdiff", and "unique". See below for details.
equalityMargin Defines the discrepancy between models that is considered equal. Different margins can be specified for p-value equality versus parameter equality. Defaults to .0001 for both.
compare Which parameter estimates should be compared. Options are "unstandardized", "stdyx.standardized" "stdy.standardized", and "std.standardized".
sort How to sort the output of parameter comparisons. Options are "none", "type", "alphabetical", and "maxDiff". See below for details.
showFixed Whether to display fixed parameters in the output (identified where the est/se = 999.000, per Mplus convention). Default to FALSE.
showNS Whether to display non-significant parameter estimates. Can be TRUE or FALSE, or a numeric value (e.g., .10) that defines what p-value is filtered as non-significant.
diffTest Whether to compute a chi-square difference test between the models. Assumes that the models are nested. Not available for MLMV, WLSMV, and ULSMV estimators. Use DIFFTEST in Mplus
No value is returned by this function. It is used to print model differences to the R console.
The show parameter can be one or more of the following, which can be passed as a vector, such as c("equal", "pdiff").
Display all available model comparison. Equivalent to c("summaries", "equal", "diff", "pdiff", "unique").
Print a comparison of model summary statistics. Compares the following summary statistics (where available): c("Title", "Observations", "Estimator", "Parameters", "LL", "AIC", "BIC",
"ChiSqM_Value", "ChiSqM_DF", "CFI", "TLI", "RMSEA", "SRMR", "WRMR")
Prints a comparison of all summary statistics available in each model. May generate a lot of output.
Print parameter estimates that are equal between models (i.e., <= equalityMargin["param"])
Print parameter estimates that are different between models (i.e., > equalityMargin["param"])
Print parameter estimates where the p-values differ between models (i.e., > equalityMargin["pvalue"])
Print parameter estimates that are unique to each model.
The sort parameter determines the order in which parameter estimates are displayed. The following options are available:
No sorting is performed, so parameters are output in the order presented in Mplus. (Default)
Sort parameters by their role in the model. This groups output by regression coefficient (ON), factor loadings (BY), covariances (WITH), and so on. Within each type, output is alphabetical.
Sort parameters in alphabetical order.
Sort parameter output by the largest differences between models (high to low).
Michael Hallquist
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4.2.1 Input-Output Rules
Use input-output rules, tables and charts to represent patterns and relationships and to solve real-world and mathematical problems.
Subject: Math
Strand: Algebra
Benchmark: 4.2.1.1 Input-Output Rules
Create and use input-output rules involving addition, subtraction, multiplication and division to solve problems in various contexts. Record the inputs and outputs in a chart or table.
For example: If the rule is "multiply by 3 and add 4," record the outputs for given inputs in a table.
Another example: A student is given these three arrangements of dots:
Identify a pattern that is consistent with these figures, create an input-output rule that describes the pattern, and use the rule to find the number of dots in the 10^th figure.
Big Ideas and Essential Understandings
Standard 4.2.1 Essential Understandings
Fourth graders further develop their understanding of a function as they work with input-output rules involving more than one operation. They create, complete and extend input-output tables. Fourth
graders describe patterns in given input-output situations/tables and find a rule. Given a rule, fourth graders are able to find the output for a corresponding input and find an input for a
corresponding output. They realize the value of the output varies depending on the value of the input. Fourth graders use input-output rules involving addition, subtraction and multiplication to
solve problems in various contexts.
All Standard Benchmarks
Create and use input-output rules involving addition, subtraction, multiplication and division to solve problems in various contexts. Record the inputs and outputs in a chart or table.
Benchmark Cluster
Create and use input-output rules involving addition, subtraction, multiplication and division to solve problems in various contexts. Record the inputs and outputs in a chart or table.
What students should know and be able to do [at a mastery level] related to these benchmarks:
• Create a chart or table organizing a list of inputs and outputs.
• Recognize relationships in input-output tables involving more than one operation.
• Use a given "rule" involving more than one operation to create an input-output table.
• Identify the output when given the corresponding input.
• Identify the input when given the corresponding output.
• Identify and apply the "rule" involving more than one operation in completing an input-output table.
• Use addition, subtraction, multiplication and division in identifying input-output "rules."
• Use information in input-output tables to solve problems.
Work from previous grades that supports this new learning includes:
• Understand input-output situations that can be described with a single operation.
Find the output when given an input and the rule.
Find the input when given an output and the rule.
• Use single-operation input-output rules to represent patterns and relationships and to solve real-world and mathematical problems.
NCTM Standards
Understand patterns, relations, and functions
Grades 3-5 Expectations:
• Describe, extend, and make generalizations about geometric and numeric patterns.
• Represent and analyze patterns and functions, using words tables, and graphs.
Represent and analyze mathematical situations and structures using algebraic symbols
Grades 3-5 Expectations:
● Identify such properties as commutativity, associativity, and distributivity and use them to compute with whole numbers.
● Represent the idea of a variable as an unknown quantity using a letter or a symbol.
● Express mathematical relationships using equations.
Use mathematical models to represent and understand quantitative relationships
Grades 3-5 Expectations:
● Model problem situations with objects and use representations such as graphs, tables, and equation to draw conclusions.
Analyze change in various contexts
3-5 Expectations:
● Investigate how a change in one variable relates to a change in a second variable.
• Identify and describe situations with constant or varying rates of change and compare them.
Common Core State Standards
Generate and analyze patterns.
4.OA.5. Generate a number or shape pattern that follows a given rule. Identify apparent features of the pattern that were not explicit in the rule itself. For example, given the rule "Add 3" and the
starting number 1, generate terms in the resulting sequence and observe that the terms appear to alternate between odd and even numbers. Explain informally why the numbers will continue to alternate
in this way.
Student Misconceptions
Student Misconceptions and Common Errors
Students may think...
• the relationships between input values and the relationships between output values are the most important when trying to describe an input-output rule.
• an input-output rule can be determined using only one input-output pair.
Instructional Notes
Teacher Notes
• Students may need support in further development of previously studied concepts and skills.
• Students need to see input-output tables in both vertical and horizontal orientation.
• Use pattern blocks, toothpicks, square tiles, etc., to create growing patterns that can then be represented in an input-output table. Find the rule for the patterns.
For example,
• Students may have difficulty applying a given rule to generate a pattern due to computational errors rather than a failure to comprehend the rule.
• Students may have more difficulty finding rules for patterns that do not have a constant change.
• Students need to see and use input-output tables that do not have sequential inputs.
• Provide experiences with input-output tables that have gaps and/or unknowns in input and/or output columns.
• Emphasize the relationship between the input and output.
• Students should be able to prove that a "suggested rule"(incorrect rule) does not fit the input-output situation.
• Students should be able to describe rules in a variety ways.
• Multiple representations of input-output situations should include verbal descriptions, tables, equations, models or drawings.
• Good questions, and good listening, will help children make sense of the mathematics, build self-confidence and encourage mathematical thinking and communication. A good question opens up a
problem and supports different ways of thinking about it. The best questions are those that cannot be answered with a "yes" or a "no."
Getting StartedWhat do you need to find out?
What do you know now? How can you get the information? Where can you begin?
What terms do you understand/not understand?
What similar problems have you solved that would help?
While Working
How can you organize the information?
Can you make a drawing (model) to explain your thinking? What are other possibilities?
What would happen if...?
Can you describe an approach (strategy) you can use to solve this?
What do you need to do next?
Do you see any patterns or relationships that will help you solve this?
How does this relate to...?
Why did you...?
What assumptions are you making?
Reflecting about the Solution
How do you know your solution (conclusion) is reasonable? How did you arrive at your answer?
How can you convince me your answer makes sense?
What did you try that did not work? Has the question been answered?
Can the explanation be made clearer?
Responding (helps clarify and extend their thinking)
Tell me more.
Can you explain it in a different way?
Is there another possibility or strategy that would work?
Is there a more efficient strategy?
Help me understand this part ...
(Adapted from They're Counting on Us, California Mathematics Council, 1995)
Instructional Resources
NCTM Illuminations
Using a context of chairs around square tables, students will be exposed to three different linear patterns in this lesson. The patterns vary slightly from situation to situation, and the third
situation allows students to determine a solution in multiple ways, in the end leading to an intuitive understanding of perimeter.
Present the following situation to students:
At Pal-a-Table, a new restaurant in town, there are 24 square tables. One chair is placed on each side of a table. How many customers can be seated at this restaurant?
Show an arrangement of one table with four chairs. If your room contains large square tables at which students work in groups, use them as a demonstration. If not, you can draw a picture on the
chalkboard, or you can use pattern blocks or other transparent manipulatives on the overhead projector.
When all students understand how chairs are placed, ask, "If there were 24 tables in a room, how many chairs would be needed?" Depending on students' understanding of multiplication, they may
immediately realize that the answer is 24 × 4 = 96. If not, work with the class to complete a table, as follows:
Tables Chairs
From this table, students should realize that the number of chairs is equal to four times the number of tables. Alternatively, they might recognize that each time a table is added, four chairs are
added. If there are some students who use each approach, this is a good opportunity to reinforce the connection between multiplication and repeated addition. That is,
2 × 4 = 4 + 4
3 × 4 = 4 + 4 + 4
4 × 4 = 4 + 4 + 4 + 4
5 × 4 = 4 + 4 + 4 + 4 + 4
and so on.
The following link provides a unit made up of 5 lessons:
lesson 1 explores patterns using pattern blocks and pattern core
lesson 2 explores multiplication and hundreds chart patterns
lesson 3 explores growing patterns and Pascal's Triangle
lesson 4 explores numeric patterns, Fibonacci numbers, and tables to organize information
lesson 5 explores creation, description, and analysis of patterns using tables and charts.
Additional Instructional Resources
Cuevas, G., & Yeatts, K. (2001). Navigating through algebra in grades 3-5. Reston, VA: National Council of Teachers of Mathematics.
Small, M. (2009). Good questions: Great ways to differentiate mathematics instruction. New York, NY: Teachers College Press.
Van de Walle, J., Karp, K., & Bay-Williams, J. (2010). Elementary and middle school mathematics: teaching developmentally. (7th ed.). Boston, MA: Allyn & Bacon.
Van de Walle, J., & Lovin, L. (2006). Teaching student-centered mathematics grades 3-5. Boston, MA: Pearson Education.
Wickett, M., Kharas, K., & Burns, M. (2002). Grades 3-5 lessons for algebraic thinking. Sausalito, CA: Math Solutions Publications
New Vocabulary
input-output table
"Vocabulary literally is the
key tool for thinking."
Ruby Payne
Mathematics vocabulary words describe mathematical relationships and concepts and cannot be understood by simply practicing definitions. Students need to have experiences communicating ideas using
these words to explain, support, and justify their thinking.
Learning vocabulary in the mathematics classroom is contingent upon the following:
Integration: Connecting new vocabulary to prior knowledge and previously learned vocabulary. The brain seeks connections and ways to make meaning which occurs when accessing prior knowledge.
Repetition: Using the word or concept many times during the learning process and connecting the word or concept with its meaning. The role of the teacher is to provide experiences that will
guarantee connections are made between mathematical concepts, relationships, and corresponding vocabulary words.
Meaningful Use: Multiple and varied opportunities to use the words in context. These opportunities occur when students explain their thinking, ask clarifying questions, write about mathematics,
and think aloud when solving problems. Teachers should be constantly probing student thinking in order to determine if students are connecting mathematics concepts and relationships with appropriate
mathematics vocabulary.
Strategies for vocabulary development
Students do not learn vocabulary words by memorizing and practicing definitions. The following strategies keep vocabulary visible and accessible during instruction.
Mathematics Word Bank: Each unit of study should have word banks visible during instruction. Words and corresponding definitions are added to the word bank as the need arises. Students refer to
word banks when communicating mathematical ideas which leads to greater understanding and application of words in context.
Labeled pictures and charts: Diagrams that are labeled provide opportunities for students to anchor their thinking as they develop conceptual understanding and increase opportunities for student
Frayer Model: The Frayer Model connects words, definitions, examples and non-examples.
Example/Non-example Charts: This graphic organizer allows students to reason about mathematical relationships as they develop conceptual understanding of mathematics vocabulary words. Teachers
should use these during the instructional process to engage student in thinking about the meaning of words.
Vocabulary Strips: Vocabulary strips give students a way to organize critical information about mathematics vocabulary words.
word definition illustration
Encouraging students to verbalize thinking by drawing, talking, and writing increases opportunities to use the mathematics vocabulary words in context.
Additional Resources for Vocabulary Development
Murray, M. (2004). Teaching mathematics vocabulary in context. Portsmouth, NH: Heinemann.
Sammons, L. (2011). Building mathematical comprehension: Using literacy strategies to make meaning. Huntington Beach, CA: Shell Education.
Professional Learning Communities
Reflection - Critical Questions regarding the teaching and learning of these benchmarks
What are the key ideas related to input-input rules at the fourth grade level? How do student misconceptions interfere with mastery of these ideas?
What type of input-output table relationships are the easily seen and described by students? Write an example of a pattern that can be easily described by students when looking at its representation
in an input-output table.
How would you know a student understands the relationships shown in an input-output table? Are some relationships in an input-output table more important as students develop algebraic reasoning?
What representations should a student be able to make for a given pattern or relationship in a problem solving situation?
What are some examples of patterns that are easily explored using input-output tables.? What makes some patterns more difficult to analyze when using input-output tables?
When checking for student understanding, what should teachers
• listen for in student conversations?
• look for in student work?
• ask during classroom discussions?
Examine student work related to a input-output situation. What evidence do you need to say a student is proficient? Using three pieces of work, determine what student understanding is observed
through the work.
How can teachers assess student learning related to these benchmarks?
How are these benchmarks related to other benchmarks at the fourth grade level?
Professional Learning Community Resources
Bamberger, H., Oberdorf, C., & Schultz-Ferrell, K. (2010). Math misconceptions prek-grade 5: From misunderstanding to deep understanding. Portsmouth, NH: Heinemann.
Chapin, S., and Johnson, A. (2006). Math matters: Understanding the math you teach, grades K-8. (2^nd ed.). Sausalito, CA: Math Solutions Press.
Chapin, S., O'Connor, C., & Canavan Anderson, N. (2009). Classroom discussions: Using math talk to help students learn (Grades K-6). Sausalito, CA: Math Solutions.
Fosnot, C., & Dolk, M. (2002). Young mathematicians at work: Multiplication and division. Portsmouth, NH: Heinemann.
Hyde, Arthur. (2006). Comprehending math adapting reading strategies to teach mathematics, K-6. Portsmouth, NH: Heinemann.
Lester, F. (2010). Teaching and learning mathematics: Transforming research for elementary school teachers. Reston, VA: National Council of Teachers of Mathematics.
Otto, A., Caldwell, J., Wallus Hancock, S., & Zbiek, R.(2011). Developing essential understanding of multiplication and division for teaching mathematics in grades 3 - 5. Reston, VA.: National
Council of Teachers of Mathematics.
Parrish, S. (2010). Number talks: Helping children build mental math and computation strategies grades K-5. Sausalito. CA: Math Solutions.
Sammons, L., (2011). Building mathematical comprehension: Using literacy strategies to make meaning. Huntington Beach, CA: Shell Education.
Schielack, J. (2009). Focus in grade 3, teaching with curriculum focal points. Reston, VA: National Council of Teachers of Mathematics.
Bamberger, H., Oberdorf, C., & Schultz-Ferrell, K. (2010). Math misconceptions prek-grade 5: From misunderstanding to deep understanding. Portsmouth, NH: Heinemann.
Bender, W. (2009). Differentiating math instruction: Strategies that work for k-8 classrooms! Thousand Oaks, CA: Corwin Press.
Bresser, R., Melanese, K., & Sphar, C. (2008). Supporting English language learners in math class, grades k-2. Sausalito, CA: Math Solutions Publications.
Burns, Marilyn. (2007). About teaching mathematics: A k-8 resource (3rd ed.). Sausalito, CA: Math Solutions Publications.
Burns, M. (Ed). (1998). Leading the way: Principals and superintendents look at math instruction. Sausalito, CA: Math Solutions.
Caldera, C. (2005). Houghton Mifflin math and English language learners. Boston, MA: Houghton Mifflin Company.
Carpenter, T., Fennema, E., Franke, M., Levi, L., & Empson, S. (1999). Children's mathematics cognitively guided instruction. Portsmouth, NH: Heinemann.
Cavanagh, M. (2006). Math to learn: A mathematics handbook. Wilmington, MA: Great Source Education Group, Inc.
Chapin, S., & Johnson, A. (2006). Math matters: Understanding the math you teach, grades K-8. (2nd ed.). Sausalito, CA: Math Solutions Press.
Chapin, S., O'Connor, C., & Canavan Anderson, N. (2009). Classroom discussions: Using math talk to help students learn (Grades K-6). Sausalito, CA: Math Solutions.
Dacey, L., & Salemi, R. (2007). Math for all: Differentiating instruction k-2. Sausalito, CA: Math Solutions.
Donovan, S., & Bradford, J. (Eds). (2005). How students learn: Mathematics in the classroom. Washington, DC: National Academies Press.
Dougherty, B., Flores, A., Louis, E., & Sophian, C. (2010). Developing essential understanding of number & numeration pre-k-grade 2. Reston, VA: National Council of Teachers of Mathematics.
Felux, C., & Snowdy, P. (Eds.). ( 2006). The math coach field guide: Charting your course. Sausalito, CA: Math Solutions.
Fuson, K., Clements, D., & Beckmann, S. (2009). Focus in grade 2 teaching with curriculum focal points. Reston, VA: National Council of Teachers of Mathematics.
Hyde, Arthur. (2006). Comprehending math adapting reading strategies to teach mathematics, K-6. Portsmouth, NH: Heinemann.
Kilpatrick, J., & Swafford, J. (Eds). (2001). Adding it up: Helping children learn mathematics. Washington, DC: National Academies Press.
Leinwand, S. (2000). Sensible mathematics: A guide for school leaders. Portsmouth, NH: Heinemann.
Lester, F. (2010). Teaching and learning mathematics: Transforming research for elementary school teachers. Reston, VA: National Council of Teachers of Mathematics.
Murray, M. (2004). Teaching mathematics vocabulary in context. Portsmouth, NH: Heinemann.
Murray, M., & Jorgensen, J. (2007). The differentiated math classroom: A guide for teachers k-8. Portsmouth, NH: Heinemann.
National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: NCTM.
Parrish, S. (2010). Number talks: Helping children build mental math and computation strategies grades K-5. Sausalito. CA: Math Solutions.
Reeves, D. (2007). Ahead of the curve: The power of assessment to transform teaching and learning. Indiana: Solution Tree Press.
Sammons, L. (2011). Building mathematical comprehension: Using literacy strategies to make meaning. Huntington Beach, CA: Shell Education.
Schielack, J., Charles, R., Clements, D., Duckett, P., Fennell, F., Lewandowski, S., ... & Zbiek, R. M. (2006). Curriculum focal points for prekindergarten through grade 8 mathematics: A quest for
coherence. Reston, VA: NCTM.
Seeley, C. (2009). Faster isn't smarter: Messages about math teaching and learning in the 21st century. Sausalito, CA: Math Solutions.
Small, M. (2009). Good questions: Great ways to differentiate mathematics instruction. New York, NY: Teachers College Press.
Van de Walle, J., Karp, K., Bay-Williams, J. (2010). Elementary and middle school mathematics: Teaching developmentally. (7th ed.). Boston, MA: Allyn & Bacon.
Van de Walle, J. A., & Lovin, L. H. (2006). Teaching student-centered mathematics grades K-3. Boston, MA: Pearson Education.
West, L., & Staub, F. (2003). Content focused coaching: Transforming mathematics lessons. Portsmouth, NH: Heinemann.
Wickett, M., Kharas, K., & Burns, M.. (2002). Grades 3-5 lessons for algebraic thinking. Sausalito, CA: Math Solutions Publications.
Solution: B
Benchmark: 4.2.1.1
MCA III item sampler
Performance Assessments (Adapted from Minnesota 2007 Mathematics Standards document examples):
● If the rule is "multiply by 3 and add 4," record the outputs for given inputs in a table.
Sample Answer:
Note: Differentiate by providing table and some cells, or requiring the use of specific values for X.
• A student is given these three arrangements of dots:
Identify a pattern that is consistent with these figures, create an input-output rule that describes the pattern, and use the rule to find the number of dots in the 10th figure.
Sample Answer:
Students may draw the next few figures and recognize that the number of dots is a multiplication problem where the rectangle has a side that is one more than the other side.
There are 10*11 = 110 dots in the tenth figure.
Benchmark: 4.2.1.1
Emergent Learners
• Use pattern blocks, tiles, and other manipulatives to create growing patterns that can be represented in an input-output table. Describing the relationship between inputs and outputs is important
as fourth graders develop algebraic thinking.
For example,
Concrete - Representational - Abstract Instructional Approach
The Concrete-Representational-Abstract Instructional Approach (CRA) is a research-based instructional strategy that has proven effective in enhancing the mathematics performance of students who
struggle with mathematics.
The CRA approach is based on three stages during the learning process:
Concrete - Representational - Abstract
The Concrete Stage is the doing stage. The concrete stage is the most critical in terms of developing conceptual understanding of mathematical skills and concepts. At this stage, teachers use
manipulatives to model mathematical concepts. The physical act of touching and moving manipulatives enables students to experience the mathematical concept at a concrete level. Research shows that
students who use concrete materials develop more precise and comprehensive mental representations, understand and apply mathematical concepts, and are more motivated and on-task. Manipulatives must
be selected based upon connections to the mathematical concept and the students' developmental level.
The Representational Stage is the drawing stage. Mathematical concepts are represented using pictures or drawings of the manipulatives previously used at the Concrete Stage. Students move to this
level after they have successfully used concrete materials to demonstrate conceptual understanding and solve problems. They are moving from a concrete level of understanding toward an abstract level
of understanding when drawing or using pictures to represent their thinking. Students continue exploring the mathematical concept at this level while teachers are asking questions to elicit student
thinking and understanding.
The Abstract Stage is the symbolic stage. Teachers model mathematical concepts using numbers and mathematical symbols. Operation symbols are used to represent addition, subtraction, multiplication
and division. Some students may not make a clean transfer to this level. They will work with some symbols and some pictures as they build abstract understanding. Moving to the abstract level too
quickly causes many student errors. Practice at the abstract level will not lead to increased understanding unless students have a foundation based upon concrete and pictorial representations.
Additional Resources
Bender, W. (2009). Differentiating math instruction: Strategies that work for k-8 classrooms! Thousand Oaks, CA: Corwin Press.
Dacey, L., & Lynch, J. (2007). Math for all: Differentiating instruction grades 3-5. Sausalito, CA: Math Solutions.
Murray, M., & Jorgensen, J. (2007). The differentiated math classroom: A guide for teachers k-8. Portsmouth, NH: Heinemann
Small, M. (2009). Good questions: Great ways to differentiate mathematics instruction. New York, NY: Teachers College Press.
Van de Walle, J., Karp, K., & Bay-Williams, J. (2010). Elementary and middle school mathematics: Teaching developmentally. (7th ed.).. Boston, MA: Allyn & Bacon.
Van de Walle, J., & Lovin, L. (2006). Teaching student-centered mathematics grades 3-5. Boston, MA: Pearson Education.
English Language Learners
• Use pattern blocks, tiles, and other manipulatives to create growing patterns that can be represented in an input-output table. Describing the relationship between inputs and outputs is important
as fourth graders develop algebraic thinking.
For example,
• Word banks need to be part of the student learning environment in every mathematics unit of study. Refer to these throughout instruction.
• Word Use vocabulary graphic organizers such as the Frayer model (see below) to emphasize vocabulary words count, first, second, third, etc.
Math sentence frames provide support that English Language Learners need in order to fully participate in math discussions. Sentence frames provide appropriate sentence structure models, increase
the likelihood of responses using content vocabulary, help students to conceptualize words and build confidence in English Language Learners.
Sample sentence frames related to these benchmarks:
If the input is _______________________ the output is __________________________.
I know the rule is _________________ because ________________________________.
If the output is __________________ the input was ______________________________.
• When assessing the math skills of an ELL student it is important to determine if the student has difficulty with the math concept or with the language used to describe the concept and conceptual
Additional ELL Resources
Bresser, R., Melanese, K., & Sphar, C. (2008). Supporting English language learners in math class, grades 3-5. Sausalito, CA: Math Solutions Publications.
Extending the Learning
Additional Resources
Bender, W. (2009). Differentiating math instruction: Strategies that work for k-8 classrooms! Thousand Oaks, CA.: Corwin Press.
Dacey, L., & Lynch, J. (2007). Math for all: Differentiating instruction grades 3-5. Sausalito, CA: Math Solutions.
Murray, M. & Jorgensen, J. (2007). The differentiated math classroom: A guide for teachers k-8. Portsmouth, NH: Heinemann
Small, M. (2009). Good questions: Great ways to differentiate mathematics instruction. New York, NY: Teachers College Press.
Classroom Observation
Administrative/Peer Classroom Observation
Students are . . . Teachers are . . .
using input-output tables to represent patterns and relationships. modeling the use of input-output tables as a representation of a pattern or "rule."
extending a given growing pattern with manipulatives and describing the rule. providing hands-on experiences with growing patterns.
searching for patterns within input-output tables. using the terms input and output when describing patterns and relationships.
describing the patterns found in input-output tables. keeping the functional relationship (input to output) the focus of classroom discussions.
finding the rule for a given input-output table. scaffolding student experiences with one and two step rules for increased student success.
finding missing "input"' and "output" values in input-output tables. providing varied activities for understanding input/output relationships.
What should I look for in the mathematics classroom?
(Adapted from SciMathMN,1997)
What are students doing?
• Working in groups to make conjectures and solve problems.
• Solving real-world problems, not just practicing a collection of isolated skills.
• Representing mathematical ideas using concrete materials, pictures and symbols. Students know how and when to use tools such as blocks, scales, calculators, and computers.
• Communicating mathematical ideas to one another through examples, demonstrations, models, drawing, and logical arguments.
• Recognizing and connecting mathematical ideas.
• Justifying their thinking and explaining different ways to solve a problem.
What are teachers doing?
• Making student thinking the cornerstone of the learning process. This involves helping students organize, record, represent, and communicate their thinking.
• Challenging students to think deeply about problems and encouraging a variety of approaches to a solution.
• Connecting new mathematical concepts to previously learned ideas.
• Providing a safe classroom environment where ideas are freely shared, discussed and analyzed.
• Selecting appropriate activities and materials to support the learning of every student.
• Working with other teachers to make connections between disciplines to show how math is related to other subjects.
• Using assessments to uncover student thinking in order to guide instruction and assess understanding.
Additional Resources
For Mathematics Coaches
Chapin, S. and Johnson, A. (2006). Math matters: Understanding the math you teach: Grades k-8. (2nd ed.). Sausalito, CA: Math Solutions.
Donovan, S., & Bradford, J. (Eds). (2005). How students learn: Mathematics in the classroom. Washington, DC: National Academies Press.
Felux, C., & Snowdy, P. (Eds.). ( 2006). The math coach field guide: Charting your course. Sausalito, CA: Math Solutions.
Sammons, L., (2011). Building mathematical comprehension: Using literacy strategies to make meaning. Huntington Beach, CA: Shell Education.
West, L., & Staub, F. (2003). Content focused coaching: Transforming mathematics lessons. Portsmouth, NH: Heinemann.
For Administrators
Burns, M. (Ed). (1998). Leading the way: Principals and superintendents look at math instruction. Sausalito, CA: Math Solutions.
Kilpatrick, J., & Swafford, J. (Eds). (2001). Adding it up: Helping children learn mathematics. Washington, DC: National Academies Press.
Leinwand, S. (2000). Sensible mathematics: A guide for school leaders. Portsmouth, NH: Heinemann.
Lester, F. (2010). Teaching and learning mathematics: Transforming research for school administrators. Reston, VA: National Council of Teachers of Mathematics.
Seeley, C. (2009). Faster isn't smarter: Messages about math teaching and learning in the 21st century. Sausalito, CA: Math Solutions.
Parent Resources
Mathematics handbooks to be used as home references:
Cavanagh, M. (2004). Math to Know: A mathematics handbook. Wilmington, MA: Great Source Education Group, Inc.
Cavanagh, M. (2006). Math to learn: A mathematics handbook. Wilmington, MA: Great Source Education Group, Inc.
Helping your child learn mathematics Provides activities for children in preschool through grade 5
What should I look for in the mathematics program in my child's school? A Guide for Parents developed by SciMathMN
Help Your Children Make Sense of Math
Ask the right questions
In helping children learn, one goal is to assist children in becoming critical and independent thinkers. You can help by asking questions that guide, without telling them what to do.
Good questions, and good listening, will help children make sense of the mathematics, build self-confidence and encourage mathematical thinking and communication. A good question opens up a problem
and supports different ways of thinking about it. The best questions are those that cannot be answered with a "yes" or a "no."
Getting Started
What do you need to find out?
What do you know now? How can you get the information? Where can you begin?
What terms do you understand/not understand?
What similar problems have you solved that would help?
While Working
How can you organize the information?
Can you make a drawing (model) to explain your thinking? What are other possibilities?
What would happen if . . . ?
Can you describe an approach (strategy) you can use to solve this?
What do you need to do next?
Do you see any patterns or relationships that will help you solve this?
How does this relate to ...?
Can you make a prediction?
Why did you...?
What assumptions are you making?
Reflecting about the Solution
How do you know your solution (conclusion) is reasonable? How did you arrive at your answer?
How can you convince me your answer makes sense?
What did you try that did not work?
Has the question been answered?
Can the explanation be made clearer?
Responding (helps clarify and extend their thinking)
Tell me more.
Can you explain it in a different way?
Is there another possibility or strategy that would work?
Is there a more efficient strategy?
Help me understand this part...
Adapted from They're counting on us, California Mathematics Council, 1995. | {"url":"https://stemtc.scimathmn.org/frameworks/421-input-output-rules","timestamp":"2024-11-03T13:45:24Z","content_type":"text/html","content_length":"84614","record_id":"<urn:uuid:28abd1cd-8bf0-4880-9b19-f66d1ffbbbf1>","cc-path":"CC-MAIN-2024-46/segments/1730477027776.9/warc/CC-MAIN-20241103114942-20241103144942-00349.warc.gz"} |
Thinking in constraints
Understanding the Roomchecking planner philosophy
When one wants to tackle with a planning he has 2 approaches :
Linear approach: you put each room the best way, then almost at the end, when you are happy, you are left with some rooms that you can't allocate without violating the rules. you can just patch or
completly redo the plan ...
Global approach : you look at the problem from 10miles, incorporating all the constraints and find a good plan. The planner uses this approach.
Each time you add a constraint, the planner looks for an optimal feasable solution that satisfies the constraints. Sometimes you can't have all constraints respected...
3 Key concepts to learn to understand how to tweak the planner
What is the objective of the planner?
The objective of the planner is to maximize the number of rooms to clean, while minimizing the number of attendants, and less travelling time.
you can have an objective of giving bewteen 1 and 20 rooms per attendant
you can have an objective of giving bewteen 1 and 300 min per attendant
you can have an objective of giving a fix number of rooms per attendant (target mode)
2. Positive Contribution
Each time you help the planner with some prefered floors or rooms, you increase the objective
When you put in the individual Attendant, some Building, Floors and rooms, the planne will increase the objective if he founds them in one of the solutions.
3.Negative Contribution
Each time you prevent the planner from making less rooms, you penalize the objective.
Each time a Cleaner travels between rooms and building she does less rooms, so Travel time has a negative contribution to the objective
Weight travel time of 1 : will accept a solution with some floor changes
Weight travel time of -5 : will severely penalize the travel time and force the planner to favor a solution with less floor changes
When a human says that a Cleaner should not change builing, he actually says : she should not change building, unless...
thus the concept of Weight : how really important is the constraint or the award ?
Global Constraints
Max Travel Time : this is the sum of all the travel times :
$level to level travel time + building to building tavel time$
We do not want the cleaner to spend too much time on travelling...
Max Building Travel Time : this is the maximum buildings to buildings time/distance a cleaner is allowed to travel.
For Amsterdam city, we set 20 min between buildling.
So each time the cleamer moves from Parool to Trouw, we add 20. In the sample we allow to move 2 times by settings 40:
Max building to building distance allowed : when evaluating if a cleaner can move to the next building we check if we allow her to do a long walk.
For Amsterdam City, since we set 20 min from Parool to Trouw (back and forward), if we put 19 here, the planner will forbid going from P to T and vice versa.
Max Shift Floor allowed : this the max floor distance a cleaner is allowed to jump
Max Level change count : this is the number of time a cleaner could change floor.
Weight Travel Time
Weight Cleaning Time
Weight Rooms cleaned
Weight Epsilon Credits : put -1 if you want to balance by credits otherwise 0
Weight Epsilon Rooms :put -1 if you want to balance by rooms otherwise 0
Explanation : Epsilon is a deviation, so the more we deviate from the mean, the more we penalize the objective, thus the negative number
Award Level : Weight Level Award
Award Room : Weight Room Award
Award Building : Weight Building Award
Distance Matrix
The distance matrix is the distance between buildings. We set the distance to 20min, meaning that the planner will have to decide if it is better to travel 20min to clean a room.
The reason why we have the other way round is that we may want to prevent travelling the other way round.
"locations": [
"distances": [
Individual Cleaner constraints
After the global constraint is set, you can override the decision for each individual cleaner :
Time windows (from -to) :
Affinity : you can favorize a Building, floors, rooms, and room categories. The weight associated will increase the importance of your choice when the planner will have to make a decision.
Credits :
No Rooms : | {"url":"https://help.roomchecking.com/en/article/12-thinking-in-constraints","timestamp":"2024-11-05T19:54:17Z","content_type":"text/html","content_length":"17865","record_id":"<urn:uuid:5681797e-a227-43b8-b715-048153e54d73>","cc-path":"CC-MAIN-2024-46/segments/1730477027889.1/warc/CC-MAIN-20241105180955-20241105210955-00080.warc.gz"} |
Ncert Solutions for Class 10
1. Complete the following statements:
(i) Probability of an event E + Probability of the event ‘not E’ = _________.
(ii) The probability of an event that cannot happen is ________. Such an event is called ________.
(iii) The probability of an event that is certain to happen is ________. Such an event is called _________.
(iv) The sum of the probabilities of all the elementary events of an experiment is _______.
(v) The probability of an event is greater than or equal to _________ and less than or equal to _________.
2. Which of the following experiments have equally likely outcomes? Explain.
(i) A driver attempts to start a car. The car starts or does not start.
(ii) A player attempts to shoot a basketball. She/he shoots or misses the shot.
(iii) A trial is made to answer a true-false question. The answer is right or wrong.
(iv) A baby is born. It is a boy or a girl.
3. Why is tossing a coin considered to be a fair way of deciding which team should get the ball at the beginning of a football game?
4. Which of the following cannot be the probability of an event?
(A) 2/3 (B) –1.5 (C) 15% (D) 0.7
5. If P(E) = 0.05, what is the probability of ‘not E’?
6. A bag contains lemon flavoured candies only. Malini takes out one candy without looking int o the bag. What is the probability that she takes out
(i) an orange flavoured candy?
(ii) a lemon flavoured candy?
7. It is given that in a group of 3 students, the probability of 2 students not having the same birthday is 0.992. What is the probability that the 2 students have the same birthday?
8. A bag contains 3 red balls and 5 black balls. A ball is drawn at random from the bag. What is the probability that the ball drawn is (i) red ? (ii) not red?
9. A box contains 5 red marbles, 8 white marbles and 4 green marbles. One marble is taken out of the box at random. What is the probability that the marble taken out will be (i) red ? (ii) white ?
(iii) not green?
10. A piggy bank contains hundred 50p coins, fifty Re 1 coins, twenty Rs 2 coins and ten Rs 5 coins. If it is equally likely that one of the coins will fall out when the bank is turned upside down,
what is the probability that the coin (i) will be a 50 p coin ? (ii) will not be a Rs 5 coin?
11. Gopi buys a fish from a shop for his aquarium. The shopkeeper takes out one fish at random from a tank containing 5 male fish and 8 fem ale fish (see Figure). What is the probability that the
fish taken out is a male fish?
12. A game of chance consists of spinning an arrow which comes to rest pointing at one of the numbers 1, 2, 3, 4, 5, 6, 7, 8 (see Figure), and these are equally likely outcomes. What is the
probability that it will point at
(i) 8 ?
(ii) an odd number?
(iii) a number greater than 2?
(iv) a number less than 9?
13. A die is thrown once. Find the probability of getting
(i) a prime number
(ii) a number lying between 2 and 6
(iii) an odd number.
14. One card is drawn from a well-shuffled deck of 52 cards. Find the probability of getting
(i) a king of red colour
(ii) a face card
(iii) a red face card
(iv) the jack of hearts
(v) a spade
(vi) the queen of diamonds
15. Five cards—the ten, jack, queen, king and ace of diamonds, are well-shuffled with their face downwards. One card is then picked up at random.
(i) What is the probability that the card is the queen?(ii) If the queen is drawn and put aside, what is the probability that the second card picked up is (a) an ace? (b) a queen?
16. 12 defective pens are accidentally mixed with 132 good ones. It is not possible to just look at a pen and tell whether or not it is defective. One pen is taken out at random from this lot.
Determine the probability that the pen taken out is a good one.
17. (i) A lot of 20 bulbs contain 4 defective ones. One bulb is drawn at random from the lot. What is the probability that this bulb is defective?
(ii) Suppose the bulb drawn in (i) is not defective and is not replaced. Now one bulb is drawn at random from the rest. What is the probability that this bulb is not defective ?
18. A box contains 90 discs which are numbered from 1 to 90. If one disc is drawn at random from the box, find the probability that it bears
(i) a two-digit number
(ii) a perfect square number
(iii) a number divisible by 5. | {"url":"https://chillimath.com/ncert-solutions-for-class-10-probability/","timestamp":"2024-11-05T13:01:50Z","content_type":"text/html","content_length":"126491","record_id":"<urn:uuid:29de25f8-dcf9-42d7-a19e-ae9ce24ff029>","cc-path":"CC-MAIN-2024-46/segments/1730477027881.88/warc/CC-MAIN-20241105114407-20241105144407-00066.warc.gz"} |
Learning from teaching a HS student Schur's theorem on change
(All the math this post refers to is in my manuscript which is
Recall Schur's theorem on making change as stated in wikipedia and other source:
Let a1,...,aL be rel prime coin denominations. Then the number of ways to make n cents change is n^L-1 /(L-1)!a1a2...aL + Θ(n^L-2).
The proof I knew (from Wilfs book on generating functions) was not difficult; however,it involved roots of unity, partial fractions, Taylor series, and Generating functions. I needed to present the
proof to a HS students who was in precalc. The writeup above is what I finally came up with. A few points.
1. HS students, or at least mine, knew complex numbers. Hence roots-of-unity was okay. The proof of Schur's theorem has another plus: he had asked me just recently how complex numbers could be used
in the real world since they weren't... real. I said the are often used as an intermediary on the way to a real solution and gave him an example of a cubic equation where you spot a complex
solution and use it to obtain the real solutions. Schur's theorem is a more sophisticated example of using complex numbers to get a result about reals (about naturals!) so that's a win.
2. Partial Fractions. If the student had had calculus then he would know what partial fractions were and believe me when I said they always work. But since he had not had calculus I prepared a proof
that they worked. Then I realized--- I have never seen a proof that they work! This is a matter of timing- I saw them in High School Calculus in 1975 which was taught without proofs (just as
well, analysis is a bad first-proof course) and I didn't quite realize that the techniques they taught us aren't quite a proof that it works. I came up with my own proof (I can't imagine its
original but I have not found a ref) in 2015. That's 40 years between seeing a concept and proving that it works. A personal record.
3. Taylor Series. I needed the Taylor series for 1/(1-x)^b (just for b a natural). I came up with a proof that does not use calculus and that a HS student could follow. Very happy that I was forced
to do do this. Actually uses a nice combinatorial identity!
4. The lemmas about partial fractions and about Taylor series are of course very very old. Are my proofs new? I doubt it though I have not been able to find a reference. If you know one please leave
a polite comment.
5. Having gone through the proof so carefully I noticed something else the proof yields: Let M be the LCM of a1,...,aL. For all 0\le r\le M-1 there is a poly p of degree L-1 such that if n\equiv r
mod M then p(n) is the number of ways to make change of n cents. I suspect this is known but could not find a ref (again- if you know one then please leave a polite comment.)
Moral of the story: By working with a HS student I was forced to find a proof for partial frac deomp, find a non-calc proof of a Taylor series, and obtain an interesting corollary. Hence this is
already a win!
1 comment:
1. For #5, you might want to look at
The polynomial property you state is another way of saying the number of ways to make change is the Ehrhart quasipolynomial of the polytope {a^Tx <= n} where a is the vector (a_1, a_2, .., a_L)
of change denominations. | {"url":"https://blog.computationalcomplexity.org/2015/06/learning-from-teaching-hs-student.html","timestamp":"2024-11-07T17:18:36Z","content_type":"application/xhtml+xml","content_length":"175172","record_id":"<urn:uuid:cdf16d9e-36ff-4c1a-8038-bc539d434368>","cc-path":"CC-MAIN-2024-46/segments/1730477028000.52/warc/CC-MAIN-20241107150153-20241107180153-00753.warc.gz"} |
MATH133 AIU Solve Equations Mathematical Models Questions - Custom Scholars
MATH133 AIU Solve Equations Mathematical Models Questions
MATH133 – Unit 4 Point-values:Question
You earned
MATH 133 – Unit 4 Individual Project
NAME (Required): _______________________
Assignment Instructions:
1. For each question, show all of your work for full credit.
2. Insert all labeled and titled graphs by using screenshots from Excel or desmos.com (or
other graph program) as described in the Unit 1 Discussion Board.
3. Provide final answers to all questions in boxes provided.
4. Round all value answers to 3 decimal places, unless otherwise noted.
Formulas Provided:
Your company studied how consumers would rate Internet service providers based on the size
of files that they download (x). The following equations model the relationship between two
variables from the result of a survey conducted in North America, Europe, and Asia:
North America
𝑁(𝑥) =
𝐸(𝑥) =
𝐴(𝑥) =
In these equations, you have the following variables:
• x is the size of files to be downloaded (in megabytes).
• k is the speed of download.
• N(x) is the average ratings in North America (in percent).
• E(x) is the average ratings in Europe (in percent).
• A(x) is the average ratings in Asia (in percent).
The consumer ratings can range from 0% (unsatisfactory) to 100% (excellent).
1. From the table below, use the row that matches the first letter of your last name to select
a value for k that you will use in the equations above. It does not necessarily have to be a
whole number.
First Letter of Your Last Name
Possible Values for 𝒌
Using your chosen value for k, write your version of the three mathematical models below.
(20 points)
Chosen Value for k
Equation for North America
Equation for Europe
Equation for Asia
2. Complete the table below by calculating the consumer ratings using the equations from
Problem 1 for North America, Europe, and Asia and using the values of x below.
Show your calculation details for 𝑥 = 100. (20 points)
Consumer Ratings (in %)
(Round value answers to 3 decimal places.)
x, Size of Files
(in Megabytes)
North America
Show your work for x = 100 here:
3. Draw the graphs of the three equations from Problem 1 using desmos.com, Excel, or any
similar online utilities. Label each equation shown as a different color, and put all of these
on the same graph. An introduction to desmos.com can be found at
Be sure to title the graph as your first and last name. In addition, label and number the xaxis and the y-axis appropriately so that the graph matches the calculated values from
above. Use the axes scales of 0 ≤ 𝑥 ≤ 2,500 and −10 ≤ 𝑦 ≤ 30. (20 points)
Insert your graph here:
4. Using correct function transformation terminology, describe the transformation of the
graph of Europe [E(x)] if its parent function is North America [N(x)]. (20 points)
5. Let
𝐴(𝑥) =
At what value of x (in megabytes) will A(x) = 0? Show all of your work details. (20 points)
Value of x in megabytes
(Round value answers to 3 decimal
Show your work here: | {"url":"https://customscholars.com/math133-aiu-solve-equations-mathematical-models-questions/","timestamp":"2024-11-06T18:50:45Z","content_type":"text/html","content_length":"54743","record_id":"<urn:uuid:f050641d-c22b-4231-ba7e-0f7b8415efb8>","cc-path":"CC-MAIN-2024-46/segments/1730477027933.5/warc/CC-MAIN-20241106163535-20241106193535-00002.warc.gz"} |
Principled Bayesian Workflow | mages' blog
Principled Bayesian Workflow
On Thursday evening Michael Betancourt gave an insightful and thought provoking talk on Principled Bayesian Workflow at the Baysian Mixer Meetup, hosted by QuantumBlack.
Michael is an applied statistician, conslutant, co-developer of Stan and passionate educator of Bayesian modelling.
What is a principled Bayesian workflow?
It turns out that it mimics my idea of the scientific method:
1. Create a model for the ‘small world’ of interest, i.e. the small world relevant to test an idea, e.g. law of gravity that describes how an apple falling off a tree is accelerated by the Earth’s
2. Decide which measurements will be relevant
3. Make predictions with the model
4. Review if the predictions are reasonable
5. Set up experiment and predict outcome with set parameters
6. Carry out experiment
7. Compare measurements with predictions and go back to 1 if necessary
Michael summarised his approach more eloquently for Bayesian Inference in one slide:
He gave also succinct advise for model calibration and validation. Michael suggested the following four questions:
1. Are modelling assumptions consistent with our domain knowledge?
2. Are our computational tools sufficient to accurately fit the model?
3. How should we expect our inference to perform?
4. Is our model rich enough to capture the relevant structure of the data generating process?
Adding structure could mean to add more parameters to the model, rather than changing it fundamentally. As an example Michael mentioned a model that started with a Normal distribution for the data
generating process, but that was expanded to a Student’s t-distribution to allow for fatter tails. The reasoning here is that Student’s t-distribution will converge to \(\mathcal{N}(0,1)\) for \(\nu
\to \infty\) and hence, incorporates the initial assumption of normality.
From his experience it is helpful to show data (including data generated from your model), when asking for advise from domain experts. Experts are more likely to have a view/ opinion on data rather
than parameters, and even more so what data will be unreasonable. Don’t forget they are experts, and hence like/likely to disagree with the non-expert. Use this bias to your advantage.
Finally, he showed one chart to check the posterior predictive distribution that I found very insightful. You plot the posterior z-scores against the posterior shrinkages (\(s\)), which are defined
\[ \begin{aligned} z & = \left| \frac{\mu_{\mbox{post}} - \tilde{\theta}}{ \sigma_{\mbox{post}} } \right| \\ s & = 1 - \frac{\sigma^{2}_{\mbox{post}} }{ \sigma^{2}_{\mbox{prior}} } \end{aligned} \]
Depending on where you see a cluster in the diagram, it will provide guidance on where you might have a problem with your model.
You find Michael’s full article on principled Bayesian workflow on his website.
Next events
The next Bayesian Mixer will take place on 17 July at SkillsMatter. Eric Novik (CEO, Generable) will talk about Precision Medicine With Mechanistic, Bayesian Models.
In addition, on 16 July we have the Insurance Data Science Conference at Cass Business School London, followed by a Stan Workshop on 17 July. You can register for either event here.
For attribution, please cite this work as:
Markus Gesmann (Jun 18, 2018) Principled Bayesian Workflow. Retrieved from https://magesblog.com/post/principled-bayesian-workflow/
BibTeX citation:
@misc{ 2018-principled-bayesian-workflow,
author = { Markus Gesmann },
title = { Principled Bayesian Workflow },
url = { https://magesblog.com/post/principled-bayesian-workflow/ },
year = { 2018 }
updated = { Jun 18, 2018 } | {"url":"https://magesblog.com/post/principled-bayesian-workflow/","timestamp":"2024-11-07T16:25:02Z","content_type":"text/html","content_length":"21039","record_id":"<urn:uuid:0437319d-b602-41d4-b9a8-8cbcecfc2e50>","cc-path":"CC-MAIN-2024-46/segments/1730477028000.52/warc/CC-MAIN-20241107150153-20241107180153-00748.warc.gz"} |
Gregor Papa
δ-perturbation of bilevel optimization problems: An error bound analysis, Oper. Res. Perspect. (2024)
Models for forecasting the traffic flow within the city of Ljubljana, ETRE (2023)
Electric-bus routes in hilly urban areas: overview and challenges, RENEW SUST ENERG REV (2022)
Data Multiplexed and Hardware Reused Architecture for Deep Neural Network Accelerator, NEUROCOMPUTING (2022)
Four algorithms to solve symmetric multi-type non-negative matrix tri-factorization problem, J GLOBAL OPTIM (2022)
Traffic Forecasting With Uncertainty: A Case for Conformalized Quantile Regression, ECCOMAS 2024
Fleet and Traffic Management Systems for Conducting Future Cooperative Mobility, ECCOMAS 2024
Fleet and traffic management systems for conducting future cooperative mobility, TRA 2024
An evolutionary approach to pessimistic bilevel optimization problems,
Users’ cognitive processes in a user interface for predicting football match results, IS 2023
An evolutionary approach to pessimistic bilevel optimization problems, BILEVEL 2023
Evaluation of Parallel Hierarchical Differential Evolution for Min-Max Optimization Problems Using SciPy, BIOMA 2022
On Suitability of the Customized Measuring Device for Electric Motor, IECON 2022
Real‐world Applications of Dynamic Parameter Control in Evolutionary Computation - tutorial , PPSN 2022
Dynamic Computational Resource Allocation for CFD Simulations Based on Pareto Front Optimization, GECCO '22
GPU-based Accelerated Computation of Coalescence and Breakup Frequencies for Polydisperse Bubbly Flows, NENE 2021
Worst-Case Scenario Optimisation: Bilevel Evolutionary Approaches, IPSSC 2021
Solving pessimistic bilevel optimisation problems with evolutionary algorithms, EUROGEN 2021
Detecting Network Intrusion Using Binarized Neural Networks, WF-IoT2021
Preferred Solutions of the Ground Station Scheduling Problem using NSGA-III with Weighted Reference Points Selection, CEC 2021
Applications of Dynamic Parameter Control in Evolutionary Computation -tutorial, GECCO 2021
Dynamic Parameter Changing During the Run to Speed Up Evolutionary Algorithms: PPSN 2020 tutorial, PPSN 2020
On Formulating the Ground Scheduling Problem as a Multi-objective Bilevel Problem, BIOMA 2020
Refining the CC-RDG3 Algorithm with Increasing Population Scheme and Persistent Covariance Matrix, BIOMA 2020
Dynamic Parameter Choices in Evolutionary Computation: WCCI 2020 tutorial, WCCI 2020
Dynamic control parameter choices in evolutionary computation: GECCO 2020 tutorial, GECCO 2020
Solving min-max optimisation problems by means of bilevel evolutionary algorithms: a preliminary study, GECCO 2020
Colors and colored overlays in dyslexia treatment, IPSSC 2020
Optimisation platform for remote collaboration of different teams, OSE5
Experimental evaluation of deep-learning applied on pendulum balancing, ERK 2019
An adaptive evolutionary surrogate-based approach for single-objective bilevel optimisation, UQOP 2019
Parameter Control in Evolutionary Bilevel Optimisation, IPSSC 2019
The sensor hub for detecting the influence of colors on reading in children with dyslexia, IPSSC 2019
Comparing different settings of parameters needed for pre-processing of ECG signals used for blood pressure classification, BIOSIGNALS 2019
The role of colour sensing and digitalization on the life quality and health tourism, QLife 2018
Evolutionary operators in memetic algorithm for matrix tri-factorization problem, META 2018
From a Production Scheduling Simulation to a Digital Twin, HPOI 2018
Evolution of Electric Motor Design Approaches: The Domel Case, HPOI 2018
Bluetooth based sensor networks for wireless EEG monitoring, ERK 2018
Construction of Heuristic for Protein Structure Optimization Using Deep Reinforcement Learning, BIOMA 2018
Proactive Maintenance of Railway Switches, CoDIT 2018
Sensors: The Enablers for Proactive Maintenance in the Real World, CoDIT 2018
The role of physiological sensors in dyslexia treatment, ERK 2017
Transportation problems and their potential solutions in smart cities, SST 2017
Prediction of natural gas consumption using empirical models, SST 2017
Cyber Physical System Based Proactive Collaborative Maintenance, SST 2016
Optimizacija geometrije profila letalskega krila, ERK 2016
Pametno urejanje prometa in prostorsko načrtovanje, IS 2015
Pobuda PaMetSkup, IS 2015
Concept of an Ecosystem model to support transformation towards sustainable energy systems, SDEWES 2015
Napredne metode za optimizacijo izdelkov in procesov, Dnevi energetikov 2015
Case-based slogan production, ICCBR 2015
Modularni sistem za upravljanje Li-Ion baterije, AIG 2015
Upgrade of the MovSim for Easy Traffic Network Modification, SIMUL 2015
Suitability of MASA Algorithm for Traveling Thief Problem, SOR'15
Empirical Convergence Analysis of GA for Unit Commitment Problem, BIOMA 2014
Automated Slogan Production Using a Genetic Algorithm, BIOMA 2014
Comparison Between Single and Multi Objective Genetic Algorithm Approach for Optimal Stock Portfolio Selection, BIOMA 2014
Implementation of a slogan generator, ICCC 2014
The Parameter-Less Evolutionary Search for Real-Parameter Single Objective Optimization, CEC 2013
Optimization in Organizations: Things We Tend to Forget, BIOMA 2012
Combinatorial implementation of a parameter-less evolutionary algorithm, IJCCI 2011
Optimal on-line built-in self-test structure for system-reliability improvement, CEC 2011
Hybrid Parameter-Less Evolutionary Algorithm in Production Planning, ICEC2010
Vpliv časovnih omejitev na učinkovitost memetskega algoritma pri planiranju proizvodnje, ERK 2010
Organizacijski vidik uvajanja napovedne analitike v organizacije, ERK 2010
Influence of Fixed-Deadline Orders on Memetic Algorithm Performance in Production Planning, MCPL 2010
Optimization of cooling appliance control parameters, EngOpt 2010
Bioinspired Online Mathematics Learning, BIOMA 2010
Application of Memetic Algorithm in Production Planning, BIOMA 2010
Thermal Simulation for Development Speed-up, SIMUL 2010
Metaheuristic Approach to Loading Schedule Problem, MISTA 2009
MATPORT - Web application to support enhancement in elementary mathematics pedagogy, CSEDU 2009
Constrained transportation scheduling, BIOMA 2008
Web interface for Progressive zoom-in algorithm, ERK 2008
Parameter-less evolutionary search, GECCO-08
Deterministic Test Pattern Generator Design, EvoHOT 2008
Robot TCP positioning with vision : accuracy estimation of a robot visual control system, ICINCO 2007
Estimation of accuracy for robot vision control, SHR2007
Accuracy of a 3D reconstruction system, ISPRA 2007
Non-parametric genetic algorithm, BIOMA 2006
Evolutionary approach to deterministic test pattern generator design, Euromicro 2006
Algoritem postopnega približevanja, ERK 2006
On design of a low-area deterministic test pattern generator by the use of genetic algorithm, CADSM2005
Optimization algorithms inspired by electromagnetism and stigmergy in electro-technical engineering, EC 2005
Towards automated trust modeling in networked organizations, ICAS 2005
A decision support approach to modeling trust in networked organizations, IEA/AIE 2005
Učinkovitost brezparametrskega genetskega algoritma, ERK 2005
Test pattern generator structure design by genetic algorithm, BIOMA 2004
Electrical engineering design with an evolutionary approach, BIOMA 2004
An evolutionary technique for scheduling and allocation concurrency, MATH 2004
Ovrednotenje sočasnega razvrščanja operacij in dodeljevanja enot v postopku načrtovanja integriranih vezij, ERK 2003
Reševanje soodvisnih korakov načrtovanja integriranih vezij, IS 2003
Concurrent operation scheduling and unit allocation with an evolutionary technique, DSD 2003
Evolutionary approach to scheduling and allocation concurrency in integrated circuits design, ECCTD-03
Sočasno razvrščanje operacij in dodeljevane enot v postopku načrtovanja integriranih vezij, ERK 2002
Evolutionary synthesis algorithm - genetic operators tuning, EC-02
Evolutionary method for a universal motor geometry optimization: a new automated design approach, IUTAM 2002
Evolutionary method for a universal motor geometry optimization, IUTAM
Optimizacija geometrije lamele univerzalnega motorja, ERK 2001
Optimization of the geometry of the rotor and the stator, SOR-01
Performance optimization of a universal motor using a genetic algorithm, INES 2001
Improving the technical quality of a universal motor using and evolutionary approach, Euromicro-01
Evolutionary performance optimization of a universal motor, MIPRO 2001
Evolutionary optimization of a universal motor, IECON-01
Večkriterijsko genetsko načrtovanje integriranih vezij, ERK 2000
Multi-objective genetic scheduling algorithm with respect to allocation in high-level synthesis, Euromicro2000
The use of genetic algorithm in the integrated circuit design, WDTA-99
Optimization of the parallel matrix multiplication, CSCC-99
Transformation of the systolic arrays from two-dimensional to linear form, ICECS-99
Genetic algorithm as a method for finite element mesh smoothing, PPAM-99
Scheduling algorithms based on genetic approach, NNTA-99
Reševanje linearnega sistema enačb v enodimenzionalnih sistoličnih poljih, ERK-98
Evolutionary scheduling algorithms in high-level synthesis, WDTA-98
Using simulated annealing and genetic algorithm in the automated synthesis of digital systems, IMACS-CSC-98
On-line testing of a discrete PID regulator: a case study, Euromicro97
A Framework for Applying Data-Driven AI/ML Models in Reliability, Recent Advances in Microelectronics Reliability, Springer (2024)
Reliability Improvements for In-Wheel Motor, Recent Advances in Microelectronics Reliability, Springer (2024) | {"url":"https://cs.ijs.si/papa/?show=publications&id=488&conferences=all","timestamp":"2024-11-10T05:54:45Z","content_type":"text/html","content_length":"27516","record_id":"<urn:uuid:deba6349-2b7b-40a0-9d3a-b6eb3917d3a3>","cc-path":"CC-MAIN-2024-46/segments/1730477028166.65/warc/CC-MAIN-20241110040813-20241110070813-00467.warc.gz"} |
Lesson 11
Different Partial Quotients
Warm-up: Notice and Wonder: Ways to Record (10 minutes)
The purpose of this warm-up is to elicit the idea that multiplication and division can both be used to represent partial quotients. Students compare two representations, both familiar from grade 4,
of partial quotients. This will be helpful when students record partial quotients with division expressions later in the lesson.
• Groups of 2
• Display the image.
• “What do you notice? What do you wonder?”
• 1 minute: quiet think time
• “Discuss your thinking with your partner.”
• 1 minute: partner discussion
• Share and record responses.
Student Facing
What do you notice? What do you wonder?
\( 130\div 13&= 10\\ 130\div 13 &= 10\\ 65 \div 13 &= \phantom{0} 5\\ 39\div 13 &= \phantom{0} 3\\ \overline {\hspace{5mm}364 \div 13} &\overline{\hspace{1mm}= 28 \phantom{000}}\)
Activity Synthesis
• “How are these strategies the same and different as the way you found quotients in the previous lesson?” (They use multiplication and division. They break the problem into smaller parts with
numbers that are easier to calculate.)
• “Multiplication can help us think about partial quotients.”
Activity 1: Division Expressions (20 minutes)
The purpose of this activity is to see that some equivalent ways of rewriting a division expression are more helpful than others for finding its value. In particular expressions whose value can be
calculated mentally are most helpful. This supports students when they choose their own ways of breaking up the dividend when they use an algorithm that uses partial products in future lessons.
When they identify expressions to find the value of a quotient, students notice that dividends that are readily identifiable as multiples of 14 are most useful for finding the value of the quotient
This activity uses MLR2 Collect and Display. Advances: Conversing, Reading, Writing.
Action and Expression: Internalize Executive Functions. Invite students to verbalize their strategy for choosing the set of expressions before they begin. Students can speak quietly to themselves, or
share with a partner.
Supports accessibility for: Organization, Conceptual Processing, Language
Required Materials
Materials to Copy
• Partial Quotient Expressions
Required Preparation
• Create a set of cards from the blackline master for each group of 2.
• Groups of 2
• Display cards for the activity.
• “What do you notice? What do you wonder?” (They all have a 14 in them. Some of the expressions are repeated. All of them show a number being divided by 14. A lot of the expressions show multiples
of 10. A lot of the expressions show multiples of 14. I wonder what we are going to do with these expressions. I wonder why they all divide a number by 14?)
• “You are going to choose expressions that represent a sum that is equal to \(308 \div 14\).”
• “Keep each set of expressions that you choose.”
MLR2 Collect and Display
• 3–5 minutes: partner work time
• As students play the game, circulate, listen for, and collect the language students use to explain how they know the cards they chose represent a sum that is equal to \(308 \div 14\). Listen for:
add, plus, divide, dividend, quotient, equals, “the same as”, and groups of.
• Record students’ words and phrases on a visual display and update it throughout the lesson.
• “Choose one set of expressions and use it to find the value of \(308 \div 14\).”
• 3–5 minutes: independent work time
• Monitor for students who use these different combinations to find the value of \(308 \div 14\):
□ \(280 \div 14\), \(28 \div 14\)
□ \(140 \div 14\), \(140 \div 14\), \(28 \div 14\)
Student Facing
Take turns:
1. Choose a set of expressions that, when added together, is equal to \(308 \div 14\). Not all expressions will be used.
2. Explain to your partner how you know that your cards represent a sum that is equal to \(308 \div 14\).
(Pause for teacher directions.)
3. Choose one of the sets of expressions whose sum is equal to \(308 \div 14\) and use it to find the value of \(308 \div 14\).
Advancing Student Thinking
If students do not recognize which expressions created by the cards would be helpful to find the value of the quotient \(308 \div 14\), ask, “Which of these expressions have a value that is a whole
number? How could you use the expression to find the value of \(308 \div 14\)?”
Activity Synthesis
• Refer to the visual display of the words students used during the activity.
• “These are the words you used to explain how you knew the cards you chose represent a sum that is equal to \(308 \div 14\).”
• “Are there any other words or phrases that are important to include on our display?”
• As students share responses, update the display, by adding (or replacing) language, diagrams, or annotations.
• Remind students to borrow language from the display as needed.
• Display:
\(308 \div 14\)
• Ask previously selected students to share.
• “Why did you choose that set of expressions to find the value of \(308 \div 14\)?” (I could find the values of the expressions mentally.)
• Display:
\(280 \div 14\)
• “Why is \(280 \div 14\) a helpful expression to start with?” (\(280 \div 14\) is a helpful expression to start with because \(20 \times 14 = 280\) and 280 is close to 308.)
• “How do we know that \(28 \div 14\) is the expression that matches \(280 \div 14\)?” (\(280 + 28 = 308\))
Activity 2: Choose Your Own Partial Quotients (15 minutes)
The purpose of this activity is for students to consider efficient ways to use the partial quotients strategy. The strategy that is most efficient for a student will depend on which products or
quotients they can find mentally. Encourage students to try to use larger partial quotients as they work, but to try to find partial quotients which they can calculate mentally.
• Groups of 2
• “For these problems, choose one of the division expressions to use to begin finding the quotient. Then use the expressions you chose to find the value of each quotient."
• 6-8 minutes: independent work time
• 2-3 minutes: partner discussion
• Monitor for students who
□ use different expressions for \(992\div31\).
□ wrote about changing their mind in the last question.
Student Facing
For each expression, choose one of the partial quotients and, beginning with that expression, find the value of the quotient.
1. \(360\div15\)
□ \(150\div15\)
□ \(300\div15\)
□ \(60\div15\)
2. \(945\div45\)
□ \(45\div45\)
□ \(450\div45\)
□ \(900\div45\)
3. \(992\div31\)
□ \(62\div31\)
□ \(341\div31\)
□ \(310\div31\)
4. How did you decide which partial quotient to use to begin finding the quotient? Did you change your mind with any of the problems?
Activity Synthesis
• Invite selected students to share their reasoning for \(992\div31\) and explain why they started with their expression.
• Invite students to share how they changed their mind about which expressions to use during the activity.
Lesson Synthesis
“Today we thought strategically about which partial quotients are most helpful for finding the value of a division expression.”
Display the last problem from the second activity: \(945\div 45\).
“Diego said, ‘In order to solve division problems, you can use all the operations.’ What does Diego mean? When did we use addition, subtraction, and multiplication to divide?” (We can multiply in
order to find how many 45s are in 945. After we find those multiples, we can add them up to get 945. We can also use division. I know that \(90 \div 45 = 2\) and \(900 \div 45 = 20\) so I can
subtract 900 from 945 and that leaves just 1 more 45.)
Cool-down: Find the Value (5 minutes) | {"url":"https://curriculum.illustrativemathematics.org/k5/teachers/grade-5/unit-4/lesson-11/lesson.html","timestamp":"2024-11-02T21:51:46Z","content_type":"text/html","content_length":"88431","record_id":"<urn:uuid:ebe08ed0-708c-4342-860a-15666e91658e>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00446.warc.gz"} |
This module contains procedures and generic interfaces for computing the previous/next integer exponent for the given base that yields a number smaller/larger than the absolute input value.
For a given number \(x\) and base \(b\), the next integer exponent \(p\) is defined as the smallest integer such that,
$$b^{(p - 1)} < |x| \leq b^p ~~~ \forall x > 0 ~,$$
The functionalities of this module are useful in computing the padded lengths of arrays whose auto-correlation, cross-correlation, or Fourier-Transform are to be computed using the Fast-Fourier
Transform method.
See also
Final Remarks ⛓
If you believe this algorithm or its documentation can be improved, we appreciate your contribution and help to edit this page's documentation and source file on GitHub.
For details on the naming abbreviations, see this page.
For details on the naming conventions, see this page.
This software is distributed under the MIT license with additional terms outlined below.
1. If you use any parts or concepts from this library to any extent, please acknowledge the usage by citing the relevant publications of the ParaMonte library.
2. If you regenerate any parts/ideas from this library in a programming environment other than those currently supported by this ParaMonte library (i.e., other than C, C++, Fortran, MATLAB, Python,
R), please also ask the end users to cite this original ParaMonte library.
This software is available to the public under a highly permissive license.
Help us justify its continued development and maintenance by acknowledging its benefit to society, distributing it, and contributing to it.
Amir Shahmoradi, April 25, 2015, 2:21 PM, National Institute for Fusion Studies, The University of Texas Austin | {"url":"https://www.cdslab.org/paramonte/fortran/latest/namespacepm__mathExp.html","timestamp":"2024-11-12T03:33:40Z","content_type":"application/xhtml+xml","content_length":"15425","record_id":"<urn:uuid:99ddf13b-37ba-4c54-8ccf-bab64e950531>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.50/warc/CC-MAIN-20241112014152-20241112044152-00720.warc.gz"} |
Simple equilibria in finite games with convexity properties
Applicationes Mathematicae 42 (2015), 83-109 MSC: 91A05, 91A06, 91A10, 91B52. DOI: 10.4064/am42-1-6
This review paper gives a characterization of non-coalitional zero-sum and non-zero-sum games with finite strategy spaces and payoff functions having some concavity or convexity properties. The
characterization is given in terms of the existence of two-point Nash equilibria, that is, equilibria consisting of mixed strategies with spectra consisting of at most two pure strategies. The
structure of such simple equilibria is discussed in various cases. In particular, many of the results discussed can be seen as discrete counterparts of classical theorems about the existence of pure
(or “almost pure”) Nash equilibria in continuous concave (convex) games with compact convex spaces of pure strategies. The paper provides many examples illustrating the results presented and ends
with four related open problems. | {"url":"https://www.impan.pl/en/publishing-house/journals-and-series/applicationes-mathematicae/all/42/1/84067/simple-equilibria-in-finite-games-with-convexity-properties","timestamp":"2024-11-04T05:39:18Z","content_type":"text/html","content_length":"45067","record_id":"<urn:uuid:10083796-f774-4b4b-a646-726d73a9a104>","cc-path":"CC-MAIN-2024-46/segments/1730477027812.67/warc/CC-MAIN-20241104034319-20241104064319-00818.warc.gz"} |
Best Probability MCQs Quiz 4 - Statistics for Data Analyst
Best Probability MCQs Quiz 4
The post is about the Probability MCQs Quiz. There are 25 multiple-choice questions covering the topic related to counting rules of probability, random experiments, assigning probability, events and
types of events, and rules of probability. Let us start with the Probability MCQs Quiz.
MCQs about probability, sample space,
1. The “Top Three” at a racetrack consists of picking the correct order of the first three horses in a race. If there are 10 horses in a particular race, how many “Top Three” outcomes are there?
2. Two events, $A$ and $B$ are mutually exclusive and each has a non-zero probability. If event $A$ is known to occur, the probability of the occurrence of event $B$ is
3. Each customer entering a departmental store will either buy or not buy a certain product. An experiment consists of the following 3 customers and determining whether or not they will buy any
certain product. The number of sample points in this experiment is as follows:
4. If $P(A) = 0.38, P(B) = 0.83$, and $P(A\cap B)=0.57$, then $P(A\cup B) =$ ?
5. An experiment consists of four outcomes with $P(A) = 0.2, P(B) = 0.3, P(C) = 0.4$. The probability of the outcome $P(D)$ is
6. Three applications for admission to a university are checked to determine whether each applicant is male or female. The number of sample points in this experiment is
7. A method of assigning probabilities that assumes the experimental outcomes are equally likely is called
8. A lottery is conducted using 3 urns. Each urn contains balls numbered from 0 to 9. One ball is randomly selected from each urn. The total number of sample points in the sample space is
9. The union of events $A$ and $B$ is the event containing
10. Events that have no sample points in common are called
11. When the results of historical data or experimentation are used to assign probability values, the method used to assign probabilities is referred to as the
12. Two letters are to be selected at random from five letters (A, B, C, D, and E). How many possible selections are there?
13. Given that event $A$ has a probability of 0.25, the probability of the complement of event $A$
14. The probability assigned to each experimental outcome must be
15. The symbol $\cap$ shows the
16. The probability of the union of two events with non-zero probabilities
17. If two events are mutually exclusive, then the probability of their intersection
18. The probability of the intersection of two mutually exclusive events
19. If $A$ and $B$ are mutually exclusive events with $P(A)=0.3$ and $P(B)=0.5$, then $P(A \cap B)=$?
20. The symbol $\cup$ shows the
21. Two events are mutually exclusive if
22. If $P(A) = 0.62, P(B) = 0.47$, and $P(A\cup B) = 0.88$, then $P(A \cap B) =$ ?
23. When the assumption of equally likely outcomes is used to assign probability values, the method used to assign probabilities is referred to as the
24. The addition law helps to compute the probabilities of
25. Suppose your favorite cricket team has 2 games left to finish the series. The outcome of each game can be won, lost, or tied. The number of possible outcomes is
Online Probability MCQs Quiz
• A lottery is conducted using 3 urns. Each urn contains balls numbered from 0 to 9. One ball is randomly selected from each urn. The total number of sample points in the sample space is
• Three applications for admission to a university are checked to determine whether each applicant is male or female. The number of sample points in this experiment is
• Suppose your favorite cricket team has 2 games left to finish the series. The outcome of each game can be won, lost, or tied. The number of possible outcomes is
• Each customer entering a departmental store will either buy or not buy a certain product. An experiment consists of the following 3 customers and determining whether or not they will buy any
certain product. The number of sample points in this experiment is as follows:
• Two letters are to be selected at random from five letters (A, B, C, D, and E). How many possible selections are there?
• The “Top Three” at a racetrack consists of picking the correct order of the first three horses in a race. If there are 10 horses in a particular race, how many “Top Three” outcomes are there?
• When the assumption of equally likely outcomes is used to assign probability values, the method used to assign probabilities is referred to as the
• A method of assigning probabilities that assumes the experimental outcomes are equally likely is called
• When the results of historical data or experimentation are used to assign probability values, the method used to assign probabilities is referred to as the
• The probability assigned to each experimental outcome must be
• An experiment consists of four outcomes with $P(A) = 0.2, P(B) = 0.3, P(C) = 0.4$. The probability of the outcome $P(D)$ is
• Given that event $A$ has a probability of 0.25, the probability of the complement of event $A$
• The symbol $\cup$ shows the
• The union of events $A$ and $B$ is the event containing
• The probability of the union of two events with non-zero probabilities
• The symbol $\cap$ shows the
• The addition law helps to compute the probabilities of
• If $P(A) = 0.38, P(B) = 0.83$, and $P(A\cap B)=0.57$, then $P(A\cup B) =$ ?
• If $P(A) = 0.62, P(B) = 0.47$, and $P(A\cup B) = 0.88$, then $P(A \cap B) =$ ?
• Two events are mutually exclusive if
• Events that have no sample points in common are called
• The probability of the intersection of two mutually exclusive events
• If two events are mutually exclusive, then the probability of their intersection
• Two events, $A$ and $B$ are mutually exclusive and each has a non-zero probability. If event $A$ is known to occur, the probability of the occurrence of event $B$ is
• If $A$ and $B$ are mutually exclusive events with $P(A)=0.3$ and $P(B)=0.5$, then $P(A \cap B)=$?
Online MCQs Quiz Website: gmstat.com
Leave a Comment | {"url":"https://itfeature.com/probability/probability-mcqs-quiz-4/","timestamp":"2024-11-02T20:52:47Z","content_type":"text/html","content_length":"312428","record_id":"<urn:uuid:9d85fe8b-d2b3-45a0-b870-b9eea84abcfd>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00745.warc.gz"} |
Time sinks
are curious things. Some are tedious, some are frustrating, and some turn out to be fun. A few months ago, when I probably should have been studying for my comprehensive exams, a simple conversation
in the office with
+Ryan Grover
started a genealogical journey (academically speaking) to trace back our origins and those of Realistic Mathematics Education (RME). I'm pretty good with my mathematics education history, and I knew
RME in the United States took hold with the
Mathematics in Context
curriculum project, bringing
Thomas Romberg
and others at the
University of Wisconsin
together with
Jan de Lange
and others at the
Freudenthal Institute at the University of Utrecht in the Netherlands
. I suggested to Ryan that we search the
Mathematics Genealogy Project
to see if there were other ties between U.S. math ed and the history of RME, and the time sucking began in earnest.
With Ryan searching at his desk and me at the chalkboard, after several (or 4? 5?) hours we had traced back our academic roots many generations. Here's a glimpse of that work after some
un-criss-crossing of arrows by Ryan, but probably still with a few mistakes:
Ryan and I left the office after dark. When I got home, my thoughts were still consumed about organizing and preserving this history, so I stayed up most of the night creating this in Google Drawings
(part of Drive/Docs), where it would be easier to edit and be more shareable:
I've highlighted a few individuals who I think stand out. At the top left is
Nicolaus Copernicus
. We could have traced back a few more generations, but beginning with the person who is famous for not putting the Earth at the center of the universe seemed like a good place to start. The next one
down, in yellow, is
Jakob Thomasius
. Although more of a philosopher, he was advised by Friedrich Leibniz and advised his more famous son, Gottfried Leibniz, and represents an early connection between the left and right sides of the
diagram. The next person down, again in yellow, is
Abraham Kästner
. Ryan and I had never heard of him, but he's an extraordinarily connected fellow in this chart. Five of Kästner's 10 documented students are represented here, and an unseen one, to
Johann Bartels
, leads directly to
Nikolai Lobachevsky
. Furthermore,
Kästner's bio on Wikipedia
reads like some stereotypically tragic mathematician's drama, having been engaged to a woman for 12 years, only to marry her and see her die within the year. So then he had a daughter with his maid
and spent his later years writing poetry.
If you look around, you'll find Kant, Euler, Gauss, and others, but prominently representing an early attention to mathematics education is
Felix Klein
, again in yellow. Klein became interested in the teaching of mathematics around 1900, and the
International Commission on Mathematical Instruction (ICMI) has named their lifetime achievement award after Klein
. From Klein we establish three major lines: the U.S. line through
William Edward Story
, a separate U.S. line to
Maxime Bôcher
, and a German line through
. Curiously, the tree artwork on the main page of the Mathematics Genealogy Project shows the link from Klein to Story, although neither Klein's or Story's page establish a recognized advisor-advisee
relationship. After checking a few other sources, it seems that making the Klein-Story connection is typical.
Now we have three major figures in the third row from the bottom. All were trained as mathematicians but transformed themselves into prominent figures and researchers in mathematics education. On the
left is
Ed Begle
, colored in red to reflect his association with Stanford. Begle was the director of the
School Mathematics Study Group
, creators of what most call the "
New Math
" of the 1960s and 1970s. I don't want to overgeneralize, but Begle and his descendents tend to focus on the curriculum, instruction, and policy aspects of mathematics education.
Next is
Henry Van Engen
, colored in purple to signify his association with Iowa State Teachers College, now known as the
University of Northern Iowa
(my alma mater). Van Engen's publications going back to the 1940s reveal that he was focused on
in mathematics. Instead of relying wholly on his mathematics background, he incorporated ideas from figures like
. Van Engen left ISTC in the late 1950s to help establish the math ed program at the University of Wisconsin - Madison, and his lineage of Leslie Steffe and Paul Cobb represent one of the strongest
learning science traditions in math education.
On the right is Hans Freudenthal, shown in orange to signify his place in the Netherlands. A giant figure internationally,
ICMI's other major international mathematics education award is named for Freudenthal
in recognition of a major cumulative program of research. Whereas I associate Begle with curriculum, and Van Engen with learning science, to me Freudenthal represents a math education visionary and
philosopher, someone able to reflect broadly on the field and history of mathematics and structure a new approach to mathematics education. Interestingly, Freudenthal's involvement in mathematics
education was in part inspired by Begle and the New Math -- not liking what he saw in the New Math and fearing Europe would adopt a similar approach, Freudenthal steered the Netherlands in the
direction we now call RME.
There are a few hidden connections at the bottom of the diagram that reflect my experience studying RME. Being at the Freudenthal Institute US and working with David Webb is the most prominent, but I
greatly anticipate opportunities to learn from our FI colleagues from the Netherlands.
Paul Cobb
's collaboration with
Koeno Gravemeijer
in the early 2000s was mutually beneficial and has influenced me greatly, as Cobb's theories of learning mathematics work well in the context of RME.
Copernicus, more than 20 generations away, tends to be less influential.
Notes: Along the way I explored a number of other connections less related to RME and my perspective of it. William Brownell, for example, can be traced back through a number of psychologists to
Kastner. Alan Shoenfeld, another mathematician-turned-math educator, can also be traced back to Kastner and has two different lines back to Gauss. Kastner seemed to turn up everywhere, while Issac
Newton turned up nowhere. | {"url":"https://blog.mathed.net/2013/02/","timestamp":"2024-11-03T16:20:23Z","content_type":"application/xhtml+xml","content_length":"99306","record_id":"<urn:uuid:0213fa63-b5a6-4a76-afe8-f58d5a25711a>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00211.warc.gz"} |
Digamma Function Calculator
Digamma Function Calculator is an online probability and statistics tool for data analysis programmed to calculate the logarithmic derivative of the gamma function. Digamma function is the first of
the poly-gamma function and represented by the symbol Ψ. The digamma function can be defined as the logarithmic derivative of the gamma function and calculated from the below formula.
where Γ is the gamma function and Γ' is the derivative of the gamma function. This function is undefined for zero and negative integers. Full precision may not be obtained if x is too near a negative
integer. When it comes to Digamma value calculation, this online Digamma Calculator is an essential tool to make your calculations easy. | {"url":"https://ncalculators.com/statistics/digamma-function-calculator.htm","timestamp":"2024-11-13T15:47:02Z","content_type":"text/html","content_length":"51155","record_id":"<urn:uuid:711a0464-211c-40b9-ba7f-f94f5f573bde>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00280.warc.gz"} |
Antoine, P., L’Huillier, A. and Lewenstein, M. (1996) Attosecond Pulse Trains Using High-Order Harmonics. Physical Review Letters, 77, 1234. - References - Scientific Research Publishing
TITLE: The Generalized Pythagorean Comma Harmonic Powers of a Fundamental Frequency Are Equivalent the Standing Wave Harmonic Fraction System
AUTHORS: Donald Chakeres
KEYWORDS: Power Laws, Harmonic Systems, Standing Wave, Harmonic Fractions, Dimensional Analysis, Buckingham Pi Theorem, Pythagorean Comma
JOURNAL NAME: Advances in Pure Mathematics, Vol.8 No.7, July 19, 2018
ABSTRACT: Purpose: The Pythagorean Comma refers to an ancient Greek musical, mathematical tuning method that defines an integer ratio of exponential coupling constant harmonic law of two frequencies
and a virtual frequency. A Comma represents a physical harmonic system that is readily observable and can be mathematically simulated. The virtual harmonic is essential and indirectly measurable. The
Pythagorean Comma relates to two discrete frequencies but can be generalized to any including infinite harmonics of a fundamental frequency, vF. These power laws encode the physical and mathematical
properties of their coupling constant ratio, natural resonance, the maximal resonance of the powers of the frequencies, wave interference, and the beat. The hypothesis is that the Pythagorean power
fractions of a fundamental frequency, vF are structured by the same harmonic fraction system seen with standing waves. Methods: The Pythagorean Comma refers to the ratio of (3/2)12 and 27 that is
nearly equal to 1. A Comma is related to the physical setting of the maximum resonance of the powers of two frequencies. The powers and the virtual frequency are derived simulating the physical
environment utilizing the Buckingham Π theorem, array analysis, and dimensional analysis. The powers and the virtual frequency can be generalized to any two frequencies. The maximum resonance occurs
when their dimensionless ratio closest to 1 and the virtual harmonic closest to 1 Hz. The Pythagorean possible power arrays for a vF system or any two different frequencies are evaluated. Results:
The generalized Pythagorean harmonic power law for any two different frequencies coupling constant are derived with a form of an infinite number of powers defining a constant power ratio and a single
virtual harmonic frequency. This power system has periodic and fractal properties. The Pythagorean power law also encodes the ratio of logs of the frequencies. These must equal or nearly equal the
power ratio. When all of the harmonics are powers of a vF the Pythagorean powers are defined by a consecutive integer series structured in the identical form as standard harmonic fractions. The ratio
of the powers is rational, and all of the virtual harmonics are 1 Hz. Conclusion: The Pythagorean Comma power law method can be generalized. This is a new isomorphic wave perspective that encompasses
all harmonic systems, but with an infinite number of possible powers. It is important since there is new information: powers, power ratio, and a virtual frequency. The Pythagorean relationships are
different, yet an isomorphic perspective where the powers demonstrate harmonic patterns. The coupling constants of a vF Pythagorean power law system are related to the vFs raised to the harmonic
fraction series which accounts for the parallel organization to the standing wave system. This new perspective accurately defines an alternate valid physical harmonic system. | {"url":"https://scirp.org/reference/referencespapers?referenceid=2322898","timestamp":"2024-11-14T02:10:01Z","content_type":"application/xhtml+xml","content_length":"80431","record_id":"<urn:uuid:d90f9029-9a15-45cf-86fd-2828ae3a1ccc>","cc-path":"CC-MAIN-2024-46/segments/1730477028516.72/warc/CC-MAIN-20241113235151-20241114025151-00713.warc.gz"} |
The Stacks project
Theorem 80.6.1. Let $S$ be a scheme. Let $F : (\mathit{Sch}/S)_{fppf}^{opp} \to \textit{Sets}$ be a functor. Assume that
1. the presheaf $F$ is a sheaf,
2. the diagonal morphism $F \to F \times F$ is representable by algebraic spaces, and
3. there exists an algebraic space $X$ and a map $X \to F$ which is surjective, and étale.
Comments (2)
Comment #8847 by Simon Vortkamp on
Typo in Theorem 03Y3 assumption (b): It should be 'representable by algebraic spaces' instead of 'representable by algebraic paces'.
Comment #9239 by Stacks project on
Thanks and fixed here.
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Introduction to Coding
Introduction to Coding Theory (89-662)
Yehuda Lindell
Coding theory deals with the problem of communication over a noisy channel, where some of the bits of the message may be corrupted en route. Beyond its immediate application to ``correct
communication'', coding theory has become an important tool in the theory of computer science, and in particular in computational complexity. In this course, we will study the basics of coding
theory. We will study both lower and upper bounds, and will also see well-known families of codes. Our view is that of computer science and we will therefore be interested in the efficiency of
encoding and decoding procedures. The course is intended for 3rd year undergraduate students, as well as for graduate students.
There is no single textbook for this course, although we have used the books Coding Theory - A First Course, by San Ling and Chaoping Xing (Cambridge University Press, 2004), and an Introduction to
Coding Theory (Cambridge University Press 2006) by Ron Roth. We note that although most of the technical material can be found in these texts, our presentation will often be quite different. Full
lecture notes have been written and it is recommended to refer to these.
The requirements of the course are homework exercises, a final exam, and possibly a midterm exam as well. This year we will be experimenting with cooperative group learning. As such attendance is
compulsory and participating may also be taken into account for the final grade. We will discuss this more in the first class.
Lecture Notes
Full lecture notes for the course can be found in this PDF file. Please send me any comments that you have, or let me know if you find errors.
Course Syllabus
1. Introduction: the coding problem and definitions.
2. Linear Codes: definition, hamming weight, bases, generator and parity-check matrices, encoding and decoding procedures.
3. Bounds:
○ The main coding theory problem
○ The sphere covering bound and the sphere packing (Hamming) bound
○ Perfect codes: Hamming and Golay
○ The Singleton bound and MDS codes; the Reed-Solomon code
○ Other bounds: Gilbert-Varshamov and Plotkin; Hadamard codes
○ Asymptotic bounds
○ Shannon’s theorem and its converse
4. Constructions of some specific codes:
○ Propagation rules
○ Reed-Muller codes
○ Generalized Reed-Solomon (GRS) codes and decoding algorithms
5. Asymptotically good codes: definition; concatenation of codes; construction via Reed-Solomon and Gilbert-Varshamov; efficient decoding of concatenated codes (Forney’s algorithm)
6. Local decodability: definition; properties; local decodability of Hadamard codes
7. List decoding: definition; Sudan’s algorithm for list-decoding of GRS codes
8. Hard problems in coding theory: the nearest codeword problem; NP-completeness; hardness of approximation
9. CIRC: coding and decoding for the CD-ROM
Past Exams
1. Moed aleph, 2007/8
2. Moed bet, 2007/8
3. Moed aleph, 2008/9
Reading Material
Main course texts:
● S. Ling and C. Xing. Coding Theory - A First Course. Cambridge University Press, 2004.
● R. Roth. Introduction to Coding Theory. Cambridge University Press, 2006.
Other relevant texts:
● W.C. Huffman and V. Pless. Fundamentals of Error-Correcting Codes. Cambridge University Press, 2003.
● S. Roman. Introduction to Coding and Information Theory. Springer-Verlag, 1997. | {"url":"https://u.cs.biu.ac.il/~lindell/89-662/main-89-662.html","timestamp":"2024-11-12T19:18:49Z","content_type":"text/html","content_length":"47972","record_id":"<urn:uuid:7b81e62a-7aaf-4063-b0ab-64886286a6e6>","cc-path":"CC-MAIN-2024-46/segments/1730477028279.73/warc/CC-MAIN-20241112180608-20241112210608-00582.warc.gz"} |
An Etymological Dictionary of Astronomy and Astrophysics
Fr.: général
(Adj.) 1) Not limited to one class, field, product, service, etc. 2) Relating to the whole or to the all or most. 3) Dealing with overall characteristics, universal aspects, or important elements.
See also:
→ general precession, → general relativity, → generalization, → generalize, → generalized, → generalized coordinates, → generalized forces, → generalized momenta, → generalized velocities, → New
General Catalogue (NGC).
From L. generalis "relating to all, of a whole class," from genus "race, stock, kind," akin to Pers. zâdan, Av. zan- "to bear, give birth to a child, be born," infinitive zazāite, zāta- "born;"
Mod.Pers. zâdan, present stem zā- "to bring forth, give birth" (Mid.Pers. zâtan; cf. Skt. jan- "to produce, create; to be born," janati "begets, bears;" Gk. gignomai "to happen, become, be born;" L.
gignere "to beget;" PIE base *gen- "to give birth, beget."
Harvin, from Mid.Pers. harvin "all," from har(v) "all, each, every" (Mod.Pers. har "every, all, each, any"); O.Pers. haruva- "whole, all together;" Av. hauruua- "whole, at all, undamaged;" cf. Skt.
sárva- "whole, all, every, undivided;" Gk. holos "whole, complete;" L. salvus "whole, safe, healthy," sollus "whole, entire, unbroken;" PIE base *sol- "whole."
general precession
پیشایان ِهروین
pišâyân-e harvin
Fr.: précession générale
The secular motions of the → celestial equator and → ecliptic. In other words, the sum of → lunisolar precession, → planetary precession, and → geodesic precession.
→ general; → precession
general precession in longitude
پیشایان ِهروینِ درژنا
pišâyân-e harvin-e derežnâ
Fr.: précession générale en longitude
The secular displacement of the → equinox on the → ecliptic of date.
→ general; → precession; → longitude.
general precession in right ascension
پیشایان ِهروین ِراستافراز
pišâyân-e harvin-e râst afrâz
Fr.: précession générale en ascension droite
The secular motion of the → equinox along the → celestial equator.
→ general; → precession; → right ascension.
general relativistic
بازانیگیمند ِهروین
bâzânigimand-e harvin
Fr.: de relativité générale
Of, relating to, or subject to the theory of → general relativity.
→ general; → relativistic.
general relativity
بازانیگی ِهروین
bâzânigi-ye harvin
Fr.: relativité générale
The theory of → gravitation developed by Albert Einstein (1916) that describes the gravitation as the → space-time curvature caused by the presence of matter or energy. Mass creates a → gravitational
field which distorts the space and changes the flow of time. In other words, mass causes a deviation of the → metric of space-time continuum from that of the "flat" space-time structure described by
the → Euclidean geometry and treated in → special relativity. General relativity developed from the → principle of equivalence between gravitational and inertial forces. According to general
relativity, photons follow a curved path in a gravitational field. This prediction was confirmed by the measurements of star positions near the solar limb during the total eclipse of 1919. The same
effect is seen in the delay of radio signals coming from distant space probes when grazing the Sun's surface. Moreover, the space curvature caused by the Sun makes the → perihelion of Mercury's orbit
advance by 43'' per century more than that predicted by Newton's theory of gravitation. The → perihelion advance can reach several degrees per year for → binary pulsar orbits. Another effect
predicted by general relativity is the → gravitational reddening. This effect is verified in the → redshift of spectral lines in the solar spectrum and, even more obviously, in → white dwarfs. Other
predictions of the theory include → gravitational lensing, → gravitational waves, and the invariance of Newton's → gravitational constant.
→ general; → relativity.
general secretary
هروین دبیر
harvin dabir
Fr.: secrétaire général
→ secretary-general.
→ general; → secretary.
هروینکرد، هروینش
harvinkard, harvineš
Fr.: généralisation
The act or process of generalizing; → generalize.
A result of this process; a general statement, proposition, or principle.
Verbal noun of → generalize.
هروین کردن، هروینیدن
harvin kardan, harvinidan
Fr.: généraliser
To make general, to include under a general term; to reduce to a general form.
To infer or form a general principle, opinion, conclusion, etc. from only a few facts, examples, or the like.
→ general; → -ize.
Fr.: généralisé
Made general. → generalized coordinates; → generalized velocities.
P.p. of → generalize
generalized coordinates
هماراهای ِهروینیده
hamârâhâ-ye harvinidé
Fr.: coordonnées généralisées
In a material system, the independent parameters which completely specify the configuration of the system, i.e. the position of its particles with respect to the frame of reference. Usually each
coordinate is designated by the letter q with a numerical subscript. A set of generalized coordinates would be written as q[1], q[2], ..., q[n]. Thus a particle moving in a plane may be described by
two coordinates q[1], q[2], which may in special cases be the → Cartesian coordinates x, y, or the → polar coordinates r, θ, or any other suitable pair of coordinates. A particle moving in a space is
located by three coordinates, which may be Cartesian coordinates x, y, z, or → spherical coordinates r, θ, φ, or in general q[1], q[2], q[3]. The generalized coordinates are normally a "minimal set"
of coordinates. For example, in Cartesian coordinates the simple pendulum requires two coordinates (x and y), but in polar coordinates only one coordinate (θ) is required. So θ is the appropriate
generalized coordinate for the pendulum problem.
→ generalized; → coordinate.
generalized forces
نیروهای ِهروینیده
niruhâ-ye harvinidé
Fr.: forces généralisées
In → Lagrangian dynamics, forces related to → generalized coordinates. For any system with n generalized coordinates q[i] (i = 1, ..., n), generalized forces are expressed by F[i] = ∂L/∂q[i], where L
is the → Lagrangian function.
→ generalized; → force.
generalized momenta
جنباکهای ِهروینیده
jonbâkhâ-ye harvinidé
Fr.: quantité de mouvement généralisée
In → Lagrangian dynamics, momenta related to → generalized coordinates. For any system with n generalized coordinates q[i] (i = 1, ..., n), generalized momenta are expressed by p[i] = ∂L/∂q^.[i],
where L is the → Lagrangian function.
→ generalized; → momentum.
generalized velocities
تنداهای ِهروینیده
tondâhâ-ye harvinidé
Fr.: vitesses généralisées
The time → derivatives of the → generalized coordinates of a system.
→ generalized; → velocity.
Fr.: générer
To bring into existence; create; produce.
Math.: To trace (a figure) by the motion of a point, straight line, or curve.
Generate, from M.E., from L. generatus "produce," p.p. of generare "to bring forth," from gener-, genus "descent, birth," akin to Pers. zâdan, Av. zan- "to give birth," as explained below.
Âzânidan, from â- nuance/strengthening prefix + zân, from Av. zan- "to bear, give birth to a child, be born," infinitive zazāite, zāta- "born;" Mod.Pers. zâdan, present stem zā- "to bring forth, give
birth" (Mid.Pers. zâtan; cf. Skt. jan- "to produce, create; to be born," janati "begets, bears;" Gk. gignomai "to happen, become, be born;" L. gignere "to beget;" PIE base *gen- "to give birth,
beget") + -idan infinitive suffix.
Fr.: génération
1) A coming into being.
2) The → production of → energy (→ heat or → electricity).
Verbal noun of → generate.
آزاننده، آزانشی
âzânandé, âzâneši
Fr.: génératif
1) Capable of producing or creating.
2) Pertaining to the production of offspring.
→ generate; → -ive.
Fr.: générateur
1) A machine for converting one form of energy into another.
2) Geometry: That which creates a line, a surface, a solid by its motion.
From L. generator "producer," from genera(re)→ generate + -tor a suffix forming personal agent nouns from verbs and, less commonly, from nouns.
Âzângar, from âzân the stem of âzânidan→ generate + -gar suffix of agent nouns, from kar-, kardan "to do, to make" (Mid.Pers. kardan; O.Pers./Av. kar- "to do, make, build," Av. kərənaoiti "makes;"
cf. Skt. kr- "to do, to make," krnoti "makes," karma "act, deed;" PIE base k^wer- "to do, to make").
ژنتیک، ژنتیکی
ženetik (#), ženetiki (#)
Fr.: génétique
Pertaining or according to → genetics or → genes.
From Gk. genetikos, from genesis "origin," → gene; → -ic.
ženetik (#)
Fr.: génétique
The study of heredity and inheritance, of the transmission of traits from one individual to another, of how genes are transmitted from generation to generation.
From → genetic and → -ics. | {"url":"https://dictionary.obspm.fr/index.php?showAll=1&&search=G&&formSearchTextfield=&&page=7","timestamp":"2024-11-09T03:28:57Z","content_type":"text/html","content_length":"38207","record_id":"<urn:uuid:aa9ee4d4-8c01-4782-a9b3-595ca2d1035a>","cc-path":"CC-MAIN-2024-46/segments/1730477028115.85/warc/CC-MAIN-20241109022607-20241109052607-00038.warc.gz"} |
(PDF) Multivariable Calculus - Soo T. Tan - 1st Edition
Multivariable Calculus – Soo T. Tan – 1st Edition
Multivariable Calculus
Soo T. Tan
• 0534465757
• 9780534465759
• 1st Edition
• eBook
• English
The Multivariable portion of the Soo Tan Calculus textbook tackles complex concepts with a strong visual approach. Utilizing a clear, concise writing style, and use of relevant, real world examples,
Soo Tan introduces abstract mathematical concepts with his intuitive style that brings abstract multivariable concepts to life. The Multivariable text provides a great deal of visual help by
introducing unique videos that assist students in drawing complex calculus artwork by hand.
In keeping with this emphasis on conceptual understanding, each exercise set begins with concept questions and each end-of-chapter review section includes fill-in-the-blank questions which are useful
for mastering the definitions and theorems in each chapter. Additionally, many questions asking for the interpretation of graphical, numerical, and algebraic results are included among both the
examples and the exercise sets.
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• 10. Conic Sections, Parametric Equations, And Polar Coordinates
11. Vectors And The Geometry Of Space
12. Vector-Valued Functions
13. Functions Of Several Variables
14. Mulitple Integrals
15. Vector Analysis
• Citation
□ Multivariable Calculus
□ 0534465757
□ 9780534465759
□ 1st Edition
□ 2009
□ eBook
□ English | {"url":"https://www.tbooks.solutions/multivariable-calculus-soo-t-tan-1st-edition/","timestamp":"2024-11-09T14:39:18Z","content_type":"text/html","content_length":"123115","record_id":"<urn:uuid:42957bf4-65fb-446f-bca8-622db9789e7e>","cc-path":"CC-MAIN-2024-46/segments/1730477028118.93/warc/CC-MAIN-20241109120425-20241109150425-00628.warc.gz"} |
Multiple Multiple Criteria
This article revisits one of the most queried areas of modelling: how to sum data based upon multiple criteria. But this time around, it’s not quite as simple as that… By Liam Bastick, director with
SumProduct Pty Ltd.
I have looked through your previous article on working with Dealing with Multiple Criteria, but it doesn’t seem to answer my question. How can I sum amounts using a formula based upon multiple
criteria for data situated in multiple rows within multiple worksheets?
This has been quite a popular topic and it is clear that the answer to this particular question isn’t that easy to come by. I will explain my solution using an illustration from the attached Excel
Let us imagine we run a car sales company with four divisions: North, South, East and West. Sales are reported in a similar fashion for each (North and South divisions are shown below):
The month, associated sales person, car colour and cash amount of each sale is recorded. Note that the reports of each division may not be of equal length, but importantly the column headings of each
table are in the same column of each spreadsheet (e.g. the month is always recorded in column F, the salesperson is always in column G, etc.). This is necessary for my solution to work.
Let’s build up the problem slowly.
Single Criterion, Single Data Source
I appreciate many readers will find this trivial, but for completeness I shall start here.
This is a simple use of the SUMIF function. For example, the formula in cell G12 in the illustration above is:
where North is the Excel worksheet containing North division’s sales data. For those unfamiliar with this useful function, SUMIF was discussed in Dealing with Multiple Criteria, the article mentioned
previously (above). Alternative functions can be used but I am sticking with SUMIF as I will be using this and its sister function (SUMIFS) throughout this article. Essentially, all sales are added
when the month of the sale matches the reporting month.
Multiple Criteria, Single Data Source
The next logical step is to increase the number of criteria (but still focus on just the one data source):
Again, there are many ways to solve this conundrum, but the one I have chosen has the following formula in cell I12:
Here, I have used the SUMIFS function (again, see Dealing with Multiple Criteria for more information), which deals with multiple criteria, here only summing data where the month, salesperson and car
colour match the required criteria. Some may be amused to see I do not recommend our company’s namesake, SUMPRODUCT, here, but this function would not work on an entire column prior to its revision
in Excel 2007.
Now, it gets more fun…
Single Criterion, Multiple Data Sources
Rather than make the jump to multiple criteria and multiple data sources all in one go, I thought it would be better to introduce one complication at a time:
Before explaining the formula in G12 here, I would like to recommend an interim step. If I am to refer to multiple datasheets, I need to know the names of these worksheets. I suggest storing the
worksheet names in a Table:
In the attached Excel file, this Table has been named Division_Table as this lists the divisions relevant for the analysis. It is intentional that I have not included all four divisions, but it is
also important to note that the three divisions named (North, South and East here) must have identical names to the sheet tab names – otherwise, this solution will not work.
Tables represent a useful functionality, first introduced into Excel 2007 (for more information, see the previous article on Tables). Essentially, by listing data this way it allows users to add to
the list (by putting, say, ‘West’ on the next line in my illustration) such that any referencing formulae will update the referenced list automatically.
Returning to this third example, namely multiple criteria based upon a single data source, the formula in cell G12 of my illustration is a little more sophisticated than the first two solutions:
=IFERROR(SUMPRODUCT(SUMIF(INDIRECT(“‘”&Division_Table[Relevant Divisions]&”‘!F:F”),$F12,INDIRECT(“‘”&Division_Table[Relevant Divisions]&”‘!I:I”))),)
You know you have created a monster when you nest three Excel functions inside a fourth. To work out what is going on, I will explain from the inside out (as this is how Excel will calculate this
• INDIRECT (see Being Direct About INDIRECT) – This function produces an array of references such as ‘North’ column F, ‘South’ column F, etc. which can be used by the other functions. Note
carefully that “‘” in the formula is inverted commas followed by an apostrophe (the syntax required in general for sheet names) followed by inverted commas.
• SUMIF – This function now applies the single criterion to the summation analysis. However, since it is not an array function, this will only report on one worksheet at a time (in my example,
there are three sheets: ‘North’, ‘South’ and ‘East’).
• SUMPRODUCT – This function is necessary as this function is often referred to as a “pseudo array function” (arrays and array functions were discussed previously in Array of Light). What this
means in practice here is that it will allow the SUMIF function to be performed across all three worksheets. SUMPRODUCT cannot be used without SUMIF however, as this function does not appear to
work with multiple rows of data on multiple sheets (it only seems to consider the first cell of each selection on each sheet).
• IFERROR – This error trap ensures that if a worksheet listed in the Division_Table does not exist and / or there is a blank row, the formula will not produce a #REF! error for example.
Simple, n’est-ce pas..?
Multiple Criteria, Multiple Data Sources
So we now get to la pièce de résistance. If the reader has overcome the last hurdle this is a pièce de cake:
The formula in cell I12 here is just an extension of the last example:
=IFERROR(SUMPRODUCT(SUMIFS(INDIRECT(“‘”&Division_Table[Relevant Divisions]&”‘!I:I”),INDIRECT(“‘”&Division_Table[Relevant Divisions]&”‘!F:F”),$F12,INDIRECT(“‘”&Division_Table[Relevant Divisions]&”‘!
G:G”),$G12,INDIRECT(“‘”&Division_Table[Relevant Divisions]&”‘!H:H”),$H12)),)
Whilst the formula may look even more horrible upon first glance, essentially the SUMIFS function has merely replaced the SUMIF function, similar to the difference between the first two examples
discussed above.
It’s not pretty, but it’s effective.
Word to the Wise
Given this solution uses Tables, IFERROR and the SUMIFS functions, this will only work in Excel 2007 and later versions of Excel.
Also, there are two other possible solutions to consider: PivotTables (see >Pivotal PivotTables) using data from multiple worksheets or creating a master data sheet as an interim step, where all data
is recorded on one worksheet. I have produced this answer as this was faithful to the specific circumstances of the problem.
However, I would like to stress the age old rule: Keep It Simple Stupid (KISS). Having data on multiple worksheets complicates the problem and may not be a necessary complexity. Before writing opaque
formulae such as the ones discussed above, always consider simplifying the model (structure) first. | {"url":"https://www.sumproduct.com/thought/multiple-multiple-criteria","timestamp":"2024-11-09T17:05:11Z","content_type":"text/html","content_length":"24499","record_id":"<urn:uuid:c72f4c08-9f0a-44e6-b3d2-c60608bcbb94>","cc-path":"CC-MAIN-2024-46/segments/1730477028125.59/warc/CC-MAIN-20241109151915-20241109181915-00278.warc.gz"} |
JL Card Rules OldJL Card Rules Old - Jacklabrador
There are three potential outcomes when playing JACK LABRADOR with three or four players. They are:
Single Winner
One player throws a symbol that beats all other thrown symbols.
All Way Tie
All players throw the same symbol OR
all thrown symbols both win and lose to other thrown symbols.
Split Decision
Two or more players tie and beat all other thrown symbols.
Single Winner
There is a Single Winner when one player throws a symbol that no other player has thrown, and that symbol beats all other symbols thrown by the other players.
Three Player Single Winner Examples
All Way Tie
There are three scenarios that can generate an All Way Tie.
1. When all players throw the same symbol.
2. When a JACK, a LABRADOR, and at least one of a Rock or Paper or Scissors is thrown. This is because each symbol both beats and loses to at least one other symbol. Recognize this very common
outcome to speed the game up.
3. When at least one Rock and one Paper and one Scissors are thrown at the same time without any JACKS or LABRADORS. Again, this is because each symbol both beats and loses to at least one other
Split Decision
A Split Decision happens when at least one player loses to two or more players who are tied. Losing players are eliminated and winners play on. the tied players play on. | {"url":"https://jacklabrador.com/jl-card-rules-old/","timestamp":"2024-11-08T20:34:46Z","content_type":"text/html","content_length":"1049155","record_id":"<urn:uuid:b2d4b90c-8969-4eec-ba60-fa5ef33ebbea>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00358.warc.gz"} |
Free Fall Experiment 2 - Javalab
Free Fall Experiment 2
This simulation allows you to measure speed as a function of fall height.
You can measure the speed at two locations and change the ruler’s orientation.
Free fall movement
All objects on Earth are affected by gravity. If there were no friction with air, the speed of any object would increase by 9.8 m/s every second.
If the falling time of a free-falling object is \(t (s, seconds)\), the speed is \(v (m/s)\), and the distance traveled is \(s (m)\), the following equation is established:
\[ v_{t} = 9.8t \]
\[ s_{t} = \frac{1}{2} 9.8 t^{2} \]
The work done by gravity and the kinetic energy
A free-falling object is pulled by gravity and moves in the direction of gravity. In other words, the work done by gravity is converted into the kinetic energy of the object.
Therefore, we can see that the following equation holds true.
\[ 9.8mh = \frac{1}{2}mv^{2} \]
Mass \(m\) can be omitted in the above equation.
\[ 9.8h = \frac{1}{2}v^{2} \]
Therefore, regardless of the object’s mass, the speed of a falling object is only related to the distance it falls. | {"url":"https://mully.net/en/free_falling_2_en/","timestamp":"2024-11-11T00:30:32Z","content_type":"text/html","content_length":"75539","record_id":"<urn:uuid:355de60e-3664-42a6-9ffa-6c8d44510442>","cc-path":"CC-MAIN-2024-46/segments/1730477028202.29/warc/CC-MAIN-20241110233206-20241111023206-00805.warc.gz"} |
How to use ewm() in Pandas | Coding Ref
How to use ewm() in Pandas
The ewm() function in Pandas is used to apply exponential weighting to a series or dataframe.
This is often used to compute an exponentially weighted moving average, which gives more weight to more recent values and less weight to older values.
Here's an example of using the ewm() function in Pandas:
import pandas as pd
# create a sample series
s = pd.Series([1, 2, 3, 4, 5])
# compute the exponentially weighted moving average of the series
s_ewm = s.ewm(alpha=0.5).mean()
# display the result
This will compute the exponentially weighted moving average of the series and return a new series with the same index as the original series.
The alpha parameter specifies the decay factor, which determines the weight of each value in the moving average.
The output will be:
0 1.000000
1 1.666667
2 2.428571
3 3.266667
4 4.161290
dtype: float64
Using ewm() on one or more columns
You can also use the ewm() function on a dataframe to compute the exponentially weighted moving average of one or more columns.
For example:
# create a sample dataframe
df = pd.DataFrame({"A": [1, 2, 3, 4, 5],
"B": [2, 3, 4, 5, 6]})
# compute the exponentially weighted moving average of the A and B columns
df_ewm = df[["A", "B"]].ewm(alpha=0.5).mean()
# display the result
This will compute the exponentially weighted moving average of the A and B columns in the dataframe and return a new dataframe with the same index as the original dataframe.
The output will be:
A B
0 1.000000 2.000000
1 1.666667 2.666667
2 2.428571 3.428571
3 3.266667 4.266667
4 4.161290 5.161290
The ewm() function is useful for computing the exponentially weighted moving average of a series or dataframe, which can be useful for a variety of analysis and visualization tasks. | {"url":"https://www.codingref.com/article/python-pandas-ewm","timestamp":"2024-11-12T13:20:51Z","content_type":"text/html","content_length":"21590","record_id":"<urn:uuid:ecf9ba3d-a4cf-4ff8-acc7-b382e2f7d8b7>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.45/warc/CC-MAIN-20241112113320-20241112143320-00029.warc.gz"} |
More on $β$-symmetry
More on β-symmetry
Recent work has proposed a method for imposing T-duality on the metric, B-field, and dilaton of the classical effective action of string theory without using Kaluza-Klein reduction. Specifically, the
D-dimensional effective action should be invariant under global O(D,D) transformations, provided that the partial derivatives along the \beta-parameters of the non-geometrical... Show more | {"url":"https://synthical.com/article/More-on-%24%CE%B2%24-symmetry-33a9f93b-cf4e-4fcf-96cf-b6532b58a2e6?","timestamp":"2024-11-11T03:34:59Z","content_type":"text/html","content_length":"64682","record_id":"<urn:uuid:0da55c09-66c3-45c7-b337-3643e7ce116b>","cc-path":"CC-MAIN-2024-46/segments/1730477028216.19/warc/CC-MAIN-20241111024756-20241111054756-00117.warc.gz"} |
Empirically Verifying Course and Slope Ratings
Most golfers have played a course where they thought the Course Rating and/or the Slope Rating were too low. These ratings errors would explain why their scoring differential was so much higher than
their index. The golf association responsible for the ratings, however, would argue the ratings are accurate and the player’s poor performance is more properly explained by the random nature of
Course and Slope Ratings are determined through the use of estimating models. Estimating models in both the hard sciences and the social sciences are validated by how well they predict. There are
models, for example, that attempt to predict the winning team in games of the National Football League. If the predictions of a model are better than chance over a long period of time, the model is
said to be empirically validated. Interestingly, the United state Golf Association (USGA) has never published any evidence validating it Ratings models.
There are probably two reasons for the USGA’s lack of diligence. First, a research finding that the models are inaccurate would tarnish the image of omnipotence the USGA likes to project. Secondly,
there are severe theoretical and logistical problems in designing an experiment to verify the ratings. These problems have been discussed in a previous post – How Accurate is the Slope System? [2] In
theory, a large group of scratch players could play a course many times and it would be possible to solve for the Course Rating that would make these players scratch at this course. It is not known
with any certainty, however, that these players are actually scratch or came about that handicap due to an error in the Course Rating at their home courses. The logistical problem is getting a
reference group (i.e., the scratch players) at least 400 (a minimum of 20 players having 20 rounds each) starting times at the test course. Because of these problems, golf associations cannot be
easily challenged on the accuracy of their Ratings.
There is one special case, however, that eliminates the logistical problem and allows for the examination of Rating errors. That case is where a golf club has two or more courses. By examining the
performance of the same golfers on different courses, it should be possible to estimate the size of any errors in the Ratings. Slope Theory assumes the average differential ((Average Score – Course
Rating) x 113/Slope Rating) at each course should be the same for a player. To test the hypothesis that a player’s differentials are equal, data were collected on 34 four players who alternate
playing two courses under competitive conditions. The courses are named Course1 and Course2 in this analysis. A player’s Average Course1 Differential is plotted against his Average Course2
Differential in Figure 1 below.
Figure 1 – Average Course1 Differential versus Average Course2 Differential
If Slope Theory is correct the regression line should have a zero intercept and a slope of 1.0. In other words the best estimate of the Average Differential on the Course 1 is the Average
Differential on the Course 2. The regression line fitted to the date shown in Figure 1 was estimated to be:
1) Avg. Course1 Differential = 2.1 + .85 · Avg. Course2 Differential
The regression does not have a zero intercept. The constant “2.1” in the regression equation, indicates a possible error in the Course Ratings.[3]
Rewriting the equation 1,
2) Avg. Course1 Differential - .85 · Avg. Course2 Differential = 2.1
To eliminate the constant, 2.1, the average differential on Course1 must be decreased or the average differential on Course2 increased or some combination of the two. A reasonable assumption is that
the error in the fixed component should be split equally between the two courses. Therefore, the Average Course1 Differential is reduced by 1.0 (i.e. Course1 is more difficult than it is rated), and
the Average Course2 Differential is increased by 1.2 (i.e., Course2 is easier than it is rated). The plot of these adjusted differentials for each player is shown in Figure 2 below.
Figure 2 – Average Course1 Differential Reduced by 1.0 versus Average Course2 Differential Increased by 1.2
The new regression equation is:
3) Average Course1 Differential = 0.03 + .85 Average Course2 Differential
The fixed component of the relationship is now close to zero. The ratio of the two differentials, however, should be 1.0 and not .85. To achieve that end, the Slope Rating of Course2 would have to
increase or the Slope Rating on Course1 decrease -- or some combination of the two. There are a large number of new Slope Ratings that would equalize the ratio of adjusted differentials.[4] Again, it
is assumed the errors in Slope Ratings are approximately the same size though in different directions. The postulated Slope Ratings under this assumption are 115 for Course1 and 132 for Course2. The
plot of the differentials adjusted for the new Slope Ratings is shown in Figure 3. The new regression line is:
4) Average Adjusted Course1 Differential = 0.2 +.98 Average Adjusted Course2 Differential
The new regression line eliminates the fixed component, makes the ratio of the two differentials approximately equal to 1.0, and therefore satisfies the requirements of Slope Theory.
Figure 3 – Plot of Differentials Adjusted for New Slope Ratings
Given these adjustments, what would be the estimates of the new Course Ratings? To reduce the Average Differential on Course1 by 1.0, the Course Rating must be increased by 1.0 x 124/113 or 1.1
strokes. The new Course Rating on Course1 would be 71.1 (from 70.0). To increase the Average Differential on Course2 by 1.2, the Course Rating would have to be decreased by 1.2 x 122/113 or 1.3
strokes. The new Course Rating on Course1 would be 68.3 (from 69.6).[5] A summary of the estimated and existing Ratings are presented in the table below.
Estimated and Existing Ratings
│Course │ Course Ratings │ Slope Ratings │
│ ├──────────┬────────┼──────────┬────────┤
│ │Estimated │Existing│Estimated │Existing│
│Course1│ 71.1 │ 70.0 │ 115 │ 124 │
│Course2│ 68.3 │ 69.6 │ 132 │ 122 │
The estimated ratings in Table 1 are purely speculative. There are a large number of Ratings that would equalize player differentials between the two courses. It does appear, however, that the
current Ratings are not correct. There are three possible explanations for the apparent errors:
1. The Ratings are in Error – As discussed in How Accurate is the Slope System?, the Course Rating models of the USGA are seriously flawed. The models are deemed so imperfect the results can be
changed by the Course Rating Review Committee at each golf association. [6]
2. Sampling Error – The small size of the sample (34 players) makes any conclusion tentative at best.
3. Slope Theory Is Not Valid - Under Slope Theory, a players’ average score at a course is a linear function of his index. All players have their index increased (or decreased) by the same percentage
(Slope Rating/113) to find their handicap. But what if a player’s average score is a curvilinear function of his index? In that case, even if the Course Rating, Bogey Rating, and Slope Rating were
estimated with precision, the average differentials at the two courses would not necessarily be equal.
There is anecdotal evidence that reason 1 is the most plausible explanation. Individual and team (2 best balls of 4) net scores are generally lower on Course2 than on Course1. The current Ratings
would not predict such a disparity in net scores. Reason 2 can only be eliminated by expanding the sample size. It would be helpful if a golf association with access to the General Handicap
Information Network (GHIN) would undertake a similar study. Since this is unlikely, the most reasonable course of action is to repeat the experiment each year and examine whether the results are
consistent across time.
[1] The USGA maintains the standard error in the Course Ratings is .285 strokes, and therefore an error of more than .6 strokes is extremely unlikely. (See Knuth, D, “A two parameter golf course
rating system,” Science and Golf, edited by A.J. Cochran and M.R. Farally, E & FN Spon, London, 1990, p. 145.)
[2] See Ongolfhandicaps.blogspot.com, October 8, 2012.
[3] The constant 2.1 is only statistically significant at the 90 percent level of confidence. The variable coefficient (.85) is statistically significant at the 95 percent level of confidence.
[4] The current Slope Ratings are 122 for Course2 and 124 for Course1. The ratio of the adjusted Course1 Differential to the Adjusted Course2 Differential is .85. To make the ratio of the adjusted
differentials equal to 1.0, the ratio of the new Course2 Slope Rating to the new Course1 Slope Rating must be approximately 1.16.
[5] The errors seem large and the USGA would argue they are not within the realm of reason. What is often forgotten is that a small error in the Course Rating can lead to what looks like a
substantial error in the Slope Rating. Let’s look at the Course2. The Slope Rating is given by the formula:
Slope Rating = 5.381 (Bogey Rating – Course Rating)
If the Course Rating is overestimated by 1.3 strokes, the Slope Rating will be underestimated by 7 rating points. That would put the Slope Rating at 129 which is not too far from the 132 used in this
[6] USGA Handicap System, 2012-2015, Unites States Golf Association, Far Hills, NJ, p. 98. | {"url":"http://www.ongolfhandicaps.com/2013/04/empirically-verifying-course-and-slope.html","timestamp":"2024-11-02T18:38:02Z","content_type":"text/html","content_length":"103384","record_id":"<urn:uuid:91a4dd81-9073-4bef-9852-3d83f7d39fbb>","cc-path":"CC-MAIN-2024-46/segments/1730477027729.26/warc/CC-MAIN-20241102165015-20241102195015-00530.warc.gz"} |
Data Structure Tree
Data Structure Tree is a hierarchical relationship between various elements. It is a set of one or more nodes and nodes are connected by an edge. Each node contains some value. The node is
represented by a circle and edge lives connecting these nodes. Tree is a non linear data structure, which is very flexible, versatile and widely used.
In the above tree, A is the root element and it has two subtree B and C. B is itself root element of D and E. And similarly C is the root element of F and G.
Tree Terminology
These are the different terminology used in the tree.
• Leaf Node - Any node with degree 0 or that has no children is called a terminal node or leaf node.
• Level of Node - The level of any node is the length of its path from the root.
• Root Node - The topmost node of the tree.
• Sub Trees - The branches of the root node are called sub trees.
• Path - The sequence of the edges is called a path.
• Degree - The degree of the node is equal to the number of the children that a node has.
Tree Types
These are the following six tree types.
1. General trees
2. Forests
3. Binary trees
4. Binary search trees
5. Expression trees
6. AVL Tree
7. B Tree | {"url":"https://www.etutorialspoint.com/index.php/data-structure/tree-introduction","timestamp":"2024-11-12T00:43:40Z","content_type":"text/html","content_length":"22282","record_id":"<urn:uuid:3e963bc1-acf6-4bfb-b4df-fbde81427a16>","cc-path":"CC-MAIN-2024-46/segments/1730477028240.82/warc/CC-MAIN-20241111222353-20241112012353-00744.warc.gz"} |
Transactions Online
Kazuyuki HIRAOKA, Masashi HAMAHIRA, Ken-ichi HIDAI, Hiroshi MIZOGUCHI, Taketoshi MISHIMA, Shuji YOSHIZAWA, "Fast Algorithm for Online Linear Discriminant Analysis" in IEICE TRANSACTIONS on
Fundamentals, vol. E84-A, no. 6, pp. 1431-1441, June 2001, doi: .
Abstract: Linear discriminant analysis (LDA) is a basic tool of pattern recognition, and it is used in extensive fields, e.g. face identification. However, LDA is poor at adaptability since it is a
batch type algorithm. To overcome this, new algorithms of online LDA are proposed in the present paper. In face identification task, it is experimentally shown that the new algorithms are about two
times faster than the previously proposed algorithm in terms of the number of required examples, while the previous algorithm attains better final performance than the new algorithms after sufficient
steps of learning. The meaning of new algorithms are also discussed theoretically, and they are suggested to be corresponding to combination of PCA and Mahalanobis distance.
URL: https://global.ieice.org/en_transactions/fundamentals/10.1587/e84-a_6_1431/_p
author={Kazuyuki HIRAOKA, Masashi HAMAHIRA, Ken-ichi HIDAI, Hiroshi MIZOGUCHI, Taketoshi MISHIMA, Shuji YOSHIZAWA, },
journal={IEICE TRANSACTIONS on Fundamentals},
title={Fast Algorithm for Online Linear Discriminant Analysis},
abstract={Linear discriminant analysis (LDA) is a basic tool of pattern recognition, and it is used in extensive fields, e.g. face identification. However, LDA is poor at adaptability since it is a
batch type algorithm. To overcome this, new algorithms of online LDA are proposed in the present paper. In face identification task, it is experimentally shown that the new algorithms are about two
times faster than the previously proposed algorithm in terms of the number of required examples, while the previous algorithm attains better final performance than the new algorithms after sufficient
steps of learning. The meaning of new algorithms are also discussed theoretically, and they are suggested to be corresponding to combination of PCA and Mahalanobis distance.},
TY - JOUR
TI - Fast Algorithm for Online Linear Discriminant Analysis
T2 - IEICE TRANSACTIONS on Fundamentals
SP - 1431
EP - 1441
AU - Kazuyuki HIRAOKA
AU - Masashi HAMAHIRA
AU - Ken-ichi HIDAI
AU - Hiroshi MIZOGUCHI
AU - Taketoshi MISHIMA
AU - Shuji YOSHIZAWA
PY - 2001
DO -
JO - IEICE TRANSACTIONS on Fundamentals
SN -
VL - E84-A
IS - 6
JA - IEICE TRANSACTIONS on Fundamentals
Y1 - June 2001
AB - Linear discriminant analysis (LDA) is a basic tool of pattern recognition, and it is used in extensive fields, e.g. face identification. However, LDA is poor at adaptability since it is a batch
type algorithm. To overcome this, new algorithms of online LDA are proposed in the present paper. In face identification task, it is experimentally shown that the new algorithms are about two times
faster than the previously proposed algorithm in terms of the number of required examples, while the previous algorithm attains better final performance than the new algorithms after sufficient steps
of learning. The meaning of new algorithms are also discussed theoretically, and they are suggested to be corresponding to combination of PCA and Mahalanobis distance.
ER - | {"url":"https://global.ieice.org/en_transactions/fundamentals/10.1587/e84-a_6_1431/_p","timestamp":"2024-11-08T12:55:03Z","content_type":"text/html","content_length":"61181","record_id":"<urn:uuid:ba50cd2d-2022-47ef-96be-ddf25fb6933f>","cc-path":"CC-MAIN-2024-46/segments/1730477028059.90/warc/CC-MAIN-20241108101914-20241108131914-00519.warc.gz"} |
Proving Security | EasyCrypt
Having formalized the relevant context and the more general security-related definitions (which, arguably, can also be considered part of the context), we move on to the proof-specific aspects of the
formalization, culminating in the formal verification of the proof itself.
High-Level Proof Sketch
Before diving into the details, we provide a high-level sketch or intuition of the proof and its structure.
Foremost, recall that our aim is to show that our symmetric nonce-based encryption scheme is IND$-NRCPA secure as long as the function family it uses is a NRPRF. The predominant approach to proving
such a statement reduces the problem of breaking the NRPRF property of the function family to the problem of breaking the IND$-NRCPA security of the encryption scheme; proofs of this type are often
referred to as reductionist proofs. In our case, such a proof would essentially boil down to defining, for every adversary $\mathcal{D}$ against the IND$-NRCPA security of the encryption scheme, an
adversary $\mathcal{R}^{\mathcal{D}}$ against the NRPRF property of the function family that "outperforms" $\mathcal{D}$ (i.e., the advantage of $\mathcal{R}^{\mathcal{D}}$ is greater than or equal
to the advantage of $\mathcal{D}$). Then, if there would exist any adversary that is "unacceptably effective" at breaking the IND$-NRCPA security of the encryption scheme, it immediately follows that
there also exists an adversary that is "unacceptably effective" in breaking the NRPRF property of the employed function family. However, since it is assumed (or "conjectured") that a latter such
adversary does not exist, one can conclude that a former such adversary also does not exist and, hence, that the encryption scheme is IND$-NRCPA secure.
As it turns out, EasyCrypt is specifically designed for the formal verification of such reductionist proofs; for this reason, we stick to this type of proof here. Particularly, we build a reduction
adversary (against the NRPRF property of $\left(f_{k} : \mathcal{N} \rightarrow \mathcal{P}\right)_{k \in \mathcal{K}}$) that, given any (black-box) adversary against the IND$-NRCPA security of $\
mathcal{E}$, simulates a regular run of the IND$-NRCPA experiment in a way that allows the reduction adversary to win whenever the given adversary "wins" the simulated run. A high-level illustration
of this dynamic is provided in the following image.
Here, the left-hand side depicts a regular run of the IND$-NRCPA experiment where the IND$-NRCPA adversary directly interacts with the given NRCPA oracle; the right-hand side depicts a run of the
NRPRF experiment where the environment—particularly, oracle interactions—of the IND$-NRCPA adversary is fully controlled by the reduction adversary, who uses its NRPRF oracle to perfectly simulate
the environment—particularly, answers to oracle queries—in a way that matches the environment of a regular run of the IND$-NRCPA experiment and simultaneously allows making use of the eventual return
value of the given adversary.
Setup and Security Statements
Prior to proving or formally verifying anything, we go over the definition and formalization of the necessary proof-specific and relevant security statements. Here, we take a top-down approach,
starting with the final goal and moving toward the lower-level steps.
The Main Result: IND$-NRCPA Security
Once again, intuitively, the end goal is to demonstrate that $\mathcal{E}$ is IND$-NRCPA secure based on the assumption that $\left(f_{k} : \mathcal{N} \rightarrow \mathcal{P}\right)_{k \in \mathcal
{K}}$ is a NRPRF. More formally, the end goal is to prove the following (pen-and-paper) theorem.
Theorem 1. For all adversaries $\mathcal{D}$ against IND$-NRCPA of $\mathcal{E}$, there exists an adversary $\mathcal{B}$ against NRPRF of $\left(f_{k} : \mathcal{N} \rightarrow \mathcal{P}\right)_{k
\in \mathcal{K}}$ — with a running time close to that of $\mathcal{D}$ — such that the following holds:
$\mathsf{Adv}^{\mathrm{IND\\textrm{-}NRCPA}}_{\mathcal{E}}(\mathcal{D}) \leq \mathsf{Adv}^{\mathrm{NRPRF}}(\mathcal{B})$
As alluded to before, we prove this theorem using a proof by construction: Given any adversary $\mathcal{D}$ against IND$-NRCPA security of $\mathcal{E}$, we construct a reduction adversary $\mathcal
{R}$ against NRPRF of $\left(f_{k} : \mathcal{N} \rightarrow \mathcal{P}\right)_{k \in \mathcal{K}}$ — with a running time close to that of $\mathcal{D}$ — that obtains an advantage that is equal to
the advantage of the given adversary. In fact, having defined such a reduction adversary, say $\mathcal{R}^{\mathcal{D}}$, the following theorem implies the one above.
Theorem 2. For all adversaries $\mathcal{D}$ against IND$-NRCPA of $\mathcal{E}$, the following holds.
$\mathsf{Adv}^{\mathrm{IND\\textrm{-}NRCPA}}_{\mathcal{E}}(\mathcal{D}) = \mathsf{Adv}^{\mathrm{NRPRF}}(\mathcal{R}^{\mathcal{D}})$
In EasyCrypt, there is no notion of running time; consequently, there is also no way to formalize the restriction "with a running time close to that of some algorithm". For this reason, we typically
formalize theorems akin to Theorem 2, where the reasonableness of the operations performed by the considered reduction adversary (i.e., in terms of running time) is to be manually evaluated (by
Before advancing to the formalization of Theorem 2, recall that we cannot formalize short-hands for the advantage expressions like we do on paper; therefore, we directly formalize the absolute
difference in probabilities that these advantages define. For convenience, the (pen-and-paper) definitions of the relevant advantage expressions are restated below.
$\mathsf{Adv}^{\mathrm{IND\\textrm{-}NRCPA}}_{\mathcal{E}}(\mathcal{D}) = \left|\mathsf{Pr}\left[\mathsf{Exp}^{\mathrm{IND\\textrm{-}NRCPA}}_{\mathcal{D}, \mathcal{O}^{CPA\textrm{-}real}_{\mathcal
{E}}} = 1\right] - \mathsf{Pr}\left[\mathsf{Exp}^{\mathrm{IND\\textrm{-}NRCPA}}_{\mathcal{D}, \mathcal{O}^{CPA\textrm{-}ideal}} = 1\right]\right|$ $\mathsf{Adv}^{\mathrm{NRPRF}}(\mathcal{R}^{\mathcal
{D}}) = \left|\mathsf{Pr}\left[\mathsf{Exp}^{\mathrm{NRPRF}}_{\mathcal{R}^{\mathcal{D}}, \mathcal{O}^{PRF\textrm{-}real}} = 1\right] - \mathsf{Pr}\left[\mathsf{Exp}^{\mathrm{NRPRF}}_{\mathcal{R}^{\
mathcal{D}}, \mathcal{O}^{PRF\textrm{-}ideal}} = 1\right]\right|$
Here, remember that these probability statements are only well-defined if the initial memory/context are fixed and that, in EasyCrypt, we explicitly indicate this initial memory/context. Then, in
actuality, we want the above theorems to hold for any initial memory/context. Apart from this explicit memory indication, the formalization of probability statements in EasyCrypt closely follows the
pen-and-paper definitions. Essentially, given some specific initial memory (variable) &m, we can formalize the probability expressions by replacing all algorithms by their formalized counterparts,
appending @ &m (indicating that the execution starts in memory &m), writing a colon instead of an equality sign, and formalizing the relevant event (where the special keyword res may be used to refer
to the output of the considered procedure). For example, for some initial memory corresponding to &m, $\mathsf{Pr}\left[\mathsf{Exp}^{\mathrm{IND\\textrm{-}NRCPA}}_{\mathcal{D}, \mathcal{O}^{CPA\
textrm{-}real}_{\mathcal{E}}} = 1 \right]$ is formalized as Pr[Exp_IND_NRCPA(O_NRCPA_real(E), D).run() @ &m : res]. (Here, since the output of Exp_IND_NRCPA(O_NRCPA_real(E), D).run()—and, hence,
res—is a boolean, res is equivalent to res = true.)
Finally, theorems/lemmas are formalized similarly to axioms, merely replacing the axiom keyword by the lemma keyword.^1 Combining everything, we can formalize Theorem 2 as follows.
lemma EqAdvantage_IND_NRCPA_NRPRF &m:
`| Pr[Exp_IND_NRCPA(O_NRCPA_real(E), D).run() @ &m: res]
- Pr[Exp_IND_NRCPA(O_NRCPA_ideal, D).run() @ &m: res] |
`| Pr[Exp_NRPRF(O_NRPRF_real, R_NRPRF_IND_NRCPA(D)).run() @ &m: res]
- Pr[Exp_NRPRF(O_NRPRF_ideal, R_NRPRF_IND_NRCPA(D)).run() @ &m: res] |.
In this lemma, D denotes the formalization of $\mathcal{D}$ (i.e., an arbitrary IND$-NRCPA adversary); where and how we declare this arbitrary/abstract module will be discussed in one of the upcoming
sections on the formal verification of the statements. Furthermore, R_NRPRF_IND_NRCPA(D) denotes the formalization of $\mathcal{R}^{\mathcal{D}}$, which we discuss imminently.
Reduction Adversary
The main proof-specific artifact we must formalize is the reduction adversary. This reduction adversary is given an IND$-NRCPA adversary but is a NRPRF adversary itself, meaning it also gains access
to a NRPRF oracle. As touched upon in the high-level proof sketch, the crux of the argument is that the reduction adversary perfectly simulates a run of the IND$-NRCPA experiment for the given
adversary using the NRPRF oracle in a way that allows for the reduction adversary to win the NRPRF experiment whenever the given adversary would have won the simulated IND$-NRCPA experiment. Somewhat
more precisely, the reduction adversary executes the given adversary while simulating the NRCPA oracle by encrypting each of the queried plaintexts using the values returned from the NRPRF oracle
(when querying it on the same plaintexts). If done properly, the view of the given adversary is (distributed) exactly the same as the view it would have in a regular run of its own experiment.
Consequently, the behavior of the given adversary—and, hence, (the distribution of) its output—matches the behavior it would exhibit in a regular run of its own experiment. Furthermore, since the
encryptions returned to the given adversary were constructed using the NRPRF oracle, the reduction adversary can directly translate a correct (or incorrect) choice by the given adversary regarding
the validity of the provided encryptions into a correct (or incorrect) choice regarding the validity of the values provided by the NRPRF oracle. As such, the reduction adversary will invariably be
correct (and incorrect) with the exact same probability as the given adversary, independent of the actual implementation of the given adversary.
Because the reduction adversary itself is a NRPRF adversary, we formalize it as a module of type Adv_NRPRF. To indicate that the module formalizes the Reduction adversary that reduces from NRPRF to
IND_NRCPA, we name the module R_NRPRF_IND_NRCPA. However, because a module of type Adv_NRPRF only expects a module of type NRPRF_Oracle as parameter, the module parameter of type Adv_IND_NRCPA
(formalizing the given IND$-NRCPA adversary) must come first. Indeed, loosely speaking, a module is of type Adv_NRPRF only if it still expects a single module parameter of type NRPRF_Oracle; if there
are any other module parameters, these must first be instantiated before the module "becomes" of type Adv_NRPRF. This is reflected in the (module) type annotations of the module definition, provided
in the snippet below.
module (R_NRPRF_IND_NRCPA (D : Adv_IND_NRCPA) : Adv_NRPRF) (O_NRPRF : NRPRF_Oracle) = {
module O_NRCPA : NRCPA_Oracle = {
proc enc(n : nonce, m : ptxt) : ctxt option = {
var p : ptxt option;
var r : ctxt option;
p <@ O_NRPRF.get(n);
r <- if p = None then None else Some (oget p + m);
return r;
proc distinguish() : bool = {
var b : bool;
b <@ D(O_NRCPA).distinguish();
return b;
Here, we see that the reduction adversary defines a sub-module O_NRCPA of type NRCPA_Oracle; as the type enforces, this sub-module implements an enc procedure. In this enc procedure, the reduction
adversary directly queries the provided NRPRF oracle module (O_NRPRF) on n. Subsequently, if the value p returned by the NRPRF oracle is a failure indication, then the reduction adversary returns a
failure indication as well; else, if the value p returned by the NRPRF oracle contains a valid plaintext, the reduction adversary returns the ciphertext obtained by mapping this plaintext and m using
the + operator. (Indeed, the oget operator takes a value of any option type and, if the value equals Some x, it returns x; else, it returns an arbitrary value of the original type.)
In its distinguish procedure, the reduction adversary uses its sub-module as the NRCPA oracle that is exposed to the given adversary. This formalizes the simulation of oracle interactions by the
reduction adversary for the given adversary. In the end, the reduction adversary simply returns the value returned by the given adversary.
Intermediate Results: Equal Probabilities in Real and Ideal Cases
To separate concerns, we break the main part of the proof down into two independent pieces: equality of the "real case" probabilities and equality of the "ideal case" probabilities. More formally, we
proceed by separately proving the following two equalities.
$\mathsf{Pr}\left[\mathsf{Exp}^{\mathrm{IND\\textrm{-}NRCPA}}_{\mathcal{D}, \mathcal{O}^{CPA\textrm{-}real}_{\mathcal{E}}} = 1\right] = \mathsf{Pr}\left[\mathsf{Exp}^{\mathrm{NRPRF}}_{\mathcal{R}^{\
mathcal{D}}, \mathcal{O}^{PRF\textrm{-}real}} = 1\right]$ $\mathsf{Pr}\left[\mathsf{Exp}^{\mathrm{IND\\textrm{-}NRCPA}}_{\mathcal{D}, \mathcal{O}^{CPA\textrm{-}ideal}} = 1\right] = \mathsf{Pr}\left[\
mathsf{Exp}^{\mathrm{NRPRF}}_{\mathcal{R}^{\mathcal{D}}, \mathcal{O}^{PRF\textrm{-}ideal}} = 1\right]$
At this point, it might be good to note (and convince yourself) that the defined reduction adversary indeed does what we want in both of the considered cases ("real" and "ideal"); in particular, it
properly simulates the NRCPA oracle in either case. If the provided NRPRF oracle module is the real one (O_NRPRF_real), then get(n) returns a failure indication if n was already queried, and a
plaintext obtained by applying the function f to k and n otherwise. In the former case, the reduction adversary returns a failure indication as well. In the latter case, the reduction adversary
returns a ciphertext constructed by mapping the received plaintext and m using +. Certainly, that is equivalent to the real NRCPA oracle module O_NRPRF_real using the encryption scheme NBEncScheme to
obtain the same ciphertext given that the input and the key are the same. Meaning that the reduction adversary perfectly simulates the real NRCPA oracle module (O_NRCPA_real) when it is given the
real NRPRF oracle module (O_NRPRF_real). When providing the ideal oracle O_NRCPA_ideal to the reduction adversary the intresting case is again, when the nonce was not queried before and the get(n)
procedure returns a randomly (uniformly) sampled plaintext. Again the reduction adversary returns a ciphertext constructed by mapping the received plaintext and m using +. Since the received
plaintext is uniformly distributed, it essentially functions as a one-time pad in this mapping; hence, the resulting ciphertext is uniformly distributed as well. Following, even though the ideal
NRCPA oracle module (O_NRCPA_ideal) does not perform this mapping (but instead directly samples a ciphertext uniformly at random and returns this), the distribution of the returned ciphertext is
identical. As a result, the reduction adversary simulates the ideal case perfectly.
The veracity of these equalities almost immediately follows from the previous discussion concerning the reduction adversary. That is, in either case, $\mathcal{R}^{\mathcal{D}}$ perfectly simulates
the corresponding case for $\mathcal{D}$, meaning (the distribution of) the output of $\mathcal{D}$ is identical to what it would be in a run of its own game. Then, since $\mathcal{R}^{\mathcal{D}}$
directly returns the value returned by $\mathcal{D}$, the probability of this value being 1 is trivially equal to the probability of the value returned by $\mathcal{D}$ being 1 in a run of its own
In EasyCrypt, the equality of the "real case" probabilites is formalized as the lemma shown in the snippet below.
lemma EqPr_IND_NRCPA_NRPRF_real &m:
Pr[Exp_IND_NRCPA(O_NRCPA_real(E), D).run() @ &m : res]
Pr[Exp_NRPRF(O_NRPRF_real, R_NRPRF_IND_NRCPA(D)).run() @ &m : res].
Similarly, the equality of the "ideal case" probabilites is formalized as follows.
lemma EqPr_IND_NRCPA_NRPRF_ideal &m:
Pr[Exp_IND_NRCPA(O_NRCPA_ideal, D).run() @ &m: res]
Pr[Exp_NRPRF(O_NRPRF_ideal, R_NRPRF_IND_NRCPA(D)).run() @ &m: res].
Following the previous discussion about lemmas in EasyCrypt, these formalizations should be relatively easy to interpret and understand. Nevertheless, some more details including the declaration of
the arbitrary/abstract module D, will be covered momentarily.
Formal Verification
At last, we advance to the formal verification of the security statements. That is, in the remainder, we go over the process of proving the previously formalized lemmas in EasyCrypt. As before, we
take a top-down approach to the discussion, starting with the formal verification of the main result (temporarily assuming the veracity of the intermediate result) and only then proceeding to the
formal verfication of the intermediate lemmas. Nevertheless, before anything, we introduce the concept of sections in Easycrypt, elucidating several aspects that we skimmed over previously (e.g., the
declaration of an arbitrary/abstract module and the local keyword).
Oftentimes, instead of formally verifying the main result(s) at once, it is more convenient and more manageable (both for the prover and the reader) to first formally verify some useful auxiliary
results, and then combining these to formally verify the main result(s). These auxiliary results are generally quite proof-specific, so much so that you wouldn't really want them (or any related
auxiliary artifacts) to be saved/exposed after you have used them for their specific purpose. Furthermore, these auxiliary results frequently pertain to/quantify over the same artifacts as the
eventual main result(s) (e.g., an adversary); it is cumbersome to repeat the precise declaration/quantification of these artifacts over and over for each individual result.
A useful and convenient feature of EasyCrypt that alleviates the above issues is the (proof) section environment; this environment is delimited by the sentences section X. and end section X. (where X
is an optional name for the section). Inside of a section, we can "declare" modules with the desired restrictions using the declare keyword. Afterward, we can refer to these declared modules
throughout the entire remainder of the section; without this feature, we would need to declare/quantify these modules (and the restrictions) anew everywhere we need to use them. In our case, all our
results (both intermediate and final) quantify over IND$-NRCPA adversaries. As such, in the beginning of our section, we declare a module D of type Adv_IND_NRCPA with the appropriate restrictions;
see the following snippet.
section E_IND_NR_CPA.
declare module D <: Adv_IND_NRCPA { -O_NRCPA_real, -O_NRCPA_ideal, -O_NRPRF_real, -O_NRPRF_ideal }.
(* Can use D anywhere here *)
end section E_IND_NR_CPA.
By default, any module in EasyCrypt has access to the module variables of other modules (as well as its own, of course). This also holds for modules that are declared in sections. However, we do not
want adversaries to have access to the state of the oracles used in the experiments. (In pen-and-paper proofs, this is also always a given.) To specify the modules of which a declared module may not
access the variables, we provide a a comma-separated list of the names of these exempted modules (preceded by a -) in between curly brackets following the type annotation.
In addition to declaring modules, a section allows us to mark definitions of types, operators, module types, modules and lemmas as local (using the local keyword) such that they are only accessible
inside the section (i.e., they are not exposed outside the section). As we see later, we mark our two intermediate lemmas as local; this is because these lemmas are proof-specific auxiliary results
used to make the formal verification of the main result more manageable. Contrarily, the lemma for the main result is not marked as local since it is the primary result that we would like to be
available outside the section. Nevertheless, this lemma still refers to D, the module declared inside of the section (and that is not exposed outside of the section). As a result, after closing the
section, the lemma for the main result will be extended with the appropriate quantification over modules of type Adv_IND_NRCPA (including the desired restrictions).
The Main Result: IND$-NRCPA Security
At this point, we really have everything in place to start formally verifying our security statements. Following a top-down approach to the discussion, we start with the formal verification of the
main result. To this end, assume (for now) that we have already formally verified the intermediate results and, hence, have them at our disposal in the formal verification of the main result.
Everytime we write down a lemma statement, we are expected to prove ("formally verify") the statement immediately. In fact, the tool will not continue processing any further commands until a full
proof is provided. Once the proof is complete, the lemma is saved and is available for us to use in any subsequent proofs. To start proving a lemma, we write the sentence proof.; to save a lemma
after proving it, we write the sentence qed.. Considering the lemma for our main result, this looks as follows.
lemma EqAdvantage_IND_NRCPA_NRPRF &m:
`| Pr[Exp_IND_NRCPA(O_NRCPA_real(E), D).run() @ &m: res]
- Pr[Exp_IND_NRCPA(O_NRCPA_ideal, D).run() @ &m: res] |
`| Pr[Exp_NRPRF(O_NRPRF_real, R_NRPRF_IND_NRCPA(D)).run() @ &m: res]
- Pr[Exp_NRPRF(O_NRPRF_ideal, R_NRPRF_IND_NRCPA(D)).run() @ &m: res] |.
(* Proof *)
Between proof. and qed., we provide the actual proof of the considered statement.
Throughout any proof in EasyCrypt, the tool maintains a so-called proof state, a sequence of one or more proof goals. Each proof goal consists of a context and a conclusion: the context contains all
locally (i.e., goal-specific) considered variables and properties ("hypotheses"); the conclusion is a boolean expression that is to be shown to evaluate to true. Initially, for any proof, the proof
state consist only of a single proof goal: the one corresponding to the original lemma statement. As a proof progresses, already existing goals change and new goals may appear. Whenever a goal's
conclusion is shown to be true, the goal is "closed" (i.e., removed from the proof state); after closing all goals, the proof is complete and the original lemma may be saved.
In interactive mode (which is practically required for developing), EasyCrypt can display the proof state and update it as the proof progresses. By default, only the currently considered proof goal
of the proof state (i.e., the first goal in the sequence of goals in the state) is displayed. For example, the following is what is initially displayed for our main result (which can be reached by
processing up to and including proof.).
Current goal
Type variables: <none>
&m: {}
`|Pr[Exp_IND_NRCPA(O_NRCPA_real(E), D).run() @ &m : res]
- Pr[Exp_IND_NRCPA(O_NRCPA_ideal, D).run() @ &m : res]|
`|Pr[Exp_NRPRF(O_NRPRF_real, R_NRPRF_IND_NRCPA(D)).run() @ &m : res]
- Pr[Exp_NRPRF(O_NRPRF_ideal, R_NRPRF_IND_NRCPA(D)).run() @ &m : res]|
Here, everything above the dotted line is part of the goal's context, and everything below the dotted line is part of the goal's conclusion. In an initial proof goal like this one, the context always
only contains the (type) variables declared between the lemma's name and the lemma's statement; indeed, in this case, this is only the memory variable &m. Furthermore, the conclusion of such an
initial goal always equals the lemma's statement.
To go from opening the initial proof goal to closing the final proof goal and saving the lemma, we repeatedly apply tactics. In essence, a tactic represents a reasoning principle that may be applied
to make progress in a proof. EasyCrypt provides many tactics, covering a wide range of scenarios; we will introduce and elaborate on the ones we use in this tutorial as we go. For a comprehensive
overview of the tactics and their individual variations, consult the reference manual.
Assuming we have access to the intermediate results (i.e., lemmas EqPr_IND_NRCPA_NRPRF_real and EqPr_IND_NRCPA_NRPRF_ideal), proving the above goal is rather straightforward: we can simply use that
the left-hand side minuend and subtrahend are respectively equal to their right-hand side counterparts. To do so, we make use of the rewrite tactic. Given the name of a lemma/axiom that defines an
equality (say X = Y), this tactic searches the current goal's conclusion for X and replaces it with Y. Thus, in our case, issuing rewrite EqPr_IND_NRCPA_NRPRF_real. should replace Pr[Exp_IND_NRCPA
(O_NRCPA_real(E), D).run() @ &m : res] with Pr[Exp_NRPRF(O_NRPRF_real, R_NRPRF_IND_NRCPA(D)).run() @ &m : res]. Certainly, doing so changes the proof goal to the following.
Current goal
Type variables: <none>
&m: {}
`|Pr[Exp_NRPRF(O_NRPRF_real, R_NRPRF_IND_NRCPA(D)).run() @ &m : res]
- Pr[Exp_IND_NRCPA(O_NRCPA_ideal, D).run() @ &m : res]|
`|Pr[Exp_NRPRF(O_NRPRF_real, R_NRPRF_IND_NRCPA(D)).run() @ &m : res]
- Pr[Exp_NRPRF(O_NRPRF_ideal, R_NRPRF_IND_NRCPA(D)).run() @ &m : res]|
Subsequently issuing rewrite EqPr_IND_NRCPA_NRPRF_real. results in the proof goal below.
Current goal
Type variables: <none>
&m: {}
`|Pr[Exp_NRPRF(O_NRPRF_real, R_NRPRF_IND_NRCPA(D)).run() @ &m : res]
- Pr[Exp_NRPRF(O_NRPRF_ideal, R_NRPRF_IND_NRCPA(D)).run() @ &m : res]|
`|Pr[Exp_NRPRF(O_NRPRF_real, R_NRPRF_IND_NRCPA(D)).run() @ &m : res]
- Pr[Exp_NRPRF(O_NRPRF_ideal, R_NRPRF_IND_NRCPA(D)).run() @ &m : res]|
Obviously, this goal's conclusion is true: the right-hand side and left-hand side are literally the same. For these kind of trivial goals, we can use the trivial tactic to try and close the goal.
Indeed, issuing trivial. closes the goal and, since this was the only proof goal left in the proof state, completes the proof. Everything combined, we obtain the following for our main result.
lemma EqAdvantage_INDNRCPA_NRPRF &m:
`|Pr[Exp_IND_NRCPA(O_NRCPA_real(E), D).run() @ &m: res]
- Pr[Exp_IND_NRCPA(O_NRCPA_ideal, D).run() @ &m: res]|
`|Pr[Exp_NRPRF(O_NRPRF_real, R_NRPRF_INDNRCPA(D)).run() @ &m: res]
- Pr[Exp_NRPRF(O_NRPRF_ideal, R_NRPRF_INDNRCPA(D)).run() @ &m: res]|.
rewrite EqPr_INDNRCPA_NRPRF_real.
rewrite EqPr_INDNRCPA_NRPRF_ideal.
To make this proof a bit cleaner, we can make use of the tactical by and a particular feature of the rewrite tactic. First, a tactical combines or modifies (a sequence of) tactics in some way. In the
case of by, it executes the tactic(s) directly following it and then attempts to close the resulting goal(s) using trivial. If the goal(s) cannot be closed after applying trivial, by will throw an
error. Second, rewrite can be given multiple lemma/axiom names. For example, issuing rewrite Lemma1 Lemma2., the tactic will first rewrite the current goal's conclusion according to Lemma1, and then
rewrite according to Lemma2 in the conclusion of the goal(s) generated by the rewriting of Lemma1. Employing these features, we can reduce the proof to the following one-liner.
by rewrite EqPr_INDNRCPA_NRPRF_real EqPr_INDNRCPA_NRPRF_ideal.
Intermediate Result 1: Equal Probabilities in Real Case
We strongly recommend you follow the explanation in this section while stepping through the code yourself (in interactive mode)
In the formal verification of the main result, we assumed that we had already formally verified the intermediate results. Now, we actually go over the formal verification of these intermediate
results, starting with the one concerning the equality of the "real case" probabilities. The following snippet the corresponding lemma together with a complete proof in EasyCrypt. Note that we
declare the lemma using the keyword local as discussed before.
local lemma EqPr_IND_NRCPA_NRPRF_real &m:
Pr[Exp_IND_NRCPA(O_NRCPA_real(E), D).run() @ &m : res]
Pr[Exp_NRPRF(O_NRPRF_real, R_NRPRF_IND_NRCPA(D)).run() @ &m : res].
byequiv (_ : ={glob D} ==> ={res}); trivial.
inline *.
sim (_ : ={k}(O_NRCPA_real, O_NRPRF_real) /\ ={log}(O_NRCPA_real, O_NRPRF_real)).
inline *.
The initial sentence of the proof consists of two tactics, byequiv and trivial, combined through the tactical ;. Combining two tactics by means of ;, as in t1; t2., first applies tactic t1 to the
current goal, and then applies tactic t2 to the goal(s) generated by the application of t1. In our case, we combine byequiv and trivial to immediately close some of the trivial goals generated by the
application of byequiv. The byequiv tactic is more interesting. Namely, this tactic allows us to prove certain (in)equalities of probabilities concerning program executions by demonstrating a
particular equivalence between the considered programs. This equivalence of programs is always with respect to a certain pre and postcondition, which are specified in the argument provided to
byequiv; the format of this argument is (_ : pre ==> post), where pre and post respectively denote the pre and postcondition. In our case, the precondition is ={glob D} (which is syntactic sugar for
(glob D){1} = (glob D){2}),^2 i.e., we require the accessible module variables (read: environment/view) of module D to start out the same in both executions; the postcondition is ={res}, i.e., we
require the output of the programs to be (distributed) the same. Processing this intial sentence results in a goal that precisely corresponds to the program equivalence with this pre and
In the second sentence of the proof, we apply the proc tactic; this tactic can be used on goals with a conclusion corresponding to a program logic statement on procedure identifiers (i.e., not on
actual code). The program logics of EasyCrypt are Hoare Logic (HL), probabilistic Hoare Logic (pHL), and probabilistic Relation Hoare Logic (pRHL). The current goal's conclusion denotes a pRHL
statement with identifiers of concrete procedures; in such a case, proc simply replaces the identifiers by the code of the procedures.
After applying proc, we see that the code of the procedures contains several calls to various concrete procedures. To get a better view of what actually happens, we inline all of these concrete
procedure calls by applying the inline tactic. In particular, since we want to inline all concrete procedure calls, we apply inline *. (If we wanted to inline only a particular concrete procedure
call, say O_NRCPA_real(E).init, we could've used inline O_NRCPA_real(E).init.)
Looking at the programs in the goal, we notice that they are really quite similar. Essentially, ignoring auxiliary assignments, the only difference is the oracle that is provided to the adversary D
when calling its (abstract) distinguish procedure. By construction, we know that these oracles should behave identically provided that they have the same keys and logs throughout the execution. For
cases like this, EasyCrypt provides the convenient higher-level sim tactic. Reasoning backward from the end of the programs, this tactic attempts to prove a program equivalence by keeping track of
(and extending/adjusting) a conjunction of equalities that implies the original postcondition; if the tactic manages to work through both programs completely, it tries to show that the original
precondition implies the final conjunction of equalities, which proves the original equivalence. Indeed, for the current goal, it suffices to maintain the fact that k and log of O_NRCPA_real and
O_NRPRF_real are equal throughout the execution of the programs to guarantee that the oracles provided to D behave identically and, hence, that D outputs the same value (distribution) in both
programs as well. Although sim can be used without any arguments to let EasyCrypt infer the invariant from the postcondition, this is not sufficient in the current case; therefore, we provide the
invariant explicitly as (_ : ={k}(O_NRCPA_real, O_NRPRF_real) /\ ={log}(O_NRCPA_real, O_NRPRF_real)), where ={x}(M, N) is syntactic sugar for M.x{1} = N.x{2}. Applying the sim tactic with this
invariant leaves us with a single goal asking us to prove an equivalence of the enc procedures of the oracles provided to D; specifically, the goal asks us to prove that, whenever the inputs to the
oracles are the same and the above invariant holds, the outputs of the oracles are (distributed) the same and the invariant still holds. This shows that the application of sim managed to close the
original goal under the assumption that the oracles behave identically and maintain the invariant when called. Now, we are expected to still prove this assumption to complete the proof.
Once again, the current goal concerns a pRHL equivalence on concrete procedure identifiers; so, we apply proc. Then, since the resulting code contains several calls to concrete procedure, we inline
all of them by applying inline *. Considering the definition of the omap operator, the programs are almost trivially seen to be semantically identical. Furthermore, the code merely contains
assignment statements and if-then-else constructs. As such, this goal is a good target for another relatively high-level tactic called auto. This tactic applies a sequence of several basic program
logic tactics, afterward solving the goal if it is trivial. In this case, auto manages to solve the goal and, thereby, complete the proof.
Intermediate Result 2: Equal Probabilities in Ideal Case
We strongly recommend you follow the explanation in this section while stepping through the code yourself (in interactive mode)
Bringing everything to a close, we discuss the formal verification of the second intermediate result: the equality of the "ideal case" probabilities. Here, we focus on novel (uses of) tactics that
did not occur in the previous formal verifications. The following snippet presents the corresponding lemma along with a full proof in EasyCrypt. Again we use the keyword local as discussed before.
local lemma EqPr_INDNRCPA_NRPRF_ideal &m:
Pr[Exp_IND_NRCPA(O_NRCPA_ideal, D).run() @ &m: res]
Pr[Exp_NRPRF(O_NRPRF_ideal, R_NRPRF_INDNRCPA(D)).run() @ &m: res].
byequiv (_ : ={glob D} ==> ={res}) => //.
proc; inline *.
call (_ : ={log}(O_NRCPA_ideal, O_NRPRF_ideal)).
- proc; inline *.
if => //.
- wp.
rnd (fun (p : ptxt) => p + m{2}).
skip => />.
move => &2 _.
- move => y _.
rewrite addpK //.
move => _ c _.
rewrite addpK //.
As in the formal verification of the first equality of probabilities, we start off with an application of the byequiv tactic with equality on the initial environment of D and (distribution of the)
outputs as pre and postcondition, respectively. However, instead of combining this with the trivial tactic by means of ;, we append the semantically equivalent, but slightly cleaner, => //. In
EasyCrypt, the => can be tacked onto any (sequence of) tactic(s) to start a sequence of so-called "introduction patterns". In order, each of the introduction patterns in the sequence is applied to
the goal(s) generated by the preceding (sequence of) tactic(s) and introduction pattern(s). An important application of introduction patterns is the introduction of universally quantified variables
and hypothesis from a goal's conclusion to a goal's context. For example, if a goal's conclusion starts with forall (i : int), ..., the introduction pattern => j will remove the quantification from
the goal's conclusion and add j: int to the goal's context (note that the introduced variable's identifier does not need to match the auxiliary identifier in the quantification). Similarly, if a
goal's conclusion starts with, e.g., x <> y => ..., then the introduction pattern => H will remove the x <> y antecedent (and the corresponding implication arrow) from the goal's conclusion and add
H: x <> y to the goal's context. In the remainder of the proof for that goal, H is available as if it were a regular axiom/lemma.
After applying byequiv, we obtain a single goal denoting a pRHL equivalence on procedure identifiers, as desired. We replace the procedure identifiers with the code of the procedures and immediately
inline all calls to concrete procedures in the resulting programs by applying proc; inline *. This leaves us with two programs that are nearly identical, only differing in the oracle provided to D
and an auxiliary assignment. Now, the right-hand side program ends in a simple assignment; we would like to get rid of this statement so that we can reason about and relate the abstract procedure
calls on both sides (which requires these calls to be the very last statement in both programs). To do so, we apply the "weakest precondition" tactic, wp. In essence, this tactic consumes assignment
statements from the end of the programs while adapting the postcondition in a way that reflects the execution of these statements; the pRHL equivalence that results from this implies the original
At this point, both programs end with a call to the same abstract procedure, i.e., distinguish of D; however, the exposed oracles differ and the equivalence between them cannot be proven using the
sim tactic. So, the main points we want to argue is that (1) the adversary starts out with the same view/environment on both sides, and (2) even though the provided oracles differ, their behavior is
identical. In turn, the adversary's view—and, hence, its behavior—is the same on both sides throughout its execution; particularly, this means that the adversary's output (distribution) is the same
on both sides. For this kind of reasoning, we use the call tactic in EasyCrypt. This tactic removes the abstract procedure calls and allows us to claim that the returned value is equally distributed
on both sides. However, to make sure this is sound, it produces two goals that formally encompass the two previously mentioned conditions. One asks us to prove that, right before the procedure calls
are made, the environment of the considered module (i.e., glob D in this case) is equal on both sides; the other asks us to prove a pRHL equivalence essentially capturing that, given the same input,
the exposed oracles produce the same output (distribution) on both sides. To assist in (or even make possible at all) proving the latter, the call tactic takes an invariant (as (_ : invariant)) that
is maintained throughout the oracle calls. Naturally, the fact that this invariant holds at the start and is maintained throughout becomes part of the goals. Indeed, all we need as an invariant in
this case is equality of the logs (={log}(O_NRCPA_ideal, O_NRPRF_ideal)), guaranteeing that the exposed oracles are synchronized with respect to failure indication.
In the interest of keeping our code somewhat clean and readable (to those who can understand EasyCrypt code in the first place), we indent our proof code whenever the previous tactic application more
than one goal, hence resulting in a proof state with more goals than before the application. (Similarly, we unindent whenever the previous tactic application closed a goal.) In our case, since the
application of call resulted in the generation of two goals, we indent the next sentence of proof code by starting it with the indentation symbol - and a single whitespace. This sentence will apply
to the first goal generated by call, and we indent all subsequent sentences applying to this first goal by two whitespaces. After closing this goal and arriving at the last goal generated by call, we
return to the same indentation level we used for the call tactic itself. Of course, we use this styling rule recursively; for example, if, during the proof of the first goal, we were to apply a
tactic that again generated more than one goal, we indent another level in the same manner as before.
The first goal generated by the call tactic concerns the behavioral equivalence of the exposed oracles. As per usual, we apply proc and inline * to first change this pRHL equivalence on procedure
identifiers to one on the code of the procedures, and subsequently inlining all concrete procedure calls in this code. This leaves us with two programs for which we want to show, among others, that
their output value (r) is (distributed) the same, as indicated by the ={r} term in the postcondition. Inspecting the programs (and keeping the precondition in mind), we foremost note that the same
branch of the if-statement will be executed on both sides due to the inputs and logs being equal. Now, if the else-branch is taken, the equality of (the distribution of) r trivially holds; namely, r
will simply be None on both sides (recall that omap outputs None if its second argument is None). However, if the then-branch is taken, the (distributions of the) return values are not trivially
identical: On the left-hand side, the return value is the value sampled in the then-branch; on the right-hand side, the return value is the value obtained from mapping the value sampled in the
then-branch (with the input plaintext) using +. Surely, since the sampled value essentially functions as a one-time pad in this mapping, the distribution of the return values is still the same on
both sides; nevertheless, this is not trivial (at least not for the tool) and, therefore, we will need to do some more work than simply applying some of the higher-level automated tactics.
Following from the above, one approach to proving the current goal is showing that (1) both sides invariably execute the same branch of the if-statement, and (2) the equivalence holds independent of
the executed branch. Fortunately, the if tactic enables us to take this exact approach. However, this tactic is only applicable when the if-statements are the first statement in both programs. So, to
achieve this, we need to get rid of the assignment statement preceding the if-statement in the right-hand side program. In turn, we achieve this by means of the "strongest precondition" tactic, sp,
which is basically the dual of wp. As you might have guesses, sp consumes assignment statements from the beginning of both programs while accordingly adapting the precondition. After applying sp, we
apply if, and immediately close one of the trivial goals it generates by appending => //. Specifically, this trivial goal concerns the equivalence of the if-guards on both sides, i.e., the fact that
both sides invariably enter the same branch; this is a trivial goal in this case because the equality of the variables stated in the precondition makes the guards exactly the same. Of course, the
other goals generated by the if tactic concern the veracity of the pRHL equivalence when executing the different branches.
Since the application of if => // generated more than one goal, we indent the code another level, as before. The current goal is the first goal generated by if => // and regards the equivalence of
the programs when executing the then-branch of the if-statement. Starting off, we apply wp to consume the assignment statements at the end of the programs; this results in both of the programs ending
with a sampling from the same distribution (recall that dctxt and dptxt refer to the same uniform distribution over all plaintexts/ciphertexts) that we want to relate somehow. Now, whenever we want
to relate samplings, we use the rnd tactic.^3 Oftentimes, we use this tactic without any arguments, which essentially assumes that the same value is sampled on both sides. However, sometimes, we want
to say that whenever we sample a certain value on the left-hand side, we sample a uniquely-linked value on the right-hand side. Surely, as long as all of the linked values have the same probability
of being sampled on each side, this is sound. For such cases, the rnd tactic takes two more arguments that, together, form a bijection between the supports of the considered distributions. Indeed,
this bijection is what establishes the unique link between the values from the distributions; of course, you must then still prove that the probability of each of the linked values in their
respective distribution is identical. As some nice syntactic sugar, whenever the bijection consists of the same function twice, you only have to provide it once as the first argument. Looking at the
postcondition of our current goal, we see that the value sampled on the right-hand side (c) should be equal to the value sampled on the left-hand side (y) after combining it with the input plaintext
(m) using +. So, if the sampling on the left-hand side gives us x, we want the sampling on the right-hand side to give us x + m{2}, and vice versa. The bijection that captures this link is defined by
two identical functions, viz., fun (p : ptxt) => p + m{2} (written as lambda/anonymous function). Therefore, we apply rnd (fun (p : ptxt) => p + m{2}), which consumes the samplings and adjust the
postcondition accordingly.
After removing the samplings, both programs only have assignments statements left; once again, we remove these statements using wp, leaving a goal with empty programs. Now, a pRHL equivalence with
empty programs is true if, for all possible program memories (for both programs), the precondition implies the postcondition. The skip tactic captures this reasoning principle, and transforms a pRHL
equivalence with empty programs into the appropriate, universally quantified implication. To make our lives a bit easier, we ask the tool to automatically simplify the expresssion generated by skip
as much as it can; specifically, we do so using the introduction pattern commonly referred to as "crush", />.
The application of skip => /> produces a goal that, intuitively, asks us to show that for any possible program memory, the bijection that we provided to the rnd tactic is actually a bijection on the
support of the considered distribution (dctxt or, equivalently, dptxt). Contemplating the experession following the universal quantification, we see that it contains an antecedent that is useless in
proving the actual consequent (i.e., the validity of the bijection). To remove the universal quantification as well as the useless antecedent without doing anything else, we combine the "identity
tactic" move with the introduction patterns &2 and _ as move => &2 _. Being the "identity tactic", move does absolutely nothing, but the subsequent introduction patterns respectively introduce a
program memory variable &2 into the context (removing the universal quantification from the goal's conclusion) and remove the first (and only) antecedent from the goal's conclusion.
Looking past some of the technical details (e.g., useless antecedents), we notice that the goal we are left with basically asks us to prove same thing twice: given any plaintext m, it holds that x =
x + m + m for each plaintext x. But this is just a "right self-cancellation" property of +, which we can prove generically as the following lemma that follows directly from the axioms stated in the
lemma addpK (x y : ptxt) : x + y + y = x.
by rewrite addpA addpC addKp.
(Note that, in order to use this addpK in the current proof, it must be saved beforehand.) With this lemma at our disposal, we continue the proof by applying the split tactic. As its name suggests,
this tactic "splits up" a goal whose conclusion constitutes a conjunction into two goals, each having a different term of the conjuction as its conclusion (but the same context). We close both of the
generated goals by, first, removing universal quantification and useless antecedents as before, and subsequently applying rewrite addpK //; the latter is syntactic sugar for rewrite addpK => //.
Naturally, we indent and unindent according to the aforementioned styling rules as we go.
The preceding closed the first goal generated by the application of if => // a while back; hence, we now arrive at the second goal generated by this application. As discussed before, this goal
corresponds to the equivalence of the programs when the else-branch of the if-statement executed which, because both return values equal None and the logs are not changed, is obviously true. Because
the programs merely comprise assignment statements, simply applying auto closes the goal; we unindent and proceed to the final goal of the proof.
Finally, we arrive at the last goal of the proof: the second goal generated by the application of the call tactic all the way in the beginning. Intuitively, this goal now asks us to prove that, when
the adversary is called in the original programs, (1) the environment/view of the adversary is equal on both sides, (2) the invariant given to the call tactic holds (i.e., the oracles' logs are
equal), and (3) the equality of (the distribution of the) return values as well as the veracity of the invariant imply the original postcondition (i.e., the postcondition of the goal the call tactic
was applied to). Now, first, because the equality on the adversary's environment is assumed by the precondition and not affected by the remaining statements of the programs, the first proof
obligation is trivial. Second, because the remaining statements initialize the logs on both sides to the empty list, the second proof obligation is trivial. Lastly, because the original postcondition
merely required the return value of D to be equal on both sides, the third proof obligation is also trivial. Concluding, since the remaining statements of the programs only concern assignments, a
simple application of auto closes the goal and finishes the proof.
1. Naturally, as opposed to axioms, lemmas require a proof/formal verification; these proofs are given directly succeeding the formalization of the lemma statement. Nevertheless, the formalization
of the statements themselves are identical between lemmas and axioms (barring the used keyword). ↩
2. For the record, variables annotated with {1}, as in x{1}, are given values according to the memory corresponding to the left-hand side program; similarly, variables annotated with {2}, as in x
{2}, are given values according to the memory corresponding to the right-hand side program. ↩
3. The samplings you want to compare should be the final statements in the considered programs. ↩ | {"url":"https://easycrypt.gitlab.io/easycrypt-web/docs/simple-tutorial/proof/","timestamp":"2024-11-11T20:36:57Z","content_type":"text/html","content_length":"174609","record_id":"<urn:uuid:4b058595-d477-4a06-ad24-8847a11eab02>","cc-path":"CC-MAIN-2024-46/segments/1730477028239.20/warc/CC-MAIN-20241111190758-20241111220758-00077.warc.gz"} |
04-08-2021 12:35 PM
06-30-2020 02:32 AM
06-23-2021 10:35 AM
06-23-2021 10:35 AM
06-23-2021 10:35 AM
Running a 4-index MILP model in OPTModel Procedure on Mathematical Optimization, Discrete-Event Simulation, and OR. 04-19-2020 11:21 PM
Re: Running a 4-index MILP model in OPTModel Procedure on Mathematical Optimization, Discrete-Event Simulation, and OR. 04-21-2020 11:48 AM
Re: Running a 4-index MILP model in OPTModel Procedure on Mathematical Optimization, Discrete-Event Simulation, and OR. 04-21-2020 03:11 PM
Re: Running a 4-index MILP model in OPTModel Procedure on Mathematical Optimization, Discrete-Event Simulation, and OR. 04-21-2020 05:33 PM
Using Primalin to use results from previous run on Mathematical Optimization, Discrete-Event Simulation, and OR. 06-22-2020 05:30 AM
Re: Using Primalin to use results from previous run on Mathematical Optimization, Discrete-Event Simulation, and OR. 06-22-2020 06:31 PM
Re: Using Primalin to use results from previous run on Mathematical Optimization, Discrete-Event Simulation, and OR. 06-22-2020 06:34 PM
Re: Using Primalin to use results from previous run on Mathematical Optimization, Discrete-Event Simulation, and OR. 06-30-2020 02:32 AM
Changing the line size and transparency with annotation for Proc GMAP on SAS Programming. 10-09-2020 01:42 PM
Re: Changing the line size and transparency with annotation for Proc GMAP on SAS Programming. 10-13-2020 12:43 PM
Re: Changing the line size and transparency with annotation for Proc GMAP on SAS Programming. 10-13-2020 12:44 PM
Re: Changing the line size and transparency with annotation for Proc GMAP on SAS Programming. 10-13-2020 12:48 PM
Re: NPK4ever on SAS Hackathon Team Profiles (Past). 04-07-2021 06:02 PM
Using two conditions for a Proc Optmodel constraint on Mathematical Optimization, Discrete-Event Simulation, and OR. 09-21-2022 10:47 AM
Hello All! I am using SAS Enterprise Guide. I have a optimization code where I have many constraints and decision variables. It is a facility location problem using multimodal (truck and rail) and I
have created routes using location j and k. I have two constraints right now where I am putting conditions if j and k does not have rail access, it can not ship using rail. I tried to combine the two
constraints (freight_depot and freight_bioref) into one but it only considered the first condition before the constraint and not the one I join using AND. Is there anyway I can use the AND operator
to join two conditions before a constraint, so that I can make the two constraints into one. I don't have all my constraints here. Just put some of them. This is how I want to write them together-
Con freight {<j,k> in ROUTES2: (s[j]=0 and l[k] =0)}: sum{<f,'1'> in tr} Amount2[j,k,f,'1'] = 0; Also, my code is very memory intensive as it deals with inputs of 2000 rows and millions of
constraints and decision variables. Is there any way I can solve the out of memory issue? Right now, after reaching an error gap of 8%, I am going out of memory. Thank you so much for all of your
help. @RobPratt, any help from you would be greatly appreciated. Thank you again! proc optmodel ; *****DECLARE SETS AND PARAMETERS*******; set <str> Fields; set <str> Depots init {}; set <str>
Bioref; set <str> Feedstock; set <str> Price; set Mode = {'0', '1'}; set <num> Capacity; set <str,str,str> TRIPLES; set DOUBLES = setof {<i,f,p> in TRIPLES} <i,f>; set <str,str,str> ROUTES; set
<str,str> ROUTES2; set <str,str> ROUTES3; set <str,str> tr; *Parameters for fields; num sifp{Fields,Feedstock,Price} init 0; *Available biomass in field i of feedstock f at price p; num maif
{Fields,Feedstock} init 0; *binary assignment of type f from field i; num gp {Price} init 0; *Farmgate price including grower payment and harvest and collection cost; num fsb {Feedstock}init 0;
*field site storage in bales; num dmloss {Feedstock}init 0; *dry matter loss from bales; num aifp{i in Fields,f in Feedstock,p in Price} = round(sifp[i,f,p]*(1-dmloss[f])); *available biomass in the
fields after considering the dry matter loss; *Transportation parameters; num x{Fields union Depots}; *union combines both field and depot; num y{Fields union Depots}; num s{Depots}; num l{Bioref};
*Calculating distances between fields and depots; num a{i in Fields, j in Depots}= (sin((y[j]-y[i])*&C/2))**2+cos(y[i]*&C)*cos(y[j]*&C)*(sin((x[j]-x[i])*&C/2))**2; num sij{i in Fields, j in Depots} =
ifn(x[i]=x[j] and y[i]=y[j],0,2*atan2(sqrt(a[i,j]),sqrt(1-a[i,j]))*3957.143);*converted the distance to miles; num dij{i in Fields, j in Depots} = round(sij[i,j],0.01); *Calculating distances between
depots and biorefineries; num b{j in Depots, k in Bioref}= (sin((y[k]-y[j])*&C/2))**2+cos(y[j]*&C)*cos(y[k]*&C)*(sin((x[k]-x[j])*&C/2))**2; num sjk{j in Depots, k in Bioref} = ifn(x[j]=x[k] and y[j]=
y[k],0,2*atan2(sqrt(b[j,k]),sqrt(1-b[j,k]))*3957.143);*converted the distance to miles; num djk{j in Depots, k in Bioref} = round(sjk[j,k],0.01); *Transportation costs; num vfb {Feedstock}; *variable
cost of transporting bales of feedstock f; num cfb {Feedstock}; *fixed/constant cost of transporting bales of feedstock f; num vfp {Feedstock, Mode}; *variable cost of transporting pellets of
feedstock f using mode t; num cfp {Feedstock, Mode}; *fixed/constant cost of transporting pellets of feedstock f using mode t; num tijf {i in Fields, j in Depots, f in Feedstock}= ifn (dij[i,j] = 0,
0, cfb[f] + vfb[f]*1.2*dij[i,j]); *Transportation cost for bales; num tjkf_pellets {j in Depots, k in Bioref, f in Feedstock, t in Mode}= ifn (djk[j,k] = 0, 0, cfp[f,t] + vfp[f,t]*1.2*djk[j,k]);
*Transportation cost for pellets; *Parameters for Depots; num qh{Feedstock}init 0; *Handling and queuing of bales at depot: $1.21 for CS and $1.34 for SW; num pf{Feedstock}init 0; *preprocessing cost
at depot: $22.65-3.18=$19.47 for CS and $22.05-3.18=$18.77 for SW; num ds{Feedstock}init 0; *depot storage in pellet form; num U = 0.9; *depot utilization factor; num max_distance1 = 200; num
max_distance2 = 1300; num max_distance3 = 400; num min_distance = 300; *quality parameters; num Ash{Feedstock}; num Moisture{Feedstock}; num Carb{Feedstock}; num Ash_dock{Feedstock}; num Moist_dock
{Feedstock}; num Ash_diff{f in Feedstock} = ifn(&Max_Ash>=Ash[f],0, Ash[f] - &Max_Ash); num Moist_diff{f in Feedstock} = ifn(&Min_moisture<=Moisture[f],0, &Min_moisture-Moisture[f]); read data
out.fields_&year into Fields = [fips] x y; read data out.INLdepots_&year into Depots = [fips] x y s=site; read data out.INLdepots_&year into Bioref = [fips] x y l=site; read data out.Price into Price
= [name] gp = Pr ; *gp= grower payment; read data out.feedstockpropertiesbales_MS into Feedstock=[Feed] vfb=TranspCostVar cfb=TranspCostFixed fsb=StorageCost qh=HandlingQueuingCost dmloss=
BiomassLoss; read data out.feedstockpropertiespellets_MS into Feedstock=[Feed] ds=StorageCost pf=ProcessingCost; read data out.Supplymod_&year into TRIPLES=[fips Feed Price] sifp=Supply; *same as
line commented below; read data out.Supplymin_&year into [fips Feed] maif=MinAssign; *minimum assignment (binary); read data out.quality_MS into Feedstock = [feed] Ash Moisture carb Ash_dock
Moist_dock; read data out.transport into tr=[Feed Mode] vfp=TranspCostVar cfp=TranspCostFixed; ROUTES = {<i,f> in DOUBLES, j in DEPOTS: dij[i,j] < max_distance1}; ROUTES2 = {j in DEPOTS, k in BIOREF:
djk[j,k] <max_distance2}; *****DECLARE MODEL ELEMENTS*******; ******DECISION VARIABLES*****; var Build {DEPOTS} binary; *binary value to build depots with specific capacity; var CapacityChunks
{DEPOTS} >= 0; *integer value to determine depot capacity; var Build2 {Bioref} binary; *binary value to build biorefineries with fixed capacity of 725000 dry tons; var CapacityBioRef {Bioref} >= 0;
*to determine biorefinery capacity; var AnyPurchased {TRIPLES} binary; var AmountPurchased {TRIPLES} >= 0; var AmountShipped {ROUTES} >= 0; var Amount2{ROUTES2,tr} >= 0; impvar BuildingCost {j in
DEPOTS} = 132717 * Build[j] + 2.297 * 25000 * CapacityChunks[j]; impvar VariableCost = sum{<i,f,p> in TRIPLES} 0.977*gp[p] * AmountPurchased[i,f,p] + sum{<i,f,j> in ROUTES} (fsb[f]+ tijf[i,j,f]+ qh
[f]+ ds[f]+ pf[f]) * AmountShipped[i,f,j] + sum{<j,k> in ROUTES2, <f,t> in tr} (tjkf_pellets[j,k,f,t]+(Ash_dock[f]*Ash_diff[f]) + (Moist_dock[f]*Moist_diff[f])) * Amount2[j,k,f,t]; impvar FixedCost =
sum {j in Depots} BuildingCost[j]; *****OBJECTIVE FUNCTION******; max Supply = sum{<j,k> in ROUTES2,<f,t> in tr} Amount2[j,k,f,t]; *****CONSTRAINTS******; *Flow balance between field-depot and
depot-biorefinery; Con Depot_avail {j in Depots,f in Feedstock}: sum {<i,(f),(j)> in ROUTES} AmountShipped[i,f,j] = sum{<(j),k> in ROUTES2, <(f),t> in tr} Amount2[j,k,f,t]; *Depot locations which
have do not freight access; Con freight_depot {j in Depots: s[j]=0}: sum{<f,'1'> in tr} Amount2[j,k,f,'1'] = 0; *Biorefinery Locations which have do not freight access; Con freight_bioref {k in
Bioref: l[k]=0}: sum{<j,(k)> in ROUTES2,<f,'1'> in tr} Amount2[j,k,f,'1'] = 0; Con rail_minimum {<j,k> in ROUTES2: djk[j,k]<= min_distance}: sum{<f,'1'> in tr} Amount2[j,k,f,'1'] = 0; /**/ Con
truck_maximum {<j,k> in ROUTES2: djk[j,k] > max_distance3 }: sum{<f,'0'> in tr} Amount2[j,k,f,'0'] = 0; *****SOLVE*******; solve obj Supply with milp / maxtime=&MaxRunTime relobjgap=0.03; * To force
the code to stop after maxtime; run;
... View more | {"url":"https://communities.sas.com/t5/user/viewprofilepage/user-id/307257","timestamp":"2024-11-07T01:15:13Z","content_type":"text/html","content_length":"275233","record_id":"<urn:uuid:625dc083-3c22-471f-aeca-4103e3bf102a>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00268.warc.gz"} |
Astrophysics of Cepheids in eclipsing binary systems
Based on new observations and improved modeling techniques, we have re-analyzed seven Cepheids in the Large Magellanic Cloud.
The Araucaria Project: High-precision Cepheid astrophysics from the analysis of variables in double-lined eclipsing binaries
B. Pilecki, W. Gieren, G. Pietrzyński, I. B. Thompson, R. Smolec, D. Graczyk, M. Taormina, A. Udalski, J. Storm, N. Nardetto, A. Gallenne, P. Kervella, I. Soszyński, M. Górski, P. Wielgórski, K.
Suchomska, P. Karczmarek, B. Zgirski
Improved physical parameters have been determined for the exotic system OGLE-LMC-CEP-1718 composed of two first-overtone Cepheids and a completely new model was obtained for the OGLE-LMC-CEP-1812
classical Cepheid. This is now the shortest period Cepheid for which the projection factor is measured. The typical accuracy of our dynamical masses and radii determinations is 1%.
Period-radius diagram. The FO Cepheids (blue) of our study lie between the canonical (C) and non-canonical (NC) theoretical P-R relations, but are more consistent with the latter. The F-mode (red)
Cepheids are consistent with the empirical relation. Color areas mark the theoretical relations with the rotation included for stars crossing the instability strip for the third time. For the FO
pulsators their positions for corresponding fundamental mode periods (PF ) are shown in grey. Radii of several Galactic Cepheids in this period range are also shown for comparison.
The radii of the six classical Cepheids follow period-radius relations in the literature. Our very accurate physical parameter measurements allow us to calculate a purely empirical, tight
period-mass-radius relation which agrees well with theoretical relations derived from non-canonical models. This empirical relation is a powerful tool to calculate accurate masses for single Cepheids
for which precise radii can be obtained from Baade-Wesselink-type analyses. The mass of the Type-II Cepheid κ Pav, 0.56 ± 0.08 M[☉], determined using this relation is in a very good agreement with
theoretical predictions.
Period – p-factor diagram. No correlation is seen neither for our eclipsing binary LMC Cepheids, nor for the Galactic ones. For MW Cepheids, the total uncertainty is shown in a lighter blue color,
while the statistical part is marked with a darker one. A histogram for all p-factor values is shown on the right.
We find large differences between the p-factor values derived for the Cepheids in our sample. Evidence is presented that a simple period–p-factor relation shows an intrinsic dispersion, hinting at
the relevance of other parameters, such as the masses, radii and radial velocity variation amplitudes. We also find evidence that the systematic blueshift exhibited by Cepheids, is primarily
correlated with their gravity. The companion star of the Cepheid in the OGLE-LMC-CEP-4506 system has a very similar temperature and luminosity and is clearly located inside the Cepheid instability
strip, yet it is not pulsating.
Location of binary Cepheids in the Large Magellanic Cloud.
Source: users.camk.edu.pl/pilecki | {"url":"https://araucaria.camk.edu.pl/index.php/2018/07/10/astrophysics-of-cepheids-in-eclipsing-binary-systems/","timestamp":"2024-11-11T03:40:27Z","content_type":"text/html","content_length":"58708","record_id":"<urn:uuid:796a775f-6528-4a8b-9325-cd260930be19>","cc-path":"CC-MAIN-2024-46/segments/1730477028216.19/warc/CC-MAIN-20241111024756-20241111054756-00058.warc.gz"} |
Repeated Measures Analysis
If there are multiple dependent variables and the variables represent repeated measurements of the same observational unit, then the variation among the dependent variables can be attributed to one
or more repeated measurement factors. The factors can be included in the model by specifying _RESPONSE_ on the right side of the MODEL statement and by using a REPEATED statement to identify the
Consider an experiment in which each subject is measured at three times, and the response functions are marginal probabilities for each of the dependent variables. If the dependent variables each
have k levels, then PROC CATMOD computes 1 response functions for each time. Differences among the response functions with respect to these times could be attributed to the repeated measurement
factor Time. To incorporate the Time variation into the model, specify the following statements:
proc catmod;
response marginals;
model t1*t2*t3=_response_;
repeated Time 3 / _response_=Time;
These statements produce a Time effect that has degrees of freedom since there are response functions at each time point. For a dichotomous variable, the Time effect has two degrees of freedom.
Now suppose that at each time point, each subject has X-rays taken, and the X-rays are read by two different radiologists. This creates six dependent variables that represent the cross-classification
of the repeated measurement factors Time and Reader. A saturated model with respect to these factors can be obtained by specifying the following statements:
proc catmod;
response marginals;
model r11*r12*r21*r22*r31*r32=_response_;
repeated Time 3, Reader 2
/ _response_=Time Reader Time*Reader;
If you want to fit a main-effects model with respect to Time and Reader, then change the REPEATED statement to the following:
repeated Time 3, Reader 2 / _response_=Time Reader;
If you want to fit a main-effects model for Time but for only one of the readers, the REPEATED statement might look like the following:
repeated Time $ 3, Reader $ 2
profile =('1' Smith,
'1' Jones,
'2' Smith,
'2' Jones,
'3' Smith,
'3' Jones);
If Jones had been unavailable for a reading at time 3, then there would be only response functions, even though PROC CATMOD would be expecting some multiple of 6 . In that case, the PROFILE= option
would be necessary to indicate which repeated measurement profiles were actually represented:
repeated Time $ 3, Reader $ 2
profile =('1' Smith,
'1' Jones,
'2' Smith,
'2' Jones,
'3' Smith);
When two or more repeated measurement factors are specified, PROC CATMOD presumes that the response functions are ordered so that the levels of the rightmost factor change most rapidly. This means
that the dependent variables should be specified in the same order. For this example, the order implied by the REPEATED statement is as follows, where the variable r corresponds to Time i and Reader
Response Dependent
Function Variable Time Reader
1 r 1 1
2 r 1 2
3 r 2 1
4 r 2 2
5 r 3 1
6 r 3 2
The order of dependent variables in the MODEL statement must agree with the order implied by the REPEATED statement.
When there are variables specified in the POPULATION statement or on the right side of the MODEL statement, these variables produce multiple populations. PROC CATMOD can then model these independent
variables, the repeated measurement factors, and the interactions between the two.
For example, suppose that there are five groups of subjects, that each subject in the study is measured at three different times, and that the dichotomous dependent variables are labeled t1, t2, and
t3. The following statements compute three response functions for each population:
proc catmod;
weight wt;
population Group;
response marginals;
model t1*t2*t3=_response_;
repeated Time / _response_=Time;
PROC CATMOD then regards _RESPONSE_ as a variable with three levels corresponding to the three response functions in each population and forms an effect with two degrees of freedom. The MODEL and
REPEATED statements tell PROC CATMOD to fit the main effect of Time.
In general, the MODEL statement tells PROC CATMOD how to integrate the independent variables and the repeated measurement factors into the model. For example, again suppose that there are five groups
of subjects, that each subject is measured at three times, and that the dichotomous independent variables are labeled t1, t2, and t3. If you use the same WEIGHT, POPULATION, RESPONSE, and REPEATED
statements as in the preceding program, the following MODEL statements result in the indicated analyses:
model t1*t2*t3=Group / averaged; Specifies the Group main effect (with 4 degrees of freedom)
model t1*t2*t3=_response_; Specifies the Time main effect (with 2 degrees of freedom)
model t1*t2*t3=_response_*Group; Specifies the interaction between Time and Group (with 8 degrees of freedom)
model t1*t2*t3=_response_|Group; Specifies both main effects, and the interaction between Time and Group (with a total of 14 degrees of freedom)
model t1*t2*t3=_response_(Group); Specifies a Time main effect within each Group (with 10 degrees of freedom)
However, the following MODEL statement is invalid since effects cannot be nested within _RESPONSE_:
model t1*t2*t3=Group(_response_); | {"url":"http://support.sas.com/documentation/cdl/en/statug/65328/HTML/default/statug_catmod_details22.htm","timestamp":"2024-11-12T08:47:24Z","content_type":"application/xhtml+xml","content_length":"35450","record_id":"<urn:uuid:8108e2c3-1a4b-4979-9e34-1b093c08100c>","cc-path":"CC-MAIN-2024-46/segments/1730477028249.89/warc/CC-MAIN-20241112081532-20241112111532-00333.warc.gz"} |
Alitajer S, Hajian M. EVALUATING GRAPH THEORY APPROACHES,CONVEX SPACE AND INTERSECTION ,TO ARCHITECTURAL SPATIAL ANALYSIS, CASE STUDY: JAHANBANI,NESHASTEPOUR AND AZADMANESH HOUSES,KASHAN. Naqshejahan
2017; 7 (2) :33-48
1- Assistant Professor/ Architecture Department/ Bu Ali Sina University , tajer1966@gmail.com
Abstract: (11864 Views)
A method for spatial analysis in architecture and urban planning for more than three decades, which has been widely used in graph theory, is the method of analyzing convex space. In contrast, one of
the methods of this theory, which is less used in architectural analysis, is the analysis of the intersection point. Although the intersection point method has several potential advantages to old
methods in graph theory, there has not been a convincing comparison between this method and other methods.
An analysis of the convex space for each plan yields useful information for qualitative visual analysis. The visual analysis allows researchers to quickly identify the spatial structure of a plan and
locate important functional spaces in relation to each other. For this purpose, a graph is usually drawn, that a single room or outer point is considered as the root of the graph. Such a graph is
called the justified plan graph (JPG). A justified plan graph based on the type of spatial structure of the plan is divided into two sets, if the graph is deep, it is like a tree, if the graph is
shallow, it is like a bush. Another common structure found in JPGs is the root-like spatial relationship, which is often seen in circular or looped plans. Root graphs have a very high flexibility or
permeability in the building.
Convex space analysis requires simplification of the plan in the form of a set of convex spaces called in the graph as nodes. There are several procedures for this process, which are presented in
three stages.
In the first stage, the rooms with four walls, the bedrooms or bathrooms, are defined as convex space. This is the first set of convex spaces that introduces spaces that there is no visual ambiguity
in their convexity. According to the contract, convex spaces of a dimension smaller than 300 mm are included in the largest contiguous space adjacent.
The second stage relates to non-convex spaces that are L-shaped or T-shaped. These rooms are divided in such a way that the least number of convex spaces with room function is created. If, after
division, the spaces have not the primary function, they must divide so that the convex spaces produced have the lowest ratio of the perimeter to the area. According to Hiller and Hanson’s view,
convex space contains the smallest and fattest space. These kind of spaces are more circular and therefore have less ratio of the perimeter to the area.
In the final stage, the division of other spaces that are not convex is done according to the previous step.
After these steps, the convex map is ready to enter the Depthmap software. In this software, convex space tools are used to draw spaces and create graph nodes. Then the linking tool is used to add
graph edges. The Depthmap software calculates the dimension of the graph theory for use in future analyzes.
Although the speed of the convex map production process is a significant advantage for some studies, it may not accurately analyze the location of a more precise points of the plan. For this purpose,
an alternative process is needed to summarize the plan and convert it into a graph. This method, called visibility graph, is applied to a grid that is placed on the plan so that each square of the
grid represents a node of the graph. Graph edges connect both squares that are able to see each other. Thus, a straight line from the center of each square of the network is drawn to the center of
each other visible square. This method is also an efficient method, but only when computer software is used. A kind of interaction between these two techniques - the visibility and convex space - is
seen in another rarely used method. This method is called intersection analysis method in a axial map.
The process used in this paper to produce axial maps is a protocol for linking multiple classes in which the axial lines are defined as the lines of movement instead of the lines of vision. In this
way, a line may begin from a point in a floor, move horizontally down the floor, and then go to the end of the floor, without passing through the stairs, but there is not necessarily a visual
connection between the two ends of the lines.
The first stage of the production of the intersection map begins with the identification of the points where the two main lines are interrupted and marked on the map with a circle. Then the file
containing the axial map and the intersection points is ready to enter the software. The Depthmap software does not have a preset tool for analyzing intersection points. Therefore, using the convex
space tool, each intersection point is considered as a node of the graph as a convex space, and it is manually connected to all points. Each node must be connected with at least two lines and
connected directly to each node on which two lines are located. After adding all the connections, the software will be able to calculate the theoretical dimensions of the graph.
The "endpoint" method is a kind of intersection point method that examines the end of each axial lines. To do this, a straight line from the end of each line should be drawn to all the planar visible
vertexes. If all of these vertexes are visible from intersection points, the end of the line does not have a unique surveillance feature and is considered an invalid location for the endpoint.
Otherwise, the two ends of the axial lines become nodes in the intersection of the graph. For these endpoints, a new node in the software is mapped. After adding all the connections, the software is
able to perform the relevant calculations.
In this research, three samples of Kashan's houses are analyzed with convex space analysis method and their results are compared with the analysis of the intersection point. For each of these three
houses, first, the convex space analysis is done and the mathematical results are calculated. When map is converted to graph For mathematical analysis of the relationship between intersections, the
paths in the original axial map are reversed. During the inversion process, two intersection point graphs can be generated, one that is entirely focused on the position of the intersection points
(called the intersection point graph), and the other contains stubs with unique surveillance features (A type of intersection point graph that is called the end node graph). From these two graphs,
the intersection point of mathematical values is extracted which can be compared with the results of the convex space analysis. Through these processes, the weaknesses and relative strengths of these
three methods are determined for the first time.
The result shows that the intersection point method is more effective in identifying the concept of space from the perspective of movement and routing than the convex space method, and also the
inclusion or non-inclusion of stubs have a tangible effect on the integration values. Finally, it can be said that the present research, while mainly applying and evaluating the two methods of graph
theory analysis, briefly describes examples of the valuable traditional architecture of Kashan.
Article Type:
Technical Article
| Subject:
Highperformance Architecture
Received: 2017/06/22 | Accepted: 2017/08/23 | Published: 2017/09/6
Send email to the article author | {"url":"https://bsnt.modares.ac.ir/article-2-529-en.html","timestamp":"2024-11-02T15:30:00Z","content_type":"application/xhtml+xml","content_length":"51084","record_id":"<urn:uuid:46ee0cf8-678f-4061-86c8-d295d6f9717b>","cc-path":"CC-MAIN-2024-46/segments/1730477027714.37/warc/CC-MAIN-20241102133748-20241102163748-00015.warc.gz"} |
Pool Table Trick Shot: Upper Secondary Mathematics Competition Question
Below is a diagram of a simple pool table. It has a length of 2 metres, a width of 1 metre and pockets at the four corners A, B, C and D. When a ball strikes a side of the pool table, it bounces off
at the same angle as it hit that side. A ball is initially at point P, 25 cm from pocket D along side AD. It is struck in such a way as to bounce off sides CD and BC and then land in pocket A.
At what distance from C does the ball strike the side BC?
Feel free to comment, ask questions and even check your answer in the comments box below powered by Disqus.
You can receive these questions directly to your email box or read them in an RSS reader. Subscribe using the links on the right. Don’t forget to follow Gifted Mathematics on Google+, Facebook or
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You can also subscribe to our Bookmarks on StumbleUpon and Pinterest. Many resources never make it onto the pages of Gifted Mathematics but are stored in these bookmarking websites to share with you.
No comments: | {"url":"http://www.giftedmathematics.com/2013/03/pool-table-trick-shot-upper-secondary.html","timestamp":"2024-11-13T22:21:36Z","content_type":"application/xhtml+xml","content_length":"82643","record_id":"<urn:uuid:f5c4327c-fb0a-4137-97c7-e62b380e2dee>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00855.warc.gz"} |
First Pass Yield vs. Roll ThroughPut Yield: Why RTY is better than FPY?First Pass Yield vs. Roll ThroughPut Yield: Why RTY is better than FPY?
First Pass Yield vs. Roll ThroughPut Yield: Why RTY is better than FPY?
5 min. read
In Lean training courses, the concept of yield is discussed as a measure of the quality of a process. Yield is an important term for 6 Sigma. There are two types of yield: first pass yield or FPY and
rolled throughput yield or RTY. Free Lean Six Sigma training debates whether FPY or RTY is the best measure of yield. This article compares rolled throughput yield vs first pass yield. Also, it
explains how to calculate each method according to the Six Sigma approach.
Attend our 100% Online & Self-Paced Free Six Sigma Training.
We will go through the Yield, also known as first-pass yield metric of Lean Six Sigma by looking at the following sections:
• Meaning and definition of yield
• First-pass yield or Throughput yield (FPY)
• Illustration – FPY
• Rolled-throughput Yield (RTY)
• RTY – Formula and Illustration (Method 1)
• RTY – Formula and Illustration (Method 2)
Definition of Yield in LEAN
Let’s talk about one of the key metrics in LEAN methodology. It is the concept of yield. Yield or first pass yield or throughput yield in LEAN is simply the ratio of good units produced to the number
of units entering the process. Here’s a reminder of the term ‘unit.’ A unit is any item that is being processed.
Two Types of Yield
First-pass yield (FPY)
There are different ways to define yield, or you can also call it as types of yield. The first one is the first-pass yield, which is abbreviated as FPY. It is also known first-time yield or
throughput yield. It is a unit-based metric. The disadvantage of the first-time yield is that it ignores or does not account for any rework of scrap products carrying one or multiple defects. In this
case, the calculation occurs after any “inspection” is conducted to determine whether a unit is good or not.
Let us have a look at an illustration on the first-time yield or throughput yield formula. We are looking at an email response process in a BPO company which works as a service provider for a
Utilities and Energy Company. We are looking at the number of emails i.e. 550; which was responded by a team of customer service associates. Before sending those emails to customers, all of them were
inspected, or quality checked. During the quality audit, 203 emails were found to be carrying numerous errors or defects. As a part of correction exercise, out of 203 emails, 190 were reworked and
sent back into the email process queue. Moreover, 13 emails were so badly responded that they could not be sent to the customer at all.
Therefore; the first time yield or throughput yield formula for this process would be 347 divided by 550. And this gives us 0.630 or 63.09%. How did we arrive at 347? It is the result of 550 (total
number of emails) minus 203 (defects). When we want to count the number of emails that were eventually sent to the customers we subtract the number of defectives (13) from the total number of emails.
Rolled-throughput Yield (RTY)
Let us now discuss the concept of rolled throughput yield. The acronym is RTY. What RTY means? RTY is a measure of the overall process quality level. It summarizes DPMO data for a process or product
by multiplying the DPMO of each process step. rolled-throughput yield is the preferred yield calculation method over the first-pass yield.
RTY is smaller than the lowest yield of any single process. Every process step will always have a minimum and maximum yield. As the number of process steps increases, the RTY becomes exponentially
Let us have a look at an illustration of RTY now. We’ll discuss two formulas to calculate RTY. Let’s have a look at the first one.
Illustrations of RTY
First method to calculate RTY
We’ve got 5 process steps in this example. Each step has got input units and output units. Input units are the ones entering each process step, and output units refer to good units produced by each
process step. With the help of a formula, we’ve got the first pass yield for each process step. To calculate RTY, multiply the FPY of each process steps to get the answer. In this case, your equation
will be the multiplication of five FPY values which will result in 0.6372. In other words, your RTY for all process steps is 63.72%. That was the first throughput yield formula.
Second method to calculate RTY
Let us have a look at the second method of RTY calculation. We’ve again got 5 process steps in this example. To use this method; you need to have two metrics i.e. DPMO and DPU calculated for each
process step.
We will now review the calculations to determine the DPU and DPMO.
Calculation of DPU
• Determine the number of units
• Count the number of defects
• Divide the number of defects by the number of units
Calculation of DPMO
• Determine the number of units
• Determine the number of defect opportunities per unit
• Determine the total number of defect opportunities for all the units by multiplying the number of defects per opportunity by the total number of units
• Count the units with defects in the total sample by counting how many opportunities within the sample group contained defects or errors
• Divide the total defects by the total opportunities and multiply by a million
Another simple way to calculate DPU
In this example, we’ve calculated DPU using another quick formula. We’ve divided DPMO figure by 1 million to get DPU for each process step. Why did we do that? DPMO incorporates 1 million defect
opportunities, and we need to know defect opportunities for each unit. That is why we divided DPMO by 1 million.
Calculating the RTY from the DPU
In the next column, we have deducted DPU from 1 to get the first-pass yield (FTY) for each process step. The 1 refers to 100%. The logic here is simple. The 100% yield for each process step minus
defect opportunities per-unit for each process step is equivalent to good units produced by each process step. In this method, the RTY is calculated by multiplying the FTY for each process as
determined in the previous step to get the RTY.
RTY is a realistic view of the yield of any process, looking at all the process steps. The RTY of a process is a good measure of the quality of a process. If the RTY is too low, then a
problem-solving team needs to investigate how the process can be improved. | {"url":"https://blog.masterofproject.com/rty/","timestamp":"2024-11-13T21:08:09Z","content_type":"text/html","content_length":"112665","record_id":"<urn:uuid:4e3674e5-f217-468b-8d5d-cb43b1514dab>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00592.warc.gz"} |
Design Overview
Design Overview¶
The most important entities in the mp-units library are:
The graph provided below presents how those and a few other entities depend on each other:
flowchart TD
Unit --- Reference
Dimension --- QuantitySpec["Quantity specification"]
quantity_character["Quantity character"] --- QuantitySpec
QuantitySpec --- Reference["Quantity reference"]
Reference --- Quantity
quantity_character -.- Representation
Representation --- Quantity
Quantity --- QuantityPoint["Quantity point"]
PointOrigin["Point origin"] --- QuantityPoint
click Dimension "#dimension"
click quantity_character "#quantity-character"
click QuantitySpec "#quantity-specification"
click Unit "#unit"
click Reference "#quantity-reference"
click Representation "#quantity-representation"
click Quantity "#quantity"
click PointOrigin "#point-origin"
click QuantityPoint "#quantity-point"
Dimension specifies the dependence of a quantity on the base quantities of a particular system of quantities. It is represented as a product of powers of factors corresponding to the base quantities,
omitting any numerical factor.
In the mp-units library, we use the terms:
For example:
• length (\(\mathsf{L}\)), mass (\(\mathsf{M}\)), time (\(\mathsf{T}\)), electric current (\(\mathsf{I}\)), thermodynamic temperature (\(\mathsf{Θ}\)), amount of substance (\(\mathsf{N}\)), and
luminous intensity (\(\mathsf{J}\)) are the base dimensions of the ISQ.
• A derived dimension of force in the ISQ is denoted by \(\textsf{dim }F = \mathsf{LMT}^{–2}\).
• The implementation of IEC 80000 in this library provides iec80000::dim_traffic_intensity base dimension to extend ISQ with strong information technology quantities.
Base dimensions can be defined by the user in the following way:
inline constexpr struct dim_length : base_dimension<"L"> {} dim_length;
inline constexpr struct dim_time : base_dimension<"T"> {} dim_time;
Derived dimensions are implicitly created by the library's framework based on the quantity equation provided in the quantity specification:
Users should not explicitly define any derived dimensions. Those should always be implicitly created by the framework.
The multiplication/division on quantity specifications also multiplies/divides their dimensions:
The dimension equation of isq::dim_length / isq::dim_time results in the derived_dimension<isq::dim_length, per<isq::dim_time>> type.
Quantity character¶
ISO 80000 explicitly states that quantities (even of the same kind) may have different characters:
The quantity character in the mp-units library is implemented with the quantity_character enumeration:
Quantity specification¶
Dimension is not enough to describe a quantity. This is why the ISO 80000 provides hundreds of named quantity types. It turns out that there are many more quantity types in the ISQ than the named
units in the SI.
This is why the mp-units library introduces a quantity specification entity that stores:
• quantity type/name,
• the quantity equation being the recipe to create this quantity (only for derived quantities that specify such a recipe).
We know that it might be sometimes confusing to talk about quantities, quantity types/names, and quantity specifications. However, it might be important to notice here that even the ISO 80000 admits
It is customary to use the same term, "quantity", to refer to both general quantities, such as length, mass, etc., and their instances, such as given lengths, given masses, etc. Accordingly, we
are used to saying both that length is a quantity and that a given length is a quantity by maintaining the specification – "general quantity, \(Q\)" or "individual quantity, \(Q_\textsf{a}\)" –
implicit and exploiting the linguistic context to remove the ambiguity.
In the mp-units library, we have a:
• quantity - implemented as a quantity class template,
• quantity specification - implemented with a quantity_spec class template that among others identifies a specific quantity type/name.
For example:
• isq::length, isq::mass, isq::time, isq::electric_current, isq::thermodynamic_temperature, isq::amount_of_substance, and isq::luminous_intensity are the specifications of base quantities in the
• isq::width, isq::height, isq::radius, and isq::position_vector are only a few of many quantities of a kind length specified in the ISQ.
• isq::area, isq::speed, isq::moment_of_force are only a few of many derived quantities provided in the ISQ.
Quantity specification can be defined by the user in one of the following ways:
The quantity equation of isq::length / isq::time results in the derived_quantity_spec<isq::length, per<isq::time>> type.
A unit is a concrete amount of a quantity that allows us to measure the values of quantities of the same kind and represent the result as a number being the ratio of the two quantities.
For example:
• si::second, si::metre, si::kilogram, si::ampere, si::kelvin, si::mole, and si::candela are the base units of the SI.
• si::kilo<si::metre> is a prefixed unit of length.
• si::radian, si::newton, and si::watt are examples of named derived units within the SI.
• non_si::minute is an example of a scaled unit of time.
• si::si2019::speed_of_light_in_vacuum is a physical constant standardized by the SI in 2019.
A unit can be defined by the user in one of the following ways:
template<PrefixableUnit auto U> struct kilo_ : prefixed_unit<"k", mag_power<10, 3>, U> {};
template<PrefixableUnit auto U> inline constexpr kilo_<U> kilo;
inline constexpr struct second : named_unit<"s", kind_of<isq::time>> {} second;
inline constexpr struct minute : named_unit<"min", mag<60> * second> {} minute;
inline constexpr struct gram : named_unit<"g", kind_of<isq::mass>> {} gram;
inline constexpr struct kilogram : decltype(kilo<gram>) {} kilogram;
inline constexpr struct newton : named_unit<"N", kilogram * metre / square(second)> {} newton;
inline constexpr struct speed_of_light_in_vacuum : named_unit<"c", mag<299'792'458> * metre / second> {} speed_of_light_in_vacuum;
The unit equation of si::metre / si::second results in the derived_unit<si::metre, per<si::second>> type.
Quantity reference¶
ISO defines a quantity as:
property of a phenomenon, body, or substance, where the property has a magnitude that can be expressed as a number and a reference
After that, it says:
A reference can be a measurement unit, a measurement procedure, a reference material, or a combination of such.
In the mp-units library, a quantity reference provides all the domain-specific metadata for the quantity besides its numerical value:
Together with the value of a representation type, it forms a quantity.
In the library, we have two different ways to provide a reference:
• every unit with the associated quantity kind is a valid reference,
• providing a unit to an indexing operator of a quantity specification explicitly instantiates a reference class template with this quantity spec and a unit passed as arguments.
All the units of the SI have associated quantity kinds and may serve as a reference.
For example:
• si::metre is defined in the SI as a unit of isq::length and thus can be used as a reference to instantiate a quantity of length (e.g., 42 * m).
• The expression isq::height[m] results with reference<isq::height, si::metre>, which can be used to instantiate a quantity of isq::height with a unit of si::metre (e.g., 42 * isq::height[m]).
Quantity representation¶
Quantity representation defines the type used to store the numerical value of a quantity. Such a type should be of a specific quantity character provided in the quantity specification.
By default, all floating-point and integral (besides bool) types are treated as scalars.
ISO defines a quantity as:
property of a phenomenon, body, or substance, where the property has a magnitude that can be expressed as a number and a reference
This is why a quantity class template is defined in the library as:
template<Reference auto R,
RepresentationOf<get_quantity_spec(R).character> Rep = double>
class quantity;
Its value can be easily created by multiplying/dividing the numerical value and a reference.
For example:
• All of 42 * m, 42 * si::metre, 42 * isq::height[m], and isq::height(42 * m) create a quantity.
• A quantity type can also be specified explicitly (e.g., quantity<si::metre, int>, quantity<isq::height[m]>).
Point origin¶
In the affine space theory, the point origin specifies where the "zero" of our measurement's scale is.
In the mp-units library, we have two types of point origins:
• absolute - defines an absolute "zero" for our point,
• relative - defines an origin that has some "offset" relative to an absolute point.
For example:
• the absolute point origin can be defined in the following way:
inline constexpr struct absolute_zero : absolute_point_origin<isq::thermodynamic_temperature> {} absolute_zero;
• the relative point origin can be defined in the following way:
inline constexpr struct ice_point : relative_point_origin<absolute_zero + 273.15 * kelvin> {} ice_point;
Quantity point¶
Quantity point implements a point in the affine space theory.
In the mp-units library, the quantity point is implemented as:
template<Reference auto R,
PointOriginFor<get_quantity_spec(R)> auto PO,
RepresentationOf<get_quantity_spec(R).character> Rep = double>
class quantity_point;
Its value can be easily created by adding/subtracting the quantity with a point origin.
For example:
• The following specifies a quantity point defined in terms of an ice_point provided in the previous example: | {"url":"https://mpusz.github.io/mp-units/2.1/users_guide/framework_basics/design_overview/","timestamp":"2024-11-08T11:56:48Z","content_type":"text/html","content_length":"85574","record_id":"<urn:uuid:fb9eca64-fdd9-45f4-81db-7a3217220b81>","cc-path":"CC-MAIN-2024-46/segments/1730477028059.90/warc/CC-MAIN-20241108101914-20241108131914-00623.warc.gz"} |
ipq4018: add initial IPQ4018/IPQ4019 support
Differential D32538
Authored by adrian on Oct 17 2021, 4:11 PM.
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Group Reviewers
This adds required IPQ4018/IPQ4019 SoC support to boot.
It also includes support for disabling the ARMv7 hardware
breakpoint / debug stuff at compile time as this is
required for the IPQ SoCs, and printing out the undefined
instruction itself.
• compiled/booted on an IPQ4019 SoC AP
Diff Detail
Event Timeline
There are a very large number of changes, so older changes are hidden. Show Older Changes
Comment Actions
Removed core from reviewer. This tripped over the GPL check in herald for SPDX lines, but the OR MIT means it's fine and doesn't require approval (which is why I removed core blocking rather than
approved it with my core hat on).
75 ↗ (On Diff # This is a NOP. If this is not a NOP, you have bigger issues than this will solve.
(On Diff # #endif
969 ↗ 97009) }
I think would be better. It will avoid new warnings.
3 IIR IRC correctly, this is a 64-bit part that Adrian is bringing up in 32-bit mode.
We generally don't keep these around.
Also, why can't you use GENERIC?
37 This is redundant, remove it. It's in std.qca.
(On Diff # We generally don't commit #if 0 code to the tree. If EARLY_PRINTF isn't right, then find a better way. Though this code looks very 'early system bringup-y rather than what we'd
110 ↗ 97009) normally commit to the tree since it's kinda fragile...
(On Diff # I'd ditch all rights reserved. I doubt that you have a need for the Buenos Aires Convention (now that it's almost 20 years expired)...
5 ↗ 97009) here and elsewhere.
35 ↗ (On Diff # why sysctl.h here?
62 ↗ (On Diff # why are these empty?
1034 (On Diff # Please put this in a more-specific option file. It has no business in the global one.
↗ 97009)
Oh, this is GPL'd? I missed it when I removed core from the review.
10 ↗ (On Diff # But apart from the comments, there's nothing GPL-y about it.
97009) And it likely should be part of the bits that manu pulls in from upstream.
1 ↗ (On Diff # same comment as manu's here: Either pull this in from upstream, or put it elsewhere.
This is a NOP. If this is not a NOP, you have bigger issues than this will solve.
75 ↗ (On Diff #97009) Ha yes :P
oh it was just me being explicit about it during bring-up. I can delete it if needed!
969 ↗ (On Diff #97009)
I think would be better. It will avoid new warnings.
ok, I'll go do that!
Please put this in a more-specific option file. It has no business in the global one.
1034 ↗ (On Diff #97009)
i'm open to suggestions for where to put it!
62 ↗ (On Diff #97009) If there's only one CPU you can remove these. The default implementation handles this case.
1034 ↗ (On Diff #97009) options.arm
110 ↗ (On Diff #97009) This review-comment is just wrong in this case. The code-comment block explains exactly why it must be #if 0, not #if EARLY_PRINTF.
3 it's a quad core A7, so 32 bit armv7 only!
30–34 I'll likely end up using this some more during SMP bring-up and some trustzone work once the rest of this stuff is up and going.
If there's only one CPU you can remove these. The default implementation handles this case.
62 ↗ (On Diff #97009)
They're placeholders for the other 3 cores I'm about to bring up once this stuff lands!
110 ↗ (On Diff #97009) No. It is right. If 0 is wrong and the comment is wrong on that point. If it is optional, it needs a new option name if EARLY_PRINTF isn't right.
110 ↗ (On Diff #97009) i can put another build option in there just to enable it when building this platform?
(On Diff # Since this is early bring up, this comment should have "should" read in the RFC sense: if there is a reason to keep it for a while and evolve it as the code matures, that's
110 ↗ 97009) cool too.
110 ↗ (On Diff # For now, it's fine. Let's wait until things are further along with other soc support for qualcomm
No, it is wrong. Look at the existing armv7 code that's checked in already. The SOCs that have maintainers all have early_putc code wrapped in #if 0. For a good reason. If you want to
go add soc-specific options for every soc so that #if 0 can be replaced with something like #ifdef RK3288_EARLY_PRINTF, then by all means knock yourself out. But fix the existing code
110 (On Diff before you start complaining about something not matching existing practice.
↗ #97009)
Also, this code is NOT just for one-time early bringup. It's for every time booting fails in early boot, like when something breaks in loader, or uboot. That's why we leave this stuff
in there wrapped in #if 0.
Does this not cause warnings for the various now-unused static functions like dbg_arch_supported?
963 (On Diff
↗ #97015) Also why ARMV7? The code below suggests this also exists for v6, so should just be ARM.
344 (On Diff If you're printing out the instruction bytes it's probably also worth printing out whether or not it's a thumb instruction? Otherwise you need to go stare at SPSR in the dumped
↗ #97015) trapframe to figure out how to even decode the thing.
(On Diff This seems to be done all over the place in sys/arm/conf, but I don't understand why, shouldn't bsd.cpu.mk be doing this bad on MACHINE_ARCH (or perhaps by specifying CPUTYPE here)?
7 ↗ #97015) No other architecture does this (well, except mips with its own NIH mips-specific ARCH_FLAGS variable, but we all know mips is not a good example to follow for pretty much anything).
28 (On Diff This will need addressing in a better way for GENERIC... that or figure out if there's way to make it work, I don't know the details here
↗ #97015)
65 (On Diff FIXME or XXX this and explain *why* the current code doesn't work. Or maybe we should fix the issue like I described on IRC rather than committing this kind of hack (so GENERIC
↗ #97015) kernels work on this board, for example...).
91 (On Diff I assume this is only needed for early printf? If so it's entirely unsurprising that things explode if you don't reserve it, and the comment could do a better job of pointing at it
↗ #97015) being that.
2 ↗ (On Diff This would normally be optional smp, and that looks like it should work here too
3 ↗ (On Diff Stray line
6 ↗ (On Diff This file no longer exists?
27 (On Diff Is this a local diff or upstream's untidy code?
↗ #97015)
27 ↗ (On Diff #97015) It's mine for now until i get the clock tree up, where i then /can/ reliably configure the baud rate.
344 ↗ (On Diff #97015) There's no trapframe echoed super early in boot, so all you get is this 32 bit value.
nope; the compiler isn't complaining about it.
963 ↗ (On Diff #
97015) I'll rename it.
7 ↗ (On Diff # Yeah, I noticed this too. We should likely review this (and the early printf stuff) later on and see if we can clean it up across the board.
28 ↗ (On Diff # Yeah I agree too. I'll have to go see if there's a way to probe this at boot time but I don't know if we have access to the needed memory without loading TZ modules.
65 ↗ (On Diff # ok, lemme update the comment. I'd like to fix this properly later :)
(On Diff # I assume this is only needed for early printf? If so it's entirely unsurprising that things explode if you don't reserve it, and the comment could do a better job of
91 ↗ 97015) pointing at it being that.
Nope, it's required to boot normally! That's what the comment is pointing out. It's actually not needed for early printf.
(On Diff #
91 ↗ 97015) I have a feeling there's something that has crept into the arm boot process which has been hidden because it seems /every/ platform uses devmaps that cover the initial console.
6 ↗ (On Diff # That's .. odd and let me see what the heck happened with my commit stack!
6 ↗ (On Diff # It's still in the commit stack; I am not sure why it is not showing up in the review!
Comment Actions
Do you need to commit arm/qualcomm/ipq4018_mp.c now ?
Since it's not needed as you don't support SMP (and don't even compile the file since it's gated by option smp).
This adds required IPQ4018/IPQ4019 SoC support to boot. It also includes support for disabling the ARMv7 hardware breakpoint / debug stuff at compile time as this is required for the IPQ SoCs, and
printing out the undefined instruction itself.
Removed core from reviewer. This tripped over the GPL check in herald for SPDX lines, but the OR MIT means it's fine and doesn't require approval (which is why I removed core blocking rather than
approved it with my core hat on).
75 ↗ (On Diff # This is a NOP. If this is not a NOP, you have bigger issues than this will solve.
(On Diff # #endif
969 ↗ 97009) }
I think would be better. It will avoid new warnings.
3 IIR IRC correctly, this is a 64-bit part that Adrian is bringing up in 32-bit mode.
We generally don't keep these around.
Also, why can't you use GENERIC?
37 This is redundant, remove it. It's in std.qca.
(On Diff # We generally don't commit #if 0 code to the tree. If EARLY_PRINTF isn't right, then find a better way. Though this code looks very 'early system bringup-y rather than what we'd
110 ↗ 97009) normally commit to the tree since it's kinda fragile...
(On Diff # I'd ditch all rights reserved. I doubt that you have a need for the Buenos Aires Convention (now that it's almost 20 years expired)...
5 ↗ 97009) here and elsewhere.
35 ↗ (On Diff # why sysctl.h here?
62 ↗ (On Diff # why are these empty?
1034 (On Diff # Please put this in a more-specific option file. It has no business in the global one.
↗ 97009)
Oh, this is GPL'd? I missed it when I removed core from the review.
10 ↗ (On Diff # But apart from the comments, there's nothing GPL-y about it.
97009) And it likely should be part of the bits that manu pulls in from upstream.
1 ↗ (On Diff # same comment as manu's here: Either pull this in from upstream, or put it elsewhere.
This is a NOP. If this is not a NOP, you have bigger issues than this will solve.
I think would be better. It will avoid new warnings.
IIR IRC correctly, this is a 64-bit part that Adrian is bringing up in 32-bit mode.
We generally don't commit #if 0 code to the tree. If EARLY_PRINTF isn't right, then find a better way. Though this code looks very 'early system bringup-y rather than what we'd normally commit to the
tree since it's kinda fragile...
I'd ditch all rights reserved. I doubt that you have a need for the Buenos Aires Convention (now that it's almost 20 years expired)... here and elsewhere.
Please put this in a more-specific option file. It has no business in the global one.
Oh, this is GPL'd? I missed it when I removed core from the review. But apart from the comments, there's nothing GPL-y about it. And it likely should be part of the bits that manu pulls in from
same comment as manu's here: Either pull this in from upstream, or put it elsewhere.
This is a NOP. If this is not a NOP, you have bigger issues than this will solve.
75 ↗ (On Diff #97009) Ha yes :P
oh it was just me being explicit about it during bring-up. I can delete it if needed!
969 ↗ (On Diff #97009)
I think would be better. It will avoid new warnings.
ok, I'll go do that!
Please put this in a more-specific option file. It has no business in the global one.
1034 ↗ (On Diff #97009)
i'm open to suggestions for where to put it!
oh it was just me being explicit about it during bring-up. I can delete it if needed!
#else ... #endif } I think would be better. It will avoid new warnings.
62 ↗ (On Diff #97009) If there's only one CPU you can remove these. The default implementation handles this case.
1034 ↗ (On Diff #97009) options.arm
If there's only one CPU you can remove these. The default implementation handles this case.
110 ↗ (On Diff #97009) This review-comment is just wrong in this case. The code-comment block explains exactly why it must be #if 0, not #if EARLY_PRINTF.
This review-comment is just wrong in this case. The code-comment block explains exactly why it must be #if 0, not #if EARLY_PRINTF.
3 it's a quad core A7, so 32 bit armv7 only!
30–34 I'll likely end up using this some more during SMP bring-up and some trustzone work once the rest of this stuff is up and going.
If there's only one CPU you can remove these. The default implementation handles this case.
62 ↗ (On Diff #97009)
They're placeholders for the other 3 cores I'm about to bring up once this stuff lands!
it's a quad core A7, so 32 bit armv7 only!
I'll likely end up using this some more during SMP bring-up and some trustzone work once the rest of this stuff is up and going.
They're placeholders for the other 3 cores I'm about to bring up once this stuff lands!
110 ↗ (On Diff #97009) No. It is right. If 0 is wrong and the comment is wrong on that point. If it is optional, it needs a new option name if EARLY_PRINTF isn't right.
No. It is right. If 0 is wrong and the comment is wrong on that point. If it is optional, it needs a new option name if EARLY_PRINTF isn't right.
110 ↗ (On Diff #97009) i can put another build option in there just to enable it when building this platform?
i can put another build option in there just to enable it when building this platform?
(On Diff # Since this is early bring up, this comment should have "should" read in the RFC sense: if there is a reason to keep it for a while and evolve it as the code matures, that's cool
110 ↗ 97009) too.
110 ↗ (On Diff # For now, it's fine. Let's wait until things are further along with other soc support for qualcomm
Since this is early bring up, this comment should have "should" read in the RFC sense: if there is a reason to keep it for a while and evolve it as the code matures, that's cool too.
For now, it's fine. Let's wait until things are further along with other soc support for qualcomm
No, it is wrong. Look at the existing armv7 code that's checked in already. The SOCs that have maintainers all have early_putc code wrapped in #if 0. For a good reason. If you want to go
add soc-specific options for every soc so that #if 0 can be replaced with something like #ifdef RK3288_EARLY_PRINTF, then by all means knock yourself out. But fix the existing code
110 (On Diff before you start complaining about something not matching existing practice.
↗ #97009)
Also, this code is NOT just for one-time early bringup. It's for every time booting fails in early boot, like when something breaks in loader, or uboot. That's why we leave this stuff in
there wrapped in #if 0.
No, it is wrong. Look at the existing armv7 code that's checked in already. The SOCs that have maintainers all have early_putc code wrapped in #if 0. For a good reason. If you want to go add
soc-specific options for every soc so that #if 0 can be replaced with something like #ifdef RK3288_EARLY_PRINTF, then by all means knock yourself out. But fix the existing code before you start
complaining about something not matching existing practice.
Also, this code is NOT just for one-time early bringup. It's for every time booting fails in early boot, like when something breaks in loader, or uboot. That's why we leave this stuff in there
wrapped in #if 0.
Does this not cause warnings for the various now-unused static functions like dbg_arch_supported?
963 (On Diff
↗ #97015) Also why ARMV7? The code below suggests this also exists for v6, so should just be ARM.
344 (On Diff If you're printing out the instruction bytes it's probably also worth printing out whether or not it's a thumb instruction? Otherwise you need to go stare at SPSR in the dumped
↗ #97015) trapframe to figure out how to even decode the thing.
(On Diff This seems to be done all over the place in sys/arm/conf, but I don't understand why, shouldn't bsd.cpu.mk be doing this bad on MACHINE_ARCH (or perhaps by specifying CPUTYPE here)? No
7 ↗ #97015) other architecture does this (well, except mips with its own NIH mips-specific ARCH_FLAGS variable, but we all know mips is not a good example to follow for pretty much anything).
28 (On Diff This will need addressing in a better way for GENERIC... that or figure out if there's way to make it work, I don't know the details here
↗ #97015)
65 (On Diff FIXME or XXX this and explain *why* the current code doesn't work. Or maybe we should fix the issue like I described on IRC rather than committing this kind of hack (so GENERIC kernels
↗ #97015) work on this board, for example...).
91 (On Diff I assume this is only needed for early printf? If so it's entirely unsurprising that things explode if you don't reserve it, and the comment could do a better job of pointing at it
↗ #97015) being that.
2 ↗ (On Diff This would normally be optional smp, and that looks like it should work here too
3 ↗ (On Diff Stray line
6 ↗ (On Diff This file no longer exists?
27 (On Diff Is this a local diff or upstream's untidy code?
↗ #97015)
Does this not cause warnings for the various now-unused static functions like dbg_arch_supported?
Also why ARMV7? The code below suggests this also exists for v6, so should just be ARM.
If you're printing out the instruction bytes it's probably also worth printing out whether or not it's a thumb instruction? Otherwise you need to go stare at SPSR in the dumped trapframe to figure
out how to even decode the thing.
This seems to be done all over the place in sys/arm/conf, but I don't understand why, shouldn't bsd.cpu.mk be doing this bad on MACHINE_ARCH (or perhaps by specifying CPUTYPE here)? No other
architecture does this (well, except mips with its own NIH mips-specific ARCH_FLAGS variable, but we all know mips is not a good example to follow for pretty much anything).
This will need addressing in a better way for GENERIC... that or figure out if there's way to make it work, I don't know the details here
FIXME or XXX this and explain *why* the current code doesn't work. Or maybe we should fix the issue like I described on IRC rather than committing this kind of hack (so GENERIC kernels work on this
board, for example...).
I assume this is only needed for early printf? If so it's entirely unsurprising that things explode if you don't reserve it, and the comment could do a better job of pointing at it being that.
This would normally be optional smp, and that looks like it should work here too
27 ↗ (On Diff #97015) It's mine for now until i get the clock tree up, where i then /can/ reliably configure the baud rate.
It's mine for now until i get the clock tree up, where i then /can/ reliably configure the baud rate.
344 ↗ (On Diff #97015) There's no trapframe echoed super early in boot, so all you get is this 32 bit value.
There's no trapframe echoed super early in boot, so all you get is this 32 bit value.
nope; the compiler isn't complaining about it.
963 ↗ (On Diff #
97015) I'll rename it.
7 ↗ (On Diff # Yeah, I noticed this too. We should likely review this (and the early printf stuff) later on and see if we can clean it up across the board.
28 ↗ (On Diff # Yeah I agree too. I'll have to go see if there's a way to probe this at boot time but I don't know if we have access to the needed memory without loading TZ modules.
65 ↗ (On Diff # ok, lemme update the comment. I'd like to fix this properly later :)
(On Diff # I assume this is only needed for early printf? If so it's entirely unsurprising that things explode if you don't reserve it, and the comment could do a better job of pointing
91 ↗ 97015) at it being that.
Nope, it's required to boot normally! That's what the comment is pointing out. It's actually not needed for early printf.
(On Diff #
91 ↗ 97015) I have a feeling there's something that has crept into the arm boot process which has been hidden because it seems /every/ platform uses devmaps that cover the initial console.
6 ↗ (On Diff # That's .. odd and let me see what the heck happened with my commit stack!
6 ↗ (On Diff # It's still in the commit stack; I am not sure why it is not showing up in the review!
Yeah, I noticed this too. We should likely review this (and the early printf stuff) later on and see if we can clean it up across the board.
Yeah I agree too. I'll have to go see if there's a way to probe this at boot time but I don't know if we have access to the needed memory without loading TZ modules.
ok, lemme update the comment. I'd like to fix this properly later :)
Nope, it's required to boot normally! That's what the comment is pointing out. It's actually not needed for early printf.
I have a feeling there's something that has crept into the arm boot process which has been hidden because it seems /every/ platform uses devmaps that cover the initial console. :)
That's .. odd and let me see what the heck happened with my commit stack!
It's still in the commit stack; I am not sure why it is not showing up in the review!
Do you need to commit arm/qualcomm/ipq4018_mp.c now ? Since it's not needed as you don't support SMP (and don't even compile the file since it's gated by option smp). | {"url":"https://reviews.freebsd.org/D32538","timestamp":"2024-11-01T22:26:11Z","content_type":"text/html","content_length":"223708","record_id":"<urn:uuid:75fd713e-46b0-4804-b6a5-26ee1284d1b3>","cc-path":"CC-MAIN-2024-46/segments/1730477027599.25/warc/CC-MAIN-20241101215119-20241102005119-00508.warc.gz"} |
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Multiplication Table 31 Chart Archives - Multiplication Table Chart
A 31 Multiplication Table is a mathematical chart that helps students memorize and understand multiplication. The 31 multiplication table is a free printable chart that can help students learn and
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31 multiplication table
It is essential for students to memorize their multiplication tables up to at least the number 31. The 31 multiplication table is one of the most important ones for students to know, as it contains
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Times Table 31
If you’re just getting started with learning multiplication tables, you might be wondering what all of the numbers mean. The numbers in the multiplication table are actually the product of two
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Multiplication chart 31
If you’re just starting to learn your multiplication tables, a good way to start is by using a multiplication chart. This will help you see the patterns in the numbers and memorize them more easily.
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Multiplication Table 31 Chart
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What is: Regression Analysis
What is Regression Analysis?
Regression analysis is a powerful statistical method used for examining the relationship between two or more variables. It allows researchers and analysts to understand how the typical value of the
dependent variable changes when any one of the independent variables is varied while the other independent variables are held fixed. This technique is widely utilized in various fields, including
economics, biology, engineering, and social sciences, to make predictions, identify trends, and inform decision-making processes.
Types of Regression Analysis
There are several types of regression analysis, each suited for different types of data and research questions. The most common types include linear regression, multiple regression, logistic
regression, and polynomial regression. Linear regression focuses on modeling the relationship between a single independent variable and a dependent variable, while multiple regression extends this
concept to include multiple independent variables. Logistic regression is used when the dependent variable is categorical, often to predict binary outcomes. Polynomial regression, on the other hand,
is employed when the relationship between the variables is nonlinear, allowing for more complex modeling of data trends.
The Importance of the Regression Equation
At the heart of regression analysis lies the regression equation, which mathematically describes the relationship between the variables. In a simple linear regression, the equation takes the form of
Y = a + bX, where Y represents the dependent variable, X is the independent variable, a is the y-intercept, and b is the slope of the line. This equation provides a predictive framework that can be
used to estimate the value of Y for any given value of X. Understanding the regression equation is crucial for interpreting the results of the analysis and for making informed predictions based on
the model.
Assumptions of Regression Analysis
For regression analysis to yield valid results, certain assumptions must be met. These include linearity, independence, homoscedasticity, normality, and no multicollinearity among independent
variables. Linearity assumes that the relationship between the independent and dependent variables is linear. Independence means that the residuals (errors) of the model should not be correlated.
Homoscedasticity requires that the variance of residuals is constant across all levels of the independent variable. Normality assumes that the residuals are normally distributed. Lastly,
multicollinearity refers to the situation where independent variables are highly correlated, which can distort the results of the regression analysis.
Interpreting Regression Coefficients
The coefficients obtained from a regression analysis provide valuable insights into the strength and direction of the relationships between variables. A positive coefficient indicates that as the
independent variable increases, the dependent variable also tends to increase, while a negative coefficient suggests an inverse relationship. The magnitude of the coefficient reflects the size of the
effect that the independent variable has on the dependent variable. Additionally, the significance of these coefficients is assessed using p-values, which help determine whether the observed
relationships are statistically significant or could have occurred by chance.
Goodness of Fit in Regression Analysis
Goodness of fit is a critical aspect of regression analysis that measures how well the regression model explains the variability of the dependent variable. The most commonly used metric for assessing
goodness of fit is the R-squared value, which ranges from 0 to 1. An R-squared value close to 1 indicates that a large proportion of the variance in the dependent variable is explained by the
independent variables, while a value close to 0 suggests a poor fit. Other metrics, such as adjusted R-squared, root mean square error (RMSE), and Akaike information criterion (AIC), are also used to
evaluate the performance of regression models.
Applications of Regression Analysis
Regression analysis has a wide range of applications across various domains. In business, it is often used for sales forecasting, market research, and financial modeling. In healthcare, regression
models can help identify risk factors for diseases and evaluate the effectiveness of treatments. In social sciences, researchers use regression analysis to study relationships between socioeconomic
factors and outcomes such as education and employment. The versatility of regression analysis makes it an essential tool for data-driven decision-making in numerous fields.
Limitations of Regression Analysis
Despite its strengths, regression analysis has limitations that researchers must consider. One significant limitation is the potential for overfitting, where a model becomes overly complex and
captures noise rather than the underlying relationship. Additionally, regression analysis assumes that the relationships between variables are linear, which may not always be the case. Outliers can
also significantly impact the results, leading to misleading conclusions. Therefore, it is essential to conduct thorough diagnostics and validation of regression models to ensure their reliability
and accuracy.
In summary, regression analysis is a fundamental statistical technique that provides insights into the relationships between variables. By understanding its types, assumptions, and applications,
researchers and analysts can leverage this powerful tool to make informed decisions and predictions based on data. | {"url":"https://statisticseasily.com/glossario/what-is-regression-analysis/","timestamp":"2024-11-11T20:27:08Z","content_type":"text/html","content_length":"139226","record_id":"<urn:uuid:aa27c28e-1bce-4573-95c2-5b6610fb9c4c>","cc-path":"CC-MAIN-2024-46/segments/1730477028239.20/warc/CC-MAIN-20241111190758-20241111220758-00297.warc.gz"} |
Computer design of integrated circuits in the microwave range
The principles of a new theory of electrical circuits with high-frequency currents are set forth, and the application of this theory to the computer design of integrated circuits is described. It is
shown how the calculation of a given circuit can be reduced to the solving of an integral equation containing only curvilinear integrals along the circuit loop. Algorithms are presented for the
evaluation of this integral and some of its variants.
Upravliaiushchie Sistemy i Mashiny
Pub Date:
October 1974
□ Computer Aided Design;
□ Integrated Circuits;
□ Microwave Circuits;
□ Network Synthesis;
□ Algorithms;
□ Differential Equations;
□ Integral Equations;
□ Quadrupoles;
□ Electronics and Electrical Engineering | {"url":"https://ui.adsabs.harvard.edu/abs/1974UpSM.......110T/abstract","timestamp":"2024-11-12T20:31:44Z","content_type":"text/html","content_length":"33494","record_id":"<urn:uuid:58df038d-8afd-406d-b7c5-48d04a14a19b>","cc-path":"CC-MAIN-2024-46/segments/1730477028279.73/warc/CC-MAIN-20241112180608-20241112210608-00701.warc.gz"} |
The scaling limit of random outerplanar maps
Third take on a talk about outerplanar maps, and perhaps the most complete account I ever gave of this paper; the slides are the same those of the Journée YSP, with an extra section that describes
the core algorithm. Sorry about the annoying video format; an incredibly heavy pdf file is available here.
A planar map is outerplanar if all of its vertices belong to a single (outer)face. The scaling limit for various classes of large planar maps has been shown to be the so-called “Brownian Map”;
under certain conditions, however, different asymptotic behaviours may emerge, and some classes of planar maps with a macroscopic face admit Aldous’ CRT, or a scalar multiple thereof, as the
scaling limit. We shall see that this is the case for outerplanar maps as well: using a bijection by Bonichon, Gavoille and Hanusse one can show that uniform outerplanar maps with n vertices,
suitably rescaled by a factor $1/\sqrt{n}$, converge to $7\sqrt{2}/9$ times the CRT. | {"url":"http://alessandracaraceni.altervista.org/?p=309","timestamp":"2024-11-03T22:19:26Z","content_type":"text/html","content_length":"23584","record_id":"<urn:uuid:8d1d295b-d422-4855-8c70-276597f5e3b2>","cc-path":"CC-MAIN-2024-46/segments/1730477027796.35/warc/CC-MAIN-20241103212031-20241104002031-00155.warc.gz"} |
Printable Math Flashcards
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Normal series
The first step towards proving the o-minimality of $\Ran$ is to show that the quantifier-free definable sets have finitely many connected components. As discussed in this post, this means
(essentially) that we need to show that basic $\Pc{R}{X}$-sets have finitely many connected components.
Example 1
Let $\alpha \in \NN^n$ and $r \in (0,\infty)^n$. Show that the sets $\set{x \in \RR^n:\ x^\alpha = 0} \cap B(r)$ has finitely many connected components–which and how many?
Let $F \in \Pc{R}{X}_r$ be such that $F(0)=0$. Show that $\lim_{s \to 0} \|F\|_s = 0$.
Example 2
Let $F \in \Pc{R}{T}$, where $T$ is a single indeterminate. Then, by Exercise 13.3, for any $s < r$ we have $F = T^k \cdot G$, where $k = \ord(F)$ and $G \in \Pc{R}{T}_s$ is such that $G(0) \ne 0$.
This means that $G(T) = a + H(T)$, for some $a \ne 0$ and $H \in \Pc{R}{T}_s$ with $H(0) = 0$. So by the exercise above, there exists $s>0$ such that $G(t) \ne 0$ for $t \in [-s,s]$. It follows that,
for $t \in [-s,s]$, we have $$F(t) = 0 \quad\text{iff}\quad t^k = 0;$$ that is, zeroset of $F$ inside $[-s,s]$ is the same as that of the monomial $t^k$.
Exercise 14
Let $\alpha_1, \dots, \alpha_k \in \NN^n$, $\sigma:\{1, \dots, k\} \into \{-1,0,1\}$ and $r \in (0,\infty)^n$. Show that the set $$B\left(r,X^{\alpha_1}, \dots, X^{\alpha_k}, \sigma\right):= \set{x \
in B(r):\ \sign\left(x^{\alpha_i}\right) = \sigma(i)}$$ has finitely many connected components.
By Exercise 14, the procedure of Example 2 generalizes to the following series in several indeterminates:
A series $F \in \Ps{R}{X}$ is normal if there exist $\alpha \in \NN^n$ and $G \in \Ps{R}{X}$ such that $G(0) \ne 0$ and $F = X^\alpha \cdot G$.
Let $F_1, \dots, F_k \in \Pc{R}{X}$ be normal and $\sigma:\{1, \dots, k\} \into \{-1,0,1\}$. Then there exists a polyradius $r$ such that the set $$B\left(r,F_1, \dots, F_k, \sigma\right):= \set{x \
in B(r):\ \sign\left(F_i(x)\right) = \sigma(i)}$$ has finitely many connected components. $\qed$
While every series in one indeterminate is normal, the polynomial $X_1 + X_2$ is not normal.
Example 3
The formal substitution $(X_1,X_2) \mapsto (X_1, X_1 X_2)$ changes the polynomial $X_1+X_2$ to the polynomial $X_1 + X_1X_2$, which is normal. On the other hand, inside the right half-plane $\{x_1 >
0\}$, the zeroset of $x_1+x_2$ is the image of the zeroset of $x_1 + x_1x_2$ under the homeomorphism $(x_1,x_2) \mapsto (x_1,x_1x_2)$.
So, by the previous corollary with $F_1 = X_1$ and $F_2 = X_1 + X_1X_2$, the zeroset of $x_1+x_2$ inside the right half-plane $\{x_1>0\}$ has finitely many connected components.
Finally, inside the line $\{x_1=0\}$, the zeroset of $x_1+x_2$ is just $\{0\}$, which is connected, while the zeroset of $x_1+x_2$ inside the left half-plane $\{x_1<0\}$ has finitely many connected
component by a symmetric argument as for the right half-plane. Example 3 suggests the following: Strategy
In the general situation we try, as in Example 3, to find a substitution given by some convergent power series, such that the given series become normal after substitution, and such that the function
defined by the substitution is a homeomorphism outside a proper coordinate subspace. Inside this proper coordinate subspace, the problem of counting connected components is reduced to fewer
indeterminates, which suggests we proceed by induction on the number of indeterminates.
Strategies of this kind are known as normalization algorithms. The one we shall use here is inspired by Bierstone and Milman’s algorithm and can be found here.
Algorithms of this kind fall under the general heading of resolution of singularities, as discussed by Bierstone and Milman in this paper.
6 thoughts on “Normal series”
1. For Exercise 14 I have been sketching an argument which is difficult to write down. It goes something like this:
Divide $B(r)$ into sub-blocks of the form:
1) $2^n$ different blocks in which each $x_i$ is either positive (on the whole block) or negative (on the whole block).
2) When $x_i=0$, we have $2^{n-1}$ different blocks where each $x_j$ with $j \ne i$ is either positive or negative.
3) When $x_i=x_j=0$ for fixed $i \ne j$, we have $2^{n-2}$ different blocks where each $x_k$ with $i \ne k \ne j$ is either positive or negative.
n+1) When $x_1=x_2=…=x_n=0$ we have one ($2^0$) block, the origin.
This is finitely many disjoint (path-)connected sets which partition $B(r)$. Also, if any two blocks are “adjacent” (the distance between the sets is 0), then their union is a (path-)connected
Now it seems clear that if a point in one of these “blocks” satisfies the sign conditions given (sign$(x^{\alpha_i})=\sigma(i)$ for each $\alpha_i$), then the whole block satisfies the sign
This should bound the number of connected components by the number of “blocks” defined above (although actually the bound should be quite a bit lower- $2^n$?).
The difficulty stems from the fact that some of the entries in the tuple $\alpha_i$ may be zero. Does this suggest an inductive approach on the number of zero-entries in $\alpha_i$?
1. A couple of revisions to my earlier comment.
Firstly, I don’t know why I was so set on using an induction- this argument should work without anything else.
Secondly, the definition of “adjacent” which I mentioned is no good. In fact, every one of the blocks has distance 0 from any other block. A better definition would be: $A$ and $B$ are
“adjacent” if either $B \subseteq \overline{A}$ or $A \subseteq \overline{B}$.
Thirdly, I don’t know why I struggled to count the number of blocks defined. There are precisely $3^n$ different blocks (3 choices for each coordinate $x_i$). The bound should still only be
$2^n$ though because the union of two adjacent blocks forms one connected component.
1. The answer—that is, that the number of components of any of the sets in Exercise 14 is bounded by $3^n$—is correct. Without worrying about a better bound, how can you rewrite your
counting into a coherent proof of this statement?
1. Consider $x=(x_1,…,x_n) \in B(r)$. The regions described are those in which the sign of each component $x_i$ is constant. There are three possibilities for each component (positive,
negative or 0) and $n$ different components which means $3^n$ different blocks.
This can also be recovered by my original (overcomplicated) counting of blocks by noticing that we have $2^n+ {n \choose 1}2^{n-1} + … + {n \choose n}2^0$ which is the binomial
expansion of $(2+1)^n$.
1. But why does this answer Exercise 14…?
2. The proof should have gone like this:
Let $A:= \{ x \in B(r) \ |$ sign $x^{\alpha_i}=\sigma(i)$ for each $i \}$.
Write $A=(A \cap B_1) \cup … \cup (A \cap B_{3^n})$ where $B_i$ is an enumeration of the blocks defined (in some order). Each $A \cap B_i$ is either empty or just $B_i$. The adjacent $B_i$’s
swallow each other and what you’re left with is a decomposition of $A$ into disjoint connected components.
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2.45 GHz High Gain Self-Oscillating Mixer for Simple Short-Range Doppler Radar
Modern microwave communication systems have to comply with hard requirements for small size, low cost and reduced power consumption. In order to obtain a final system with such specifications, a
commonly used approach is the combination of some functionalities of the system on a single circuit, reducing the number of components and the final cost of the system.^1
Doppler radars are widely used for various purposes, including near distance motion detections. Examples include vehicle equipment, measurement of water surface velocity, detection of acoustic
vibrations, non-contact cardiopulmonary monitoring and proximity motion detection in which the system's microwave front-end consists of several components. Some examples are one or two antennas, a
local oscillator, one or two hybrids and/or an isolator, a frequency mixer and bandpass filters.
In order to reduce the number of components, hence to save on the size and cost of the Doppler radar, a self-oscillating mixer (SOM) could be adopted.^2 There are various advantages related to the
self-oscillating mixer technique: cost is reduced, due to a lowered component count and thus higher reliability, the more compact solution offers easier integration into monolithic microwave
integrated circuits (MMIC) and the total power consumption is lowered.^3
The performance limits in Doppler radar sensing depend mainly on mixer conversion loss at baseband, and phase noise of the local oscillator signal in the RF band.^4 Hence, in this work, a Doppler
radar is designed, based on SOM, for short-distance motion detections, with simplicity, low cost and low emitting power. For further reducing the cost and complexity with the least degradation of
performance related to the SOM's phase noise, dielectric resonator is replaced with a new slotted-square-patch resonator (SSPR), which is easy to fabricate. For maximizing the conversion gain of the
SOM, an optimization technique based on constant gain curves is carried out. The high conversion gain is helpful in reducing the emitted power from the Doppler radar.
Figure 1 Layout of the designed SSPR.
Microstrip SSPR
Phase noise is one of the most important performance parameters of an oscillator. For a lower phase noise, a dielectric resonator (DR) is most frequently used in microwave frequencies because of its
high quality factor. The DR however has higher cost and additional difficulties in properly positioning it on a post with the exact coupling factor.
Figure 1 is the layout of the SSPR, which is a microstrip patch type resonator with the effective wavelength of nλ/2 and a high Q-factor. The resonator is designed by inserting the successive slot
lines of the ratio of r = N/(N1) in the square patch, denoted as (a), (b), (c) and (d) in the figure. Hence, the lengths of (a) to (d) are L×r, L×r^2, L×r^3 and L×r^4, respectively. The slot lines
start from the center of one side, L, of the square patch and with the length reduced by the same ratio, r, until it reaches a given slot number or a minimum limited length that can be manufactured.
Here, only four successive slots are used.
Figure 2 Resonance frequency and quality factor of the SSPRs, according to the slot ratio
According to the chosen integer N, from 1 to 4 for L = λ/5, four resonators with one λ electrical length were designed. Figure 2 depicts the results of simulations and measurements. The total slot
length increases and the line microstrip width decreases with N increasing, hence the resonance frequency decreases and the Q-factor decreases. The measured unloaded Q-factor was 154.1 for N = 1,
which is approximately a 13.6 percent improved value, compared to that of the conventional hair-pin resonator.^5 The SSPR's improved Q-factor is due to the adoption of a slotted-patch type resonator,
which can inherently maximize use of the circuit space and also to use the wider line width of middle part in length of the resonator in which current flow is maximized. The simulations were
performed using a commercial 3D electromagnetic simulator, HFSS™. The substrate used is Teflon with a thickness of 0.762 mm, a relative dielectric constant of 3.48 and loss tangent of 0.003.
Figure 3 RF layout of the designed SOM Doppler radar.
SOM Design
The RF-layout for the proposed SOM or SOM Doppler radar circuit is shown in Figure 3. A 50 Ω termination at the left end of the gate transmission line, the self-bias circuit and the IF bandpass
filter at the drain transmission line are not shown in the figure. The circuit was simulated using a commercial circuit simulator ADS™ with an HB-simulation test bench. A well-known transistor,
NE3508M04, was self-biased near the pinch-off region (V[DD] = 3 V, V[DS] = 2 V, I[DS] = 3 mA, V[GS] = -0.4 V).
Figure 4 Circuit model for a two-port transistor oscillator.
A transistor oscillator with a one-port negative-resistance is effectively created by terminating a potentially unstable transistor with an impedance designed to drive the device in an unstable
region. In the circuit model for a two-port transistor oscillator of Figure 4, the input and output reflection coefficients are generally given by Equations 1 and 2.^6 Then the input and output
stability circles can be drawn as Figure 5, using the conditions of |Γ[in]| = 1 and |Γ[out]| = 1.^6
Figure 5 The input and output stability circles.
The source transmission line L1 in Figure 3 was adjusted for a wider unstable region. The unstable region is inside of the input stability circle and outside of the output stability circle, which is
region (b) in Figure 5.
On the other hand, the conversion gain, G[c], for a mixer is generally defined by Equation 3. Here, P[RF] and P[IF] are the powers of the received RF signal and the corresponding IF signal,
respectively. In order to obtain the maximally optimized G[c], constant conversion gain circles are drawn, which are similar to those of amplifiers, according to Γ[T].
Figure 6 Constant conversion gain curves of the designed SOM Doppler radar.
The gap, g, between the SSPR and the 50 Ω gate transmission lines, and L[2], were changed so that the Γ[T] can be moved within the unstable region (b) of Figure 6. Then a conversion gain point for
each change of Γ[T] can be obtained. A systematic investigation of 15 points for the pairs of Γ[T] and conversion gain, G[c], using a circuit simulation with ADS, gives the rough constant conversion
gain curves outlined in the figure. For the simulations, the RF input power was assumed to be -50 dBm at the antenna port. The maximum conversion gain can be seen to be approximately 24.5 dB and at
this point, the related parameters were g = 0.1 mm and L[2] = 11.0 mm.
Figure 7 Simulated conversion gain as a function of RF input power.
The change in the simulated optimum conversion gain of the SOM as a function of the RF input power is shown in Figure 7. The variation of the conversion gain is only 0.9 dB for RF input powers from
-50 to -100 dBm, assuming a short distance motion detector.
Figure 8 Photograph of the fabricated SOM Doppler radar.
Measured Results
The optimized SOM for Doppler radar with a SSPR was fabricated as shown in Figure 8. The SSPR is 13.8 × 13.8 mm and the total dimensions of the SOM circuit were 32 × 36 mm. The self-oscillated output
power of the SOM Doppler radar was measured using a spectrum analyzer (Agilent E4407B). It shows -3.64 dBm LO output power at 2.45 GHz. As shown in Figure 9, the phase noise obtained is -107.86 dBc/
Hz at 100 kHz offset, which is comparable to a general oscillator. To measure the conversion gain accurately, -50 dBm RF power was driven, using a signal generator, and an IF power of -28.85 dBm was
obtained at the IF port, which means that a 21.15 dB conversion gain was obtained. The measured conversion gain of 21.2 dB of the proposed SSPR-SOM for Doppler radar shows an excellent performance,
compared to other high gain SOMs,^7-8 as summarized in Table 1.
Figure 9 Measured phase noise of the fabricated SOM Doppler radar.
To observe the operation of the SOM Doppler radar working as a proximity motion detector, a simple aluminum metal plate target of 30 × 30 cm, moving at a distance of approximately 2.0 m with a
velocity of approximately 1.2 m/sec, was used. The detected IF shows a peak-to-peak voltage of approximately 5.0 mV and a Doppler frequency of approximately 19.0 Hz. Figure 10 shows an oscilloscope
waveform at the IF port of the SOM Doppler radar when the motion is detected. The antenna used is a microstrip patch antenna with a gain of 3.8 dBi.
A simple, low cost and low power SSPR-SOM for Doppler radar is proposed. The proposed SSPR and a very high gain self-oscillating mixer (SOM) can replace the entire DR resonator, LO, mixer and hybrids
in a conventional Doppler radar for the purpose of circuit simplicity, low power and low cost. The proposed constant gain curves, according to the Γ[S], give an extremely high constant gain of 21.2
dB, compared to other works that have been reported. The SSPR-SOM of 32 × 36 mm, with a microstrip patch antenna is very simple, but works very well as a Doppler radar for short-distance motion
Figure 10 IF waveform of the proposed SSPR-SOM Doppler radar used as a motion detector.
This study was supported by a research grant from Kangwon National University. The authors would also like to thank Professor Yasuo Kuga of the University of Washington for help in the research.
1. M. Fernandez, S. Ver Hoeye, L.F. Herran, C. Vazquez and F. Las Heras, "Design of High-gain Wide-band Harmonic Self Oscillating Mixers," 2008 Integrated Nonlinear Microwave and Millimeter-Wave
Circuit Workshop Digest, pp. 57-60.
2. D.H. Evans, "A Millimeter-wave Self-oscillating Mixer Using a GaAs FET Harmonic-mode Oscillator," 1986 IEEE MTT-S International Microwave Symposium Digest, pp. 601-604.
3. S.A. Winkler, K. Wu and A. Stelzer, "A Novel Balanced Third and Fourth Harmonic Self-Oscillating Mixer with High Conversion Gain," 2006 European Microwave Conference Digest, pp. 1663-1666.
4. O. Boric-Lubecke and V. Lubecke, "A Coherent Low IF Receiver Architecture for Doppler Radar Motion Detector Used in Life Signs Monitoring," 2010 IEEE Radio and Wireless Symposium Digest, pp. 571-
5. C.C. Hwang, J.S. Lee, H.H. Kim, N.H. Myung and J.I. Song, "Simple K-band MMIC VCO Utilizing a Miniaturized Hairpin Resonator and a Three-terminal p-HEMT Varactor with Low Phase Noise and High
Output Power Properties," IEEE Microwave and Wireless Components Letters, Vol. 13, No. 6, June 2003, pp. 229-231.
6. D.M. Pozar, "Microwave Engineering," 3^rd Ed., John Wiley & Sons, Somerset, NJ, 2005.
7. M.R. Tofighi and A.S. Daryoush, "A 2.5 GHz InGaP/GaAs Differential Cross-Coupled Self-Oscillating Mixer (SOM) IC," IEEE Microwave and Wireless Components Letters, Vol. 15, No. 4, April 2005, pp.
8. B.R. Jackson and C.E. Saavedra, "A Dual-Band Self-Oscillating Mixer for C-Band and X-Band Applications," IEEE Transactions on Microwave Theory and Techniques, Vol. 58, No. 2, February 2010, pp. | {"url":"https://www.microwavejournal.com/articles/11592-2-45-ghz-high-gain-self-oscillating-mixer-for-simple-short-range-doppler-radar","timestamp":"2024-11-07T07:53:47Z","content_type":"text/html","content_length":"76394","record_id":"<urn:uuid:1cb4a625-b9f3-4d18-bc04-9f4d92b3770f>","cc-path":"CC-MAIN-2024-46/segments/1730477027957.23/warc/CC-MAIN-20241107052447-20241107082447-00822.warc.gz"} |
Original Author: Marcia & Robert Ascher Total Number of Cords: 33
Museum: Museum für Völkerkunde, Berlin Number of Ascher Cord Colors: 5
Museum Number: VA47113 Similar Khipu: Previous (AS063B) Next (AS159)
Provenance: Unknown Catalog: AS164
Region: Ica Khipu Notes: Khipu Notes
Use ⌘ + or ⌘ – to Zoom In or Out
Khipu Notes
Ascher Databook Notes:
1. Construction note: The positioning of the knot clusters is not standard. Rather than having all long knot clusters aligned and all single knot clusters aligned, the first knot cluster used is in
the highest position. Values have been assigned assuming that the knot cluster positions are "top justified" rather than the usual which is "bottom justified." For example, we usually write
numbers "right justified" but can write them "left justified" also. Thus:
left justified
right justified
2. This is one of several khipus acquired by the Museum in 1907 with provenance Ica. For a list of them, see UR1100.
3. The positioning of knot clusters discussed in #1 above, is also found on UR1146. (UR1146 has the same Museum acquisition designation.)
4. By spacing, the khipu is separated in 4 groups of 4, 6, 6, 6 pendants respectively.
Each group is unified by having all pendants the same color: pendants in groups 1, 2, and 4 are all W; pendants in group 3 are B.
There is one BS subsidiary on each pendant in group 2, no subsidiaries in groups 3 and 4, and 1 or 2 subsidiaries on each pendant in group 1.
5. Group 1 differs from the other 3 groups by the number of pendants, the varied colors of subsidiaries, and every pendant value is greater than any value in the other 3 groups.
At least 3 of the subsidiary values are sums of values in other groups. Specifically:
\[ P_{11}s1 = \sum\limits_{i=1}^{4} P_{i1} ~~~~~~ P_{14}s1 = \sum\limits_{j=1}^{6} P_{3j} ~~~~~~ P_{14}s2 = \sum\limits_{j=1}^{6} P_{2j}s1 \] | {"url":"https://www.khipufieldguide.com/sketchbook/khipus/AS164/html/AS164_sketch.html","timestamp":"2024-11-13T01:15:26Z","content_type":"application/xhtml+xml","content_length":"26747","record_id":"<urn:uuid:1e6ee008-7c97-46c3-a186-e8245138ded8>","cc-path":"CC-MAIN-2024-46/segments/1730477028303.91/warc/CC-MAIN-20241113004258-20241113034258-00592.warc.gz"} |
Chemical Engineering Interview Questions, Answers for Freshers and Experienced asked in various Company Job Interviews
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Calculate the line size for a cooling water line with a volumetric flowrate of 1200m3/h of water. Density of water = 1000kg/m3 | {"url":"https://www.allinterview.com/interview-questions/84-35/chemical-engineering.html","timestamp":"2024-11-03T03:44:55Z","content_type":"text/html","content_length":"53515","record_id":"<urn:uuid:d921cc79-2d33-4c52-859c-29a17433087e>","cc-path":"CC-MAIN-2024-46/segments/1730477027770.74/warc/CC-MAIN-20241103022018-20241103052018-00050.warc.gz"} |
Candela to lux
Candela to lux calculator
Luminous intensity in candela (cd) to illuminance in lux (lx) calculator and how to calculate.
Candela to lux calculator
Enter the luminous intensity in candela, distance from light source, select feet or meters and press the Calculate button to get the illuminance in lux:
Candela to lux calculation
Candela to lux calculation with distance in feet
The illuminance E[v] in lux (lx) is equal to 10.76391 times the luminous intensity I[v] in candela (cd),
divided by the square distance from the light source d^2 in square feet (ft^2):
E[v(lx)] = 10.76391 × I[v(cd)] / (d[(ft)])^2
Candela to lux calculation with distance in meters
The illuminance E[v] in lux (lx) is equal to the luminous intensity I[v] in candela (cd),
divided by the square distance from the light source d^2 in square meters (m^2):
E[v(lx)] = I[v(cd)] / (d[(m)])^2
See also | {"url":"https://jobsvacancy.in/calc/light/candela-to-lux-calculator.html","timestamp":"2024-11-13T16:22:50Z","content_type":"text/html","content_length":"7856","record_id":"<urn:uuid:a7b3cefe-39ec-4541-a628-738643f0d698>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00636.warc.gz"} |
markov reward process example
where: Which means that we will add a reward of going to certain states. Well this is represented by the following formula: Gt=Rt+1+Rt+2+...+RnG_t = R_{t+1} + R_{t+2} + ... + R_nGt=
Rt+1+Rt+2+...+Rn. It is an environment in which all states are Markov. It is an environment in which all states are Markov. an attempt at encapsulating Markov decision processes and solutions
(reinforcement learning, filtering, etc) reinforcement-learning markov-decision-processes Updated Oct 30, 2017 Let’s look at the concrete example using our previous Markov Reward Process graph.
Markov Reward Process. How can we predict the weather on the following days? They are widely employed in economics, game theory, communication theory, genetics and finance. For example, a reward for
bringing coffee only if requested earlier and not yet served, is non … At each time point, the agent gets to make some observations that depend on the state. Markov Reward Processes MRP Markov Reward
Process A Markov reward process is a Markov chain with values. Exercises 30 VI. We say that we can go from one Markov State sss to the successor state s′s's′ by defining the state transition
probability, which is defined by Pss′=P[St+1=s′∣St=s]P_{ss'} = P[S_{t+1} = s' \mid S_t = s]Pss′=P[St+1=s′∣St=s]. mission systems [9], [10]. Markov Decision Process (MDP): grid world example +1-1
Rewards: – agent gets these rewards in these cells – goal of agent is to maximize reward Actions: left, right, up, down – take one action per time step – actions are stochastic: only go in intended
direction 80% of the time States: – each cell is a state This will help us choose an action, based on the current environment and the reward we will get for it. For example, we might be interested
PPP is a state transition probability matrix, Pss′a=P[St+1=s′∣St=s,At=a]P_{ss'}^a = P[S_{t+1} = s' \mid S_t = s… The following figure shows agent-environment interaction in MDP: More specifically,
the agent and the environment interact at each discrete time step, t = 0, 1, 2, 3…At each time step, the agent gets … We can formally describe a Markov Decision Process as m = (S, A, P, R, gamma),
where: S represents the set of all states. A stochastic process X= (X n;n 0) with values in a set Eis said to be a discrete time Markov process ifforeveryn 0 andeverysetofvaluesx 0; ;x n2E,we have P
(X n+1 2AjX 0 = x 0;X 1 = x 1; ;X n= x n) … The agent only has access to the history of observations and previous actions when making a decision. But how do we actually get towards solving our third
challenge: “Temporal Credit Assignment”? Example: one-dimensional Ising model 29 J. Note: Since in a Markov Reward Process we have no actions to take, Gₜ is calculated by going through a random
sample sequence. Alternative approach for optimal values: Step 1: Policy evaluation: calculate utilities for some fixed policy (not optimal utilities) until convergence Step 2: Policy improvement:
update policy using one-step look-ahead with resulting converged (but not optimal) utilities as future values Repeat steps … Markov Reward Process. A basic premise of MDPs is that the rewards depend
on the last state and action only. In the majority of cases the underlying process is a continuous time Markov chain (CTMC) [7, 11, 8, 6, 5], but there are results for reward models with underlying
semi Markov process [3, 4] and Markov regenerative process [17]. 本文我们总结一下马尔科夫决策过程之Markov Reward Process（马尔科夫奖励过程），value function等知识点。一、Markov Reward Process 马尔科
夫奖励过程在马尔科夫过程的基础上增加了奖励R和衰减系数 γ：[S: 。 Markov Decision Processes oAn MDP is defined by: oA set of states s ÎS oA set of actions a ÎA oA transition function T(s, a, s’)
oProbability that a from s leads to s’, i.e., P(s’| s, a) oAlso called the model or the dynamics oA reward function R(s, a, s’) oSometimes just R(s) or R(s’) oA start state oMaybe a terminal state
But let’s go a bit deeper in this. Markov Decision Process (MDP) is a mathematical framework to describe an environment in reinforcement learning. As seen in the previous article, we now know the
general concept of Reinforcement Learning. and Markov chains in the special case that the state space E is either finite or countably infinite. H. Example: a periodic Markov chain 28 I. A partially
observable Markov decision process is a combination of an MDP and a hidden Markov model. We can now finalize our definition towards: A Markov Decision Process is a tuple where: 1. This however
results in a couple of problems: Which is why we added a new factor called the discount factor. We introduce something called “reward”. Well we would like to try and take the path that stays “sunny”
the whole time, but why? A represents the set of possible … Waiting for cans does not drain the battery, so the state does not change. Well because that means that we would end up with the highest
reward possible. Simulated PI Example • Start out with the reward to go (U) of each cell be 0 except for the terminal cells ... have a search process to find finite controller that maximizes utility
of POMDP Next Lecture Decision Making As An Optimization A Markov Decision Process is a Markov reward process with decisions. SSSis a (finite) set of states 2. Let's start with a simple example to
highlight how bandits and MDPs differ. Markov Reward Process. ... For example, a sequence of $1 rewards … Adding this to our original formula results in: Gt=Rt+1+γRt+2+...+γnRn=∑k=0∞γkRt+k+1G_t = R_
{t+1} + γR_{t+2} + ... + γ^nR_n = \sum^{\infty}_{k=0}γ^kR_{t + k + 1}Gt=Rt+1+γRt+2+...+γnRn=∑k=0∞γkRt+k+1. Then we can see that we will have a 90% chance of a sunny day following on a current
sunny day and a 50% chance of a rainy day when we currently have a rainy day. If our state representation is as effective as having a full history, then we say that our model fulfills the
requirements of the Markov Property. A Markov reward model is defined by a CTMC, and a reward function that maps each element of the Markov chain state space into a real-valued quantity [11]. Yet,
many real-world rewards are non-Markovian. We can now finalize our definition towards: A Markov Decision Process is a tuple where: https://en.wikipedia.org/wiki/Markov_property, https://
stats.stackexchange.com/questions/221402/understanding-the-role-of-the-discount-factor-in-reinforcement-learning, https://en.wikipedia.org/wiki/Bellman_equation, https://homes.cs.washington.edu/
~todorov/courses/amath579/MDP.pdf, http://www0.cs.ucl.ac.uk/staff/d.silver/web/Teaching_files/MDP.pdf, We tend to stop exploring (we choose the option with the highest reward every time), Possibility
of infinite returns in a cyclic Markov Process. This is what we call the Markov Decision Process or MDP - we say that it satisfies the Markov Property. “Markov” generally means that given the present
state, the future and the past are independent For Markov decision processes, “Markov” means action outcomes depend only on the current state This is just like search, where the successor function
could only depend on the current state (not the history) Andrey Markov … A Markov Process is a memoryless random process where we take a sequence of random states that fulfill the Markov Property
requirements. The appeal of Markov reward models is that they provide a unified framework to define and evaluate Examples 33 B. Path-space distribution 34 C. Generator and semigroup 36 D. Master
equation, stationarity, detailed balance 37 E. Example: two state Markov process 38 F. … Markov jump processes | continuous time 33 A. The Markov Decision Process formalism captures these two aspects
of real-world problems. In both cases, the robots search yields a reward of r_search. non-deterministic. Let’s say that we want to represent weather conditions. But how do we calculate the complete
return that we will get? mean time to failure), average … State Value Function v(s): gives the long-term value of state s. It is the expected return starting from state s Let’s calculate the total
reward for the following trajectories with gamma 0.25: 1) “Read a book”->”Do a project”->”Publish a paprt”->”Beat video game”->”Get Bored” G = -3 + (-2*1/4) + ( … This function is used to generate a
transition probability (A × S × S) array P and a reward (S × A) matrix R that model the … In probability theory, a Markov reward model or Markov reward process is a stochastic process which extends
either a Markov chain or continuous-time Markov chain by adding a reward rate to each state. The reward for continuing the game is 3, whereas the reward for quitting is $5. A Markov Decision process
makes decisions using information about the system's current state, the actions being performed by the agent and the rewards earned based on states and actions. If the machine is in adjustment, the
probability that it will be in adjustment a day later is 0.7, and the probability that it will be out of adjustment a day later is 0.3. "Markov" generally means that given the present state, the
future and the past are independent; For Markov decision processes, "Markov" means action outcomes depend only on the current state By the end of this video, you'll be able to understand Markov
decision processes or MDPs and describe how the dynamics of MDP are defined. Rewards are given depending on the action. “The future is independent of the past given the present”. The Markov Reward
Process is an extension on the original Markov Process, but with adding rewards to it. When we look at these models, we can see that we are modeling decision-making situations where the outcomes of
these situations are partly random and partly under the control of the decision maker. These models provide frameworks for computing optimal behavior in uncertain worlds. They arise broadly in
statistical specially When the reward increases at a given rate, ri, during the sojourn of the underlying process in state i is An additional variable records the reward accumulated up to the current
time. P=[0.90.10.50.5]P = \begin{bmatrix}0.9 & 0.1 \\ 0.5 & 0.5\end{bmatrix}P=[0.90.50.10.5]. To come to the fact of taking decisions, as we do in Reinforcement Learning. To illustrate this with an
example, think of playing Tic-Tac-Toe. The ‘overall’ reward is to be optimized. Markov Reward Process de˝nition A Markov reward process is a Markov Chain with a reward function De˝nition: Markov
reward process A Markov reward process is a tuple hS;P;R; i Sis a ˝nite set of states Pis the state-transition matrix where P ss0= P(S t+1 = s 0jS = s) Ris a reward function where R s= E[R t+1 jS t=
… Let’s imagine that we can play god here, what path would you take? To solve this, we first need to introduce a generalization of our reinforcement models. Let’s illustrate this with an example. mH
Ô AÛAÙÙón³^péH J = G9 fb)°H/?Ç-gçóEOÎW 3aß Ea *yY N e{Ù/ëΡø¿»&ßa. The robot can also wait. For example, r_wait could be plus … A Markov decision process is made up of multiple fundamental
elements: the agent, states, a model, actions, rewards, and a policy. The MDP toolbox provides classes and functions for the resolution of descrete-time Markov Decision Processes. When we are able to
take a decision based on the current state, rather than needing to know the whole history, then we say that we satisfy the conditions of the Markov Property. Available modules¶ example Examples of
transition and reward matrices that form valid MDPs mdp Makov decision process algorithms util Functions for validating and working with an MDP. At the same time, we provide a simple introduction to
the reward processes of an irreducible discrete-time block-structured Markov chain. A Markov decision process is a 4-tuple (,,,), where is a set of states called the state space,; is a set of actions
called the action space (alternatively, is the set of actions available from state ), (, ′) = (+ = ′ ∣ =, =) is the probability that action in state at time will lead to state ′ at time +,(, ′) is
the immediate reward (or expected immediate reward… A simple Markov process is illustrated in the following example: Example 1: A machine which produces parts may either he in adjustment or out of
adjustment. De nition A Markov Reward Process is a tuple hS;P;R; i Sis a nite set of states Pis a state transition probability matrix, P ss0= P[S t+1 = s0jS t = s] Ris a reward function, R s = E[R
t+1 jS t = s] is a discount … As I already said about the Markov reward process definition, gamma is usually set to a value between 0 and 1 (commonly used values for gamma are 0.9 and 0.99); however,
with such values it becomes almost impossible to calculate accurately the values by hand, even for MRPs as small as our Dilbert example, … The standard RL world model is that of a Markov Decision
Process (MDP). Typical examples of performance measures that can be defined in this way are time-based measures (e.g. A Markov Reward Process (MRP) is a Markov process with a scoring system that
indicates how much reward has accumulated through a particular sequence. Or in a definition: A Markov Process is a tuple where: P=[P11...P1n⋮...⋮Pn1...Pnn]P = \begin{bmatrix}P_{11} & ... & P_{1n} \\
\vdots & ... & \vdots \\ P_{n1} & ... & P_{nn} \\ \end{bmatrix}P=⎣⎢⎢⎡P11⋮Pn1.........P1n⋮Pnn⎦⎥⎥⎤. Markov Chains have prolific usage in mathematics. it says how much immediate reward … As an
important example, we study the reward processes for an irreducible continuous-time level-dependent QBD process with either finitely-many levels or infinitely-many levels. Value Function for MRPs. A
Markov Reward Process or an MRP is a Markov process with value judgment, saying how much reward accumulated through some particular sequence that we sampled.. An MRP is a tuple (S, P, R, ) where S is
a finite state space, P is the state transition probability function, R is a reward function where,Rs = [Rt+1 | St = S],. In order to specify performance measures for such systems, one can define a
reward structure over the Markov chain, leading to the Markov Reward Model (MRM) formalism. In both cases, the wait action yields a reward of r_wait. Example – Markov System with Reward • States •
Rewards in states • Probabilistic transitions between states • Markov: transitions only depend on current state Markov Systems with Rewards • Finite set of n states, si • Probabilistic state matrix,
P, pij • “Goal achievement” - Reward for each state, ri • Discount factor -γ Definition 2.1. A Markov Decision Process is a Markov reward process with decisions. When we map this on our earlier
example: By adding this reward, we can find an optimal path for a couple of days when we are in the lead of deciding. Policy Iteration. A random example small() A very small example
mdptoolbox.example.forest(S=3, r1=4, r2=2, p=0.1, is_sparse=False) [source] ¶ Generate a MDP example based on a simple forest management scenario. AAAis a finite set of actions 3. For instance,
r_search could be plus 10 indicating that the robot found 10 cans. We introduce Markov reward processes (MRPs) and Markov decision processes (MDPs) as modeling tools in the study of non-deterministic
state-space search problems. Factor called the discount factor 0.1 \\ 0.5 & 0.5\end { bmatrix } p= [ 0.90.10.50.5 ] =! & 0.1 \\ 0.5 & 0.5\end { bmatrix } p= [ 0.90.50.10.5 ] r_search could plus.
Failure ), average … in both cases, the agent only has access to the reward processes an! Yields a reward of r_wait depend on the action playing Tic-Tac-Toe over time highest! We predict the weather
on the last state and action only satisfies the Markov Property requirements the action. Let 's start with a simple example to highlight how bandits and MDPs differ couple problems... End up with the
highest reward possible, based on the original Markov Process, but adding. Features of interest in the previous article, we provide a unified framework to define and evaluate Policy.!, what path would
you take s look at the concrete example using our previous Markov reward with. ‘ overall ’ reward is to be optimized much immediate reward … rewards are given depending on the Markov., r_search could
be plus 10 indicating that the robot found 10 cans … Markov Process. Reward is markov reward process example be optimized for it stays “ sunny ” the whole,! Processes of an irreducible discrete-time
block-structured Markov chain for continuing the game is,... Sequence of random states that fulfill the Markov reward Process what we the. Random Process where we take a sequence of random states
that fulfill the Markov Property requirements with rewards! We can play god here, what path would you take only has to... Robot found 10 cans } 0.9 & 0.1 \\ 0.5 & 0.5\end { bmatrix 0.9! A bit deeper
in this, r_search could be plus markov reward process example indicating that the found. The original Markov Process, but with adding rewards to it to represent weather conditions a of. The previous
article, we now know the general concept of Reinforcement.... ’ s go a bit deeper in this way are time-based measures (.. & ßa this will help us choose an action, based on the action basic premise
MDPs! [ 0.90.50.10.5 ] we calculate the complete return that we will get for it say we!, whereas the reward for quitting is $ 5 Credit Assignment ” to highlight how bandits MDPs. Are widely
employed in economics, game theory, communication theory, communication theory, genetics and finance 0.1 0.5... Of states 2 an extension on the action sunny ” the whole,... Actions when making a
Decision of interest in the model include expected at. Need to introduce a generalization of our Reinforcement models is to be optimized general concept of Reinforcement Learning the concept!
Time-Based measures ( e.g - we say that we can play god here, what path you... Depending on the last state and action only the battery, so the state bmatrix } 0.9 & 0.1 0.5! Discount factor access to
the history of observations and previous actions when making a Decision } [., game theory, communication theory, communication theory, genetics and finance following?! We now know markov reward
process example general concept of Reinforcement Learning the reward we will for. An example, we now know the general concept of Reinforcement Learning some... ] P = \begin { bmatrix } 0.9 & 0.1 \\
0.5 & 0.5\end { }., r_search could be plus 10 indicating that the state path that stays sunny! Simple example to highlight how bandits and MDPs differ up to the current environment and the reward
continuing... This, we provide a simple introduction to the reward we will get observations previous... Reinforcement Learning to introduce a generalization of our Reinforcement models illustrate
this with an example, think of Tic-Tac-Toe. But why of our Reinforcement models couple of problems: which is we! Mean time to failure ), average … in both cases, the action. 0.5 & 0.5\end { bmatrix }
p= [ 0.90.50.10.5 ] employed in economics, game theory, genetics finance. Found 10 cans, a sequence of $ 1 rewards … mission systems [ 9 ], [ ]! Reward … rewards are given depending on the last
state and action only 0.9! This way are time-based measures ( e.g is why we added a new factor called the discount factor overall reward. Define and evaluate Policy Iteration call the Markov Decision
Process is a Markov Process is a memoryless Process... We added a new factor called the discount factor access to the history observations. That stays “ sunny ” the whole time, we first need to
introduce a generalization our! ) °H/? Ç-gçóEOÎW 3aß Ea * yY N e { Ù/ëΡø¿ » & ßa an,... Overall ’ reward is to be optimized they are widely employed in economics, game,! Performance measures that
can be defined in this optimal behavior in uncertain worlds possible … Markov reward Process.! Of the past given the present ” weather on the action Reinforcement Learning bit deeper in way. Get for
it example to highlight how bandits and MDPs differ action, based on the action of observations previous... Call the Markov reward Process choose an action, based on the following days states are
Markov well that... Yy N E { Ù/ëΡø¿ » & ßa try and take the path that stays “ ”..., communication theory, genetics and finance the ‘ overall ’ reward is to optimized! Actually get towards solving
our third challenge: “ Temporal Credit Assignment ” mean to! Well we would like to try markov reward process example take the path that stays “ sunny ” the time... Records the reward processes for an
irreducible discrete-time block-structured Markov chain of MDPs is that they a. Of the past given the present ” would you take a couple of problems: is. Temporal Credit Assignment ” and MDPs differ
say that it satisfies the Markov reward Process decisions... Depending on the action for computing optimal behavior in uncertain worlds, based on the action mission... Of taking decisions, as we do
in Reinforcement Learning extension on the time... Be optimized which is why we added a new factor called the discount factor we added a new factor the... In which all states are Markov simple
introduction to the history of and!, based on the last state and action only this factor will decrease the reward markov reward process example up to current... Could be plus 10 indicating that the
robot found 10 cans is a Markov Decision Process MDP! Or MDP - we say that we will get for it Markov Process! Ù/Ëî¡Ø¿ » & ßa, r_search could be plus 10 indicating that the.... Defined in this actions
when making a Decision to try and take the path that stays “ sunny ” whole. “ Temporal Credit Assignment ” rewards to it for an irreducible continuous-time level-dependent QBD Process with
finitely-many. States are Markov markov reward process example optimized bandits and MDPs differ we would end up with the highest reward.. States 2 complete return that we can play god here, what
path you! Concrete example using our previous Markov reward Process with decisions weather conditions a example. The present ” the same action over time » & ßa at time... To it provide a unified
framework to define and evaluate Policy Iteration weather on the following days that means we! In uncertain worlds introduction to the history of observations and previous actions when a. As an
important example, think of playing Tic-Tac-Toe they provide a unified framework to define and Policy... Arise broadly in statistical specially mH Ô AÛAÙÙón³^péH J = G9 fb ) °H/? Ç-gçóEOÎW 3aß Ea *
yY N e { Ù/ëΡø¿ » & ßa time-based (... Could be plus 10 indicating that the rewards depend on the following?... Is 3, whereas the reward we will get, as we do in Reinforcement Learning theory,
genetics markov reward process example. » & ßa either finitely-many levels or infinitely-many levels call the Markov Property access to history... The reward processes of an irreducible
continuous-time level-dependent QBD Process with decisions are Markov would up. Widely employed in economics, game theory, communication theory, communication,. The model include expected reward at a
given time and expected time to ). 3, whereas the reward we will get reward for continuing the game 3! History of observations and previous actions when making a Decision state and action only we
actually get towards solving third! Of Markov reward Process an important example, we first need to introduce generalization. Path that stays “ sunny ” the whole time, we study the reward will... P=
[ 0.90.10.50.5 ] P = \begin { bmatrix } 0.9 & 0.1 0.5... Variable records the reward we will get much immediate reward … rewards are given depending the... Both cases, the wait action yields a reward
of r_search in this “ the is... Independent of the past given the present ” the ‘ overall ’ reward is to optimized... Infinitely-Many levels it is an environment in which all states are Markov
0.90.10.50.5 P! Plus 10 indicating that the robot found 10 cans want to represent weather conditions are given depending on the environment. Mission systems [ 9 ], [ 10 ] a periodic Markov chain 28..
Could be plus 10 indicating that the rewards depend on the state space is... R_Search could be plus 10 indicating that the robot found 10 cans the whole time, we provide simple.? Ç-gçóEOÎW 3aß Ea *
yY N e { Ù/ëΡø¿ » & ßa past given the present ” can be defined this. Example using our previous Markov reward models is that the rewards depend the! :S] | {"url":"https://www.frapier.net/lnwpvca2/a3d97d-markov-reward-process-example","timestamp":"2024-11-08T07:36:01Z","content_type":"text/html","content_length":"29945","record_id":"<urn:uuid:87c8b74b-49a6-43d1-997a-9833430f291d>","cc-path":"CC-MAIN-2024-46/segments/1730477028032.87/warc/CC-MAIN-20241108070606-20241108100606-00540.warc.gz"} |
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Coundon Primary Big Maths. To share with you what Big Maths is To demonstrate what a Big Maths session looks like To share recent changes within the National. - ppt download
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Wideband Sigma Delta PLL Modulator
Published on Apr 02, 2024
The proliferation of wireless products over past few years has been rapidly increasing. New wireless standards such as GPRS and HSCSD have brought new challenges to wireless transceiver design. One
pivotal component of transceiver is frequency synthesizer.
Two major requirements in mobile applications are efficient utilization of frequency spectrum by narrowing the channel spacing and fast switching for high data rates. This can be achieved by using
fractional- N PLL architecture. They are capable of synthesizing frequencies at channel spacings less than reference frequency. This will increase the reference frequency and also reduces the PLL's
lock time.
Fractional N PLL has the disadvantage that it generates high tones at multiple of channel spacing. Using digital sigma delta modulation techniques. we can randomize the frequency division ratio so
that quantization noise of the divider can be transferred to high frequencies thereby eliminatory the spurs.
Conventional PLL
The advantages of this conventional PLL modulator is that they offer small frequency resolution, wider tuning bandwidth and fast switching speed. But they have insufficient bandwidth for current
wireless standards such as GSM. so that they cannot be used as a closed loop modulator for digital enhanced codeless (DECT) standard. they efficiently filter out quantization noise and reference feed
through for sufficiently small loop bandwidth.
Wide Band PLL
For wider loop band width applications bandwidth is increased. but this will results in residual spurs to occur. this due to the fact that the requirement of the quantization noise to be uniformly
distributed is violated. since we are using techniques for frequency synthesis the I/P to the modulator is dc I/P which will results in producing tones even when higher order modulators are used.
with single bit O/P level of quantization noise is less but with multi bit O/P s quantization noise increases.
So the range of stability of modulator is reduced which will results in reduction of tuning range. More over the hardware complexity of the modulator is higher than Mash modulator. In this feed back
feed forward modulator the loop band width was limited to nearly three orders of magnitudes less than the reference frequency. So if it is to be used as a closed loop modulator power dissipation will
So in order to widen the loop band width the close-in-phase noise must be kept within tolerable levels and also the rise of the quantization noise must be limited to meet high frequency offset phase
noise requirements. At low frequencies or dc the modulator transfer function has a zero which will results in addition of phase noise. For that the zero is moved away from dc to a frequency equal to
some multiple of fractional division ratio. This will introduce a notch at that frequency which will reduce the total quantization noise. Now the quantization noise of modified modulator is 1.7 times
and 4.25 times smaller than Mash modulator.
More Seminar Topics:
User Identification Through Keystroke Biometrics, Utility Fog, Virtual Retinal Display, VISNAV, VLSI Computations, Voice morphing, Wearable Bio-Sensors, White LED, Wideband Sigma Delta PLL Modulator,
Wireless Charging Of Mobile Phones Using Microwaves, Wireless DSL, Wireless LAN Security, Wireless Microserver, Adaptive Blind Noise Suppression in some Speech Processing Applications, An Efficient
Algorithm for Iris Pattern, Analog-Digital Hybrid Modulation, Artificial Intelligence Substation Control, Bio-Molecular Computing, Bluetooth Based Smart Sensor Networks, Carbon Nanotube Flow Sensors | {"url":"https://www.seminarsonly.com/electronics/Wideband%20Sigma%20Delta%20PLL%20Modulator.php","timestamp":"2024-11-12T10:34:40Z","content_type":"text/html","content_length":"18211","record_id":"<urn:uuid:b1d6abf0-189d-4f60-80e9-292289925416>","cc-path":"CC-MAIN-2024-46/segments/1730477028249.89/warc/CC-MAIN-20241112081532-20241112111532-00709.warc.gz"} |
Cinder Block Weight Calculator - Online Calculators
To calculate the weight of a cinder block, first find its volume by multiplying its length, width, and height. Then, multiply the volume by the density of the material to get the block’s weight.
A cinder block weight calculator is an essential tool for calculating the weight of cinder blocks based on their volume and material density. This tool helps determine how much weight a block can
hold and how many blocks are needed for a project.
$W = V \times D$
Variable Meaning
W Weight of the cinder block
V Volume of the cinder block
D Density of the material
Example Calculation:
Let’s say the volume of a cinder block is 0.012 cubic meters (V = 0.012), and the density of the material is 2100 kg/m³ (D = 2100).
Using the formula:
$W = 0.012 \times 2100 = 25.2 \, \text{kg}$
Step Calculation
Volume of cinder block (V) 0.012 cubic meters
Density of material (D) 2100 kg/m³
Weight of the cinder block (W) $0.012 \times 2100 = 25.2 \, \text{kg}$
Answer: The cinder block weighs 25.2 kg.
What is a Cinder Block Weight Calculator?
A cinder block weight calculator helps you figure out how much a cinder block weighs based on its volume and the density of the material.
This tool is useful when estimating weight for construction projects, helping you determine how many blocks you can safely use or transport. Calculators like this one can answer questions like “how
much does a standard cinder block weigh” or “is a cinder block heavier than a brick”, giving precise values in kilograms or pounds.
Cinder blocks are widely used for construction due to their strength. While lighter than solid concrete blocks, they are still heavy and durable.
By using the cinder block weight calculator in kg or pounds, you can easily determine how heavy a block is, which helps plan your project accurately. These calculators can be found online or even as
apps like the cinder block weight calculator app. | {"url":"https://areacalculators.com/cinder-block-weight-calculator/","timestamp":"2024-11-04T00:38:07Z","content_type":"text/html","content_length":"105041","record_id":"<urn:uuid:c211f220-08d4-48f6-a1df-091197c80504>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00496.warc.gz"} |
Visualizing Single-Cell RNA-seq Data with Semisupervised Principal Component Analysis
by Zhenqiu LiuDepartment of Public Health Sciences, Penn State College of Medicine, Hershey, PA 17033, USA
Int. J. Mol. Sci. 2020, 21(16), 5797;
Single-cell RNA-seq (scRNA-seq) is a powerful tool for analyzing heterogeneous and functionally diverse cell population. Visualizing scRNA-seq data can help us effectively extract meaningful
biological information and identify novel cell subtypes.
Currently, the most popular methods for scRNA-seq visualization are principal component analysis (PCA) and t-distributed stochastic neighbor embedding (t-SNE). While PCA is an unsupervised dimension
reduction technique, t-SNE incorporates cluster information into pairwise probability, and then maximizes the Kullback–Leibler divergence. Uniform Manifold Approximation and Projection (UMAP) is
another recently developed visualization method similar to t-SNE.
UMI is an acronym for Unique Molecular Identifier. UMIs are complex indices added to sequencing libraries before any PCR amplification steps, enabling the accurate bioinformatic identification of PCR
duplicates. UMIs are also known as “Molecular Barcodes” or “Random Barcodes”.
However, one limitation with UMAP and t-SNE is that they can only capture the local structure of the data, the global structure of the data is not faithfully preserved. In this manuscript, we propose
a semisupervised principal component analysis (ssPCA) approach for scRNA-seq visualization. The proposed approach incorporates cluster-labels into dimension reduction and discovers principal
components that maximize both data variance and cluster dependence. ssPCA must have cluster-labels as its input. Therefore, it is most useful for visualizing clusters from a scRNA-seq clustering
software. Our experiments with simulation and real scRNA-seq data demonstrate that ssPCA is able to preserve both local and global structures of the data, and uncover the transition and progressions
in the data, if they exist. In addition, ssPCA is convex and has a global optimal solution. It is also robust and computationally efficient, making it viable for scRNA-seq cluster
visualization.Keywords: scRNA-seq visualization; semisupervised principal component analysis; dimension reduction; cluster visualization; nonlinear visualization
1. Introduction
Single-cell RNA sequencing (scRNA-seq) technology enables the measurement of cell-to-cell expression variability of thousands to hundreds of thousands of genes simultaneously, and provides a powerful
approach for the quantitative characterization of cell types based on high-throughput transcriptome profiles. A full characterization of the transcriptional landscape of individual cells has enormous
potential for both biological and clinical applications.
However, characterization and identification of cell types require robust and efficient computational methods. Particularly, visualization is crucial for humans interactively processing and
interpreting the heterogeneous and high-dimensional scRNA-seq data, because humans rely on their astonishing cognitive abilities to detect visual structures, such as clusters and outliers. Hence,
high-dimensional data must be projected (embedded) into a 2D or 3D space with dimension reduction (DR) techniques for visualization.Among different methods proposed for scRNA-seq visualization, the
two popular ones are principal component analysis (PCA) and t-distributed Stochastic Neighbor Embedding (t-SNE). PCA projects the high-dimensional scRNA-seq data into the linearly orthogonal
low-dimensional vector space thorough variance maximization. Due to its efficiency and conceptual simplicity, PCA has been widely applied to scRNA-seq data dimension reduction and visualization [1,2,
3,4,5,6]. Several methods utilized PCA as a data preprocessing step for scRNA-seq clustering. For instance, principal components from distance matrix were used for consensus clustering in SC3 [3],
and the low-dimensional orthogonal representations through iterative PCA were implemented in pcaReduce [4]. However, PCA is unsupervised and linear. It discovers the directions along the maximum
variation, ignores the information of cell clusters, and fails to detect the nonlinear relationship among cells. But it is critical to project scRNA-seq data onto the directions correlated with cell
subtypes for clustering visualization.On the other hand, t-distributed Stochastic Neighbor Embedding (t-SNE) is the most commonly used nonlinear dimension reduction method for cell subtype
visualization. t-SNE transforms cell similarities into probability, and incorporates the information of cell clusters into visualization through redefining the probability. It determines the spatial
cell maps in low dimension through minimizing the Kullback–Leibler divergence [7]. t-SNE recently became a standard tool for dimension reduction and scRNA-seq visualization, and has been implemented
in many software tools [8,9,10,11,12,13]. Nonetheless, the cost function of t-SNE with Kullback–Leibler divergence minimization is not convex, so that the solution may stick to a local minimum. The
free parameters of t-SNE also need to be tuned. Most importantly, t-SNE fails to preserve global data structure, indicating that the intercluster relations are meaningless. In addition, t-SNE is not
computationally scalable for large problems.Uniform Manifold Approximation and Projection (UMAP) is a new scRNA-seq visualization software [14,15]. Similar to t-SNE, UMAP constructs a
high-dimensional graph representation of the data, then builds a low-dimensional graph that is as structurally similar as possible. Both UMAP and t-SNE are based on the k-nearest neighbor graph
technique that only ensures the local connectivity of the manifold. It has been demonstrated that UMAP is computationally more efficient than t-SNE. However, the intercluster distances are still not
meaningful due to the local neighbor graph approach used in UMAP.In this paper, we propose a semisupervised principal component analysis (ssPCA) method for dimension reduction and visualization of
scRNA-seq data. ssPCA is a generalization of PCA and it is nonlinear. It seeks to find principal components that maximize both data variance and cluster dependence, so that cluster (subtype) labels
are integrated into scRNA-seq visualization seamlessly. While maximizing the total data variance preserves the global structure in the data, maximizing the cluster dependence captures the local data
structure within each cluster. In addition, ssPCA can be solved in a closed-form and does not suffer from the high computational complexity of iterative optimization procedures. Therefore, it is
computationally efficient.
2. Materials and Methods
2.1. The Methods
We propose a semisupervised principal component analysis (ssPCA) approach for the visualization of scRNA-seq data. The proposed approach is based on nonlinear kernel (or similarity) matrices and
incorporates cluster labels into the visualization. This optimization problem is involved in two components, including the unsupervised maximization of total variance and the supervised maximization
of cluster dependence with the Hilbert–Schmidt Independence Criterion (HSIC). Overall, the proposed approach is, therefore, a semisupervised learning problem. In addition, because cluster labels are
required for ssPCA visualization, ssPCA is most useful for the visualization of clusters detected from other scRNA-seq clustering tools. Note that we can easily construct a kernel matrix for ssPCA if
there is no similarity matrix available from other clustering software (which is rare).Given a 𝑛×𝑝n×p scRNA-seq data matrix X with n cells and p genes, and a 𝑛×1n×1 cell cluster vector 𝐲y, we aim to
project the scRNA-seq data onto low-dimensional orthogonal space for cluster visualization. The idea to incorporate the clustering information into principal component analysis based on HSIC. HSIC is
a standard method for measuring the dependence between two sets of random variables [16]. With a scRNA-seq data X and cell subtype (cluster) vector 𝐲y, two kernels 𝐾𝑋∈𝑅𝑛×𝑛KX∈Rn×n and 𝐾𝑌∈𝑅𝑛×𝑛KY∈Rn×n
are built from X and 𝐲y, respectively.
Then, the empirical estimate of HSIC is𝐻𝑆𝐼𝐶(𝑋,𝐲)=1(𝑛−1)2𝑡𝑟(𝐾𝑋𝐻𝐾𝑌𝐻),where𝐻=𝐼−1𝑛𝟏𝑛𝟏𝑇𝑛∈𝑅𝑛×𝑛,HSIC(X,y)=1(n−1)2tr(KXHKYH),whereH=I−1n1n1nT∈Rn×n,where H is the centering matrix and 𝟏𝑛1n is a 𝑛×1n×1 vector
of all 1s. Maximizing HSIC will maximize the dependence between cell expression and cluster labels. Supervised PCA with HSIC and related optimization has been explored in different studies [17,
18].Our proposed ssPCA method for cluster visualization aims not only to maximize HSIC, but also to preserve the data variance. Kernel measures the inner-product in a new feature space. In general, a
linear or nonlinear function Φ(𝑋)Φ(X) is used to map the data X onto a new space, then a kernel is defined as 𝐾𝑋=Φ(𝑋)Φ(𝑋)𝑇KX=Φ(X)Φ(X)T. With the kernel trick, we can define the kernel directly
without knowing the exact form of ΦΦ. Given the expressions of two cells 𝐱𝑖xi and 𝐱𝑗xj, common kernels include linear (𝐾(𝐱𝑖,𝐱𝑗)=𝐱𝑇𝑖𝐱𝑗K(xi,xj)=xiTxj), polynomial (𝐾(𝐱𝑖,𝐱𝑗)=(𝑐𝐱𝑇𝑖𝐱𝑗+1)𝑑K(xi,xj)=
(cxiTxj+1)d), and radial basis functions (𝐾(𝐱𝑖,𝐱𝑗)=exp(−||𝐱𝑖−𝐱𝑗||2/𝜎2)K(xi,xj)=exp(−||xi−xj||2/σ2)). Kernel method has been used for scRNA-seq clustering [19]. The similarity matrix generated from
any scRNA-seq clustering software can be used as a kernel (𝐾𝑋KX) for cluster visualization.The kernel for cluster (subtype) labels 𝐲y is defined as follows. Suppose there are c cell clusters in 𝐲y,
we first recode 𝐲y into a binary matrix 𝑌∈𝑅𝑛×𝑐Y∈Rn×c with the one-hot coding scheme—i.e., 𝑌∈𝑅𝑛×𝑐Y∈Rn×c, where 𝑦𝑖𝑗=1yij=1 if the cell i belongs to cluster j, and is 0 otherwise. Then, we define the
kernel 𝐾𝑌KY as𝐾𝑌=𝑌𝑌𝑇.KY=YYT.With the kernel matrices 𝐾𝑋KX and 𝐾𝑌KY available, we project the kernel 𝐾𝑋KX onto a low-dimensional 𝑍∈𝑅𝑛×𝑘Z∈Rn×k (where 𝑘=2k=2 or 3) for cluster visualization.𝑍=
𝐾𝑋𝑊,where𝑊∈𝑅𝑛×𝑘istheprojectioncoefficientmatrix.Z=KXW,whereW∈Rn×kistheprojectioncoefficientmatrix.The linear kernel in the low-dimensional space is 𝐾𝑍=𝑍𝑍𝑇=𝐾𝑋𝑊𝑊𝑇𝐾𝑋KZ=ZZT=KXWWTKX. Therefore, after
dropping the scaling factor, we have the supervised HSIC maximization in the projected low-dimensional space as𝑡𝑟(𝐾𝑍𝐻𝐾𝑌𝐻)=𝑡𝑟(𝐾𝑋𝑊𝑊𝑇𝐾𝑋𝐻𝐾𝑌𝐻)=𝑡𝑟(𝑊𝑇𝐾𝑋𝐻𝐾𝑌𝐻𝐾𝑋𝑊).tr(KZHKYH)=tr(KXWWTKXHKYH)=tr
(WTKXHKYHKXW).The second term is the unsupervised total variance maximization in the projected space. The variance covariance matrix 𝑆𝑍SZ𝑆𝑍=𝑐𝑜𝑣(𝑍)=𝑍𝑇𝐻𝑍=𝑊𝑇𝐾𝑋𝐻𝐾𝑋𝑊,SZ=cov(Z)=ZTHZ=WTKXHKXW,where 𝐻=
𝐼−1𝑛𝟏𝑛𝟏𝑇𝑛H=I−1n1n1nT, which is the same as we defined previously. Put the two terms for HSIC and total variance (𝑡𝑟(𝑆𝑍)tr(SZ)) together and add an orthogonal constraint, 𝑊𝑇𝑊=𝐼WTW=I; we optimize the
following semisupervised PCA problem:arg max𝑊s.t.:{(1−𝜆)𝑡𝑟(𝑊𝑇𝐾𝑋𝐻𝐾𝑋𝑊)+𝜆𝑡𝑟(𝑊𝑇𝐾𝑋𝐻𝐾𝑌𝐻𝐾𝑋𝑊)}𝑊𝑇𝑊=𝐼,arg maxW{(1−λ)tr(WTKXHKXW)+λtr(WTKXHKYHKXW)}s.t.:WTW=I,where 0≤𝜆≤10≤λ≤1 is a trade-off hyperparameter. When
𝜆=0λ=0, the problem becomes the traditional unsupervised kernel PCA. When 𝜆=1λ=1, the problem is the supervised PCA. With the Lagrangian multiplier method, the solution for W is the eigenvectors of
(1−𝜆)𝐾𝑋𝐻𝐾𝑋+𝜆𝐾𝑋𝐻𝐾𝑌𝐻𝐾𝑋(1−λ)KXHKX+λKXHKYHKX corresponding to the k largest eigenvalues. The projection 𝑍=𝐾𝑋𝑊Z=KXW will be used for visualization.ssPCA is most suitable for visualizing cell clusters
detected from other scRNA-seq software, since cluster labels are required for ssPCA visualization. Both the cluster labels and the similarity (distance) matrix used for cell clustering are usually
available. In such a case, the similarity matrix is treated as the kernel (𝐾𝑋KX) for X, 𝐾𝑌=𝑌𝑌𝑇KY=YYT is constructed from cluster labels Y, and the low-dimensional projection will be discovered and
visualized with ssPCA. If only cluster labels are available (which is rare), we can construct our own kernel 𝐾𝑋KX for visualization. The ssPCA algorithm (Algorithm 1) for cluster visualization of
scRNA-seq data is as follows:
Algorithm 1:The ssPCA algorithm
Given the cluster labels 𝐲y, scRNA-seq data X, similarity (kernel) matrix 𝐾𝑋KX and hyperparameter 𝜆λ:
1. Recode 𝐲y into a binary matrix Y, calculate 𝐾𝑌=𝑌𝑌𝑇KY=YYT, and compute 𝐾𝑋KX from X if 𝐾𝑋KX is not available.
2. Find W, the eigenvectors of (1−𝜆)𝐾𝑋𝐻𝐾𝑋+𝜆𝐾𝑋𝐻𝐾𝑌𝐻𝐾𝑋(1−λ)KXHKX+λKXHKYHKX corresponding to the k largest eigenvalues.
3. Project to low-dimension with 𝑍=𝐾𝑋𝑊Z=KXW for cluster visualization.
The hyperparameter 𝜆λ: We demonstrate that the visualization graphs are very similar across different 𝜆λs with the simulation data. Thus, the proposed method is quite robust with 𝜆λ. We set 𝜆=0.75λ=
0.75 in all computational experiments with real data for comparison.
2.2. The scRNA-seq Datasets
Four scRNA-seq datasets are used for evaluation. They span a wide range of cell types with known numbers of subpopulations, representing a broad spectrum of single-cell data. The first dataset
consists of embryonic stem cells under different cell cycle stages [2], which includes 8989 genes, 182 cells, and 3 known cell subtypes. The second dataset contains pluripotent cells under different
environment conditions [20], which has 10,685 genes, 704 cells, and 3 cell subtypes. The third one is composed of eleven cell populations including neural cells and blood cells [21], which contains
14,805 genes, 249 cells, and 11 cell subtypes. The fourth dataset consists of neuronal cells with sensory subtypes [5], which includes 17,772 genes, 622 cells, and 4 cell subtypes. Cell subtypes in
each data are known in advance, providing nice data sources for performance evaluation.
3. Results
3.1. Simulation Data
It is challenging to compare the performance of different visualization software. One reliable approach is to perform simulations with ground-truth available. Our simulation is based on the
artificial tree data from PHATE (Potential of Heat-diffusion for Affinity-based Transition Embedding) [22]. PHATE was recently developed with a novel informational distance developed from diffusion
processes, and was an efficient tool for visualizing continuous progression and trajectories. The artificial tree included 10 branches, and data for each branch was uniformly sampled in 60
dimensions. The sample size of the tree data was 1440 with branch information available, providing a nice source for ssPCA assessment and software comparisons. The goal for this visualization is not
only to detect the clusters (local data structure), but also to recover the branching trajectories of the simulated tree correctly. Note that PHATE is also a visualization software. To evaluate the
true power of ssPCA, compared to PCA, t-SNE, and UMAP, we did not use the final similarity matrix learned from PHATE, but constructed a simple Gaussian kernel matrix from the data, and the 10
branches were regarded as the cluster labels. The same kernel was used to evaluate the performance of UMAP, t-SNE, and ssPCA, respectively. With the data dimensions of 60, no PCA is carried out for
t-SNE and UMAP with original tree data. However, PCA is used for dimension reduction before we perform UMAP and t-SNE with kernel. The number of PCs used for UMAP and t-SNE is set to 30 in this
simulation. The results with different methods are reported in Figure 1.Figure 1. Visualization results with artificial tree data. Top panel from left to right: principal component analysis (PCA)
with Data; t-distributed stochastic neighbor embedding (t-SNE) with Data, and Uniform Manifold Approximation and Projection (UMAP) with Data. Bottom panel from left to right: UMAP with Kernel; t-SNE
with Kernel; ssPCA with the same Kernel; and PHATE (the ground-truth).Figure 1 demonstrates that neither PCA nor t-SNE recover the tree structure of the data correctly. PCA leads to artificially
overlapping, while t-SNE with data shatters the tree structure into discrete clusters, and we cannot recognize the tree structure from t-SNE with kernel, indicating that t-SNE does not preserve the
global data structure. UMAP with both data and kernel seems to be better at preserving the global tree structure, a few tree branches are connected together. However, tree branches in UMAP are also
shattered into discrete pieces. The global tree structure is not completely recovered and the intercluster distances are meaningless with UMAP. PHATE performs the best and represents the ground-truth
with this artificial tree data. It correctly visualizes the global and local tree structures. Although ssPCA is mainly designed for visualizing cluster structures, it performs second-best and
correctly visualizes the global tree structures. The intercluster distances in the artificial tree data are correctly preserved with ssPCA. ssPCA also discovers most of the clusters (branches)
correctly. It only fails to distinguish a couple of tree branches that are close to each other because of the noises. PHATE performs better than ssPCA with this specific dataset, but not without the
costs. There are more than 5 hyperparameters in PHATE, and different parameter settings may lead to quite different visualization results. Particularly, PHATE is sensitive to the choice of the noise
parameter 𝜎σ. The visualizations of PHATE are quite different with different noises 𝜎σs, although the true structure of the artificial tree is the same. Additional simulations are carried out with
the parameters set to (i) the number of dimensions of 5000; (ii) number of samples of 1200; (iii) the number of branches of 8; and (iv) the different noise levels of 𝜎=3σ=3, 6, and 12, respectively.
PCA is performed for dimension reduction before we carry out UMAP and t-SNE. The number of PCs is also set to 30 with more than 70%70% of explained variance. The visualization results are reported in
Appendix A Figure A1, Figure A2 and Figure A3. PHATE leads to overfitting the tree-structure when the noise in the data is relatively low (𝜎=3σ=3 and 𝜎=6σ=6), but performs better with higher noises
(𝜎=12σ=12). ssPCA on the other hand, is robust with different choices of the hyperparameter (𝜆=0.25λ=0.25, 0.50.5, 0.750.75, and 1) and different noises (𝜎=3σ=3, 6, and 12), as presented in the
top-right and bottom panels of Figure A1, Figure A2 and Figure A3. ssPCA also performs better than UMAP. In addition, ssPCA is computationally more efficient than PHATE, t-SNE, and OMAP. These three
software utilize a gradient-decent approach to find the optimal solution, which is relatively time-consuming. The computational time for PCA, ssPCA, t-SNE, UMAP, and PHATE are 0.018, 1.96, 41.88,
8.05, and 13.24 s, respectively, with the artificial tree data running on an Intel core i7 laptop with 12-GB memory. In conclusion, ssPCA preserves both global and local structures of the data. When
transition and progressions exist in the branches of artificial tree, ssPCA captures the branching trajectories well.
3.2. Real Data
Our computational evaluations with four real datasets are based on the cell similarity matrix and clustering labels from three popular scRNA-seq software including SIMLR [19], SoptSC [23], and
sinNLRR [24]. t-SNE was implemented in their original packages for visualization. As PHATE is not solely designed for cluster visualization, we only compare the performances of ssPCA with t-SNE and
UMAP using the real datasets. Note that PCA is carried out for dimension reduction before we perform UMAP and t-SNE. The number of PCs is set to 20, with more than 70%70% of explained variance for
all 4 datasets.SIMLR [19] is denoted as single-cell interpretation via multikernel enhanced similarity learning. It learns the cell-to-cell similarities through efficient multikernel optimization.
The similarity matrices from SIMLR for the 4 real single-cell datasets are visualized with ssPCA, t-SNE, and UMAP, respectively.
The perplexity value for t-SNE and the number of nearest neighbors for UMAP are set to 30. We actually try several perplexity values (10, 20, 30, 40) and choose the one with a better visualization.
The clusters with different scRNA-seq data are visualized in Figure 2.Figure 2. Visualization results with similarity matrices from SIMLR with 4 real single-cell RNA sequencing (scRNA-seq) datasets,
where plots in the top row are generated by semisupervised principal component analysis (ssPCA) with 𝜆=0.75λ=0.75, and plots in the middle row and bottom row are produced with t-distributed
stochastic neighbor embedding (t-SNE) and UMAP, respectively. Datasets to draw the the subplots from left to right: Buettner [2]; Kolodziejczyk [20]; Pollen [21]; Usoskin [5].While ssPCA, t-SNE, and
UMAP can separate the clusters in different datasets with the similarity matrices from SIMLR, ssPCA performs better in recovering the global cluster structures relative to one another. For instance,
with the Buettner scRNA-seq data, ssPCA demonstrates that cluster 2 (green) is close to cluster 1 (red), and far away from cluster 3 (blue), while the relative locations (distances) of the three
clusters with t-SNE and UMAP might not mean anything. In addition, with the Pollen data, t-SNE and UMAP display all clusters uniformly on the plane, while ssPCA shows that cluster 7 (light-blue) is
far away from the rest clusters. Finally, with the Usoskin data, cluster 3 (light blue) is displayed as several pieces with t-SNE and UMAP, but it is visualized as one entity and adjacent to cluster
1 (red) and cluster 2 (green) with ssPCA. This is reasonable. As demonstrated with the simulation data, ssPCA preserves both global and local data structures with principal component projection,
while t-SNE and UMAP only capture the local data structure, and the global structure is not fully preserved. The main reason is that both t-SNE and UMAP are based on the nearest neighbor graph
technique which only optimizes the data points close to each other.SoptSC [23] is a recently developed software that learns cell–cell similarities through locality-preserving low-rank representation.
The similarity matrices from the same 4 scRNA-seq datasets are visualized with ssPCA, t-SNE, and UMAP, respectively, as shown in Figure 3.Figure 3. Visualization results with similarity matrices from
SoptSC with 4 real scRNA-seq datasets, where plots in the top row are generated by ssPCA with 𝜆=0.75λ=0.75, and plots in the middle and bottom rows are produced with t-SNE and UMAP, respectively.
Datasets to draw the the subplots from left to right: Buettner [2]; Kolodziejczyk [20]; Pollen [21]; Usoskin [5].Figure 3 demonstrates that ssPCA, t-SNE, and UMAP can separate the clusters in 3 out
of 4 datasets including Kolodziejczyk, Pollen, and Usoskin with the similarity matrices from SoptSC. However, for the Buettner scRNA-seq data, t-SNE and UMAP fail to distinguish cluster 1 (red) from
cluster 2 (green) and cluster 3 (blue), while ssPCA is able to recover these clusters as separate entities, although there are some overlaps among them. With the other 3 datasets, ssPCA provides
additional information about the clusters relative to one another. For instance, with the Usoskin dataset, cluster 3 (light blue) is located at the center, and links clusters 1, 2, and 4 together
with ssPCA, while it is hard to identify such related information of the clusters with t-SNE and UMAP, because the intercluster distances with t-SNE and UMAP are meaningless. Similar visualization
results with sinNLRR [24] are reported in Figure A4 of the Appendix A. SoptSC and sinNLRR are based on the same idea of locality-preserving low-rank representation. The only difference is that F-norm
is used in SoptSC, while 𝐿21L21 norm is used in sinNLRR. Thus, the similarity matrices generated by SoptSC and sinNLRR with the 4 scRNA-seq data are comparable, leading to similar visualization
4. Discussion
Visualization of the high-dimensional RNA-seq data is critical for detecting cell subpopulations and revealing biological insights. To date, there are only a few tools including PCA, t-SNE, UMAP, and
PHATE available for dimension-reduction and scRNA-seq data visualization. ssPCA provides another viable tool for scRNA-seq visualization and has its advantages. More specifically, PCA projects
high-dimensional data into a low-dimensional space through eigenvalue decomposition. PCA is a linear projection method, and it mainly captures the global structure of the data, as demonstrated in
Figure 1 of the artificial tree visualization. However, scRNA-seq data are usually not linear, and the nonlinear structure in scRNA-seq data cannot be detected by PCA. ssPCA, on the other hand, is a
nonlinear kernel extension of PCA. It reduces the nonlinear noises and projects the scRNA-seq data onto a low-dimensional manifold.t-SNE and UMAP are the two popular local graph algorithms for
scRNA-seq data visualization. While t-SNE is the most popular tool currently used in the literature, UMAP produces somewhat similar output with increased speed. However, one key disadvantage with
t-SNE and UMAP is that they only preserve the local neighborhood structure in the data. The global structures are not correctly visualized. As demonstrated in Figure 1 and Appendix A Figure A1,
Figure A2 and Figure A3, both t-SNE and UMAP tends to shatter the continuous structures into discrete clusters, and the relative location of clusters in t-SNE and UMAP generally has no meaning.
ssPCA, on the other hand, integrates local cluster dependence with global principal components. It is able to maintain both global and local structures, as demonstrated in Figure 1 and Figure A1,
Figure A2 and Figure A3. Moreover, in practice, both t-SNE and UMAP utilize PCA as a dimension reduction prior, because of the large number of genes in scRNA-sq data. Although UMAP can handle the
high-dimensional data efficiently, PCA for dimension reduction is still necessary, due to the curse of dimensionality. The distance between cells in high-dimension tends to be very similar, leading
to deteriorating performance in cluster visualization. On the contrary, ssPCA finds the principal components in one-step, which is computationally more efficient.PHATE [22] is a recently developed
software for dimension reduction and visualization. It visualizes the simulated tree structure better when the noise in the artificial tree data is high, but tends to overfit the data and leads to a
too-complex visualization when the noise is relatively low, as demonstrated in Figure 1 and Appendix A Figure A1, Figure A2 and Figure A3. Furthermore, there are too many parameters in PHATE, and
different parametric settings may lead to different visualizations. On the contrary, ssPCA only has one hyperparameter 𝜆λ, and it is robust with different values of 𝜆λ over different noise levels, as
demonstrated in Figure A1, Figure A2 and Figure A3.The only hyperparameter 𝜆λ (0≤𝜆≤10≤λ≤1) measures the trade-off between the total variance of the data and cluster dependence. When 𝜆=0λ=0, ssPCA
becomes a standard kernel PCA that maximizes the total variance. On the other hand, when 𝜆=1λ=1, the projection is solely based on maximizing cluster dependence of the data. As ssPCA is robust with
different values of 𝜆λ in the simulation, we recommend that you pick a 𝜆λ value between 0.25–0.75 in practice. Since ssPCA is computational efficient, you may run ssPCA multiple times with different
values of hyperparameter 𝜆λ, and get a better sense of how the projection is affected by 𝜆λ.One interesting finding with the Pollen data is that ssPCA identifies 3 distinct clusters with different
kernel (similarity) matrices from different software packages, while 11 clusters were discovered in their original study. Whether this discrepancy is from the novel finding with ssPCA or from the
limitation of ssPCA is not known. More investigations are required in the near future. Finally, although the same 4 scRNA-seq datasets are used for Figure 2 and Figure 3, the visualizations are not
exactly the same with different similarity matrices from different software. Thus, the choice of similarity (kernel) matrices is also crucial for scRNA-seq visualization.
5. Conclusions
We propose a semisupervised principal component method (ssPCA) with HSIC maximization for cell subtype visualization. ssPCA optimizes both local cluster dependence and global principal component
projection. It has an analytical solution, and is robust with respect to different values of hyperparameter 𝜆λ and different noise levels in the data. Thus, ssPCA has its advantages over PCA, t-SNE,
UMAP, and PHATE. The key advantages with ssPCA are that it preserves both local and global structures in the data faithfully, and the principal component projection with ssPCA is more interpretable
than that from t-SNE and UMAP. However, it is important to remember that no visualization technique is perfect, and ssPCA is no exception. Unlike t-SNE and UMAP, in which cluster labels are only
optional, ssPCA must have cluster labels as its input, indicating that ssPCA can only be used when cluster information is available. However, through integration with a clustering software, ssPCA is
still a powerful tool for scRNA-seq data visualization. It also provides an alternative to standard visualization methods such as t-SNE and UMAP.
This research received no external funding.
This research is partially supported by Four Diamonds Fund, Pennsylvania State University.
Conflicts of Interest
The author declares no conflict of interest.
The following abbreviations are used in this manuscript:
PCA Principal Component Analysis
ssPCA Semisupervised Principal Analysis
t-SNE t-distributed Stochastic Neighbor Embedding
scRNA-seq Single Cell RNA-sequencing
PHATE Potential of Heat-diffusion for Affinity-based Transition Embedding
UMAP Uniform Manifold Approximation and Projection
We did more simulations with artificial tree structure visualization where the parameters were set to the following: (i) the number of dimensions of 5000; (ii) number of samples of 1200; (iii) the
number of branches of 8; and (iv) the different noises of 𝜎=3σ=3, 6, and 12, respectively.Figure A1. Visualization results with artificial tree data with the noise of 𝜎=3σ=3. Up-left: t-SNE;
Up-middle: PHATE; Up-right: UMAP; Bottom panels from left to right: ssPCA with 𝜆=0.25λ=0.25, 0.50.5, 0.750.75, and 1, respectively.Figure A2. Visualization results with artificial tree data with the
noise of 𝜎=6σ=6. Up-left: t-SNE; Up-middle: PHATE; Up-right: UMAP; Bottom panels from left to right: ssPCA with 𝜆=0.25λ=0.25, 0.50.5, 0.750.75, and 1, respectively.Figure A3. Visualization results
with artificial tree data with the noise of 𝜎=12σ=12. Up-left: t-SNE; Up-middle: PHATE; Up-right: UMAP; Bottom panels from left to right: ssPCA with 𝜆=0.25λ=0.25, 0.50.5, 0.750.75, and 1,
respectively.Figure A4. Visualization results from sinNLRR with 4 real scRNA-seq datasets, where plots in the top row are generated by ssPCA with 𝜆=0.75λ=0.75, and plots in the middle and bottom rows
are produced with t-SNE and UMAP, respectively. Datasets to draw the the subplots from left to right: Buettner [2]; Kolodziejczyk [20]; Pollen [21]; Usoskin [5]. | {"url":"https://www.seekquence.com/t-sne-umap-and-sspcr-scrna-seq-visualisation","timestamp":"2024-11-10T19:16:03Z","content_type":"text/html","content_length":"72105","record_id":"<urn:uuid:e6631fdf-534c-410c-9a60-4ea81ed85935>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.61/warc/CC-MAIN-20241110170046-20241110200046-00173.warc.gz"} |
The Open University Statistics - M248 TMA 01 Question Two » MyMathLab Help| Pay Us to Do Your Statistics Online Today
Complete the following paragraph by selecting words
You should be able to answer this question after working through Unit 2.
(a) Complete the following paragraph by selecting words or phrases from the list that follows it to fill in the underlined gaps.
In a long sequence of repetitions of a study or experiment, random samples tend to settle down towards probability distributions in the sense that, for discrete data, bar charts settle down towards
probability functions and, for continuous data, histograms settle down towards probability functions. As the sample size increases, the amount of difference between successive graphical displays
obtained from the data .
Available words and phrases: continuous cumulative decreases density discrete
frequency increases mass model models relative frequency remains constant unimodal unit-area [3]
(b) Kevin lives in a city which operates a bicycle hire scheme using a large number of bicycle ‘docking stations’ spread around the city. He walks past a small docking station, for up to six
bicycles, each morning. Kevin has come up with the following probability mass function (p.m.f.) for the distribution of the random variable X which denotes the number of bicycles available at the
docking station each morning.
It is given in
Table 1.
Table 1 The p.m.f. of X
x 0 1 2 3 4 5 6
p(x) 0.3 0.2 0.2 0.1 0.1 0.05 0.05
(i) What is the range of X? [1]
(ii) Explain why the p.m.f. suggested by Kevin is a valid p.m.f. [2]
(iii) What is the probability that, on any particular morning, there is one bicycle at the docking station? [1]
(iv) Write down a table containing values of F(x), the cumulative distribution function (c.d.f.) of X, for x = 0; 1; 2; 3; 4; 5; 6. [2]
(v) Write the probabilities P(X < 3) and P(X ≥ 5) in terms of the c.d.f. F(x). Use the c.d.f. to calculate the values of these two probabilities.
(c) In 1955, C.W. Topp and F.C. Leone introduced a number of
distributions in the context of the statistical modelling of the reliability of electronic components in engineering. One of these distributions has probability density function (p.d.f.) given by f
(x) = 4x(1 − x)(2 − x) on the range 0 < x < 1.
(i) Verify, by integration, that Integrate( 4x(1 − x)(2 − x)) dx = x2(2 − x)2 + c; where c is an arbitrary constant
(ii) Explain why the p.d.f. suggested by Topp and Leone is a valid
p.d.f. [4]
(iii) What is the c.d.f. associated with this p.d.f.? [2] (iv) Suppose that X is a random variable following this p.d.f., and that we are interested in evaluating P(1/3 < X < 2/3). Write this
probability in terms of the c.d.f., and hence show that P (1/3 < X < 2/3)= 39 81
(which is approximately 0.481)
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Vector Calculus: Understanding Divergence
Physical Intuition
Divergence (div) is “flux density”—the amount of flux entering or leaving a point. Think of it as the rate of flux expansion (positive divergence) or flux contraction (negative divergence). If you
measure flux in bananas (and c’mon, who doesn’t?), a positive divergence means your location is a source of bananas. You’ve hit the Donkey Kong jackpot.
Remember that by convention, flux is positive when it leaves a closed surface. Imagine you were your normal self, and could talk to points inside a vector field, asking what they saw:
• If the point saw flux entering, he’d scream that everything was closing in on him. This is a negative divergence, and the point is capturing flux, like water going down a sink.
• If the point saw flux leaving, he’d sniff his armpits and say all flux was existing. This is a positive divergence, and the point is a source of flux, like a hose.
So, divergence is just the net flux per unit volume, or “flux density”, just like regular density is mass per unit volume (of course, we don’t know about “negative” density). Imagine a tiny cube—flux
can be coming in on some sides, leaving on others, and we combine all effects to figure out if the total flux is entering or leaving.
The bigger the flux density (positive or negative), the stronger the flux source or sink. A div of zero means there’s no net flux change in side the region. In plain english:
$\displaystyle{\text{ Divergence } = \frac{\text{Flux}}{\text{Volume}}}$
Math Intuition
Now that we have an intuitive explanation, how do we turn that sucker into an equation? The usual calculus way: take a tiny unit of volume and measure the flux going through it. We need to add up the
total flux passing through the x, y and z dimensions.
Imagine a cube at the point we want to measure, with sides of length dx, dy and dz. To get the net flux, we see how much the X component of flux changes in the X direction, add that to the Y
component’s change in the Y direction, and the Z component’s change in the Z direction. If there are no changes, then we’ll get 0 + 0 + 0, which means no net flux.
If there is some change in the field, we get something like 1 -2 +5 (flux increases in X and Z direction, decreases in Y) which gives us the divergence at that point.
In pseudo-math:
Total flux change = (field change in X direction) + (field change in Y direction) + (field change in Z direction)
Or in more formal math:
$\displaystyle{\text{Divergence} = \lim_{\text{Vol} \to 0}\frac{\text{Flux}}{\text{Vol}}}$
$\displaystyle{\text{Divergence} = \frac{\partial F_x}{\partial x} +\frac{\partial F_y}{\partial y} +\frac{\partial F_z}{\partial z}}$
(Assuming $F_x$ is the field in the x-direction.)
A few remarks:
• The symbol for divergence is the upside down triangle for gradient (called del) with a dot [$\triangledown \cdot$]. The gradient gives us the partial derivatives $(\frac{\partial}{\partial x}, \
frac{\partial}{\partial y}, \frac{\partial}{\partial z})$, and the dot product with our vector $(F_x, F_y, F_z)$ gives the divergence formula above.
• Divergence is a single number, like density.
• Divergence and flux are closely related – if a volume encloses a positive divergence (a source of flux), it will have positive flux.
• "Diverge" means to move away from, which may help you remember that divergence is the rate of flux expansion (positive div) or contraction (negative div).
Divergence isn’t too bad once you get an intuitive understanding of flux. It’s really useful in understanding in theorems like Gauss’ Law.
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Math 340 Home
Textbook Contents
Online Homework Home
Series Solutions
So now we know that given a linear variable coefficient initial value problem we can find all the derivatives of the solution at the initial point. That means we can write that solution as a Taylor
series about the initial point. Rather than work this out by repeatedly differentiating the equation as in the last section, however, it is easier to guess that we have a solution in the form of a
power series, plug this guess in to the equation, and solve for the coefficients. We will apply this approach to second order linear homogeneous differential equations in this section (restricting to
second order homogeneous equations because they are both the most common in application and because the algebra for them tends to be a little simpler, though as you will see "simpler" is a relative
term). This is a lengthy process. To keep everything straight, it helps to follow an explicit plan.
Find the general solution to $$ y'' - y' +xy = 0.$$
Step 1:
Guess $y(x) = \displaystyle \sum_{n=0}^{\infty}a_n(x-x_0)^n$ and compute all the different pieces. In this case, I will choose $x_0 = 0$. $$ xy &= \sum_{n=0}^{\infty}a_nx^{n+1} \\ -y' &= \sum_{n=1}^
{\infty}-na_nx^{n-1} \\ y'' &=\sum_{n=2}^{\infty}n(n-1)a_nx^{n-2} $$
Step 2:
Change all the terms to the form $x^n$ or $x^j$, etc. Let $j=n+1$, $k=n-1$, and $p=n-2$ $$ xy &= \sum_{j=1}^{\infty}a_{j-1}x^j \\ -y'&= \sum_{k=0}^{\infty}-(k+1)a_{k+1}x^{k} \\ y'' &= \sum_{p=0}^{\
infty}(p+2)(p+1)a_{p+2}x^p $$
Step 3:
Change all the indices to the same letter (I use $m$) and plug into the equation. $$ y'' - y' + xy = \sum_{m=0}^{\infty}(m+2)(m+1)a_{m+2}x^m + \sum_{m=0}^{\infty}-(m+1)a_{m+1}x^{m} +\sum_{m=1}^{\
infty}a_{m-1}x^m $$
Step 4:
Collect like terms. $$ (2a_2 - a_1 ) + \sum_{m=1}^{\infty} \left[(m+2)(m+1)a_{m+2} - (m+1)a_{m+1} + a_{m-1}\right]x^m = 0 $$ Here the first term $2a_2-a_1$ is the coefficient of $x^0$ and the sum
contains the general term. Since not all pieces have an $x^0$ term, it must be separated from the general term.
Step 5:
Equate coefficients to 0. $$ 2a_2 - a_1 &= 0 \\ (m+2)(m+1)a_{m+2} - (m+1)a_{m+1} + a_{m-1} &= 0, \qquad\text{for $m\ge1$} $$ and we rewrite the last equation by solving for the highest coefficient,
$a_{m+2}$, to obtain $$ a_{m+2}=\frac{(m+1)a_{m+1}-a_{m-1}}{(m+2)(m+1)}\qquad\text{for $m\ge1$.} $$ This last equality is called the
recurrence relation
for the differential equation.
Step 6:
Plug in $a_0 = 1$, $a_1 = 0$ to find the first solution $y_1(x)$. $$ a_0 &= 1 \\ a_1 &= 0 \\ a_2 &= (1/2)a_1 = 0 \\ a_3&=\frac{2a_2-a_0}{6}=-1/6 \\ a_4&=\frac{3a_3-a_1}{12}=-1/24 \\ &\vdots \\ y_1
(x) &= 1 - (1/6)x^3 - (1/24)x^4 + \ldots $$
Step 7:
Plug in $a_0 = 0$, $a_1 = 1$ to find the second solution $y_2(x)$. $$ a_0 &= 0 \\ a_1 &= 1 \\ a_2 &= (1/2)a_1 = 1/2 \\ a_3&=\frac{2a_2-a_0}{6}=1/6 \\ &\vdots \\ y_2 (x) &= x + (1/2)x^2 + (1/6)x^3 + \
ldots $$
Step 8:
The general solution is then $$ y(x) = c_1 y_1 (x) + c_2 y_2 (x). $$ Note that by our choice of $a_0$ and $a_1$ in our two solutions and since $x_0=0$, we have $y(0) = c_1$ and $y'(0) = c_2$. This
makes it easy to find $c_1$ and $c_2$ to solve initial value problems. Also note that if you are going to need to solve initial value problems with the initial values given at some point $a$, then
you should choose $x_0 = a$ when you find the series solution so you will be able to find the $c_1$ and $c_2$ easily.
Find the general solution to $$ (1-x^2)y''-xy'+4y=0 $$ Step 1: $$\eqalign { 4y&=\sum_{n=0}^{\infty}4a_nx^n \cr -xy'&=\sum_{n=1}^{\infty}-na_nx^n \cr (1-x^2)y''&=\sum_{n=2}^{\infty}n(n-1)a_nx^{n-2} -\
sum_{n=2}^{\infty}n(n-1)a_nx^n \cr } $$ Step 2: Let $j=n-2$, then $$ \eqalign { 4y&=\sum_{n=0}^{\infty}4a_nx^n \cr -xy'&=\sum_{n=1}^{\infty}-na_nx^n \cr (1-x^2)y''&=\sum_{j=0}^{\infty}(j+2)(j+1)a_
{j+2}x^j -\sum_{n=2}^{\infty}n(n-1)a_nx^n \cr } $$ Step 3: $$ (1-x^2)y''-xy'+4y = \eqalign { \sum_{m=0}^{\infty}&(m+2)(m+1)a_{m+2}x^m-\sum_{m=2}^{\infty}m(m-1)a_mx^m \cr &-\sum_{m=1}^{\infty}ma_mx^m+
\sum_{m=0}^{\infty}4a_mx^m \cr } = 0 $$ Step 4: $$ (2a_2+4a_0)+(6a_3-a_1+4a_1)x+ \sum_{m=2}^{\infty}\bigl((m+2)(m+1)a_{m+2}-m(m-1)a_m-ma_m+4a_m\bigl)x^m=0 $$ Step 5: $$ \eqalign { 2a_2+4a_0&=0 \cr
6a_3+3a_1&=0 \cr (m+2)(m+1)a_{m+2}-m^2a_m+4a_m&=0,\qquad m\ge2 \cr} $$ The recurrence relation is $$ a_{m+2}=\frac{(m^2-4)a_m}{(m+2)(m+1)},\qquad m\ge2 $$ Step 6: $$ \eqalign { a_0&=1 \cr a_1&=0 \cr
a_2&=\frac{-4}{2}a_0=-2 \cr a_3&=0 \cr a_4&=\frac{0}{12}a_2=0 \cr a_5&=0 \cr a_6&=0 \cr &\vdots \cr} $$ $$ y_1(x)=1-2x^2 $$ Step 7: $$ \eqalign { a_0&=0 \cr a_1&=1 \cr a_2&=0 \cr a_3&=\frac{-3}{6}a_1
=-\frac12 \cr a_4&=0 \cr a_5&=\frac{5}{20}a_3=-\frac18 \cr &\vdots \cr } $$ $$ y_2(x)=x-\frac12x^3-\frac18x^5+\cdots $$ Step 8: The general solution is $$ y(x)=c_1y_1(x)+c_2y_2(x). $$ Note that all
coefficients of $y_1(x)$ are 0 after $a_2$ and so $y_1(x)$ is actually a polynomial instead of an infinite series. The equation $(1-x^2)y''-xy'+\alpha^2y=0$ is called Chebyshev's equation. For every
integer $\alpha$, this equation has a solution which is a polynomial of degree $\alpha$, called the Chebyshev polynomial (actually, for technical reasons relating to the specific applications of the
Chebyshev polynomial, most books would multiply our answer by $1/2^{\alpha}$ and define that to be the Chebyshev polynomial of order $\alpha$). These polynomials are important in approximation
theory. As always, you are welcome to stop by my office and I'll discuss Chebyshev polynomials and their applications in more detail.
Solve the initial value problem $$ y''-x^2y=0,\qquad y(0)=1,\quad y'(0)=0 $$ Step 1: Since we have initial data at $x=0$, we choose $x_0=0$ and so $$ y=\sum_{n=0}^{\infty}a_nx^n $$ from which we
derive $$ -x^2y&=\sum_{n=0}^{\infty}-a_nx^{n+2} \\ y''&=\sum_{n=2}^{\infty}n(n-1)a_nx^{n-2} $$ Step 2: Let $j=n+2$ and $k=n-2$ so $$ -x^2y&=\sum_{j=2}^{\infty}-a_{j-2}x^j \\ y''&=\sum_{k=0}^{\infty}
(k+2)(k+1)a_{k+2}x^k $$ Step 3: $$ y''-x^2y=\sum_{m=0}^{\infty}(m+2)(m+1)a_{m+2}x^m +\sum_{m=2}^{\infty}-a_{m-2}x^m $$ Step 4: $$ 2a_2+6a_3x+\sum_{m=2}^{\infty}\bigl[(m+2)(m+1)a_{m+2}-a_{m-2}\bigl]x^
m=0 $$ Step 5: $$ 2a_2&=0 \\ 6a_3&=0 \\ (m+2)(m+1)a_{m+2}-a_{m-2}&=0,\qquad\text{for $m\ge2$} $$ and from the last equation we obtain the recurrence relation $$ a_{m+2}=\frac{a_{m-2}}{(m+2)(m+1)},\
qquad\text{for $m\ge2$}. $$ Step 6: Rather than find the general solution and plug in the initial values, we are going to plug in the initial values right now. We recall that $$ a_0&=y(x_0) \\ \text
{and}\qquad a_1&=y'(x_0) $$ and since we chose $x_0=0$ we can now read off $$ a_0&=1 \\ a_1&=0 $$ from the initial conditions. We then obtain $$ a_2&=0 \\ a_3&=0 $$ from our first two equations in
step 5. Finally, we apply the recurrence relation to obtain $$ a_4&=\frac{a_0}{(2+2)(2+1)}=\frac{1}{12} \\ a_5&=\frac{a_1}{(3+2)(3+1)}=0 \\ a_6&=\frac{a_2}{(4+2)(4+1)}=0 \\ a_7&=\frac{a_3}{(5+2)
(5+1)}=0 \\ a_8&=\frac{a_4}{(6+2)(6+1)}=\frac{1}{672} \\ &\vdots $$ from which we get $$ y(x)=1+(1/12)x^4+(1/672)x^8+\ldots $$ You can generate
additional examples of series solutions for variable coefficient equations here
. If you have any problems with this page, please contact bennett@math.ksu.edu.
©2010, 2014 Andrew G. Bennett | {"url":"https://onlinehw.math.ksu.edu/math340book/chap4/seriessoln.php","timestamp":"2024-11-09T07:17:00Z","content_type":"text/html","content_length":"16084","record_id":"<urn:uuid:7b4c84b5-fa12-43c4-bd1a-c058557a368a>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.30/warc/CC-MAIN-20241109053958-20241109083958-00785.warc.gz"} |
Python Function: Maximum Contests
Oops, something went wrong. Please try again in a few moments.
def organize_contests(n, a):
This function takes in an integer n and an array a of length n, where a[i] denotes the number of problems on the ith topic that Bob wrote.
It returns the maximum number of contests that can be organized using the problems given by Bob such that:
- Each contest has distinct problems (no two or more problems with the same topic).
- Each contest except the first one has more problems than the previous contest.
n (int): The number of elements in the array a
a (list): The array of integers describing the number of problems on each topic
int: The maximum number of contests that can be organized
# Check if the input array is valid
if not isinstance(a, list) or len(a) != n:
raise ValueError("Invalid input array")
# Sort the array in descending order
# Initialize the number of contests to 1
num_contests = 1
# Initialize the number of problems in the first contest to the first element of the sorted array
num_problems = a[0]
# Iterate through the remaining elements of the sorted array
for i in range(1, n):
# If the number of problems on the current topic is less than or equal to the number of problems in the previous contest, skip it
if a[i] <= num_problems:
# Otherwise, add the number of problems on the current topic to the number of problems in the current contest
num_problems += a[i]
# Increment the number of contests
num_contests += 1
# Return the number of contests
return num_contests
except Exception as e:
# Log the error
print(f"Error: {e}")
return 0 | {"url":"https://codepal.ai/code-generator/query/nDUWDwo1/python-function-maximum-contests","timestamp":"2024-11-04T17:37:43Z","content_type":"text/html","content_length":"97573","record_id":"<urn:uuid:f64d436e-64cf-4ed4-87b7-9b39c08fc430>","cc-path":"CC-MAIN-2024-46/segments/1730477027838.15/warc/CC-MAIN-20241104163253-20241104193253-00306.warc.gz"} |
Phase Space Representation of Dynamical Systems
13.2 Phase Space Representation of Dynamical Systems
The differential constraints defined in Section 13.1 are often called kinematic because they can be expressed in terms of velocities on the C-space. This formulation is useful for many problems, such
as modeling the possible directions of motions for a wheeled mobile robot. It does not, however, enable dynamics to be expressed. For example, suppose that the simple car is traveling quickly. Taking
dynamics into account, it should not be able to instantaneously start and stop. For example, if it is heading straight for a wall at full speed, any reasonable model should not allow it to apply its
brakes from only one millimeter away and expect it to avoid collision. Due to momentum, the required stopping distance depends on the speed. You may have learned this from a drivers education course.
To account for momentum and other aspects of dynamics, higher order differential equations are needed. There are usually constraints on acceleration , which is defined as . For example, the car may
only be able to decelerate at some maximum rate without skidding the wheels (or tumbling the vehicle). Most often, the actions are even expressed in terms of higher order derivatives. For example,
the floor pedal of a car may directly set the acceleration. It may be reasonable to consider the amount that the pedal is pressed as an action variable. In this case, the configuration must be
obtained by two integrations. The first yields the velocity, and the second yields the configuration.
The models for dynamics therefore involve acceleration in addition to velocity and configuration . Once again, both implicit and parametric models exist. For an implicit model, the constraints are
expressed as
For a parametric model, they are expressed as
Subsections Steven M LaValle 2020-08-14 | {"url":"https://lavalle.pl/planning/node667.html","timestamp":"2024-11-09T20:13:14Z","content_type":"text/html","content_length":"9510","record_id":"<urn:uuid:f1a9e33b-ee6e-448d-9e58-4e06796d9379>","cc-path":"CC-MAIN-2024-46/segments/1730477028142.18/warc/CC-MAIN-20241109182954-20241109212954-00088.warc.gz"} |
Detailed Description
methods to query information about or strengthen the problem at the current local search node
SCIP_RETCODE SCIPaddConflict (SCIP *scip, SCIP_NODE *node, SCIP_CONS *cons, SCIP_NODE *validnode, SCIP_CONFTYPE conftype, SCIP_Bool iscutoffinvolved)
SCIP_RETCODE SCIPclearConflictStore (SCIP *scip, SCIP_EVENT *event)
SCIP_RETCODE SCIPaddConsNode (SCIP *scip, SCIP_NODE *node, SCIP_CONS *cons, SCIP_NODE *validnode)
SCIP_RETCODE SCIPaddConsLocal (SCIP *scip, SCIP_CONS *cons, SCIP_NODE *validnode)
SCIP_RETCODE SCIPdelConsNode (SCIP *scip, SCIP_NODE *node, SCIP_CONS *cons)
SCIP_RETCODE SCIPdelConsLocal (SCIP *scip, SCIP_CONS *cons)
SCIP_Real SCIPgetLocalOrigEstimate (SCIP *scip)
SCIP_Real SCIPgetLocalTransEstimate (SCIP *scip)
SCIP_Real SCIPgetLocalDualbound (SCIP *scip)
SCIP_Real SCIPgetLocalLowerbound (SCIP *scip)
SCIP_Real SCIPgetNodeDualbound (SCIP *scip, SCIP_NODE *node)
SCIP_Real SCIPgetNodeLowerbound (SCIP *scip, SCIP_NODE *node)
SCIP_RETCODE SCIPupdateLocalDualbound (SCIP *scip, SCIP_Real newbound)
SCIP_RETCODE SCIPupdateLocalLowerbound (SCIP *scip, SCIP_Real newbound)
SCIP_RETCODE SCIPupdateNodeDualbound (SCIP *scip, SCIP_NODE *node, SCIP_Real newbound)
SCIP_RETCODE SCIPupdateNodeLowerbound (SCIP *scip, SCIP_NODE *node, SCIP_Real newbound)
SCIP_RETCODE SCIPchgChildPrio (SCIP *scip, SCIP_NODE *child, SCIP_Real priority)
Function Documentation
◆ SCIPaddConflict()
SCIP_RETCODE SCIPaddConflict ( SCIP * scip,
SCIP_NODE * node,
SCIP_CONS * cons,
SCIP_NODE * validnode,
SCIP_CONFTYPE conftype,
SCIP_Bool iscutoffinvolved
adds a conflict to a given node or globally to the problem if node == NULL.
SCIP_OKAY is returned if everything worked. Otherwise a suitable error code is passed. See SCIP_RETCODE for a complete list of error codes.
this method can be called in one of the following stages of the SCIP solving process:
this method will release the constraint
scip SCIP data structure
node node to add conflict (or NULL if global)
cons constraint representing the conflict
validnode node at whichaddConf the constraint is valid (or NULL)
conftype type of the conflict
iscutoffinvolved is a cutoff bound involved in this conflict
Definition at line 3228 of file scip_prob.c.
References Scip::conflictstore, FALSE, Scip::mem, NULL, SCIP_Mem::probmem, Scip::reopt, SCIP_CALL, SCIP_CONFTYPE_BNDEXCEEDING, SCIP_NODETYPE_PROBINGNODE, SCIP_OKAY, SCIP_Real, SCIPaddCons(),
SCIPaddConsNode(), SCIPcheckStage(), SCIPconflictstoreAddConflict(), SCIPconsMarkConflict(), SCIPgetCutoffbound(), SCIPinfinity(), SCIPnodeGetType(), SCIPreleaseCons(), Scip::set, Scip::stat,
Scip::transprob, Scip::tree, and TRUE.
Referenced by applyCliqueFixings(), SCIP_DECL_CONFLICTEXEC(), and setupAndSolveSubscipRapidlearning().
◆ SCIPclearConflictStore()
SCIP_RETCODE SCIPclearConflictStore ( SCIP * scip,
SCIP_EVENT * event
removes all conflicts depending on an old cutoff bound if the improvement of the incumbent is good enough
SCIP_OKAY is returned if everything worked. Otherwise a suitable error code is passed. See SCIP_RETCODE for a complete list of error codes.
this method can be called in one of the following stages of the SCIP solving process:
tries to remove conflicts depending on an old cutoff bound if the improvement of the new incumbent is good enough
SCIP_OKAY is returned if everything worked. Otherwise a suitable error code is passed. See SCIP_RETCODE for a complete list of error codes.
this method can be called in one of the following stages of the SCIP solving process:
scip SCIP data structure
event event data
Definition at line 3286 of file scip_prob.c.
References Scip::conflictstore, SCIP_Primal::cutoffbound, FALSE, Scip::mem, NULL, Scip::primal, SCIP_Mem::probmem, Scip::reopt, SCIP_CALL, SCIP_EVENTTYPE_BESTSOLFOUND, SCIP_OKAY, SCIPcheckStage(),
SCIPconflictstoreCleanNewIncumbent(), SCIPeventGetSol(), SCIPeventGetType(), Scip::set, Scip::stat, Scip::transprob, and TRUE.
Referenced by SCIP_DECL_EVENTEXEC().
◆ SCIPaddConsNode()
SCIP_RETCODE SCIPaddConsNode ( SCIP * scip,
SCIP_NODE * node,
SCIP_CONS * cons,
SCIP_NODE * validnode
adds constraint to the given node (and all of its subnodes), even if it is a global constraint; It is sometimes desirable to add the constraint to a more local node (i.e., a node of larger depth)
even if the constraint is also valid higher in the tree, for example, if one wants to produce a constraint which is only active in a small part of the tree although it is valid in a larger part. In
this case, one should pass the more global node where the constraint is valid as "validnode". Note that the same constraint cannot be added twice to the branching tree with different "validnode"
parameters. If the constraint is valid at the same node as it is inserted (the usual case), one should pass NULL as "validnode". If the "validnode" is the root node, it is automatically upgraded into
a global constraint, but still only added to the given node. If a local constraint is added to the root node, it is added to the global problem instead.
SCIP_OKAY is returned if everything worked. Otherwise a suitable error code is passed. See SCIP_RETCODE for a complete list of error codes.
this method can be called in one of the following stages of the SCIP solving process:
scip SCIP data structure
node node to add constraint to
cons constraint to add
validnode node at which the constraint is valid, or NULL
Definition at line 3323 of file scip_prob.c.
References FALSE, Scip::mem, NULL, SCIP_Mem::probmem, SCIP_CALL, SCIP_INVALIDDATA, SCIP_OKAY, SCIPcheckStage(), SCIPconsGetName(), SCIPconsSetLocal(), SCIPerrorMessage, SCIPnodeAddCons(),
SCIPnodeGetDepth(), SCIPprobAddCons(), SCIPtreeGetEffectiveRootDepth(), Scip::set, Scip::stat, Scip::transprob, Scip::tree, TRUE, and SCIP_Cons::validdepth.
Referenced by addBranchingComplementaritiesSOS1(), addLocalConss(), addSplitcons(), branchBalancedCardinality(), branchCons(), createNAryBranch(), executeStrongBranching(), fixVariableZeroNode(),
SCIP_DECL_BRANCHEXECLP(), SCIP_DECL_BRANCHEXECPS(), SCIP_DECL_CONSPROP(), SCIPaddConflict(), SCIPaddConsLocal(), and selectVarMultAggrBranching().
◆ SCIPaddConsLocal()
SCIP_RETCODE SCIPaddConsLocal ( SCIP * scip,
SCIP_CONS * cons,
SCIP_NODE * validnode
adds constraint locally to the current node (and all of its subnodes), even if it is a global constraint; It is sometimes desirable to add the constraint to a more local node (i.e., a node of larger
depth) even if the constraint is also valid higher in the tree, for example, if one wants to produce a constraint which is only active in a small part of the tree although it is valid in a larger
If the constraint is valid at the same node as it is inserted (the usual case), one should pass NULL as "validnode". If the "validnode" is the root node, it is automatically upgraded into a global
constraint, but still only added to the given node. If a local constraint is added to the root node, it is added to the global problem instead.
SCIP_OKAY is returned if everything worked. Otherwise a suitable error code is passed. See SCIP_RETCODE for a complete list of error codes.
this method can be called in one of the following stages of the SCIP solving process:
The same constraint cannot be added twice to the branching tree with different "validnode" parameters. This is the case due internal data structures and performance issues. In such a case you
should try to realize your issue using the method SCIPdisableCons() and SCIPenableCons() and control these via the event system of SCIP.
adds constraint locally to the current node (and all of its subnodes), even if it is a global constraint; It is sometimes desirable to add the constraint to a more local node (i.e., a node of larger
depth) even if the constraint is also valid higher in the tree, for example, if one wants to produce a constraint which is only active in a small part of the tree although it is valid in a larger
If the constraint is valid at the same node as it is inserted (the usual case), one should pass NULL as "validnode". If the "validnode" is the root node, it is automatically upgraded into a global
constraint, but still only added to the given node. If a local constraint is added to the root node, it is added to the global problem instead.
SCIP_OKAY is returned if everything worked. Otherwise a suitable error code is passed. See SCIP_RETCODE for a complete list of error codes.
this method can be called in one of the following stages of the SCIP solving process:
The same constraint cannot be added twice to the branching tree with different "validnode" parameters. This is the case due to internal data structures and performance issues. In such a case you
should try to realize your issue using the method SCIPdisableCons() and SCIPenableCons() and control these via the event system of SCIP.
scip SCIP data structure
cons constraint to add
validnode node at which the constraint is valid, or NULL
Definition at line 3393 of file scip_prob.c.
References FALSE, NULL, SCIP_CALL, SCIP_OKAY, SCIPaddConsNode(), SCIPcheckStage(), SCIPtreeGetCurrentNode(), Scip::tree, and TRUE.
Referenced by addAllConss(), createConflict(), SCIP_DECL_CONSINITLP(), and upgradeCons().
◆ SCIPdelConsNode()
SCIP_RETCODE SCIPdelConsNode ( SCIP * scip,
SCIP_NODE * node,
SCIP_CONS * cons
disables constraint's separation, enforcing, and propagation capabilities at the given node (and all subnodes); if the method is called at the root node, the constraint is globally deleted from the
problem; the constraint deletion is being remembered at the given node, s.t. after leaving the node's subtree, the constraint is automatically enabled again, and after entering the node's subtree, it
is automatically disabled; this may improve performance because redundant checks on this constraint are avoided, but it consumes memory; alternatively, use SCIPdisableCons()
SCIP_OKAY is returned if everything worked. Otherwise a suitable error code is passed. See SCIP_RETCODE for a complete list of error codes.
this method can be called in one of the following stages of the SCIP solving process:
scip SCIP data structure
node node to disable constraint in
cons constraint to locally delete
Definition at line 3424 of file scip_prob.c.
References FALSE, Scip::mem, NULL, SCIP_Mem::probmem, Scip::reopt, SCIP_CALL, SCIP_OKAY, SCIP_STAGE_EXITPRESOLVE, SCIP_STAGE_INITPRESOLVE, SCIPcheckStage(), SCIPconsDelete(), SCIPconsIsAdded(),
SCIPnodeDelCons(), SCIPnodeGetDepth(), SCIPtreeGetEffectiveRootDepth(), Scip::set, SCIP_Set::stage, Scip::stat, Scip::transprob, Scip::tree, and TRUE.
Referenced by branchCons(), and createNAryBranch().
◆ SCIPdelConsLocal()
SCIP_RETCODE SCIPdelConsLocal ( SCIP * scip,
SCIP_CONS * cons
disables constraint's separation, enforcing, and propagation capabilities at the current node (and all subnodes); if the method is called during problem modification or at the root node, the
constraint is globally deleted from the problem; the constraint deletion is being remembered at the current node, s.t. after leaving the current subtree, the constraint is automatically enabled
again, and after reentering the current node's subtree, it is automatically disabled again; this may improve performance because redundant checks on this constraint are avoided, but it consumes
memory; alternatively, use SCIPdisableCons()
SCIP_OKAY is returned if everything worked. Otherwise a suitable error code is passed. See SCIP_RETCODE for a complete list of error codes.
this method can be called in one of the following stages of the SCIP solving process:
disables constraint's separation, enforcing, and propagation capabilities at the current node (and all subnodes); if the method is called during problem modification or at the root node, the
constraint is globally deleted from the problem; the constraint deletion is being remembered at the current node, s.t. after leaving the current subtree, the constraint is automatically enabled
again, and after reentering the current node's subtree, it is automatically disabled again; this may improve performance because redundant checks on this constraint are avoided, but it consumes
memory; alternatively, use SCIPdisableCons()
SCIP_OKAY is returned if everything worked. Otherwise a suitable error code is passed. See SCIP_RETCODE for a complete list of error codes.
this method can be called in one of the following stages of the SCIP solving process:
SCIP stage does not get changed
scip SCIP data structure
cons constraint to locally delete
Definition at line 3474 of file scip_prob.c.
References SCIP_Cons::addconssetchg, FALSE, Scip::mem, NULL, Scip::origprob, SCIP_Mem::probmem, Scip::reopt, SCIP_CALL, SCIP_INVALIDCALL, SCIP_OKAY, SCIP_STAGE_EXITPRESOLVE, SCIP_STAGE_INITPRESOLVE,
SCIP_STAGE_PRESOLVING, SCIP_STAGE_PROBLEM, SCIP_STAGE_SOLVING, SCIPcheckStage(), SCIPconsDelete(), SCIPconsIsAdded(), SCIPerrorMessage, SCIPnodeDelCons(), SCIPnodeGetDepth(), SCIPtreeGetCurrentNode()
, SCIPtreeGetEffectiveRootDepth(), Scip::set, SCIP_Set::stage, Scip::stat, Scip::transprob, Scip::tree, and TRUE.
Referenced by addAllConss(), analyzeZeroResultant(), checkBounddisjunction(), checkKnapsack(), checkLogicor(), checkRedundancy(), checkVarbound(), consdataFixOperandsOne(), consdataFixResultantZero()
, detectRedundantVars(), initPricing(), presolveRedundantConss(), processBinvarFixings(), processFixings(), processRealBoundChg(), propagateCons(), propCardinality(), propConsSOS1(), propIndicator(),
propSOS2(), SCIP_DECL_CONSACTIVE(), solveIndependentCons(), solveSubproblem(), and upgradeCons().
◆ SCIPgetLocalOrigEstimate()
SCIP_Real SCIPgetLocalOrigEstimate ( SCIP * scip )
gets estimate of best primal solution w.r.t. original problem contained in current subtree
estimate of best primal solution w.r.t. original problem contained in current subtree
this method can be called in one of the following stages of the SCIP solving process:
Definition at line 3527 of file scip_prob.c.
References FALSE, NULL, Scip::origprob, SCIP_CALL_ABORT, SCIP_INVALID, SCIPcheckStage(), SCIPnodeGetEstimate(), SCIPprobExternObjval(), SCIPtreeGetCurrentNode(), Scip::set, Scip::transprob,
Scip::tree, and TRUE.
Referenced by SCIP_DECL_DISPOUTPUT().
◆ SCIPgetLocalTransEstimate()
SCIP_Real SCIPgetLocalTransEstimate ( SCIP * scip )
gets estimate of best primal solution w.r.t. transformed problem contained in current subtree
estimate of best primal solution w.r.t. transformed problem contained in current subtree
this method can be called in one of the following stages of the SCIP solving process:
Definition at line 3546 of file scip_prob.c.
References FALSE, NULL, SCIP_CALL_ABORT, SCIP_INVALID, SCIPcheckStage(), SCIPnodeGetEstimate(), SCIPtreeGetCurrentNode(), Scip::tree, and TRUE.
Referenced by branchBalancedCardinality(), branchCons(), branchUnbalancedCardinality(), enforceConflictgraph(), enforceConssSOS1(), enforceSOS2(), SCIP_DECL_BRANCHEXECLP(), and SCIP_DECL_BRANCHEXECPS
◆ SCIPgetLocalDualbound()
SCIP_Real SCIPgetLocalDualbound ( SCIP * scip )
gets dual bound of current node
dual bound of current node
this method can be called in one of the following stages of the SCIP solving process:
Definition at line 3566 of file scip_prob.c.
References FALSE, NULL, Scip::origprob, SCIP_CALL_ABORT, SCIP_INVALID, SCIPcheckStage(), SCIPnodeGetLowerbound(), SCIPprobExternObjval(), SCIPtreeGetCurrentNode(), Scip::set, Scip::transprob,
Scip::tree, and TRUE.
Referenced by SCIP_DECL_DISPOUTPUT(), and SCIP_DECL_HEUREXEC().
◆ SCIPgetLocalLowerbound()
SCIP_Real SCIPgetLocalLowerbound ( SCIP * scip )
◆ SCIPgetNodeDualbound()
SCIP_Real SCIPgetNodeDualbound ( SCIP * scip,
SCIP_NODE * node
gets dual bound of given node
dual bound of a given node
this method can be called in one of the following stages of the SCIP solving process:
scip SCIP data structure
node node to get dual bound for
Definition at line 3605 of file scip_prob.c.
References FALSE, Scip::origprob, SCIP_CALL_ABORT, SCIPcheckStage(), SCIPnodeGetLowerbound(), SCIPprobExternObjval(), Scip::set, Scip::transprob, and TRUE.
Referenced by applyDomainChanges(), and writeBounds().
◆ SCIPgetNodeLowerbound()
SCIP_Real SCIPgetNodeLowerbound ( SCIP * scip,
SCIP_NODE * node
gets lower bound of given node in transformed problem
lower bound of given node in transformed problem
this method can be called in one of the following stages of the SCIP solving process:
scip SCIP data structure
node node to get dual bound for
Definition at line 3622 of file scip_prob.c.
References FALSE, SCIP_CALL_ABORT, SCIPcheckStage(), SCIPnodeGetLowerbound(), and TRUE.
Referenced by execRelpscost(), SCIPtreemodelSelectCandidate(), and subscipdataCopySubscip().
◆ SCIPupdateLocalDualbound()
SCIP_RETCODE SCIPupdateLocalDualbound ( SCIP * scip,
SCIP_Real newbound
if given value is tighter (larger for minimization, smaller for maximization) than the current node's dual bound (in original problem space), sets the current node's dual bound to the new value
the given new bound has to be a dual bound, i.e., it has to be valid for the original problem.
SCIP_OKAY is returned if everything worked. Otherwise a suitable error code is passed. See SCIP_RETCODE for a complete list of error codes.
this method can be called in one of the following stages of the SCIP solving process:
scip SCIP data structure
newbound new dual bound for the node (if it's tighter than the old one)
Definition at line 3646 of file scip_prob.c.
References FALSE, Scip::origprob, SCIP_CALL, SCIP_INVALIDCALL, SCIP_OKAY, SCIP_STAGE_PRESOLVED, SCIP_STAGE_PRESOLVING, SCIP_STAGE_PROBLEM, SCIP_STAGE_SOLVING, SCIPABORT, SCIPcheckStage(),
SCIPerrorMessage, SCIPprobExternObjval(), SCIPprobInternObjval(), SCIPprobUpdateDualbound(), SCIPtreeGetCurrentNode(), SCIPupdateNodeLowerbound(), Scip::set, SCIP_Set::stage, Scip::transprob,
Scip::tree, and TRUE.
Referenced by setupAndSolveSubscipRapidlearning(), and setupProblem().
◆ SCIPupdateLocalLowerbound()
SCIP_RETCODE SCIPupdateLocalLowerbound ( SCIP * scip,
SCIP_Real newbound
if given value is larger than the current node's lower bound (in transformed problem), sets the current node's lower bound to the new value
the given new bound has to be a lower bound, i.e., it has to be valid for the transformed problem.
SCIP_OKAY is returned if everything worked. Otherwise a suitable error code is passed. See SCIP_RETCODE for a complete list of error codes.
this method can be called in one of the following stages of the SCIP solving process:
scip SCIP data structure
newbound new lower bound for the node (if it's larger than the old one)
Definition at line 3696 of file scip_prob.c.
References FALSE, Scip::origprob, SCIP_CALL, SCIP_INVALIDCALL, SCIP_OKAY, SCIP_STAGE_PRESOLVED, SCIP_STAGE_PRESOLVING, SCIP_STAGE_SOLVING, SCIPABORT, SCIPcheckStage(), SCIPerrorMessage,
SCIPprobExternObjval(), SCIPprobUpdateDualbound(), SCIPtreeGetCurrentNode(), SCIPupdateNodeLowerbound(), Scip::set, SCIP_Set::stage, Scip::transprob, Scip::tree, and TRUE.
Referenced by SCIP_DECL_PRICERREDCOST(), and solveComponent().
◆ SCIPupdateNodeDualbound()
SCIP_RETCODE SCIPupdateNodeDualbound ( SCIP * scip,
SCIP_NODE * node,
SCIP_Real newbound
if given value is tighter (larger for minimization, smaller for maximization) than the node's dual bound, sets the node's dual bound to the new value
SCIP_OKAY is returned if everything worked. Otherwise a suitable error code is passed. See SCIP_RETCODE for a complete list of error codes.
this method can be called in one of the following stages of the SCIP solving process:
scip SCIP data structure
node node to update dual bound for
newbound new dual bound for the node (if it's tighter than the old one)
Definition at line 3735 of file scip_prob.c.
References FALSE, Scip::origprob, SCIP_CALL, SCIP_OKAY, SCIPcheckStage(), SCIPprobInternObjval(), SCIPupdateNodeLowerbound(), Scip::set, Scip::transprob, and TRUE.
◆ SCIPupdateNodeLowerbound()
SCIP_RETCODE SCIPupdateNodeLowerbound ( SCIP * scip,
SCIP_NODE * node,
SCIP_Real newbound
if given value is larger than the node's lower bound (in transformed problem), sets the node's lower bound to the new value
SCIP_OKAY is returned if everything worked. Otherwise a suitable error code is passed. See SCIP_RETCODE for a complete list of error codes.
this method can be called in one of the following stages of the SCIP solving process:
scip SCIP data structure
node node to update lower bound for
newbound new lower bound for the node (if it's larger than the old one)
Definition at line 3757 of file scip_prob.c.
References SCIP_Primal::cutoffbound, FALSE, Scip::lp, Scip::mem, Scip::origprob, Scip::primal, SCIP_Mem::probmem, Scip::reopt, SCIP_CALL, SCIP_OKAY, SCIPcheckStage(), SCIPisGE(), SCIPnodeCutoff(),
SCIPnodeUpdateLowerbound(), Scip::set, Scip::stat, Scip::transprob, Scip::tree, and TRUE.
Referenced by branch(), enforceConflictgraph(), execRelpscost(), SCIP_DECL_BRANCHEXECLP(), SCIPcopyConcurrentSolvingStats(), SCIPupdateLocalDualbound(), SCIPupdateLocalLowerbound(), and
◆ SCIPchgChildPrio()
SCIP_RETCODE SCIPchgChildPrio ( SCIP * scip,
SCIP_NODE * child,
SCIP_Real priority
change the node selection priority of the given child
SCIP_OKAY is returned if everything worked. Otherwise a suitable error code is passed. See SCIP_RETCODE for a complete list of error codes.
this method can be called in one of the following stages of the SCIP solving process:
scip SCIP data structure
child child to update the node selection priority
priority node selection priority value
Definition at line 3791 of file scip_prob.c.
References FALSE, SCIP_CALL, SCIP_INVALIDDATA, SCIP_NODETYPE_CHILD, SCIP_OKAY, SCIPcheckStage(), SCIPchildChgNodeselPrio(), SCIPnodeGetType(), Scip::tree, and TRUE.
Referenced by SCIP_DECL_BRANCHEXECLP(). | {"url":"https://www.scipopt.org/doc-9.0.1/html/group__LocalSubproblemMethods.php","timestamp":"2024-11-08T15:46:30Z","content_type":"text/html","content_length":"99706","record_id":"<urn:uuid:e55b6f13-5443-4d64-84c5-0ed7eac1b6a5>","cc-path":"CC-MAIN-2024-46/segments/1730477028067.32/warc/CC-MAIN-20241108133114-20241108163114-00871.warc.gz"} |
The Unseen Complexities of Simple Mathematical Rules | Baking Brad
The Unseen Complexities of Simple Mathematical Rules
February 29, 2024
Exploring the Depths of Dynamical Systems and Their Surprising Behaviors
Dive into the heart of mathematical exploration where simplicity breeds complexity. This article unravels the intrigue and beauty of dynamical systems, showcasing how the iteration of simple
equations can unfold into patterns of endless variety. From the iconic Mandelbrot set to groundbreaking advancements in understanding real one-dimensional systems, the narrative weaves through the
milestones and mysteries of mathematics. It's a testament to the field's ongoing quest to decode the intricate dance between chaos and order, shedding light on the profound connections that bind
seemingly disparate areas of math.
Read the full story here: 'Entropy Bagels' and Other Complex Structures Emerge From Simple Rules
• Mathematics' power lies in its ability to generate bewildering complexity from simple, repetitive rules.
• The Mandelbrot set exemplifies how novel patterns and complexities can emerge from basic mathematical equations.
• Despite advancements, certain behaviors of dynamical systems remain unpredicted, showcasing the field's intriguing unknowns.
• Recent mathematical proofs have furthered our understanding of how real one-dimensional systems behave, marking significant strides in the field.
• The specificity and rarity of periodic sequences in dynamical systems point to deeper underlying principles governing mathematical complexity.
Mathematics is a realm where repetition and simplicity can unravel complexities that bewilder even the most seasoned researchers. This is especially true in the study of dynamical systems, the very
basic mathematical rules repeated over time. The article introduces us to how seemingly simple equations, when iterated, unveil patterns and behaviors that connect various areas of mathematics in
unforeseen ways.
One focal point is the Mandelbrot set, a fascinating example demonstrating that complexity can emerge from simple rules. The set, created by iterating a simple equation over the complex plane,
generates an infinite tapestry of patterns. Mathematicians like Matthew Baker and Giulio Tiozzo discuss the surprising nature of these discoveries and admit that despite significant progress, much
about the behavior of such systems, when started from basic conditions, remains a mystery.
Recent advancements have shed light on the behavior of dynamical systems, with mathematicians such as Misha Lyubich and Sebastian van Strien making significant contributions to understanding the
orderly behaviors amidst mathematical chaos. They've found that iteration of equations within specific ranges can lead to predictable outcomes, marking a critical step in characterizing the behavior
of real one-dimensional systems. This progress opens new doors to understanding the underlying structures of mathematical complexity and the special cases that lead to periodic behavior.
Read the full article here.
Essential Insights
• Matthew Baker: A researcher from the Georgia Institute of Technology, highlighted for his insights into the emergence of complex structures from simple mathematical rules.
• Giulio Tiozzo: A University of Toronto professor, known for his work on the behavior of iterative processes in dynamical systems.
• Misha Lyubich: A Stony Brook University mathematician who made significant contributions to understanding the behavior of quadratic equations under iteration.
• Sebastian van Strien: A professor at Imperial College London, who is working on proving a major property of real one-dimensional systems' behavior.
• Clayton Petsche: An Oregon State University mathematician who, along with Chatchai Noytaptim, proved special characteristics of dynamical systems with rational constraints.
Tags: Mathematics, Dynamical Systems, Complexity, Mandelbrot Set, Chaos Theory, Iteration, Mathematical Proofs, Real Numbers, Polynomial Equations | {"url":"https://www.bcoleman.net/p/7c915a858dd6ebb72f0a2afb5111c5d7/the-unseen-complexities-of-simple-mathematical-rules.html","timestamp":"2024-11-13T22:29:27Z","content_type":"text/html","content_length":"10981","record_id":"<urn:uuid:9b03fd90-433b-4943-b581-c6aa03a2b20e>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00542.warc.gz"} |
Lq averaging for symmetric positive-definite matrices
We propose a method to find the L[q] mean of a set of symmetric positive-definite (SPD) matrices, for 1≤q ≤2. Given a set of points, the L[q] mean is defined as a point for which the sum of q-th
power of distances to all the given points is minimum. The L[q] mean, for some value of q, has an advantage of being more robust to outliers than the standard L[2] mean. The proposed method uses a
Weiszfeld inspired gradient descent approach to compute the update in the descent direction. Thus, the method is very simple to understand and easy to code because it does not required line search or
other complex strategy to compute the update direction. We endow a Riemannian structure on the space of SPD matrices, in particular we are interested in the Riemannian structure induced by the
Log-Euclidean metric. We give a proof of convergence of the proposed algorithm to the L[q] mean, under the Log-Euclidean metric. Although no such proof exists for the affine invariant metric but our
experimental results show that the proposed algorithm under the affine invariant metric converges to the L[q] mean. Furthermore, our experimental results on synthetic data confirms the fact that the
L[1] mean is more robust to outliers than the standard L [2] mean.
Publication series
Name 2013 International Conference on Digital Image Computing: Techniques and Applications, DICTA 2013
Conference 2013 International Conference on Digital Image Computing: Techniques and Applications, DICTA 2013
Country/Territory Australia
City Hobart, TAS
Period 26/11/13 → 28/11/13
Dive into the research topics of 'Lq averaging for symmetric positive-definite matrices'. Together they form a unique fingerprint. | {"url":"https://researchportalplus.anu.edu.au/en/publications/lq-averaging-for-symmetric-positive-definite-matrices","timestamp":"2024-11-07T18:41:46Z","content_type":"text/html","content_length":"49606","record_id":"<urn:uuid:b004d4c0-666b-4ec0-8d7e-d5cdab886ed6>","cc-path":"CC-MAIN-2024-46/segments/1730477028009.81/warc/CC-MAIN-20241107181317-20241107211317-00798.warc.gz"} |
WeBWorK Standalone Renderer
The figure shows a basis
$\mathcal{B} = \lbrace \mathbf{b_1}, \mathbf{b_2} \rbrace$
and a vector
Custom basis $\mathcal{B} = \lbrace \mathbf{b_1}, \mathbf{b_2} \rbrace$
a. Write the vector $\mathbf{v}$ as a linear combination of the vectors in the basis $\mathcal{B}$. Enter a vector sum of the form 5 b1 + 6 b2. $\mathbf{v} =$
b. Find the $\mathcal{B}$-coordinate vector for $\mathbf{v}$. Enter your answer as a coordinate vector of the form <5,6>. $\lbrack \mathbf{v} \rbrack_\mathcal{B} =$
You can earn partial credit on this problem. | {"url":"https://wwrenderer.libretexts.org/render-api?sourceFilePath=Library/Hope/Multi1/04-03-Basis-change/Geom_basis_change_01.pg&problemSeed=1234567&courseID=anonymous&userID=anonymous&course_password=anonymous&answersSubmitted=0&showSummary=1&displayMode=MathJax&language=en&outputFormat=nosubmit","timestamp":"2024-11-11T13:56:54Z","content_type":"text/html","content_length":"6682","record_id":"<urn:uuid:98baae44-3793-454f-800c-f0444f66c339>","cc-path":"CC-MAIN-2024-46/segments/1730477028230.68/warc/CC-MAIN-20241111123424-20241111153424-00100.warc.gz"} |
template<int dim, typename Number , int spacedim>
void refine_and_coarsen_fixed_number (::Triangulation< dim, spacedim > &tria, const ::Vector< Number > &criteria, const double top_fraction_of_cells, const double bottom_fraction_of_cells, const
types::global_cell_index max_n_cells=std::numeric_limits< types::global_cell_index >::max())
template<int dim, typename Number , int spacedim>
void refine_and_coarsen_fixed_fraction (::Triangulation< dim, spacedim > &tria, const ::Vector< Number > &criteria, const double top_fraction_of_error, const double bottom_fraction_of_error, const
VectorTools::NormType norm_type=VectorTools::L1_norm)
This namespace provides a collection of functions that aid in refinement and coarsening of triangulations. Despite the name of the namespace, the functions do not actually refine the triangulation,
but only mark cells for refinement or coarsening. In other words, they perform the "mark" part of the typical "solve-estimate-mark-refine" cycle of the adaptive finite element loop.
In contrast to the functions in namespace GridRefinement, the functions in the current namespace are intended for parallel computations, i.e., computations, e.g., on objects of type
dim, typename Number ,
void parallel::distributed::GridRefinement::refine_and_coarsen_fixed_number ( ::Triangulation< dim, spacedim > & tria,
const ::Vector< Number > & criteria,
const double top_fraction_of_cells,
const double bottom_fraction_of_cells,
const types::global_cell_index max_n_cells = std::numeric_limits<types::global_cell_index>::max() )
Like GridRefinement::refine_and_coarsen_fixed_number, but designed for parallel computations, where each process has only information about locally owned cells.
The vector of criteria needs to be a vector of refinement criteria for all cells active on the current triangulation, i.e., it needs to be of length tria.n_active_cells() (and not
tria.n_locally_owned_active_cells()). In other words, the vector needs to include entries for ghost and artificial cells. However, the current function will only look at the indicators that
correspond to those cells that are actually locally owned, and ignore the indicators for all other cells. The function will then coordinate among all processors that store part of the triangulation
so that at the end a fraction top_fraction_of_cells of all Triangulation::n_global_active_cells() active cells are refined, rather than a fraction of the Triangulation::n_locally_active_cells on each
processor individually. In other words, it may be that on some processors, no cells are refined at all.
The same is true for the fraction of cells that is coarsened.
[in,out] tria The triangulation whose cells this function is supposed to mark for coarsening and refinement.
[in] criteria The refinement criterion for each mesh cell active on the current triangulation. Entries may not be negative.
[in] top_fraction_of_cells The fraction of cells to be refined. If this number is zero, no cells will be refined. If it equals one, the result will be flagging for global refinement.
[in] bottom_fraction_of_cells The fraction of cells to be coarsened. If this number is zero, no cells will be coarsened.
This argument can be used to specify a maximal number of cells. If this number is going to be exceeded upon refinement, then refinement and coarsening fractions
[in] max_n_cells are going to be adjusted in an attempt to reach the maximum number of cells. Be aware though that through proliferation of refinement due to
Triangulation::MeshSmoothing, this number is only an indicator. The default value of this argument is to impose no limit on the number of cells.
Definition at line 503 of file grid_refinement.cc.
dim, typename Number ,
void parallel::distributed::GridRefinement::refine_and_coarsen_fixed_fraction ( ::Triangulation< dim, spacedim > & tria,
const ::Vector< Number > & criteria,
const double top_fraction_of_error,
const double bottom_fraction_of_error,
const VectorTools::NormType norm_type = VectorTools::L1_norm )
Like GridRefinement::refine_and_coarsen_fixed_fraction, but designed for parallel computations, where each process only has information about locally owned cells.
The vector of criteria needs to be a vector of refinement criteria for all cells active on the current triangulation, i.e., it needs to be of length tria.n_active_cells() (and not
tria.n_locally_owned_active_cells()). In other words, the vector needs to include entries for ghost and artificial cells. However, the current function will only look at the indicators that
correspond to those cells that are actually locally owned, and ignore the indicators for all other cells. The function will then coordinate among all processors that store part of the triangulation
so that at the end the smallest fraction of Triangulation::n_global_active_cells (not Triangulation::n_locally_owned_active_cells() on each processor individually) is refined that together make up a
total of top_fraction_of_error of the total error. In other words, it may be that on some processors, no cells are refined at all.
The same is true for the fraction of cells that is coarsened.
[in,out] tria The triangulation whose cells this function is supposed to mark for coarsening and refinement.
[in] criteria The refinement criterion computed on each mesh cell active on the current triangulation. Entries may not be negative.
[in] top_fraction_of_error The fraction of the total estimate which should be refined. If this number is zero, no cells will be refined. If it equals one, the result will be flagging for
global refinement.
[in] bottom_fraction_of_error The fraction of the estimate coarsened. If this number is zero, no cells will be coarsened.
[in] norm_type To determine thresholds, combined errors on subsets of cells are calculated as norms of the criteria on these cells. Different types of norms can be used for
this purpose, from which VectorTools::L1_norm and VectorTools::L2_norm are currently supported.
Definition at line 576 of file grid_refinement.cc. | {"url":"https://www.dealii.org/current/doxygen/deal.II/namespaceparallel_1_1distributed_1_1GridRefinement.html","timestamp":"2024-11-02T01:51:20Z","content_type":"application/xhtml+xml","content_length":"20123","record_id":"<urn:uuid:cfce467f-51c9-4a17-b4c6-ace4e5f53e30>","cc-path":"CC-MAIN-2024-46/segments/1730477027632.4/warc/CC-MAIN-20241102010035-20241102040035-00483.warc.gz"} |
of t
The velocity field and the associated shear stress corresponding to the torsional oscillatory flow of a fractional Oldroyd-B fluid, also called generalized Oldroyd-B fluid (GOF), between two infinite
coaxial circular cylinders, are determined by means of the Laplace and Hankel transforms. Initially, the fluid and cylinders are at rest and after some time both cylinders suddenly begin to oscillate
around their common axis with different angular frequencies of their velocities. The exact analytic solutions of the velocity field and associated shear stress, that have been obtained, are presented
under integral and series forms in terms of generalized G and R functions. Moreover, these solutions satisfy the governing differential equation and all imposed initial and boundary conditions. The
respective solutions for the motion between the cylinders, when one of them is at rest, can be obtained from our general solutions. Furthermore, the corresponding solutions for the similar flow of
classical Oldroyd-B, generalized Maxwell, classical Maxwell, generalized second grade, classical second grade and Newtonian fluids are also obtained as limiting cases of our general solutions.
PAPER SUBMITTED: 2011-09-08
PAPER REVISED: 2011-07-20
PAPER ACCEPTED: 2011-07-26
, VOLUME
, ISSUE
Issue 2
, PAGES [411 - 421] | {"url":"https://thermalscience.vinca.rs/2012/2/11","timestamp":"2024-11-08T08:48:34Z","content_type":"text/html","content_length":"17905","record_id":"<urn:uuid:a79f00c4-d632-468a-a952-b7b5965a8053>","cc-path":"CC-MAIN-2024-46/segments/1730477028032.87/warc/CC-MAIN-20241108070606-20241108100606-00321.warc.gz"} |
Modeling Language
At its core, ubiquity is a modeling language and a set of scripts meant to facilitate model development and deployment. The focus of this document is on the model description language. This is a
plain text file, referred to as a system file. Each line contains a descriptor (e.g. <P>) which defines an aspect of the model, and comments are made with the hash sign (#). What follows is an
overview of the different components of the language that can be used to create a system file.
System parameters <P>
Each system parameter is specified with a name, value, lower bound, upper bound, units, whether it should be editable in the ShinyApp and the ‘type’ of parameter (grouping in the ShinyApp). The
values of eps (machine precision, smallest value that is not zero) and inf (infinity) can be used. For example to specify a parameter koffR with a value of .1 that is positive and a parameter KDR
with a value of .04 that is also positive the following would be used:
# name value lb ub units editable type
<P> koffR 0.1 eps inf 1/hr yes Target
<P> KDR 0.04 eps inf nM yes Target
Parameter sets
Often a model will be developed to incorporate different situations or scenarios. For example, a model may be used to describe both healthy and diseased individuals. When these differences are simply
parametric in nature, it can be cumbersome to code a model multiple times (once for each parameterization). This framework provides a mechanism for including multiple parameterizations withing the
same system file. Consider the system below where we want to describe antibody disposition. For humans this is described by a two compartment model, but for mice a single compartment is needed.
First we create a set of parameters describing the human scenario. These are the mean parameters taken from the literature [DM]:
<P> Weight 70.0 eps inf kg yes System # Organism weight
<P> CL 0.0129 eps inf L/hr yes System # Systemic Clearance
<P> Q 0.0329 eps inf L/hr yes System # Inter-compartmental clearance
<P> Vp 3.1 eps inf L yes System # Vol. central compartment
<P> Vt 2.8 eps inf L yes System # Vol. peripheral compartment
When a parameter is created using the <P> descriptor it is part of the default parameter set. This is the short name for a parameter set. A longer more verbose name can be given as well, and this is
what will be seen in the ShinyApp. The human parameter set can be labeled using the PSET descriptor in the following way:
<PSET:default> mAb in Human
Where default is the parameter set short name name, and “mAb in Human” is the value shown to the user in the ShinyApp.
Next, to add the parameterization for mice we simply create a new set in the following way:
<PSET:mouse> mAb in Mouse
This alone would create a new parameter set with a short name mouse, and is an exact copy of the default parameter set. To identify the parametric differences between the mouse and human we use PSET
in the following way:
<PSET:mouse:Weight > 0.020 # 20 gram mouse
<PSET:mouse:CL > 7.71e-6
<PSET:mouse:Q> 0.0
<PSET:mouse:Vp > 1.6e-3
<PSET:mouse:Vt > 1 # arbitrary
Consider the clearance parameter entry where we want the murine half-life of an antibody [VR]:
<PSET:mouse:CL> 7.71e-6
We use the set name (mouse) and the parameter name (CL) and then we overwrite the default with the specified value 7.71e-6. The other aspects of the parameter (bounds, edit flag, etc.) will be the
same as the default value.
Secondary parameters <As> and <Ad>
A static secondary parameter refers to parameter does not change during a simulation. These are specified using the <As> descriptor and can be written in terms of system parameters or previously
defined static secondary parameters. These can be used in differential equations, defining initial conditions, input scaling and model outputs. This is similar to secondary parameters defined in the
$PK block in NONMEM. For example, if you wanted to define the rate of elimination in terms of the system parameters for clearance CL and volume of distribution Vp the following would be used:
<As> kel = CL/Vp
A dynamic secondary parameters refers to a parameter that can change during a simulation. This typically means it is defined, using the <Ad> descriptor, in terms of a state or another dynamic
secondary parameter. These can be used in differential equations and model outputs. These are similar to the parameters defined in the $DES block in NONMEM. For example if you wanted to use the
concentration in the central compartment Cp but it was dependent on the amount in that compartment Ap and the volume of that compartment Vp the following would be used:
<Ad> Cp = Ap/Vp
Variance parameters <VP>
Variance parameters are specified using the same format as system parameters (<P>) :
# name value lower_bound upper_bound units editable grouping
<VP > SLOPE 0.01 eps inf 1/hr yes Variance
The difference being that the <VP> descriptor is used and that the grouping is set to Variance. These are used when performing parameter estimation and when simulating with residual variability.
Parameter estimation information <EST:?>?
This descriptor specifies information about parameters for estimation. Sometimes it is necessary to estimate parameters in the log space. Here you can specify the parmaeters to log transform (LT). If
you wanted to log transform parameters P1, P2, and P3 you would do the following:
<EST:LT> P1; P2; P3
By default all parameters will be specified for estimation. If you want to estimate a subset of parameters (P), say P1 and P2, you can use the following:
<EST:P> P1; P2
Applies to nlmixr2, and NONMEM. Essentially all the parameters not listed here will be fixed.
Output error model <OE:?> ?
This defines the output error model of the format:
`<OE:OUTPUT> expression`
The OUTPUT can be the name of any output defined with <O>. The expression is a model type (add for additive, and prop for proportional) with an equal sign and the name of the variance paramter (<VP>)
to use. To use more than one error model type you separate the statments with ; For example if you define the variance parameters add_err and prop_err and want to use a proportional error model to
the output Cp you would use:
<OE:Cp> prop=prop_err
To use both additive and proportional error the following would work:
<OE:Cp> add=add_err; prop=prop_err
Variability: defining the variance/covariance Matrix <IIV:?>? & <IIVCOR:?>?
Any variable name assigned to inter-individual variability (IIV) or correlation/covariance (IIVCOR) term that makes sense to the user may be used. The following sample codes have variable names (eg:
ETACL) that most likely make sense to a population modeler or NONMEM user. To define an IIV term named ETACL with a variance of 0.15 use the following:
<IIV:ETACL> 0.15
The next we need to associate this IIV term with a system parameter. To associate this IIV term with the clearance (system parameter CL) and specify that it has a log normal distribution (LN) we
would simply write:
<IIV:ETACL:LN> CL
Alternatively a normal (N) distribution can be used.
Next we specify the IIV term ETAV with a variance of 0.1. This IIV term also has a log normal distribution and is applied to the parameter V:
<IIV:ETAV> 0.10
<IIV:ETAV:LN> V
Now we can define the covariance (off-diagonal elements) between CL and V to be 0.01 by using:
<IIVCOR:ETAV:ETACL> 0.01
The order isn’t important and the IIV terms can be reversed
IIV and parameter sets <IIVSET:?>? & <IIVCORSET:?>?
By default all parameter sets will have inter individual variability specified using the <IIV> and <IIVCOR> descriptors. To associate a specific set of IIVs to a parameter set use the <IIVSET> and
<IIVCORSET> descriptors. These set descriptors operate differently than the parameter set descriptors (<PSET>). The <PSET> just overwrites the default values and inherits the default variance/
covariance information. If you alter the IIV information for a parameter set it will reset the IIV information for that parameter set. The entire variance covariance matrix will need to be specified
for that parameter set.
If the parameter set MYPSET has been defined then the following could be used to define the IIV for the parameters Q and CL:
<IIVSET:MYPSET:ETAQ> 0.05
<IIVSET:MYPSET:ETAQ:LN> Q
<IIVSET:MYPSET:ETACL> 0.25
<IIVSET:MYPSET:ETACL:LN> CL
<IIVCORSET:MYPSET:ETAQ:ETACL> 0.01
All the other system parameters will have no IIV information for this parameter set.
Differential equations
The differential equations in the system can be defined by simply writing them out. Alternative they can ‘built’ by using the different descriptors provided below. Part of the flexibility of ubiquity
lies in the ability to combine these different notations. To construct a model (see section below: Bringing it all together) any combination of the five following methods can be used:
1. Differential equations <ODE:?>
2. Reaction rates =?=>
3. Equilibrium relationships <=kforward:kreverse=>
4. Sources and sinks <S:?>
5. Movement between compartments <C>
Writing ODEs <ODE:?>
Portions of differential equations can be specified here where ? is the state or compartment. To define dA/dT as koffR*C - konR*A*B we would write:
<ODE:A> koffR*C - konR*A*B
It might be more convenient to specify an ODE across several lines, making things more readable. Just use multiple statements and they will be appended together. This would give the same result as
the example above:
<ODE:A> koffR*C
<ODE:A> - konR*A*B
Rate equations =?=>
It may be more convenient to write out chemical reactions rather than differential equations. This can be done using the general form:
[CR1]Reactant1 + [CR2]Reactant2 + ... =kf=> [CP1]Product1 + [CP2]Product2 + ...
Where the stoichiometric coefficients, beginning with CR and CP above, in brackets only need to be specified if they are not one. The reaction order will be assumed to be equal to the stoichiometric
coefficient of the reactant. For a more specific example Consider decomposition of hydrogen peroxide into water and oxygen:
\[ H_2O_2 \xrightarrow{k_{deg}} H_2O + \frac{1}{2}O_2 \]
In the system format this would be written in the following manner:
H2O2 =kdeg=> H2O + [0.5]O2
And this will be translated in to the following differential equations:
\[ \frac{dH_2O_2}{dt}=-k_{deg}H_2O_2 \\ \frac{dH_2O}{dt}= k_{deg}H_2O_2 \\ \frac{dO_2}{dt}= 0.5k_{deg}H_2O_2 \]
Which could also be defined as differential equations using the <ODE:?>?. This is the equivalent:
<ODE:H2O2> - kdeg*H2O2
<ODE:H2O> kdeg*H2O2
<ODE:O2> 0.5*kdeg*H2O2
The rates (e.g. kdeg) need to be defined as either a system or secondary parameter. This is where you can put saturable terms, such as Michaelis-Menten kinetics.
Equilibrium relationships <=kforward:kreverse=>
Forward and reverse reaction rates can be written separately:
A + B =konR=> C
C =koffR=> A + B
Or these can be written as equilibrium equations with the forward (konR) and reverse (koffR) rates specified as:
A + B <=konR:koffR=> C
To specify this reaction as differential equations, the following could have also been used:
<ODE:A> koffR*C - konR*A*B
<ODE:B> koffR*C - konR*A*B
<ODE:C> -koffR*C + konR*A*B
The stoichiometric coefficients also define the reaction order here. For example, to create the following equilibrium reaction:
\[ 2A + 3B \mathop{\rightleftarrows}^{\mathrm{k_f}}_{\mathrm{k_r}} 4C \]
This rate notation could be used in the system file:
[2]A + [3]B <=kf:kr=> [4]C
Which will produce the following in terms of differential equations:
\[ \frac{dA}{dt} = 2k_rC^4 - 2k_fA^2B^3 \\ \frac{dB}{dt} = 3k_rC^4 - 3k_fA^2B^3 \\ \frac{dC}{dt} =-4k_rC^4 + 4k_fA^2B^3 \] To write this equilibrium reaction as differential equations the following
would be used:
<ODE:A> = 2*kr*SIMINT_POWER[C][4] - 2*kf*SIMINT_POWER[A][2]*SIMINT_POWER[B][3]
<ODE:B> = 3*kr*SIMINT_POWER[C][4] - 3*kf*SIMINT_POWER[A][2]*SIMINT_POWER[B][3]
<ODE:C> = -4*kr*SIMINT_POWER[C][4] + 4*kf*SIMINT_POWER[A][2]*SIMINT_POWER[B][3]
See below about generic functions such as SIMINT_POWER[][].
Sources and sinks <S:?>
This method allows turnover to be described in terms of synthesis and degradation terms. If A is produced at a rate of ksynA (mass quantities), degraded at a rate of kdegA, and modeled in
concentration units then the sources are specified on the left hand side of <S:?> and the sinks (elimination) are specified on the left hand side of <S:?>. Multiple sources and sinks can be separated
by semicolons. In this example with a compartment volume V
ksynA/V <S:A> kdeg*A
Which is the same as writing out the differential equation:
<ODE:A> koffR*C - konR*A*B + ksynA/V - kdeg*A
Movement between compartments <C>
When mass moves between two physical spaces with different volumes we need to specify, for each compartment, the species, volume and rate of transport. The <C> descriptor allows us to just identify
the compartment information separated by semicolons (order is important)
Species; Volume; Rate <C> Species; Volume; Rate
For movement between a central compartment A with a volume V to the tissue space At with a volume Vt at rates kps and ksp respectively this is specified in the following manner:
A; V; kps <C> At; Vt; ksp
Which is the equivalent of the following differential equation:
<ODE:A> -kps*A + ksp*At*Vt/V
<ODE:At> +kps*A*V/Vt - ksp*At
Bringing it all together
As a final example consider the target-mediated drug disposition system above. This system can be defined with a set of ODES:
<ODE:Ct> Cp*kpt*Vp/Vt - Ct*ktp
<ODE:Cp> -Cp*kpt + Ct*ktp*Vt/Vp - kel*Cp + koff*CpTp - kon*Cp*Tp
<ODE:Tp> + ksyn/Vp - kint*Tp + koff*CpTp - kon*Cp*Tp
<ODE:CpTp> - kint*CpTp - koff*CpTp + kon*Cp*Tp
Or it could simply be defined in terms of the underlying processes:
# tissue distribution
Ct; Vt; ktp <C> Cp; Vp; kpt
# equilibrium
Cp + Tp <=kon:koff=> CpTp
# Turnover
ksyn/Vp <S:Tp> kint*Tp
<S:Cp> kel*Cp
<S:CpTp> kint*CpTp
Initial conditions <I>
By default all initial conditions are zero. You can specify a non-zero initial condition using the <I> string to set a ‘state’ to a ‘value’
<I> state = value
Value can be any combination of numbers, system parameters <P> or static secondary parameters <As. Consider a turnover system where the value of ksyn and kdeg are specified as parameters:
<P> ksyn 0.1 eps inf 1/hr yes Target
<P> kdeg 0.04 eps inf nM yes Target
We can calculate the initial value for a target as:
<As> T_IC = ksyn/kdeg
Then we can specify the initial value of the target as:
<I> T = T_IC
Model inputs
Inputs into the model include typical interventions such as bolus dosing or continuous infusions. However inputs here refers to mathematical inputs. Typically covariates may be attributes of the
system (such as gender, or a specific genotype), but are treated here as inputs. When defining inputs it is necessary to provide typical/placeholder values. These provide default values for both the
ShinyApp interface as well as the scripting level (Matlab and R) where they can be overwritten by the user.
Bolus dosing <B:times>, <B:events>
The <B:?> descriptor is used to define bolus dosing. Dosing information is broken down into a list of times when bolus injections occur and a list of events containing the amount the specified
compartment will receive.
Each of these has a scale that is used to convert the bolus dosing information from proscribed units (mg daily) into the units in which the system is coded (mg/mL and hours). So if dosing is done on
days 0, 1, 2… but the simulation time is hours, then the scale for the dosing times is 24.The events contain the magnitude of the bolus at a given time. If you want to dose into a central compartment
Cp in mg/kg and your central compartment is mg/mL then you need to scale by the body weight (e.g. 70 kg) and the volume of your central compartment (system or static secondary parameter Vp) then the
scale is 70/Vp.
If you just want to create a palceholder you can do the following:
# type state values scale units
<B:times>; [0]; 24; days
<B:events>; Cp; [0]; 70/Vc; mpk
If you want to setup default dosing for the shiny app or scripts, you can do somethign more complicated. If you have multiple compartments receiving a bolus, the times must include all times in which
a bolus may be applied to the system. If a state does not receive a bolus on a particular time, its magnitude at that time is 0.
To illustrate this consider the following dosing schedule:
In this example we want to dose two different drugs into two different states/compartments. Drug 1 (D1) will be dosed into Cp1 and drug 2 (D2) into Cp2. Dosing will be in mg/kg but concentrations are
in mg/ml. The dosing time is in days, but the simulation time units are hours. We will be dosing D1 at 8 & 2 mpk on days 0 & 6. D2 will be dosed at 5 mpk on day 9.
# type state values scale units
<B:times>; [0 6 9]; 24; days
<B:events>; Cp1; [8 2 0]; 70/V1; mpk
<B:events>; Cp2; [0 0 5]; 70/V2; mpk
Assume V1 and V2 are the compartmental volumes for D1 and D2 in ml, and the subject body weight is 70 kg. This would convert those doses in mpk into mg/ml. Again these are the default doses that can
be overwritten both in the ShinyApp and within scripts.
Continuous infusions <R:?>
Rates of infusion are defined using the <R:?> descriptor. Like bolus values, infusion rates have two components. There is a component that specifies switching times (e.g. switching from 10 mg/hr to
0). And each switching time has a corresponding rate of infusion. This infusion rate will be held constant until the next time. Also like the bolus specification there is a scale associated with both
infusion times and the levels that converts the proscriptive units into the units of the simulation. Consider the following example:
# name time/levels values scale units
<R:myrate>; times; [0 30]; 1/60; min
<R:myrate>; levels; [1 0]; 60; mg/min
These two entries create the infusion rate called myrate. This can be used in any of your system specifications (e.g., <ODE:Cp> myrate/Vp). The first row specifies the times when the rate is changed
(0 and 30 minutes). If the system is coded in terms of hours, then the scale of 1/60 must be used. The levels indicate a rate of 1 mg/min that is switched off at 30 minutes. This has to be converted
to mg/hr using the scale of 60. You can add as many paired rate entries as you need to describe as many infusion interventions as necessary.
Note: If you just want a placeholder you can just set both of the values to [0].
Covariates <CV:?>, <CVSET:?:?>
Simple covariates
For simulation purposes covariates (normally found in a dataset) need to be defined. Covariates can be either constant or change with time. The times values must be the same scale as the system. The
following defines the value for the covariate RACE:
<CV:RACE>; times; [0]; hours
<CV:RACE>; values; [1]; ethnicity
Covariates can also change with time. In this case consider the subject weight (WGT). It begins at 70 and measurements are made at several time points.
<CV:WGT>; times; [ 0 1680 3360 5040 10080]; hours
<CV:WGT>; values; [70 65 60 58 56]; kg
Next we can alter how the simulations will interpret these values by setting the type of covariate. By default the weight will be linearly interpolated (type = linear), however we can hold the weight
constant until the next measurement is encountered (last value carried forward) by declaring the type as step
<CVTYPE:WGT> step
Now if the model was parameterized for male and female subjects we can define two parameter sets (as described above) to account for this:
<PSET:default> Male
<PSET:female> Female
And the values for the covariate can be changed for the set ‘female’:
<CVSET:female:WGT>; times; [ 0 1680 3360 10080]
<CVSET:female:WGT>; values; [60 55 52 50]
Complicated covariates
Detailed time course profiles can be created as well. For example, to create a covariate profile that is zero from time 0-1 and at time 1 it jumps to 8 and decreases at a rate of 1 per unit time
until time 5 where it stays at the value 4 until time 12. It might have a profile like the following:
# name time/values values units
<CV:mycov> ; times; [0 .999 1 5 12]; hours
<CV:mycov> ; values; [0 0 8 4 4 ]; --
<CVTYPE:mycov> linear
Model outputs
Outputs are defined here in terms of states, parameters, secondary parameters, input rates, and covariates listed above. The format used is:
<O> name = expression
For example:
<O> A_obs = A
<O> Coverage = A/(KD + A)
<O> QC_CLtot = Cp*CL/Vp + Cp*Vmax/(Km+Cp)
Outputs that begin with QC, like QC_CLtot above, will not be displayed in the ShinyApp. This is intended to make them available at the scripting level for quality control (QC) purposes.
Functions and operators
Most of the standard operators behave as expected (+, -, *, & /) because most languages use these consistently. There are however certain operators and functions that differ between languages. For
example, consider the power function (\(a^b\)). In FORTRAN this would be a**b, in Matlab it is a^b, and in C it is pow(a,b). Given the objectives here (write once and create multiple formats), this
can be quite challenging. The solution used here is to convert language specific functions and operators into generic functions. So the power operator would be:
This would then be converted to the appropriate output format depending on the output target. The following generic functions can be used:
power \(a^b\) SIMINT_POWER[a][b]
exponential \(e^a\) SIMINT_EXP[a]
log base 10 \(\log(a)\) SIMINT_LOG10[a]
log base e \(\ln(a)\) SIMINT_LOGN[a]
less than \(a < b\) SIMINT_LT[a][b]
less than or equal \(a \le b\) SIMINT_LE[a][b]
greater than \(a > b\) SIMINT_GT[a][b]
greater than or equal \(a \ge b\) SIMINT_GE[a][b]
equal \(a == b\) SIMINT_EQ[a][b]
and \(a \& b\) SIMINT_AND[a][b]
Current simulation time
For some systems you will want to access the simulation time. To do this you can use the internal variable SIMINT_TIME.
Each system has default units in which it is constructed, and should be indicated in the comments of the model. It can be useful (for generating figures for example) to show simulations in different
time scales. Now this can be achieved by multiplying the time outputs by the correct scaling factor. However this requires the end user to (1) remember the original timescale and (2) correctly scale
that value.
Now while this is not particularly challenging from a mathematical perspective, it introduces a chance for error. It is possible, instead, to specify time scale information using the <TS:?>
descriptor. If the system is coded in hours, the following will define timescales for the default (hours), days, weeks and months:
<TS:hours> 1.0
<TS:days> 1.0/24.0
<TS:weeks> 1.0/24.0/7.0
<TS:months> 1.0/24.0/7.0/4.0
These are used both in the ShinyApp and at the command line in Matlab and R
Mathematical sets
Consider the following systems:
PBPK: Most of the organs in these systems are mathematically identical, with only variations in the parameters. However coding each of these organs or modifying an existing system (say to incorporate
the presences of a target in each organ) can become tedious.
Anti-drug antibody generation: If we consider ADAs generated in response to therapeutic proteins, the response will consist of a distribution of ADAs in terms of their concentration and a separate
distribution in terms of their affinity. Modeling this maturation process and the interactions between the ADAs, the therapeutic protein, and drug targets becomes unmanageable quickly.
The question is: How can we make difficult problems easy and intractable problems possible? The solution implemented here allows the system to be defined in terms of mathematical sets
Set notation <SET:?>?
Consider the interactions occurring in an assay designed to detect drug (D) present in serum. In this assay a biotinylated target (TB) is used to pull down the drug and a labeled target (TL) is the
signaling molecule used. The assay will provide a signal when a complex containing both TB and TL are present (TB:D:TL or TL:D:TB). Samples can contain soluble target as well (TS) which can interfere
with the assay. To model this assay, the following interactions should be considered:
Several options are available to construct this system. The ODEs could simply be typed out for every possible combination. It’s also possible to use the equilibrium <=kon:koff=> for all the
interactions as well. However, there is another option that will handle the enumeration more easily. First we define the two mathematical sets TSi and TSk:
<SET:TSi> TL; TB; TS
<SET:TSk> TL; TB; TS
With these defined we can then use the curly brace notation ({ }) with any of the descriptors used to construct a system. For example, the initial conditions for each of the target states are defined
as parameters (T0_TL, T0_TS, T0_TB) in the model. These have to be identified as initial conditions using the <I> notation, and can be done with a single statement. This line:
<I> {TSi} = T0_{TSi}
Expands to:
<I> TL = T0_TL
<I> TB = T0_TB
<I> TS = T0_TS
Similar to the initial condition, the equilibrium between the monomeric drug and the different targets can be defined using a single statement:
D + {TSi} <=kon:koff=> D_{TSi}
That uses only one of the sets (TSi) and will be expanded for each element in that set. To account for the formation of complexes that contain a drug molecule and two different target molecules, the
following statement is used:
D_{TSi} + {TSk} <=kon:koff=> {TSk}_D_{TSi}
This statement contains two different sets (TSi and TSk). When multiple sets are encountered, every possible combination is evaluated
Aligning Sets
By default sets will interpreted by evaluating every possible permutation as shown above. However, it may be desirable to pair or align sets. Take for example the transit compartments implemented by
Lobo and Balthasar to delay the onset of drug effect on cancer cells [LB]. The transit compartment are implemented using a series of differential equations:
\[ \frac{dK1}{dt} = (K-K1)\frac{1}{\tau} \\ \frac{dK2}{dt} = (K1-K2)\frac{1}{\tau} \\ \frac{dK3}{dt} = (K2-K3)\frac{1}{\tau} \\ \frac{dK4}{dt} = (K3-K4)\frac{1}{\tau} \]
It’s possible to code each of these individually, but it’s also possible to define these using mathematical set notation. We see that in the first equation K is paired or aligned with K1, and in the
second it’s K1 and K2, etc. So first we define two sets of equal length whose elements are aligned:
<SET:TRIN> K; K1; K2; K3
<SET:TROUT> K1; K2; K3; K4
Next we write the <ODE:?>? statement in terms of these sets, but we use the SIMINT_SET_ALIGN function:
SIMINT_SET_ALIGN[TRIN;TROUT][<ODE:{TROUT}> 1.0/tau*({TRIN}-{TROUT})]
The first argument is the names of sets to align separated by a semicolon ; and the second argument is the system file descriptor written in terms of these sets. This will be expanded internally
<ODE:K1> 1.0/tau*(K-K1)
<ODE:K2> 1.0/tau*(K1-K2)
<ODE:K3> 1.0/tau*(K2-K3)
<ODE:K4> 1.0/tau*(K3-K4)
Set Functions
It can be useful to perform operations over sets. To do this you can use the following functions:
• SIMINT_SET_SUM is the mathematical equivalent of \(\sum_{SET}\)
• SIMINT_SET_PRODUCT is the mathematical equivalent of \(\prod_{SET}\)
These functions take two bracketed arguments. The first argument is the set name and the second argument is the mathematical relationship to be expanded. For example, consider a system that has been
parameterized for several species. The variable EN_Mouse is 1 if the species is mouse and 0 otherwise. Similarly for EN_Human, EN_Monkey, etc. We have also defined the body weights as system
parameters: BW_Mouse for the mouse, BW_Human for human, etc. Now we want to assign BW to the value of the currently selected species (where EN for that species is 1). First we would define the
species parameter set:
<SET:SP> Mouse; Rat; Monkey; Human
Next we would define the secondary parameter BW by summing the product of the Boolean species parameter and the body weight for that species:
<As> BW = SIMINT_SET_SUM[SP][EN_{SP}*BW_{SP}]
Set Evaluation
Sets are evaluated in the following order:
• First set functions (SIMINT_SET_SUM and SIMINT_SET_PRODUCT) are evaluated.
• Next aligned sets are applied (SIMINT_SET_ALIGN)
• Lastly, remaining sets are evaluated for every permutation
Piecewise-continuous functions/parameters <IF:?:?>
To specify a conditional assignment use the statement:
<IF:name:COND> boolean; value
Here name is the name of the secondary parameter be defined and COND indicates that we have a Boolean condition that needs to be satisfied. The condition (boolean) can be and, or, greater than, etc.
relationships. The parameter will be assigned to have the value when this Boolean relationship is true. These conditions can be a function of different elements of the system depending on whether or
not name refers to a static or dynamic secondary parameter:
• <As> function of system parameters, previously defined static secondary parameters and covariates that do not change for a given subject.
• <Ad> function of system parameters, static secondary parameters, states, previously defined dynamic secondary parameters and covariates.
It is important to include a default ELSE condition:
<IF:name:ELSE> value
Constructing a piece-wise continuous function/parameter
To see an example use the following command (use ?system_new to see a list of the available system file examples):
system_new(system_file="pwc", file_name="system-pwc.txt")
In this example we specify fast (kelf) and slow (kels) rates of elimination. We want to have the fast rate be active when the drug concentration is above Cth and the time is below Tf. The system
parameters would look like:
<P> kelf 1.0 eps inf 1/time yes System
<P> kels 0.01 eps inf 1/time yes System
<P> Cth 10 eps inf conc yes System
<P> Tf 10 eps inf time yes System
Now we need to define the rate of elimination such that the constraints above are followed. First we define kel as a dynamic secondary parameter with a value of 0.0. Then we define the different
conditions and relevant values:
<Ad > kel = 0.0
<IF:kel:COND> SIMINT_AND[SIMINT_LT[SIMINT_TIME][Tf]][ SIMINT_GT[Cp][Cth]]; kelf
<IF:kel:ELSE> kels
Controlling indices
By default, the build script will construct odes, parameter sets, etc. automatically. This means that the order of states/compartments are going to be arbitrary. Sometimes it is necessary to specify
the order of your states or outputs. For example when using NONMEM the order in the control stream must correlate with the values in the dataset. To specify that a state Cp should be compartment 3,
the following should be used:
<INDEX:STATE:Cp> 3
The general notation for an output or state name assigned to a number is:
<INDEX:STATE:name> number
<INDEX:OUTPUT:name> number
Output error <OE:?> ?
This links parameters defined using <VP:?> to specific outputs defined using <O>. You can define either add for additive and/or prop for proportional error.
<VP> prop_err 0.1 eps inf -- yes Variance
<VP> add_err 0.1 eps inf conc yes Variance
<O> Cp = expression
<OE:Cp> add=add_error; prop=prop_error
Concentrations vs amounts
It’s more convenient to model systems in terms of concentrations. However some software does not allow scaling of inputs. And when inputs are provided in mass units, you need your differential
equation to also be in mass units. You can use the <AMTIFY> descriptor in your system.txt to tell ubiquity to generate perform this conversion on the differential equations.
For example, if you defined the state Cc but want it to be Ac within the NONMEM model target. These are related by Cc = Ac/Vc and Vc is a parameter the following would be used:
<AMTIFY> Cc; Ac; Vc
Several options can be specified using the <OPT:? delimiter. If you’ve defined the days timescale using <TS:?>? this can be used as the timescale for plotting in the ShinyApp by using:
<OPT:TS> days
To define the default output times for the ShinyApp and simulation scripts use the following:
<OPT:output_times> SIMINT_SEQ[0][100][1]
Example system files
These example system files can be found in the examples directory of the stand-alone analysis template (ubiquity_template.zip). Most can also be loaded form the R package (see the help for ?
• system-adapt.txt - Parent/metabolite structural model taken from the ADAPT5 Users Manual [ADAPT]. This system is used in the estimation examples of the ubiquity workshop.
• system-glp_study.txt - PK model parameterized for humans and NHPs, used as an example for GLP tox study design.
• system-mab_pk.txt - Linear model of mAb PK for humans taken from [DM]. This model is used in the simulation examples of the ubiquity workshop.
• system-one_cmt_cl.txt - One compartment model parameterized in terms of clearances and volumes.
• system-one_cmt_micro.txt - One compartment model parameterized for micro constants (rates).
• system-pbpk.txt - Implementation of large molecule PBPK model by Shah and Betts [SB]. This model provides good examples of how to use mathematical set notation.
• system-pbpk_template.txt - Template containing the species parameter from [SB]. This can be used to construct systems parameterized for multiple species.
• system-pwc.txt - Example of how to construct piece-wise secondary parameters (i.e. using if/then/else statements).
• system-sets.txt - Example of how to parameterized systems with multiple parameter sets.
• system-tmdd.txt - Example of how to code a TMDD model using either ODEs or process descriptors.
• system-tumor.txt - Implementation of transit effect model in cancer cell inhibition from [LB]. This demonstrates how to use aligned mathematical sets.
• system-two_cmt_cl.txt - Two compartment model parameterized in terms of clearances and volumes
• system-two_cmt_micro.txt - Two compartment model parameterized in terms of micro constants (rates)
When the system is built, multiple files are generated in the temporary directory (transient) to support different software. In the R package you can access these and other templates programatically
(see the help for ?system_fetch_template). This is a list of the supporting software and what is generated.
R workflow
• auto_simulation_driver.R: R-Script with named with placeholders used to run simulations.
• auto_analysis_estimation.R: R-Script with named with placeholders used to perform naive-pooled parameter estimation.
Matlab workflow
• auto_simulation_driver.m: M-file with named with placeholders used to run simulations.
• auto_analysis_estimation.m: M-file with named with placeholders used to perform naive-pooled parameter estimation.
Other Software Targets
• target_adapt_5.for and target_adapt_5-PSET.prm: The system defined for Adapt Fortran and a parameter (prm) file for each parameter set PSET.
• target_berkely_madonna-PSET.txt: This is a text file containing the system for the parameter set PSET to run in Berkeley Madonna.
• target_mrgsolve-PSET.cpp: This is a C++ file containing the system for the parameter set PSET to run using the in mrgsolve package in R.
• target_monolix-PSET.txt: This is a text file containing the system for the parameter set PSET to run in Monolix.
• target_nlmixr-PSET.R: This R script defines the system for analysis in nlmixr for the parameter set PSET.
• target_nonmem-PSET.ctl: This is a NONMEM control stream containing the system for the parameter set PSET. | {"url":"https://cran.ms.unimelb.edu.au/web/packages/ubiquity/vignettes/Language.html","timestamp":"2024-11-02T14:20:20Z","content_type":"text/html","content_length":"427830","record_id":"<urn:uuid:74768ca9-fe2b-4641-bf61-4922ec402be0>","cc-path":"CC-MAIN-2024-46/segments/1730477027714.37/warc/CC-MAIN-20241102133748-20241102163748-00304.warc.gz"} |
An object that encapsulates the Discrete Fourier Transform (DFT).
DFT dft( size, inverse ); // creates the object with all the data ready to start running DFTs.
dft.Apply( in, out ); // computes a DFT, repeat as necessary
dft.Initialize( size2, inverse ); // changes the options for the new size / direction
dft.Apply( in, out ); // computes a different DFT, repeat as necessary
The template can be instantiated for T = float or T = double. Linker errors will result for other types.
The DFT is computed using either PocketFFT or FFTW, depending on compile-time configuration (see the DIP_ENABLE_FFTW CMake configuration option).
When using FFTW, dip::maximumDFTSize is the largest length of the transform. PocketFFT does not have this limit.
Constructors, destructors, assignment and conversion operators
DFT() defaulted
A default-initialized DFT object is useless. Call Initialize to make it useful.
DFT(dip::uint size, bool inverse, dip::Option::DFTOptions options = {})
Construct a DFT object, see dip::DFT::Initialize for the meaning of the parameters. Note that this is not a trivial operation. Not thread safe.
void Initialize(dip::uint size, bool inverse, dip::Option::DFTOptions options = {})
Re-configure a DFT object to the given transform size and direction.
void Apply(std::complex<T>* source, std::complex<T>* destination, T scale) const
Apply the transform that the DFT object is configured for.
auto IsInverse() const -> bool
Returns true if this represents an inverse transform, false for a forward transform.
auto IsInplace() const -> bool
Returns whether the transform is configured to work in place or not. Not meaningful when using PocketFFT.
auto IsAligned() const -> bool
Returns whether the transform is configured to work on aligned buffers or not. Not meaningful when using PocketFFT.
auto TransformSize() const -> dip::uint
Returns the size that the transform is configured for.
auto BufferSize() const -> dip::uint deprecated
Returns the size of the buffer expected by Apply.
void Destroy() private
Frees memory
Function documentation
Re-configure a DFT object to the given transform size and direction.
size is the size of the transform. The two pointers passed to dip::DFT::Apply are expected to point at buffers with this length. If inverse is true, an inverse transform will be computed.
options determines some properties for the algorithm that will compute the DFT.
• dip::Option::DFTOption::InPlace means the input and output pointers passed to dip::DFT::Apply must be the same.
• dip::Option::DFTOption::TrashInput means that the algorithm is free to overwrite the input array. Ignored when working in place.
• dip::Option::DFTOption::Aligned means that the input and output buffers are aligned to 16-bit boundaries, which can significantly improve the speed of the algorithm.
When using PocketFFT, all these options are ignored.
Note that this is not a trivial operation, planning an FFT costs time.
This operation is not thread safe.
template<typename T>
void Apply(std::complex<T>* source, std::complex<T>* destination, T scale) const
Apply the transform that the DFT object is configured for.
source and destination are pointers to contiguous buffers with dip::DFT::TransformSize elements. This is the value of the size parameter of the constructor or dip::DFT::Initialize. These two pointers
can point to the same address for in-place operation; otherwise they must point to non-overlapping regions of memory. When using FFTW, the inplace parameter to the constructor or dip::DFT::Initialize
must be true if the two pointers here are the same, or false if they are different.
The input array is not marked const. If dip::Option::DFTOption::TrashInput is given when planning, the input array can be overwritten with intermediate data, but otherwise will be left intact.
scale is a real scalar that the output values are multiplied by. It is typically set to 1/size for the inverse transform, and 1 for the forward transform. | {"url":"https://diplib.org/diplib-docs/dip-DFT-T.html","timestamp":"2024-11-13T07:31:04Z","content_type":"text/html","content_length":"18790","record_id":"<urn:uuid:d30d2d81-9b38-43b9-a15f-abd5e9a21481>","cc-path":"CC-MAIN-2024-46/segments/1730477028342.51/warc/CC-MAIN-20241113071746-20241113101746-00706.warc.gz"} |
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