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Enhanced diagnostic method for rolling bearings using time-reassigned multi-synchro squeezing transform
In response to the significant challenges posed by strong non-stationarity and the vulnerability to intense background noise in rolling bearing signals, as well as the inherent limitations of
conventional convolutional neural networks (CNN) when processing one-dimensional (1D) signals without fully leveraging the inter-data relationships, this study introduces an innovative diagnostic
approach for rolling bearings. The method employs the Time-Reassigned Multi-Synchro Squeezing Transform (TMSST) to preprocess 1D vibration signals. By harnessing the temporal correlations across
various intervals, TMSST generates a set of time-frequency feature maps that are subsequently fed into a CNN to adaptively extract and classify the fault characteristics of rolling bearings. To
substantiate the efficacy of the proposed model, the Case Western Reserve University's bearing dataset serves as the benchmark for the fault diagnosis analysis. Moreover, the study incorporates
several alternative data processing techniques for comparative evaluation of the classification accuracy. The findings reveal that the proposed model, when juxtaposed with other image encoding
methods, consistently delivers superior diagnostic performance across a spectrum of load conditions and noise environments. It achieves an impressive global accuracy of 95.67 %, thereby facilitating
robust end-to-end fault pattern recognition in rolling bearings.
• This paper presents a groundbreaking TMSST-CNN model that achieves a remarkable diagnostic accuracy of 95.67% for rolling bearing faults.
• The research demonstrates significant performance improvements in the model's diagnostic precision through the application of reinforcement learning.
• Comparative analysis shows that the TMSST image encoding technique surpasses other methods in fault diagnosis.
1. Introduction
Rolling bearings, integral to the rotating machinery, exert a pivotal influence on the operational stability and longevity of the mechanism under diverse loading and positional scenarios. The
real-time surveillance of the vibration signals emanating from the machinery is of paramount importance for its stable function, offering maintenance personnel an all-encompassing assessment of the
equipment's operational status [1]. However, the conventional fault diagnosis techniques, which are predominantly dependent on the manual analysis by experts, have proven to be insufficient in
tackling the challenges of voluminous, heterogeneous, and rapid data streams characteristic of the modern machinery industry. Specifically, in the context of vast datasets from mechanical equipment
under fluctuating operational conditions, the traditional methods often encounter limitations in their monitoring capabilities and generalization performance, particularly when faced with intricate
and mutable fault information [2]. Consequently, the integration of mechanical equipment data with intelligent algorithms to forge intelligent fault diagnosis technologies has emerged as an essential
strategy to surmount these challenges [3].
The conventional intelligent fault diagnosis process is typically structured around three fundamental stages: Initially, signal acquisition is executed through sensors and related devices to gather
foundational data on the machinery's operational status. Subsequently, signal processing methodologies are applied to distill features from the acquired signals, thereby uncovering the characteristic
information indicative of equipment faults. Ultimately, leveraging the extracted feature data, machine learning (ML) or deep learning (DL) algorithms are engaged for fault identification,
ascertaining the nature and severity of the equipment's faults [4]. By amalgamating intelligent algorithms with mechanical equipment data, intelligent fault diagnosis methods not only enhance the
precision and efficiency of fault diagnosis but also promote predictive maintenance, thereby providing a solid foundation for the secure and stable operation of mechanical equipment [5].
The 1D vibration signals of rolling bearings encapsulate a wealth of information regarding their operational status, characterized by their inherent nonlinearity and non-stationarity. Consequently,
the extraction of fault features stands as an indispensable step in the realm of fault diagnosis [6]. Time-frequency analysis emerges as a robust signal processing technique that concurrently
examines both temporal and spectral aspects of a signal. The spectrum of common time-frequency analysis methods encompasses Empirical Mode Decomposition (EMD) [7], Short-Time Fourier Transform (STFT)
[8], and Wavelet Transform (WT) [9]. EMD offers the capability to adaptively decompose signals into a series of Intrinsic Mode Functions (IMFs) that represent different scale-specific components.
However, the process may encounter the problem of mode mixing, leading to inaccurate decomposition and affecting subsequent analysis and judgment. STFT, while adept at conducting time-frequency
analysis, is constrained by its fixed time resolution, which may not adequately capture the abrupt transitions present in vibration signals. This characteristic results in STFT losing some feature
quantities, leading to misjudgment of bearing fault signals. Conversely, WT is distinguished by its variable time window that contracts with increasing signal frequency and expands otherwise, thereby
extending the capabilities of STFT and mitigating its inherent limitations, which has led to its broad adoption in various applications. However, when processing signals with complex spectra, it may
not provide accurate analysis results, which can lead to misjudgment or missed judgment when analyzing bearing fault characteristics.
Yan et al. have provided a comprehensive review of the applications of Continuous Wavelet Transform (CWT), Discrete Wavelet Transform (DWT), Wavelet Packet Transform (WPT), and Second-Generation
Wavelet Transform (SGWT) within the domain of fault diagnosis [10]. TMSST is a signal processing technique particularly suited for processing nonlinear and non-stationary signals. It is an
improvement based on the traditional SST. Through time reassignment technology, TMSST further processes signals to improve the accuracy and resolution of time-frequency analysis. Time reassignment
can reduce cross-terms in time-frequency representation, thereby more clearly revealing the intrinsic structure of the signal. By adjusting the time axis of the signal, the time-frequency
representation of the signal becomes more focused, making it easier to identify and extract fault characteristics. In parallel, fault recognition is equally pivotal in the diagnostic process, as mere
feature extraction is insufficient for handling the demands of voluminous data processing. Traditional fault recognition tools encompass Bayesian classifiers [11], Artificial Neural Networks (ANNs)
[12, 13], and Support Vector Machines (SVMs). Both Bayesian classifiers and ANNs are adept at discerning fault types, provided that a substantial number of training samples are at their disposal.
However, procuring an ample dataset of fault samples in practical scenarios can be quite challenging. SVMs, endowed with robust generalization capabilities, commendable versatility, and high
classification precision, are capable of achieving effective classification even with a modest number of samples, which has propelled their widespread application in the research of mechanical fault
diagnosis [14-17]. Nonetheless, SVMs may underperform with redundant data due to their inherent limitations in learning deep features, attributed to their shallow architecture [18].
The unprecedented success of CNN in the domain of image classification has spurred significant interest in transforming sensor-collected signals into image-based representations through specialized
encoding techniques, a topic that is currently at the forefront of research [19]. Tao et al. have pioneered a method that amalgamates Short-Time Fourier Transform (STFT) with Classification
Generative Adversarial Networks (cGAN) to transmute 1D signals into two-dimensional (2D) time-frequency images, thereby achieving commendable diagnostic accuracy [20]. Yuan et al. have harnessed the
Hilbert-Huang Transform (HHT) to translate the temporal sequences of vibration signals into time-frequency images, subsequently employing a CNN to discern fault-sensitive features within the
time-frequency spectrum from these images for fault classification [21]. Zheng et al. introduced a novel Multi-Synchronous Compression S-Transform, integrating the S-Transform within a
multi-synchronous compression framework, and substantiated the efficacy of this approach through both simulated and field signals [22]. In addition, Zhou et al. have presented a rolling bearing
diagnosis methodology predicated on the Wigner-Ville Distribution (WVD) [23]. These methodologies underscore the potential of transmuting vibration signals into image representations for fault
diagnosis via CNNs. By capitalizing on the prodigious feature extraction process of CNNs, it becomes feasible to distill meaningful insights from time-frequency images and classify various fault
types with precision. This investigative trajectory is replete with promise for augmenting the fidelity and expedience of mechanical fault diagnosis systems.
This paper introduces an innovative TMSST-CNN model. which offers significant improvements in signal transformation and fault recognition. TMSST is used to convert raw data into feature-rich images
without relying on preset parameters, effectively extracting useful information from these complex signals. Subsequently, these feature maps can be combined with CNN to further enhance the accuracy
of fault diagnosis by leveraging CNN's powerful feature extraction capabilities after transforming the signals into images. The proposed methodology’s efficacy is corroborated using rolling bearing
data procured from the Case Western Reserve University Bearing Data Center and Fault Diagnosis Prototype Rig. Moreover, the model’s generalization capability is rigorously tested across a variety of
load conditions and noisy environments. The findings indicate that the TMSST-CNN model surpasses alternative 2D image encoding techniques in the realm of rolling bearing fault diagnosis, attaining an
accuracy of 95.67 %.
2. Time-reassigned multi-synchro squeezing transform
2.1. Time-reassigned synchro squeezing transform
In this section, we first briefly introduce the theoretical basis of the TSST [24]. A single-component signal with varying frequency can be described in Eq. (1):
$\stackrel{^}{s}\left(\omega \right)=A\left(\omega \right){e}^{i\phi \left(\omega \right)},$
where, $A\left(\omega \right)$ and $\phi \left(\omega \right)$ represent the amplitude and phase of the signal in the frequency domain, while $-{\phi }^{"}\left(\omega \right)$ denotes the group
delay (GD). When a signal is represented in the time-frequency domain, the Ideal Time-frequency Representation (ITFR) can be expressed as Eq. (2):
$ITFR\left(t,\omega \right)=A\left(\omega \right){e}^{i\phi \left(\omega \right)}\delta \left(t+{\phi }^{"}\left(\omega \right)\right),$
where, $\delta \left(\right)$ represents the Dirac delta function. According to Eq. (2), ideally, the time-frequency characteristics of a signal should only appear on the GD trajectory. The STFT can
be used to extend the signal given by Eq. (1) into the time-frequency domain. In the frequency domain, the STFT of signal $\stackrel{^}{g}\left(\xi \right)$ using a moving window function can be
expressed as Eq. (3):
$G\left(t,\omega \right)=\left(2\pi {\right)}^{-1}{\int }_{-\infty }^{+\infty }\stackrel{^}{s}\left(\xi \right)\stackrel{^}{g}\left(\xi -\omega \right){e}^{i\left(\xi -\omega \right)t}d\xi .$
Assuming that the analyzed signal exhibits slow frequency variations, which implies that the magnitude of $\exists \mathbit{\epsilon }$ is sufficiently small, and given that $\forall \omega$
satisfies conditions $\left|A"\left(\omega \right)\right|\le \epsilon$ and $\left|\phi "\text{'}\left(\omega \right)\right|\le \epsilon$, it is possible to derive a first-order expansion of the
signal. This expansion provides a simplified representation of the signal’s behavior, capturing its essential characteristics in the time-frequency domain. The first-order approximation allows for a
more tractable analysis while preserving key information about the signal's dynamics, as show in Eq. (4):
$\stackrel{^}{s}\left(\xi \right)=A\left(\omega \right){e}^{i\left(\phi \left(\omega \right)+{\phi }^{"}\left(\omega \right)\left(\xi -\omega \right)\right)}.$
Substituting Eq. (4) into Eq. (3), we can obtain:
$\begin{array}{l}G\left(t,\omega \right)={\left(2\pi \right)}^{-1}{\int }_{-\infty }^{+\infty }A\left(\omega \right){e}^{i\left(\phi \left(\omega \right)+\phi "\left(\omega \right)\left(\xi -\omega \
right)\right)}\stackrel{^}{g}\left(\xi -\omega \right){e}^{i\left(\xi -\omega \right)t}d\xi \\ ={\left(2\pi \right)}^{-1}A\left(\omega \right){e}^{i\phi \left(\omega \right)}{\int }_{-\infty }^{+\
infty }{e}^{i\left(t+\phi "\left(\omega \right)\right)\left(\xi -\omega \right)}\stackrel{^}{g}\left(\xi -\omega \right)d\xi \\ =A\left(\omega \right){e}^{i\phi \left(\omega \right)}g\left(t+{\phi }^
{"}\left(\omega \right)\right),\end{array}$
where, $g\left(t\right)$ represents the window function in the time domain. According to Eq. (5), the time-frequency energy propagates along the GD trajectory. To enhance the energy concentration of
Eq. (5), the 2D GD estimation is as follows:
$\stackrel{^}{t}\left(t,\omega \right)=\mathrm{R}\mathrm{e}\left(\frac{i{\partial }_{\omega }G\left(t,\omega \right)}{G\left(t,\omega \right)}\right).$
Substituting Eq. (5) into Eq. (6), we can obtain:
$\stackrel{^}{t}\left(t,\omega \right)=-{\phi }^{"}\left(\omega \right).$
Next, we perform a 1D integration of A along the time direction to compress the ambiguous time-frequency energy into the GD trajectory. This process can be expressed as Eq. (8):
$Ts\left(u,\omega \right)={\int }_{-\infty }^{+\infty }G\left(t,\omega \right)\delta \left(u-\stackrel{^}{t}\left(t,\omega \right)\right)dt.$
Combining Eq. (7) and Eq. (8), we can obtain:
$\begin{array}{l}Ts\left(u,\omega \right)={\left(2\pi \right)}^{-1}{\int }_{-\infty }^{+\infty }{\int }_{-\infty }^{+\infty }\stackrel{^}{s}\left(\xi \right)\stackrel{^}{g}\left(\xi -\omega \right)
{e}^{i\left(\xi -\omega \right)t}d\xi dt\delta \left(u+\phi "\left(\omega \right)\right)\\ ={\int }_{-\infty }^{+\infty }\stackrel{^}{s}\left(\xi \right)\stackrel{^}{g}\left(\xi -\omega \right)\delta
\left(\xi -\omega \right)\delta \left(\omega -\xi \right)d\xi \delta \left(u+{\phi }^{"}\left(\omega \right)\right)=\stackrel{^}{s}\left(\omega \right)\stackrel{^}{g}\left(0\right)\delta \left(u+{\
phi }^{"}\left(\omega \right)\right).\end{array}$
Eq. (9) illustrates that for weakly frequency-varying signals as described by Eq. (4), the TSST is capable of producing an optimal time-frequency representation. This is achieved by compressing the
ambiguous time-frequency energy onto the group delay (GD) trajectory. Nonetheless, in practical scenarios, mechanical failure-induced vibration signals are frequently tainted with noise and exhibit a
high degree of complexity. To augment the energy concentration within the time-frequency representation (TFR) for signals that are both strongly frequency-varying and strongly time-varying, the
subsequent section will introduce the technique of time-reassigned multi-synchro squeezing transform. This method is designed to enhance the clarity and precision of the TFR, thereby facilitating
more accurate analysis and diagnosis of mechanical faults.
2.2. Time-reassigned multi-synchro squeezing transform
For a strongly frequency-varying signal, where $\exists \epsilon$ is small enough and $\forall \omega$ satisfies conditions $\left|A"\left(\omega \right)\right|\le \epsilon$ and $\left|\phi "\text{'}
\left(\omega \right)\right|\le \epsilon$, the signal given by Eq. (1) can be extended as Eq. (10):
$\stackrel{^}{s}\left(\omega \right)=A\left(\omega \right){e}^{i\left(\phi \left(\omega \right)+\phi "\left(\omega \right)\left(\xi -\omega \right)+0.5\phi \text{'}\text{'}\left(\omega \right)\left(\
xi -\omega {\right)}^{2}\right)}.$
The Fourier transform of the Gaussian window function used in STFT can be expressed as Eq. (11):
$\stackrel{^}{g}\left(\omega \right)=\sqrt{2\sigma \pi }{e}^{-0.5\sigma {\omega }^{2}}.$
Substituting Eq. (10) into Eq. (3), we obtain Eq. (12):
$\begin{array}{l}G\left(t,\omega \right)={\left(2\pi \right)}^{-1}{\int }_{-\infty }^{+\infty }A\left(\omega \right){e}^{i\left(\phi \left(\omega \right)+{\phi }^{"}\left(\omega \right)\left(\xi -\
omega \right)+0.5{\phi }^{\text{'}\text{'}}\left(\omega \right)\left(\xi -\omega {\right)}^{2}\right)}\sqrt{2\sigma \pi }{e}^{-\frac{\sigma \left(\xi -\omega {\right)}^{2}}{2}}{e}^{i\left(\xi -\omega
\right)t}d\xi \\ ={\left(2\pi \right)}^{-1}\sqrt{2\sigma \pi }A\left(\omega \right){e}^{i\phi \left(\omega \right)}{\int }_{-\infty }^{+\infty }{e}^{i\left(t+{\phi }^{"}\left(\omega \right)\right)\
left(\xi -\omega \right)}{e}^{0.5\left(i{\phi }^{"\text{'}}\left(\omega \right)-\sigma \right)\left(\xi -\omega {\right)}^{2}}d\xi \\ =A\left(\omega \right){e}^{i\phi \left(\omega \right)}\sqrt{\frac
{\sigma }{\sigma -i{\phi }^{"\text{'}}\left(\omega \right)}}{e}^{-\frac{{\left(t+{\phi }^{"}\left(\omega \right)\right)}^{2}}{2\sigma -2i{\phi }^{"\text{'}}\left(\omega \right)}}.\end{array}$
According to Eq. (6), we can obtain the 2D GD estimation as Eq. (13):
$\stackrel{^}{t}\left(t,\omega \right)=-{\phi }^{"}\left(\omega \right)+\frac{{\phi }^{"\text{'}}\left(\omega {\right)}^{2}}{{\sigma }^{2}+{\phi }^{\text{'}\text{'}}\left(\omega {\right)}^{2}}\left
(t+{\phi }^{"}\left(\omega \right)\right).$
According to Eq. (13), for signals with strong frequency variations, the expression given by Eq. (6) cannot provide an accurate estimation of the true GD of the signal. Now, substituting A into Eq.
(13), we obtain Eq. (14):
$\stackrel{^}{t}\left(-{\phi }^{"}\left(\omega \right),\omega \right)=-{\phi }^{"}\left(\omega \right).$
Eq. (14) indicates that the group delay $-\phi "\left(\omega \right)$ is a fixed point of $\stackrel{^}{t}\left(t,\omega \right)$, implying that a fixed-point iteration algorithm can be employed to
reduce the error between C and D. The first iteration can be expressed as Eq. (15):
$\stackrel{^}{t}\left(\stackrel{^}{t}\left(t,\omega \right),\omega \right)=-{\phi }^{"}\left(\omega \right)+{\left(\frac{{\phi }^{"\text{'}}{\left(\omega \right)}^{2}}{{\sigma }^{2}+{\phi }^{"\text
{'}}{\left(\omega \right)}^{2}}\right)}^{2}\left(t+{\phi }^{"}\left(\omega \right)\right).$
As can be seen from Eq. (15), the fixed-point iteration algorithm effectively constructs a new 2D GD estimation $\stackrel{^}{t}\left(\stackrel{^}{t}\left(t,\omega \right),\omega \right)$. Then, we
can derive (16):
$\stackrel{^}{t}\left(\stackrel{^}{t}\left(t,\omega \right),\omega \right)+{\phi }^{"}\left(\omega \right)<\stackrel{^}{t}\left(t,\omega \right)+{\phi }^{"}\left(\omega \right).$
Eq. (16) implies that after one iteration, the new 2D GD estimation $\stackrel{^}{t}\left(\stackrel{^}{t}\left(t,\omega \right),\omega \right)$ is already closer to $-\phi "\left(\omega \right)$ than
$\stackrel{^}{t}\left(t,\omega \right)$. By performing a second iteration, we can further obtain Eq. (17):
$\stackrel{^}{t}\left(\stackrel{^}{t}\left(\stackrel{^}{t}\left(t,\omega \right),\omega \right),\omega \right)=-{\phi }^{"}\left(\omega \right)+{\left(\frac{{\phi }^{"\text{'}}\left(\omega {\right)}^
{2}}{{\sigma }^{2}+{\phi }^{\text{'}\text{'}}\left(\omega {\right)}^{2}}\right)}^{3}\left(t+{\phi }^{"}\left(\omega \right)\right).$
From Eq. (17), we can obtain Eq. (18):
$\stackrel{^}{t}\left(\stackrel{^}{t}\left(\stackrel{^}{t}\left(t,\omega \right),\omega \right),\omega \right)+{\phi }^{"}\left(\omega \right)<\stackrel{^}{t}\left(\stackrel{^}{t}\left(t,\omega \
right),\omega \right)+{\phi }^{"}\left(\omega \right).$
Comparing Eq. (18) with Eq. (16), the results indicate that with each iteration, the newly constructed 2D GD estimation becomes closer to the true $-\phi "\left(\omega \right)$. Denoting ${\stackrel
{^}{t}}^{\left[N\right]}\left(t,\omega \right)$ as the newly constructed 2D GD estimation after the $N$th iteration, we obtain Eq. (19):
${\stackrel{^}{t}}^{\left[N\right]}\left(t,\omega \right)=-{\phi }^{"}\left(\omega \right)+{\left(\frac{{\phi }^{"\text{'}}{\left(\omega \right)}^{2}}{{\sigma }^{2}+{\phi }^{"\text{'}}{\left(\omega \
right)}^{2}}\right)}^{N}\left(t+{\phi }^{"}\left(\omega \right)\right).$
Eq. (19) indicates that when the number of iterations is sufficiently large, ${\stackrel{^}{t}}^{\left[N\right]}\left(t,\omega \right)$ will approach $-\phi "\left(\omega \right)$ indefinitely, i.e.,
as show in Eq. (20):
$\underset{N\to \infty }{\mathrm{l}\mathrm{i}\mathrm{m}}{\stackrel{^}{t}}^{\left[N\right]}\left(t,\omega \right)=-{\phi }^{"}\left(\omega \right).$
Replacing ${\stackrel{^}{t}}^{\left[N\right]}\left(t,\omega \right)$ with $\stackrel{^}{t}\left(t,\omega \right)$ in Eq. (8), we obtain:
$T{s}^{\left[N\right]}\left(u,\omega \right)={\int }_{-\infty }^{+\infty }G\left(t,\omega \right)\delta \left(u-{\stackrel{^}{t}}^{\left[N\right]}\left(t,\omega \right)\right)dt.$
After sufficient iterations, we can obtain Eq. (22):
$\underset{N\to \infty }{\mathrm{l}\mathrm{i}\mathrm{m}}Ts\left(u,\omega \right)=\stackrel{^}{s}\left(\omega \right)\stackrel{^}{g}\left(0\right)\delta \left(u+{\phi }^{"}\left(\omega \right)\
Eq. (22) demonstrates that after sufficient iterations, the time-frequency energy of Eq. (21) can be effectively compressed onto the GD trajectory, even for signals with strong frequency variations.
3. Convolutional neural network structure
CNN, as quintessential exemplars of feedforward neural networks, are renowned for their distinctive features in image analysis, including local receptive fields, weight sharing, and spatial
subsampling. A canonical CNN architecture is composed of three principal layers: the Convolutional Layer (CL), the Subsampling Layer (SL), and the Fully Connected Layer (FL). In the subsequent
sections, we will explore the foundational principles and operational functions of these layers within the context of CNNs, elucidating their individual contributions to the network's overall
3.1. Convolutional Layer
The CL executes a sliding convolution operation on the input data using its set of kernels, adhering to a predefined stride. This process effectively captures features from localized regions of the
input. The output of the convolution is subsequently subjected to an activation function, yielding the resultant feature maps. In contemporary practice, the ReLU has emerged as the activation
function of choice, favored for its merits such as minimal computational overhead and accelerated training kinetics. To encapsulate the essence of the convolutional layer's mathematical framework,
its model is articulated in Eq. (23):
${x}_{j}^{l}=f\left({\sum }_{i\in {M}_{j}}{x}_{i}^{l-1}\mathrm{*}{k}_{ij}^{l}+{b}_{j}^{l}\right),$
where, * denotes the convolution operation; ${M}_{j}$ represents the selected input mapping; $l$ is the $l$th layer in the network; $k$ is the kernel matrix with a size of $S$×$S$; and $f$ is the
nonlinear activation function.
3.2. Subsampling layer
After each convolutional layer, a single subsampling layer is applied. The purpose of this layer is to reduce the size of the input features and the number of network parameters. The mathematical
model can be described as Eq. (24):
${x}_{j}^{l}=f\left({\beta }_{j}^{l}\mathrm{d}\mathrm{o}\mathrm{w}\mathrm{n}\left(2{x}_{j}^{l-1}+{b}_{j}^{l}\right)\right),$
where, down(·) represents the subsampling function. Typically, this function sums up each distinct $n$×$n$ block in the input image, resulting in an output image that is smaller by a factor of $n$ in
both spatial dimensions. Each output mapping has its own multiplicative bias $\beta$ and additive bias $b$. The subsampling function chosen in this paper is max pooling. Its main principle is to
divide the input image into a set of non-overlapping rectangles, and for each such subregion, the maximum value is output.
3.3. Subsampling layer
The fully connected layer is a traditional feedforward neural network where all neurons are connected to all activations of the previous layer. Its purpose is to collect and classify all features.
The output layer uses the Softmax function as the activation function. The Softmax function takes an arbitrary real-valued vector and compresses it to values between 0 and 1. The Softmax function is
defined as follows Eq. (25):
$\sigma \left({\begin{array}{l}z\right)\end{array}}_{j}=\frac{{e}^{{e}_{j}}}{\sum _{k=1}^{K}{e}^{{z}_{k}}},j=1,...,K.$
The ADAM optimization algorithm is employed to train the CNN, thereby optimizing the network parameters, specifically the weights and biases. ADAM's prowess lies in its ability to dynamically adjust
the learning rate for each parameter by leveraging the first-order moment estimate (mean) and the second-order moment estimate (variance) of the gradient. This adaptive approach has been instrumental
in enhancing the optimization process of CNN. Within the scope of this study, ADAM is utilized as both the feature extractor and classifier for the diagnosis of rolling bearing faults. To mitigate
the risk of model overfitting, dropout operations are strategically incorporated into the FCL. The detailed architecture of the network is delineated in Table 1 and illustrated in Fig. 1.
Fig. 1The Structure of CNN
In Fig. 1, the input is a two-dimensional image. Firstly, four layers of convolution operations are performed to extract the features of the frequency domain image of the fault vibration signal.
Then, the dimension is reduced through the pooling layer. After undergoing the above-mentioned convolution and pooling processes, all features are combined through the fully connected layer, and
softmax is used to classify different features, thereby obtaining different bearing fault categories.
Table 1CNN structure parameters
Net layer Conv kernel Number of layers
C1 Conv layer 1 7×7 4
P1 Pooling layer 1 2×2 4
C2 Conv layer 2 7×7 6
P2 Pooling layer 2 2×2 6
F1 Fully connected layer 1×1 256
3.4. Reverse parameter update
For a specific classification task, the training objective of a CNN is to minimize the loss function of the network, thus it is crucial to select an appropriate loss function. Common loss functions
include mean squared error, cross-entropy, and negative log-likelihood. In this paper, we choose the cross-entropy loss function, which has proven to be effective, and its expression is as follows
Eq. (26):
${E}^{2}=-\frac{1}{n}{\sum }_{k=1}^{n}\left[{\gamma }_{k}\mathrm{l}{\mathrm{n}}^{2}{t}_{k}+\left(1-{y}_{k}\right)\mathrm{l}\mathrm{n}\left(1-{t}_{k}{\right)}^{2}\right],$
where, $n$ represents the number of samples for a specific fault category; $t$ is the predicted value; and $y$ is the true value. During the training process, the method used to minimize the loss
function is gradient descent. By taking the first-order partial derivative of Eq. (26), the learnable parameters ($w$ and $b$) of the CNN can be updated layer by layer, as show in Eq. (27) and Eq.
${w}^{"}=w-\eta \frac{\partial E}{\partial w},$
${b}^{"}=b-\eta \frac{\partial E}{\partial b},$
where, $w\mathrm{"}$ and $b\mathrm{"}$ represent the updated weights and biases, respectively; $w$ and $b$ are the current weights and biases; $\eta$ is the learning rate parameter, which controls
the step size of weight updates. If $\eta$ is too large, it can cause the network to converge to a local optimum; if $\eta$ is too small, it will increase the training time of the network.
3.5. Reverse parameter update
The fault diagnosis method based on CNN can integrate signal preprocessing, fault feature extraction, and fault pattern classification to achieve the specific process of adaptive extraction of fault
features and intelligent diagnosis, as shown in Fig. 2. The collected vibration signals are divided into training and testing sets after TMSST. Firstly, the training set is input into the CNN for
parameter learning, and the weights ($w$) and biases ($b$) are continuously updated using the gradient descent method. Then, the trained parameters are applied to the testing set to obtain the fault
diagnosis results.
4. Experimental validation
This section aims to validate the feasibility and effectiveness of the proposed method using the measured vibration signals from rolling bearings. Furthermore, the robustness of the method under
various fault conditions will be discussed.
Fig. 2Fault diagnosis flowchart based on TMSST-CNN
4.1. Dataset description
Fig. 3 shows the experimental platform, The testbed comprises a motor, torque sensor, power meter, and electronic controller. The bearing vibration signals are measured by sensors, and the amplitudes
of these signals are represented by acceleration (https://engineering.case.edu/bearingdatacenter/apparatus-and-procedures).
To evaluate the performance of the proposed method, real bearing data were employed, which originated from the Bearing Fault Database of Case Western Reserve University [25]. This bearing fault
database is a widely used resource that contains bearing vibration data under different operating conditions and fault modes, and is extensively employed in research on fault diagnosis and
prediction. The testbed comprises a motor, torque sensor, power meter, and electronic controller. The bearing vibration signals are measured by sensors, and the amplitudes of these signals are
represented by acceleration. The database includes both normal data and data from various fault modes, such as inner race faults, outer race faults, and rolling element faults. Each fault mode has
multiple samples under different operating conditions, including varying parameters like rotational speed, load, and operating time. SKF's 6205-2RS deep groove ball bearing was taken as an example,
and the drive-end bearing data were selected for verification. Single-point faults were arranged on the inner ring, outer ring, and rolling elements of the rolling bearing using electrical discharge
machining techniques. Three fault diameters of 0.18, 0.36, and 0.54 mm were considered, with all faults having a depth of 0.28 mm. In total, nine fault types were examined. In this experiment, the
length of each segment was determined to be 300 samples. 400 samples were constructed for each type of signal feature, and One-hot encoding was adopted to label ten different bearing operating
conditions. The dataset was then divided into a training set and a test set in a 7:3 ratio. The construction of rolling bearing samples is summarized in Table 2. It includes ten different operating
conditions, including normal state and nine different fault states, and the same proportion of data sets is taken.
Fig. 3Fault bearing vibration signal acquisition platform
Table 2Sample structure of rolling bearing
0.17 0.36 0.54 0
Diameter (mm)
Rolling Inner Outer Rolling Inner Outer Rolling Inner Outer Normal Load
Label 1 2 3 4 5 6 7 8 9 10
Train 280 280 280 280 280 280 280 280 280 280
0.746 kW
Test 120 120 120 120 120 120 120 120 120 120
Fig. 4TMSST time-frequency diagrams for different types of faults: a) Normal; b) 0.17 mm rolling fault; c) 0.17 mm inner fault; d) 0.17 mm outer fault; e) 0.36 mm rolling fault; f) 0.36 mm inner
fault; g) 0.36 mm outer fault; h) 0.54 mm rolling fault; i) 0.54 mm inner fault; j) 0.54 mm outer fault
Traditional time-domain analysis has difficulties in accurately representing the damage severity and fault type characteristics of rolling bearings. Therefore, by leveraging the uniqueness of TMSST
encoding in mapping time series, the original vibration signals are encoded to generate distinct fault patterns, as shown in Fig. 4. Subsequently, these patterns are classified using CNN for the
identification of 10 types of rolling bearing features.
Fig. 4 presents the TMMST diagrams for ten distinct fault types. It is evident that traditional time-domain analysis of fault signals struggles to precisely articulate the extent of deterioration and
the distinctive characteristics of various fault types in rolling bearings. Consequently, employing the time reassignment multi-synchronous compression transformation to convert the time-domain
signals of rolling bearings into 2D time-frequency images can significantly amplify the discernible features of different fault types. As depicted in Fig. 4, signals characterized by dissimilar
damage features and fault types are challenging to discern in the time domain, whereas 2D images are adept at effectively extracting their fault characteristics. Furthermore, this study conducts a
comparative analysis of TMMST with several other signal processing techniques, including the STFT, HHT, WVD, Synchronous Compression Transform (SCT), Multiscale Synchronous Compression Transform
(MS-SCT), Time-Reassigned Multisynchronous Compression Transform (TR-MS-SCT), and Time-Reassigned Synchronous Compression Transform (TR-SCT). For instance, in fault type 2, the corresponding 2D image
is displayed in Fig. 5. Post these transformations, the CNN is integrated to classify the feature maps corresponding to the 10 types of faults.
Fig. 5Result chart of frequency domain variation methods for 0.17 mm inner fault at different times
4.2. Experimental result
To further verify the reliability of the proposed method, TMSST-CNN was used to identify ten types of faults in rolling bearings. There are a total of 4000 samples in the training and testing sets,
divided into ten types of faults. In this section, the dataset was shuffled and the fault diagnosis model of TMMST-CNN was used to verify the model with different proportions of training and testing
sets. The verification results are shown in Fig. 6.
From the confusion matrices presented in Figs. 5-6, it is evident that when the proportion of the training set is set to 70 %, a higher accuracy is achieved compared to when it is set to 60 %.
Additionally, it can be effectively observed that the majority of misclassifications occur primarily in the categorization of Fault 6 and Fault 9. Examination of Fig. 4 reveals that the faults prone
to misclassification exhibit insufficiently distinct characteristics in terms of energy distribution and varying fluctuation durations. However, upon undergoing the TMSST transformation, these
differences become more prominent, resulting in a richer set of characteristics. To further validate the superiority of the method proposed in this study, a comparison was conducted between TMMST and
other methods.
4.3. Experimental comparison
To verify the superiority of the proposed method, in this section, TMMST was compared with Short Time Fourier Transform, Hilbert Huang Transform, Wigner Ville Distribution, Synchronous Compression
Transform, and Multiscale Synchronous Compression Transform. To highlight the superiority of TMSST-CNN, the accuracy, precision, and recall in the confusion matrix were used as evaluation indicators.
Accuracy is the overall evaluation of the identification effect of all types of load appliances in the test set. The values of these evaluative metrics are bounded within the interval [0,1], where a
higher value indicates superior identification capabilities of the algorithmic model [21]. These metrics are computed using the formulas presented in Eq. (29-32):
Fig. 6Performance of the method proposed in this article on different training sets: a) the accuracy of each type of fault when the training set is 60 %, b) the accuracy of each type of fault when
the training set is 70 %
In Table 3, ${R}_{Acc}$ represents the proportion of correctly predicted samples out of the total number of samples. In fault diagnosis, a high relative accuracy indicates that the model is able to
reliably identify faulty and non-faulty states. ${R}_{Pre}$ is the proportion of actual positive samples (faulty state) among the predicted positive samples, reflecting the model’s ability to avoid
misdiagnosing non-faulty states as faulty. ${R}_{Rec}$, also known as recall, signifies the model's capability to capture the majority of faulty states, thereby reducing the risk of missed
detections. The ${F}_{1-score}$, on the other hand, provides a balanced consideration of both precision and recall, making it highly useful for evaluating the overall performance of the model in
specific applications.
In the formula, $G$ represents the number of correctly classified fault types in the test set; $N$ represents the total number of fault types in the test set; ${T}_{P}$ is a true positive,
representing the number of faults in this article where the predicted label of the fault type in the test set matches the true label; ${F}_{N}$ is a false negative, which represents the number of
faults in the test set that were mistakenly identified as other types of faults for a certain type of fault; ${F}_{P}$ is false positive. In this article, it represents the number of faults in the
test set that were incorrectly identified as a certain type of fault, while other types of faults were identified as such.
Through the analysis of Table 3, it can be seen that compared to directly using time-domain signals for evaluation in CNN models, time-frequency transformation can effectively improve the diagnostic
rate of fault types. Meanwhile, compared to traditional time-frequency transformation methods, the method proposed in this paper has better performance. ${R}_{Acc}$, ${R}_{Pre}$, ${R}_{Rec}$, and $F$
all have the best performance, with a global accuracy of 95.67 %. They have very high accuracy for 10 types of faults in rolling bearings and have a certain degree of robustness. Therefore, TMSST's
processing of time-domain signals can effectively enhance data features, and TMSST-CNN is a method with a good diagnostic success rate for bearings.
Table 3Comparison results of different methods
CNN STFT-CNN HHT-CNN WVD-CNN SST-CNN MSST-CNN TMSST-CNN
${R}_{Acc}$ 85.92 % 91.06 % 88.32 % 90.67 % 93.21 % 93.44 % 95.67 %
${R}_{Pre}$ 86.44 % 82.14 % 89.28 % 91.55 % 93.94 % 94.27 % 96.34 %
${R}_{Rec}$ 86.06 % 91.42 % 89.10 % 91.39 % 93.88 94.02 % 95.88 %
F -score 87.33 92.74 91.02 92.61 95.35 96.12 98.12
5. Conclusions
This paper introduces a novel TMSST-CNN model for the diagnosis of rolling bearing faults. The TMSST component of the model takes into account the comprehensive integration of correlations across
various time intervals during the encoding of rolling bearing signals. Consequently, when employed in conjunction with a CNN for the adaptive extraction of signal features and fault classification,
it facilitates a more nuanced analysis, culminating in an impressive diagnostic accuracy of 95.67 %. To ascertain the model’s generalization capability, training was conducted using different ratios
of training to testing data sets. The outcomes demonstrate that the model's performance has been markedly enhanced through the application of reinforcement learning techniques, consistently
sustaining high diagnostic precision. A comparative analysis was undertaken across various image encoding methodologies and network architectures. The findings reveal that the TMSST image
transformation technique outperforms alternative approaches in diagnosing rolling bearing faults. The methodology presented in this paper is capable of deeper learning, thereby attaining superior
accuracy in fault diagnosis.
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About this article
rolling bearing
convolutional neural networks
diagnosis method
time-reassigned multi-synchro squeezing transform
time-frequency feature maps
The authors have not disclosed any funding.
Data Availability
The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.
Author Contributions
Yunxiu Zhang: conceptualization, methodology, formal analysis, writing-original draft, writing-review and editing, supervision. Bingxian Li: methodology, data collection, data analysis, resources,
writing-review and editing. Zhiyin Han: literature review, conceptualization, writing-review and editing, visualization.
Conflict of interest
The authors declare that they have no conflict of interest.
Copyright © 2024 Yunxiu Zhang, et al.
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Calculate the missing term. take g as 10m/s2 m = 1000g, h = 1-Turito
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Calculate the missing term. Take g as 10m/s^2
m = 1000g, h = 10m, GPE =? (Take g = 10 m/s^2)
A. 1000J
B. 1000KJ
C. 100J
D. 100KJ
When the m is 1 kg and h is 10 m, the GPE will be 100 J.
The correct answer is: 100J
• The gravitational potential energy can be calculated as GPE=mgh.
• When the m is 1000g, and h is 10m, the formula can be applied by using the gravity value.
• m= 1000 g= 1 Kg
h= 10 m
GPE=1x 10x 10=100 J.
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dariusdan's 8-dir movement
This is my code snippet for 8-dir movement in PICO-8. There are some in BBS like this but I think I have something to add here because:
1 - It doesn't move faster when going diagonals;
2 - It can move at speeds below 1 pixel/frame (sub-pixel movement);
3 - And most importantly: IT DOESN'T HAVE ANY ANNOYING JITTERING (the staircase effect)!!!
Feel free to scrutinize my code and give me any constructive tips. I am still getting started with PICO-8 and Lua.
Either my memory is wrong or browser/pico8 changed, but now the movement is jittery. Not an up-and-down staircase effect but a back and forth along the diagonal.
"Either my memory is wrong or browser/pico8 changed, but now the movement is jittery. Not an up-and-down staircase effect but a back and forth along the diagonal."
Not back and forth, but maybe stuttering. This is not an issue. It happens because the speed on the demo is 0.75. If speed is below 1, it means sometimes you don't move this frame. The position is
drawn to the rounded values of x and y, which are floating point numbers.
In fact, if speed is equal to any non-integer value, you might think the character's movement is stuttering, like the framerate is inconsistent, when in fact a non-integer speed only means that you
don't move a fixed amount of pixels each frame, which can make you feel unconfortable, but mathematically is correct.
I probably don’t know the right technical terms but that’s not what matters — I see is a visual glitch where the movement goes to some pixel, then back, then there again.
(also if we’re being pedantic, numbers are fixed-point not floating-point! :)
@merwok, I dunno what to tell you. I even recorded the movement of the pixel in video and watched it frame by frame to try to detect any backsteps.
The demo below is the exact same but here speed is equal to sqrt(2). This makes xspd and yspd equals to 1 when going diagonals, which is very pleasant to the eye, and equals to 1.41 when going
horizontal and vertical, which causes a stuttering effect.
I think this is an intractable problem, because of the resolution and have decimal position values.
Perhaps it would be possible to smooth both vertical / horizontal movement and diagonal movement simultaneously, if you normalized the movement based on frames, so the movement rounding comes out
right, but perhaps not. It seems like you'd be limited to a specific speed to get the numbers to round right.
My conclusion is that it is just a limitation of p8 and should be used in a game as a trick for gamers to use to their advantage.
If you set the speed = 1
and spdx and spdy = .75
it creates an even horizontal / vertical and diagonal movement, though vertical movement is not as smooth as the unnormalized movement.
while .75 is not precise in terms of normalization, i think it's close enough and helps create smoother pixel location rounding.
at 30fps though, i still think the unnormalized movement looks better.
How about this, @dariusdan and co ? Instead of moving in floating point, use the timer to handle diagonal moves. 100% U D L R and 75% for UL UR DL DR.
@dariusdan this works great! I tried making my own code to normalize diagonals and get rid of jittering, but it only works most of the time. Sometimes when going SW or NW, the jittering returns, but
not always. I don't know why. Here is my code:
function _init()
function _update()
if (btn(⬆️)) dy-=p.spd
if (btn(⬇️)) dy+=p.spd
if (btn(⬅️)) dx-=p.spd
if (btn(➡️)) dx+=p.spd
--normalized diagonals
if dx*dy !=0 then
--anti-jittering/smooth diags
--idea from lazy devs academy
--on first diagonal frame
--set subpixel position
--to center of the pixel
if diag and not
prev_diag then
--update position
function _draw()
I'm jealous of your consistent diagonals. I'm trying to reverse engineer your code, but I'm pretty new to coding and can't quite understand it. Would you consider creating a bare-bones version that
optimizes for readability for noobs? It could be a valuable resource for us new developers.
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Boolean algebra
logical operations
in the same way that elementary algebra describes numerical operations.
Boolean algebra was introduced by George Boole in his first book The Mathematical Analysis of Logic (1847),^[1] and set forth more fully in his An Investigation of the Laws of Thought (1854).^[2]
According to Huntington, the term Boolean algebra was first suggested by Henry M. Sheffer in 1913,^[3] although Charles Sanders Peirce gave the title "A Boolian [sic] Algebra with One Constant" to
the first chapter of his "The Simplest Mathematics" in 1880.^[4] Boolean algebra has been fundamental in the development of digital electronics, and is provided for in all modern programming
languages. It is also used in set theory and statistics.^[5]
A precursor of Boolean algebra was
algebra of concepts. The usage of binary in relation to the
I Ching
was central to Leibniz's
characteristica universalis
. It eventually created the foundations of algebra of concepts.
Leibniz's algebra of concepts is deductively equivalent to the Boolean algebra of sets.
Boole's algebra predated the modern developments in
In the 1930s, while studying
switching circuits,
Claude Shannon
observed that one could also apply the rules of Boole's algebra in this setting,
and he introduced
switching algebra
as a way to analyze and design circuits by algebraic means in terms of
logic gates
. Shannon already had at his disposal the abstract mathematical apparatus, thus he cast his switching algebra as the
two-element Boolean algebra
. In modern circuit engineering settings, there is little need to consider other Boolean algebras, thus "switching algebra" and "Boolean algebra" are often used interchangeably.
Logic sentences that can be expressed in classical
first order logic
Although the development of
Whereas expressions denote mainly
truth values
. These values are represented with the
, 0 and 1. They do not behave like the
0 and 1, for which
1 + 1 = 2
, but may be identified with the elements of the
two-element field GF(2)
, that is,
integer arithmetic modulo 2
, for which
1 + 1 = 0
. Addition and multiplication then play the Boolean roles of XOR (exclusive-or) and AND (conjunction), respectively, with disjunction
x ∨ y
(inclusive-or) definable as
x + y − xy
and negation
1 − x
. In
may be replaced by
, since they denote the same operation; however, this way of writing Boolean operations allows applying the usual arithmetic operations of integers (this may be useful when using a programming
language in which
is not implemented).
Boolean algebra also deals with
sequence of bits is a commonly used example of such a function. Another common example is the totality of subsets of a set
: to a subset
, one can define the
indicator function
that takes the value
, and
. The most general example is the set elements of a
Boolean algebra
, with all of the foregoing being instances thereof.
As with elementary algebra, the purely equational part of the theory may be developed, without considering explicit values for the variables.^[17]
This section
needs additional citations for verification
(April 2019)
Basic operations
While Elementary algebra has four operations (addition, subtraction, multiplication, and division), the Boolean algebra has only three basic operations:
binary operators
${\displaystyle \land }$
) and OR (
${\displaystyle \lor }$
) and the
unary operator
${\displaystyle eg }$
), collectively referred to as
Boolean operators
Boolean variables. They are used to store either true or false values.
The basic operations on Boolean variables
are defined as follows:
Logical operation Operator Notation Alternative notations Definition
Conjunction AND x ∧ y x AND y, Kxy x ∧ y = 1 if x = y = 1, x ∧ y = 0 otherwise
Disjunction OR x ∨ y x OR y, Axy x ∨ y = 0 if x = y = 0, x ∨ y = 1 otherwise
Negation NOT ¬x NOT x, Nx, x̅, x', !x ¬x = 0 if x = 1, ¬x = 1 if x = 0
Alternatively, the values of x ∧ y, x ∨ y, and ¬x can be expressed by tabulating their values with
truth tables as follows:
${\displaystyle x}$ ${\displaystyle y}$ ${\displaystyle x\wedge y}$ ${\displaystyle x\vee y}$
0 0 0 0 ${\displaystyle x}$ ${\displaystyle eg x}$
When used in expressions, the operators are applied according to the precedence rules. As with elementary algebra, expressions in parentheses are evaluated first, following the precedence rules.^[21]
If the truth values 0 and 1 are interpreted as integers, these operations may be expressed with the ordinary operations of arithmetic (where x + y uses addition and xy uses multiplication), or by the
minimum/maximum functions:
{\displaystyle {\begin{aligned}x\wedge y&=xy=\min(x,y)\\x\vee y&=x+y-xy=x+y(1-x)=\max(x,y)\\eg x&=1-x\end{aligned}}}
One might consider that only negation and one of the two other operations are basic because of the following identities that allow one to define conjunction in terms of negation and the disjunction,
and vice versa (De Morgan's laws):^[22]
{\displaystyle {\begin{aligned}x\wedge y&=eg (eg x\vee eg y)\\x\vee y&=eg (eg x\wedge eg y)\end{aligned}}}
Secondary operations
Operations composed from the basic operations include, among others, the following:
Material conditional: ${\textstyle x\rightarrow y=eg {x}\vee y}$
Material biconditional: ${\textstyle x\leftrightarrow y=(x\land y)\lor (eg x\land eg y)=(x\lor eg y)\land (eg x\lor y)}$
Exclusive OR (XOR): ${\textstyle x\oplus y=eg (x\leftrightarrow y)=(x\vee y)\wedge eg (x\wedge y)=(x\vee y)\wedge (eg x\vee eg y)=(x\wedge eg y)\vee (eg x\wedge y)}$
These definitions give rise to the following truth tables giving the values of these operations for all four possible inputs.
Secondary operations. Table 1
${\displaystyle x}$ ${\displaystyle y}$ ${\displaystyle x\rightarrow y}$ ${\displaystyle x\oplus y}$ ${\displaystyle x\leftrightarrow y,x\equiv y}$
Material conditional
The first operation, x → y, or Cxy, is called material implication. If x is true, then the result of expression x → y is taken to be that of y (e.g. if x is true and y is false, then x → y is
also false). But if x is false, then the value of y can be ignored; however, the operation must return some Boolean value and there are only two choices. So by definition, x → y is true when x is
false. (relevance logic suggests this definition, by viewing an implication with a false premise as something other than either true or false.)
Exclusive OR (XOR)
The second operation, x ⊕ y, or Jxy, is called exclusive or (often abbreviated as XOR) to distinguish it from disjunction as the inclusive kind. It excludes the possibility of both x and y being
true (e.g. see table): if both are true then result is false. Defined in terms of arithmetic it is addition where mod 2 is 1 + 1 = 0.
Logical equivalence
The third operation, the complement of exclusive or, is equivalence or Boolean equality: x ≡ y, or Exy, is true just when x and y have the same value. Hence x ⊕ y as its complement can be
understood as x ≠ y, being true just when x and y are different. Thus, its counterpart in arithmetic mod 2 is x + y. Equivalence's counterpart in arithmetic mod 2 is x + y + 1.
A law of Boolean algebra is an
model of the Boolean laws, and as a means for deriving new laws from old as in the derivation of
x ∨ (y ∧ z) = x ∨ (z ∧ y)
y ∧ z = z ∧ y
(as treated in
§ Axiomatizing Boolean algebra
Monotone laws
Boolean algebra satisfies many of the same laws as ordinary algebra when one matches up ∨ with addition and ∧ with multiplication. In particular the following laws are common to both kinds of
Associativity of ∨: ${\displaystyle x\vee (y\vee z)}$ ${\displaystyle =(x\vee y)\vee z}$
Associativity of ∧: ${\displaystyle x\wedge (y\wedge z)}$ ${\displaystyle =(x\wedge y)\wedge z}$
Commutativity of ∨: ${\displaystyle x\vee y}$ ${\displaystyle =y\vee x}$
Commutativity of ∧: ${\displaystyle x\wedge y}$ ${\displaystyle =y\wedge x}$
Distributivity of ∧ over ∨: ${\displaystyle x\wedge (y\vee z)}$ ${\displaystyle =(x\wedge y)\vee (x\wedge z)}$
Identity for ∨: ${\displaystyle x\vee 0}$ ${\displaystyle =x}$
Identity for ∧: ${\displaystyle x\wedge 1}$ ${\displaystyle =x}$
Annihilator for ∧: ${\displaystyle x\wedge 0}$ ${\displaystyle =0}$
The following laws hold in Boolean algebra, but not in ordinary algebra:
Annihilator for ∨: ${\displaystyle x\vee 1}$ ${\displaystyle =1}$
Idempotence of ∨: ${\displaystyle x\vee x}$ ${\displaystyle =x}$
Idempotence of ∧: ${\displaystyle x\wedge x}$ ${\displaystyle =x}$
Absorption 1: ${\displaystyle x\wedge (x\vee y)}$ ${\displaystyle =x}$
Absorption 2: ${\displaystyle x\vee (x\wedge y)}$ ${\displaystyle =x}$
Distributivity of ∨ over ∧: ${\displaystyle x\vee (y\wedge z)}$ ${\displaystyle =(x\vee y)\wedge (x\vee z)}$
Taking x = 2 in the third law above shows that it is not an ordinary algebra law, since 2 × 2 = 4. The remaining five laws can be falsified in ordinary algebra by taking all variables to be 1. For
example, in absorption law 1, the left hand side would be 1(1 + 1) = 2, while the right hand side would be 1 (and so on).
All of the laws treated thus far have been for conjunction and disjunction. These operations have the property that changing either argument either leaves the output unchanged, or the output changes
in the same way as the input. Equivalently, changing any variable from 0 to 1 never results in the output changing from 1 to 0. Operations with this property are said to be monotone. Thus the axioms
thus far have all been for monotonic Boolean logic. Nonmonotonicity enters via complement ¬ as follows.^[5]
Nonmonotone laws
The complement operation is defined by the following two laws.
{\displaystyle {\begin{aligned}&{\text{Complementation 1}}&x\wedge eg x&=0\\&{\text{Complementation 2}}&x\vee eg x&=1\end{aligned}}}
All properties of negation including the laws below follow from the above two laws alone.^[5]
In both ordinary and Boolean algebra, negation works by exchanging pairs of elements, hence in both algebras it satisfies the double negation law (also called involution law)
{\displaystyle {\begin{aligned}&{\text{Double negation}}&eg {(eg {x})}&=x\end{aligned}}}
But whereas ordinary algebra satisfies the two laws
{\displaystyle {\begin{aligned}(-x)(-y)&=xy\\(-x)+(-y)&=-(x+y)\end{aligned}}}
Boolean algebra satisfies De Morgan's laws:
{\displaystyle {\begin{aligned}&{\text{De Morgan 1}}&eg x\wedge eg y&=eg {(x\vee y)}\\&{\text{De Morgan 2}}&eg x\vee eg y&=eg {(x\wedge y)}\end{aligned}}}
The laws listed above define Boolean algebra, in the sense that they entail the rest of the subject. The laws complementation 1 and 2, together with the monotone laws, suffice for this purpose and
can therefore be taken as one possible complete set of laws or
Writing down further laws of Boolean algebra cannot give rise to any new consequences of these axioms, nor can it rule out any model of them. In contrast, in a list of some but not all of the same
laws, there could have been Boolean laws that did not follow from those on the list, and moreover there would have been models of the listed laws that were not Boolean algebras.
This axiomatization is by no means the only one, or even necessarily the most natural given that attention was not paid as to whether some of the axioms followed from others, but there was simply a
choice to stop when enough laws had been noticed, treated further in § Axiomatizing Boolean algebra. Or the intermediate notion of axiom can be sidestepped altogether by defining a Boolean law
directly as any tautology, understood as an equation that holds for all values of its variables over 0 and 1.^[25]^[26] All these definitions of Boolean algebra can be shown to be equivalent.
Duality principle
Principle: If {X, R} is a partially ordered set, then {X, R(inverse)} is also a partially ordered set.
There is nothing special about the choice of symbols for the values of Boolean algebra. 0 and 1 could be renamed to α and β, and as long as it was done consistently throughout, it would still be
Boolean algebra, albeit with some obvious cosmetic differences.
But suppose 0 and 1 were renamed 1 and 0 respectively. Then it would still be Boolean algebra, and moreover operating on the same values. However, it would not be identical to our original Boolean
algebra because now ∨ behaves the way ∧ used to do and vice versa. So there are still some cosmetic differences to show that the notation has been changed, despite the fact that 0s and 1s are still
being used.
But if in addition to interchanging the names of the values, the names of the two binary operations are also interchanged, now there is no trace of what was done. The end product is completely
indistinguishable from what was started with. The columns for x ∧ y and x ∨ y in the truth tables have changed places, but that switch is immaterial.
When values and operations can be paired up in a way that leaves everything important unchanged when all pairs are switched simultaneously, the members of each pair are called dual to each other.
Thus 0 and 1 are dual, and ∧ and ∨ are dual. The duality principle, also called De Morgan duality, asserts that Boolean algebra is unchanged when all dual pairs are interchanged.
One change not needed to make as part of this interchange was to complement. Complement is a self-dual operation. The identity or do-nothing operation x (copy the input to the output) is also
self-dual. A more complicated example of a self-dual operation is (x ∧ y) ∨ (y ∧ z) ∨ (z ∧ x). There is no self-dual binary operation that depends on both its arguments. A composition of self-dual
operations is a self-dual operation. For example, if f(x, y, z) = (x ∧ y) ∨ (y ∧ z) ∨ (z ∧ x), then f(f(x, y, z), x, t) is a self-dual operation of four arguments x, y, z, t.
The principle of duality can be explained from a
acting on the set of Boolean polynomials.
Walter Gottschalk
remarked that consequently a more appropriate name for the phenomenon would be the
of quaternality
Diagrammatic representations
Venn diagrams
A Venn diagram^[27] can be used as a representation of a Boolean operation using shaded overlapping regions. There is one region for each variable, all circular in the examples here. The interior and
exterior of region x corresponds respectively to the values 1 (true) and 0 (false) for variable x. The shading indicates the value of the operation for each combination of regions, with dark denoting
1 and light 0 (some authors use the opposite convention).
The three Venn diagrams in the figure below represent respectively conjunction x ∧ y, disjunction x ∨ y, and complement ¬x.
Figure 2. Venn diagrams for conjunction, disjunction, and complement
For conjunction, the region inside both circles is shaded to indicate that x ∧ y is 1 when both variables are 1. The other regions are left unshaded to indicate that x ∧ y is 0 for the other three
The second diagram represents disjunction x ∨ y by shading those regions that lie inside either or both circles. The third diagram represents complement ¬x by shading the region not inside the
While we have not shown the Venn diagrams for the constants 0 and 1, they are trivial, being respectively a white box and a dark box, neither one containing a circle. However, we could put a circle
for x in those boxes, in which case each would denote a function of one argument, x, which returns the same value independently of x, called a constant function. As far as their outputs are
concerned, constants and constant functions are indistinguishable; the difference is that a constant takes no arguments, called a zeroary or nullary operation, while a constant function takes one
argument, which it ignores, and is a unary operation.
Venn diagrams are helpful in visualizing laws. The commutativity laws for ∧ and ∨ can be seen from the symmetry of the diagrams: a binary operation that was not commutative would not have a symmetric
diagram because interchanging x and y would have the effect of reflecting the diagram horizontally and any failure of commutativity would then appear as a failure of symmetry.
Idempotence of ∧ and ∨ can be visualized by sliding the two circles together and noting that the shaded area then becomes the whole circle, for both ∧ and ∨.
To see the first absorption law, x ∧ (x ∨ y) = x, start with the diagram in the middle for x ∨ y and note that the portion of the shaded area in common with the x circle is the whole of the x circle.
For the second absorption law, x ∨ (x ∧ y) = x, start with the left diagram for x∧y and note that shading the whole of the x circle results in just the x circle being shaded, since the previous
shading was inside the x circle.
The double negation law can be seen by complementing the shading in the third diagram for ¬x, which shades the x circle.
To visualize the first De Morgan's law, (¬x) ∧ (¬y) = ¬(x ∨ y), start with the middle diagram for x ∨ y and complement its shading so that only the region outside both circles is shaded, which is
what the right hand side of the law describes. The result is the same as if we shaded that region which is both outside the x circle and outside the y circle, i.e. the conjunction of their exteriors,
which is what the left hand side of the law describes.
The second De Morgan's law, (¬x) ∨ (¬y) = ¬(x ∧ y), works the same way with the two diagrams interchanged.
The first complement law, x ∧ ¬x = 0, says that the interior and exterior of the x circle have no overlap. The second complement law, x ∨ ¬x = 1, says that everything is either inside or outside the
x circle.
Digital logic gates
Digital logic is the application of the Boolean algebra of 0 and 1 to electronic hardware consisting of
logic gates connected to form a
circuit diagram
. Each gate implements a Boolean operation, and is depicted schematically by a shape indicating the operation. The shapes associated with the gates for conjunction (AND-gates), disjunction
(OR-gates), and complement (inverters) are as follows:
From left to right: AND, OR, and NOT gates.
The lines on the left of each gate represent input wires or ports. The value of the input is represented by a voltage on the lead. For so-called "active-high" logic, 0 is represented by a voltage
close to zero or "ground," while 1 is represented by a voltage close to the supply voltage; active-low reverses this. The line on the right of each gate represents the output port, which normally
follows the same voltage conventions as the input ports.
Complement is implemented with an inverter gate. The triangle denotes the operation that simply copies the input to the output; the small circle on the output denotes the actual inversion
complementing the input. The convention of putting such a circle on any port means that the signal passing through this port is complemented on the way through, whether it is an input or output port.
The duality principle, or De Morgan's laws, can be understood as asserting that complementing all three ports of an AND gate converts it to an OR gate and vice versa, as shown in Figure 4 below.
Complementing both ports of an inverter however leaves the operation unchanged.
More generally, one may complement any of the eight subsets of the three ports of either an AND or OR gate. The resulting sixteen possibilities give rise to only eight Boolean operations, namely
those with an odd number of 1s in their truth table. There are eight such because the "odd-bit-out" can be either 0 or 1 and can go in any of four positions in the truth table. There being sixteen
binary Boolean operations, this must leave eight operations with an even number of 1s in their truth tables. Two of these are the constants 0 and 1 (as binary operations that ignore both their
inputs); four are the operations that depend nontrivially on exactly one of their two inputs, namely x, y, ¬x, and ¬y; and the remaining two are x ⊕ y (XOR) and its complement x ≡ y.
Boolean algebras
The term "algebra" denotes both a subject, namely the subject of algebra, and an object, namely an algebraic structure. Whereas the foregoing has addressed the subject of Boolean algebra, this
section deals with mathematical objects called Boolean algebras, defined in full generality as any model of the Boolean laws. We begin with a special case of the notion definable without reference to
the laws, namely concrete Boolean algebras, and then give the formal definition of the general notion.
Concrete Boolean algebras
A concrete Boolean algebra or field of sets is any nonempty set of subsets of a given set X closed under the set operations of union, intersection, and complement relative to X.^[5]
(Historically X itself was required to be nonempty as well to exclude the degenerate or one-element Boolean algebra, which is the one exception to the rule that all Boolean algebras satisfy the same
equations since the degenerate algebra satisfies every equation. However, this exclusion conflicts with the preferred purely equational definition of "Boolean algebra", there being no way to rule out
the one-element algebra using only equations— 0 ≠ 1 does not count, being a negated equation. Hence modern authors allow the degenerate Boolean algebra and let X be empty.)
Example 1. The
Example 2. The empty set and X. This two-element algebra shows that a concrete Boolean algebra can be finite even when it consists of subsets of an infinite set. It can be seen that every field of
subsets of X must contain the empty set and X. Hence no smaller example is possible, other than the degenerate algebra obtained by taking X to be empty so as to make the empty set and X coincide.
Example 3. The set of finite and
sets of integers, where a cofinite set is one omitting only finitely many integers. This is clearly closed under complement, and is closed under union because the union of a cofinite set with any set
is cofinite, while the union of two finite sets is finite. Intersection behaves like union with "finite" and "cofinite" interchanged. This example is countably infinite because there are only
countably many finite sets of integers.
Example 4. For a less trivial example of the point made by example 2, consider a Venn diagram formed by n closed curves partitioning the diagram into 2^n regions, and let X be the (infinite) set of
all points in the plane not on any curve but somewhere within the diagram. The interior of each region is thus an infinite subset of X, and every point in X is in exactly one region. Then the set of
all 2^2^n possible unions of regions (including the empty set obtained as the union of the empty set of regions and X obtained as the union of all 2^n regions) is closed under union, intersection,
and complement relative to X and therefore forms a concrete Boolean algebra. Again, there are finitely many subsets of an infinite set forming a concrete Boolean algebra, with example 2 arising as
the case n = 0 of no curves.
Subsets as bit vectors
A subset Y of X can be identified with an indexed family of bits with index set X, with the bit indexed by x ∈ X being 1 or 0 according to whether or not x ∈ Y. (This is the so-called characteristic
function notion of a subset.) For example, a 32-bit computer word consists of 32 bits indexed by the set {0,1,2,...,31}, with 0 and 31 indexing the low and high order bits respectively. For a smaller
example, if ${\displaystyle X=\{a,b,c\}}$ where a, b, c are viewed as bit positions in that order from left to right, the eight subsets {}, {c}, {b}, {b,c}, {a}, {a,c}, {a,b}, and {a,b,c} of X can
be identified with the respective bit vectors 000, 001, 010, 011, 100, 101, 110, and 111. Bit vectors indexed by the set of natural numbers are infinite sequences of bits, while those indexed by the
reals in the unit interval [0,1] are packed too densely to be able to write conventionally but nonetheless form well-defined indexed families (imagine coloring every point of the interval [0,1]
either black or white independently; the black points then form an arbitrary subset of [0,1]).
From this bit vector viewpoint, a concrete Boolean algebra can be defined equivalently as a nonempty set of bit vectors all of the same length (more generally, indexed by the same set) and closed
under the bit vector operations of
∧, ∨, and ¬, as in
1010∧0110 = 0010
1010∨0110 = 1110
, and
¬1010 = 0101
, the bit vector realizations of intersection, union, and complement respectively.
Prototypical Boolean algebra
The set {0,1} and its Boolean operations as treated above can be understood as the special case of bit vectors of length one, which by the identification of bit vectors with subsets can also be
understood as the two subsets of a one-element set. This is called the prototypical Boolean algebra, justified by the following observation.
The laws satisfied by all nondegenerate concrete Boolean algebras coincide with those satisfied by the prototypical Boolean algebra.
This observation is proved as follows. Certainly any law satisfied by all concrete Boolean algebras is satisfied by the prototypical one since it is concrete. Conversely any law that fails for some
concrete Boolean algebra must have failed at a particular bit position, in which case that position by itself furnishes a one-bit counterexample to that law. Nondegeneracy ensures the existence of at
least one bit position because there is only one empty bit vector.
The final goal of the next section can be understood as eliminating "concrete" from the above observation. That goal is reached via the stronger observation that, up to isomorphism, all Boolean
algebras are concrete.
Boolean algebras: the definition
The Boolean algebras so far have all been concrete, consisting of bit vectors or equivalently of subsets of some set. Such a Boolean algebra consists of a set and operations on that set which can be
shown to satisfy the laws of Boolean algebra.
Instead of showing that the Boolean laws are satisfied, we can instead postulate a set X, two binary operations on X, and one unary operation, and require that those operations satisfy the laws of
Boolean algebra. The elements of X need not be bit vectors or subsets but can be anything at all. This leads to the more general abstract definition.
A Boolean algebra is any set with binary operations ∧ and ∨ and a unary operation ¬ thereon satisfying the Boolean laws.^[29]
For the purposes of this definition it is irrelevant how the operations came to satisfy the laws, whether by fiat or proof. All concrete Boolean algebras satisfy the laws (by proof rather than fiat),
whence every concrete Boolean algebra is a Boolean algebra according to our definitions. This axiomatic definition of a Boolean algebra as a set and certain operations satisfying certain laws or
axioms by fiat is entirely analogous to the abstract definitions of group, ring, field etc. characteristic of modern or abstract algebra.
Given any complete axiomatization of Boolean algebra, such as the axioms for a complemented distributive lattice, a sufficient condition for an algebraic structure of this kind to satisfy all the
Boolean laws is that it satisfy just those axioms. The following is therefore an equivalent definition.
A Boolean algebra is a complemented distributive lattice.
The section on axiomatization lists other axiomatizations, any of which can be made the basis of an equivalent definition.
Representable Boolean algebras
Although every concrete Boolean algebra is a Boolean algebra, not every Boolean algebra need be concrete. Let n be a square-free positive integer, one not divisible by the square of an integer, for
example 30 but not 12. The operations of greatest common divisor, least common multiple, and division into n (that is, ¬x = n/x), can be shown to satisfy all the Boolean laws when their arguments
range over the positive divisors of n. Hence those divisors form a Boolean algebra. These divisors are not subsets of a set, making the divisors of n a Boolean algebra that is not concrete according
to our definitions.
However, if each divisor of n is represented by the set of its prime factors, this nonconcrete Boolean algebra is
isomorphic to the concrete Boolean algebra consisting of all sets of prime factors of
, with union corresponding to least common multiple, intersection to greatest common divisor, and complement to division into
. So this example, while not technically concrete, is at least "morally" concrete via this representation, called an
. This example is an instance of the following notion.
A Boolean algebra is called representable when it is isomorphic to a concrete Boolean algebra.
The next question is answered positively as follows.
Every Boolean algebra is representable.
That is, up to isomorphism, abstract and concrete Boolean algebras are the same thing. This result depends on the Boolean prime ideal theorem, a choice principle slightly weaker than the axiom of
choice. This strong relationship implies a weaker result strengthening the observation in the previous subsection to the following easy consequence of representability.
The laws satisfied by all Boolean algebras coincide with those satisfied by the prototypical Boolean algebra.
It is weaker in the sense that it does not of itself imply representability. Boolean algebras are special here, for example a relation algebra is a Boolean algebra with additional structure but it is
not the case that every relation algebra is representable in the sense appropriate to relation algebras.
Axiomatizing Boolean algebra
The above definition of an abstract Boolean algebra as a set together with operations satisfying "the" Boolean laws raises the question of what those laws are. A simplistic answer is "all Boolean
laws", which can be defined as all equations that hold for the Boolean algebra of 0 and 1. However, since there are infinitely many such laws, this is not a satisfactory answer in practice, leading
to the question of it suffices to require only finitely many laws to hold.
In the case of Boolean algebras, the answer is "yes": the finitely many equations listed above are sufficient. Thus, Boolean algebra is said to be finitely axiomatizable or finitely based.
Moreover, the number of equations needed can be further reduced. To begin with, some of the above laws are implied by some of the others. A sufficient subset of the above laws consists of the pairs
of associativity, commutativity, and absorption laws, distributivity of ∧ over ∨ (or the other distributivity law—one suffices), and the two complement laws. In fact, this is the traditional
axiomatization of Boolean algebra as a complemented distributive lattice.
By introducing additional laws not listed above, it becomes possible to shorten the list of needed equations yet further; for instance, with the vertical bar representing the Sheffer stroke
operation, the single axiom ${\displaystyle ((a\mid b)\mid c)\mid (a\mid ((a\mid c)\mid a))=c}$ is sufficient to completely axiomatize Boolean algebra. It is also possible to find longer single
axioms using more conventional operations; see Minimal axioms for Boolean algebra.^[30]
Propositional logic
logical system that is intimately connected to Boolean algebra.
Many syntactic concepts of Boolean algebra carry over to propositional logic with only minor changes in notation and terminology, while the semantics of propositional logic are defined via Boolean
algebras in a way that the tautologies (theorems) of propositional logic correspond to equational theorems of Boolean algebra.
Syntactically, every Boolean term corresponds to a propositional formula of propositional logic. In this translation between Boolean algebra and propositional logic, Boolean variables x, y, ...
become propositional variables (or atoms) P, Q, ... Boolean terms such as x ∨ y become propositional formulas P ∨ Q; 0 becomes false or ⊥, and 1 becomes true or T. It is convenient when referring to
generic propositions to use Greek letters Φ, Ψ, ... as metavariables (variables outside the language of propositional calculus, used when talking about propositional calculus) to denote propositions.
The semantics of propositional logic rely on
Boolean-valued semantics arbitrary Boolean algebras are considered. A
is a propositional formula that is assigned truth value 1 by every truth assignment of its propositional variables to an arbitrary Boolean algebra (or, equivalently, every truth assignment to the two
element Boolean algebra).
These semantics permit a translation between tautologies of propositional logic and equational theorems of Boolean algebra. Every tautology Φ of propositional logic can be expressed as the Boolean
equation Φ = 1, which will be a theorem of Boolean algebra. Conversely, every theorem Φ = Ψ of Boolean algebra corresponds to the tautologies (Φ ∨ ¬Ψ) ∧ (¬Φ ∨ Ψ) and (Φ ∧ Ψ) ∨ (¬Φ ∧ ¬Ψ). If → is in
the language, these last tautologies can also be written as (Φ → Ψ) ∧ (Ψ → Φ), or as two separate theorems Φ → Ψ and Ψ → Φ; if ≡ is available, then the single tautology Φ ≡ Ψ can be used.
One motivating application of propositional calculus is the analysis of propositions and deductive arguments in natural language.^[31] Whereas the proposition "if x = 3, then x + 1 = 4" depends on
the meanings of such symbols as + and 1, the proposition "if x = 3, then x = 3" does not; it is true merely by virtue of its structure, and remains true whether "x = 3" is replaced by "x = 4" or "the
moon is made of green cheese." The generic or abstract form of this tautology is "if P, then P," or in the language of Boolean algebra, P → P.
Replacing P by x = 3 or any other proposition is called instantiation of P by that proposition. The result of instantiating P in an abstract proposition is called an instance of the proposition.
Thus, x = 3 → x = 3 is a tautology by virtue of being an instance of the abstract tautology P → P. All occurrences of the instantiated variable must be instantiated with the same proposition, to
avoid such nonsense as P → x = 3 or x = 3 → x = 4.
Propositional calculus restricts attention to abstract propositions, those built up from propositional variables using Boolean operations. Instantiation is still possible within propositional
calculus, but only by instantiating propositional variables by abstract propositions, such as instantiating Q by Q → P in P → (Q → P) to yield the instance P → ((Q → P) → P).
(The availability of instantiation as part of the machinery of propositional calculus avoids the need for metavariables within the language of propositional calculus, since ordinary propositional
variables can be considered within the language to denote arbitrary propositions. The metavariables themselves are outside the reach of instantiation, not being part of the language of propositional
calculus but rather part of the same language for talking about it that this sentence is written in, where there is a need to be able to distinguish propositional variables and their instantiations
as being distinct syntactic entities.)
Deductive systems for propositional logic
An axiomatization of propositional calculus is a set of tautologies called
axioms and one or more inference rules for producing new tautologies from old. A
in an axiom system
is a finite nonempty sequence of propositions each of which is either an instance of an axiom of
or follows by some rule of
from propositions appearing earlier in the proof (thereby disallowing circular reasoning). The last proposition is the
proved by the proof. Every nonempty initial segment of a proof is itself a proof, whence every proposition in a proof is itself a theorem. An axiomatization is
when every theorem is a tautology, and
when every tautology is a theorem.
Sequent calculus
Propositional calculus is commonly organized as a
of the succedent by the antecedent.
Entailment differs from implication in that whereas the latter is a binary operation that returns a value in a Boolean algebra, the former is a binary relation which either holds or does not hold. In
this sense, entailment is an external form of implication, meaning external to the Boolean algebra, thinking of the reader of the sequent as also being external and interpreting and comparing
antecedents and succedents in some Boolean algebra. The natural interpretation of ⊢ is as ≤ in the partial order of the Boolean algebra defined by x ≤ y just when x ∨ y = y. This ability to mix
external implication ⊢ and internal implication → in the one logic is among the essential differences between sequent calculus and propositional calculus.^[33]
Boolean algebra as the calculus of two values is fundamental to computer circuits, computer programming, and mathematical logic, and is also used in other areas of mathematics such as set theory and
In the early 20th century, several electrical engineers intuitively recognized that Boolean algebra was analogous to the behavior of certain types of electrical circuits. Claude Shannon formally
proved such behavior was logically equivalent to Boolean algebra in his 1937 master's thesis, A Symbolic Analysis of Relay and Switching Circuits.
Today, all modern general-purpose computers perform their functions using two-value Boolean logic; that is, their electrical circuits are a physical manifestation of two-value Boolean logic. They
achieve this in various ways: as voltages on wires in high-speed circuits and capacitive storage devices, as orientations of a magnetic domain in ferromagnetic storage devices, as holes in punched
cards or paper tape, and so on. (Some early computers used decimal circuits or mechanisms instead of two-valued logic circuits.)
Of course, it is possible to code more than two symbols in any given medium. For example, one might use respectively 0, 1, 2, and 3 volts to code a four-symbol alphabet on a wire, or holes of
different sizes in a punched card. In practice, the tight constraints of high speed, small size, and low power combine to make noise a major factor. This makes it hard to distinguish between symbols
when there are several possible symbols that could occur at a single site. Rather than attempting to distinguish between four voltages on one wire, digital designers have settled on two voltages per
wire, high and low.
Computers use two-value Boolean circuits for the above reasons. The most common computer architectures use ordered sequences of Boolean values, called bits, of 32 or 64 values, e.g.
01101000110101100101010101001011. When programming in
data registers. These registers operate on voltages, where zero volts represents Boolean 0, and a reference voltage (often +5 V, +3.3 V, or +1.8 V) represents Boolean 1. Such languages support both
numeric operations and logical operations. In this context, "numeric" means that the computer treats sequences of bits as
binary numbers
(base two numbers) and executes arithmetic operations like add, subtract, multiply, or divide. "Logical" refers to the Boolean logical operations of disjunction, conjunction, and negation between two
sequences of bits, in which each bit in one sequence is simply compared to its counterpart in the other sequence. Programmers therefore have the option of working in and applying the rules of either
numeric algebra or Boolean algebra as needed. A core differentiating feature between these families of operations is the existence of the
operation in the first but not the second.
Two-valued logic
Other areas where two values is a good choice are the law and mathematics. In everyday relaxed conversation, nuanced or complex answers such as "maybe" or "only on the weekend" are acceptable. In
more focused situations such as a court of law or theorem-based mathematics, however, it is deemed advantageous to frame questions so as to admit a simple yes-or-no answer—is the defendant guilty or
not guilty, is the proposition true or false—and to disallow any other answer. However, limiting this might prove in practice for the respondent, the principle of the simple yes–no question has
become a central feature of both judicial and mathematical logic, making
two-valued logic
deserving of organization and study in its own right.
A central concept of set theory is membership. An organization may permit multiple degrees of membership, such as novice, associate, and full. With sets, however, an element is either in or out. The
candidates for membership in a set work just like the wires in a digital computer: each candidate is either a member or a nonmember, just as each wire is either high or low.
Algebra being a fundamental tool in any area amenable to mathematical treatment, these considerations combine to make the algebra of two values of fundamental importance to computer hardware,
mathematical logic, and set theory.
Two-valued logic can be extended to
. In these interpretations, a value is interpreted as the "degree" of truth – to what extent a proposition is true, or the probability that the proposition is true.
Boolean operations
The original application for Boolean operations was mathematical logic, where it combines the truth values, true or false, of individual formulas.
Natural language
Natural languages such as English have words for several Boolean operations, in particular conjunction (and), disjunction (or), negation (not), and implication (implies). But not is synonymous with
and not. When used to combine situational assertions such as "the block is on the table" and "cats drink milk", which naïvely are either true or false, the meanings of these logical connectives often
have the meaning of their logical counterparts. However, with descriptions of behavior such as "Jim walked through the door", one starts to notice differences such as failure of commutativity, for
example, the conjunction of "Jim opened the door" with "Jim walked through the door" in that order is not equivalent to their conjunction in the other order, since and usually means and then in such
cases. Questions can be similar: the order "Is the sky blue, and why is the sky blue?" makes more sense than the reverse order. Conjunctive commands about behavior are like behavioral assertions, as
in get dressed and go to school. Disjunctive commands such love me or leave me or fish or cut bait tend to be asymmetric via the implication that one alternative is less preferable. Conjoined nouns
such as tea and milk generally describe aggregation as with set union while tea or milk is a choice. However, context can reverse these senses, as in your choices are coffee and tea which usually
means the same as your choices are coffee or tea (alternatives). Double negation, as in "I don't not like milk", rarely means literally "I do like milk" but rather conveys some sort of hedging, as
though to imply that there is a third possibility. "Not not P" can be loosely interpreted as "surely P", and although P necessarily implies "not not P," the converse is suspect in English, much as
with intuitionistic logic. In view of the highly idiosyncratic usage of conjunctions in natural languages, Boolean algebra cannot be considered a reliable framework for interpreting them.
Digital logic
Boolean operations are used in
digital logic to combine the bits carried on individual wires, thereby interpreting them over {0,1}. When a vector of
identical binary gates are used to combine two bit vectors each of
bits, the individual bit operations can be understood collectively as a single operation on values from a
Boolean algebra
with 2
Naive set theory
Naive set theory interprets Boolean operations as acting on subsets of a given set X. As we saw earlier this behavior exactly parallels the coordinate-wise combinations of bit vectors, with the union
of two sets corresponding to the disjunction of two bit vectors and so on.
Video cards
The 256-element free Boolean algebra on three generators is deployed in
video cards
offer all
2^2^3 = 256
ternary operations for this purpose, with the choice of operation being a one-byte (8-bit) parameter. The constants
SRC = 0xaa
DST = 0xcc
, and
MSK = 0xf0
allow Boolean operations such as
(meaning XOR the source and destination and then AND the result with the mask) to be written directly as a constant denoting a byte calculated at compile time,
in the
if just
, etc. At run time the video card interprets the byte as the raster operation indicated by the original expression in a uniform way that requires remarkably little hardware and which takes time
completely independent of the complexity of the expression.
Modeling and CAD
computer aided design offer a variety of methods for building objects from other objects, combination by Boolean operations being one of them. In this method the space in which objects exist is
understood as a set
(the three-dimensional analogue of pixels in two-dimensional graphics) and shapes are defined as subsets of
, allowing objects to be combined as sets via union, intersection, etc. One obvious use is in building a complex shape from simple shapes simply as the union of the latter. Another use is in
sculpting understood as removal of material: any grinding, milling, routing, or drilling operation that can be performed with physical machinery on physical materials can be simulated on the computer
with the Boolean operation
x ∧ ¬y
x − y
, which in set theory is set difference, remove the elements of
from those of
. Thus given two shapes one to be machined and the other the material to be removed, the result of machining the former to remove the latter is described simply as their set difference.
Boolean searches
Search engine queries also employ Boolean logic. For this application, each web page on the Internet may be considered to be an "element" of a "set." The following examples use a syntax supported by
Google.^[NB 1]
• Doublequotes are used to combine whitespace-separated words into a single search term.^[NB 2]
• Whitespace is used to specify logical AND, as it is the default operator for joining search terms:
"Search term 1" "Search term 2"
• The OR keyword is used for logical OR:
"Search term 1" OR "Search term 2"
• A prefixed minus sign is used for logical NOT:
"Search term 1" −"Search term 2"
See also
1. ^ Not all search engines support the same query syntax. Additionally, some organizations (such as Google) provide "specialized" search engines that support alternate or extended syntax. (See
e.g., Syntax cheatsheet, Google codesearch supports regular expressions).
2. ^ Doublequote-delimited search terms are called "exact phrase" searches in the Google documentation.
1. ^ Boole, George (2011-07-28). The Mathematical Analysis of Logic - Being an Essay Towards a Calculus of Deductive Reasoning.
2. .
3. ^ "The name Boolean algebra (or Boolean 'algebras') for the calculus originated by Boole, extended by Schröder, and perfected by Whitehead seems to have been first suggested by Sheffer, in 1913."
Edward Vermilye Huntington, "New sets of independent postulates for the algebra of logic, with special reference to Whitehead and Russell's Principia mathematica", in Transactions of the American
Mathematical Society 35 (1933), 274-304; footnote, page 278.
4. .
5. ^ .
6. .
7. ^ Lenzen, Wolfgang. "Leibniz: Logic". Internet Encyclopedia of Philosophy.
8. ^ .
9. ^ Weisstein, Eric W. "Boolean Algebra". mathworld.wolfram.com. Retrieved 2020-09-02.
11. .
12. .
13. .
14. .
15. .
16. .
17. Halmos, Paul Richard
(1963). Lectures on Boolean Algebras. van Nostrand.
18. ^ Bacon, Jason W. (2011). "Computer Science 315 Lecture Notes". Archived from the original on 2021-10-02. Retrieved 2021-10-01.
19. ^ "Boolean Algebra - Expression, Rules, Theorems, and Examples". GeeksforGeeks. 2021-09-24. Retrieved 2024-06-03.
20. ^ "Boolean Logical Operations" (PDF).
21. ^ "Boolean Algebra Operations". bob.cs.sonoma.edu. Retrieved 2024-06-03.
22. ^ "Boolean Algebra" (PDF).
23. .
24. ^ "Elements of Boolean Algebra". www.ee.surrey.ac.uk. Retrieved 2020-09-02.
25. ^ McGee, Vann, Sentential Calculus Revisited: Boolean Algebra (PDF)
28. .
29. .
31. .
32. .
33. .
Further reading
• Mano, Morris; Ciletti, Michael D. (2013). Digital Design. Pearson. .
• Whitesitt, J. Eldon (1995). Boolean algebra and its applications. .
• Dwinger, Philip (1971). Introduction to Boolean algebras. Würzburg, Germany: Physica Verlag.
• .
• Bocheński, Józef Maria (1959). A Précis of Mathematical Logic. Translated from the French and German editions by Otto Bird. Dordrecht, South Holland: D. Reidel.
Historical perspective
• Cambridge and Dublin Mathematical Journal
. III: 183–198.
• Hailperin, Theodore (1986). Boole's logic and probability: a critical exposition from the standpoint of contemporary algebra, logic, and probability theory (2 ed.). .
• Gabbay, Dov M.; Woods, John, eds. (2004). The rise of modern logic: from Leibniz to Frege. Handbook of the History of Logic. Vol. 3. ., several relevant chapters by Hailperin, Valencia, and
• Badesa, Calixto (2004). "Chapter 1. Algebra of Classes and Propositional Calculus". The birth of model theory: Löwenheim's theorem in the frame of the theory of relatives. .
• . Retrieved 2022-10-25.
• "The Algebra of Logic Tradition" entry by Burris, Stanley in the Stanford Encyclopedia of Philosophy, 21 February 2012
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Digital Math Resources
Display Title
Math Example: Ratios with Double Number Lines: Example 9
Math Example: Ratios with Double Number Lines: Example 9
This example features a three-part ratio of 5:2:1 for orange, lemon, and lime juice. Given 6 cups of lemon juice, students need to determine the amounts of orange and lime juice required. The
solution demonstrates that 15 cups of orange juice and 3 cups of lime juice are needed to maintain the ratio.
By presenting a ratio with a wider range of values between components, this example challenges students to think more critically about proportional relationships. It illustrates how multiple number
lines can effectively represent ratios with larger differences between quantities, providing a clear visual tool for solving more complex ratio problems.
Providing multiple worked-out examples is essential for students to develop a comprehensive understanding of ratios and the application of number lines in problem-solving. Each new example introduces
additional complexity, helping students to recognize patterns and relationships across various scenarios. This approach enhances their ability to apply ratio concepts in diverse situations and
strengthens their analytical skills.
Teacher Script: "Now we have a juice mixture with a 5:2:1 ratio of orange, lemon, and lime. If we have 6 cups of lemon juice, how can we use our multiple number lines to determine the amounts of
orange and lime juice needed? Pay attention to how the intervals on each line differ. Can you explain why the orange juice line increases by 5s, the lemon juice by 2s, and the lime juice by 1s? How
does this help us solve more complex ratio problems efficiently?"
For a complete collection of math examples related to Ratios click on this link: Math Examples: Double Number Lines Collection. | {"url":"https://www.media4math.com/library/math-example-ratios-double-number-lines-example-9","timestamp":"2024-11-14T23:39:06Z","content_type":"text/html","content_length":"51330","record_id":"<urn:uuid:11bc1c53-ecb0-4ada-a8fe-90581993a192>","cc-path":"CC-MAIN-2024-46/segments/1730477397531.96/warc/CC-MAIN-20241114225955-20241115015955-00886.warc.gz"} |
Factorial Numbers: Write a C program to find Factorial Numbers of a given numbers. -
Factorial Numbers: Write a C program to find Factorial Numbers of a given numbers.
C language practice programs
Factorial numbers are a mathematical concept that plays a crucial role in various branches of mathematics and computer science. The factorial of a non-negative integer is the product of all positive
integers less than or equal to that number. It is denoted by the symbol !.
For example:
• The factorial of 5, denoted as 5!, is calculated as 5×4×3×2×1=1205×4×3×2×1=120.
• The factorial of 0 is defined to be 1, denoted as 0!=10!=1.
Factorials often find applications in permutations, combinations, and probability, making them an essential concept in algorithm design and mathematical calculations.
Now, let’s explore how to calculate the factorial of a given number using the C programming language. Below is a simple C program that takes user input for a number and calculates its factorial:
Recursive function
#include <stdio.h>
// Function to calculate factorial recursively
int factorial(int n) {
if (n == 0 || n == 1) {
return 1;
} else {
return n * factorial(n - 1);
int main() {
int num;
// Input
printf("Enter a number: ");
scanf("%d", &num);
// Check for negative input
if (num < 0) {
printf("Factorial is not defined for negative numbers.\n");
} else {
// Calculate and display the factorial
printf("Factorial of %d = %d\n", num, factorial(num));
return 0;
This program defines a recursive function factorial to calculate the factorial of a given number. The main function takes user input, checks if it’s non-negative, and then calls the factorial
function to compute and display the result.
Iterative method:
#include <stdio.h>
// Function to calculate factorial iteratively
int factorial(int n) {
int result = 1;
for (int i = 1; i <= n; i++) {
result *= i;
return result;
int main() {
int num;
// Input
printf("Enter a number: ");
scanf("%d", &num);
// Check for negative input
if (num < 0) {
printf("Factorial is not defined for negative numbers.\n");
} else {
// Calculate and display the factorial
printf("Factorial of %d = %d\n", num, factorial(num));
return 0;
In this program, the factorial function uses a loop to calculate the factorial iteratively. The loop runs from 1 to the given number (n), and the result is updated at each iteration. The rest of the
program is similar to the previous one. It takes user input, checks for negative values, and then calculates and displays the factorial.
Remember, factorials can grow very quickly, and for large input values, the program may encounter integer overflow. In such cases, you may need to use a larger data type, such as long long, to handle
larger results.
Understanding factorial numbers and being able to calculate them is a fundamental skill in the world of mathematics and computer science. Whether you are working on algorithms, probability, or
combinatorics, the concept of factorials will likely play a role in your problem-solving journey.
Read my other blogs:
Embedded C language Interview Questions.
Automotive Interview Questions
Understanding AUTOSAR Architecture: A Guide to Automotive Software Integration
Big Endian and Little Endian in Memory
Zero to Hero in C language Playlist
Embedded C Interview Questions
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I. Statistical mechanics of bubbly liquids. II. Behavior of sheared suspensions of non-Brownian particles
Yurkovetsky, Yevgeny (1998) I. Statistical mechanics of bubbly liquids. II. Behavior of sheared suspensions of non-Brownian particles. Dissertation (Ph.D.), California Institute of Technology.
doi:10.7907/NMJQ-2X32. https://resolver.caltech.edu/CaltechETD:etd-06222005-110302
NOTE: Text or symbols not renderable in plain ASCII are indicated by [...]. Abstract is included in .pdf document. I. Statistical mechanics of bubbly liquids. The dynamics of bubbles at high Reynolds
numbers is studied from the viewpoint of statistical mechanics. Individual bubbles are treated as dipoles in potential flow. A virtual mass matrix of the system of bubbles is introduced, which
depends on the instantaneous positions of the bubbles, and is used to calculate the energy of the bubbly flow as a quadratic form of the bubbles' velocities. The energy is shown to be the system's
Hamiltonian and is used to construct a canonical ensemble partition function, which explicitly includes the total impulse of the suspension along with its energy. The Hamiltonian is decomposed into
an effective potential due to the bubbles' collective motion and a kinetic term due to the random motion about the mean. An effective bubble temperature - a measure of the relative importance of the
bubbles' relative to collective motion--is derived with the help of the impulse-dependent partition function. Two effective potentials are shown to operate: one, due to the mean motion of the
bubbles, dominates at low bubble temperatures where it leads to their grouping in flat clusters normal to the direction of the collective motion, while the other, temperature invariant, is due to the
bubbles' position-dependent virtual mass and results in their mutual repulsion. Numerical evidence is presented for the existence of the effective potentials, the condensed and dispersed phases and a
phase transition. II. Behavior of sheared suspensions of non-Brownian particles. Suspensions of non-Brownian particles in simple shear flow of a Newtonian solvent in the range of particle phase
concentration, [...], from 0.05 to 0.52, are studied numerically by Stokesian Dynamics. The simulations are a function of [...] and the dimensionless shear rate, [...], which measures the relative
importance of the shear and short-ranged interparticle forces. The pair-distribution functions, shear viscosity, normal stress differences, suspension pressure, long-time self-diffusion coefficients,
and mean square of the particle velocity fluctuations in the velocity-gradient and vorticity directions are computed, tabulated and plotted. In concentrated suspensions ([...] > 0.45), two distinct
microstructural patterns are shown to exist at the highest and lowest shear rates. At [...] = 0.1 the particles form hexagonally packed strings in the flow direction. As [...] increases, the strings
are gradually being replaced by non-compact clusters of particles kept together by strong lubrication forces while the particle pair-distribution displays a broken fore-aft symmetry. These changes in
the microstructure are accompanied by increases in the shear viscosity, normal stress differences, suspension pressure, longtime self-diffusion coefficients, and fluctuational motion. Agreement is
found between the simulation results and the theoretical predictions of Brady and Morris (1997).
Item Type: Thesis (Dissertation (Ph.D.))
Subject Keywords: bubbles in ideal fluid; rheology of non-Brownian suspensions
Degree Grantor: California Institute of Technology
Division: Chemistry and Chemical Engineering
Major Option: Chemical Engineering
Thesis Availability: Public (worldwide access)
Research Advisor(s): • Brady, John F.
Thesis Committee: • Brady, John F. (chair)
• Brennen, Christopher E.
• Gavalas, George R.
• Kornfield, Julia A.
• Hunt, Melany L.
• Wang, Zhen-Gang
Defense Date: 24 July 1996
Non-Caltech Author Email: yyurkovetsky (AT) ccny.cuny.edu
Record Number: CaltechETD:etd-06222005-110302
Persistent URL: https://resolver.caltech.edu/CaltechETD:etd-06222005-110302
DOI: 10.7907/NMJQ-2X32
Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code: 2684
Collection: CaltechTHESIS
Deposited By: Imported from ETD-db
Deposited On: 22 Jun 2005
Last Modified: 21 Dec 2019 04:09
Thesis Files
PDF (Yurkovetsky_Y_1998.pdf) - Final Version
See Usage Policy.
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How can I define a parameter in relation to some elements in the variable | AIMMS Community
For all t, H(t) = Z(t+1) + Z(t).
Z(t) is the variable. And H(t) is the parameter calculated with the adjacent elements in the variable. How can I do this?
BTW, in this instance, what’s the ‘index domain’ of the parameter H, as the index t=1,2,……,24, i.e. the maximum element in Z is Z(24), if I define the ‘index domain’ of H as t directly, which means
that there exists H(24) = Z(25)+Z(24), which should not be existed. Would this cause error or how can I limit the index domain of H to t = 1,2,…..,23? | {"url":"https://community.aimms.com/math-or-optimization-modeling-39/how-can-i-define-a-parameter-in-relation-to-some-elements-in-the-variable-1408","timestamp":"2024-11-02T12:42:50Z","content_type":"text/html","content_length":"135091","record_id":"<urn:uuid:e80a5b89-b5d9-4da7-a007-c908d4f33f2e>","cc-path":"CC-MAIN-2024-46/segments/1730477027710.33/warc/CC-MAIN-20241102102832-20241102132832-00428.warc.gz"} |
List of Deep Learning Layers
This page provides a list of deep learning layers in MATLAB^®.
To learn how to create networks from layers for different tasks, see the following examples.
Deep Learning Layers
Use the following functions to create different layer types. Alternatively, use the Deep Network Designer app to create networks interactively.
To learn how to define your own custom layers, see Define Custom Deep Learning Layers.
Input Layers
Layer Description
imageInputLayer An image input layer inputs 2-D images to a neural network and applies data normalization.
image3dInputLayer A 3-D image input layer inputs 3-D images or volumes to a neural network and applies data normalization.
sequenceInputLayer A sequence input layer inputs sequence data to a neural network and applies data normalization.
featureInputLayer A feature input layer inputs feature data to a neural network and applies data normalization. Use this layer when you have a data set of numeric scalars representing
features (data without spatial or time dimensions).
inputLayer An input layer inputs data into a neural network with a custom format.
pointCloudInputLayer (Lidar A point cloud input layer inputs 3-D point clouds to a network and applies data normalization. You can also input point cloud data such as 2-D lidar scans.
Convolution and Fully Connected Layers
Layer Description
convolution1dLayer A 1-D convolutional layer applies sliding convolutional filters to 1-D input.
convolution2dLayer A 2-D convolutional layer applies sliding convolutional filters to 2-D input.
convolution3dLayer A 3-D convolutional layer applies sliding cuboidal convolution filters to 3-D input.
groupedConvolution2dLayer A 2-D grouped convolutional layer separates the input channels into groups and applies sliding convolutional filters. Use grouped convolutional layers for channel-wise
separable (also known as depth-wise separable) convolution.
transposedConv1dLayer A transposed 1-D convolution layer upsamples one-dimensional feature maps.
transposedConv2dLayer A transposed 2-D convolution layer upsamples two-dimensional feature maps.
transposedConv3dLayer A transposed 3-D convolution layer upsamples three-dimensional feature maps.
fullyConnectedLayer A fully connected layer multiplies the input by a weight matrix and then adds a bias vector.
Sequence Layers
Layer Description
sequenceInputLayer A sequence input layer inputs sequence data to a neural network and applies data normalization.
lstmLayer An LSTM layer is an RNN layer that learns long-term dependencies between time steps in time-series and sequence data.
lstmProjectedLayer An LSTM projected layer is an RNN layer that learns long-term dependencies between time steps in time-series and sequence data using projected learnable weights.
bilstmLayer A bidirectional LSTM (BiLSTM) layer is an RNN layer that learns bidirectional long-term dependencies between time steps of time-series or sequence data. These
dependencies can be useful when you want the RNN to learn from the complete time series at each time step.
gruLayer A GRU layer is an RNN layer that learns dependencies between time steps in time-series and sequence data.
gruProjectedLayer A GRU projected layer is an RNN layer that learns dependencies between time steps in time-series and sequence data using projected learnable weights.
convolution1dLayer A 1-D convolutional layer applies sliding convolutional filters to 1-D input.
transposedConv1dLayer A transposed 1-D convolution layer upsamples one-dimensional feature maps.
maxPooling1dLayer A 1-D max pooling layer performs downsampling by dividing the input into 1-D pooling regions, then computing the maximum of each region.
averagePooling1dLayer A 1-D average pooling layer performs downsampling by dividing the input into 1-D pooling regions, then computing the average of each region.
globalMaxPooling1dLayer A 1-D global max pooling layer performs downsampling by outputting the maximum of the time or spatial dimensions of the input.
flattenLayer A flatten layer collapses the spatial dimensions of the input into the channel dimension.
wordEmbeddingLayer (Text
Analytics Toolbox) A word embedding layer maps word indices to vectors.
peepholeLSTMLayer (Custom A peephole LSTM layer is a variant of an LSTM layer, where the gate calculations use the layer cell state.
layer example)
Activation Layers
Layer Description
reluLayer A ReLU layer performs a threshold operation to each element of the input, where any value less than zero is set to zero.
leakyReluLayer A leaky ReLU layer performs a threshold operation, where any input value less than zero is multiplied by a fixed scalar.
clippedReluLayer A clipped ReLU layer performs a threshold operation, where any input value less than zero is set to zero and any value above the clipping ceiling is set to that clipping
eluLayer An ELU activation layer performs the identity operation on positive inputs and an exponential nonlinearity on negative inputs.
geluLayer A Gaussian error linear unit (GELU) layer weights the input by its probability under a Gaussian distribution.
tanhLayer A hyperbolic tangent (tanh) activation layer applies the tanh function on the layer inputs.
swishLayer A swish activation layer applies the swish function on the layer inputs.
softplusLayer A softplus layer applies the softplus activation function Y = log(1 + e^X), which ensures that the output is always positive. This activation function is a smooth continuous
(Reinforcement Learning version of reluLayer. You can incorporate this layer into the deep neural networks you define for actors in reinforcement learning agents. This layer is useful for creating
Toolbox) continuous Gaussian policy deep neural networks, for which the standard deviation output must be positive.
softmaxLayer A softmax layer applies a softmax function to the input.
sigmoidLayer A sigmoid layer applies a sigmoid function to the input such that the output is bounded in the interval (0,1).
functionLayer A function layer applies a specified function to the layer input.
preluLayer A PReLU layer performs a threshold operation, where for each channel, any input value less than zero is multiplied by a scalar learned at training time.
sreluLayer (Custom layer A SReLU layer performs a thresholding operation, where for each channel, the layer scales values outside an interval. The interval thresholds and scaling factors are
example) learnable parameters.
(Deep Learning HDL A mish activation layer applies the mish function on layer inputs.
Normalization Layers
Layer Description
batchNormalizationLayer A batch normalization layer normalizes a mini-batch of data across all observations for each channel independently. To speed up training of the convolutional neural
network and reduce the sensitivity to network initialization, use batch normalization layers between convolutional layers and nonlinearities, such as ReLU layers.
groupNormalizationLayer A group normalization layer normalizes a mini-batch of data across grouped subsets of channels for each observation independently. To speed up training of the
convolutional neural network and reduce the sensitivity to network initialization, use group normalization layers between convolutional layers and nonlinearities, such
as ReLU layers.
instanceNormalizationLayer An instance normalization layer normalizes a mini-batch of data across each channel for each observation independently. To improve the convergence of training the
convolutional neural network and reduce the sensitivity to network hyperparameters, use instance normalization layers between convolutional layers and nonlinearities,
such as ReLU layers.
layerNormalizationLayer A layer normalization layer normalizes a mini-batch of data across all channels for each observation independently. To speed up training of recurrent and multilayer
perceptron neural networks and reduce the sensitivity to network initialization, use layer normalization layers after the learnable layers, such as LSTM and fully
connected layers.
crossChannelNormalizationLayer A channel-wise local response (cross-channel) normalization layer carries out channel-wise normalization.
Utility Layers
Layer Description
dropoutLayer A dropout layer randomly sets input elements to zero with a given probability.
spatialDropoutLayer A spatial dropout layer randomly selects input channels with a given probability, and sets all its elements to zero during training.
crop2dLayer A 2-D crop layer applies 2-D cropping to the input.
crop3dLayer A 3-D crop layer crops a 3-D volume to the size of the input feature map.
identityLayer An identity layer is a layer whose output is identical to its input. You can use an identity layer to create a skip connection, which allows the input to skip one or more
layers in the main branch of a neural network. For more information about skip connections, see More About.
networkLayer A network layer contains a nested network. Use network layers to simplify building large networks that contain repeating components.
complexToRealLayer A complex-to-real layer converts complex-valued data to real-valued data by splitting the data in a specified dimension.
realToComplexLayer A real-to-complex layer converts real-valued data to complex-valued data by merging the data in a specified dimension.
scalingLayer (Reinforcement A scaling layer linearly scales and biases an input array U, giving an output Y = Scale.*U + Bias. You can incorporate this layer into the deep neural networks you define
Learning Toolbox) for actors or critics in reinforcement learning agents. This layer is useful for scaling and shifting the outputs of nonlinear layers, such as tanhLayer and sigmoid.
(Reinforcement Learning A quadratic layer takes an input vector and outputs a vector of quadratic monomials constructed from the input elements. This layer is useful when you need a layer whose
Toolbox) output is a quadratic function of its inputs. For example, to recreate the structure of quadratic value functions such as those used in LQR controller design.
stftLayer (Signal
Processing Toolbox) An STFT layer computes the short-time Fourier transform of the input.
istftLayer (Signal
Processing Toolbox) An ISTFT layer computes the inverse short-time Fourier transform of the input.
cwtLayer (Wavelet Toolbox) A CWT layer computes the continuous wavelet transform of the input.
icwtLayer (Wavelet Toolbox) An ICWT layer computes the inverse continuous wavelet transform of the input.
modwtLayer (Wavelet A MODWT layer computes the maximal overlap discrete wavelet transform (MODWT) and MODWT multiresolution analysis (MRA) of the input.
Resizing Layers
Layer Description
resize2dLayer (Image Processing Toolbox) A 2-D resize layer resizes 2-D input by a scale factor, to a specified height and width, or to the size of a reference input feature map.
resize3dLayer (Image Processing Toolbox) A 3-D resize layer resizes 3-D input by a scale factor, to a specified height, width, and depth, or to the size of a reference input feature map.
dlhdl.layer.reshapeLayer (Deep Learning HDL Toolbox) A reshape layer reshapes layer activation data.
Pooling and Unpooling Layers
Layer Description
averagePooling1dLayer A 1-D average pooling layer performs downsampling by dividing the input into 1-D pooling regions, then computing the average of each region.
averagePooling2dLayer A 2-D average pooling layer performs downsampling by dividing the input into rectangular pooling regions, then computing the average of each region.
averagePooling3dLayer A 3-D average pooling layer performs downsampling by dividing three-dimensional input into cuboidal pooling regions, then computing the average values of each region.
adaptiveAveragePooling2dLayer A 2-D adaptive average pooling layer performs downsampling to give you the desired output size by dividing the input into rectangular pooling regions, then computing the
average of each region.
globalAveragePooling1dLayer A 1-D global average pooling layer performs downsampling by outputting the average of the time or spatial dimensions of the input.
globalAveragePooling2dLayer A 2-D global average pooling layer performs downsampling by computing the mean of the height and width dimensions of the input.
globalAveragePooling3dLayer A 3-D global average pooling layer performs downsampling by computing the mean of the height, width, and depth dimensions of the input.
maxPooling1dLayer A 1-D max pooling layer performs downsampling by dividing the input into 1-D pooling regions, then computing the maximum of each region.
maxPooling2dLayer A 2-D max pooling layer performs downsampling by dividing the input into rectangular pooling regions, then computing the maximum of each region.
maxPooling3dLayer A 3-D max pooling layer performs downsampling by dividing three-dimensional input into cuboidal pooling regions, then computing the maximum of each region.
globalMaxPooling1dLayer A 1-D global max pooling layer performs downsampling by outputting the maximum of the time or spatial dimensions of the input.
globalMaxPooling2dLayer A 2-D global max pooling layer performs downsampling by computing the maximum of the height and width dimensions of the input.
globalMaxPooling3dLayer A 3-D global max pooling layer performs downsampling by computing the maximum of the height, width, and depth dimensions of the input.
maxUnpooling2dLayer A 2-D max unpooling layer unpools the output of a 2-D max pooling layer.
Combination Layers
Layer Description
additionLayer An addition layer adds inputs from multiple neural network layers element-wise.
multiplicationLayer A multiplication layer multiplies inputs from multiple neural network layers element-wise.
depthConcatenationLayer A depth concatenation layer takes inputs that have the same height and width and concatenates them along the channel dimension.
concatenationLayer A concatenation layer takes inputs and concatenates them along a specified dimension. The inputs must have the same size in all dimensions except the
concatenation dimension.
weightedAdditionLayer (Custom layer A weighted addition layer scales and adds inputs from multiple neural network layers element-wise.
Transformer Layers
Layer Description
selfAttentionLayer A self-attention layer computes single-head or multihead self-attention of its input.
attentionLayer A dot-product attention layer focuses on parts of the input using weighted multiplication operations.
positionEmbeddingLayer A position embedding layer maps sequential or spatial indices to vectors.
sinusoidalPositionEncodingLayer A sinusoidal position encoding layer maps position indices to vectors using sinusoidal operations.
embeddingConcatenationLayer An embedding concatenation layer combines its input and an embedding vector by concatenation.
indexing1dLayer A 1-D indexing layer extracts the data from the specified index of the time or spatial dimensions of the input data.
patchEmbeddingLayer (Computer Vision Toolbox) A patch embedding layer maps patches of pixels to vectors.
Neural ODE Layers
Layer Description
neuralODELayer A neural ODE layer outputs the solution of an ODE.
Object Detection Layers
Layer Description
roiMaxPooling2dLayer (Computer An ROI max pooling layer outputs fixed size feature maps for every rectangular ROI within the input feature map. Use this layer to create a Fast or Faster R-CNN object
Vision Toolbox) detection network.
roiAlignLayer (Computer Vision
Toolbox) An ROI align layer outputs fixed size feature maps for every rectangular ROI within an input feature map. Use this layer to create a Mask R-CNN network.
ssdMergeLayer (Computer Vision
Toolbox) An SSD merge layer merges the outputs of feature maps for subsequent regression and classification loss computation.
yolov2TransformLayer (Computer A transform layer of the you only look once version 2 (YOLO v2) network transforms the bounding box predictions of the last convolution layer in the network to fall
Vision Toolbox) within the bounds of the ground truth. Use the transform layer to improve the stability of the YOLO v2 network.
spaceToDepthLayer (Image A space to depth layer permutes the spatial blocks of the input into the depth dimension. Use this layer when you need to combine feature maps of different size
Processing Toolbox) without discarding any feature data.
depthToSpace2dLayer (Image
Processing Toolbox) A 2-D depth to space layer permutes data from the depth dimension into blocks of 2-D spatial data.
dlhdl.layer.sliceLayer (Deep A slice layer divides the input to the layer into an equal number of groups along the channel dimension of the image.
Learning HDL Toolbox)
See Also
trainnet | trainingOptions | dlnetwork | Deep Network Designer
Related Topics | {"url":"https://fr.mathworks.com/help/deeplearning/ug/list-of-deep-learning-layers.html","timestamp":"2024-11-10T17:10:10Z","content_type":"text/html","content_length":"132338","record_id":"<urn:uuid:dd473a56-765a-4397-853d-7ddb66e2c72e>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.61/warc/CC-MAIN-20241110170046-20241110200046-00796.warc.gz"} |
6152 Setting Up Transition Matrices
• 1.
Create a transition matrix, T, to represent the network diagram above. Use columns to define the starting point. What is the value of ?
• 2.
Create a transition matrix, T, to represent the network diagram above. Use columns to define the starting point. What is the value of ?
The value of 0.55 is the answer to the question, but without any context or further information provided, it is not possible to determine the specific meaning or significance of this value. It
could represent a probability, a percentage, or some other numerical measurement within the context of the transition matrix and network diagram.
• 3.
Create a transition matrix, T, to represent the network diagram above. Use columns to define the starting point. What is the value of ?
The value of 0.6 is the answer to the question, but without any context or additional information about the network diagram or transition matrix, it is difficult to provide a specific explanation
for why 0.6 is the correct answer. It is possible that the transition matrix T has been calculated or provided elsewhere and the value of 0.6 corresponds to a specific element or calculation
within that matrix. However, without further information, a more detailed explanation cannot be provided.
• 4.
Create a transition matrix, T, to represent the network diagram above. Use columns to define the starting point. What is the value of ?
The value of 0.45 is the answer to the question asking for the value of a specific element in the transition matrix T. However, since the network diagram and the specific element in question are
not provided, it is not possible to determine the exact explanation for why the value is 0.45.
• 5.
Create a transition matrix, T, to represent the network diagram above. Use columns to define the starting point. What is the value of ?
• 6.
Create a transition matrix, T, to represent the network diagram above. Use columns to define the starting point. What is the value of ?
The value of 0.7 represents the probability of transitioning from one state to another in the network diagram. In a transition matrix, each element represents the probability of moving from one
state (defined by the row) to another state (defined by the column). Therefore, the value of 0.7 indicates that there is a 70% chance of transitioning from the starting point to a specific point
in the network.
• 7.
Create a transition matrix, T, to represent the network diagram above. Use columns to define the starting point. What is the value of ?
The transition matrix, T, represents the probabilities of transitioning from one state to another in a network diagram. In this case, the starting point is defined by the columns of the matrix.
Since the question does not provide any information about the network diagram or the specific values in the matrix, it is not possible to determine the value of 0.25 or provide a meaningful
• 8.
Create a transition matrix, T, to represent the network diagram above. Use columns to define the starting point. What is the value of ?
The given answer indicates that both the starting points in the transition matrix have a value of 0.3. This means that there is an equal probability of transitioning from either starting point to
the next state in the network diagram.
• 9.
Create a transition matrix, T, to represent the network diagram above. Use columns to define the starting point. What is the value of ?
The value of 0.6 is the answer to the question.
• 10.
Create a transition matrix, T, to represent the network diagram above. Use columns to define the starting point. What is the value of ?
The given answer of 0.45 suggests that the value being referred to is the probability of transitioning from one state to another in the network diagram. It implies that there is a 45% chance of
moving from the starting point to the next state in the diagram.
• 11. | {"url":"https://www.proprofs.com/quiz-school/story.php?title=nzgwnjywongu","timestamp":"2024-11-09T03:59:50Z","content_type":"text/html","content_length":"1049571","record_id":"<urn:uuid:4ea2b7f6-0318-40f2-8309-e3abf7cd8eca>","cc-path":"CC-MAIN-2024-46/segments/1730477028115.85/warc/CC-MAIN-20241109022607-20241109052607-00707.warc.gz"} |
Diffusive N-waves and metastability in the burgers equation
We study the effect of viscosity on the large time behavior of the viscous Burgers equation by using a transformed version of Burgers (in self-similar variables) that captures efficiently the
mechanism of transition to the asymptotic states and allows us to estimate the time of evolution from an N-wave to the final stage of a diffusion wave. Then we construct certain special solutions of
diffusive N-waves with unequal masses. Finally, using a set of similarity variables and a variant of the Cole-Hopf transformation, we obtain an integrated Fokker-Planck equation. The latter is
solvable and provides an explicit solution of the viscous Burgers equation in a series of Hermite polynomials. This format captures the long-time-small-viscosity interplay, as the diffusion wave and
the diffusive N-waves correspond, respectively, to the first two terms in the Hermite polynomial expansion.
• Convection-diffusion
• Diffusion waves
• Diffusive N-waves
• Metastability
ASJC Scopus subject areas
• Analysis
• Computational Mathematics
• Applied Mathematics
Dive into the research topics of 'Diffusive N-waves and metastability in the burgers equation'. Together they form a unique fingerprint. | {"url":"https://academia.kaust.edu.sa/en/publications/diffusive-n-waves-and-metastability-in-the-burgers-equation","timestamp":"2024-11-12T13:48:02Z","content_type":"text/html","content_length":"54111","record_id":"<urn:uuid:38292c9e-ef7c-49a8-a58d-b18f21fc5d05>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.45/warc/CC-MAIN-20241112113320-20241112143320-00334.warc.gz"} |
What do you mean by non degenerate?
What do you mean by non degenerate?
Nondegenerate forms A nondegenerate or nonsingular form is a bilinear form that is not degenerate, meaning that is an isomorphism, or equivalently in finite dimensions, if and only if for all implies
that . The most important examples of nondegenerate forms are inner products and symplectic forms.
What does degenerate function mean?
In mathematics, a degenerate distribution is a probability distribution in a space (discrete or continuous) with support only on a space of lower dimension. The probability mass function equals 1 at
this point and 0 elsewhere.
What does it mean for a distribution to be degenerate?
probability distribution
In mathematics, a degenerate distribution is the probability distribution of a discrete random variable whose support consists of only one value. Examples include a two-headed coin and rolling a die
whose sides all show the same number.
What are the 3 types of random variable?
A random variable, usually written X, is a variable whose possible values are numerical outcomes of a random phenomenon. There are two types of random variables, discrete and continuous.
What do you mean by non-degenerate solution?
A basic feasible solution is non-degenerate if there are exactly n tight constraints. Definition 3. A basic feasible solution is degenerate if there are more than n tight constraints. We say that a
linear programming problem is degenerate if it contains degenerate vertices or basic feasible solutions.
What is a non-degenerate triangle?
Non-degenerate triangle − it is a triangle that has a positive area. The condition for a non-degenerate triangle with sides a, b, c is − a + b > c a + c > b b + c > a. Let’s take an example to
understand the problem better −
How do you know if a solution is degenerate?
Definition: An LP is degenerate if in a basic feasible solution, one of the basic variables takes on a zero value. Degeneracy is a problem in practice, because it makes the simplex algorithm slower.
Standard form. Note that one of the basic variables is 0.
What does it mean for a random variable to be degenerate?
The formal definition of a degenerate random variable is that it’s a distribution assigning all of the probability to a single point: A random variable, X, is degenerate if, for some a constant, c, P
(X = c) = 1. If a random variable does not meet the above definition, then it is non-degenerate.
Can CDF be a constant?
The cumulative distribution function (CDF) of a random variable X is denoted by F(x), and is defined as F(x) = Pr(X ≤ x). Notice also that the CDF of a discrete random variable will remain constant
on any interval of the form .
What is difference between the two types of random variables?
Random variables are classified into discrete and continuous variables. The main difference between the two categories is the type of possible values that each variable can take. In addition, the
type of (random) variable implies the particular method of finding a probability distribution function.
When a problem is called a non-degenerate?
Is there such a thing as a degenerate random variable?
NO. Let X be any variable and Y independent such that Y = 0 with probability 1. Then X Y is degenerate, but X need not be. This was already answered in comments: No. Only one of them needs to be. Let
X be zero with probability 1 and let Y be any finite-valued random variable.
Do you need to make X a degenerate variable?
NO. Let X be any variable and Y independent such that Y = 0 with probability 1. Then X Y is degenerate, but X need not be. This was already answered in comments:
What is the cumulative function of a degenerate distribution?
The cumulative distribution function of the univariate degenerate distribution is: Constant random variable. In probability theory, a constant random variable is a discrete random variable that takes
a constant value, regardless of any event that occurs.
What makes a singular distribution not a degenerate distribution?
Singular distributions (those that don’t have a density or a mass function) are not degenerate; but if some variables depend deterministically on others then a change of variable can make some
marginals degenerate. Bill Bell’s answer pointed this out in the case of a singular normal distribution. | {"url":"https://forwardonclimate.org/various-papers/what-do-you-mean-by-non-degenerate/","timestamp":"2024-11-07T09:30:15Z","content_type":"text/html","content_length":"57265","record_id":"<urn:uuid:d1e1fc9d-1484-4396-a71a-0187988be7b0>","cc-path":"CC-MAIN-2024-46/segments/1730477027987.79/warc/CC-MAIN-20241107083707-20241107113707-00822.warc.gz"} |
Liking Diring a Car: How to determine the cost of a product or service
The component skill would be how to calculate the cost of a product or service.
To do this you must:
• Understand how cost is measured (basis of accounting)
• Understand the theory of cost categories like direct and direct costs, raw material, labour, direct overhead, indirect overhead etc..
• Understand how to apply that theory to real-world data through analyzing accounting data and through other information such as conversations with various departments such as manufacturing etc..
to apply these categories to costs; for example, which costs are direct, indirect or mix
• Understanding how to perform various math calculations such as per unit and covert; between units of measure (ex. Lbs and KG); break down a fixed cost into variable and fixed components
• Understand and perform calculations around overhead rates
Once these skills are understood, you can then begin to complete costing calculations.
Example for "Liking Diring a Car: How to determine the cost of a product or service": | {"url":"https://bank.ecampusontario.ca/response/liking-diring-a-car-how-to-determine-the-cost-of-a-product-or-service/","timestamp":"2024-11-14T08:11:06Z","content_type":"text/html","content_length":"30876","record_id":"<urn:uuid:5232f602-c1cc-4fac-aa0b-ace513c81922>","cc-path":"CC-MAIN-2024-46/segments/1730477028545.2/warc/CC-MAIN-20241114062951-20241114092951-00422.warc.gz"} |
Extensions of the Euler-Lagrange Equations
13.4.3 Extensions of the Euler-Lagrange Equations
Several extensions of the Euler-Lagrange equation can be constructed to handle complications that arise in addition to kinetic energy and potential energy in a conservative field. Each extension
usually involves adding more terms to (13.129) to account for the new complication. Problems that can be handled in this way are closed kinematic chains, nonholonomic constraints, and nonconservative
forces (such as friction).
Steven M LaValle 2020-08-14 | {"url":"https://lavalle.pl/planning/node704.html","timestamp":"2024-11-03T06:43:45Z","content_type":"text/html","content_length":"3905","record_id":"<urn:uuid:1452ef56-6031-443f-a3ad-8d6ff69536ad>","cc-path":"CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00180.warc.gz"} |
Meaning of Interest in Grade 12 - Theory of Factor Pricing | Online Notes Nepal
Meaning of Interest
Meaning of Interest refers to a payment for the use of a certain sum of money for an agreed period of time. The payment made by a borrower for the use of loan for a year expressed as the ratio which
that payment bears to the loan is called interest. And this term is known to representing the money equivalent while income is derived from capital. Thus interest is the price paid for the sorting
period for the loanable funds or money capital.
Interest is commonly expressed as a certain percentage on the capital sum of the loan. If a person borrows Rs 1000 at the rate of interest of 10% per annum, he will have to repay Rs 1100 in the next
year. Here Rs 1000 is the amount and Rs 100 is the interest.
Interest is divided into two types.
• Net interest.
• Gross interest.
Net interest
The net interest is the price paid for the services of capital employed. Net interest is that part of the gross interest which paid for the use of capital. According to Chapman net interest is the
payment for the loan of capital, where there is no risk, no inconvenience and no work entailed on the lender.
Gross interest
The payment made by the borrower or for the use of capital to the lender comprises gross interest. It involves payment for the work risk and supervision of investment. According to Chapman gross
interest includes payment for the loan of capital, payment cover risk of loss payment for the inconvenience of the investment and payment for the work and worry involved in watching investment
calling them and investing. | {"url":"https://onlinenotesnepal.com/meaning-of-interest","timestamp":"2024-11-09T10:24:03Z","content_type":"text/html","content_length":"75205","record_id":"<urn:uuid:0880d556-dbc3-4400-9e23-507e7dbe9e13>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.75/warc/CC-MAIN-20241109085148-20241109115148-00590.warc.gz"} |
Insertion Sort in JavaScript | Implementing Insertion Sort in JavaScript
Introduction to Insertion Sort in JavaScript
Sorting is one of the important concepts that programmers learn to begin their journey in computer science irrespective of the programming language selected to learn. Sorting helps us to locate the
target data we want to search in a faster and convenient manner, by sorting them in either ascending or descending order.
Sorting algorithms are used to reorder elements, where an element can be a number or a string. There are many types of sorting algorithms based on their method of sorting and the approach they follow
to sort the elements, and each type has its advantages and disadvantages.
In this blog, we will be focusing on insertion sort, a common sort that is easy to understand and implement.
What is Insertion Sort in JavaScript?
Insertion Sort is a simple, easy to understand the algorithm that works best with a small data list by sorting each element in the data list one by one from left to right direction. It is also known
as a comparison sort where it compares the current value with the other values within the same data list that’s being sorted. It follows an iterative approach to place each element in the right order
in the data list.
The more time an algorithm takes to sort, its performance is said to be bad and need to consider another algorithm to sort out the data. Insertion sort has a time complexity of O(n²) or runs
quadratic time to sort the data list in the worst-case scenario. This typically isn’t very effective and should not be used for large lists. However, it usually outperforms advanced algorithms such
as quicksort or mergesort on smaller lists.
Insertion sort, most of the time is more efficient than other quadratic sorting algorithms such as bubble sort or selection sort. Its best-case scenario, time is O(n), or linear, which occurs if the
input array is already sorted. On average, insertion sort’s run time is still quadratic.
In the below example we will have an easy high-level approach to sort data stored in an array data structure and use its sort method to sort the data without implementing any algorithms.
Example – Insertion Sort Algorithm
<!DOCTYPE html>
// Declaring unsorted data and storing it in array data structure
var dataArray = [96,5,42,1,6,37,21]
// Function - Insertion Sort Algo.
function insertSort(unsortedData) {
for (let i = 1; i < unsortedData.length; i++) {
let current = unsortedData[i];
let j;
for(j=i-1; j >= 0 && unsortedData[j] > current;j--) {
unsortedData[j + 1] = unsortedData[j]
unsortedData[j + 1] = current;
return unsortedData;
// print sorted array
Explanation: In the algorithm, we have implemented 2 for loops, the outer for loop is to iterate over the array elements and the inner for loop is used to sort the array elements in the ascending
order of their value. The current variable holds the current value of the array and variable j is set to one value less than the current index position of the array. We check whether the current
element (current) is smaller than the array value at j^th position (unsortedData[j])and if it is true then we sort those values.
Iteration 1 – current (96) : [96,5,42,1,6,37,21]
Iteration 2 – current (5) : [5,96,42,1,6,37,21]
Iteration 3 – current (42) : [5,42,96,1,6,37,21]
Iteration 4 – current (1) : [1,5,42,96,6,37,21]
Iteration 5 – current (6) : [1,5,6,42,96,37,21]
Iteration 6 – current (37) : [1,5,6,37,42,96,21]
Iteration 7 – current (21) : [1,5,6,21,37,42,96]
The outer for loop iteration starts at 1^st index position since we want to move the smallest element to the left hand side so we are comparing whether the current element is smaller than the
elements on its left hand side.
Types of Sorting
The types of algorithms that are used for sorting data encompasses the following concepts or ideas in their approach to sorting the data:
• Comparison versus non-comparison-based strategies,
• Iterative versus Recursive implementation,
• Divide-and-Conquer paradigm (this or that),
• Randomize Approach.
Let’s consider a few examples:
1. Merge sort uses a divide-and-conquer approach to sort elements in an array.
2. Insertion Sort, Bubble Sort is a comparison-based sort.
When data is sorted, it becomes easier to come up with an optimal solution to complex problems. for example,
• Searching for a specific value,
• Finding the minimum or maximum value,
• Testing for uniqueness and deleting duplicates,
• Counting how many times a specific value has appeared, etc.
In this article, we have gone through the definition of insertion sort and its time complexity and various other sorting algorithm types based on their approach. Studying various sorting algorithms
helps us to identify which one is better suited at certain circumstances or use cases that help us to sort the data at a faster rate.
Recommended Articles
This is a guide to Insertion Sort in JavaScript. Here we discuss what is insertion sort in javascript and its types with example respectively. You may also look at the following articles to learn
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System Expectancy
A lot of people want to compare their trading system with that of other people. To achieve this we need to calculate the expectancy of the system over the long term rather than just saying I made 100
pts last week and x made 50 pts therefore my system is better.
The best way to calculate expectancy is take a reasonable number of actual trades (at least 100) and calculate the following based on those results:
Average Win Size (AW) = The average of all trades which won (do not count the biggest win and do not count breakeven trades)
Average Loss Size (AL) = The average of all trades which lost (again do not count any breakeven trades)
Average Win Percentage (W%) = total winners/total number of trades excluding breakeven trades and the biggest winner
Average Loss Percentage (L%) = 100%-W%
Average number of trades per year (T) = number of trades in the sample size/number of weeks the sample covers * 52
Expectancy = ((AW*W% + AL*L%) / - AL ) * T
Attached is an Excel S/S with the formula in to help you.
Here are the figures for my trading system that I use on the E-mini S&P:
My expectancy is 232. So for each $1 I risk on every trade I expect to make $232 in a year.
Obviously the higher the expectancy the better, if it is negative then you will be losing money and should reevaluate your system.
I hope this helps people when they are designing their own systems and looking for comparisons.
Hi Tim,
But you haven't enclosed the system rules for us to compare to
All will be revealed in my book: 'How I made a million trading index futures'. I've only got about $999,000 to go before I can start writing it!
I was hoping that others might let us know the expectancy of their own trading for comparison.
This post is intended to be mildly humorous, no book actually exists. No free advertising or exaggerated hype intended.
Does Metastock calculates automatically these numbers?I think so...
i'll try to get an expectancy result for my system,although i think the best number to put here..
is ..how much money,in how much time , with how much capital
You get almost 2 trades a day, and on average you make 10 points on your winners and lose 5 points on
your losers?
I didnt think the S&P was that volatile?
I'm afraid your expectancy calculation doesn't seem right to me.
Expectancy is calculated on a PER TRADE basis as opposed to a PER YEAR basis, but aside from that, I still don't see where you get your $232 dollars per year from. Let me explain the way I see it,
and then maybe you can either agree with me or re-clarify your calculations.
The official calculation for expectancy is: (PW*AW) - (PL*AL)
...........where PW = Probability of a winning trade
AW = Average Win
PL = Probability of losing trade
AL = Average Loss
If we put your figures into this formula, then we get: (0.51*10.79) - (0.49*5.14) = 2.98pts per trade
Now here's where I don't get your figures. The average win and average loss that you have given are in points NOT dollars. Therefore, the annual expectancy of your system seems to be 2.98pts * 400
trades per year = 1193 pts over a 12 month period.
I don't see where your $232 comes from. If you know what your AW and AL are in dollars (not points) then we could work out the true expectancy of your system.
Hope all this makes sense.
I prefer to analyse chances and possible account drawdowns of trading systems and concepts using monte carlo simulator software.
I've taken your input values (in your posting) and get the following mcs results (25.000 simulation runs with always 400 trades = 10.000.000 trades are simulated):
a) min.,average and max. profits and drawdowns (mcs1.gif attached)
b) distribution of simulation runs (mcs2.gif attached)
If your system results (and or the market conditions) remain constant and assuming 400 trades a year, you can estimate yearly profits between $24.000 and $92.000 (average $59.000). The max. account
drawdown or negative balance of your trading account was only -$5.600...
...very good results.
But the conditions seldom remain constant, so it may be useful to test your system also with additional synthetic data (via data simulation, data scrambling procedures), which can also be build with
mcs software.
Good luck,
zentrader (Volker Butzlaff, developer of Zen Monte Carlo Simulator v5.0)
I suppose I should point out that sidinuk has posted only once since last December, and that was in July. So anyone wanting to attract his attention may want to send him an email . . .
Thanks dbphoenix,
but I think the discussion concerning system expectancy is interesting for every (system) trader, not only for sidinuk...
zentrader22 said:
Thanks dbphoenix,
but I think the discussion concerning system expectancy is interesting for every (system) trader, not only for sidinuk...
I suspect so. But sometimes people get upset if they don't get a response from the person they've addressed.
I found the following information on correctly analysing system expectancy in the TradeStation strategy chat forum, posted by ghkramer. It really is very good info:
Last edited:
A lot of people want to compare their trading system with that of other people. To achieve this we need to calculate the expectancy of the system over the long term rather than just saying I made
100 pts last week and x made 50 pts therefore my system is better.
The best way to calculate expectancy is take a reasonable number of actual trades (at least 100) and calculate the following based on those results:
Average Win Size (AW) = The average of all trades which won (do not count the biggest win and do not count breakeven trades)
Average Loss Size (AL) = The average of all trades which lost (again do not count any breakeven trades)
Average Win Percentage (W%) = total winners/total number of trades excluding breakeven trades and the biggest winner
Average Loss Percentage (L%) = 100%-W%
Average number of trades per year (T) = number of trades in the sample size/number of weeks the sample covers * 52
Expectancy = ((AW*W% + AL*L%) / - AL ) * T
Attached is an Excel S/S with the formula in to help you.
Here are the figures for my trading system that I use on the E-mini S&P:
My expectancy is 232. So for each $1 I risk on every trade I expect to make $232 in a year.
Obviously the higher the expectancy the better, if it is negative then you will be losing money and should reevaluate your system.
I hope this helps people when they are designing their own systems and looking for comparisons.
I think this is your true expectancy.
10.79 points is 10.79* 50= $539.5 for the average win
5.19 * 50= $259.5
((.51*539.5)-(49*259.5)) = $147.99
Therefore you will make an average of $147.99 per trade.. sure you prob already know this now.
1+ (147.99/ 259.5)= 1.57.. so you'll be making about $ 0.57 for each $1.00 you risk.
Looks a nice system...
This calculation is obvioulsy assuming theat u are risking 5.19 points per trade!!
Cheers Charlie
Why expectancy?
Why would anyone apply expectancy to actual trades that have occurred? Expectancy is applied to projected or anticipated outcomes. If you have actual trades (100) with actual profits and actual
losses, then the outcomes are already known and the expectancy application is simply useless.
Let's assume that a Stock Trader with $300,000 executed x number of trades and made a profit of $75,000 in one year. His roic or return on investment capital is 75/300 or 25%.
Now a Commodity Spread Trader with $100,000 executed y number of trades and made a profit of $150,000 for the year. Her roic is 150/100 or 150%.
Why make a simple problem complex? It does not matter how much of the investment capital you are trading with at any one time. The stock trader can use only $100,000 of the $300,000 while the
Commodity Spread Trader might use only $15,000 of his $150,000 at any one time. The bottom line is that they are both tying up $300,000 and $100,000 respectively that could have been invested
The main point is that once you have the actual outcomes or results, your roic should be the basis for comparison and not expectancy which is a probabilistic methodology for anticipating the
direction of the market through weighting.
Why would anyone apply expectancy to actual trades that have occurred? Expectancy is applied to projected or anticipated outcomes. If you have actual trades (100) with actual profits and actual
losses, then the outcomes are already known and the expectancy application is simply useless.
Let's assume that a Stock Trader with $300,000 executed x number of trades and made a profit of $75,000 in one year. His roic or return on investment capital is 75/300 or 25%.
Now a Commodity Spread Trader with $100,000 executed y number of trades and made a profit of $150,000 for the year. Her roic is 150/100 or 150%.
Why make a simple problem complex? It does not matter how much of the investment capital you are trading with at any one time. The stock trader can use only $100,000 of the $300,000 while the
Commodity Spread Trader might use only $15,000 of his $150,000 at any one time. The bottom line is that they are both tying up $300,000 and $100,000 respectively that could have been invested
The main point is that once you have the actual outcomes or results, your roic should be the basis for comparison and not expectancy which is a probabilistic methodology for anticipating the
direction of the market through weighting.
I would hardly call the expectancy calculation a complex one. I'm confused with your statement of saying that expectancy is a probabilistic way of trying to anticaipate the direction of the market, i
have never seen it used in this manner. If one wanted to try and do the more 'complex' method of Monte Carlo Simulation this could be used to help anticiapte the 'probability' of where inflation and
interest rates might go in the future (Central banks uses this). This would still be caculated from past data, its all we have to use. We're better off with past data than no data at all otherwise
what are we trying to calculate.. our own wishful thinking, our hope, dreams, fears... what?
Oohh and expectancy is useful as a form of benchmarking, it is another way of measuring how likely a system might be profitable in the future, just like how people use Profit Factor as a important
guage of a system.
I would hardly call the expectancy calculation a complex one. I'm confused with your statement of saying that expectancy is a probabilistic way of trying to anticaipate the direction of the
market, i have never seen it used in this manner. If one wanted to try and do the more 'complex' method of Monte Carlo Simulation this could be used to help anticiapte the 'probability' of where
inflation and interest rates might go in the future (Central banks uses this). This would still be caculated from past data, its all we have to use. We're better off with past data than no data
at all otherwise what are we trying to calculate.. our own wishful thinking, our hope, dreams, fears... what?
Oohh and expectancy is useful as a form of benchmarking, it is another way of measuring how likely a system might be profitable in the future, just like how people use Profit Factor as a
important guage of a system.
The original premise of using expectancy to compare two different trading systems with completed trades and known results of profits and losses is wrong and here it is again:
"A lot of people want to compare their trading system with that of other people. To achieve this we need to calculate the expectancy of the system over the long term rather than just saying I made
100 pts last week and x made 50 pts therefore my system is better.
The best way to calculate expectancy is take a reasonable number of actual trades (at least 100) and calculate the following based on those results:"
If for example, both of us have traded for one year and we have our profits and losses statements for the year, we can only compare our trading system and methodology using roic or return on our
investment capital, not Expectancy or Expected Value.
You, SCFX can use Expectancy or Expected Value Calculation to project the performance of your trading system (only) into the future. There are too many variables and unknowns for you to compare my
Expectancy to your Expectancy .....from emotional trading to human errors to broker fraud to you name it.
Expectancy or Expected Value Calculation in itself is not complex but the assumptions you make in calculating it for your own trades create the complexity. Wrong assumptions create wrong Expectation
of market direction and profitability (Expectancy) with resultant losses.
I hope my explanation is more explicit this time around.
If traders are wanting to compare systems, then expectancy is only half the story. You also have to take into consideration the opportunity factor.
For example, system A might have an expectancy of $1.46 per trade and system B might have an expectancy of $0.57 per trade. On the face of it, system A looks the better system, until you learn that
system A averages 100 trades a year whilst system B averages 1000 trades a year, meaning that system B is actually nearly 4 times as profitable as system A over a 12 month period.
Sure, but then just use a % value over a period of time and you get an easily comparable expectancy measure.
I posted something similar of ET not so long ago and I think the idea of knowing the expectancy of any trading system (whether it's some complex formula or a simple visual backtest - doesn't matter
it's what works for you) is vital to trading psychology.
For example, I have a few strategies and for each one I have the trade entry expectancy fully documented in my trading plan so that I know the probability of loss after pulling the trigger. I know
entries are like only 5% of the trade, and the other 95% is made up of trade management and psychology, but if you know the chance of success against loss before you pull the trigger, then it helps
deal with losses psychologically and it removes some of the emotion out of trading (well it does for me anyway). My evaluation of entry expectancy also takes into account drawdowns which I find helps
out in trade management (i.e. know what drawdown is likely or to be expected over a set period of time) which in turn lends itself to knowing when a trade has gone wrong and what level a suitable
stop loss should be.
I probably haven't explained myself very well but just my two cents worth. | {"url":"https://www.trade2win.com/threads/system-expectancy.6103/","timestamp":"2024-11-12T15:50:30Z","content_type":"text/html","content_length":"160441","record_id":"<urn:uuid:89484aea-4852-48b9-8580-bd1fabedebac>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.63/warc/CC-MAIN-20241112145015-20241112175015-00578.warc.gz"} |
Maths Quiz For Class 4 With Answers
4th Grade Math Quiz Questions and Answers
This "Maths Quiz for Class 4 With Answers" is designed to challenge what you have learned so far, from basic arithmetic to problem-solving skills, in a way that is both engaging and educational.
Whether you are a math whiz or just sharpening your skills, this "4th Grade Math Quiz" covers key topics including addition, subtraction, multiplication, and division, ensuring that you stay on top
of your math game.
With easy questions and clear explanations, this "Maths Quiz for Class 4" helps you understand where you stand and what you might need to review. It is perfect for students looking to Read more
practice before exams or just for a quick math refresher. You will find a variety of questions that test not only your math knowledge but also your ability to think critically.
• 1.
What is the Roman numeral for 18?
□ A.
□ B.
□ C.
□ D.
Correct Answer
B. XVIII
The Roman numeral for 18 is XVIII. Roman numerals are represented by combinations of letters from the Latin alphabet, primarily I, V, X, L, C, D, and M. To form the numeral for 18, 'X' (which
stands for 10) is followed by 'VIII' (which stands for 8). When these values are added together—10 + 8—you get 18. This numeral system does not use repetitive or consecutive characters that
would sum negatively when placed before larger values, except in specific cases (such as IV for 4 or IX for 9), thus XVIII is the correct notation for 18.
• 2.
What is the smallest 4-digit number formed using the digits 0, 3, 5, and 6?
□ A.
□ B.
□ C.
□ D.
Correct Answer
D. 3056
To find the smallest 4-digit number using the digits 0, 3, 5, and 6, we need to consider the place values:
Thousand's place: Since it's a 4-digit number, we cannot start with 0, so the smallest possible digit for the thousand's place is 3.
Remaining digits: After placing 3 in the thousand's place, the remaining digits are 0, 5, and 6.
Ordering for smallest number: Arrange the remaining digits in ascending order after 3, which would be 0, 5, and 6.
Final number: Combining these, the smallest number is 3056.
Thus, the smallest 4-digit number you can form with the digits 0, 3, 5, and 6 is: 3056
• 3.
What is the predecessor of the smallest 5-digit number?
□ A.
□ B.
□ C.
□ D.
Correct Answer
A. 9999
The smallest 5-digit number is 10000. The predecessor of a number is the number that comes immediately before it. Therefore, the predecessor of 10000 is 9999. This number, 9999, is the largest
4-digit number and directly precedes the smallest 5-digit number, completing the transition from four digits to five.
• 4.
What is the smallest single-digit composite number?
Correct Answer
B. 4
A composite number is defined as a positive integer that has at least one positive divisor other than one or itself. The smallest single-digit composite number is 4. This is because 4 can be
divided evenly by 1, 2, and 4, making it the first single-digit number that fits the definition of a composite number. Numbers like 2 and 1 are not composite; 2 is a prime number and 1 is neither
prime nor composite as it only has one divisor, which is itself.
• 5.
What is the sum of the angles in a triangle?
□ A.
□ B.
□ C.
□ D.
Correct Answer
A. 180 degrees
The sum of the interior angles of any triangle is always 180 degrees. This is a fundamental principle in geometry, applicable to all triangles, regardless of their type (scalene, isosceles, or
equilateral). Understanding this property is crucial for solving various geometric problems, and constructing accurate figures, and is foundational in proving more complex theorems in the field
of geometry.
• 6.
What is the product of 8 and 12?
□ A.
□ B.
□ C.
□ D.
Correct Answer
A. 96
Multiplying 8 by 12 gives the product of 96. This basic arithmetic operation, multiplication, is used to calculate the total when a particular item or value is grouped into equal parts, such as
calculating the total number of objects in multiple groups or determining the total cost when multiple items are priced the same. Mastery of multiplication is essential for both everyday tasks
and more complex mathematical calculations.
• 7.
How many one-sixths are there in 2?
Correct Answer
C. 12
To find how many one-sixths make up the number 2, you divide 2 by 1/6. This is equivalent to multiplying 2 by the reciprocal of 1/6, which is 6. So, 2 times 6 equals 12. This means there are
twelve one-sixths in 2. Understanding how to convert whole numbers to fractions and vice versa is crucial in fractional arithmetic, helping clarify relationships between different quantities and
making calculations more straightforward.
• 8.
How can 3 hundredths be expressed as a decimal?
□ A.
□ B.
□ C.
□ D.
Correct Answer
D. 0.03
Three hundredths, expressed as a decimal, is written as 0.03. In decimal notation, the term "hundredths" refers to the second place after the decimal point, where each place value represents a
power of ten. The first place after the decimal is tenths (1/10), and the second is hundredths (1/100). Therefore, three-hundredths means three parts out of one hundred, which translates directly
to 0.03 in decimal form. Understanding decimal place values is essential for accurate representation of fractions and decimals in mathematical calculations.
• 9.
What fraction is equivalent to 0.07?
□ A.
□ B.
□ C.
□ D.
Correct Answer
B. 7/100
The decimal 0.07 represents seven hundredths, which can be directly converted into a fraction as 7/100. This fraction reflects the decimal point's position, indicating that the number is seven
parts out of a hundred. Understanding decimal-to-fraction conversions involves recognizing the place value of each digit in a decimal. Here, the 7 is in the hundredths place, meaning it is the
second digit to the right of the decimal point, thus making the equivalent fraction 7/100. This skill is fundamental in math for accurately interpreting and using decimal values in various
• 10.
How many minutes are there in 3 hours and 40 minutes?
□ A.
□ B.
□ C.
□ D.
Correct Answer
C. 220 mins
To find the total number of minutes in 3 hours and 40 minutes, you first convert the hours into minutes and then add the remaining minutes. Since there are 60 minutes in an hour, 3 hours equals
180 minutes (3 hours x 60 minutes/hour). Adding the 40 minutes gives a total of 220 minutes. This calculation is straightforward and essential for converting time units, allowing for easier
computation and understanding of time durations in different formats. | {"url":"https://www.proprofs.com/quiz-school/story.php?title=maths-quiz-grade-4","timestamp":"2024-11-11T01:09:11Z","content_type":"text/html","content_length":"608111","record_id":"<urn:uuid:4c02211d-f5b4-4b7d-88c9-66461f3029a3>","cc-path":"CC-MAIN-2024-46/segments/1730477028202.29/warc/CC-MAIN-20241110233206-20241111023206-00716.warc.gz"} |
Title : Complexity theoretic Aspects of L-reachability
Speaker : Sunil.K.S (IITM)
Details : Tue, 22 Sep, 2015 2:00 PM @ BSB 361
Abstract: : Reachability problems in mathematical structures are a well-studied problem in space complexity. An
important example is the graph reachability problem where given a graph G and two special vertices s and
t, is there a path from s to t in the graph G. This problem exactly captures the space complexity of
problems solvable in nondeterministic logarithmic space.
A natural extension of the problem using formal language theory is the L-reachability problem : Fix a
language L defined over a finite alphabet A. Given a graph whose edges are labelled by alphabet symbols
and two special vertices s and t, test if there is path from s to t in the graph such that the
concatenation of the symbols seen from s to t in the path, forms a string in the language L. Although
the original motivation to study this problem is its application in the optimisation and parallelisation
phases of compiler design, in this work, we take a complexity theoretic view.
By restricting the language using formal language theory, we show that the complexity of L-reachability
increases with the power of the formal language class. We show that there is a regular language for
which the L-reachability problem is NLOG-complete even for undirected graphs. In the case of linear
languages, the complexity of L-reachability does not go beyond the complexity of L itself. Further,
there is a deterministic context-free language L for which L-DAGREACH is LogCFL-complete.
We use L-reachability as a lens to study structural complexity. In this direction we show that there is
a language A in LOG for which A-DAGREACH is NP-complete. Using this we show that P vs NP question is
equivalent to P vs DAGREACH^{-1}(P) question. This leads to the intriguing possibility that by proving
DAGREACH^{-1}(P) is contained in some subclass of P, we can prove an upward translation of separation of
complexity classes. Note that, currently a way to upward translate the separation of complexity classes
is not known. | {"url":"https://cse.iitm.ac.in/seminar_details.php?arg=ODU=","timestamp":"2024-11-07T03:16:28Z","content_type":"application/xhtml+xml","content_length":"14922","record_id":"<urn:uuid:9b3c1a26-a3e4-4f38-a9db-a41cc9813c47>","cc-path":"CC-MAIN-2024-46/segments/1730477027951.86/warc/CC-MAIN-20241107021136-20241107051136-00643.warc.gz"} |
What Does Standard Deviation Tell Us? (4 Things To Know) | jdmeducational
What Does Standard Deviation Tell Us? (4 Things To Know)
Standard deviation is used often in statistics to help us describe a data set, what it looks like, and how it behaves. However, this raises the question of how standard deviation helps us to
understand data.
So, what does standard deviation tell us? Standard deviation tells us about the variability of values in a data set. It is a measure of dispersion, showing how spread out the data points are around
the mean. Together with the mean, standard deviation can also indicate percentiles for a normally distributed population.
Of course, standard deviation can also be used to benchmark precision for engineering and other processes. It can also tell us how accurate predictions have been in the past, and how likely they are
to be accurate in the future.
In this article, we’ll talk about standard deviation and what it can tell us. We’ll also mention what “N standard deviations from the mean” refers to in a normal distribution.
Let’s get started.
(You can also watch a video summary of this article on YouTube).
What Does Standard Deviation Tell Us?
Standard deviation is a number that tells us about the variability of values in a data set. That is, standard deviation tells us how data points are spread out around the mean.
(You can learn more about what affects standard deviation in my article here).
The formula for standard deviation takes into account the mean of the data set by calculating the square difference between each data point and the mean.
Standard deviation is a measure of dispersion, telling us about the variability of values in a data set. Compare this to the mean, which is a measure of central tendency, telling us where the
average value lies.
Standard deviation tells us how far, on average, each data point is from the mean:
• A large value for standard deviation means that the data is spread far out, with some of it far away from the mean. That is, on average, a given data point is far from the mean.
• A small value for standard deviation means that the data is clustered near the mean. That is, on average, a given data point is close to the mean.
• A zero value for standard deviation means that all of the data has the same value (which is also the value of the mean). This situation is rare, but it is possible.
Together with the mean, standard deviation can also tell us where percentiles of a normal distribution are. Remember that a percentile tells us that a certain percentage of the data values in a set
are below that value.
For example, let’s say the 80^th percentile of IQ test scores is 113. This means that 80 percent of people have an IQ below 113.
Alternatively, it means that 20 percent of people have an IQ of 113 or above. So, if your IQ is 113 or higher, you are in the top 20% of the sample (or the population if the entire population was
For a normal distribution, the following table summarizes some common percentiles based on standard deviations above the mean (M = mean, S = standard deviation).
Standard Percentile
Deviations (Percent
From Below
Mean Value)
M – 3S 0.15%
M – 2S 2.5%
M – S 16%
M 50%
M + S 84%
M + 2S 97.5%
M + 3S 99.85%
For a normal distribution, this
table summarizes some common
percentiles based on standard
deviations above the mean
(M = mean, S = standard deviation).
In practical terms, standard deviation can also tell us how precise an engineering process is. For example, a small standard deviation in the size of a manufactured part would mean that the
engineering process has low variability.
We can also decide on a tolerance for errors (for example, we only want 1 in 100 or 1 in 1000 parts to have a “defect”, which we could define as having a size that is 2 or more standard deviations
above or below the desired mean size.
You can learn more about the difference between mean and standard deviation in my article here.
Example: Two Data Sets With The Same Mean & Sample Size, But Different Standard Deviations
Consider the following two data sets with N = 10 data points:
• A = {2, 4, 6, 8, 10, 12, 14, 16, 18, 20}
• B = {9, 10, 11, 11, 11, 11, 11, 12, 13}
For the first data set A, we have a mean of 11 and a standard deviation of 6.06.
For the second data set B, we have a mean of 11 and a standard deviation of 1.05.
Both data sets have the same sample size and mean, but data set A has a much higher standard deviation. This is due to the fact that there are more data points in set A that are far away from the
mean of 11.
Data set B, on the other hand, has lots of data points exactly equal to the mean of 11, or very close by (only a difference of 1 or 2 from the mean).
As you can see from the graphs below, the values in data in set A are much more spread out than the values in data in set B.
The graph for data set A has data that is spread out from the mean of 11 (the standard deviation is 6.06).
The graph for data set B has less data that is spread out from the mean of 11 (the standard deviation is 1.05).
Why Use Standard Deviation Instead Of Variance?
Remember that standard deviation is the square root of variance. This raises the question of why we use standard deviation instead of variance.
Since we add and subtract standard deviation from mean, it makes sense for these two measures to have the same units.
When we calculate variance, we take the difference between a data point and the mean (which gives us linear units, such as feet or pounds).
When we square these differences, we get squared units (such as square feet or square pounds). To get back to linear units after adding up all of the square differences, we take a square root.
Why Use Standard Deviation Instead Of Range?
Remember that the range of a data set is the difference between the maximum and the minimum values. Using the range of a data set to tell us about the spread of values has some disadvantages:
• Range only takes into account two data values from the set: the maximum and the minimum. The rest of the data values are ignored.
• Range does not tell us anything about how far the average data point is from the mean. In fact, range does not take the mean into account at all.
• Range is highly susceptible to outliers, regardless of sample size. Whenever the minimum or maximum value of the data set changes, so does the range – possibly in a big way.
Standard deviation, on the other hand, takes into account all data values from the set, including the maximum and minimum.
Standard deviation also tells us how far the average value is from the mean of the data set.
Finally, when the minimum or maximum of a data set changes due to outliers, the mean also changes, as does the standard deviation. However, for larger sample sizes, this effect is less pronounced.
How Do You Interpret Standard Deviation?
A low standard deviation means that the data in a set is clustered close together around the mean. A high standard deviation means that the data in a set is spread out, some of it far from the mean.
The best way to interpret standard deviation is to think of it as the spacing between marks on a ruler or yardstick, with the mean at the center.
The width between marks on a ruler can be thought of as the value of standard deviation.
Every time we travel one standard deviation from the mean of a normal distribution, we know that we will see a predictable percentage of the population within that area.
The width of a standard deviation is always the same, and it can help us to find percentiles in a population with a normal distribution.
What Is A Low Standard Deviation?
A low standard deviation is one where the coefficient of variation (CV) is less than 1. The coefficient of variation is defined as
• CV = (standard deviation of data set) / (mean of data set)
Note that CV < 1 implies that the standard deviation of the data set is less than the mean of the data set.
It is also important to note that a mean close to zero will skew the coefficient of variation to a high value. Even worse, a mean of zero implies an undefined coefficient of variation (due to a zero
In the example from earlier, we have coefficients of variation of:
• Data Set A: CV = (standard deviation of data set) / (mean of data set) = 6.06 / 11 = 0.55.
• Data Set B: CV = (standard deviation of data set) / (mean of data set) = 1.05 / 11 = 0.10.
What Is A High Standard Deviation?
A high standard deviation is one where the coefficient of variation (CV) is greater than 1. Note that CV > 1 implies that the standard deviation of the data set is greater than the mean of the data
This is more likely to occur in data sets where there is a great deal of variability (high standard deviation) but an average value close to zero (low mean).
What Is 1 Standard Deviation From The Mean?
When we say “1 standard deviation from the mean”, we are talking about the following range of values:
where M is the mean of the data set and S is the standard deviation.
We know that any data value within this interval is at most 1 standard deviation from the mean. We could say that this data is relatively close to the mean.
For example, if we have a data set with mean 200 (M = 200) and standard deviation 30 (S = 30), then the interval
• (M – S, M + S)
• =(200 – 30, 200 + 30)
• =(170 , 230)
Is the range of values that are one standard deviation (or less) from the mean.
What Percentage Is 1 Standard Deviation From The Mean?
For a data set that follows a normal distribution, approximately 68% (just over 2/3) of values will be within one standard deviation from the mean.
So, for every 1000 data points in the set, 680 will fall within the interval (S – E, S + E).
Going back to our example above, if the sample size is 1000, then we would expect 680 values (68% of 1000) to fall within the range (170, 230).
What Is 2 Standard Deviations From The Mean?
When we say “2 standard deviations from the mean”, we are talking about the following range of values:
where M is the mean of the data set and S is the standard deviation.
We know that any data value within this interval is at most 2 standard deviations from the mean. Some of this data is close to the mean, but a value 2 standard deviations above or below the mean is
somewhat far away.
For example, if we have a data set with mean 200 (M = 200) and standard deviation 30 (S = 30), then the interval
• (M – 2S, M + 2S)
• =(200 – 2(30), 200 + 2(30))
• =(140 , 260)
Is the range of values that are 2 standard deviations (or less) from the mean.
What Percentage Is 2 Standard Deviations From The Mean?
For a data set that follows a normal distribution, approximately 95% (19 out of 20) of values will be within 2 standard deviations from the mean.
So, for every 1000 data points in the set, 950 will fall within the interval (S – 2E, S + 2E).
Going back to our example above, if the sample size is 1000, then we would expect 950 values (95% of 1000) to fall within the range (140, 260).
What Is 3 Standard Deviations From The Mean?
When we say “3 standard deviations from the mean”, we are talking about the following range of values:
where M is the mean of the data set and S is the standard deviation.
We know that any data value within this interval is at most 3 standard deviations from the mean. Some of this data is close to the mean, but a value 3 standard deviations above or below the mean is
very far away from the mean (and this happens rarely).
For example, if we have a data set with mean 200 (M = 200) and standard deviation 30 (S = 30), then the interval
• (M – 3S, M + 3S)
• =(200 – 3(30), 200 + 3(30))
• =(110 , 290)
Is the range of values that are 3 standard deviations (or less) from the mean.
What Percentage Is 3 Standard Deviations From The Mean?
For a data set that follows a normal distribution, approximately 99.7% (997 out of 1000) of values will be within 3 standard deviations from the mean.
So, for every 1000 data points in the set, 997 will fall within the interval (S – 3E, S + 3E).
Going back to our example above, if the sample size is 1000, then we would expect 997 values (99.7% of 1000) to fall within the range (110, 290).
What Is 4 Standard Deviations From The Mean?
When we say “4 standard deviations from the mean”, we are talking about the following range of values:
where M is the mean of the data set and S is the standard deviation.
We know that any data value within this interval is at most 4 standard deviations from the mean. Some of this data is close to the mean, but a value that is 4 standard deviations above or below the
mean is extremely far away from the mean (and this happens very rarely).
For example, if we have a data set with mean 200 (M = 200) and standard deviation 30 (S = 30), then the interval
• (M – 4S, M + 4S)
• =(200 – 4(30), 200 + 4(30))
• =(80 , 320)
Is the range of values that are 4 standard deviations (or less) from the mean.
What Percentage Is 4 Standard Deviations From The Mean?
For a data set that follows a normal distribution, approximately 99.99% (9999 out of 10000) of values will be within 4 standard deviations from the mean.
So, for every 10000 data points in the set, 9999 will fall within the interval (S – 4E, S + 4E).
Going back to our example above, if the sample size is 10000, then we would expect 9999 values (99.99% of 10000) to fall within the range (80, 320).
What Is 5 Standard Deviations From The Mean?
When we say “5 standard deviations from the mean”, we are talking about the following range of values:
where M is the mean of the data set and S is the standard deviation.
We know that any data value within this interval is at most 5 standard deviations from the mean. Some of this data is close to the mean, but a value that is 5 standard deviations above or below the
mean is extremely far away from the mean (and this almost never happens).
For example, if we have a data set with mean 200 (M = 200) and standard deviation 30 (S = 30), then the interval
• (M – 5S, M + 5S)
• =(200 – 5(30), 200 + 5(30))
• =(50 , 350)
Is the range of values that are 5 standard deviations (or less) from the mean.
What Percentage Is 5 Standard Deviations From The Mean?
For a data set that follows a normal distribution, approximately 99.9999% (999999 out of 1 million) of values will be within 5 standard deviations from the mean.
So, for every 1 million data points in the set, 999,999 will fall within the interval (S – 5E, S + 5E).
Going back to our example above, if the sample size is 1 million, then we would expect 999,999 values (99.9999% of 10000) to fall within the range (50, 350).
The probability of a person being outside of this range would be 1 in a million.
Now you know what standard deviation tells us and how we can use it as a tool for decision making and quality control. You also know how it is connected to mean and percentiles in a sample or
You can learn more about standard deviation (and when it is used) in my article here.
You can also learn about the factors that affects standard deviation in my article here.
You can learn about the difference between standard deviation and standard error here.
You can learn about when standard deviation is a percentage here.
You might also want to check out my article on how statistics are used in business.
You can learn about how to use Excel to calculate standard deviation in this article.
You might also want to learn about the concept of a skewed distribution (find out more here).
I hope you found this article helpful. If so, please share it with someone who can use the information.
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If tan−1x+tan−1y+tan−1z=2π, then value of xy+yz+zx is... | Filo
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If , then value of is
a. 1
b. 0
d. none of these
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Question Text If , then value of is
Updated On Aug 26, 2023
Topic Trigonometry
Subject Mathematics
Class Class 12
Answer Type Video solution: 2
Upvotes 204
Avg. Video Duration 5 min | {"url":"https://askfilo.com/user-question-answers-mathematics/if-then-value-of-is-33343239353636","timestamp":"2024-11-07T15:54:10Z","content_type":"text/html","content_length":"299595","record_id":"<urn:uuid:723a6af8-ad2d-4321-a76d-a643109ddbcf>","cc-path":"CC-MAIN-2024-46/segments/1730477028000.52/warc/CC-MAIN-20241107150153-20241107180153-00166.warc.gz"} |
Talk Abstract: 2001b
Title: Generalised splines via the maximum principle (8 pages)
Detail: SIAM Conference on Control and its Applications, San Diego, CA, 2001/07/11
The maximum principle of Pontryagin is applied to control systems where the drift vector field is the geodesic spray corresponding to an affine connection. The result is a second-order differential
equation whose right-hand side is the ``adjoint Jacobi equation.'' By choosing the cost function to be the square norm of the input with respect to a Riemannian metric, one generates equations which
generalise spline equations in two directions: (1) the setting is that of manifolds with a general affine connection and (2) it is allowed to impose the cost only on those accelerations which live in
a subbundle of the tangent bundle.
94K pdf
Last Updated: Tue Jun 18 09:11:10 2024
Andrew D. Lewis (andrew at mast.queensu.ca) | {"url":"https://mast.queensu.ca/~andrew/talks/abstracts/2001b.html","timestamp":"2024-11-03T13:19:14Z","content_type":"text/html","content_length":"1683","record_id":"<urn:uuid:adc4d020-22b4-45fa-a7ef-74e9d595dc4c>","cc-path":"CC-MAIN-2024-46/segments/1730477027776.9/warc/CC-MAIN-20241103114942-20241103144942-00382.warc.gz"} |
Derivation of the posterior model probability
Proof: Derivation of the posterior model probability
Index: The Book of Statistical Proofs
Model Selection
Bayesian model selection
Posterior model probability
▷ Derivation
Theorem: Let there be a set of generative models $m_1, \ldots, m_M$ with model evidences $p(y \vert m_1), \ldots, p(y \vert m_M)$ and prior probabilities $p(m_1), \ldots, p(m_M)$. Then, the posterior
probability of model $m_i$ is given by
\[\label{eq:PMP} p(m_i|y) = \frac{p(y|m_i) \, p(m_i)}{\sum_{j=1}^{M} p(y|m_j) \, p(m_j)}, \; i = 1, \ldots, M \; .\]
Proof: From Bayes’ theorem, the posterior model probability of the $i$-th model can be derived as
\[\label{eq:PMP-s1} p(m_i|y) = \frac{p(y|m_i) \, p(m_i)}{p(y)} \; .\]
Using the law of marginal probability, the denominator can be rewritten, such that
\[\label{eq:PMP-s2} p(m_i|y) = \frac{p(y|m_i) \, p(m_i)}{\sum_{j=1}^{M} p(y,m_j)} \; .\]
Finally, using the law of conditional probability, we have
\[\label{eq:PMP-s3} p(m_i|y) = \frac{p(y|m_i) \, p(m_i)}{\sum_{j=1}^{M} p(y|m_j) \, p(m_j)} \; .\]
Sources: Metadata:
| shortcut:
| author:
| date: 2020-07-28, 03:58. | {"url":"https://statproofbook.github.io/P/pmp-der","timestamp":"2024-11-05T15:52:19Z","content_type":"text/html","content_length":"9201","record_id":"<urn:uuid:d34de468-3b6d-4a7f-89dd-9abfaf0cb117>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00205.warc.gz"} |
Sumerian Coefficients in the Pottery Factory and Calculator Demo Example
This page is under development. Comments are welcome, but please load any comments in the comments section at the bottom of the page. Please include your wiki MONIKER and date in your comment with
the same courtesy that I will give you. Aside from your courtesy, your wiki MONIKER and date as a signature and minimal good faith of any internet post are the rules of this TCL-WIKI. Its very hard
to reply reasonably without some background of the correspondent on his WIKI bio page. Thanks, gold 12DEC2018
gold Here is some starter code for calculating fuel and dimensions of ancient pottery kilns. The impetus for these calculations was checking rated kiln capacity in some excavation reports and modern
replicas. Most of the testcases involve replicas or models, using assumptions and rules of thumb .
In the Sumerian coefficient lists, there are coefficients which were used in determining the fuel capacity of a Sumerian kiln and the daily work rates of the pottery workers. One coefficient is
called "sa esir had ina kiriti" (coefficient pitch in kiln) which has a value of ~10. Culled from different lists, the kiln coefficients range 10/20 in base 60, and possibly represent different style
kilns. The math problem is how this kiln coefficient was used in estimating kiln capacity and work rates. One difficulty is determining the effective power of the coefficient in base 60. For example,
20 could represent either 20*3600,20,20/60, 20/3600, or even 1/20. A complete math problem or explanation on kiln dimensions (L,W,H) is lacking.
However, the Sumerian coefficient lists show the Sumerians busy with coefficients for computing the horizontal area of kiln figures or the volumes of possible kilns.. In general terms, some
coefficients were used to convert a surface area sar (eg. square nindans in Sumerian usage) into a volume sar with units of gurs. A nindan equals 6 meters, a surface area sar about 36 square meters,
and a volume gar about 0.3 cubic meters. The generic formula would be surface sars in nindan*nindan times coefficient equals volume sar in units of gurs. After a considerable amount of scratch paper,
the Sumerians appear to be using daily fuel packing rate in volume gars. A complete and period text available on the dimensions of the various Sumerian kilns is lacking, but the various coefficients
have been examined based on modeling the traditional native kilns and scaling available replicas.
While the Sumerians usually measured horizontal areas in square nindans for the kilns and grain bins, the problem is essentially the same: fitting circles or volume units into a fixed area or
geometric figure. In modern terms, this is called a circle packing problem. One nindan equals 6 meters and a square nindan equals 36 square meters. A volume gur equals 0.3 cubic meters. One common
size of kiln in the excavations was diameter 0.5 meters and height 0.5 meters, figures for below ground combustion chamber. The volume would be (1/4)*PI*D*D*H,.25*3.14*.5*.5*.5 or 0.098 cubic meters.
In Sumerian terms, the volume would be 0.098/.3, decimal 0.3266 or 20/60 + 36/3600 volume gurs in base 60. The estimated coefficient 20:36 is within range of the 0;20:48:06 (in base 60) number on the
clay tablets. One possibility is that the pitch and reed kiln coefficients represent the fraction of (fuel volume) / (burning chamber volume). At least some kilns had support pillars or walls within
the burning chamber which would subtract from the total volume. Other kilns were single chamber kilns which filled up some of their volume with support shelves, wares, and ware props. In other words,
the total volume of the chamber was not occupied with fuel. Also, it was found that coefficient*circumference* circumference* circumference equals nearly the amount of gurs in the burning chamber.
The circumference cubed is not a likely formula, since use of cubed value has not been associated with a coefficient before. From other coefficients on the tablets, the daily work rate of excavations
was 10/60 gurs and if the kiln fuel loading problem is analogous, the number of man work days to load the kiln fuel was (20.8/60)/ (10/60) or roughly 2 work days.
There is more than one theory on the kiln used by the reed workers. The reed workers cooked the reeds with lime to soften and bleach the reeds before weaving mats. In some cases, the reed workers may
have dried their baskets in an oven to assure a dry and insect free product. Possibly, the reed workers made a sort of charcoal briquet or cooking fuel, baking green reeds in a special oven (eg. a
charcoal kiln). Lets return to the original Sumarian phrase, "20_48_06 sa gir4 ad kup4". As others have remarked, the Sumerian/Akkadian script on the tablets is the most homonym filled language on
earth. The Sumerian word "ad" could either be associated with reed (addatu, reed pith) or day's work (adu, sun come), or village (aduru, wall of reeds). Since the jobs as diverse as potters, ditch
diggers, and basket weavers have their daily rate of work in the coefficient lists, there is at least some possibility that "ad" modifies "kupru" and refers to a daily task on the construction liquid
asphalt. Also, its easier and closer to the meaning of a Sumerian noun-phrase to develop a full sentence, sic. "20_48_06 sa gir4 ad kup4" could mean: The 20_48_06 coefficient of? (kiln that burns
reeds) is? a? (day's work) on? (construction liquid asphalt pitch). There are traditional kilns and replicas which still use the Sumerian and Greek designs, so the calculator empirical formulas are
based on the known wood fuel/volume ratios.
From the contemporary and traditional kilns, the amount of heat energy needed to fire the known dimensions of Sumerian kilns can be estimated. Also from modern tests, the heat constants of burning
wood/reeds/asphalt are 13.5/17/40 megajoules per (dry) kilogram. The 13.5 MJ/kg for wood is an average number based on scrap wood and uncovered woodlots, not completely dry wood. Rounding the
Sumerian coefficient for burning reeds is 20/60 and the Sumerian coefficient for burning asphalt is 10/60. The Sumerians are probably multiplying the volume of the kiln burning chamber times the
coefficient fraction to estimate the needed volume of fuel. Since the ratio of the heat constants (17/40 or 0.425) indicates that reeds supply roughly half the heat of asphalt, the mass of reeds to
fire a kiln should be twice <(20/60)/(10/60)> that of asphalt. While there is more current documentation about burning wood in kilns than reeds or asphalt, some tentative calculations can be made
with the Sumerian coefficients for reeds and asphalt. A kiln of 3 cubic meters volume would be roughly 3/.3 or 10 volume gurs. Using the asphalt coefficient, the needed asphalt fuel would be 10 gurs
* (10/60) or 1.66 gurs. Assuming the specific gravity of 1 for asphalt, this would be 1.66*.3*1.*300. , 149.39 kilograms of asphalt. In Sumerian mass units, this would be 149.9/.4977 or 301 mannas.
For the fuel reeds, the calculation would be 10 gurs * (20/60), 300 kilograms, or 600 mannas of reeds. From other tablets, the Sumerians counted a bundle of reeds or manload of reeds as 20 manas, so
the amount of reeds would be 600 manas/20 or 30 bundles.
One topic of interest is the efficiency of kilns, Sumerian or otherwise. For example, one tenth of the energy used in contemporary China is burned making bricks in kilns and similar proportions in
other industrial countries. The efficiency of a kiln is measured by the applied heat on the clay mass over the total heat used, usually in percent of applied heat over total heat or in megajoules per
1000 kilograms of pottery. While the numbers are not complete in the Sumerian case, one can look at the features of the Sumerian cylindrical kilns and estimate an ideal heat budget. Starting with an
equal diameter and height burning chamber, then find the ideal proportion of heat sharing in a cylinder. As in a Sumerian math problem, the clay mass will be a central pillar with a radius and
diameter at 1/4 that of the surrounding kiln. The diameter and height of the kiln are both 1. The height of the clay mass will be 1/2 the diameter of the kiln. In an ideal heat engine, 1/2 the total
heat will remain in the kiln and 1/2 will leave up the chimney. The next math problem is finding the proportions of the heat that remains in the kiln.
Of the heat that remains in the kiln, the transfer of heat to the pottery is ideally proportional to the surface area of the clay mass over the internal surface area of the cylindrical kiln. So
estimate the ratio clay surface area over ( top + bottom + wall), 2*pi*r2*h2/ ((pi*r1*r1)+(pi*r1*r1)+ (2*pi*r1*r1)),substituting r2= r1/2 and h2=h1/2, leading to 2*pi*(r1/2)*(h1/2) over 2*pi*r1*
(r1+h1), reducing to (1/4+h1/2)/(r1+h1). Since h1=2r1, the ratio is (1/4+(2r1/2))/(r1+2r1), (1+4r1)/12r1 evaluated at r1=.5 or 3/6. We can fraction with 3*x+6*x = 1/2 total heat or x=1/18, so we can
develop fractions 6/18, 3/18, 9/18 for the terms of total heat. For heat budgeting, the total heat equals 6/18 (walls+top+bottom) +3/18(clay mass) + 9/18 (outlet air). Similarly, the (top + bottom)
over inner wall surface are (pi*r1*r1+pi*r1*r1)/(2*pi*r1*h1), where h1=1,(r1*r1/r1) where r1 evals at .5, and leads to a 1:2 ratio for cylinder ends to inner wall. Accounting for the walls and top,
total heat = 4/18 (walls) + top(1/18) + bottom(1/18) +3/18(clay mass) + 9/18 (outlet air). The proportion 3/18, decimal 0.166 or 17 percent is the maximum efficiency expected from a circular batch
kiln. Most of the circular kilns fall short of this ideal because the clay mass or wares do not receive the best exposure to the hot air flow. Still, any more sophisticated analysis or suggested
improvements will have account for heat budget or reduce the terms and losses. ( The heat flux and budget of the kiln has nonlinear terms over time. )
The price of fired brick in Sumerian times might also be indicative of energy costs per brick. In Ur III, 1 gur of barley brought 288 bricks from the brickmaker. In the equivalence texts, the
brickmaker was paid 1 ban or (1/4) gur of barley a day and daily work quota was 288 bricks a day. So some equations can be set up. 1 gur barley = (1/4 gur)payday + ( price of fuel for 288 bricks), 1
gur = price of 600 manas of asphalt, and 1 mana = 0.4977 kg. Subtraction gives 3/4 gur = price of fuel for 288 bricks. Substituting , 3/4 gur will buy (3/4)*600 or 450 mannas of asphalt = 288 *
bricks. So, 1 brick = 450/288 or 1.56 manas of asphalt per brick or 1.56*.4977, 0.776 kilograms of asphalt per brick. 0.776 kilograms * 40 MJ/kg fuel gives 31.04 MJ per brick. A brick weighed 7.4
kilograms, so the applied heat of the brick was 31.4/7.4 or 4.3 MJ/kg.
Possible ratios and fractions for the unfired pottery clay mass in Sumerian circular kilns can be developed from the traditional kilns and rules of thumb. One style of clamp kilns uses a formula
ratio of fuel to clay mass as 400 kilograms of coal to 1000 green bricks, each green brick weighing 3.6 kilograms. So the ratio of fuel is 400/(1000*3.6), 0.11 coal kilograms per clay mass. For
comparison, the British equivalent formulas from the 1880s was 454 kilograms coal per 1000 soft bricks and 907 kilograms coal per 1000 hard vitrified bricks, figuring green bricks at 3.8 kilograms.
The coal has a heating value of 25 Megajoules per kilogram, so the process heat per kilogram is (400*25)/(1000*3.6) or 2.7 MJ/kg. The finished brick has lost moisture and weighs about 3 kg. Most of
the traditional kilns have a range of 4 to 6 MJ/kg. Actually, the MJ/kg per brick is the most consistent parameter reported on traditional kilns, since the fuels, ceramic shapes, and brick weights
vary so much. The traditional circular kilns usually fire from 500 to 1200 kilograms of green clay for a kiln of 3 cubic meters. Figuring the volume of the green clay as 2500 kilograms per cubic
meter, the traditional fractions of clay volume to kiln volume would be from (500/2500)/3 to (1200/2500)/3, decimal 0.066 to 0.16, or 4/60 to 10/60 in base sixty. Another approach is (fuel & energy)
/clay ratio. Previously it was found that the reed burning kiln (gir4) of 3 cubic meters would need 300 kilograms of reeds, which has a heat equivalent of 17 MJ/kg * 300 kg or 5100 MJ. The asphalt
kiln (kirim) would need 149.4 kilograms of asphalt fuel, which is 40MJ/kg *149.4 or 5976 MJ. Taking the clay heat ratio and the 6 MJ/kg from the traditional kilns, the reed kiln would need (5100 MJ)/
(6 MJ per kg) or 850 kg of green clay. The asphalt kiln would need (5976 MJ) / (6 MJ per kg) or 1000 kg of green clay. Returning to volume equivalents, the volume fraction of clay over kiln volume
would range from (850/2500)/3 to (1000/2500)/3, decimal 0.111 to 0.133, or 6.6/60 (reeds) to 8/60 (asphalt). Of course, these brick calculations are really order of magnitude calculations. But we
realized that the coefficients "20_48_06 sa gir4 ad kup4" for the reed kiln and "10_45_06 esir2 had5 a sa in-na ki-ri-im" for the asphalt kiln share a common 06 digit. Possibly 06 is really a
separate coefficient and represents the fraction of clay volume over the kiln volume as 6/60. If the kiln coefficient lines are three separate numbers, then possibly coefficient values 45/48 refers
to the top end area of a cylindrical kiln (3 for pi, (45/60) *D*D, in area sars) or the volume calculations for the cylindrical kiln (3 for pi, (45/60)*D*D*H, in volume gurs).
Some parameters from the modern clamp kiln can be estimated for some scale considerations on Sumerian volumes of brick piles or possible field kilns. The 1880’s British clamp kiln was LWH 60/11/12
feet or 18.29/3.353/3.657 meters and contained 8E4 English green bricks of about 8.6 kg each. The volume of the British clamp kiln here would be roughly 224 cubic meters, 18.29*3.353*3.657. The clay
mass was about (8E4*3.8) or 3.04E5 kilograms and would require 3.63E4 kg of coal. In the British clamp kiln, there are upper limits to width at 4 meters to pitch wood or fuel through tunnels to the
inside of the kiln and upper limits to height at 4 meters as to avoid extensive scaffolding. There is no particular upper limit to the British clamp length and volume, but given a width of about 4
meters, a kiln length below 4 meters would probably waste fuel.
For the Sumerian evidence of possible field and clamp kilns, there are a number of tablets which discuss a volume of 9 sar bricks and other possible volumes from 0.44 to 810 cubic meters. The tablet
VAT6598 discusses a wall (e-gar) that was 4 volume sars and contained 9 sar bricks or 1.2 cubic meters. Tablet YBC4708 is a problem text for a truncated brick pyramid of 1.5 volume sar or 0.44 cubic
meters. The tablet YBC4708 concerns a brick pile of 30/9/3 meters and a volume of 810 cubic meters (uncertain if kiln?). Another text mentions an im-gir4 (mud kiln) of 9 brick sar or 6480 bricks,
which appears to be a clamp kiln. The volume of the mud kiln would be 6480 *.33*.33*.08 or 56.4 cubic meters. One text mentions a truncated triangular prism that amounted to 9 sar baked bricks. The
truncated triangular prism would be a particularly good form for a clamp kiln, since the offset or sloping inward walls offer better stability while the clamp kiln burns and loses some supporting
volume. The value of 9 brick sar alone has shown up in several brick porting and receiving texts and math problems, though not always in obvious context with kilns, prism shapes, or fuel. One text
mentions receiving a total of 36000 bricks and 9 sar bricks, implying 9 sar bricks was a special unit. Based mostly on the problem text of the truncated triangular prism, the 9 sar values in the
tablets appear to endorse the best candidate volume for a field or clamp kiln.
A wage budget for a Sumerian im-gir4 (mud kiln) can be estimated from the given volume and the daily wage figures the Sumerians were computing in the coefficient and equivalency lists. The third
month of the Sumerian calendar was iti sig4.ga ( month of forming bricks). The third month roughly the month of May, after the month of the yoked ox when the hired men came back from plowing and
sowing fields. The mud bricks were normally set out before and dried during the hot season. Also, straw, reeds, and brush would be available for fuel. While building a clamp kiln would not be a job
for a single man or single work day, some of the coefficients for digging clay of 6 cubic meters daily, bringing mud of 4 cubic meters daily, mixing earth of 3 cubic meters daily, and especially the
mud kiln of 6480 bricks are approaching the magnitude of a small brick clamp. According to the metrology lists, a volume sar would contain 1944 baked bricks of the square top type (LWH .33/.33/.08
meters). We can try to set up a budget for a small clamp kiln. The volume of the mud kiln would be 6480 *.33*.33*.08 or 56.4 cubic meters. The total daily work on the clay mass would be (56.4/6) 9.4
days digging clay > (56.4/4) 14.1 days porting clay> (56.4/3) 18.8 days mixing clay> (56.4/(288*.33*.33*.08)) 24.47 days forming bricks > (4?) days loading kiln and firing bricks. The sum of these
days times the minimum wage ( 9.4+14.1+18.8+24.47+4)*(1/8 gur barley)+ 2.3 gurs worth of fuel or 11.15 barley gurs. 11.15 barley gurs for 6480 bricks should be near the price of baked bricks. .In
UR111, 1 gur barley bought 8_24 bricks or 504 burnt bricks, proportionally the bricks from the mud kiln should bring (6480*1. gur)/504 or 12.86 barley gurs. The delta price is 12.86-11.15 or 1.7
barley gurs, but some of the skilled workmen and foremen received 0.5 - 1 gur a day, so the budget is reasonably close. Actually, the daily task of brick firing and daily wage at the kiln as opposed
to general “making brick” needs to be firmed up better.
For the possible pitch/clay ratio of a field kiln, some trial calculations for equivalent fuels can be made based on the British clamp kiln of the 1880's. The British equivalent formulas from the
1880s was 454 kilograms coal per 1000 soft bricks and 907 kilograms coal per 1000 hard vitrified bricks, figuring green bricks at 3.8 kilograms. These British equivalent formulas were still being
used in former commonwealth countries as late as 2000 CE, prior to the Kyoto Protocols on Greenhouse Gases. These equivalent calculations have some interest for replica kilns, since some historic
fuels are no longer available or effectively outlawed. By mass, the British formula is 454 kilograms coal over 1000*3.8 kilograms, 454/3800, 0.11947 or 7.16/60. By mass the British formula is 907
kilograms coal over 1000*3.8 kilograms, 907/3800, 0.2386 or 14/60 mass fraction. The volume fractions can be calculated using specific gravity, 1.3 for coal and 1.9 for clay which are equivalent to
coal =1300kg/cubic meters and clay=1900kg/cubic meters. The coal/clay volume fraction for the soft bricks would be 454*1300/3800*1900= .0817, or 4.9/60 volume fraction. The coal/clay volume fraction
for the hard bricks would be 907*1300/3800*1900= 0.1633, 9.7/60 volume fraction. These coal/clay fractions can be converted into equivalent heating units of bunker oil, which is very close to dry
bitumen in density and heating value. The clay mass (3800 kg) can be converted into a volume, (3800 kg/1900 kg/cm) *1000 liters/cm, 2000 liters of clay. Using 794.77 liters of bunker oil for 1000 kg
of coal, the soft brick volume fraction for bunker oil liters/ clay liters would be (360 liters of oil) over (2000 liters of clay), 360/2000, decimal 0.18, or 10.79/60, volume fraction. The hard
brick volume fraction for bunker oil liters/ clay liters would be (720 liters of oil) over (2000 liters of clay), 720/2000, decimal 0.36, or 21.6/60 volume fraction. The majority of the extant UR111
clay bricks probably could have been fired with the soft brick fuel ratio, but none of the calculations here rule out higher fuel rates for other ceramics, metals, frits, and artificial stones.
There was not much accessible on renewable fuel reeds (gi bar) or acacia firewood, but some considerations can be extrapolated from published tablets and equivalency lists. The daily norm for reed
fodder was 3 bales per day or 15 bundles per day. If the collecting fuel reeds is similar to other norms, the daily norm was 3 bales * 5 bundles*20 manas * (mana/.4977kg) or 602.7 kilograms of reeds.
The fuel equivalent per day was 602.7*16 MJ/kg or 9643 MJ per day. Reed collection was unskilled labor, so the firewood or firereed collector was probably paid 1/8 gur of barley , 5 sila of barley,
or equivalent 1/8 silver piece a day. There was provision (eg. pay norms) for porters to carry the reeds to the user. A silver shekel bought 300 bundles of reeds at a weight of 20 manas per bundle or
(20/.4977) 40.2 kg per bundle. In proportion, the 15 bundles of the collector should bring 15/300 or 1/20 shekel, other costs aside. There are accounts of taxes on reeds and some reed beds were noted
as controlled by the royal household, priest, or princes. In energy terms, a silver shekel bought 300 bundles * 40.2 * 16 MJ/kg or 1.92E5 Megajoules. For comparison with the asphalt fuel, one shekel
brought (600 manas/0.4977)*40 MJ/kg or 4.82E4 Megajoules.
Probably the greatest improvements that the Ubaidians, Sumerians, and Greeks made was the buried combustion chamber for better insulation and the separate burning chamber and grate to force the hot
air past the wares. The buried combustion chamber and separate burning chamber was rated at 7 percent efficiency over the original pit kilns (1-2%). Although difficult to interpret, some Ubaidian and
Sumerian kilns had interior support walls, interior baffles, concave prism support bricks, and brick floor channels, which moderated the air flow inside the kiln (possibly efficiency of 9%, downdraft
flow over the wares?). At least, the Greeks had some manual chokes or obstructions in the kiln chimney to dampen the air during preburn and moderate the air flow from the kiln exit (probably a 2
percent gain, if present). Some of the Sumerian kilns had extended or roofed stoking channels (1-2 meters long), which may have been to improve air draft at the entry and reduce wind eddies at the
stoking chamber door. Some of the roofed stoking channels are sloping downwards to a below ground combustion chamber, which could be interpreted as gravity feed of fuel. Rarely, air flues of pebble
beds under a raised kiln subfloor have been found at Tell Ziyada. The pebble beds in the Tell Ziyada kiln may have preheated the air and distributed air to the rear of the kiln ( 4700 BCE?, Ubaid
culture). Its not uncommon on the contemporary traditional kilns to see flaming backdrafts and smoke at the stoking door, which surely indicates poor draft and a heat loss. Some of the tablets fired
in Sumeria had pinholes or firing holes in the tablets, possibly for improving the evenness of fired clay. There are a number of Sumerian fired vessels and unfired vessels that were coated with pitch
for waterproofing, which may have saved on fuel. Some of the traditional Indian and Mexican kilns have blackened pottery or black core pottery from added carbon or organics, but this is uncommonly
reported in Sumerian pottery or difficult to interpret as an improvement. Most of the Sumerian above ground structure and kiln roofs have not survived, but some traditional Indian kilns have roofs of
layered mudplaster, shards, and straw, which would serve for insulation. The use of renewable reeds for fuel probably has some lessons in the modern era. One reason for studing the Sumerian kilns is
the improvement of contemporary design in that the Sumerians faced and solved many of the same problems in kiln design.
Ideal Heat budget of a circular kiln
terms of heat budget percent fraction rounded fraction in base 60 comment or improvement
top kiln 5.5 1/18 4/60 insulation, mud plaster on top
bottom kiln 5.5 1/18 4/60 insulation, few excavation comments
walls kiln 22. 4/18 13/60 insulation, mud plaster on walls
heat loss to chimney 50. 9/18 30/60 use waste heat, preheat entry air
wares, clay mass 16.6 3/18 10/60 holes in bricks for maximum exposure
hollow core bricks, not found so far
lead more hot air onto wares
design wares for maximum exposure
ware props and stands for better exposure
design wares for least needed heat
incorporate fuel into clay mass
use waste heat to dry (next batch) wares and fuel
sum of total heat 100.0 18/18 60/60 price tab for pottery and bricks
Common sizes or ranges of circular kilns, burning chamber
diameter meters height meters vol. cubic meters circumference nindans cfe pitch vol gurs wood kg manas of wood decimal workday loading gur formula????
0.5 0.5 0.0981 1.57 0.2616 0.327 83.385 167.540 0.0327 0.358
1.0 1.0 0.7853 3.14 0.523 2.617 667.505 1341.179 0.261 2.866
1.5 1.5 2.6507 4.71 0.785 8.835 2253.095 4527.014 0.8833 9.674
2. 2. 6.283 6.283 1.047 20.943 5340.55 10730.460 2.0933 22.965
Sumerian coefficients at the pottery factory
daily work of one man
in base 60 transliterated name english decimal / reciprocal comment
0:12 sa has-as-bi coefficient it's 12/60 5 possibly making 5 ration bowls a day , pottery
4 sa dug bi coefficient rate 60/15 15/60 making 4 ration bowls a day , pottery
3:45 sa pi-ti-iq-ti coefficient wall high 3/60+45/3600 16 raising mud wall, 3/60+45/3600 surface a day
20/60 sa sag coefficient making bricks 20/60 60/20 making 240 bricks a day or 20/60 sar
1:30? igi.gub gis coefficient wood funiture 1+30/60 90/60 1.5 days on task, pegging planks, making door, bed or chair a day
40/60 sa gis-ig coefficient wood door 40/60 60/40 2/3 door a day, planing wood and pegging planks
3 sa gis pannum coefficient wood crates 3 20/60 3 crates a day, pegging planks, pannum measure crates
7:26 sa gis coefficient making wood plank ~7/60 ~ 60/7 planks & crates ,daily 3crates*6sides*.497*.497= 4.44 sq.m., 4.44/36 = decimal 0.123 sar or 7
surface /60+ 23/3600 sar daily
12 u4, u4-1-se hours of workday 12 6/60 common to several accounts and math problems
20:48:06 sa gir4 ad kup4 reed worker's kiln coefficient ~20/60 ~60/20 possibly fraction for volume of combustion chamber fuel over total chamber volume
10:45:06 esir2 had5 a sa in-na pitch in kiln coefficient ~10/60 ~60/10 possibly fraction for volume of combustion chamber fuel over total chamber volume
20 sa3 sahar ha-ba-ti coefficient sar digging clay 20/60 60/20 20/60 gur for clay mining (river banks?), equiv. 6 cubic meters
10 sa3 sahar ba-la-lu coef. sar mixing earth of one 10/60 60/10 3 cubic meters of clay mixing
24 sa3 sig4 sa3-ha-a-ti coefficient brick forming earth 24/60 60/24 another daily norm on forming brick possiby 24/60 (brick) sar= 24/60 * 720 or 288 (baked?)
bricks a day
20 sa3 sig4 coefficient making bricks 20/60 60/20 another daily norm equals 240 bricks a day, 8*60 = 240, 20/60 * 720 =240 bricks
6_40 sa3? im lag coefficient making mud squares 6/60 +40/3600 ~60/6 another daily norm,decimal 0.111*720 equals 80 mud?? bricks a day,
4_16 sa3? im lag coefficient making mud squares 4/60 +16/3600 ~60/4 another daily norm,decimal 0.0711*720 equals 51 mud?? bricks a day,
1_41_15 igi-gub sig4 ib2 si8 recip. coeff. brick square mass 0.02875 34.78 recip. mass of mud? brick, decimal 0.02875 1/manas,34.78 mannas, 69.88 kg mass
3_22_30 igi-gub sig4 in recip. coeff. brick in xxx? 0.0575 17.39 recip. mass of mud? brick , decimal 0.0575 1/manas,17.39 mannas, 34.94 kg mass
5_24 igi-gub pes sig4 recip. coeff. thirds brick 0.09 11.11 recip. mass of fired brick , decimal 0.09 1/manas,11.11 mannas, 22.32 kg mass
al-ur-ra fired
2_42 igi-gub sig4 al-ur-ra recip. coeff. brick fired 0.045 22.22 recip. mass of fired brick, decimal 0.045 1/manas,22.22 mannas, 44.64 kg mass
1_20 igi-gub sahar il2 sig4 constant of carrying (of 1/60+20/3600 ~ 60/1 daily task bringing mud, 1/60+20/3600 gurs,decimal 0.222 gurs,.222gurs*18.= 4 cubic meters of
du3 bricks) mud clay
9 im-gir4 mud kiln 9 1/9 9 sar bricks, contents of an im-gir4, truncated triangular prism, possible clamp kiln,
9sar*720= 6480 bricks
table: Wet, Sun Dried, Fired printed in tcl wiki
Brick Ratios format
ratio alternate ratio for decimal numerator denominator comment, if any
base60 base10
3/2 90/60 1.5 wet bricks fired bricks Old Babylonian
2/3 60/90 0.666 fired bricks wet bricks Old Babylonian
6/5 72/60 1.2 sun dried brick fired bricks Old Babylonian
5/6 50/60 0.8333 fired brick sun dried bricks Old Babylonian
1/1 60/60 1. fired bricks fired bricks Old Babylonian
5/1 300/60 5. price of fired bricks gur/sar price of fired bricks gur/ more common, Old Babylonian
10/1 600/60 10. price of fired bricks gur/sar price of fired bricks gur/ less common, redundant, Old
sar Babylonian
1/5 12/60 0.20 market rate of fired bricks sar/gur market rate of unfired more common , Old Babylonian
bricks sar/gur
1/10 6/60 0.10 market rate of fired bricks sar/gur market rate of unfired less common , redundant, Old
bricks sar/gur Babylonian
6.3/5 76/60 1.2666 sun dried brick fired bricks modern estimate in analysis
O.B. derived mostly from calculations in texts, not necessarily coefficient
Alternate or redundant values intended for ranging calculations , not hard and
fast coefficients by themselves
Testcases Section
In planning any software, it is advisable to gather a number of testcases to check the results of the program. The math for the testcases can be checked by pasting statements in the TCL console.
Aside from the TCL calculator display, when one presses the report button on the calculator, one will have console show access to the capacity functions (subroutines).
Testcase 1
kiln dimension
meters or other testcase 1 testcase 2 testcase 3 testcase 4 comment
diameter of combustion chamber .5 1. 1.5 2. optional,many traditional kilns have single chamber
height of combustion chamber .5 1. 1.5 2. diameter and height
base length central support .25 .25 .4 .5 circular support base
width central support .25 .25 .4 .5 circular support base
vertical height of firing chamber .5 1. 1.5 2. some are single chambers
width of firing chamber .5 1. 1.5 2. equals combustion chamber
height of firing chamber dome (if) .2 .3 .4 .5 optional,many traditional kilns do not have domes or tops
thickness of floor on f. chamber .07 .07 .1 .1 optional,many traditional kilns do not have brick floors
length of stoking channel .75 .75 .75 .8 should be small or closeable
width of stoking channel .25 .30 .40 .4 should be small or closeable
chimney height 6 9 12 15 many traditional kilns do not have chimneys
air valve, chimney damper .25 .25 .25 .25 advanced feature, not usually on traditional kilns
Testcase 4
quantity value units
wood kiln is fired for 5 days
fuel 1 cord wood a day.
firing temperature 1200 degrees centigrade.
combustion chamber 4 cubic meters
Testcase 5
quantity value units comment
wood kiln is fired for 5 hours
diameter of combustion chamber 2 meters
internal diameter of c. chamber 1.7 meters
height of combustion chamber 1.5 meters
fuel 400 kilograms wood a day.
ambient temperature 30 centigrade.
firing temperature 1000 degrees centigrade .
combustion chamber 4 cubic meters
chimney height 6 meters
white oak 3700 lb/cord
white oak 24 MMBTU/cord
white oak specific gravity 850 kilograms per cubic meter
One MMBtu = 1,000,000 Btu
mass_flow_rate = burning_area* fuel_burn_rate*fuel_density
where fuel_burn_rate= initial_rate + (e**+c*kelvin)
Testcase 6
testcase number: 6
max room fuel kg 2705.181
estimated fuel,oak kg 369.708
total heat megajoules 4991.059
kiln style single chamber, circular
testcase number: 6
calculator inputs known values comment
kiln diameter meters 1.83 inner diameter
kiln height meters 1.21 really single chamber kiln
volume cubic meters 3.18
kiln firing seconds 8100
outputs < < <
calculator outputs calc. value measured comment
estimated fuel,oak kg 369.708 350 kg calculator based on oak
total heat megajoules 4991.05 4971.0 very close match
firing max temperature 750 660 accuracy very sensitive to firing time
man workdays to load fuel 1.060 est., not reported
test of offsite image retrival
Screenshots Section
figure 1.
figure 1b.
figure 2.
figure 3.
figure 4.
figure 5.
figure 6.
figure 7.
figure 8.
figure 9.
figure 10.
figure 11.
• Kenyan Ceramic Jiko cooking stove, by Hugh Allen
• DELCROIX, G. et HUOT, J.L., 1972, « Les fours dits « de potier » dans l’Orient ancien
• Michio: Anagama: Building Kilns and Firing
• Saraswati, B. and N.B. Behura. 1966. Pottery Techniques of Peasant India.
• Traditional Potters of India, [L1 ]
• Planting and Growing Miscanthus Reed [L2 ]
• Brick and Ceramic Sectors [L3 ]
• Energy Measurements and Conversions [L4 ]
• Mani Kiln (google >> mani kiln efficient)
• Village-Level Brickmaking [L5 ]
• Technical problems of brick production, prepared by Kelvin Mason (June 1998) [L6 ]
• Energy Used to Fire Clay Bricks, prepared by Kelvin Mason, June1998 [L7 ]
• Energy Used ... good simple math for bricks, much used [L8 ]
• Building the Mani Kiln, Drawings by Manny Hernandez (google >> mani kiln efficient)
• Ten Rules for Brick Firing,prepared by Theo Schilderman (June 1998) [L9 ]
• CFD Simulation of Flue Gas Flow in Traditional Pottery , Cecilia Schotte, thesis
• CFD Simulation of Flue Gas Flow in Pottery Furnace, Kristina Nilenius, thesis
• Equivalency values of the UR III period, Robert K. Englund, CDLI Library[L10 ]
• Equivalency values page & CDLI MySQL search engine , CDLI Library [L11 ]
Appendix Code
appendix TCL programs and scripts
# TCL source code follows
# pretty print from autoindent and ased editor
# Sumerian Circular Kiln calculator V2
# written on Windows XP on TCL
# working under TCL version 8.6
# gold on TCL Club, 12Dec2018
package require Tk
package require math::numtheory
namespace path {::tcl::mathop ::tcl::mathfunc math::numtheory }
set tcl_precision 17
frame .frame -relief flat -bg aquamarine4
pack .frame -side top -fill y -anchor center
set names {{} {kiln diameter meters:} }
lappend names {burning chamber height:}
lappend names {firing time seconds: }
lappend names {answer: volume cubic meters: }
lappend names {total heat units, megajoules:}
lappend names {optional: }
lappend names {optional: }
lappend names {firing max temperature centigrade: }
foreach i {1 2 3 4 5 6 7 8} {
label .frame.label$i -text [lindex $names $i] -anchor e
entry .frame.entry$i -width 35 -textvariable side$i
grid .frame.label$i .frame.entry$i -sticky ew -pady 2 -padx 1 }
proc about {} {
set msg "Calculator for Sumerian Circular Kiln
from TCL
# gold on TCL Club, 12Dec2018 "
tk_messageBox -title "About" -message $msg }
proc self_help {} {
set msg " Rectangular Radio Antenna V2
from TCL ,
# self help listing
# problem, Sumerian Circular Kiln V2
# 1 given follows.
1) frequency megahertz:
# Recommended procedure is push testcase and fill frame,
# change first three entries etc, push solve,
# and then push report. Report allows copy and paste
# from console to conventional texteditor. For testcases
# testcase number is internal to the calculator and
# will not be printed until the report button is pushed
# for the current result numbers.
# >>> copyright notice <<<
# This posting, screenshots, and TCL source code is
# copyrighted under the TCL/TK license terms.
# Editorial rights and disclaimers
# retained under the TCL/TK license terms
# and will be defended as necessary in court.
Conventional text editor formulas or grabbed from internet
screens can be pasted into green console.
# gold on TCL Club, 12Dec2018 "
tk_messageBox -title "Self_Help" -message $msg }
proc pi {} {expr acos(-1)}
proc kilnvolumex { d h } {
set kilnvolumexxx [* .25 [pi] $d $d $h ]
return $kilnvolumexxx
proc heatx { vol density } {
set fuel_mass [* $vol $density]
set total_heat [* $fuel_mass 13.54]
return $total_heat
proc firingtempxx { total_heat burn_time cross_section temp_ambient temp_flame} {
set d $cross_section
set firing_temp 1
set heat_flux [/ $total_heat $burn_time]
set heat_flux_persec [/ $heat_flux $burn_time]
set cross_section [ / [* [pi] $d $d ] 4.]
set item 25
set temp_ambient 25
set temp_flame 1488.
set tx $temp_ambient
set taxx $temp_flame
set t 25
set h [/ $heat_flux_persec $cross_section]
while {$item <= 4000} {
incr item
set t [+ $t 1 ]
set term1 [* 1. $t $t $t $t]
set term2 [* 1. $tx $tx $tx $tx]
set term1 [* .000000000056703 [ - $term1 $term2]]
set term2 [* $h 1.1 [/ [- $taxx $t] [ - $taxx $tx ]]]
set difference [abs [- $term1 $term2]]
if {$difference < 2.} { set temp_answer $t }
return $temp_answer
proc calculate { } {
global side1 side2 side3 side4 side5
global side6 side7 side8
global testcase_number
global kiln_volume kiln_temperature_exp
global workdays total_heat massfromvolume
global total_heat firing_temp fuel
incr testcase_number
set side1 [* $side1 1. ]
set side2 [* $side2 1. ]
set side3 [* $side3 1. ]
set side4 [* $side4 1. ]
set side5 [* $side5 1. ]
set side6 [* $side6 1. ]
set side7 [* $side7 1. ]
set side8 [* $side8 1. ]
set kiln_diameter $side1
set kiln_height $side2
set kiln_firing_time $side3
set ktime $side3
set kiln_volume 1
set kiln_volume [kilnvolumex $kiln_diameter $kiln_height]
set fuel_density 850
set massfromvolume [* $kiln_volume $fuel_density]
set fuel [* [ / 8.2 60. ] $massfromvolume]
set total_heat [* 13.5 $fuel]
set kiln_temperature_exp [+ 300. [ exp [* 1. .001 $ktime]]]
set temp_amb 25.
set temp_flame 1488.
set total_heat_joules [* $total_heat 1.0E6]
set firing_temp [firingtempxx $total_heat_joules $kiln_firing_time $kiln_diameter $temp_amb $temp_flame]
# possible type change error in original, 900. ?=>? 900
# following clamp kiln temp to 900.
if { $kiln_temperature_exp > 900. } { set kiln_temperature_exp 900. }
set workdays [/ $kiln_volume 3. ]
set side4 $kiln_volume
set side5 $total_heat
set side8 $firing_temp
proc fillup {aa bb cc dd ee ff gg hh} {
.frame.entry1 insert 0 "$aa"
.frame.entry2 insert 0 "$bb"
.frame.entry3 insert 0 "$cc"
.frame.entry4 insert 0 "$dd"
.frame.entry5 insert 0 "$ee"
.frame.entry6 insert 0 "$ff"
.frame.entry7 insert 0 "$gg"
.frame.entry8 insert 0 "$hh"
proc clearx {} {
foreach i {1 2 3 4 5 6 7 8 } {
.frame.entry$i delete 0 end } }
proc reportx {} {
global answer2
global side1 side2 side3 side4 side5
global side6 side7 side8
global testcase_number
global wavelength wavelength2
global wavelength6 wavelength10
global wavelengthsq surfacearea
global megafrequency
console eval {.console config -bg palegreen}
console eval {.console config -font {fixed 20 bold}}
console eval {wm geometry . 40x20}
console eval {wm title . " Sumerian Circular Kiln Report V2, screen grab and paste from console 2 to texteditor"}
console eval {. configure -background orange -highlightcolor brown -relief raised -border 30}
console show;
puts "%|table $testcase_number |printed in| tcl format|% "
puts "&| quantity| value| comment, if any|& "
puts "&| $testcase_number :|testcase_number | |&"
puts "&| $side1 :|kiln diameter meters: | |&"
puts "&| $side2 :|burning chamber height: | |& "
puts "&| $side3 :|firing time seconds: | |& "
puts "&| $side4 :|answer: volume cubic meters | |&"
puts "&| $side5 :|total heat units, megajoules: | |&"
puts "&| $side6 :|optional: | |&"
puts "&| $side7 :|optional: | |&"
puts "&| $side8 :|firing max temperature centigrade: | |&"
frame .buttons -bg aquamarine4
::ttk::button .calculator -text "Solve" -command { calculate }
::ttk::button .test2 -text "Testcase1" -command {clearx;fillup .5 .5 6000. 0.0981 153.9 0. 0. 700. }
::ttk::button .test3 -text "Testcase2" -command {clearx;fillup 1.5 1.5 6000. 2.65 4156. 0. 0. 870. }
::ttk::button .test4 -text "Testcase3" -command {clearx;fillup 2. 2. 6000. 6.28 9853. 0. 0. 900. }
::ttk::button .clearallx -text clear -command {clearx }
::ttk::button .about -text about -command {about}
::ttk::button .self_help -text self_help -command { self_help }
::ttk::button .cons -text report -command { reportx }
::ttk::button .exit -text exit -command {exit}
pack .calculator -in .buttons -side top -padx 10 -pady 5
pack .clearallx .cons .self_help .about .exit .test4 .test3 .test2 -side bottom -in .buttons
grid .frame .buttons -sticky ns -pady {0 10}
. configure -background aquamarine4 -highlightcolor brown -relief raised -border 30
wm title . "Sumerian Circular Kiln Calculator V2"
For the push buttons, the recommended procedure is push testcase and fill frame, change first three entries etc, push solve, and then push report. Report allows copy and paste from console. While the
testcases are in meters, the units either cancel out or are carried through in the calculator equations. So the units could be entered as English feet, Egyptian royal cubits, Sumerian gars, or
Chinese inches and the outputs of volume will in the same (cubic) units. This is an advantage since the units in the ancient Sumerian, Indian, and Chinese texts are open to question. In some benign
quarters of the globe, feet and cubic feet were still being used for design in the 1970's.
For testcases in a computer session, the eTCL calculator increments a new testcase number internally, eg. TC(1), TC(2) , TC(3) , TC(N). The testcase number is internal to the calculator and will not
be printed until the report button is pushed for the current result numbers (which numbers will be cleared on the next solve button.) The command { calculate; reportx } or { calculate ; reportx;
clearx } can be added or changed to report automatically. Another wrinkle would be to print out the current text, delimiters, and numbers in a TCL wiki style table as
puts " %| testcase $testcase_number | value| units |comment |%"
puts " &| volume| $volume| cubic meters |based on length $side1 and width $side2 |&"
initial console program
# pretty print from autoindent and ased editor
# volume of Sumerian kiln combustion chamber
# written on Windows XP on eTCL
# working under TCL version 8.5.6 and eTCL 1.0.1
# gold on TCL WIKI , 24jul2013
package require Tk
namespace path {::tcl::mathop ::tcl::mathfunc}
console show
set diameter .5
set height .5
set kiln_volume [ * .25 3.14 $diameter $diameter $height]
set kiln_volume_gurs [ / $kiln_volume .3 ]
set workdays [ / [* $kiln_volume_gurs 60] 10 ]
puts "kiln_volume $kiln_volume"
puts "kiln_volume_gurs $kiln_volume_gurs"
puts "man workdays $workdays"
Comments Section
Please place any comments here with your wiki MONIKER and date, Thanks, gold 12DEC2018 | {"url":"https://wiki.tcl-lang.org/page/Sumerian+Coefficients+in+the+Pottery+Factory+and+Calculator+Demo+Example","timestamp":"2024-11-12T15:57:47Z","content_type":"text/html","content_length":"95419","record_id":"<urn:uuid:c1a80e07-901b-4c8c-bf52-82d9fee7c421>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.63/warc/CC-MAIN-20241112145015-20241112175015-00099.warc.gz"} |
Statistics is FUN!
This summer I am taking a basic statistics course and auditing an advanced one. If the basic one is increasing my knowledge and giving me the confidence the advanced one is killing me!(in a very very
good way!) After a very long time have I have had to spend nights understanding things from various textbooks and the joy in doing it is awesome! I am writing this post to share few thoughts and
resources related to the topic.
1) In the past many of my friends have asked me for soft copies of ebooks. I download most of them from the following site:
I found almost 90% of the textbooks & novels I searched for from this website. I am using the following four statistics books to keep myself busy! Each book uses a different notation which gets to me
more often than not but as of now the patience is paying off.
a) Probability & Statistics by Athanasios Paupoulis
b) Probability & Statistics in Engineering by Hines et. al.
c) Theory and problems of probability & statistics by Schaum series
d) All of statistics by Weisserman
2) I hate calculating expecation, covariance and correlation and all that crap for discrete distributions. Firstly, the calculations are cumbersome and secondly I hate Algebra! So, I put together the
following script that calculates these. Just update the pdf and x, y vectors and you will get all the answers you need.
pdf=[11/50, 4/50, 2/50, 1/50, 1/50, 1/50;
8/50, 3/50, 2/50, 1/50, 1/50, 0;
4/50, 3/50, 2/50, 1/50, 0, 0;
3/50, 1/50, 0, 0, 0, 0;
1/50, 0, 0, 0, 0, 0;]
x = [0:5]
y = [0:4]
fx = sum(pdf)
fy = sum(pdf,2)'
ex = sum(x.*fx)
ex2 = sum(x.*x.*fx)
varx = ex2 - ex*ex
ey = sum(y.*fy)
ey2 = sum(y.*y.*fy)
vary = ey2 - ey*ey
[xgrid ygrid] = meshgrid(x,y)
exy = sum(sum(xgrid.*ygrid.*pdf))
covxy = exy - ex*ey
corr = covxy/(sqrt(varx)*sqrt(vary))
3) Proofs have always intrigued me. Although there is lot of fun in using a formula to get the answer and feel satisfied, I always found real satisfaction in deriving the formula. Being a computer
science major has deprived me of such opportunities. But thanks to the stats course I get to sit and prove stuff again :-)
I am sharing two of the proofs for which I did not find straightforward solutions online. I hope it helps others who are trying to understand the proofs. | {"url":"https://techtidings.blogspot.com/2011/07/statistics-is-fun.html","timestamp":"2024-11-15T04:25:00Z","content_type":"text/html","content_length":"48668","record_id":"<urn:uuid:7592ca9c-3aec-4577-9995-a2ec80a62683>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00870.warc.gz"} |
Complexity of problems concerning reset words for some partial cases of automata
Complexity of problems concerning reset words for some partial cases of automata
A word w is called a reset word for a deterministic finite automaton A if it maps all states of A to one state. A word w is called a compressing to M states for a deterministic finite automaton A if
it maps all states of A to at most M states. We consider several subclasses of automata: aperiodic, D-trivial, monotonic, partially monotonic automata and automata with a zero state. For these
subclasses we study the computational complexity of the following problems. Does there exist a reset word for a given automaton? Does there exist a reset word of given length for a given automaton?
What is the length of the shortest reset word for a given automaton? Moreover, we consider complexity of the same problems for compressing words.
Download data is not yet available.
How to Cite
Martyugin, P. (2009). Complexity of problems concerning reset words for some partial cases of automata. Acta Cybernetica, 19(2), 517-536. Retrieved from https://cyber.bibl.u-szeged.hu/index.php/ | {"url":"https://cyber.bibl.u-szeged.hu/index.php/actcybern/article/view/3780","timestamp":"2024-11-12T08:55:57Z","content_type":"text/html","content_length":"25842","record_id":"<urn:uuid:690cd81f-3872-4fc9-82d9-931f3d289dc3>","cc-path":"CC-MAIN-2024-46/segments/1730477028249.89/warc/CC-MAIN-20241112081532-20241112111532-00702.warc.gz"} |
The Progressive Recovering of Einstein’s Determinism under Strong Interactions • scientia.global
The Progressive Recovering of Einstein’s Determinism under Strong Interactions
Quantum mechanics relies on probabilities and uncertainties – for example, we cannot work out the outcome of a quantum system, but instead, we can suggest probabilities of certain outcomes. This has
been troublesome for determinists, who instead believe that all outcomes are governed by a set of laws. Sir Professor Ruggero Maria Santilli from The Institute of Basic Research argues that if we
extend our picture of quantum mechanics to his idea of hadronic mechanics, we can recover hidden variables and progressively recover determinism.
Einstein, Determinism and its Consequences in Quantum Mechanics
The formulation of a theory of quantum mechanics in the 1900s was not solely a pursuit of scientific thought – numerous philosophical approaches and arguments were made as well. For example, Einstein
believed in determinism, the philosophical belief that the outcome of every event is governed by the preceding events and causal laws. During his lifetime, there were no deterministic interpretations
of quantum mechanics. For example, the Born rule uses probabilities to suggest what the measurement outcome in a quantum system will be, and the Heisenberg uncertainty principle suggests that we
cannot know about some pairs of properties of a quantum particle, such as position and momentum, at the same time.
This belief led Einstein, along with his students Podolsky and Rosen, to write their famous paper in 1935. This suggested that quantum mechanics was an incomplete theory and that, eventually, there
would be some completion of the theory that would still allow for classical determinism. Sir Professor Ruggero Maria Santilli from The Institute for Basic Research in Florida has worked on his
suggestion for this completion, working on extending quantum mechanics into his theory of hadronic mechanics.
Sir Santilli suggests that there is proof for Einstein, Podolsky and Rosen’s (EPR) argument. He believes that the objections to the EPR argument are under the assumption that particles are considered
to be point-like, meaning that their mass and charge are considered to be at a single point in space, in a vacuum and under Hamiltonian interactions – interactions at a distance that tell us about
the kinetic and potential energy of the particles.
However, Santilli argues that this does not apply to structures such as sub-atomic particles like positively charged protons and neutrons, and structures made up of these hadrons like nuclei. Here,
he argues that the charge is spread within the structure rather than at a single point, giving an extended charge distribution. Also, as the volume of the nucleus is less than the sum of the volumes
of its constituent protons and neutrons, he argues that they are in a condition of mutual penetration. This leads to additional interactions, such as contact non-Hamiltonian interactions extended
over the volume of hadron overlapping, due to the dynamics within the system itself.
By studying these interior conditions, Santilli uses isomathematics and isomechanics, which preserve the core axioms of quantum mechanics but realises them to account for the extended character of
particles and their new interactions. Santilli refers to this extension of quantum mechanics as hadronic mechanics, namely, a mechanics intended for strongly interacting particles called hadrons.
Within his hadronic mechanics, Santilli suggests that this can progressively recover Einstein’s determinism with the increase of the density in the transition from hadrons to nuclei and stars and
fully recover Einstein’s determinism at the limit of gravitational collapse. Santilli believes that this can complete quantum mechanics according to the EPR argument. To do this, Santilli introduces
what is now known as the Santillian , sandwiched in between all the multiplications while maintaining the quantum axiom of associativity, to characterise the actual size of hadrons and their
additional interactions, such as the non-Hamiltonian interactions in extended particles.
Bohm’s Hidden Variables in Hadronic Mechanics
In support of the EPR argument, David Bohm suggested in 1952 that there would need to be some hidden variable. For example, instead of just having a quantum particle, there is also a hidden guiding
wave that dictates the motion of the particle. In quantum systems, we often have quantum particles that are entangled, which means that there is some correlation between the two particles. These
particles can be separated over a large distance, and there can still be entanglement between them. Until we measure the particles, we don’t know what state they are in, but by measuring one particle
in the entangled pair, as they are correlated, the outcome of the first measurement will often give us information about the second particle as well.
As an example, for Bohm’s hidden variable theory, for entangled particle two to gain information about entangled particle one when measured, there would need to be some vector travelling faster than
the speed of light to transfer this information to the second particle or the theory should be non-local in the sense of being defined on volumes, thus being outside the consistent representational
capabilities of quantum mechanics.
Santilli argues that, through the application of hadronic mathematics and mechanics, we can explain entanglement in a hidden variables theory without needing to rely on faster-than-light
communication between the particles. To explain this entanglement, Sir Santilli suggests that between these particles, there is an overlap of their wave packets, the wavelike functions that determine
the probability of the particle being measured in some state. This leads to non-linear, non-local, non-potential, and non-Hamiltonian interactions between the two particles at arbitrary distances
that are represented by the Santillian . Santilli suggests that because of this, there is continuous and instantaneous contact between the two particles at arbitrary distances, and there is no need
for the faster-than-light communication suggested in Bohm’s theory.
These non-linear, non-local, non-potential, and thus, non-Hamiltonian interactions fit into Santilli’s model of hadronic mechanics, where he introduces the aforementioned Santillian, , which can be
solely treated via isomathematics. This gives a mathematical function that can tell us about the local variables of the particle, such as coordinates and momentum, with uncertainties progressively
smaller than their quantum mechanical counterparts depending on the local density and the strength of the short-range interactions.
Furthermore, Santilli argues that is hidden because it represents contact interactions without any potential energy, and is sandwiched in between all products while conserving the quantum
mechanical axiom of associativity. As a result, he argues that we can identify with Bohm’s hidden variable in hadronic mechanics.
An additional test of Bohm’s hidden variables is also seen through Bell’s inequalities. Bell found a set of mathematical inequalities, which link to the correlations seen between entangled particles
when they are measured. If Bell’s inequalities are verified, this suggests that quantum mechanics must describe the system. It is generally assumed that if Bell’s inequalities are violated, then the
correlations can be explained by classical physics. Bell’s inequalities are often used to disprove hidden variables by stating that if there is a mathematical constraint between the two entangled
particles, this would satisfy the Bell inequality, and the system would not need quantum mechanics to explain it.
However, Santilli argues that this is not the case for the extended particles in his hadronic mechanics. He highlights how Bell inequalities are valid for point like particles under the
electromagnetic force. However, he aims to consider extended particles under the strong force and believes that Bell’s original inequalities are no longer applicable here. Sir Santilli uses
isomathematics to suggest generalised Bell’s inequalities for strong interactions, which incorporates the hidden variable, . This allows him to retain a hidden variable theory whilst also retaining
quantum correlations between particles.
The Progressive Recovering of Einstein’s Determinism
Santilli’s work on extended particles, with their constant contact with each other through mutual penetration, affects how we can consider atomic structure. As the proton and the neutron are confined
in the nucleus of the atom, whilst the electrons orbit outside it, this means that their contact interactions differ. Furthermore, as protons and neutrons are made up of further subatomic particles
called quarks, they are affected by strong force, whereas electrons aren’t. Santilli’s work highlights how, because of their confinement, the errors we associate with the proton and neutron in the
nucleus must be smaller than those we associate with the electron. This can also be incorporated in his theory of isomechanics by showing that as the density increases here, becomes smaller.
By beginning to reconcile this internal structure of the nuclei through hadronic mechanics, Santilli’s work began to progressively move towards a recovery of Einstein’s determinism. His theory
highlighted how the increasing density of particles, nuclei and even stars links with a decrease of . This led Santilli to look at the limit of gravitational collapse, for example, the mass at which
astronomical objects become so massive that they collapse under their own gravity.
Santilli’s theory of hadronic mechanics can account for the limit of gravitational collapse – at this moment when the density of the star is extremely large, the Santillian becomes zero. This
suggests a full regaining of Einstein’s determinism, as Santilli’s model suggests how these astronomical events can occur through his description of the strong interaction.
Santilli continued to work on this, forming a general principle to account for uncertainties within the strong interaction in hadronic mechanics, which he called Einstein’s isodeterminism. Through
this theory, he can overcome the potential divergences of quantum mechanics surrounding the forces and interactions of these extended particles, which allows for a generalisation of quantum mechanics
through his formulation of hadronic mechanics.
The Neutron Synthesis from Hydrogen and its Applications
The stars in our night sky begin life as clouds of dust and hydrogen, which grow progressively larger as they travel through space. Once suitably large and hot enough, they undergo a fusion process,
where the protons and electrons from the hydrogen atoms are fused to become neutrons. If certain conditions are then reached, these neutrons can be fused with protons to create a bound state called
deuteron. Fusion continues, fusing two deuterons to become helium, and the star begins to emit light due to excess energy in this process.
However, there is a suggestion that Heisenberg’s uncertainty principle implies that the fusion of this neutron in a forming star cannot occur within quantum mechanics. If we consider energy and
position in the Heisenberg uncertainty principle, the energy of the electron might have an uncertainty much larger than the energy of the neutron yet to be formed, or its coordinate might have an
uncertainty much larger than where the fusion process will occur. We can’t measure both quantities accurately, and it becomes difficult to account for this fusion process here.
Santilli can account for this with his theory of hadronic mechanics. Due to the inclusion of , in his theory, the proton and the electron are almost static in their location due to the large
pressures around them in the forming star. The proton and electron are in a condition of total mutual penetration, and by non-Hamiltonian interactions, we get an incredibly strong Coulomb attraction
between the oppositely charged proton and electron, allowing for the neutron to be synthesised. Santilli has worked on characterising this, achieving representations that are in accordance with
measured values.
Santilli has also begun to run experiments on how we can synthesise a neutron from hydrogen in the lab. By engineering an electric arc that can be submerged into hydrogen gas, the gas can be ionised
and produces a beam of neutrons. This has been developed into a commercial product called the Directional Neutron Source and can be used for different detection processes, such as locating precious
metals in mines or testing for defects in welded structures used in shipbuilding.
Furthermore, as these neutrons naturally decay back to their constituent proton and electron in about 15 minutes, Santilli suggests that his work on neutrons and hadronic mechanics could be used to
predict mechanisms to recycle radioactive waste in nuclear power plants. Exposing waste to photons of a certain energy could provide the additional energy needed to cause some neutrons to decay,
reducing the lifetime of these radioactive byproducts. For example, atoms of Molybdenum-100, an unstable isotope, have an exceptionally long lifetime of 10^19 years. With this process, Santilli
suggests these could be recycled into Technetium-100 if just one of its neutrons decays, reducing its lifetime to just 18.5 seconds.
Institute for Basic Research, Palm Harbor, Florida, FL, USA
Sir Professor Ruggero Maria Santilli originally studied Physics at the University of Naples before completing his PhD in Theoretical Physics at the University of Torino in Italy. Following this, he
has held numerous different academic positions in both Italy and the USA, culminating in his current roles as President and Professor of Physics at the Institute for Basic Research in Florida, which
he has held since 1981, and Editor-in-Chief of the Hadronic Journal and Algebras, Groups and Geometries. Throughout his academic career, he has written and published extensively in a wide range of
papers and monographs in mathematics, physics, chemistry and biology. His work focuses on areas such as hadron mechanics and Lie algebra, as well as the applications of this science in technologies,
such as in generating clean energy. Sir Santilli has been recognised through numerous awards, including medals for scientific achievements and his knighthoods from the Republic of San Marino, Italy.
E: research@i-b-r.org
W: https://www.i-b-r.org/
Professors H Ahmar, AOE Animalu, AK Aringazin, A Bayoumi, S Beghella-Bartoli, T Bhadra Man, A Bhalekar, R Brenna, C Burande, W Cai, P Caldirola, I B Das Sarma, B Davvaz, SS Dhondge, J Dunning-Davies,
I Gandzha, RMF Ganfornina, S Georgiev, T Gill, V de Haan, C-X Jiang, A Jannussis, E Johansen, J V Kadeisvili, T Kuliczkowski, J Lohmus, R Mignani, A P Mills, R Miron, R Perez-Enriquez, MR Molaei, A
Muktibodh, HC Myung, AA Nassikas, M Nishioka, R Norman, Z Oziewicz, J Rak, E Recami, A Shoeber, DS Sourlas, JN Valdez, E Trell, B Veljanoski, Gr T Tsagas, T Vougiouklis, HE Wilhelm, Y Yang, L Ying,
and others.
NASA, USAFOSR, USDOE, and the R. M. Santilli Foundation
RM Santilli, Reduction of Matter in the Universe to Protons and Electrons via the Lie-isotopic Branch of Hadronic Mechanics, Progress in Physics, 2023, 19, 73–99.
RM Santilli, Lie-isotopic representation of stable nuclei I: Apparent insufficiencies of quantum mechanics in nuclear physics, Ratio Mathematica, 2024, 52. DOI: http://dx.doi.org/10.23755/
RM Santilli, Lie-isotopic representation of stable nuclei II: Exact and time invariant representation of the Deuteron data, Ratio Mathematica, 2024, 52. DOI: http://dx.doi.org/10.23755/rm.v52i0.1608
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Professor Darin Acosta’s research at the CMS experiment utilises advanced muon detection, sophisticated trigger systems, and machine learning to deepen our understanding of the Higgs boson and
explore the potential existence of dark matter. Based at Rice University in the USA, Professor Acosta’s work has long-reaching implications that are fundamental to our understanding of the universe.
Controlling partial differential equations (PDEs) with parametric uncertainties is vital for system stability in science and engineering. Professor Francisco Jurado from Tecnológico Nacional de
México (TecNM)/La Laguna addresses this challenging problem with a novel approach using adaptive backstepping boundary control for a one-dimensional modified Burgers’ equation. His innovative work
tackles uncertainties in reactive and viscosity terms and considers Robin and Neumann boundary conditions, offering valuable insights into nonlinear PDE control.
Dr Christopher Singh at the Los Alamos National Laboratory in the USA conducts groundbreaking research into photonic devices. We find out here about the potential impacts of his work exploring
quantum physics in radiation-heavy environments. His work informs progress across excitingly diverse sectors, including space exploration, nuclear energy, and healthcare.
Through a detailed analysis of mathematical frameworks, Professor Jean-François Pommaret challenges the established scientific consensus on gravitational waves, proposing that certain mathematical
interpretations could question their existence. This article delves into the professor’s examination of the founding principles of general relativity, offering an insightful, alternative perspective
on the ongoing dialogue between mathematics and physics. | {"url":"https://www.scientia.global/the-progressive-recovering-of-einsteins-determinism-under-strong-interactions/","timestamp":"2024-11-10T15:15:22Z","content_type":"text/html","content_length":"265761","record_id":"<urn:uuid:b6392c29-c7f0-49c2-a7c3-7dcd0edc8dd3>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.60/warc/CC-MAIN-20241110134821-20241110164821-00462.warc.gz"} |
DPMO – Defects Per Million Opportunities - Lean Sigma Corporation
DPMO – Defects Per Million Opportunities
DPMO is one of a few important Six Sigma metrics that you should get comfortable with if you’re associated with Six Sigma. In order to understand DPMO it's best if you first understand both the
nomenclature and the minor nuances such as the difference between defect and defective.
DPMO Nomenclature:
• Defects = D
• Unit = U
• Opportunity to have a defect = O
Defining "defect", "defective" and "Opportunities"
Defective suggests that the value or function of the entire unit or product has been compromised. Defective items will always have at least one defect. Typically however, it takes multiple defects
and/or critical defects to cause an item to be defective. To put it simply, defective is "broken", it can't be used or sold.
A defect is an error, mistake, flaw, fault or some type of imperfection that reduces the value of a product or unit. A single defect may or may not render the product or unit "defective" depending on
the specifications of the customer. Defects
Defect vs. Defective Summary:
• Defect means that part of a unit is bad
• Defective means that the whole unit is bad.
Now let’s turn our attention to defining "opportunities" so that we can fully understand Defects per Million Opportunities (DPMO).
Opportunities are the total number of possible defects. If for example a unit has 6 possible defects then each unit produced is equal to 6 defect opportunities. If we produce 100 of those units then
there are 600 defect opportunities.
Calculating Defects per Million Opportunities
• The equation is DPMO = (D/(U*O))*1,000,000
First, find your total opportunities by multiplying the # of units by the # of defect opportunities per unit, then divide defects by your total opportunities then multiply by one million. It's not a
difficult metric but what makes it unique and effective is that it considers the various possible "defects" that a product or service might have and it provides a measure to observe performance
relative to all possible mistakes a process can make.
Most organizations only measure the rate of defectives (you know, the broken ones! because they can't be sold). However, that only serves to limit the organizations ability to continuously improve
it's process and output. Let's look at a basic example:
• Assumptions:
□ Your organization produces pencils
□ There are 5 defect opportunities per pencil (lead, wood, eraser, eraser clasp and label)
□ Your organization averages of 4 defects every 100 units
• Defect Opportunities when producing 100 pencils = 5*100 or 500
• Defect rate = 4/500
• DPMO = 4/500*1,000,000 or 8000.
What is the reason or significance of 1,000,000"? Converting defect rates to a per million value becomes necessary when the performance of your process approaches Six Sigma. When this happens, the
number of defects shrinks to virtually nothing....in fact, if you recall from the ‘What is Six Sigma’ module, it's 3.4 defects per million opportunities. By using 1,000,000 opportunities as the
barometer we have the resolution in the measurement to count defects all the way up to Six Sigma.
More Articles and Free Tools - Join Today | {"url":"https://leansigmacorporation.com/dpmo/","timestamp":"2024-11-09T08:04:05Z","content_type":"text/html","content_length":"150727","record_id":"<urn:uuid:77f2c06c-4619-4991-9b73-e61d3493298b>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.30/warc/CC-MAIN-20241109053958-20241109083958-00238.warc.gz"} |
Anatomy Drawing Lessons
Drawing Faces Step By Step
Drawing Faces Step By Step - Divide the remaining space below in half and make a line there. This line will give you the eye placement. Draw a circle and make a small horizontal line at the bottom
for the chin. Web how to draw a face step by step. Draw a vertical line down the center of the face and make sure both sides of the face are symmetrical. See an easy way to draw a face in ten simple.
Lightly sketch a vertical line down the center, then draw a horizontal line halfway between the top and bottom of the oval. Begin your drawing by deciding how much of your paper you’d like the face
to take up. Draw a vertical line down the center of the face and make sure both sides of the face are symmetrical. New tutorial (side view) : See an easy way to draw a face in ten simple.
Begin your drawing by deciding how much of your paper you’d like the face to take up. In this blog post, we will learn how to draw a face for beginners, including understanding proportions and
getting them right. Draw guidelines on the face. It’s position should be roughly half of the circle’s diameter vertically. Draw a large circle and make a horizontal line below it for the chin.
Draw a vertical line down the center of the face and make sure both sides of the face are symmetrical. Web how to draw a female face step 1: See an easy way to draw a face in ten simple. This matters
because the circle was drawn freehandedly, so the horizontal diameter could be different. It’s position should be roughly.
Draw a vertical line down the center of the face and make sure both sides of the face are symmetrical. Draw guidelines on the face. Begin your drawing by deciding how much of your paper you’d like
the face to take up. Divide the remaining space below in half and make a line there. Draw a circle and make a.
In this blog post, we will learn how to draw a face for beginners, including understanding proportions and getting them right. Web when you’re drawing a face, begin by sketching in a basic oval
shape. Lightly sketch a vertical line down the center, then draw a horizontal line halfway between the top and bottom of the oval. Divide the remaining.
Web how to draw a female face step 1: Using a ruler, draw a horizontal line at the top and bottom of your paper with two vertical lines around an inch from either side. Web when you’re drawing a
face, begin by sketching in a basic oval shape. Draw a vertical line down the center of the face and make.
See an easy way to draw a face in ten simple. Draw a circle and make a small horizontal line at the bottom for the chin. Draw a vertical line down the center of the face and make sure both sides of
the face are symmetrical. In the middle, draw another horizontal line. Using a ruler, draw a horizontal line.
Draw guidelines on the face. See an easy way to draw a face in ten simple. 102k views 2 years ago learn to draw with circle line art school. Next, draw a line down the center of the face vertically,
and another one horizontally halfway between the top and bottom. Divide the remaining space below in half and make a.
Draw a vertical line down the center of the face and make sure both sides of the face are symmetrical. Using a ruler, draw a horizontal line at the top and bottom of your paper with two vertical
lines around an inch from either side. This line will give you the eye placement. In this blog post, we will learn.
Jun 9, 2021 • 4 min read. Draw an oval slightly wider at the top than bottom. New tutorial (side view) : In the middle, draw another horizontal line. Draw a large circle and make a horizontal line
below it for the chin.
Draw a large circle and make a horizontal line below it for the chin. There are 2 ways to do this step: In the middle, draw another horizontal line. Draw a circle and make a small horizontal line at
the bottom for the chin. 102k views 2 years ago learn to draw with circle line art school.
In this blog post, we will learn how to draw a face for beginners, including understanding proportions and getting them right. Web how to draw a face step by step. Using a ruler, draw a horizontal
line at the top and bottom of your paper with two vertical lines around an inch from either side. This matters because the circle.
Drawing Faces Step By Step - Divide the remaining space below in half and make a line there. Lightly sketch a vertical line down the center, then draw a horizontal line halfway between the top and
bottom of the oval. See an easy way to draw a face in ten simple. There are 2 ways to do this step: It’s position should be roughly half of the circle’s diameter vertically. 4.2m views 3 years ago
learn to draw | tutorials. 102k views 2 years ago learn to draw with circle line art school. New tutorial (side view) : Jun 9, 2021 • 4 min read. Web when you’re drawing a face, begin by sketching in
a basic oval shape.
It’s position should be roughly half of the circle’s diameter vertically. Lightly sketch a vertical line down the center, then draw a horizontal line halfway between the top and bottom of the oval.
Next, draw a line down the center of the face vertically, and another one horizontally halfway between the top and bottom. Web when you’re drawing a face, begin by sketching in a basic oval shape.
Draw an oval slightly wider at the top than bottom.
This matters because the circle was drawn freehandedly, so the horizontal diameter could be different. See an easy way to draw a face in ten simple. Web when you’re drawing a face, begin by sketching
in a basic oval shape. Begin your drawing by deciding how much of your paper you’d like the face to take up.
Draw an oval slightly wider at the top than bottom. Begin your drawing by deciding how much of your paper you’d like the face to take up. This line will give you the eye placement.
In this blog post, we will learn how to draw a face for beginners, including understanding proportions and getting them right. Web how to draw a female face step 1: 4.2m views 3 years ago learn to
draw | tutorials.
Jun 9, 2021 • 4 Min Read.
Begin your drawing by deciding how much of your paper you’d like the face to take up. Draw guidelines on the face. See an easy way to draw a face in ten simple. Draw an oval slightly wider at the top
than bottom.
Next, Draw A Line Down The Center Of The Face Vertically, And Another One Horizontally Halfway Between The Top And Bottom.
There are 2 ways to do this step: This line will give you the eye placement. Pencil portrait drawing tutorial for beginners step by step. Divide the remaining space below in half and make a line
102K Views 2 Years Ago Learn To Draw With Circle Line Art School.
This matters because the circle was drawn freehandedly, so the horizontal diameter could be different. In this blog post, we will learn how to draw a face for beginners, including understanding
proportions and getting them right. New tutorial (side view) : Draw a vertical line down the center of the face and make sure both sides of the face are symmetrical.
Draw A Circle And Make A Small Horizontal Line At The Bottom For The Chin.
Web when you’re drawing a face, begin by sketching in a basic oval shape. It’s position should be roughly half of the circle’s diameter vertically. Using a ruler, draw a horizontal line at the top
and bottom of your paper with two vertical lines around an inch from either side. Draw a large circle and make a horizontal line below it for the chin. | {"url":"https://revivalportal.goodwood.com/art/anatomy-drawing-lessons/drawing-faces-step-by-step.html","timestamp":"2024-11-06T17:25:23Z","content_type":"text/html","content_length":"33735","record_id":"<urn:uuid:45424cc9-b239-4726-be4a-43dfd812c03d>","cc-path":"CC-MAIN-2024-46/segments/1730477027933.5/warc/CC-MAIN-20241106163535-20241106193535-00206.warc.gz"} |
Variance and short cut method of math
Class 11 please fast
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Question Text Variance and short cut method of math Class 11 please fast
Updated On Dec 9, 2022
Topic Calculus
Subject Mathematics
Class Class 11
Answer Type Video solution: 1
Upvotes 141
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electrician needs wiring regs books
Hello there
We land in Adelaide on the 30th december. I've decided to do my PEER wiring course through the distance learning option. The problem is i can't find anywhere in the UK to get a copy of the AS/NZS
wiring regs from. I need this to do the course, does anyone know where i can get a copy from?
If you can help i'd much appreciate it. Also, what's the work situation like out there for electricians at the moment? As much as i'd like to have some time off when i get there, if i could start
work sooner rather than later, i'd feel happier. That could also mean that Mel doesn't have to find a job, and could spend more time shopping:biglaugh:
hey there you can get a pdf copy from here http://www.saiglobal.com/shop/script/Details.asp?DocN=AS0733783910AT
there is loads of work for electricians here in sa for comercial industrial and domesic sparky's you couls start work as soon as you want no worries
Hello there
We land in Adelaide on the 30th december. I've decided to do my PEER wiring course through the distance learning option. The problem is i can't find anywhere in the UK to get a copy of the AS/NZS
wiring regs from. I need this to do the course, does anyone know where i can get a copy from?
If you can help i'd much appreciate it. Also, what's the work situation like out there for electricians at the moment? As much as i'd like to have some time off when i get there, if i could start
work sooner rather than later, i'd feel happier. That could also mean that Mel doesn't have to find a job, and could spend more time shopping:biglaugh:
Mail me when you get here, we employ 6 electricians but we may be looking for another 2 in the new year if all goes well
Guest melissa and darren
That's excellent Tooeasy, i'll check out that pdf version, thanks for your help.
Marty, i'll definatly mail you when i get there. It's a relief to hear there's work about out there, as that's my only worry.
Thanks for your help
Mail me when you get here, we employ 6 electricians but we may be looking for another 2 in the new year if all goes well
Hey Marty do you take on adult apprentices?
marty how much work you got what sort of work do you do im always looking for a better job
Mainly commericial and domestic maintenance, house rewiring, and insurance stuff, the usual stuff I will let you know in the new year if we require any more guys.
We dont go in roofs over 32 deg, we provide a vechile (commodore utes with canopys) and pay above award.
Hey Marty do you take on adult apprentices?
We only take on school leavers as apprentices as we employ young electricians and we have found that mature apprentices have issues taking orders from a 22 year old tradesmen, not saying you would
but it is just a policy we adopt, try Peer as they may be able to help, you could enrol with TAFE as they have lists of contractors who are looking for apprentices. | {"url":"https://www.pomsinadelaide.com/topic/6647-electrician-needs-wiring-regs-books/","timestamp":"2024-11-13T14:36:53Z","content_type":"text/html","content_length":"166103","record_id":"<urn:uuid:5364cddf-0f3e-4d26-94ef-05971ad978a4>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00734.warc.gz"} |
Problem-Solving Strategies for Efficient and Elegant Solutions, Grades 6-12
Hands-on, Practical Guidance for Educators
From math, literacy, equity, multilingual learners, and SEL, to assessment, school counseling, and education leadership, our books are research-based and authored by experts on topics most relevant
to what educators are facing today.
Problem-Solving Strategies for Efficient and Elegant Solutions, Grades 6-12
A Resource for the Mathematics Teacher
Second Edition
Afterword by Nobel Laureate Herbert A. Hauptman
Help students become skilled and confident problem solvers!
This updated edition illustrates ten basic strategies for solving a wide range of mathematics problems and provides numerous examples to show how these techniques can be incorporated into a
mathematics curriculum. Teachers and mathematics specialists can develop students' creative problem-solving skills through strategies such as working backwards, finding a pattern, adopting a
different point of view, or making a visual representation. The new edition includes:
• References to current NCTM standards
• Examples of new problems for teaching the strategies
• Solutions to sample problems
• Extensive discussions of the strategies used to solve sample problems
Product Details
• Grade Level: 6-12
• ISBN: 9781412959704
• Published By: Corwin
• Year: 2008
• Page Count: 280
• Publication date: March 20, 2008
Review Copies
This book is not available as a review copy.
"The authors have provided a unique, strategy-focused resource supported by a wealth of engaging examples that mathematics teachers can readily use to help students develop a more purposeful,
systematic, and successful approach to problem solving."
—Howard W. Smith, Superintendent
Public Schools of the Tarrytowns, Sleepy Hollow, NY
"Helps both new and veteran teachers better understand the nature of problem solving as a critical mathematics process. The authors present in very simple terms the strategies that are the backbone
of mathematics instruction. This indispensable material is useful at all levels, from basic stages to advanced student work to the development of top problem solvers."
—Daniel Jaye, Principal
Bergen County Academies, Hackensack, NJ
Help students become skilled and confident problem solvers!
Demonstrating there is always more than one approach to solving a problem, well-known authors and educators Alfred S. Posamentier and Stephen Krulik present ten basic strategies that are effective
for finding solutions to a wide range of mathematics problems. These tried-and-true methods—including working backwards, finding a pattern, adopting a different point of view, solving a simpler
analogous problem, and making a visual representation—make problem solving easier, neater, and more understandable for students as well as teachers.
Providing numerous sample problems that illustrate how mathematics teachers and specialists can incorporate these techniques into their mathematics curriculum, this updated edition also includes:
• A variety of new problems that show how to use the strategies
• References to current NCTM standards
• Solutions to the problems in each chapter
• Extensive discussions of the empowering strategies used to solve sample problems
The second edition of Problem-Solving Strategies for Efficient and Elegant Solutions, Grades 6–12 helps teachers develop students' creative problem-solving skills for success in and out of school.
Key features
• Presents 10 effective problem-solving strategies
• Includes references to current mathematics standards
• Provides a generous collection of problems showing how to use the strategies
• Includes solutions to the problems and extensive discussions of the strategies used to solve them
Table of Contents
About the Authors
1. Introduction to Problem-Solving Strategies
2. Working Backwards
The Working Backwards Strategy in Everyday Life Problem-Solving Situations
Applying the Working Backwards Strategy to Solve Mathematics Problems
Problems Using the Working Backwards Strategy
3. Finding a Pattern
The Finding a Pattern Strategy in Everyday Life Problem-Solving Situations
Applying the Finding a Pattern Strategy to Solve Mathematics Problems
Problems Using the Finding a Pattern Strategy
4. Adopting a Different Point of View
The Adopting a Different Point of View Strategy in Everyday Life Problem-Solving Situations
Applying the Adopting a Different Point of View Strategy to Solve Mathematics Problems
Problems Using the Adopting a Different Point of View Strategy
5. Solving a Simpler Analogous Problem
The Solving a Simpler Analogous Problem Strategy in Everyday Life Problem-Solving Situations
Applying the Solving a Simpler Analogous Problem Strategy to Solve Mathematics Problems
Problems Using the Solving a Simpler Analogous Problem Strategy
6. Considering Extreme Cases
The Considering Extreme Cases Strategy in Everyday Life Problem-Solving Situations
Applying the Considering Extreme Cases Strategy to Solve Mathematics Problems
Problems Using the Considering Extreme Cases Strategy
7. Making a Drawing (Visual Representation)
The Making a Drawing (Visual Representation) Strategy in Everyday Life Problem-Solving Situations
Applying the Making a Drawing (Visual Representation) Strategy to Solve Mathematics Problems
Problems Using the Making a Drawing (Visual Representation) Strategy
8. Intelligent Guessing and Testing (Including Approximation)
The Intelligent Guessing and Testing (Including Approximation) Strategy in Everyday Life Problem-Solving Situations
Applying the Intelligent Guessing and Testing (Including Approximation) Strategy to Solve Mathematics Problems
Problems Using the Intelligent Guessing and Testing (Including Approximation) Strategy
9. Accounting for All Possibilities
The Accounting for All Possibilities Strategy in Everyday Life Problem-Solving Situations
Applying the Accounting for All Possibilities Strategy to Solve Mathematics Problems
Problems Using the Accounting for All Possibilities Strategy
10. Organizing Data
The Organizing Data Strategy in Everyday Life Problem-Solving Situations
Applying the Organizing Data Strategy to Solve Mathematics Problems
Problems Using the Organizing Data Strategy
11. Logical Reasoning
The Logical Reasoning Strategy in Everyday Life Problem-Solving Situations
Applying the Logical Reasoning Strategy to Solve Mathematics Problems
Problems Using the Logical Reasoning Strategy
Afterword by Herbert A. Hauptman
Sources for Problems
Readings on Problem Solving
Quote by Howard W. Smith, Superintendent, Public Schools of the Tarrytowns, Sleepy Hollow, NY: “The authors have provided a uniquely strategy-focused resource supported by a wealth of engaging
examples that mathematics teachers can readily use to help students develop a more purposeful, systematic, and successful approach to problem solving.”
“The authors have provided a uniquely strategy-focused resource supported by a wealth of engaging examples that mathematics teachers can readily use to help students develop a more purposeful,
systematic, and successful approach to problem solving.”
Howard W. Smith, Superintendent
Public Schools of the Tarrytowns, Sleepy Hollow, NY
Quote by Daniel Jaye, Principal, Bergen County Academies, Hackensack, NJ: "This terrific resource helps both new and veteran teachers better understand the nature of problem solving as a critical
mathematics process. In very simple terms, the authors present and illuminate the strategies that are the backbone of mathematics instruction. This indispensable material is useful at all levels,
from basic stages to advanced student work to the development of top problem solvers."
"This terrific resource helps both new and veteran teachers better understand the nature of problem solving as a critical mathematics process. In very simple terms, the authors present and
illuminate the strategies that are the backbone of mathematics instruction. This indispensable material is useful at all levels, from basic stages to advanced student work to the development of
top problem solvers."
Daniel Jaye, Principal
Bergen County Academies, Hackensack, NJ | {"url":"https://us.corwin.com/books/problem-solving-strategies-2e-232159","timestamp":"2024-11-07T23:21:10Z","content_type":"text/html","content_length":"86841","record_id":"<urn:uuid:6040090b-3eb0-4942-ae22-126517ee74e7>","cc-path":"CC-MAIN-2024-46/segments/1730477028017.48/warc/CC-MAIN-20241107212632-20241108002632-00480.warc.gz"} |
Logistic Regression
A type of regression model (like Linear Regression) but where the dependent variable (the thing being predicted) is categorical, rather than continuously varying over a range. For example, Logistic
Regression might be used to predict if a house split level or ranch based on the number of rooms, where linear regression would be used to predict something like the value or selling price. It
discriminates between groups of data, but does not generalize either the best seperation between the groups, nor does it find the probable center or distribution of the groups. It works well and
inexpensively when the data is spread out but has a clear and linear boundry. When the boundry is non-linear, or some (training) data may be spread away from the actual boundry, an SVM may work
better. When the data is clustered or grouped, a Bayesian Classifier may work better.
Although the prediction is binary, the model uses a Percentage Chance Prediction for each possible outcome. E.g. what is the percentage chance that a house is split level given that it has 5 rooms?
So 0 < h[S:[o]:S](x) <= 1. The probability that y=1, given x parameterized by [S:o:S], can be written as: h[S:[o]:S](x) = P(y=1:x;[S:o:S]) Assuming that y can only be 0 or 1, if P(y=1:x;[S:o:S]) =
0.7, then we know there is a 70% chance that y is 1. There is then, obviously, a 30% chance that y is zero, or P(y=0:x;[S:o:S]) = 0.3. I.e. the total probabilities must add to 1.
As expected, our job is to find [S:o:S] such that x is transformed into a valid prediction of the probability that y is 1. We can then apply a Decision Boundary which divides the probability into a 1
or 0. For example, we might say that y is 1 when the probability of y being 1 is higher than 50%. So our hypothesis function needs to translate values of [S:O:S]^Tx such that g([S:O:S]^Tx) will be >=
0.5 when [S:O:S]^Tx >= 0 (positive) and less than 0.5 when [S:O:S]^Tx < 0 (i.e. negative) but the result will never be more than 1 or less than -1.
[S:O:S]^Tx as in Linear Regression, we need to use the Logistic Function ^ g([S:O:S]^Tx) where g(z) is:
which is also called the Sigmoid Function^. In Octave:
function g = sigmoid(z)
%SIGMOID Compute sigmoid functoon
% J = SIGMOID(z) computes the sigmoid of z.
% (z can be a matrix, vector, or scalar).
g = 1 ./ ( 1 .+ exp(-z) ) ;
This sigmoid function simply limits our result into the maximum range we desire, with a rapid transition at [S:O:S]^Tx = 0.
Note: The decision boundary may not be a straight line if the hypotheses includes terms which define other terms. E.g. x^2
Cost Function: The standard Mean Squared Error (MSE)^ function doesn't work well as a cost function for sigmoid because it results in graph that is not convex, but instead is "pitted" with many local
minima. Instead we use:
-log(h[S:[o]:S](x)) if y = 1
-log(h[S:[o]:S](x)) if y = 0
This produces a nice curve that, when y = 1, slowly approaches 0 as h[S:[o]:S](x) approaches 1 yet quickly becomes infinite near zero 0 and does exactly the opposite when y = 1, i.e. it slowly
approaches 0 as h[S:[o]:S](x) approaches 0, but quickly become infinite as it approaches 1. This gives us near infinate cost as our error is large, and very little change in cost as our error
decreases toward zero, helping us to not overshoot our goal. Also, the relationship between the loss function and the parameters w still gives us a concave error function. Thus, we can rest assured
that there is only a unique minimum on the error surface.
We can combine the two cases (y=0 or y=1) into one equation by multiplying one side by y and the other side by 1-y; which ever term is unwanted, will be multiplied by zero and drop out.
h[S:[O]:S](x^(i)) = g([S:O:S]^Tx^(i))
Hypothesis Function: If we transpose each x^(i) and build a matrix X = [(x^(1))^T;(x^(2))^T;...(x^(m))^T] given [S:O:S] = [[S:O:S][0];[S:O:S][1];...[S:O:S][n]] then the matrix product X[S:O:S] is [
[S:O:S]^Tx^(1);[S:O:S]^Tx^(2);...[S:O:S]^Tx^(m)]. In other words, if we build a matrix X (note the capital X vs x) where each row is a set of input values (x^(i))^T), then [S:O:S]^Tx^(i) becomes X[S:
O:S] which is a vector with each element being the sum of the values of [S:O:S] times each input. We can then take the sigmoid function of each row to form our vector of hypothesis values. In Octave:
hyp = sigmoid(X*theta);
Calculating our cost then becomes simple matrix math by taking the transpose of the y values or the (1-y) values times the hypothesis or 1 - hypothesis vector. In Octave:
errs = -y' * log(hyp) - (1-y)' * log(1-hyp);
The cost for a given set of parameters over the training set is then simply the sum of the costs divided by the number of errors. In Octave:
J = sum(errs)/m;
Slope Function: As with the MSE in Linear Regression, the Sigmoid function has a very simple derivative h[[S:o:S]](x^(i) - y^(i)) * x[j]^(i) (it's quite difficult to take the derivative of it, but
once you do, the result is this very simple equation) which makes it easy to calculate the slope of the error. This makes the gradient descent quick, which is why we picked it in the first place.
Just like in Linear Regression we do:
[S:O:S][j] := [S:O:S][j] - a * sum i=1 to m ( (h[[S:o:S]](x^(i)) - y^(i)) * x[j]^(i) ) / m
We have already calculated our h[[S:o:S]](x^(i)) for all i as X[S:O:S] which is a vector. The error is then that value less our known values of y. ß[i ]= h[[S:o:S]](x^(i)) - y^(i) or in Matrix form ß
= X[S:O:S] - y which is still just a vector. In Octave:
err = (hyp .- y)
To multiply by x[j]^(i) and sum, understand that we are calculating sum(ß[i] x^(i)) where x^(i) is a vector (one set of inputs) and ß[i] is a scaler (one hypothesis less the actual y value for that
input). Multiplying each x^(i) vector times each ß[i] and summing the output, is the same as Matrix multiplying the transpose of X by the vector of scalers ß.
sum(ß[i] x^(i)) = [x^(1) x^(2) ... x^(m) ] * [ß[1] ; ß[2] ; ... ß[m] ] = X^Tß
Gradient Decent: Keeping in mind that we must simultaneously update all [S:O:S] and that h[[S:O:S]] is now a totally different equation based on the sigmoid function.
% Basic Logistic Regression via gradient decent with multiple parameters in Octave
alpha = 0.01; % try larger values 0.01, 0.03, 0.1, 0.3, etc...
m = length(y); % number of training examples
p = size(X,2); % number of parameters (second diminsion of X)
for iter = 1:num_iters %for some number of iterations
hyp = sigmoid(X*theta); %calculate our hypothesis using current parameters
err = (hyp .- y); %find the error between that and the real data
s = sum( err .* X )./m; %find the slope of the error.
%Note: This is the derivative of our cost function
theta = theta - alpha .* s;
%adjust our parameters by a small distance along that slope.
Not all data fits well to a straight line. This is called "underfitting" or we may say that the algorithm as a "high bias". We can try fitting a quadratic or even higher order equation. E.g. instead
of [S:O:S][0] + [S:O:S][1]x, we might use [S:O:S][0] + [S:O:S][1]x + [S:O:S][2]x^2. But, if we choose an equation of too high an order, then we might "overfit" or have an algorithm with "high
variance", which would fit any function and isn't representing the function behind this data. Overfitting can therefore result in predictions for new examples which are not accurate even though it
exactly predicts the data in the trianing set. The training data may well have some noise, or outliers, which are not actually representative of the true function.
If the data is in 2 or 3 features, it can be plotted and a human can decide if it is being over or under fit. But when there are many parameters, it can be impossible to plot. And using a human is
sort of against the purpose of Machine Learning. It may help to reduce the number of features if we can find features that don't really apply. Another means of reducing overfitting is regularization.
To keep the system from over fitting, and instead provide a more generalized fit, we can add the sum values of the theta parameters to the cost and slope of the error. Here is that new term added to
the right of our cost function:
Question: Shouldn't we use lower weight parameters (more regularization) for higher order terms?
Don't regularize [S:O:S][0].
Lambda is used as a parameter for the amount of regularization. e.g. the amount that the parameter values are multiplied by before adding them to the cost function. To large a lambda can result in
reg = lambda * sum(theta2.^2) / (2*m);
J = J + reg;
reg = lambda .* theta2 ./ m ;
S = S + reg;
Where theta2 is either:
theta2 = theta;
theta2(1) = 0;
theta2 = [0; theta(2:end)]
(the [0; and ] aren't needed for the cost calculation, only for the gradient / slope.
Find Minimum Function: fminunc( @[cost, slope] = cost(theta, X, y), theta, options )
But there are better (and more complex) means of adjusting the theta (parameter) values to minumize cost. fminuc is a common and powerful method. The fminunc function expects a reference to a
function which will return cost and (optionally) slope / gradient values for a given set of parameters, training data, and training answers. It starts from an initial set of parameters and there are
some options which can be set, such as the maximum number of iterations.
Here is a complete learning function in Octave using fminunc:
function [J, S] = costSlope(theta, X, y, lambda)
%return cost (J) and slope (S) given training data (X, y), current theta, and lambda
m = length(y);
hyp = sigmoid(X*theta);
%make a guess based on the sigmoid of our training data times our current paramaters.
costs = -y' * log(hyp) - (1-y)' * log(1-hyp); %costs with sigmoid function
%costs = -y .* log(hyp) - (1-y) .* log(1-hyp); %more general version?
J = sum(costs(:))/m; %mean cost. (:) required for higher dimensions
reg = (lambda * sum(theta(2:end).^2) / (2*m)); %mean cost + regularization
J = J + reg; %add in the regularization
err = (hyp .- y); %actual error.
%Note this happens to be the derivative of our cost function.
S = (X' * err)./m; %slope of the error
reg = lambda .* [0;theta(2:end)] ./ m; %Also regularize the slope
S = S + reg; %add in the regularization
options = optimset('GradObj', 'on', 'MaxIter', 400);
[theta, cost] = fminunc(@(t)(costSlope(t, X, y)), initial_theta, options);
For more about the @(t) () syntax see Octave: Anonymous Functions
Also: fmincg. This works similarly to fminunc, but is more efficient when we are dealing with large number of parameters.
It's very easy to turn this into a "one hot" logistic classifier
See also:
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Big Data Counting: How to count a billion distinct objects using only 1.5KB of Memory - High Scalability -
Big Data Counting: How to count a billion distinct objects using only 1.5KB of Memory
This is a guest post by Matt Abrams (@abramsm), from Clearspring, discussing how they are able to accurately estimate the cardinality of sets with billions of distinct elements using surprisingly
small data structures. Their servers receive well over 100 billion events per month.
At Clearspring we like to count things. Counting the number of distinct elements (the cardinality) of a set is challenge when the cardinality of the set is large.
To better understand the challenge of determining the cardinality of large sets let's imagine that you have a 16 character ID and you'd like to count the number of distinct IDs that you've seen
in your logs. Here is an example:
These 16 characters represent 128 bits. 65K IDs would require 1 megabyte of space. We receive over 3 billion events per day, and each event has an ID. Those IDs require 384,000,000,000 bits or 45
gigabytes of storage. And that is just the space that the ID field requires! To get the cardinality of IDs in our daily events we could take a simplistic approach. The most straightforward idea
is to use an in memory hash set that contains the unique list of IDs seen in the input files. Even if we assume that only 1 in 3 records are unique the hash set would still take 119 gigs of RAM,
not including the overhead Java requires to store objects in memory. You would need a machine with several hundred gigs of memory to count distinct elements this way and that is only to count a
single day's worth of unique IDs. The problem only gets more difficult if we want to count weeks or months of data. We certainly don't have a single machine with several hundred gigs of free
memory sitting around so we needed a better solution.
One common approach to this problem is the use of bitmaps. Bitmaps can be used to quickly and accurately get the cardinality of a given input. The basic idea with a bitmap is mapping the
input dataset to a bit field using a hash function where each input element uniquely maps to one of the bits in the field. This produces zero collisions, and reduces the space required to count
each unique element to 1 bit. While bitmaps drastically reduce the space requirements from the naive set implementation described above they are still problematic when the cardinality is very
high and/or you have a very large number of different sets to count. For example, if we want to count to one billion using a bitmap you will need one billion bits, or roughly 120 megabytes for
each counter. Sparse bitmaps can be compressed in order to gain space efficiency, but that is not always helpful.
Luckily, cardinality estimation is a popular area of research. We've leveraged this research to provide a open source implementation of cardinality estimators, set membership
detection, and top-k algorithms.
Cardinality estimation algorithms trade space for accuracy. To illustrate this point we counted the number of distinct words in all of Shakespeare's works using three different counting
techniques. Note that our input dataset has extra data in it so the cardinality is higher than the standard reference answer to this question. The three techniques we used were Java
HashSet, Linear Probabilistic Counter, and a Hyper LogLog Counter. Here are the results:
│ Counter │ Bytes Used │ Count │ Error │
│ HashSet │ 10447016 │ 67801 │ 0% │
│ Linear │ 3384 │ 67080 │ 1% │
│ HyperLogLog │ 512 │ 70002 │ 3% │
The table shows that we can count the words with a 3% error rate using only 512 bytes of space. Compare that to a perfect count using a HashMap that requires nearly 10 megabytes of space and you
can easily see why cardinality estimators are useful. In applications where accuracy is not paramount, which is true for most web scale and network counting scenarios, using a probabilistic
counter can result in tremendous space savings.
Linear Probabilistic Counter
The Linear Probabilistic Counter is space efficient and allows the implementer to specify the desired level of accuracy. This algorithm is useful when space efficiency is important but you need
to be able to control the error in your results. This algorithm works in a two-step process. The first step assigns a bitmap in memory initialized to all zeros. A hash function is then
applied to the each entry in the input data. The result of the hash function maps the entry to a bit in the bitmap, and that bit is set to 1. The second step the algorithm counts the number of
empty bits and uses that number as input to the following equation to get the estimate.
n=-m ln Vn
In the equation m is the size of the bitmap and Vn is the ratio of empty bits over the size of the map. The important thing to note is that the size of the original bitmap can be much smaller
than the expected max cardinality. How much smaller depends on how much error you can tolerate in the result. Because the size of the bitmap, m, is smaller than the total number of distinct
elements, there will be collisions. These collisions are required to be space-efficient but also result in the error found in the estimation. So by controlling the size of the original map we
can estimate the number of collisions and therefore the amount of error we will see in the end result.
Hyper LogLog
The Hyper LogLog Counter's name is self-descriptive. The name comes from the fact that you can estimate the cardinality of a set with cardinality Nmax using just loglog(Nmax) + O(1) bits. Like the
Linear Counter the Hyper LogLog counter allows the designer to specify the desired accuracy tolerances. In Hyper LogLog's case this is done by defining the desired relative standard deviation
and the max cardinality you expect to count. Most counters work by taking an input data stream, M, and applying a hash function to that set, h(M). This yields an observable result of S
= h(M) of {0,1}^∞ strings. Hyper LogLog extends this concept by splitting the hashed input stream into m substrings and then maintains m observables for each of the substreams. Taking the
average of the additional observables yields a counter whose accuracy improves as m grows in size but only requires a constant number of operations to be performed on each element of the input
set. The result is that, according to the authors of this paper, this counter can count one billion distinct items with an accuracy of 2% using only 1.5 kilobytes of space. Compare that to
the 120 megabytes required by the HashSet implementation and the efficiency of this algorithm becomes obvious.
Merging Distributed Counters
We've shown that using the counters described above we can estimate the cardinality of large sets. However, what can you do if your raw input dataset does not fit on single machine? This is
exactly the problem we face at Clearspring. Our data is spread out over hundreds of servers and each server contains only a partial subset of the the total dataset. This is where the fact that
we can merge the contents of a set of distributed counters is crucial. The idea is a little mind-bending but if you take a moment to think about it the concept is not that much different than
basic cardinality estimation. Because the counters represent the cardinality as set of bits in a map we can take two compatible counters and merge their bits into a single map. The algorithms
already handle collisions so we can still get a cardinality estimation with the desired precision even though we never brought all of the input data to a single machine. This is terribly useful
and saves us a lot of time and effort moving data around our network.
Next Steps
Hopefully this post has helped you better understand the concept and application of probabilistic counters. If estimating the cardinality of large sets is a problem and you happen to use a JVM
based language then you should check out the stream-lib project — it provides implementations of the algorithms described above as well as several other stream-processing utilities.
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Static Stiffness Models
Constant Stiffness Model
The bushing stiffness properties are approximated by a single coefficient–the stiffness at the operating point. The force generated by the bushing is:
$F=-k\ast x$
k is the stiffness.
x is the deflection.
Cubic Stiffness Model
The bushing stiffness is approximated by two cubic polynomials that are derived from the Static Force versus Deflection curve. Below, the measured static data is shown as a blue curve:
Figure 1.
The five points in the selected area of the plot above are:
│ Point │ Description │ Location on Plot │
│ O │ Operating point. │ The force value, OF, and the slope of the static curve, OS, are selected. │
│ E[p] │ End point for positive deformation. │ This is usually the maximum positive deformation in the static test. At E[P], the slope of the static curve, E[P]S, is selected. │
│ R[p] │ Reference point for positive deformation. │ As a default, R[P] = (O + E[P])/2. At R[P], the force of the static curve, R[P]F, is selected. │
│ E[N] │ End point for negative deformation. │ This is usually the maximum negative deformation in the static test. At E[N], the slope of the static curve, E[N]S, is selected. │
│ R[N] │ Reference point for negative deformation. │ As a default, R[N] = (O + E[N])/2. At R[N], the force of the static curve, R[N]F, is selected. │
Spline Stiffness Model
Spline data is derived by reducing the static data to a curve. A cubic spline is fitted through the measured static data. The spline is then used as the interpolating function for calculating the
force at any deflection. | {"url":"https://2021.help.altair.com/2021.1/hwdesktop/altair_help/topics/tools/bushing_types_static_stiffness_models_r.htm","timestamp":"2024-11-12T09:36:39Z","content_type":"application/xhtml+xml","content_length":"57482","record_id":"<urn:uuid:79c7c7ac-83d9-4bf3-97aa-7d2b7a71c353>","cc-path":"CC-MAIN-2024-46/segments/1730477028249.89/warc/CC-MAIN-20241112081532-20241112111532-00769.warc.gz"} |
How to Calculate Marginal Cost: A Comprehensive Guide
Marginal cost is a key concept in economics and business decision-making, representing the additional cost incurred when producing one more unit of a good or service. It plays a significant role in
determining pricing strategies, production efficiency, and profitability. Understanding how to calculate marginal cost is essential for businesses looking to optimize their production processes and
maintain competitive pricing.
This article will explain the concept of marginal cost, outline the steps for calculating it, and explore its significance in various industries and business operations. We’ll also look at real-world
applications and examples to help clarify how businesses can use marginal cost analysis to improve their decision-making processes.
What Is Marginal Cost?
Marginal cost refers to the increase or decrease in total production costs resulting from producing one additional unit of output. It is a variable cost because it changes depending on the level of
production. Marginal cost helps businesses determine the optimal level of production where profits are maximized, without incurring unnecessary expenses.
The formula for calculating marginal cost is simple:
Marginal Cost (MC)=ΔTotal Cost (TC)ΔQuantity (Q)\text{Marginal Cost (MC)} = \frac{\Delta \text{Total Cost (TC)}}{\Delta \text{Quantity (Q)}}Where:
• ΔTotal Cost\Delta \text{Total Cost} is the change in total production cost.
• ΔQuantity\Delta \text{Quantity} is the change in the number of goods produced (usually 1 unit).
Marginal cost plays a critical role in deciding whether to increase production, adjust pricing, or halt production when costs exceed profits.
Steps to Calculate Marginal Cost
1. Determine the Total Cost
The first step in calculating marginal cost is identifying the total cost of producing a certain number of units. Total cost includes both fixed costs and variable costs:
• Fixed Costs: These are costs that do not change with the level of production, such as rent, salaries, and equipment depreciation.
• Variable Costs: These costs fluctuate with production levels, such as raw materials, labor, and energy consumption.
For example, if a company is producing 100 units and the total cost is $10,000, this includes both fixed and variable costs at that production level.
2. Identify the Change in Total Cost
To find the marginal cost, you need to calculate how much the total cost increases when production increases by one more unit. If the total cost of producing 100 units is $10,000 and the total cost
of producing 101 units is $10,100, then the change in total cost ( ΔTotal Cost\Delta \text{Total Cost}) is $100.
ΔTotal Cost=Total Cost of 101 Units−Total Cost of 100 Units=10,100−10,000=100\Delta \text{Total Cost} = \text{Total Cost of 101 Units} – \text{Total Cost of 100 Units} = 10,100 – 10,000 = 100
3. Measure the Change in Quantity
Typically, marginal cost is calculated based on the production of one additional unit. Therefore, the change in quantity ( ΔQuantity\Delta \text{Quantity}) is 1. In this case, the change in quantity
is from 100 units to 101 units, meaning ΔQ=1\Delta Q = 1.
4. Calculate Marginal Cost
Now that we have both the change in total cost and the change in quantity, we can plug these values into the marginal cost formula.
MC=1001=100MC = \frac{100}{1} = 100In this example, the marginal cost of producing one additional unit is $100.
Significance of Marginal Cost
Marginal cost is an essential metric for businesses aiming to optimize production efficiency and profitability. Here are some ways that marginal cost can be useful:
1. Optimal Production Level
Marginal cost analysis helps businesses determine the optimal level of production. As long as the marginal cost of producing one more unit is less than or equal to the revenue generated from selling
that unit (known as marginal revenue), a company can continue to increase production and remain profitable. When marginal cost exceeds marginal revenue, it’s time to halt or reduce production.
2. Pricing Decisions
Marginal cost is crucial for setting competitive prices. If a company knows its marginal cost, it can set prices that cover costs while remaining attractive to customers. For example, a company
should not price its product below its marginal cost, as this would result in losses.
3. Cost Control
Understanding the marginal cost helps companies identify inefficiencies in their production process. If marginal cost is rising, it may indicate that a company needs to address inefficiencies, such
as labor shortages, increased material costs, or outdated production methods.
4. Economies of Scale
Marginal cost often decreases as production levels increase, due to economies of scale. When a company increases production, it can spread its fixed costs over a larger number of units, thus reducing
the marginal cost. Conversely, if a company experiences diseconomies of scale (increased costs due to inefficiencies in managing larger production levels), marginal cost will rise.
5. Business Expansion Decisions
Marginal cost plays a role in expansion decisions, helping businesses determine whether scaling up production is financially viable. For example, a company may use marginal cost analysis to decide
whether to open a new factory or invest in more efficient production equipment.
Real-World Example: Marginal Cost in Manufacturing
Let’s consider a real-world example to illustrate how marginal cost works in practice. Suppose a car manufacturer is producing 1,000 cars, and the total cost is $20 million. To produce one additional
car (the 1,001st car), the total cost increases to $20.01 million.
In this case:
\Delta \text{Total Cost} = 20.01 \text{ million} – 20 \text{ million} = 0.01 \text{ million} \text{ (or $10,000)} ΔQuantity=1 car\Delta \text{Quantity} = 1 \text{ car}Therefore, the marginal cost of
producing one additional car is $10,000.
This information allows the manufacturer to decide whether it makes sense to continue producing additional cars. If the company can sell each car for $30,000, producing more cars is profitable, as
the marginal revenue exceeds the marginal cost. However, if the market price for cars drops below $10,000, continuing production would lead to losses, and the company should consider reducing output.
Marginal Cost in Service Industries
In service industries, such as airlines or telecommunications, marginal cost can behave differently compared to manufacturing. Many service industries have high fixed costs (e.g., purchasing
airplanes or building network infrastructure) but low variable costs for serving additional customers. This often results in a low marginal cost once fixed costs are covered.
For example, an airline incurs significant costs for purchasing planes and fuel, but the cost of serving an additional passenger on an already-scheduled flight is relatively low. Understanding this
marginal cost helps airlines optimize ticket pricing to fill seats without underpricing or overbooking.
Challenges of Marginal Cost Calculation
While the concept of marginal cost is straightforward, there are challenges in calculating it accurately. Changes in variable costs, such as fluctuating raw material prices or labor costs, can make
it difficult to estimate marginal cost precisely. Additionally, certain industries, like technology, may face non-linear marginal costs, where the cost of producing each additional unit may change
unpredictably as production scales.
In some cases, businesses may need to rely on marginal cost forecasting tools that consider future costs and market trends to make more accurate calculations.
Calculating marginal cost is crucial for making informed business decisions, from pricing products to optimizing production levels. By understanding the relationship between total costs, variable
costs, and output, companies can make strategic choices that lead to higher profitability and efficiency. Whether in manufacturing or service industries, marginal cost analysis is an invaluable tool
for ensuring long-term success.
Knowing when to increase production, cut back, or adjust pricing strategies all hinges on a clear understanding of marginal cost. With this knowledge in hand, businesses can better navigate market
fluctuations, improve operational efficiency, and maximize their competitive advantage. | {"url":"https://cheerfulzone.com/how-to-calculate-marginal-cost-a-comprehensive-guide/","timestamp":"2024-11-03T10:14:03Z","content_type":"text/html","content_length":"91876","record_id":"<urn:uuid:509f932e-e371-4af9-841d-024e3f0b1cd2>","cc-path":"CC-MAIN-2024-46/segments/1730477027774.6/warc/CC-MAIN-20241103083929-20241103113929-00792.warc.gz"} |
Automating HRV analysis: RHRVEasy
RHRV Team
RHRVEasy automates all steps of a Heart Rate Variability (HRV) analysis, including data processing, indices calculation, and statistical analysis. It takes as input a list of folders, each containing
the recordings of a same population. It calculates time, frequency, and nonlinear domain HRV indices, and then it applies hypothesis test, and corrects the significance levels. If there are more than
two experimental groups and statistically significant differences are found, it performs a post-hoc analysis to find out which groups have the differences.
0. Set up required to run this tutorial
This tutorial uses the recordings of the Normal Sinus Rhythm RR Interval Database (hereinafter referred to as NSR_DB), a subset of the RR interval time series from healthy subjects (referred to as
HEALTHY_DB), and the Congestive Heart Failure RR Interval Database (referred to as CHF_DB). The former two databases comprise data from healthy individuals, while the latter consists of recordings
from patients with severe cardiac pathology. Consequently, significant disparities in numerous HRV indices are anticipated between the healthy databases and the CHF_DB.
The three databases are available in the GitHub repository for the book “Heart Rate Variability Analysis with the R package RHRV”, under the data/Chapter8 folder, within the data/Chapter8 directory.
To execute this tutorial, download this folder to your local machine and define the following variables:
basePath <- "book_data" # adjust as needed
NSR_DB <- file.path(basePath, "normal")
CHF_DB <- file.path(basePath, "chf")
HEALTHY_DB <- file.path(basePath, "healthy")
RHRVEasy permits creating an Excel spreadsheet with all the HRV indices calculated for each recording. The following variable must contain the folder on the local machine where the Excel spreadsheet
is to be saved:
1. Time and frequency analysis
RHRVEasy enables the user to carry out a full HRV analysis by just invoking a function with a single mandatory parameter: a list with the folders containing the recordings of the experimental groups.
This list must have at least two folders. Each folder must contain all the RR recordings of the same experimental group and no additional files, as RHRVEasy will try to open all the files within
these folders. The name that will be used to refer to each experimental group within RHRVEasy will be the name of the folder in which its recordings are located.
The following function call computes the time and frequency indices for the NSR_DB and CHF_DB databases, and performs a statistical comparison of each index correcting the significance level with the
Bonferroni method. Note the use of the nJobs to use several cores and parallelize the computations. With nJobs = -1, it uses all available cores; if an integer greater than 0 is indicated, it uses
the number of cores indicated by the integer.
When the returned object is displayed in the console, it shows which indices present statistically significant differences:
## Significant differences in SDNN (Kruskal-Wallis rank sum test, bonferroni p-value = 1.117154e-07):
## chf's mean95% CI: (61.91503, 94.0085) [Bootstrap CI without adjustment]
## normal's mean95% CI: (131.1187, 148.1985) [Bootstrap CI without adjustment]
## Significant differences in SDANN (Kruskal-Wallis rank sum test, bonferroni p-value = 3.799696e-07):
## chf's mean95% CI: (48.19527, 80.0444) [Bootstrap CI without adjustment]
## normal's mean95% CI: (122.0759, 139.05) [Bootstrap CI without adjustment]
## Significant differences in SDNNIDX (Kruskal-Wallis rank sum test, bonferroni p-value = 0.01426098):
## chf's mean95% CI: (29.96821, 47.6446) [Bootstrap CI without adjustment]
## normal's mean95% CI: (47.0144, 54.5201) [Bootstrap CI without adjustment]
## Significant differences in IRRR (Kruskal-Wallis rank sum test, bonferroni p-value = 1.492754e-07):
## chf's mean95% CI: (78.67064, 124.1918) [Bootstrap CI without adjustment]
## normal's mean95% CI: (189.5291, 215.7118) [Bootstrap CI without adjustment]
## Significant differences in TINN (Kruskal-Wallis rank sum test, bonferroni p-value = 1.452872e-06):
## chf's mean95% CI: (243.1949, 373.8965) [Bootstrap CI without adjustment]
## normal's mean95% CI: (511.0544, 586.6332) [Bootstrap CI without adjustment]
## Significant differences in HRVi (Kruskal-Wallis rank sum test, bonferroni p-value = 1.452872e-06):
## chf's mean95% CI: (15.96148, 23.78737) [Bootstrap CI without adjustment]
## normal's mean95% CI: (32.80169, 37.58583) [Bootstrap CI without adjustment]
## Significant differences in ULF (Kruskal-Wallis rank sum test, bonferroni p-value = 1.74099e-08):
## chf's mean95% CI: (1182.117, 4410.562) [Bootstrap CI without adjustment]
## normal's mean95% CI: (7215.618, 9824.658) [Bootstrap CI without adjustment]
## Significant differences in VLF (Kruskal-Wallis rank sum test, bonferroni p-value = 0.002535127):
## chf's mean95% CI: (52.21509, 135.5065) [Bootstrap CI without adjustment]
## normal's mean95% CI: (131.5723, 175.2834) [Bootstrap CI without adjustment]
All computed indices, as well as all p-values resulting from all comparisons, are stored in data.frames contained in the object. Two different sets of p-values are available; the ones obtained before
(p.value) and after (adj.p.value) applying the significance level correction:
## # A tibble: 6 × 16
## file group SDNN SDANN SDNNIDX pNN50 SDSD rMSSD IRRR MADRR TINN HRVi
## <chr> <fct> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
## 1 chf201_rr… chf 75.5 52.9 49.6 2.03 20.2 20.2 93.8 7.81 358. 22.9
## 2 chf202_rr… chf 88.5 75.8 39.6 6.13 34.7 34.7 117. 15.6 350. 22.4
## 3 chf203_rr… chf 38.8 30.9 21.7 1.20 17.3 17.3 46.9 7.81 170. 10.9
## 4 chf204_rr… chf 55.1 39.1 36.0 4.84 33.0 33.0 70.3 7.81 237. 15.2
## 5 chf205_rr… chf 34.9 26.1 19.5 1.97 23.7 23.7 46.9 7.81 169. 10.8
## 6 chf206_rr… chf 41.2 34.9 14.8 2.02 18.9 18.9 31.2 7.81 122. 7.79
## # ℹ 4 more variables: ULF <dbl>, VLF <dbl>, LF <dbl>, HF <dbl>
## # A tibble: 6 × 4
## HRVIndex method p.value adj.p.value
## <chr> <chr> <dbl> <dbl>
## 1 SDNN Kruskal-Wallis rank sum test 0.00000000798 0.000000112
## 2 SDANN Kruskal-Wallis rank sum test 0.0000000271 0.000000380
## 3 SDNNIDX Kruskal-Wallis rank sum test 0.00102 0.0143
## 4 pNN50 Kruskal-Wallis rank sum test 0.774 1
## 5 SDSD Kruskal-Wallis rank sum test 0.0891 1
## 6 rMSSD Kruskal-Wallis rank sum test 0.0891 1
The format parameter specifies the format in which the RR intervals are stored. All formats supported by the RHRV package can be used: WFDB, ASCII, RR, Polar, Suunto, EDFPlus or Ambit (check the RHRV
website for more information). The default format is RR, where the beat distances in seconds are stored in a single column of an ASCII file. This is the format of the three databases used in this
By default, the frequency analysis is performed using the Fourier transform. It is also possible to use the Wavelet transform pasing the value 'wavelet' to the typeAnalysis parameter (check the paper
“García, C. A., Otero, A., Vila, X., & Márquez, D. G. (2013). A new algorithm for wavelet-based heart rate variability analysis. Biomedical Signal Processing and Control, 8(6), 542-550” for details):
easyAnalysisWavelet <- RHRVEasy(
folders = c(NSR_DB, CHF_DB),
typeAnalysis = 'wavelet',
n_jobs = -1
Note that the significant indices are the same as the previous ones.
2. Correction of the significance level
Given that multiple statistical tests are performed on several HRV indices, a correction of the significance level should be applied. The Bonferroni method is used by default. This behavior can be
overridden with the parameter correctionMethod of RHRVEasy. The possible values of this parameter besides bonferroni are holm, hochberg, hommel, BH (Benjamini & Hochberg), fdr (false discovery rate),
BY (Benjamini & Yekutieli), and none (indicating that no correction is to be made). Furthermore, there is no need to recompute the HRV indices to apply a different correction method, but the
RHRVEasyStats function can be used to this end. The confidence level can also be changed using the significance parameter (in both RHRVEasy and RHRVEasyStats functions).
easyAnalysisFDR <- RHRVEasyStats(easyAnalysis, correctionMethod = 'fdr')
pValues <- merge(
by = setdiff(names(easyAnalysis$stats), "adj.p.value"),
suffixes = c(".bonf", ".fdr")
#Let us compare the p-values obtained with different correction methods
pValues[, c("HRVIndex", "p.value", "adj.p.value.bonf", "adj.p.value.fdr")]
## HRVIndex p.value adj.p.value.bonf adj.p.value.fdr
## 1 HF 5.601495e-01 1.000000e+00 6.032380e-01
## 2 HRVi 1.037766e-07 1.452872e-06 2.421454e-07
## 3 IRRR 1.066253e-08 1.492754e-07 4.975847e-08
## 4 LF 1.651479e-02 2.312071e-01 2.568968e-02
## 5 MADRR 6.319903e-02 8.847864e-01 8.847864e-02
## 6 pNN50 7.744691e-01 1.000000e+00 7.744691e-01
3. Saving the indices to an Excel spreadsheet
If the argument saveHRVindicesInPath is specified when invoking the function RHRVEasy, an Excel spreadsheet with all the HRV indices calculated for each recording will be created in the path
specified by this parameter. The name of the spreadsheet generated is “<group 1 name>Vs<group 2 name> .xlsx”:
This spreadsheet can also be generated from the object returned by RHRVEasy by calling the function SaveHRVIndices.
4. Comparing more than two experimental groups
If the analysis involves three or more groups, when statistically significant differences are found among them it does not necessarily mean that there are statistically significant differences
between all pairs of groups. In such a scenario post-hoc tests are used to find which pairs of groups present differences:
#Comparison of the three databases
easyAnalysis3 <- RHRVEasy(
folders = c(NSR_DB, CHF_DB, HEALTHY_DB),
nJobs = -1
## Significant differences in SDNN (Kruskal-Wallis rank sum test, bonferroni p-value = 3.543622e-07):
## Significant differences in the post-hoc tests (Dunn's all-pairs test + bonferroni-p-value adjustment):
## group1 group2 adj.p.value
## 1 healthy chf 0.00799
## 2 normal chf 0.000000282
## ----------------------------
## chf's mean95% CI: (63.20538, 92.2515) [Bootstrap CI without adjustment]
## healthy's mean95% CI: (123.242, 158.269) [Bootstrap CI without adjustment]
## normal's mean95% CI: (131.665, 147.9961) [Bootstrap CI without adjustment]
## Significant differences in SDANN (Kruskal-Wallis rank sum test, bonferroni p-value = 1.345688e-06):
## Significant differences in the post-hoc tests (Dunn's all-pairs test + bonferroni-p-value adjustment):
## group1 group2 adj.p.value
## 1 normal chf 0.000000403
## ---------------------------
## chf's mean95% CI: (47.61222, 81.42191) [Bootstrap CI without adjustment]
## healthy's mean95% CI: (105.1872, 134.0331) [Bootstrap CI without adjustment]
## normal's mean95% CI: (120.4753, 138.5329) [Bootstrap CI without adjustment]
## Significant differences in SDNNIDX (Kruskal-Wallis rank sum test, bonferroni p-value = 0.001063849):
## Significant differences in the post-hoc tests (Dunn's all-pairs test + bonferroni-p-value adjustment):
## group1 group2 adj.p.value
## 1 healthy chf 0.00111
## ----------------------------
## chf's mean95% CI: (29.1345, 47.73994) [Bootstrap CI without adjustment]
## healthy's mean95% CI: (56.23389, 74.9991) [Bootstrap CI without adjustment]
## normal's mean95% CI: (47.0101, 54.33106) [Bootstrap CI without adjustment]
## Significant differences in IRRR (Kruskal-Wallis rank sum test, bonferroni p-value = 3.688167e-07):
## Significant differences in the post-hoc tests (Dunn's all-pairs test + bonferroni-p-value adjustment):
## group1 group2 adj.p.value
## 1 healthy chf 0.00395
## 2 normal chf 0.000000425
## ----------------------------
## chf's mean95% CI: (77.3305, 124.7238) [Bootstrap CI without adjustment]
## healthy's mean95% CI: (179.9086, 234.5556) [Bootstrap CI without adjustment]
## normal's mean95% CI: (187.6484, 215.9975) [Bootstrap CI without adjustment]
## Significant differences in MADRR (Kruskal-Wallis rank sum test, bonferroni p-value = 0.006224158):
## Significant differences in the post-hoc tests (Dunn's all-pairs test + bonferroni-p-value adjustment):
## group1 group2 adj.p.value
## 1 healthy chf 0.00237
## ----------------------------
## chf's mean95% CI: (8.62069, 11.85345) [Bootstrap CI without adjustment]
## healthy's mean95% CI: (16.55556, 24.66667) [Bootstrap CI without adjustment]
## normal's mean95% CI: (11.28472, 14.03356) [Bootstrap CI without adjustment]
## Significant differences in TINN (Kruskal-Wallis rank sum test, bonferroni p-value = 1.350844e-06):
## Significant differences in the post-hoc tests (Dunn's all-pairs test + bonferroni-p-value adjustment):
## group1 group2 adj.p.value
## 1 healthy chf 0.000933
## 2 normal chf 0.00000519
## ----------------------------
## chf's mean95% CI: (244.0477, 371.3618) [Bootstrap CI without adjustment]
## healthy's mean95% CI: (533.6798, 701.4795) [Bootstrap CI without adjustment]
## normal's mean95% CI: (511.6379, 586.4394) [Bootstrap CI without adjustment]
## Significant differences in HRVi (Kruskal-Wallis rank sum test, bonferroni p-value = 1.350844e-06):
## Significant differences in the post-hoc tests (Dunn's all-pairs test + bonferroni-p-value adjustment):
## group1 group2 adj.p.value
## 1 healthy chf 0.000933
## 2 normal chf 0.00000519
## ----------------------------
## chf's mean95% CI: (15.85798, 23.7487) [Bootstrap CI without adjustment]
## healthy's mean95% CI: (34.45, 45.19331) [Bootstrap CI without adjustment]
## normal's mean95% CI: (32.68737, 37.61479) [Bootstrap CI without adjustment]
## Significant differences in ULF (Kruskal-Wallis rank sum test, bonferroni p-value = 5.860632e-08):
## Significant differences in the post-hoc tests (Dunn's all-pairs test + bonferroni-p-value adjustment):
## group1 group2 adj.p.value
## 1 normal chf 0.0000000162
## ----------------------------
## chf's mean95% CI: (1075.296, 4358.885) [Bootstrap CI without adjustment]
## healthy's mean95% CI: (4995.594, 8167.694) [Bootstrap CI without adjustment]
## normal's mean95% CI: (7063.468, 9898.164) [Bootstrap CI without adjustment]
## Significant differences in VLF (Kruskal-Wallis rank sum test, bonferroni p-value = 0.0005669878):
## Significant differences in the post-hoc tests (Dunn's all-pairs test + bonferroni-p-value adjustment):
## group1 group2 adj.p.value
## 1 healthy chf 0.00239
## 2 normal chf 0.00977
## ----------------------------
## chf's mean95% CI: (54.04686, 134.9712) [Bootstrap CI without adjustment]
## healthy's mean95% CI: (171.6335, 340.8925) [Bootstrap CI without adjustment]
## normal's mean95% CI: (130.0847, 177.0061) [Bootstrap CI without adjustment]
Note that the stats data.frame now contains a column named pairwise storing the results of the post-hoc analysis for those indices where the omnibus test has been significant:
## # A tibble: 6 × 5
## HRVIndex method p.value adj.p.value pairwise
## <chr> <chr> <dbl> <dbl> <list>
## 1 SDNN Kruskal-Wallis rank sum test 0.0000000253 0.000000354 <tibble>
## 2 SDANN Kruskal-Wallis rank sum test 0.0000000961 0.00000135 <tibble>
## 3 SDNNIDX Kruskal-Wallis rank sum test 0.0000760 0.00106 <tibble>
## 4 pNN50 Kruskal-Wallis rank sum test 0.0186 0.260 <NULL>
## 5 SDSD Kruskal-Wallis rank sum test 0.0301 0.421 <NULL>
## 6 rMSSD Kruskal-Wallis rank sum test 0.0301 0.421 <NULL>
## # A tibble: 3 × 6
## HRVIndex group1 group2 method p.value adj.p.value
## <chr> <chr> <chr> <chr> <dbl> <dbl>
## 1 SDNN healthy chf Dunn's all-pairs test 0.000296 0.00799
## 2 SDNN normal chf Dunn's all-pairs test 0.0000000104 0.000000282
## 3 SDNN normal healthy Dunn's all-pairs test 0.861 1
5. Overwriting default parameters
Any parameter of any RHRV function can be specified as an additional parameter of the RHRVEasy function; in this case, the default value used for that parameter will be overwritten by the one
specified for the user. The default values used in the RHRVEasy package are the same as those used in the RHRV package. For more information about the parameters available you can consult the RHRV
website. For example, the following analysis modifies the the limits of the ULF, VLF, LF and HF spectral bands, and uses an interpolation frequency (freqhr) of 2 Hz:
6. Nonlinear analysis
The calculation of the nonlinear indices requires considerable computational resources, specially the Recurrence Quantification Analysis (RQA). Whereas in a typical HRV analysis the computation of
all the time and frequency domain indices for a few dozens of recordings often completes within a few minutes, the computation of the nonlinear indices could last many hours. That’s why the boolean
parameters nonLinear and doRQA are set to FALSE by default. If these parameters are not changed, only time and frequency indices will be calculated, as in the previous sections.
Warning: the following sentence, will take several hours to execute on a medium to high performance PC. You may reproduce the results of the paper by running this chunk of code. | {"url":"https://cran.wustl.edu/web/packages/RHRV/vignettes/RHRVEasy.html","timestamp":"2024-11-12T03:15:15Z","content_type":"text/html","content_length":"38570","record_id":"<urn:uuid:7440cbc4-6437-41af-b89b-093d3d57c06d>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.50/warc/CC-MAIN-20241112014152-20241112044152-00193.warc.gz"} |
Hierarchical Optimal Transport for Document Representation
Paper abstract bibtex
The ability to measure similarity between documents enables intelligent summarization and analysis of large corpora. Past distances between documents suffer from either an inability to incorporate
semantic similarities between words or from scalability issues. As an alternative, we introduce hierarchical optimal transport as a meta-distance between documents, where documents are modeled as
distributions over topics, which themselves are modeled as distributions over words. We then solve an optimal transport problem on the smaller topic space to compute a similarity score. We give
conditions on the topics under which this construction defines a distance, and we relate it to the word mover's distance. We evaluate our technique for \$k\$-NN classification and show better
interpretability and scalability with comparable performance to current methods at a fraction of the cost.
archivePrefix = {arXiv},
eprinttype = {arxiv},
eprint = {1906.10827},
primaryClass = {cs, stat},
title = {Hierarchical {{Optimal Transport}} for {{Document Representation}}},
url = {http://arxiv.org/abs/1906.10827},
abstract = {The ability to measure similarity between documents enables intelligent summarization and analysis of large corpora. Past distances between documents suffer from either an inability to incorporate semantic similarities between words or from scalability issues. As an alternative, we introduce hierarchical optimal transport as a meta-distance between documents, where documents are modeled as distributions over topics, which themselves are modeled as distributions over words. We then solve an optimal transport problem on the smaller topic space to compute a similarity score. We give conditions on the topics under which this construction defines a distance, and we relate it to the word mover's distance. We evaluate our technique for \$k\$-NN classification and show better interpretability and scalability with comparable performance to current methods at a fraction of the cost.},
urldate = {2019-06-28},
date = {2019-06-25},
keywords = {Statistics - Machine Learning,Computer Science - Computation and Language,Computer Science - Machine Learning,Computer Science - Information Retrieval},
author = {Yurochkin, Mikhail and Claici, Sebastian and Chien, Edward and Mirzazadeh, Farzaneh and Solomon, Justin},
file = {/home/dimitri/Nextcloud/Zotero/storage/EJGKCIUG/Yurochkin et al. - 2019 - Hierarchical Optimal Transport for Document Repres.pdf;/home/dimitri/Nextcloud/Zotero/storage/EC9XIVU7/1906.html} | {"url":"https://bibbase.org/network/publication/yurochkin-claici-chien-mirzazadeh-solomon-hierarchicaloptimaltransportfordocumentrepresentation","timestamp":"2024-11-04T19:57:54Z","content_type":"text/html","content_length":"16672","record_id":"<urn:uuid:b7cb07d8-fc27-4f01-8531-aa7ad26b2c5d>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.16/warc/CC-MAIN-20241104194528-20241104224528-00309.warc.gz"} |
Fourier Analysis by Peter Woit | Download book PDF
Fourier analysis and distribution theory by Pu Zhao Kow
This PDF covers the following topics related to Fourier Analysis : Fourier series, Weak derivatives, 1-dimensional Fourier series, n-dimensional Fourier series, Pointwise convergence and
Gibbs-Wilbraham phenomenon,Absolute convergence and uniform convergence, Pointwise convergence: Dini's criterion,. Cesàro summability of Fourier series, Fourier transform, Motivations, Schwartz
space, Fourier transform on Schwartz space, The space of tempered distributions,The space of compactly supported distributions, Convolution of functions, Tensor products, Convolution of
distributions, Convolution between distributions and functions, Convolution of distributions with non-compact supports, etc.
Author(s): Pu-Zhao Kow, Department of Mathematics and Statistics, University of Jyvaskyla, Finland
67 Pages | {"url":"https://www.freebookcentre.net/maths-books-download/Fourier-Analysis-by-Peter-Woit.html","timestamp":"2024-11-08T18:32:12Z","content_type":"text/html","content_length":"37350","record_id":"<urn:uuid:36d7d220-7ac4-4279-a33c-62282915fc8d>","cc-path":"CC-MAIN-2024-46/segments/1730477028070.17/warc/CC-MAIN-20241108164844-20241108194844-00874.warc.gz"} |
A string is attached to the rim of a small hoop of radius r =
A string is attached to the rim of a small hoop of radius r = 8.00×10−2...
A string is attached to the rim of a small hoop of radius r = 8.00×10^−2 m and mass m = 0.180 kg and then wrapped several times around the rim. If the free end of the string is held in place and the
hoop is released from rest and allowed to drop, as shown in the figure (Figure 1), calculate the angular speed and the translational speed of the rotating hoop after it has descended h = 0.750 m .
Use g = 9.80 m/s2 for the acceleration due to gravity.
1. What is the translational speed of the rotating hoop after it has descended a distance 0.750 m ?
Express your answer in meters per second. | {"url":"https://justaaa.com/physics/1294164-a-string-is-attached-to-the-rim-a-small-hoop","timestamp":"2024-11-11T08:12:45Z","content_type":"text/html","content_length":"40681","record_id":"<urn:uuid:4e48c90f-365d-412f-8109-0ae87eac708e>","cc-path":"CC-MAIN-2024-46/segments/1730477028220.42/warc/CC-MAIN-20241111060327-20241111090327-00690.warc.gz"} |
Simulation Analysis on Applicability of Meta Material and PBG Based mm-Wave Planar Antenna for Advanced Cellular Technologies
Simulation Analysis on Applicability of Meta Material and PBG Based mm-Wave Planar Antenna for Advanced Cellular Technologies ()
1. Introduction
Growing demand of bandwidth has increased an interest in antenna fields to fit into new era of wireless communication. From large size antennas to very thin and slick antennas, they have been
developed according to need and users require- ment. Basically, patch antennas played an important role in communication through wireless connections [1] . Not only the variations in parameters of
patch antennas have been significant but also characteristics of antenna miniaturization have been developed and planted to improve the radiation properties. One technique, used to reduce the size of
antenna is metamaterials based antennas.
A metamaterial is a kind of artificial synthetic composite material with a specific type of shape and structure, which exhibits properties that are not found in natural materials [2] . Basically, a
metamaterial is designed on the basis of periodic structure to get novel electromagnetic properties, e.g. negative permittivity or permeability, zero refractive index, and huge chirality [3] [4] .
Like all other common natural materials, the properties of metamaterials are decided by their components and their critical arrangements. In order to achieve specific properties, the components
should be designed with specific patterns (periodic), yielding resonant structures. These components, sometimes called meta-atoms or metamolecules, which are periodically arranged in one, two, or
three dimensions and all periodic patterns, can be coupled with each other which considerably modify the properties of the metamaterial.
Metamaterials have been widely used in design of microwave devices and antennas. It is flexibility bound to use it for huge number of new devices and antennas. Fabrication and novel performance are
being used everywhere and made a vast change to design a smart and advanced antenna for new generation of cel- lular technology. From radio frequencies to optical frequencies, metamaterials have been
designed and realized for different functions, such as negative refractive index, huge chirality, anisotropy and bianisotropy [3] [4] . From the frequen- cy point of view, metamaterials are
classified into several categories such as microwave metamaterials, terahetrz metamaterials, and photonic metamaterials. Because of specific arrangement, like 1D, 2D and 3D meramaterials have specia-
lity to enhance all the properties of structures. On the basis of microwave fre- quencies, there are several important types of metamaterials which are being used in today’s era. Veselago media [5]
[6] , first kind of metamaterials, where real parts of the permittivity and permeability have positive values, is named as DPS (double positive) in upper right corner and DNG (double negative) is
mentioned in lower left corner, where both the permittivity and permeability have negative refractive indexes. Second kind of metamaterials is 2D metamaterials named as high impedance plane, which
consists of periodic structure of resonant unit cells [7] [8] . Third type of metamaterials is based on chirality which basica- lly describes an object; especially a molecule that produces a
non-superimposible mirror image of itself [9] and due to this, the property changes the polarization of an incident electromagnetic wave. This can be found in 2D as well as 3D structures according to
usability. Others metamaterials are defined with novel characteristics, (different from others which are mentioned above), e.g. high ani-sotropy [10] , large nonlinearity and high dispersion.
Metamaterials having high dispersion and nonlinearity fall into category of electromagnetic band gap arrangement, whose periodicity can be change less than calculated wavelength of given resonant
frequency. Such structures are different from LH (left handed) metamaterials and widely known as anisotropic or none resonating or PBG (photonic band gap) structures [11] [12] . Several works have
been performed based on PBG structures and are undergoing as to get more and more bandwidth at high frequency. PBG structures are basically the arrangement of crystals (especially semiconductor) to
control the propagation of electromagnetic waves [11] [12] . Electrons behave like a wave for periodic struc- tures according to the quantum mechanics. Similar to LHM, periodic structure that can
influence on the electro magentic waves was given different names: photonic crystals (PC), photonoc band gap (PBG), electromagnetic band gap (EBG), microwave band gap (MBG), or simply periodic
structure. It is also found in 1D, 2D and 3D according to the usage. The most important part of PBG structures are defect [13] , which disturbing the periodicity of periodic structures. On account of
this, the propagation of electromagnetic waves, pass through the resonant cavity, where defect treated as a cavity. It forms free frequency mode inside the forbidden band-gap during transmission.
PBGs can be used for any frequency range, start from radio frequency (RF) to X-rays. Mostly, it is being used in optics and microwaves and they have the most applicable results, but with specific
problems according to the nature of the medium and its interaction with electromagnetic waves.
Microwave PBGs have their own specifics that are different from optics. First, the longer wavelength means bigger absolute tolerances than in optics. Secondly, capacitance and inductance are specific
properties, not directly seen in optics, because that can vary shape characteristics impedance and make influence on the formation of band gaps. 3-D PBG is complicated for both simulation and
realization, due to dependency over the angle of incident and on polarization. Quasi 3-D and 2-D structures are more successful structures and are being used to improve and isolate the antennas
characteristics and photonic crystal based waveguide. PBG application improves directivity of antennas. It incorporates main- ly: suppression of harmonics, suppression of the surface waves which is
the biggest challenge for antenna designers. As surface waves radiate from the roughness of the substrate edges and can make harm to the radiation pattern.
2. Mathematical Analysis
As mentioned above in Section I, about the metamaterial and PBG based planar antenna, which has been taken here as microstrip patch antenna with inverted U-shape above the substrate. Microstrip
antenna act as resonant cavity with four walls, two are upper and lower short walls (patch and ground) and other two are open ends of antenna (left and right) in general [14] . If the antenna is
excited at a resonant frequency, a strong field is set up inside the resonant cavity, and a strong current on the surface of the patch. This produces significant radiation due to fringing fields.
Fringing field patterns can be demonstrated by Figure 1.
In the present paper, author has compared the return loss, VSWR and bandwidth for metamaterial and PBG based antenna and has tried to find out the usability of enhanced bandwidth for 5G systems.
First to define metamaterial based devices, so many specific models are pre- sent, having information about how metamaterials works and importance of their refractive index that is its permittivity
and permeability value either both are negative or positive or only one of them is showing negative and positive va- lues in coordinate systems as shown in Figure 2, where one can easily dictates the
behavior of materials.
One of the famous model to define the properties of metamaterial, is given by Lorentz, which is most usable and well known as Lorentz oscillator model. In absence of electric fields and magnetic
fields, electronic charges, ions etc. have not contained any specific direction. In the presence of electric fields or magnetic fields, they become polarized and displaced the electron clouds.
Equation (1) is valid for electromagnetic waves used here, to show the motion of electron in the material. In Equation (1), first term in left hand side is inertia, second term is loss and third term
is restoring force, whereas, in right hand side denotes the applied electric fields. Due to these terms, electrons get polarized and start working according to electrical force and combined with the
EM wave, which is out of phase, and generates oscillation.
The inverted U-Shape planar antenna produces magnetic material like response and exhibits negative permeability, which represent the plasmonic type of frequency in the form of [15] [16] .
Figure 1. Fringing field pattern in microstrip antenna [14] .
Figure 2. Classification of metamaterials on the basis of their permittivity and permeabi- lity [17] .
where, ω[pm] = Magnetic plasma frequency,
Negative permeability also comes, when, ω < ω[pm], whereas, capacitive loaded structure is responsible for negative permittivity due to strong dielectric exhibits by this structure [18] . Due to this
condition, an electric dipole moment is generated in the structure, which exhibits the plasmonic type of permittivity, function of frequency [19] [20] .
where, ω[pe] = Electric plasma frequency,
This structure will also give negative permittivity at, ω < ω[pe]. Equations (2) and (3) are derived using Lorentz oscillator model. But, besides it, one more models, Drude model for metals is being
widely used and very easy to apply for measuring out the negative permittivity and permeability [21] . Equation (4) does not have any restoring force part in left hand side, means electron are not
bound within the structure, that’s why restoring force is equal to zero.
Negative permittivity and permeability can be calculated using Equation (4), which will be function of frequency, widely known as dielectric functions.
Equation (5) is being used to calculate the permittivity of material according to Drude model, whereas plasma frequency is related to total number of charges or electron cloud, which is measured from
Equation (6).
Like other electromagnetic materials behavior, LHM is also based upon Maxwell’s equations (
For simple rectangular microstrip antenna, the resonance frequency depends on the patch size, cavity dimension, and the filling dielectric constant, given as,
where m, n = 0, 1, 2,...., k[mn] = wave number at m, n mode, c is the velocity of light, ε[r] is the dielectric constant of substrate, and
For TM[01] mode, the length of non radiating rectangular patch’s edge at a certain resonance frequency and dielectric constant according to Equation (7) becomes,
where f[r] = resonance frequency at which the rectangular microstrip antennas are to be calculated. The radiating edge W, patch width, is usually chosen such that it lies within the range[r]. Taking
into account the effect of fringing field, the effective dielectric constant for TM[01] mode is derived using [22] [23] . Using these equations, total length can be calculated as,
where, ε[eff] = effective dielectric constant and Δl = line extension which is given as,
Rectangular microstrip is three layered structure, where patch, substrate and ground start working together. A combination of parallel plate radiation conductance and capacitance susceptance loads
both radiating edges of the patch, which can be measured as,
where λ[0] is the free-space wavelength and wave number k[0] =(2πf[r])/c. The input conductance of the patch fed on the edge will be twice the conductance of one of the edge slots, which can be
obtained as.
The patch can be fed by any feedings, so the impedance will vary from zero in the center to the edge resistance approximately as.
Now, arranging PBG structure in rectangular patch antenna follows certain rules just because of periodic structure, in which period of PBG structure T is half of the guide wavelength of a general
microwave strip λ[g], can be written as,
Here, the rectangular inverted U shape patch antenna is designed in a conventional fashion by itself and then surrounded properly by the PBG lattice structure in substrate. The period of PBG with
square lattice is T = 6 mm. Several cases of return loss with different cell size a relative to the period T are simulated, and an optimum size of the hole a/T = 2/3 is measured. The mutual coupling
among rectangular conformal microstrip antenna elements and interconnection feeding scheme should be figured out in the design steps. The distance between elements of proposed array is 12.5 mm,
integrated with groove loaded microstrip feeding with the depth l[slot]. For the matching of input impedance with the antenna can be calculated [23] ,
where, Z[c] is the characteristic impedance of the microstrip,
[01] mode, Q is obtained as,
while[r] is radiation conductance of rectangular patch,
in this proposed structure.
3. Results and Discussion
Comparative analysis has been performed between metamaterial and PBG based structure for frequency approximately 44 GHz and results related to this work are properly shown using commercial software
CST microwave studio. Parameters have been decided using all equations in Section III, and have been kept same for both design. For metamaterial based antenna design, array of inverted U has been
taken to check the compatibility with 5G advance cellular system technology. For PBG, rods have been taken, putting at equal periodicity. Parameters are considered according to the resonant frequency
given in Table 1.
Design structure of planar antenna using metamaterial and PBG have been shown in Figure 3(a) and Figure 3(b) respectively. From Figure 3(a), it is clear about the shape of radiating patch which is
inverted U, forming arrays structure to radiate properly and reduces the losses such as surface radiation loss etc. Figure 3(b) reveals the PBG based structure, where all the parameters are same,
only the difference in having rods below the patch, which actually being used to propagate single mode TM[01] mode including propagation of EM waves through stop band of the structure. This structure
especially mends for to remove losses through substrate and create a channel for propagation through a particular stop bands. Substrate material FR-4 shows a better compatibility and removes
complexity, compared to other dielectric materials.
Table 1. Design specification of planar antenna.
Figure 3. Left handed metameterial based planar antenna (a); PBG based planar antenna (b).
The return loss for both types of structures is most important parameter to reveal its applicability in mobile field communication. As shown in Figure 4(a) and Figure 4(b), return loss is more
improved in case of metamaterial antenna, as compared with PBG antenna. From Figure 4(a), return loss is minimum at 44.18 GHz, has value −30 dB, whereas in Figure 4(b), return loss is minimum at
43.28 GHz as well as at 44.5 GHz, but not comparable with Figure 4(a). Values at these two frequencies are −23 dB and −24 dB respectively. Although, Figure 4(b) is giving multiband property, but
because of losses, noise signal (reflection noise) is responsible for such return loss.
Figure 4. Return loss for metamaterial antenna (a), return loss for PBG antenna (b).
One of the important parameter is VSWR, which proposed antenna compatibility with transmission from input to output and gives information about reflection from output in form of standing wave ratio.
Ideal value of VSWR is one, which can be achieved only on zero reflection, which is not possible in case planar antennas. Figure 5(a) and Figure 5(b) show the value of VSWR at the resonant
frequencies. It is very clear from plots, the VSWR is better in case of metamaterial antenna. In the plot, passes have been taken during simulation to get more accurate results under CST microwave
Figure 5. VSWR of metamaterial inverted U shape antenna (a); VSWR of PBG antenna (b).
Total efficiency and radiation intensity of planar antennas give directivity, gain, field pattern (Electric and Magnetic) and spectral power density. From Figures 6(a)-6(d), efficiency of both
antennas directly measured using simulated plots. From the data, given below in the Plots 6(a) and (b) much more informative in terms of gain, where metamaterial planar antenna is giving higher gain
as compared to PBG antenna.
Comparative analysis between metamaterial and PBG antenna, designed at same high frequency to use it for 5G advanced technology, is given in Table 2.
Figure 6. Total and radiation efficiency of metamaterial antenna (a); Total and radiation intensity of PBG antenna (b); Farfield pattern for metamaterial antenna (c); Farfield pattern of PBG antenna
Table 2. Comparison chart between metamaterial and PBG planar antenna.
4. Conclusion
From entire analysis, it has been understood that metamaterial based antennas is more useful for 5 G advanced communication systems, with proper results and its discussion. The metamaterial inverted
U shape antenna has more impact in terms of return loss, VSWR, gain and bandwidth compared with PBG planar antennas. And it is also easy to fabricate for experimental analysis. Efficiency is better
in case of metamaterial based antenna as calculated 79%. But, all results obtained here, are simulated one using CST Microwave studio (Version 2012), but not measured from VNA. This has to be done in
The author (Smrity Dwivedi) is thankful for continuous support of her supervisor Prof. P. K. Jain, who has indirectly motivated her. | {"url":"https://scirp.org/journal/paperinformation?paperid=74599","timestamp":"2024-11-04T07:17:39Z","content_type":"application/xhtml+xml","content_length":"122894","record_id":"<urn:uuid:b5838d56-7a5e-4e7e-a2aa-0214af3494f4>","cc-path":"CC-MAIN-2024-46/segments/1730477027819.53/warc/CC-MAIN-20241104065437-20241104095437-00061.warc.gz"} |
How to Rotate A Subplot By 45 Degree In Matplotlib?
To rotate a subplot by 45 degrees in Matplotlib, you can use the "transform_rot" method to apply a rotation transformation to the subplot. First, you need to import the necessary libraries by using
the following code:
import matplotlib.pyplot as plt import matplotlib.transforms as transforms
Then, create a subplot using the plt.subplots() method, and apply the rotation transformation to the subplot using the following code:
fig, ax = plt.subplots() trans = transforms.Affine2D().rotate_deg(45) ax.transData = trans + ax.transData
This code snippet will rotate the subplot by 45 degrees. You can adjust the angle of rotation by changing the value passed to the rotate_deg() method. Finally, you can add your plot elements to the
rotated subplot as usual.
How to efficiently rotate a subplot by 45 degrees in matplotlib for accurate representation?
To efficiently rotate a subplot by 45 degrees in matplotlib for accurate representation, you can use the set_xticks and set_yticks functions to set the positions of the ticks on the x and y axes.
Here's an example of how you can do this:
1 import matplotlib.pyplot as plt
3 # Create a figure and axis
4 fig, ax = plt.subplots()
6 # Plot your data
7 ax.plot([1, 2, 3, 4, 5], [1, 4, 9, 16, 25])
9 # Rotate the subplot by 45 degrees
10 ax.set_xticks([1, 2, 3, 4, 5])
11 ax.set_yticks([1, 4, 9, 16, 25])
12 ax.set_xticklabels([1, 2, 3, 4, 5], rotation=45)
13 ax.set_yticklabels([1, 4, 9, 16, 25], rotation=45)
15 plt.show()
In this example, we first create a figure and axis using plt.subplots(). We then plot some data on the axis. To rotate the subplot by 45 degrees, we use the set_xticklabels and set_yticklabels
functions to set the positions of the ticks on the x and y axes and rotate them by 45 degrees using the rotation parameter. Finally, we display the plot using plt.show().
What is the technique for rotating a subplot in matplotlib by 45 degrees without changing the aspect ratio?
One way to rotate a subplot in matplotlib by 45 degrees without changing the aspect ratio is to use the transData coordinate transformation. Here is an example of how to achieve this:
1 import matplotlib.pyplot as plt
3 # Create a subplot
4 fig, ax = plt.subplots()
6 # Rotate the subplot by 45 degrees
7 ax.set_aspect('auto', adjustable='datalim')
8 trans = ax.transData + plt.Affine2D().rotate_deg(45)
9 for line in ax.get_lines():
10 line.set_transform(trans + line.get_transform())
12 # Display the plot
13 plt.show()
In this code snippet, we first create a subplot using plt.subplots(). We then rotate the subplot by 45 degrees using the rotate_deg method of Affine2D transformation. Finally, we apply this
transformation to each line in the subplot to rotate them as well.
By using this technique, you can rotate the subplot by 45 degrees without changing the aspect ratio.
What is the easiest way to rotate a subplot by 45 degrees in matplotlib?
The easiest way to rotate a subplot by 45 degrees in matplotlib is to use the set_rotation method on the subplot object. Here is an example code snippet that demonstrates how to rotate a subplot by
45 degrees:
1 import matplotlib.pyplot as plt
3 fig, ax = plt.subplots()
4 ax.plot([1, 2, 3, 4])
5 ax.set_title('Original subplot')
7 # Rotate the subplot by 45 degrees
8 ax.set_rotation(45)
10 plt.show()
This code will create a simple subplot with a title, and then rotate it by 45 degrees using the set_rotation method. You can adjust the rotation angle as needed to achieve the desired rotation.
How do I rotate a subplot by 45 degrees in matplotlib?
To rotate a subplot by 45 degrees in matplotlib, you can use the set_rotation method on the subplot's x-axis labels. Here is an example of how you can do this:
1 import matplotlib.pyplot as plt
3 # Create a figure and subplot
4 fig, ax = plt.subplots()
6 # Plot some data
7 x = [1, 2, 3, 4, 5]
8 y = [10, 15, 13, 18, 16]
9 ax.plot(x, y)
11 # Rotate the x-axis labels by 45 degrees
12 for tick in ax.get_xticklabels():
13 tick.set_rotation(45)
15 plt.show()
In this example, we first create a figure and subplot using plt.subplots(). We then plot some data on the subplot. To rotate the x-axis labels by 45 degrees, we iterate through the x-axis labels
using ax.get_xticklabels() and set the rotation using tick.set_rotation(45). Finally, we display the plot using plt.show().
How do you rotate a subplot by 45 degrees in matplotlib?
To rotate a subplot by 45 degrees in matplotlib, you can use the set_rotation method on the subplot's tick labels. Here's an example of how to do this:
1 import matplotlib.pyplot as plt
3 fig, ax = plt.subplots()
5 # Plot your data
7 # Rotate the x-axis tick labels by 45 degrees
8 for tick in ax.get_xticklabels():
9 tick.set_rotation(45)
11 plt.show()
This code snippet will rotate the x-axis tick labels of the subplot by 45 degrees. You can adjust the angle of rotation by changing the value passed to the set_rotation method. | {"url":"https://tech-blog.us.to/blog/how-to-rotate-a-subplot-by-45-degree-in-matplotlib","timestamp":"2024-11-12T11:57:31Z","content_type":"text/html","content_length":"168579","record_id":"<urn:uuid:9c2b002e-1c63-48e7-b9c8-f18ca4b0f1b7>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.45/warc/CC-MAIN-20241112113320-20241112143320-00410.warc.gz"} |
An Improved Lower Bound for Matroid Intersection Prophet Inequalities
We consider prophet inequalities subject to feasibility constraints that are the intersection of q matroids. The best-known algorithms achieve a Θ(q)-approximation, even when restricted to instances
that are the intersection of q partition matroids, and with i.i.d. Bernoulli random variables [13, 22, 2]. The previous best-known lower bound is Θ(√q) due to a simple construction of [28] (which
uses i.i.d. Bernoulli random variables, and writes the construction as the intersection of partition matroids). We establish an improved lower bound of q^1/2+Ω(1/log log q) by writing the
construction of [28] as the intersection of asymptotically fewer partition matroids. We accomplish this via an improved upper bound on the product dimension of a graph with p^p disjoint cliques of
size p, using recent techniques developed in [5].
Publication series
Name Leibniz International Proceedings in Informatics, LIPIcs
Volume 251
ISSN (Print) 1868-8969
Conference 14th Innovations in Theoretical Computer Science Conference, ITCS 2023
Country/Territory United States
City Cambridge
Period 1/10/23 → 1/13/23
All Science Journal Classification (ASJC) codes
• Intersection of Matroids
• Prophet Inequalities
Dive into the research topics of 'An Improved Lower Bound for Matroid Intersection Prophet Inequalities'. Together they form a unique fingerprint. | {"url":"https://collaborate.princeton.edu/en/publications/an-improved-lower-bound-for-matroid-intersection-prophet-inequali","timestamp":"2024-11-05T12:57:57Z","content_type":"text/html","content_length":"53281","record_id":"<urn:uuid:f36227b7-f8cd-4df1-a63b-c841b9bd7d37>","cc-path":"CC-MAIN-2024-46/segments/1730477027881.88/warc/CC-MAIN-20241105114407-20241105144407-00767.warc.gz"} |
Read Topics On Mathematics For Smart Systems: Proceedings Of The European Conference Rome, Italy, 26 28 October,2006 2007
Read Topics On Mathematics For Smart Systems: Proceedings Of The European Conference Rome, Italy, 26 28 October,2006 2007
Read Topics On Mathematics For Smart Systems: Proceedings Of The European Conference Rome, Italy, 26 28 October,2006 2007
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4 years ago | {"url":"https://ch.mathworks.com/matlabcentral/profile/authors/18483204?detail=all","timestamp":"2024-11-10T11:16:04Z","content_type":"text/html","content_length":"104446","record_id":"<urn:uuid:990d7fb2-98d4-4a02-b295-a64a5e87423a>","cc-path":"CC-MAIN-2024-46/segments/1730477028186.38/warc/CC-MAIN-20241110103354-20241110133354-00843.warc.gz"} |
Area, Surface Area, Volume
Find the surface area and volume of a rectangular solid with length of 2 feet, width of 3 feet, and height of 6 feet.
72 square feet 36 cubic feet
Find the surface area and volume of a cylinder with radius of 3 inches and height of 10 inches.
245.0 square inches 282.7 cubic inches
A square pyramid has a height of 4 cm and a base length of 6. Find the volume.
48 cubic centimeters
Find the slant height of a cone with a height of 15 meters and a radius of 8 meters.
17 meters
If a cone and a cylinder have the same radius and height, then how many cones of water will fill the cylinder?
Find the surface area of a rectangular solid with length of 5 meters, width of 5 meters and height of 3 meters.
110 square meters
Find the surface area cylinder with radius of 5 inches and height of 4 inches.
282.7 square inches
Find the surface area of a square pyramid with base length of 12 feet and slant height of 15 feet.
504 feet squared
Find the volume of a cone with a radius of 4 centimeters and a height of 3 centimeters.
50.3 cubic centimeters
Name two differences between a prism and a pyramid.
number of bases faces are different shapes
Find the volume of a rectangular prism with length of 10 inches, width of 4 inches, and height of 3 inches.
120 cubic inches
A cylinder has a radius of 3 feet and a height of 4 feet. Find the volume.
113.1 cubic feet
Find the surface area of a square pyramid with a base length of 6 centimeters and a slant height of 15 centimeters.
216 square centimeters
Find the height for a cone with a radius of 2 meters and a slant height of 10 meters.
9.79 meters
What is the volume of a cube with side 7 cm
343 cubic cm
What is the volume of a box with length of 150 mm, width of 73 cm, and height of 1 m? Answer in cubic centimeters
109,500 cubic centimeters.
A straw has a radius of .13 inches and a length of 7 inches. How much material was used to make the straw?
5.7 inches squared.
A square pyramid has a volume of 160 centimeters cubed. What is the height if the area of the base is 16?
30 cm
Beth wants to make a metal cone with a radius of 12 inches and slant height of 13 inches. How high is the cone?
5 inches
Find the surface area of a cube with base diagonal 5.657 cm
96 square cm
Find the volume of an equilateral triangular prism with base edge 4 meters and height 10 meters. (1 d.p.)
69.3 square meters
Find the radius of a cylinder with a height of 8 cm and a volume of 508.94 cubic cm.
4.5 cm
Find the volume of a regular hexagonal pyramid with a base edge of 4 meters and a height of 7 meters.
97 cubic meters
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5 centimeters
Find the ratio of a cube's surface area to volume in terms of s. | {"url":"https://jeopardylabs.com/play/area-surface-area-volume-8","timestamp":"2024-11-12T10:00:30Z","content_type":"application/xhtml+xml","content_length":"58998","record_id":"<urn:uuid:5dacb731-d822-4fac-8c84-d60fc5c0e35c>","cc-path":"CC-MAIN-2024-46/segments/1730477028249.89/warc/CC-MAIN-20241112081532-20241112111532-00492.warc.gz"} |
Defining Tensor
Defining Tensors
Declaring Tensors
taco::Tensor objects, which correspond to mathematical tensors, form the core of the TACO C++ API. You can declare a new tensor by specifying its name, a vector containing the size of each dimension
of the tensor, and the storage format that will be used to store the tensor:
// Declare a new tensor "A" of double-precision floats with dimensions
// 512 x 64 x 2048, stored as a dense-sparse-sparse tensor
Tensor<double> A("A", {512,64,2048}, Format({Dense,Sparse,Sparse}));
The name of the tensor can be omitted, in which case TACO will assign an arbitrary name to the tensor:
// Declare another tensor with the same dimensions and storage format as before
Tensor<double> A({512,64,2048}, Format({Dense,Sparse,Sparse}));
Scalars, which are treated as order-0 tensors, can be declared and initialized with some arbitrary value as demonstrated below:
Tensor<double> alpha(42.0); // Declare a scalar tensor initialized to 42.0
Defining Tensor Formats
Conceptually, you can think of a tensor as a tree with each level (excluding the root) corresponding to a dimension of the tensor. Each path from the root to a leaf node represents a tensor
coordinate and its corresponding value. Which dimension each level of the tree corresponds to is determined by the order in which dimensions of the tensor are stored.
TACO uses a novel scheme that can describe different storage formats for any tensor by specifying the order in which tensor dimensions are stored and whether each dimension is sparse or dense. A
sparse dimension stores only the subset of the dimension that contains non-zero values and is conceptually similar to the index arrays used in the compressed sparse row (CSR) matrix format, while a
dense dimension stores both zeros and non-zeros. As demonstrated below, this scheme is flexibile enough to express many commonly-used matrix storage formats.
You can define a new tensor storage format by creating a taco::Format object. The constructor for taco::Format takes as arguments a vector specifying the type of each dimension and (optionally) a
vector specifying the order in which dimensions are to be stored, following the above scheme:
Format dm({Dense,Dense}); // (Row-major) dense matrix
Format csr({Dense,Sparse}); // Compressed sparse row matrix
Format csc({Dense,Sparse}, {1,0}); // Compressed sparse column matrix
Format dcsr({Sparse,Sparse}, {1,0}); // Doubly compressed sparse column matrix
Alternatively, you can define a tensor format that contains only sparse or dense dimensions as follows:
Format csf(Sparse); // Compressed sparse fiber tensor
Initializing Tensors
You can initialize a taco::Tensor by calling the insert method to add a non-zero component to the tensor. The insert method takes two arguments, a vector specifying the coordinate of the non-zero
component to be added and the value to be inserted at that coordinate:
A.insert({128,32,1024}, 42.0); // A(128,32,1024) = 42.0
The insert method adds the inserted non-zeros to a temporary buffer. Before a tensor can actually be used in a computation though, you must invoke the pack method to compress the tensor into the
storage format that was specified when the tensor was first declared:
A.pack(); // Construct dense-sparse-sparse tensor containing inserted non-zeros
Loading Tensors from File
Rather than manually invoking insert and pack to initialize a tensor, you can load tensors directly from file by calling taco::read as demonstrated below:
// Load a dense-sparse-sparse tensor from file A.tns
A = read("A.tns", Format({Dense, Sparse, Sparse}));
By default, taco::read returns a packed tensor. You can optionally pass a Boolean flag as an argument to indicate whether the returned tensor should be packed or not:
// Load an unpacked tensor from file A.tns
A = read("A.tns", Format({Dense, Sparse, Sparse}), false);
Currently, TACO supports loading from the following matrix and tensor file formats:
Writing Tensors to File
You can also write a (packed) tensor directly to file by calling taco::write, as demonstrated below:
write("A.tns", A); // Write tensor A to file A.tns
taco::write supports the same set of matrix and tensor file formats as taco::read. | {"url":"http://tensor-compiler.org/docs/tensors","timestamp":"2024-11-07T03:20:45Z","content_type":"text/html","content_length":"13364","record_id":"<urn:uuid:c760480e-89c1-4c5f-8ebb-d58672311a8c>","cc-path":"CC-MAIN-2024-46/segments/1730477027951.86/warc/CC-MAIN-20241107021136-20241107051136-00609.warc.gz"} |
Carlisle Rainey - Statistical Power from Pilot Data: An Example
Statistical Power from Pilot Data: An Example
statistical power
hypothesis tests
power analysis
In this post, I provide an example of how pilot data can be used to predict the standard error in a planned study.
We can think of statistical power as determined by the ratio \(\frac{\tau}{SE}\), where \(\tau\) is the treatment effect and SE is the standard error of the estimate. To reason about statistical
power, one needs to make assumptions or predictions about the treatment effect and the standard error.
And as data-oriented researchers, we often want to use data to inform these predictions and assumptions. We might want to use pilot data.^1
^1 Here’s how Leon, Davis, and Kraemer (2011) describe the purpose of a pilot study. “The fundamental purpose of conducting a pilot study is to examine the feasibility of an approach that is intended
to ultimately be used in a larger scale study. This applies to all types of research studies. Here we use the randomized controlled clinical trial (RCT) for illustration. Prior to initiating a full
scale RCT an investigator may choose to conduct a pilot study in order to evaluate the feasibility of recruitment, randomization, retention, assessment procedures, new methods, and/or implementation
of the novel intervention. A pilot study, however, is not used for hypothesis testing. Instead it serves as an earlier-phase developmental function that will enhance the probability of success in the
larger subsequent RCTs that are anticipated.”
1. Pilot data are not useful to predict the treatment effect.
2. Pilot data are useful to predict the standard error.
With a predicted standard error in hand, we can predict the minimum detectable effect, the statistical power, or the required sample size in the planned study.
In this post, I give an example of how this can work.
Predicting the SE from pilot data
Here’s how I suggest we use pilot data to predict the standard error in the planned study:
Predicting the SE in the planned study using pilot data
Conservatively, the standard error will be about \(\sqrt{\frac{n^{pilot}}{n^{planned}}} \cdot \left\lbrack \left( \sqrt{\frac{1}{n^{pilot}}} + 1 \right) \cdot {\widehat{SE}}_{\widehat{\tau}}^{pilot}
\right\rbrack\), where \(n^{pilot}\) is the number of respondents per condition in the pilot data, \(SE_{\widehat{\tau}}^{pilot}\) is the estimated standard error using the pilot data, and \(n^
{planned}\) is the number of respondents per condition in the planned study.
The factor \(\left( \sqrt{\frac{1}{n^{pilot}}} + 1 \right)\) nudges the standard error from the pilot study in a conservative direction, since it might be an under-estimate of the actual standard
error.^2 For the details, see this early paper, but this conservative standard error estimate is approximately the upper bound of a 95% confidence interval for the standard error using the pilot
^2 More generally, we can use a bootstrap to conservatively estimate the standard error, without relying on this analytical approximation.
The Robbins et al. study
As an example, let’s use half of the experiment conducted by Robbins et al. (2024).
Robbins et al. use a 2x2 factorial vignette design, randomly assigning each respondent to read one of four vignettes. The vignette describes a hypothetical covert operation ordered by the president
that ends in either success or failure. Then, the vignette describes a whistleblower coming forward and describes the president’s opposition in Congress as either amplifying or ignoring the
President's Opposition in Congress Outcome of Operation
Success Failure
Amplifies Whistleblower Vignette 1: Success & Amplify Vignette 2: Failure & Amplify
Ignores Whistleblower Vignette 3: Success & Ignore Vignette 4: Failure & Ignore
After the vignette, the respondent is asked whether they approve of the opposition in Congress’ actions on a seven-point Likert scale from strongly approve to strongly disapprove.
For a simple example, let’s focus on the effect of amplifying the whistleblower when the operation succeeds. That is, let’s compare responses after Vignette 1 and Vignette 3. How much does amplifying
a whistleblower increase approval when the opperation succeeds? We expect a small effect here, so we should pay careful attention to power.
The task
We hoped to detect an effect as small as 0.35 points on the seven-point scale and had tentatively planned on 250 respondents per condition. To test the survey instrument and data provider, we
conducted a small pilot with about 75 respondents per condition. Let’s use those pilot data to check whether 250 respondents seem sufficient.
The data
In the {crdata} package on GitHub, you can find the the pilot data we collected leading up to the main study.
Now let’s load the pilot data. To focus on observations where the operation succeeds, we’re going to keep only the observations where the vignette describes a successful observation.
# load pilot data and keep only success condition
robbins2_pilot <- crdata::robbins_pilot |>
subset(failure == "Success") |>
Rows: 147
Columns: 5
$ cong_overall <dbl> 3, 1, -2, 0, -2, -1, 0, -1, 0, 0, 0, 2, -1, 3, -3, 0, 0, …
$ failure <fct> Success, Success, Success, Success, Success, Success, Suc…
$ amplify <fct> Ignore, Ignore, Ignore, Amplify, Ignore, Ignore, Ignore, …
$ pid7 <fct> Strong Democrat, Not very strong Republican, Strong Democ…
$ pid_strength <dbl> 3, 2, 3, 2, 2, 3, 2, 2, 2, 2, 2, 1, 0, 3, 3, 3, 3, 1, 1, …
cong_overall is the respondent’s approval of Congress’ actions on a seven-point scale and amplify indicates whether Congress amplified the whistleblower (i.e., criticized the president).
Analyzing the pilot data
Now let’s analyze the pilot data as we plan to analyze the main data set that we plan to collect later. We’re interested in the average response in the Amplify and Ignore conditions, so let’s use a
Ignore the Estimated Treatment Effect
It can be really tempting to look at the estimated treatment effect. In this pilot study, it’s actually statistically significant. I intentionally don’t show the estimated treatment effect (or
quantities requiring it, like p-values). If we looked at these, we might make one of the following mistakes:
1. “The pilot got significant results, therefore even the pilot is sufficiently powered.”
2. “The estimate from the pilot is significant, therefore we can use the estimated treatment effect in the power analysis.”
Both of these claims are misleading. The estimated treatment effect is very noisy, so ignore the estimated treatment effect.
Predicting the SE in the main study
To predict the standard error in the main study, we need two pieces of information from this pilot:
1. the sample size per condition and
2. the estimated standard error.
We can get the number of observations per condition using table().
And then we need the estimated standard error, which is computed by t.test().
Now we can predict the standard error in the planned study.
For the main study, we planned on about 250 respondents per condition.
The we can conservatively predict the standard error in the full study as \(\sqrt{\frac{n^{pilot}}{n^{planned}}} \cdot \left\lbrack \left( \sqrt{\frac{1}{n^{pilot}}} + 1 \right) \cdot {\widehat{SE}}_
{\widehat{\tau}}^{pilot} \right\rbrack\).
But is this standard error small enough?
Evaluating the predicted SE in the main study
We can convert the standard error to the minimum detectable effect with 80% power using \(2.5 \times SE\).^3
^3 See Bloom (1995) for an excellent discussion of this rule. I also write about it here.
We hoped to detect an effect as small as 0.35 points on the seven-point scale, so we’re going to need more than 250 respondents per condition!
We can also compute the power to detect an effect of 0.35 points on the seven-point scale.
Note that these are conservative estimates of the minimum detectable effect and statistical power.
Here’s what things look like if we remove the conservative nudge \(\left( \sqrt{\frac{1}{n^{pilot}}} + 1 \right)\) and predict the standard error as \(\sqrt{\frac{n^{pilot}}{n^{planned}}} \cdot {\
# without the conservative nudge
pred_se <- sqrt(n_pilot/n_planned)*se_hat_pilot
2.5*pred_se # best guess of minimum detectable effect
As you can see, the minimum detectable effect and power are a little too low. We need more respondents!
Adjusting the Sample Size
Our plan of 250 respondents per condition seems too low. If we want, we can predict the sample size we need to get to 80% power using the following rule:
Predicting the required sample size in the planned study using pilot data
For 80% power to detect the treatment effect \(\widetilde{\tau}\), we will (conservatively) need about \(n^{pilot} \cdot \left\lbrack \frac{2.5}{\widetilde{\tau}} \cdot \left( \sqrt{\frac{1}{n^
{pilot}}} + 1 \right) \cdot {\widehat{SE}}_{\widehat{\tau}}^{pilot} \right\rbrack^{2}\) respondents per condition, where \(n^{pilot}\) is the number of respondents per condition in the pilot data and
\(SE_{\widehat{\tau}}^{pilot}\) is the estimated standard error using the pilot data.
Thus to get 80% power, the pilot data suggest that we (conservatively) need about 360 respondents per condition. We used 367 in the full study. Here are the conservative predictions for 367
respondents per condition.
How did we do?
We ran the full study.^4
^4 See Robbins et al. (2024) for the full results.
Rows: 735
Columns: 5
$ cong_overall <dbl> 2, -2, -1, -3, 0, -1, -2, -1, 1, 1, 0, -3, -2, 2, 2, -3, …
$ failure <fct> Success, Success, Success, Success, Success, Success, Suc…
$ amplify <fct> Ignore, Amplify, Amplify, Amplify, Ignore, Ignore, Amplif…
$ pid7 <fct> Not very strong Republican, Not very strong Republican, S…
$ pid_strength <dbl> 2, 2, 3, 3, 3, 2, 0, 2, 2, 1, 2, 0, 3, 1, 2, 3, 2, 3, 2, …
As you can see, the pilot data gave us a good, slightly conservative prediction. We conservatively predicted a standard error of 0.138 in the planned study and we estimated a standard error of 0.132
after running the study. We conservatively predicted our power would be about 82% to detected an effect of 0.35 on the seven-point scale, but after running the study, it seems like we had about 84%
A bootstrap alternative
We can also use the bootstrap as an alternative. There are a few ways one might approach it.
Here’s one:
1. Treat the pilot data as a population. Create a data set with the planned sample size by sampling with replacement from the pilot data.
2. Perform the planned analysis on each resampled data set.
3. Store the estimated standard error from each analysis.
Repeat the process above many times. For each standard error estimate, compute the implied statistical power. This gives a distribution of power estimates. Find a value near the bottom of this
distribution. The factor we used above—The factor \(\left( \sqrt{\frac{1}{n^{pilot}}} + 1 \right)\)—nudges the standard error to about the 2.5th percentile, so we can use that here, too.
# number of bootstrap iterations
n_bs <- 10000
bs_se <- numeric(n_bs) # a container
for (i in 1:n_bs){
# resample 367 observations from each condition
bs_data <- robbins2_main %>%
group_by(amplify) %>%
sample_n(size = 367, replace = TRUE)
# run planned analysis
bs_fit <- t.test(cong_overall ~ amplify, data = bs_data)
# grab se
bs_se[i] <- bs_fit$stderr
# compute 2.5th percentile of power to obtain conservative estimate
pwr <- 1 - pnorm(1.64 - 0.35/bs_se)
quantile(pwr, probs = 0.025)
Using the analytical approximation, we got 0.815 as a conservative estimate of power. The bootstrap gave us 0.820 as a conservative estimate. The actual power in the full study turned out to be about
0.843. (Remember, all of these power calculations are power to detected an effect of 0.35 points on the seven-point scale.)
The paper
I have an early draft of a paper on these (and other) ideas. Please test them out in your own work and let me know if you have questions, comments, and suggestions. I’m interested in making the paper
as clear and useful as I can.
Bloom, Howard S. 1995.
“Minimum Detectable Effects.” Evaluation Review
19 (5): 547–56.
Leon, Andrew C., Lori L. Davis, and Helena C. Kraemer. 2011.
“The Role and Interpretation of Pilot Studies in Clinical Research.” Journal of Psychiatric Research
45 (5): 626–29.
Robbins, Caroline, Alessandro Brunelli, Jose Casto, Ainsley Coty, Andrew Louis, Bryanna Major, Maria A Martinez, et al. 2024.
“Overt Consequences of Covert Actions: Success, Failure, and Voters’ Preferences for Legislative Oversight.” | {"url":"https://www.carlislerainey.com/blog/2024-06-10-pilot-power-example/index.html","timestamp":"2024-11-02T06:12:14Z","content_type":"application/xhtml+xml","content_length":"55232","record_id":"<urn:uuid:5f295047-50cd-4c21-bf75-656497e0f1e1>","cc-path":"CC-MAIN-2024-46/segments/1730477027677.11/warc/CC-MAIN-20241102040949-20241102070949-00430.warc.gz"} |
Linguistic indexicality in algebra discussions
In discussion-oriented classrooms, students create mathematical ideas through conversations that reflect growing collective knowledge. Linguistic forms known as indexicals assist in the analysis of
this collective, negotiated understanding. Indexical words and phrases create meaning through reference to the physical, verbal and ideational context. While some indexicals such as pronouns and
demonstratives (e.g. this, that) are fairly well-known in mathematics education research, other structures play significant roles in math discussions as well. We describe students' use of entailing
and presupposing indexicality, verbs of motion, and poetic structures to express and negotiate mathematical ideas and classroom norms including pedagogical responsibility, conjecturing, evaluating
and expressing reified mathematical knowledge. The multiple forms and functions of indexical language help describe the dynamic and emergent nature of mathematical classroom discussions. Because
interactive learning depends on linguistically established connections among ideas, indexical language may prove to be a communicative resource that makes collaborative mathematical learning
• Discourse
• Indexical language
• Mathematical communication
• Social learning
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Find The Sum Of The Interior Angle Measures Of Each Polygon Worksheet - Angleworksheets.com
Finding The Sum Of Interior Angles Of A Polygon Worksheet – If you have been struggling to learn how to find angles, there is no need to worry as there are many resources available for you to use.
These worksheets will help to understand the various concepts and increase your knowledge of angles. Using the … Read more
The Sum Of Polygon Angle Measures Worksheet
The Sum Of Polygon Angle Measures Worksheet – Use free printable Measure Angle Worksheets to practice measuring angles. These worksheets will help you learn how to use a protractor and avoid angles
that are not exactly right. These worksheets also provide tips for making measurements easier. For example, you can use a protractor to measure … Read more | {"url":"https://www.angleworksheets.com/tag/find-the-sum-of-the-interior-angle-measures-of-each-polygon-worksheet/","timestamp":"2024-11-10T20:40:23Z","content_type":"text/html","content_length":"53769","record_id":"<urn:uuid:20f86d67-271d-420e-9a4f-10886062fef6>","cc-path":"CC-MAIN-2024-46/segments/1730477028191.83/warc/CC-MAIN-20241110201420-20241110231420-00287.warc.gz"} |
Which Mutual Information Representation Learning Objectives are Sufficient for Control?
Processing raw sensory inputs is crucial for applying deep RL algorithms to real-world problems. For example, autonomous vehicles must make decisions about how to drive safely given information
flowing from cameras, radar, and microphones about the conditions of the road, traffic signals, and other cars and pedestrians. However, direct “end-to-end” RL that maps sensor data to actions
(Figure 1, left) can be very difficult because the inputs are high-dimensional, noisy, and contain redundant information. Instead, the challenge is often broken down into two problems (Figure 1,
right): (1) extract a representation of the sensory inputs that retains only the relevant information, and (2) perform RL with these representations of the inputs as the system state.
Figure 1. Representation learning can extract compact representations of states for RL.
A wide variety of algorithms have been proposed to learn lossy state representations in an unsupervised fashion (see this recent tutorial for an overview). Recently, contrastive learning methods have
proven effective on RL benchmarks such as Atari and DMControl (Oord et al. 2018, Stooke et al. 2020, Schwarzer et al. 2021), as well as for real-world robotic learning (Zhan et al.). While we could
ask which objectives are better in which circumstances, there is an even more basic question at hand: are the representations learned via these methods guaranteed to be sufficient for control? In
other words, do they suffice to learn the optimal policy, or might they discard some important information, making it impossible to solve the control problem? For example, in the self-driving car
scenario, if the representation discards the state of stoplights, the vehicle would be unable to drive safely. Surprisingly, we find that some widely used objectives are not sufficient, and in fact
do discard information that may be needed for downstream tasks.
Defining the Sufficiency of a State Representation
As introduced above, a state representation is a function of the raw sensory inputs that discards irrelevant and redundant information. Formally, we define a state representation $\phi_Z$ as a
stochastic mapping from the original state space $\mathcal{S}$ (the raw inputs from all the car’s sensors) to a representation space $\mathcal{Z}$: $p(Z | S=s)$. In our analysis, we assume that the
original state $\mathcal{S}$ is Markovian, so each state representation is a function of only the current state. We depict the representation learning problem as a graphical model in Figure 2.
Figure 2. The representation learning problem in RL as a graphical model.
We will say that a representation is sufficient if it is guaranteed that an RL algorithm using that representation can learn the optimal policy. We make use of a result from Li et al. 2006, which
proves that if a state representation is capable of representing the optimal $Q$-function, then $Q$-learning run with that representation as input is guaranteed to converge to the same solution as in
the original MDP (if you’re interested, see Theorem 4 in that paper). So to test if a representation is sufficient, we can check if it is able to represent the optimal $Q$-function. Since we assume
we don’t have access to a task reward during representation learning, to call a representation sufficient we require that it can represent the optimal $Q$-functions for all possible reward functions
in the given MDP.
Analyzing Representations learned via MI Maximization
Now that we’ve established how we will evaluate representations, let’s turn to the methods of learning them. As mentioned above, we aim to study the popular class of contrastive learning methods.
These methods can largely be understood as maximizing a mutual information (MI) objective involving states and actions. To simplify the analysis, we analyze representation learning in isolation from
the other aspects of RL by assuming the existence of an offline dataset on which to perform representation learning. This paradigm of offline representation learning followed by online RL is becoming
increasingly popular, particularly in applications such as robotics where collecting data is onerous (Zhan et al. 2020, Kipf et al. 2020). Our question is therefore whether the objective is
sufficient on its own, not as an auxiliary objective for RL. We assume the dataset has full support on the state space, which can be guaranteed by an epsilon-greedy exploration policy, for example.
An objective may have more than one maximizing representation, so we call a representation learning objective sufficient if all the representations that maximize that objective are sufficient. We
will analyze three representative objectives from the literature in terms of sufficiency.
Representations Learned by Maximizing “Forward Information”
We begin with an objective that seems likely to retain a great deal of state information in the representation. It is closely related to learning a forward dynamics model in latent representation
space, and to methods proposed in prior works (Nachum et al. 2018, Shu et al. 2020, Schwarzer et al. 2021): $J_{fwd} = I(Z_{t+1}; Z_t, A_t)$. Intuitively, this objective seeks a representation in
which the current state and action are maximally informative of the representation of the next state. Therefore, everything predictable in the original state $\mathcal{S}$ should be preserved in $\
mathcal{Z}$, since this would maximize the MI. Formalizing this intuition, we are able to prove that all representations learned via this objective are guaranteed to be sufficient (see the proof of
Proposition 1 in the paper).
While reassuring that $J_{fwd}$ is sufficient, it’s worth noting that any state information that is temporally correlated will be retained in representations learned via this objective, no matter how
irrelevant to the task. For example, in the driving scenario, objects in the agent’s field of vision that are not on the road or sidewalk would all be represented, even though they are irrelevant to
driving. Is there another objective that can learn sufficient but lossier representations?
Representations Learned by Maximizing “Inverse Information”
Next, we consider what we term an “inverse information” objective: $J_{inv} = I(Z_{t+k}; A_t | Z_t)$. One way to maximize this objective is by learning an inverse dynamics model – predicting the
action given the current and next state – and many prior works have employed a version of this objective (Agrawal et al. 2016, Gregor et al. 2016, Zhang et al. 2018 to name a few). Intuitively, this
objective is appealing because it preserves all the state information that the agent can influence with its actions. It therefore may seem like a good candidate for a sufficient objective that
discards more information than $J_{fwd}$. However, we can actually construct a realistic scenario in which a representation that maximizes this objective is not sufficient.
For example, consider the MDP shown on the left side of Figure 4 in which an autonomous vehicle is approaching a traffic light. The agent has two actions available, stop or go. The reward for
following traffic rules depends on the color of the stoplight, and is denoted by a red X (low reward) and green check mark (high reward). On the right side of the figure, we show a state
representation in which the color of the stoplight is not represented in the two states on the left; they are aliased and represented as a single state. This representation is not sufficient, since
from the aliased state it is not clear whether the agent should “stop” or “go” to receive the reward. However, $J_{inv}$ is maximized because the action taken is still exactly predictable given each
pair of states. In other words, the agent has no control over the stoplight, so representing it does not increase MI. Since $J_{inv}$ is maximized by this insufficient representation, we can conclude
that the objective is not sufficient.
Figure 4. Counterexample proving the insufficiency of $J_{inv}$.
Since the reward depends on the stoplight, perhaps we can remedy the issue by additionally requiring the representation to be capable of predicting the immediate reward at each state. However, this
is still not enough to guarantee sufficiency - the representation on the right side of Figure 4 is still a counterexample since the aliased states have the same reward. The crux of the problem is
that representing the action that connects two states is not enough to be able to choose the best action. Still, while $J_{inv}$ is insufficient in the general case, it would be revealing to
characterize the set of MDPs for which $J_{inv}$ can be proven to be sufficient. We see this as an interesting future direction.
Representations Learned by Maximizing “State Information”
The final objective we consider resembles $J_{fwd}$ but omits the action: $J_{state} = I(Z_t; Z_{t+1})$ (see Oord et al. 2018, Anand et al. 2019, Stooke et al. 2020). Does omitting the action from
the MI objective impact its sufficiency? It turns out the answer is yes. The intuition is that maximizing this objective can yield insufficient representations that alias states whose transition
distributions differ only with respect to the action. For example, consider a scenario of a car navigating to a city, depicted below in Figure 5. There are four states from which the car can take
actions “turn right” or “turn left.” The optimal policy takes first a left turn, then a right turn, or vice versa. Now consider the state representation shown on the right that aliases $s_2$ and
$s_3$ into a single state we’ll call $z$. If we assume the policy distribution is uniform over left and right turns (a reasonable scenario for a driving dataset collected with an exploration policy),
then this representation maximizes $J_{state}$. However, it can’t represent the optimal policy because the agent doesn’t know whether to go right or left from $z$.
Figure 5. Counterexample proving the insufficiency of $J_{state}$.
Can Sufficiency Matter in Deep RL?
To understand whether the sufficiency of state representations can matter in practice, we perform simple proof-of-concept experiments with deep RL agents and image observations. To separate
representation learning from RL, we first optimize each representation learning objective on a dataset of offline data, (similar to the protocol in Stooke et al. 2020). We collect the fixed datasets
using a random policy, which is sufficient to cover the state space in our environments. We then freeze the weights of the state encoder learned in the first phase and train RL agents with the
representation as state input (see Figure 6).
Figure 6. Experimental setup for evaluating learned representations.
We experiment with a simple video game MDP that has a similar characteristic to the self-driving car example described earlier. In this game called catcher, from the PyGame suite, the agent controls
a paddle that it can move back and forth to catch fruit that falls from the top of the screen (see Figure 7). A positive reward is given when the fruit is caught and a negative reward when the fruit
is not caught. The episode terminates after one piece of fruit falls. Analogous to the self-driving example, the agent does not control the position of the fruit, and so a representation that
maximizes $J_{inv}$ might discard that information. However, representing the fruit is crucial to obtaining reward, since the agent must move the paddle underneath the fruit to catch it. We learn
representations with $J_{inv}$ and $J_{fwd}$, optimizing $J_{fwd}$ with noise contrastive estimation (NCE), and $J_{inv}$ by training an inverse model via maximum likelihood. (For brevity, we omit
experiments with $J_{state}$ in this post – please see the paper!) To select the most compressed representation from among those that maximize each objective, we apply an information bottleneck of
the form $\min I(Z; S)$. We also compare to running RL from scratch with the image inputs, which we call ``end-to-end.” For the RL algorithm, we use the Soft Actor-Critic algorithm.
Figure 7. (left) Depiction of the catcher game. (middle) Performance of RL agents trained with different state representations. (right) Accuracy of reconstructing ground truth state elements from
learned representations.
We observe in Figure 7 (middle) that indeed the representation trained to maximize $J_{inv}$ results in RL agents that converge slower and to a lower asymptotic expected return. To better understand
what information the representation contains, we then attempt to learn a neural network decoder from the learned representation to the position of the falling fruit. We report the mean error achieved
by each representation in Figure 7 (right). The representation learned by $J_{inv}$ incurs a high error, indicating that the fruit is not precisely captured by the representation, while the
representation learned by $J_{fwd}$ incurs low error.
Increasing observation complexity with visual distractors
To make the representation learning problem more challenging, we repeat this experiment with visual distractors added to the agent’s observations. We randomly generate images of 10 circles of
different colors and replace the background of the game with these images (see Figure 8, left, for example observations). As in the previous experiment, we plot the performance of an RL agent trained
with the frozen representation as input (Figure 8, middle), as well as the error of decoding true state elements from the representation (Figure 8, right). The difference in performance between
sufficient ($J_{fwd}$) and insufficient ($J_{inv}$) objectives is even more pronounced in this setting than in the plain background setting. With more information present in the observation in the
form of the distractors, insufficient objectives that do not optimize for representing all the required state information may be “distracted” by representing the background objects instead, resulting
in low performance. In this more challenging case, end-to-end RL from images fails to make any progress on the task, demonstrating the difficulty of end-to-end RL.
Figure 8. (left) Example agent observations with distractors. (middle) Performance of RL agents trained with different state representations. (right) Accuracy of reconstructing ground truth state
elements from state representations.
These results highlight an important open problem: how can we design representation learning objectives that yield representations that are both as lossy as possible and still sufficient for the
tasks at hand? Without further assumptions on the MDP structure or knowledge of the reward function, is it possible to design an objective that yields sufficient representations that are lossier than
those learned by $J_{fwd}$? Can we characterize the set of MDPs for which insufficient objectives $J_{inv}$ and $J_{state}$ would be sufficient? Further, extending the proposed framework to partially
observed problems would be more reflective of realistic applications. In this setting, analyzing generative models such as VAEs in terms of sufficiency is an interesting problem. Prior work has shown
that maximizing the ELBO alone cannot control the content of the learned representation (e.g., Alemi et al. 2018). We conjecture that the zero-distortion maximizer of the ELBO would be sufficient,
while other solutions need not be. Overall, we hope that our proposed framework can drive research in designing better algorithms for unsupervised representation learning for RL.
This post is based on the paper Which Mutual Information Representation Learning Objectives are Sufficient for Control?, to be presented at Neurips 2021. Thank you to Sergey Levine and Abhishek Gupta
for their valuable feedback on this blog post. | {"url":"https://bair.berkeley.edu/blog/2021/11/19/mi-sufficiency-analysis/","timestamp":"2024-11-09T20:59:31Z","content_type":"text/html","content_length":"27371","record_id":"<urn:uuid:f214635d-a36b-4c83-87b2-f1d851b4be80>","cc-path":"CC-MAIN-2024-46/segments/1730477028142.18/warc/CC-MAIN-20241109182954-20241109212954-00460.warc.gz"} |
LINEAR MOTION Linear motion is moving in a
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LINEAR MOTION Linear motion is moving in a straight path. Motion is a relative thing. It depends on what you are comparing it to -While you are sitting in your seat are you moving? -Relative to your
desk – no -Relative to the sun - yes
Speed is the distance traveled by an object over a certain amount of time (mile/hr) s =d/t s means speed (in meters /second) d means distance (in meters) t means time (in seconds)
Speeds at Different Times Instantaneous Speed is the speed that an object is traveling at that moment -Do all objects in motion always travel at the same speed? Average Speed is the total distance
that an object traveled over then entire time it took to get there
Average Speed = total distance covered / time interval If you drove down to Detroit to see a Tigers game a distance of 117 miles and it took you 5 hours because you stopped for lunch and gas what is
your average speed? Average Speed = total distance covered / time interval Average Speed = 117 miles/5 hours = 23. 4 miles/hr
Velocity is speed in a given direction -Every time an object changes its direction even if it doesn’t change speed, that object is changing its velocity.
Acceleration is the rate at which an objects velocity is changing. a = ∆v/t a means acceleration (in meters / seconds 2) ∆ means change v means velocity (in meters / second) t means time (in seconds)
Calculation Acceleration A sports car can go from 0 miles/hr to 60 miles/hr in 4 seconds. What is the rate of acceleration of that car? a = ∆v/t a = (60 miles/hr – 0 miles/hr)/4 seconds a = 15 miles/
hr x seconds This means on average the car goes 15 miles/hr faster every second
Free Fall Objects accelerate when they are falling to Earth. - the acceleration of gravity is 10 m/s 2 Do all objects that fall from the sky continue to increase their speed until they hit the Earth
or is their some limit to their speed? - At some point the object will reach terminal velocity or the speed where it will not fall any faster (120 mph)
Wind Resistance If all objects accelerate because of gravity at the same rate (10 m/s 2), then why does a feather fall faster than a steel marble? The feather catches more area because it has a large
surface area for its size. This slow the feather down. Wind Resistance is the force of air particles colliding with a falling object and slowing it down
What about objects falling from the sky? Hang time is how long a projectile can hang in the air. - examples are jumping, punting the ball in football, launching a rocket v=gt v means velocity (in
meters/second) g means the acceleration due to gravity (10 m/s 2) t means time in seconds
Calculation Hang Time d = ½ g∆t 2 d means distance the object traveled (in meters) g means the acceleration due to gravity (10 m/s 2) ∆ means change t means time is seconds Solving for this equation
for time gives us the equation; t = √ 2 d/g | {"url":"https://slidetodoc.com/linear-motion-linear-motion-is-moving-in-a/","timestamp":"2024-11-06T02:44:52Z","content_type":"text/html","content_length":"66209","record_id":"<urn:uuid:0a7554cc-9b30-4bfd-97cf-c60c74cc31a5>","cc-path":"CC-MAIN-2024-46/segments/1730477027906.34/warc/CC-MAIN-20241106003436-20241106033436-00364.warc.gz"} |
A factory is shipping 9 bicycles to a store. Each bicycle has a mass of 11.6 kg and is packed in a box with a mass of 3.41kg. What is the total mass of all the bicycles and their packing boxes?
Find an answer to your question 👍 “A factory is shipping 9 bicycles to a store. Each bicycle has a mass of 11.6 kg and is packed in a box with a mass of 3.41kg. What is the ...” in 📗 Mathematics if
the answers seem to be not correct or there’s no answer. Try a smart search to find answers to similar questions.
Search for Other Answers | {"url":"https://cpep.org/mathematics/2750261-a-factory-is-shipping-9-bicycles-to-a-store-each-bicycle-has-a-mass-of.html","timestamp":"2024-11-13T18:02:06Z","content_type":"text/html","content_length":"24108","record_id":"<urn:uuid:ab295dc1-8d2a-4357-854d-4da048ca6d86>","cc-path":"CC-MAIN-2024-46/segments/1730477028387.69/warc/CC-MAIN-20241113171551-20241113201551-00120.warc.gz"} |
Namespace Syncfusion.PivotAnalysis.UWP
Namespace Syncfusion.PivotAnalysis.UWP
Displays the Average of values
Used to store the row header and column header values as BinaryList.
Calculation type defines the view for a particular computational object (or value field).
Computes the count of double or integer values.
This class defines a set of four integers that define a covered range in the zero-based coordinate system of a PivotTable.
This class is used to computes the sum of decimal values.
Displays the aggregated value in the PivotComputation column, if all the values are aggregated to be same else the value in PadString property is displayed.
Used to provide a list of display option.
This class is used to compute the average of double or integer values.
This class is used to compute the maximum of double or integer values.
This class is used to compute the minimum of double or integer values.
This class is used to compute the standard deviation of double or integer values.
This class is used to compute the sum of double or integer values.
This class is used to compute the variance of double or integer values.
Holds the Enumeration of Expandable states.
Denotes the different error types.
Class that illustrates the expression field support along with filtering in PivotEngine.
Class which has a method to determine whether the specified item is in the hidden group collection.
Gets the information about the PiovtFields.
Enum holding the field types.
This class encapsulates the information needed to define a filter.
This class encapsulates the support for computing filter values and expressions.
Class that holds different filter items.
Class that holds the collection of objects that are filtered.
Use to get the property value without using reflection.
Specifies the layout for the Pivot control.
Class used to hold the properties of each hidden group.
Index Engine is used to load the collection of data faster than the PivotEngine. It makes use of the "EnableOnDemandCalculations" property. Unlike the PivotEngine, it loads the entire collection of
data by including all the values in the list irrespective of Row and Column headers.
This class is used to compute the sum of integer values.
Class that holds primarily the information on one row in a PivotTable. Additionally, these calls can be used to hold information on the row/column header structures.
Displays the maximum value of given values
Displays the minimum value of given values
This class provides an information about a specific cell in a PivotTable.
Gets the information from the Pivot control.
Enumerates the possible Pivot cell types.
This class holds the information needed for the calculations that appear in a Pivot control. For each calculation seen, there is an associated PivotComputationInfo object that is added to the
PivotComputationInfo collection.
This class encapsulates pivoting calculation support. To use it, first populate the PivotColumns and PivotRows collections to define the properties being pivoted. Then you populate the
PivotCalculations collection to define the values you would like to see populated.
This class used for sorting the Pivoted row/column.
Class that holds the properties which is used to handle single sort and multi sort in Grid.
This class allows a PivotEngine to automatically respond to changes in the underlying data provided that data supports appropriate events.
Class that holds different Grid constants.
Enacapulates the information needed to define a PivotItem, for either a row/column Pivot.
Information on what changes have occurred in the Pivot Schema.
Event handler for the PivotSchemaChangedEventHandler event.
Enumeration provides option to select row type.
Specifies the which action is needs to be taken in PivotGrid control dynamically when perfomring any operation.
Class used to compare and sort the given collection.
Displays the standard deviation of given values
Displays the standard deviation population of given values
Displays the sum of values
This is an abstract class that defines the necessary functionality to do PivotCalculations.
Controls whether a summary calculation is to be displayed for all levels or only for the inner-most level.
This class is primarily for internal use. It is used to generate the rows and columns that hold summaries of PivotCalculations.
Enumerates the summary types available for calculations in the Pivot control. If you use the value "Custom" in a ComputationInfo object, then you are required to explicitly set the
ComputationInfo.Summary value.
Displays the Variance of given values
Displays the VarianceP of given values
Use this interface to enable shortcut calculations to adjust a summary when an underlying value changes. The idea is to avoid re-computing the summary from scratch. For example, if your summary
computes the total of a set of values, you can quickly adjust this computed total when a value changes by subtracting the old value and adding the new value without having to re-compute the total
from scratch. Not all types of calculations lend themselves to this shortcut behavior.
Provides complete functionality of the Pivot control. | {"url":"https://help.syncfusion.com/cr/uwp/Syncfusion.PivotAnalysis.UWP.html","timestamp":"2024-11-13T13:18:39Z","content_type":"text/html","content_length":"30120","record_id":"<urn:uuid:9e28923f-222b-4acb-869c-5e76c175816d>","cc-path":"CC-MAIN-2024-46/segments/1730477028347.28/warc/CC-MAIN-20241113103539-20241113133539-00261.warc.gz"} |
Computing - Hash Tables
What is the main advantage of a hash table?
You can quickly find items given a key.
A hash table is a type of what data type?
How is an element’s key calculated?
A key-value hash table is sometimes called what?
What is a hashing algorithm?
A calculation applied to a key to transform it into an address.
What is it called when a hash function generates the same value for two different inputs?
What is it called when you find a new place for a value after two inputs have the same hash?
What is the formula for load factor?
\[\frac{\text{total items}}{\text{capacity}}\]
What can load factor be used for?
Calculating when it is neccassary to allocate more space to store something.
Once a hash has been calculated, what operation is used to normalise it to the length of the array?
What is a common method of calculating a hash for a string?
Adding together the ASCII numbers of the characters.
Related posts | {"url":"https://ollybritton.com/notes/a-level/computing/topics/hash-tables/","timestamp":"2024-11-09T12:20:27Z","content_type":"text/html","content_length":"505260","record_id":"<urn:uuid:caf1ed91-1ce4-4808-aea1-4458c13b5043>","cc-path":"CC-MAIN-2024-46/segments/1730477028118.93/warc/CC-MAIN-20241109120425-20241109150425-00104.warc.gz"} |
Faculty Research | Union College
Sean Carney
applied and computational mathematics
My research field is applied and computational mathematics, where I work on problems in numerical analysis, multiscale and stochastic modeling, and optimization problems with partial differential
equations as constraints. I am most interested in conducting research that enhances our understanding, prediction, or control of physical phenomena. Recently, I have worked on projects related to
homogenization theory, multiphase fluid mixtures, and optimization under uncertainty.
Paul Friedman
representation theory, Lie groups
My research focused on the representation theory of Lie groups. In particular, I looked at how certain representations produced by cohomological induction fit into the Langlands classification
Rylan Gajek-Leonard
number theory
My research is in algebraic number theory and arithmetic geometry. I am particularly interested in Iwasawa theory, a beautiful topic which bridges analytic objects (e.g., complex L-functions such as
the classical Riemann zeta-function) and algebra (e.g., arithmetic in number fields, structure of rational points on curves). My current research is concerned with the behavior of p-adic L-functions
attached to modular forms.
Ellen Gasparovic
computational geometry and topology, differential topology
I use techniques from algebraic and differential topology that are inspired by questions in image analysis and high-dimensional data analysis. On the image and shape analysis side, one of the main
tools I have used is Blum’s medial axis, a skeleton-like topological structure that captures the shape and geometric properties of a region. On the data analysis side, the techniques I utilize fall
under the heading of applied and computational topology, a relatively new branch of mathematics that focuses on discovering global and local structure within a dataset and finding meaning in the
“shape” of the data. I have worked on applications of topology to problems arising in image recognition, machine learning, and geometric data modeling. In recent work, I have focused extensively on
topological characterizations and summaries for metric graphs and simplicial complexes associated to them.
Jeffrey Hatley
number theory, arithmetic geometry
I work in number theory, a classical branch of mathematics which is primarily focused on understanding two things: the properties of prime numbers and solutions to polynomial equations. Using special
geometric objects like elliptic curves and modular forms, for each prime number p we can construct p-adic Galois representations. Each of these Galois representations allows us to gain a little bit
of information about all polynomial equations at once. I study various properties of these Galois representations. I’m especially interested in families of Galois representations which arise from
geometric objects which are congruent mod p.
Roger Hoerl
Big Data analytics, statistical engineering, experimental design
Much of my early research was in the area of regression analysis, especially shrinkage estimators. In the private sector I developed a greater appreciation for, and interest in, experimental design
methods. I have recently investigated Big Data analytics, particularly how and why things can go wrong when analyzing massive data sets. This ties to the discipline of statistical engineering, which
emphasizes effective integration of multiple statistical and non-statistical methods in an overall approach to scientific inquiry. I am currently conducting research into how statistical engineering
can provide effective strategies for attacking Big Data problems.
Jeffrey Jauregui
geometric analysis, general relativity
I work in geometric analysis, emphasizing connections with general relativity. Einstein’s theory of general relativity describes the universe as a spacetime, which is a four-dimensional continuum
containing all points and events, past, present and future. Gravitational effects (for instance, due to a black hole) manifest through the curvature of spacetime, and thus geometry plays an important
role in the theory. My research typically involves scalar curvature and explores connections between mass and geometry, including “quasi-local” mass and the total “ADM” mass of a spacetime. My
current interests include convergence of sequences of asymptotically flat manifolds, Bartnik’s quasi-local mass conjectures, and codimension-two geometric flows within a spacetime.
Brenda Johnson
homotopy theory, homological algebra
I am interested in algebraic topology, especially homotopy theory. My work has focused primarily on the calculus of homotopy functors, and, in particular, on developing alternative models for the
calculus of homotopy functors for use in algebraic settings and more general homotopy-theoretic contexts. Recent work involves developing a general framework for understanding functor calculus that
includes the calculus of homotopy functors and orthogonal calculus and provides a means for developing new calculus theories as well. I have also worked with undergraduates on problems in knot
theory, especially questions about intrinsically linked and knotted graphs.
Leila Khatami
commutative algebra, homological algebra
My main research interests lie in the field of commutative algebra, with connections to algebraic geometry and homological algebra. Early in my career, my research was focused on the use of
homological methods in commutative algebra. More specifically, I studied Gorenstein dimensions of modules over commutative local rings. In recent years, my research has shifted directions to include
problems concerning pairs of commuting nilpotent matrices. My interest in this project originates from my commutative algebra background and my algebraic geometry interests, while the nature of the
study has invited the use of additional tools from algebraic combinatorics and representation theory.
Phanuel Mariano
probability, geometry, analysis, partial differential equations
My research lies at the intersection of probability, geometry and analysis. In particular, I am interested in using probabilistic tools to solve problems in analysis in the setting of curved spaces
with degeneracies. The operators I study are hypoelliptic and their natural setting are spaces called sub-Riemannian manifolds. A probabilistic tool used in my research involves the coupling of
diffusion processes. Coupling is a way of constructing Markov processes with prescribed laws on the same space. I have also been interested in applying probabilistic techniques in proving
inequalities for the expected lifetime of diffusion processes and the first Dirichlet eigenvalue of a domain. One can think of the first Dirichlet eigenvalue as the fundamental frequency of a drum.
These inequalities often show the beautiful connection between probability, analysis and physics.
Grant Moles
commutative algebra, algebraic number theory
My research draws techniques and questions from the field of commutative algebra and applies them to objects of interest to the field of algebraic number theory. In particular, my research has
largely focused on factorization in orders within algebraic number fields. Related to this question is the idea that, given certain properties, we can often draw conclusions about a new, unexplored
ring from what is already known about a simpler, more familiar ring. I am currently looking into certain relationships between rings and how they might lead factorization properties to be maintained
or predictably change.
Kim Plofker
history of mathematics
I study the historical development of math, astronomy and related subjects, mostly in Sanskrit, Arabic and Latin texts from before the twentieth century. Research travel takes me most often to India,
where there are tens of thousands of manuscripts of little-known early scientific works that I use in piecing together this history. Topics I’ve published on include the early history of numerical
approximation methods in Sanskrit texts, different approaches to spherical trigonometry in Islamic and Indian astronomy, and Euler’s study of Indian calendar computations. These days I’m
collaborating with a colleague in New Zealand on a study of algorithms in Indian astronomical tables; we’re hoping to break out some of the work into undergraduate research projects (no Sanskrit
knowledge required!).
Junqing Qian
differential geometry, number theory
My research interest is at the intersection of number theory and differential geometry. More precisely, I am primarily interested in capturing algebraic information to help with problems in geometry
and geometric analysis. For example, I discovered a connection between the Kähler–Einstein metric on punctured spheres, a subject in differential geometry, and modular functions, a subject in number
theory; this connection overcame the obstacle from other methods and settled the metric problem, a problem in geometry. World-leading mathematicians projected the existence and importance of further
connections between the two different fields. Some discovered ones have been applied in physics, such as string theory. Naturally, I am also interested in topics in both fields. On the algebraic
side, I am interested in studying and developing the theories of arithmetic differential equations and geometry; on the analytic side, I am interested in Calabi flow on toric manifolds and exploring
mathematical physics.
Christina W. Tønnesen-Friedman
differential geometry, Kähler geometry
The overarching motivation in my research is the classification and explicit construction of Riemannian metrics with properties that generalize the Einstein property. For the first many years of my
career I studied this in the realm of Kähler Geometry. I mostly focused on extremal Kähler metrics but I also considered constant scalar curvature metrics, (generalized) Ricci solitons, weakly
Bochner-flat metrics, and other Kähler metrics with special geometric properties.
In the last few years I have expanded this interest to include the odd-dimensional sibling of Kähler geometry, namely Sasakian Geometry.
Jue Wang
fluid dynamics and turbulence, medical image processing and analysis, mathematics in health care
Not long after joining Union College I started research in medical imaging. I have worked on attenuation compensation and signal segmentation in ultrasound images, 3D recursive Bayesian vascular
extraction, ultrasound modulated optical tomography assessment of osteoporosis, artifact correction in digital radiography, trans-abdominal ultrasound imaging in prostate cancer radiotherapy
treatment planning, tumor detection in screening breast ultrasound, and cancer classification with deep learning. My research goal is to develop effective models and methods, that help access
clinically-meaningful information embedded in complex data, in order to increase the accuracy of medical diagnoses, and enhance early cancer detection and intervention.
Fanhui Xu
stochastic analysis, partial differential equations
My research focuses on stochastic analysis and partial differential equations, with a current emphasis on the regularization phenomena induced by noise in ordinary and partial differential equations.
Differential equations are fundamental to mathematical modeling. Typically, if the coefficients of an equation are not sufficiently regular, the equation may lack a well-defined solution, even in a
weak analytic sense. However, it has been shown that the introduction of a noise term can make the equation solvable. I am particularly interested in characterizing such noise and investigating the
properties of solutions to stochastic (partial) differential equations. My research primarily employs tools such as Kolmogorov equations and harmonic analysis.
Research descriptions for faculty emeriti
Julius Barbanel
logic, set theory, fair division
I began my research career in set theory. In particular, my interests were in large cardinal theory, which is the study of very large infinite sets. After about fifteen years in this field, I moved
into game theory, focusing specifically on fair division. This involves the allocation of goods among a collection of players, where the goals include both fairness and efficiency. I worked on both
abstract existence results and on algorithms in this area. After about fifteen years in this field, I became interested in the ancient Greek foundations of modern mathematics. I developed a general
education course on this subject, called “Ancient Greek Mathematics”.
Davide P. Cervone
simplicial geometry, topology
My mathematical research centers around polyhedral geometry in three and four dimensions. I have studied immersed surfaces in space that have the fewest possible vertices, and surfaces that have a
special cutting property called “tightness”, where any plane will divide the surface into at most two pieces. My most significant result represents one of the few cases in low dimensions where the
polyhedral theory differs in a significant way from the smooth case. Much of my work involves computer software that I have developed, and I contribute to a number of open source projects, including
MathJax (for displaying math notation on the web) and WeBWorK (an on-line homework system used at Union). I have been active in exploring how to communicate mathematics in new ways since the earliest
days of the internet.
Kathryn Lesh
algebraic topology, unstable homotopy theory
I study algebraic topology, specifically unstable homotopy theory. I started out by studying the unstable Adams spectral sequence, in a problem related to the Sullivan Conjecture on maps from
projective spaces. I returned to the unstable Adams spectral sequence in two later papers on the infinite orthogonal group SO. Most of my recent work has to do with connections between the Goodwillie
calculus and the Whitehead Conjecture (proved by Kuhn and Priddy), along with the analogous connections between the orthogonal calculus and an unproved version of the Whitehead Conjecture for
connective complex K-theory.
Susan Niefield
exponentiability, double categories, toposes, locales, quantales
My research involves using category theory, especially adjoint functors, to draw analogies between different areas of mathematics. Much of this work concerns characterizing exponentiable morphisms in
non-locally cartesian closed categories (including topological spaces, locales, toposes, posets, and affine schemes), and finding relationships between these characterizations. I am also interested
in structures which capture similarities between different mathematical objects, e.g., quantales (which relate the lattice of ideals of a ring to that of the open subsets of a space) and double
categories (which relate topological spaces, locales, quantales, toposes, posets, modules, and small categories).
Kimmo Rosenthal
category theory, algebra, logic
I have been teaching mathematics at Union College for over three decades. Prior to 2000 I published two books and 27 articles on various aspects of category theory. From 2000-2008 I served as Dean of
Studies at the college, overseeing the curriculum and the academic lives of the entire student body. After resigning as Dean, I returned to the mathematics classroom with a renewed vigor and
enthusiasm. Since 2008 I have regularly taught the First-Year Preceptorial, a required seminar on critical reading and writing. At the same time my attention has turned from mathematical research to
writing. I have had a dozen publications (mostly fiction) appear in various literary journals.
Alan Taylor
set theory, logic, simple games, social choice, mathematical political science
My graduate training was in the field of mathematical logic, and I spent the first fifteen years of my career doing infinitary combinatorics. Most of my work involved ultrafilters on omega, ideals on
uncountable cardinals, and partition theory (including a bit of work with finite Ramsey theory). I spent the following fifteen years with a number of questions from the area of “fair division” and
with some topics arising from the theory of voting. Here, I was primarily studying simple games. For the past decade I have returned to set theory with somewhat of a focus on coordinated inference as
captured by so-called hat problems.
Karl Zimmermann
number theory, formal groups
Most of my research is in the area of arithmetic geometry and in particular, formal groups, but more recently, I’ve become interested in a problem related to both formal groups and local (p-adic)
dynamical systems. While thinking about a conjecture of Lubin involving power series that commute under composition, I started working with commuting polynomials that have coefficients in the complex
numbers. A complete classification of these polynomials was given around 1920 by Ritt and Julia, working independently. That result is easily stated, but the proof is topological and fairly deep. My
current work involves looking at commuting polynomials from an algebraic point of view and trying to apply formal group theoretic techniques, when possible, to gain a better understanding of the
William S. Zwicker
mathematical logic, set theory, game theory, voting and political power, social choice theory, fair division
My early research was in combinatorial set theory, especially large cardinals and ideals and filters on Pκλ. Today, I work on applications of mathematics to the social sciences: voting and social
choice, fair division, and cooperative game theory. I’ve enjoyed co-authoring papers with mathematicians, political scientists, economists, and undergraduates from Canada, Catalonia, France, Ireland,
Italy, Turkey, Venezuela, and the U.S. I’m on the editorial board of Mathematical Social Sciences and co-author, with Alan D. Taylor, of the monograph Simple Games (Princeton, 1999). I’m most
attracted to fundamental issues in the social sciences that lead to questions of independent combinatorial or geometric interest. For a taste, try our online interactive rubber band voting simulator
(with Davide Cervone).
Our faculty conduct research in fields as diverse as commutative algebra, topological data analysis, probability, number theory, statistics, algebraic topology, differential geometry, medical
imaging, the history of mathematics, and more.
Please read below for detailed descriptions of faculty research interests. | {"url":"https://www.union.edu/mathematics/faculty-research","timestamp":"2024-11-02T02:40:04Z","content_type":"text/html","content_length":"199586","record_id":"<urn:uuid:c06da35d-14cd-4674-8174-2aecc63ada81>","cc-path":"CC-MAIN-2024-46/segments/1730477027632.4/warc/CC-MAIN-20241102010035-20241102040035-00140.warc.gz"} |
CCER seminar: Menno Bokdam
Title: On-the-fly machine learning force fields with near first principles precision: Predicting phase transitions in complex Dynamic Solids
Speaker: Menno Bokdam (UT, MESA+)
Time: March 31, 2022, 10:00–11:00
Location: Hybrid: TU/e (Flux 1.124) and online (MS Teams)
Abstract: Lattice dynamics at the atomic scale is often well described by phonons in the harmonic approximation. However, it does not always suffice. For example, it does not explain a crystal’s
phase transitions or why some materials have ultra-low thermal conductivity. While a more accurate method, ab-initio molecular-dynamics (MD), captures the anharmonicity of the atomic interactions
correctly, it is computationally orders of magnitude too expensive to describe the effects of phonon scattering related to the "rattling" and "flipping" of atoms and molecules. In this talk I will
present a recently developed method that can account for these effects based on an on-the-fly Machine- Learning Force-Field (MLFF) approach [1]. It allows us to automatically ‘train’ smooth and
‘cheap’ models of the potential energy surface based on density functional theory calculations. The MLFF gives access to the nanosecond time- and tens of nanometer length-scales and opens up the
possibility to predict complex phase transitions, capture the formation and breaking of weak bonds, ion diffusion and simulate lattice thermal conductivity in complex ‘Dynamic Solids’. It enables
linking to experiments such as NMR dipolar coupling [2] and momentum resolved phonon spectroscopy [3]. I will illustrate the capabilities of the approach with several examples from the halide
1. Jinnouchi, Lahnsteiner, Karsai, Kresse and Bokdam, Phase transitions of hybrid perovskites simulated by machine- learning force fields trained on the fly with Bayesian inference, Phys. Rev. Lett.
122, 225701 (2019)
2. Grueninger, Bokdam, Leupold, Tinnemans, Moos, de Wijs, Panzer and Kentgens, Microscopic (dis)order and dynamics of cations in mixed FA/MA lead halide perovskites, J. Phys. Chem. C, 125, 1742-1753
3. Lahnsteiner and Bokdam, Anharmonic lattice dynamics in large thermodynamic ensembles with machine-learning force fields: CsPbBr[3] a phonon liquid with Cs rattlers, Phys. Rev. B 105, 024302
The CCER seminars are aimed at researchers interested in computational approaches to (energy) research. The seminar is small-scale, typically 15 participants, and interactive, offering lots of room
for discussion. If you would like to attend, just This email address is being protected from spambots. You need JavaScript enabled to view it.. | {"url":"https://www.ccer.nl/past-events/ccer-seminar-220331","timestamp":"2024-11-13T05:20:21Z","content_type":"text/html","content_length":"19111","record_id":"<urn:uuid:047c3b28-e6e6-446c-9b97-f38bcbc1b7f9>","cc-path":"CC-MAIN-2024-46/segments/1730477028326.66/warc/CC-MAIN-20241113040054-20241113070054-00051.warc.gz"} |
Network – Troels Christensen
eric.ed.gov har udgivet: This packet includes reprints of journal articles and other resources concerning the assessment of science and math in small, rural elementary schools. Articles include: (1)
“Standards, Assessment, and Educational Quality” (Lauren B. Resnick); (2) “A True Test: Toward More Authentic and Equitable Assessment” (Grant Wiggins); (3) “How World-Class Standards Will Change Us”
(Arthur L. Costa); (4) “Smart Tests” (Deborah L. Cohen); (5) “Laser Disk Portfolios: Total Child Assessment” (Jo Campbell); (6) “Portfolios Invite Reflection–from Students and Staff” (Elizabeth A.
Hebert); (7) “Portfolio Assessment in the Hands of Teachers” (Clare Forseth); (8) “Portfolio Assessment” (Susan Black); (9) “Assessing the Outcomes of Computer-Based Instruction: The Experience of
Maryland” (Gita Z. Wilder, Mary Fowles); (10) “Why Standards May Not Improve Schools” (Elliot W. Eisner); (11) “Assessing Alternative Assessment” (Gene I.… Continue Reading
Eric.ed.gov – Learning in Linguistically Diverse Middle School Classrooms: The Role of the Classroom Peer Network
eric.ed.gov har udgivet: The literature suggests there is much to be gained from exploring the role of the peer network in linguistically diverse “mainstream” middle school classrooms (i.e.,
classrooms that include English language learners alongside fluent English-speakers). The present study uses social network analysis to examine whether between-classroom and between-student variation
in cross-language-status integration in the classroom peer network may contribute to between-classroom and between-student differences in learning. Data from a larger mixed-methods study at a
linguistically diverse middle school in the southeastern United States are analyzed to test two hypotheses: (1) Classrooms with more linguistically integrated peer networks (i.e., those in which the
network of friendships in the classroom is less segregated by ELL status) will show greater growth in classroom mean standardized test scores across the school year;… Continue Reading
tandfonline.com – Examining Islamic piety at workplace via an artificial neural network
tandfonline.com har udgivet en rapport under søgningen “Teacher Education Mathematics”: Examining Islamic piety at workplace via an artificial neural network Link til kilde
tandfonline.com – Lack of Theory Building and Testing Impedes Progress in The Factor and Network Literature
tandfonline.com har udgivet en rapport under søgningen “Teacher Education Mathematics”: Lack of Theory Building and Testing Impedes Progress in The Factor and Network Literature Link til kilde
Eric.ed.gov – Math Network Curriculum Project. Project Summary; Final Report.
eric.ed.gov har udgivet: This document summarizes the work of the Math Curriculum Project at San Francisco State University. The project developed seven curriculum units for the middle school
mathematics program, using microcomputers as a problem solving tool to foster mathematical thinking and develop insights into mathematical concepts. They also created a prototype telephone network
that is both a message system and a curricular data base for activities in each unit. Finally, they developed a teacher training model from their experiences in piloting the materials. The report
describes the objectives, methods and procedures, outcomes, and dissemination activities of the project. An overview of the units, a network manager manual, a message system user manual, and a list
of talks about the project are appended. (MNS) Link til kilde
Eric.ed.gov – Sports Shorts. [A Product of] the Regional Math Network: A Teacher Invigoration and Curriculum Development Project.
eric.ed.gov har udgivet: This middle school mathematics unit is organized around themes relating to sports activities in the Boston (Massachusetts) region and has a content focus on decimals and
percents. The activities follow a story line which features a sports reporter (the student) and his/her assignments and adventures. Each activity begins with a headline, defines a task, and includes
a follow-up question. The unit is organized by categories dealing with: (1) Sullivan Stadium (and football); (2) Fenway Park (and baseball); (3) Boston Garden (and basketball and hockey); (4) the
Boston Marathon; and (5) Miscellaneous Sports. The unit could also be arranged by season, content development sequence, or activity. The materials include student worksheets, fact sheets, editor’s
notes, transparency masters and game cards. The math themes that extend throughout the activities… Continue Reading
tandfonline.com – An exploration of the DSM-5 posttraumatic stress disorder symptom latent variable network
tandfonline.com har udgivet en rapport under søgningen “Teacher Education Mathematics”: An exploration of the DSM-5 posttraumatic stress disorder symptom latent variable network Link til kilde
Eric.ed.gov – Quincy Market. [A Product of] the Regional Math Network: A Teacher Invigoration and Curriculum Development Project.
eric.ed.gov har udgivet: In this middle school mathematics unit two imaginary characters, Horatio and Portia, decide to make their fortune in Quincy Market (Boston, Massachusetts) running a Bull
Market cart. In order to solve the problems that they encounter, they need to learn ratio and proportion, map reading, estimation, area and perimeter, population sampling, problem solving, and the
collecting and processing of data. Teacher notes at the beginning of each section indicate the math objectives, materials, and whether the activity is a reinforcement or an extension of a math skill.
The unit is divided into seven modules that can be used either independently or sequentially. These are: (1) an introduction to Quincy Market; (2) the use of the ruler; (3) map exploration; (4) ratio
and proportion; (5) scale drawing; (6)… Continue Reading
Eric.ed.gov – ENLIST-Micros Teacher Network for Rural Math & Science Teachers.
eric.ed.gov har udgivet: ENLIST-Micros (ENcourage LIteracy in Science Teachers’ uses of Microcomputers) develops state networks of science and mathematics teachers providing inservice education and
support for the implementation of computers and technology in the classroom. In Alabama, the project operated from August 1990 through June 1994. Most inservice workshops were held at Auburn
University. Participants included 50 urban, 22 suburban, and 31 rural teachers from schools in Montgomery and the Auburn area. The first 2 years of the project focused on training the teacher
participants to use microcomputers and to share their knowledge with other teachers. In the third and fourth year, veteran teachers provided individual training and inservice workshops to other
teachers. Teacher reactions were overwhelmingly positive and frequently focused on the collegiality and mutual support experienced in the… Continue Reading
Eric.ed.gov – Math Space Mission. [A Product of] the Regional Math Network: A Teacher Invigoration and Curriculum Development Project.
eric.ed.gov har udgivet: This unit is intended to teach estimation skills in such a way as to be relevant and useful to students as they apply them in various problem-solving activities. The teaching
activities feature the earth, exploration into space, and the other worlds in the solar system. The teacher’s guide contains four modules. Module I suggests the use of several multi-media experiences
to set the stage for the activities that follow. Module II, “The Solar System,” incorporates teaching activities dealing with rounding numbers, estimation of sums, differences, products and
quotients, graphing, and the application of these skills in problem solving. Module III, “The Space Shuttle,” addresses the use of the space shuttle and stresses the mathematical concepts of ratio
and proportion. Module IV,”The Space Colony,” uses geometric concepts as… Continue Reading | {"url":"http://troelschristensen.dk/tag/network/","timestamp":"2024-11-04T14:50:54Z","content_type":"text/html","content_length":"158369","record_id":"<urn:uuid:bb0817e1-1c87-4942-8511-ea474a238e06>","cc-path":"CC-MAIN-2024-46/segments/1730477027829.31/warc/CC-MAIN-20241104131715-20241104161715-00042.warc.gz"} |
How far is Pangasinan from Manila via E1
The road driving distance between Pangasinan to Manila via E1 is 210 Km. Depending on the vehicle you choose to travel, you can calculate the amount of CO2 emissions from your vehicle and assess the
environment impact. Check our
Fuel Price Calculator
to estimate the trip cost. | {"url":"https://www.distancesfrom.com/ph/how-far-is-Pangasinan-Philippines-from-Manila-via-E1/HowFarHistory/46383076.aspx","timestamp":"2024-11-10T18:15:28Z","content_type":"text/html","content_length":"184130","record_id":"<urn:uuid:fe6de1f5-6848-4a0c-bcb4-a190be742ed6>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.61/warc/CC-MAIN-20241110170046-20241110200046-00136.warc.gz"} |
Perlin Noise
What am I looking at?
This is a visualization of perlin noise.
For this simulation a 3-variable implementation of the Perlin noise algorithm is used to generate a vector field on the canvas which acts upon thousands of particles. The inputs for the algorithm are
two spacial dimensions (x,y) corresponding to position on the canvas and a third variable z which increments with time. The output is a unit vector that points in a random direction varying over
time, the particles subject to the vector field in turn trace out beautiful chaotic paths.
The special feature of the algorithm is that it produces a smooth randomness so that adjacent cells do not point in completely opposite directions.
Click options above to experiment with different visual parameters controlling the simulation. | {"url":"https://romankitsela.com/perlin/index.html","timestamp":"2024-11-09T09:30:56Z","content_type":"text/html","content_length":"10778","record_id":"<urn:uuid:78aa4d57-2711-4e7e-b496-a167892815b2>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.75/warc/CC-MAIN-20241109085148-20241109115148-00738.warc.gz"} |
I need help with my geometry homework that involves proofs, theorems, and finding the value of x
Waiting for answer This question has not been answered yet. You can hire a professional tutor to get the answer.
I need help with my geometry homework that involves proofs, theorems, and finding the value of x
I need help with my geometry homework that involves proofs, theorems, and finding the value of x
Show more
Homework Categories
Ask a Question | {"url":"https://studydaddy.com/question/i-need-help-with-my-geometry-homework-that-involves-proofs-theorems-and-finding","timestamp":"2024-11-02T05:56:44Z","content_type":"text/html","content_length":"26356","record_id":"<urn:uuid:f4b95d75-0a78-4dd0-904f-487e875b077f>","cc-path":"CC-MAIN-2024-46/segments/1730477027677.11/warc/CC-MAIN-20241102040949-20241102070949-00237.warc.gz"} |
da Silva.
Assistant Professor of Mathematics
Texas A&M University - San Antonio
Department of Mathematics
Classroom Hall, Room 314 U
: gdasilva@tamusa.edu
I completed my PhD in Mathematics at Washington University in St. Louis under direction of
Prof. Matt Kerr
, after that I was a postdoc at Imperial College London under supervision of Prof.
Tom Coates
and Prof.
Alessio Corti
Research Interests
Analysis of PDE
I'm interested in nonlinear elliptic PDEs and elliptic systems, problems related to existence and regularity of solutions.
Hodge theory & Complex Algebraic Geometry
I'm interested in Algebraic cyles and its connections. Topics include cycle class maps, Hodge-D and Hodge conjectures, higher Chow groups, real regulators and related topics.
Current Teaching
Research Papers
15. Real Analysis: Functions of a real variable. (In talks with Springer UTM)
Selected Past Teaching
• Differential Equations
• Spring 2023
• The Hodge Conjecture: A million dollar problem
• Remarks on the Hodge Conjecture for Fermat Varieties
• Ph.D. thesis
• On the arithmetic of Landau-Ginzburg model of a certain class of 3-folds (My Ph.D. major oral)
Computer Code
• Complete Intersections Hodge numbers calculator
Type the multidegree (as list of numbers):
Type the dimension (as a unique number):
A rank of my 10 favorite textbooks
With respect to the didatics (how easy it is to follow) of the author not the content of the book.
1. Riemannian Geometry, M. do Carmo
2. Differential Geometry of Curves and Surfaces, M. do Carmo
3. Introduction to Real Analysis, E. Lages Lima
4. Principles of Algebraic Geometry, P. Griffiths
5. Algebraic Geometry, R. Hartshorne
6. Fourier Analysis, E. Stein
7. Complex Analysis, L. Ahlfors
8. Partial Differential Equations, L. Evans
9. Abstract Algebra, D. Dummit | {"url":"https://www.gdasilvajr.com/","timestamp":"2024-11-07T09:27:15Z","content_type":"text/html","content_length":"14116","record_id":"<urn:uuid:a09f552e-4840-4f27-adf8-a80aeb746f5c>","cc-path":"CC-MAIN-2024-46/segments/1730477027987.79/warc/CC-MAIN-20241107083707-20241107113707-00368.warc.gz"} |
Dynamical Systems Generated by Linear Maps by Ćemal B. Dolićanin, Anatolij B. Antonevich (auth.)
By Ćemal B. Dolićanin, Anatolij B. Antonevich (auth.)
The publication bargains with dynamical platforms, generated by means of linear mappings of finite dimensional areas and their purposes. those platforms have a comparatively easy constitution from
the viewpoint of the trendy dynamical structures conception. notwithstanding, for the dynamical platforms of this kind, it truly is attainable to acquire specific solutions to precise questions being
necessary in applications.
The thought of difficulties are average and glance quite easy, yet in truth during research, they confront clients with lots of sophisticated questions and their unique research wishes a considerable
The difficulties coming up are relating to linear algebra and dynamical structures thought, and hence, the booklet could be regarded as a traditional amplification, refinement and complement to
linear algebra and dynamical platforms idea textbooks.
Read Online or Download Dynamical Systems Generated by Linear Maps PDF
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Extra resources for Dynamical Systems Generated by Linear Maps
Sample text
Note that the number k(x) is the largest of values k, thus at least one of the coordinates x(k, j, i, l) is different from zero. The number s(x) is the largest of the values l, thus at least one of
the coordinates x(k(x), j, i, l) is different from zero. 26 3 Representation of the Vector Trajectory This discussion proves the following statement concerning the complete expansion of the vector
trajectory. 1 (On the complete expansion of the positive semi-trajectory of a vector). Let A be an invertible operator of a finite-dimensional vector space.
The Archimedean axiom. If x ∈= 0 and |x| < |y|, then there is a natural number n, such that |nx| > |y|. A norm | · | on a field is called archimedean, if this axiom holds. A norm | · | on a field is
called non-archimedean, if from the condition |x| < |y| it follows that |nx| < |y| for any natural number n. 1 Let |·| be a norm in a field K . The following conditions are equivalent: (1) the
stronger triangle inequality holds |x + y| ≤ max{|x|, |y|}. (2) it is non-archimedean norm; (3) the set of natural numbers is bounded; (4) the set of all natural numbers is bounded by the number 1: |
n| ≤ 1 for any n.
Since the trajectory An x of such a point tends toward the trajectory of the vector w, the set of limit points of the trajectory An x is equal to the set of limit point of the trajectory of the point
w. The point w belongs to a subspace where the operator acts as unitary. 3 If r (x) = 1 and s(x) = 1, then the trajectory An x tends toward the trajectory of the vector w and λ(x) = λ(w) = [w] ∃ Tϕ
(w) /Hω(ϕ(w)) . 3 The Action of a Linear Operator on a Sphere In the description of the trajectory of the point, it is natural to follow not only the behavior of the points of the trajectory An x,
but also the behavior of the “directional vectors” ≤A1n x≤ An x.
Rated of 5 – based on votes | {"url":"http://en.magomechaya.com/index.php/epub/dynamical-systems-generated-by-linear-maps-by-cemal-b-dolicanin-2014-07-20","timestamp":"2024-11-12T19:50:10Z","content_type":"text/html","content_length":"28862","record_id":"<urn:uuid:ff040417-d8fe-426c-84fd-9e7275becb8e>","cc-path":"CC-MAIN-2024-46/segments/1730477028279.73/warc/CC-MAIN-20241112180608-20241112210608-00122.warc.gz"} |
Create a Time Span Field
A time span field is a measure that calculates the time difference between two date/time fields (or a date/time field and a specified date).
To create a time span field:
1. Select Cross > Define Field. The Define Fields window displays.
2. Click Time Span. The Time Span Field window displays.
3. Enter the following details:
Field Description
Field Name Enter a name for the field.
Number of Select whether to measure the time difference in days, months or years.
Select the start and end points of the time span.
• Only date/time type fields will be available for selection.
• For the end point, instead of selecting a field you can optionally specify a fixed date.
4. Click OK.
Note About Time Calculations
By default, SuperSTAR uses an estimated method for calculating the difference between dates.
If you request that the difference between two dates is specified in years then the value is calculated by dividing by 365.25 (i.e. the effect of leap years is averaged out). This can result in
inaccuracies if you calculate a range UDF on top of a time span UDF for dates that are on exact year boundaries. In this case the computed value will be slightly greater, or slightly less than the
integer multiple of years depending on how many leap days fall into the range.
For example, if you calculate the days between 1/1/2001 and 1/1/2002 then SuperCROSS returns 365 / 365.25 = 0.9993, whereas a user (and the range UDF) would expect the result to be exactly 1 year.
As an alternative, SuperCROSS can use a different time calculation: discrete time calculation. When this mode is active, SuperCROSS computes, in addition to the total number of days, the integer
value of the number of whole years, whole months and remaining days. Instead of averaging from the total number of days, it uses the following calculations:
datespan in years = nYears + (nMonths / 12) + (nDays / 365.25)
datespan in months = nYears*12 + nMonths + (nDays / (365.25/12))
In the example above, this calculation results in the expected whole number result for a full year.
To enable the calculation you need to create the following environment variable and set it to true: | {"url":"https://docs.wingarc.com.au/superstar/9.9.1/create-a-time-span-field","timestamp":"2024-11-02T17:07:05Z","content_type":"text/html","content_length":"38132","record_id":"<urn:uuid:a39cee9c-1221-4c83-b47e-7d967c78f180>","cc-path":"CC-MAIN-2024-46/segments/1730477027729.26/warc/CC-MAIN-20241102165015-20241102195015-00632.warc.gz"} |
Talk:Ieee arithmetic
Revision as of 10:46, 4 February 2010 by
Manus (Talk | contribs) (→Postcondition for put)
Most probably C compilers inline functions, but just to be sure, I'd convert them into the macros:
#define to_raw_bits(d) *((EIF_NATURAL_64*)&(d))
#define eif_is_nan_bits(value) ((value & ~RTU64C(0x8000000000000000)) > RTU64C(0x7ff0000000000000))
#define eif_is_nan(v) ((*((EIF_NATURAL_64 *)&(v)) & ~RTU64C(0x8000000000000000)) > RTU64C(0x7ff0000000000000))
Does it affect the benchmarks?
--manus 17:59, 3 February 2010 (UTC) Actually it does not on Windows for sure, I've verified that it was inlined. But you are right that those could be simply defined as macros.
--manus 20:25, 3 February 2010 (UTC) I've done again some of the benchmarks and on windows at least, some of them are slower when I use a macro. I'm no sure why I haven't looked at the
generated assembly code.
--Colin-adams 14:48, 3 February 2010 (UTC) Not IEEE arithmetic, nor maths, NaN = NaN is never true. And placing NaNs in a sort order isn't write either - REAL_32/64 are not totally ordered types.
--manus 17:57, 3 February 2010 (UTC) How do you solve the problem of assertions then in ARRAY.put for example?
--Alexander Kogtenkov 20:01, 3 February 2010 (UTC)
• Does it mean that REAL_GENERAL should inherit PART_COMPARABLE rather than COMPARABLE?
• Do we need 2 equality queries: one that tells two objects represent the same value (it is used to ensure copy does what is expected, and it is used to implement ~) and the other one that tells
that the numbers are equal in terms of ordering relation of (PART_)COMPARABLE?
--Colin-adams 12:37, 4 February 2010 (UTC): Postcondition for {ARRAY}.put should read:
inserted: v = v implies (item (i) = v)
undefined_case: v /= V implies (item (i) /= item (1))
Numeric equality
I have previously suggested separating the notion of numerical equality and object equality. Eric said that we use = for three different notions, i think, but I don't remember what these were.
Certainly PART_COMPARABLE is better than COMPARABLE for IEEE math types. I'm not sure if that is sufficient or not. --Colin-adams 12:42, 4 February 2010 (UTC) | {"url":"https://dev.eiffel.com/index.php?title=Talk:Ieee_arithmetic&oldid=13655","timestamp":"2024-11-04T14:15:58Z","content_type":"text/html","content_length":"12950","record_id":"<urn:uuid:4594ff59-07b1-42b0-98a3-d01bea72f65b>","cc-path":"CC-MAIN-2024-46/segments/1730477027829.31/warc/CC-MAIN-20241104131715-20241104161715-00262.warc.gz"} |
Theory of Combinatorial Algorithms
Mittagsseminar (in cooperation with J. Lengler, A. Steger, and D. Steurer)
Mittagsseminar Talk Information
Date and Time: Thursday, October 23, 2014, 12:15 pm
Duration: 30 minutes
Location: OAT S15/S16/S17
Speaker: May Szedlák
Combinatorial Redundancy Detection
The problem of detecting (and removing) redundant constraints is fundamental in optimization. We focus on the case where we are given a set H of n halfspaces in the d-dimensional real space. The
feasible solution set is given by the intersection of all halfspaces in H and a halfspace is called redundant if its removal does not change the feasible solution set. The currently fastest known
algorithm to detect all redundancies is the one by Clarkson. This method solves n linear programs, each of them on at most s variables, where s is the number of nonredundant variables.
In this talk we study the combinatorial aspect of redundancy detection. The basic question is: What kind of information about the linear system do we need in order to detect all redundant halfspaces?
We show that it is enough to know the finitely many dictionaries of H. A dictionary is a matrix that can be thought of as an enriched version of an intersection point of d halfspaces of H. It is
enough if the dictionaries are given in a combinatorial setting, containing only signs and in particular no numerical values.
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Integral Calculus and Physics
Mathematics is omnipresent and the significance of it has its mark in every branch of science. Physics is one such area where progress would be unimaginable without the role of mathematics.
Predicting the behavior of complex systems based on simple basic laws of physics requires mathematics. Validation of laws of physics by making testable predictions is another area where mathematics
comes to play. To make testable predictions of the postulates of quantum mechanics one needs a lot of math. The legacy of mathematics in the world of physics can be discussed with hundreds
of examples, let’s confine to one simple concept of integral calculus.
In simple language, integral calculus can be defined as a mechanism to calculate the area under a curve. Let’s say you want to calculate the area of a rectangle you multiply the width with length.
Similarly to calculate the area covered by a straight-line y= 1 (for all x) from x=0 to x=1. Note that the straight line in this range looks like a rectangle with equal sides that are of one unit
length. So one can multiply 1 x 1 and obtain area as 1 square unit. Let’s say the function is changed to y=x. This is a straight line passing through origin. The area under it within the range of x=0
to x=1 can be calculated by measuring the area of the right-angled triangle formed, when a perpendicular is drawn from x=1 to the straight line. Now let’s move on to a more complicated case like y=x^
2. y=x^2 is a parabolic curve and a simple multiplication will not get us the area. So how do we calculate the area of such curve? This is where ‘integral calculus’ comes into picture.
One can assume a curve is made of a large number of straight lines. Just like the small portions on the surface of earth appear flat but on a whole earth is spherical (though not perfectly spherical)
in shape. So to calculate area within the range of x=0 and x=1, we can take a small step like x=0 to x=0.00001 and assume that in this region the curve appears like a straight line. Since the
curve appears like a straight line, a rectangle is formed with the straight line, X-axis and the perpendiculars drawn from x=0 and x=0.00001 onto the curve. Since we know that area of a rectangle is
length X width, we can calculate the area under the curve in that small region as 0.00001X perpendicular distance from X-axis to the curve. Thus one can calculate the area of such small steps and
finally get the area of the curve in the region of x (x[1])=0 and x (x[2])=0.00001. Further the step size can be pushed down to a very small value by using the limit concept (assuming limit of x[2]-x
[1] to be zero).
One simple application of integration learnt in our Physics tuition classes is the calculation of displacement in a given time when the velocity vs time curve is given. If the velocity remains
constant or is gradually increasing one can calculate the distance by simple multiplication or a formula. But if the velocity changes abruptly following certain function one can calculate the
displacement, which is nothing but area under the curve, by using the concept of integration. | {"url":"https://tuitionphysics.com/2016-jan/integral-calculus-and-physics/","timestamp":"2024-11-09T02:30:34Z","content_type":"text/html","content_length":"93998","record_id":"<urn:uuid:dfe91fca-fab7-4129-b040-37c6bcafdea8>","cc-path":"CC-MAIN-2024-46/segments/1730477028115.85/warc/CC-MAIN-20241109022607-20241109052607-00029.warc.gz"} |
TRIMMEAN() – Drawing with Numbers
Excel’s TRIMMEAN() function can be quite useful at removing outliers, essentially it removes the top and bottom Nth percent of values and then computes the mean of the rest. Here’s the equivalent
formula in Tableau that in Superstore Sales computes the TRIMMEAN() of sales at the customer level removing the top and bottom 5th percentile of customers when used with the AVG() aggregation:
{FIXED [Customer Name]: SUM(
IF {FIXED [Customer Name] : SUM([Sales])} <
{FIXED : PERCENTILE({FIXED [Customer Name] : SUM([Sales])}, .95)}
AND {FIXED [Customer Name] : SUM([Sales])} >
{FIXED : PERCENTILE({FIXED [Customer Name] : SUM([Sales])}, .05)} THEN
Read on for how to build and validate your own TRIMMEAN() equivalent in Tableau.
When building out calculations in Tableau I try to let Tableau do as much of the computation as possible for both the calculations and the validation, so I’m typing as little as I can. Starting with
Superstore, let’s identify the top and bottom 5th percentiles, here’s a view using a reference distribution:
Now we know what we’re going to have to remove. The next step is to duplicate this worksheet as a crosstab, then build out calcs that can return the 5th and 95th percentiles of Sales at the Customer
Name level. While this can be done with table calculations (here’s an example from the Tableau forums) I’m going to use FIXED Level of Detail Expressions so I’ve got a dimension I can use, so for
example I could compare the trimmed group to the non-trimmed group. Here’s the 95th percentile Level of Detail Expression:
{FIXED : PERCENTILE({FIXED [Customer Name] : SUM([Sales])}, .95)}
The inner LOD is calculating the sales at the Customer level, then the outer LOD is returning the 95th percentile as a record level value. Here’s the two calcs which have values that compare to the
reference lines above:
The next step is to filter out the values outside of the desired range, here’s the TRIMMEAN Filter formula:
{FIXED [Customer Name] : SUM([Sales])} <
{FIXED : PERCENTILE({FIXED [Customer Name] : SUM([Sales])}, .95)}
AND {FIXED [Customer Name] : SUM([Sales])} >
{FIXED : PERCENTILE({FIXED [Customer Name] : SUM([Sales])}, .05)}
This uses the 5th and 95th percentile formulas and only returns True when the Customer level sales is less than the 95th percentile or greater than the 5th percentile, we can visually validate it by
dropping it on the Color Shelf:
Now that we have this the next step is to calculate what the trimmed mean would be. Again, we can use a view with a reference line, this time it’s been filtered using the TRIMMEAN Filter calc and the
reference line is an average:
Now we can embed the TRIMMEAN Filter formula inside an IF/THEN statement to only return the sales for the filtered values, this is the Trimmed Sales calc:
IF {FIXED [Customer Name] : SUM([Sales])} <
{FIXED : PERCENTILE({FIXED [Customer Name] : SUM([Sales])}, .95)}
AND {FIXED [Customer Name] : SUM([Sales])} >
{FIXED : PERCENTILE({FIXED [Customer Name] : SUM([Sales])}, .05)} THEN
And here it is in the workout view, only returning the sales for the trimmed customers:
Now that we have the trimmed sales there are two ways we can go. If we want the trimmed mean without the Customer Name in the Level of Detail then we can validate that in our workout view by using
Tableau’s two-pass Grand Totals to get the average of the customer-level trimmed sales. This was created by:
1. Removing the TRIMMEAN Filter pill from Colors (this increases the vizLOD and is no longer necessary).
2. Clicking on the Analytics tab.
3. Dragging out a Column Grand Total.
4. Right-clicking the SUM(Trimmed Sales) pill on Measure Values and setting Total Using->Average.
Scrolling down to the bottom we can see that the overall trimmed mean matches of 2,600.79 matches the one from the reference line.
Note that we could have used the Summary Card instead, however using the Grand Total lets us see exact values.
There’s a problem, though, if we use the Trimmed Sales all on its own in a view it breaks, whether using SUM() or AVG():
The reason why is that the Trimmed Sales is a record level value and Superstore is at the level of detail of individual order items, but we’re trying to compute the trimmed mean across Customer
Names. For the true trimmed mean in this case we need to aggregate this trimmed sales to the Customer Name like we did in the workout view, here’s the Trimmed Sales (Customer Level) formula that uses
the Trimmed Sales and wraps that in an LOD to get the Customer Level sales:
{FIXED [Customer Name]: SUM(
IF {FIXED [Customer Name] : SUM([Sales])} <
{FIXED : PERCENTILE({FIXED [Customer Name] : SUM([Sales])}, .95)}
AND {FIXED [Customer Name] : SUM([Sales])} >
{FIXED : PERCENTILE({FIXED [Customer Name] : SUM([Sales])}, .05)} THEN
This returns the same results in the workout view:
And works all on its own in a view:
Now this is a case where the FIXED level of detail expression is returning different results depending on the level of detail of the view, if we want it to return the same result then we can wrap all
that in one more LOD expression, this is the TRIMMEAN Fixed calculation:
{FIXED : AVG(
{FIXED [Customer Name]: SUM(
IF {FIXED [Customer Name] : SUM([Sales])} <
{FIXED : PERCENTILE({FIXED [Customer Name] : SUM([Sales])}, .95)}
AND {FIXED [Customer Name] : SUM([Sales])} >
{FIXED : PERCENTILE({FIXED [Customer Name] : SUM([Sales])}, .05)} THEN
And here it is in the workout view and a view without any dimensions:
Final Comments
This is a good (and bad) example of how Tableau is different from Excel. In one bad sense note that I didn’t parameterize the percentage for the trimmed mean, this is because in Tableau it would
require two parameters because we can’t put calculations as the arguments to the PERCENTILE() function. In another bad sense the calculation requires understanding Level of Detail expressions and is
not wrapped into a simple formula. On the other hand we’ve got very precise control over what the calculation is computing over with those Level of Detail expressions and aren’t just limited to doing
trimmed means, we could do trimmed medians, get the Nth percentile of the trimmed values, etc.
Here’s the trimmed mean workbook on Tableau Public. | {"url":"https://drawingwithnumbers.artisart.org/tag/trimmean/","timestamp":"2024-11-14T18:26:53Z","content_type":"text/html","content_length":"110040","record_id":"<urn:uuid:f4ae4e9b-a6dd-4f1c-a5de-696001f6de00>","cc-path":"CC-MAIN-2024-46/segments/1730477393980.94/warc/CC-MAIN-20241114162350-20241114192350-00548.warc.gz"} |
ECE 515 - Control System Theory & Design
Homework 8 - Due: 03/21
This homework is slightly on the longer side to make up for Homework 5 and Homework 6 being shorter. We suggest you start early.
Problem 1
Consider the discrete-time linear system with output \[\begin{align*} x(k+1)&=Ax(k)\\ y(k)&=Cx(k) \end{align*}\] and call it observable if different initial conditions produce different output
strings. Derive a condition for observability in terms of \(A\) and \(C\). Show that if two initial conditions produce outputs that coincide for the first \(n\) steps, then these outputs are the same
for all future steps.
Problem 2
Consider the LTI system \[\begin{align*} \dot x&=Ax\\ y&=Cx \end{align*}\] and suppose that the eigenvalues of \(A\) have negative real parts. Consider the function \(V(x)=x^TMx\) where \(M\) denotes
the observability Gramian for the infinite time horizon, i.e., \(M(0,\infty)\). Show that along solutions of the system we have \[ \dot V=-|y|^2. \]
Problem 3
a. For LTI systems, show that \((A,C)\) is observable if and only if \((-A,C)\) is observable.
b. Is the same statement true for LTV systems? Prove or give a counterexample.
Problem 4
Obtain a combined controllability/observability decomposition for the LTI system \(\dot x =Ax+Bu\), \(y=Cx\) by following these steps:
a. Ignoring the control for now, write down the Kalman observability decomposition.
b. Now add the control, noting that the \(B\) matrix assumes no special structure in the coordinates that give the observability decomposition.
c. For the observable part of the system, switch coordinates to get the Kalman controllability decomposition for it. Repeat separately for the unobservable part.
In the resulting system, make sure to specify all controllability and observability properties of various subsystems. Identify four types of modes: controllable and observable, uncontrollable but
observable, controllable but unobservable, and uncontrollable and unobservable.
Problem 5
Consider the system \(\dot x=Ax+Bu\), \(y=Cx\) and suppose that it is both controllable and observable. Now consider the feedback of the form \(u=Kx+v\), which leads to the system with new control \
(v\): \[\begin{align*} \dot x&=(A+BK)x+Bv\\ y&=Cx \end{align*}\]
a. Is the new system controllable? Prove or give a counterexample.
b. Is the new system observable? Prove or give a counterexample.
Problem 6
Consider the system
\[ \begin{aligned} \dot x&=-2x+u\\ y&=x+u \nonumber \end{aligned} \tag{1}\]
Construct a system of the form
\[ \begin{aligned} \dot z&=az+by\\ u&=cz+dy \nonumber \end{aligned} \tag{2}\]
which serves as an inverse to (1), in the sense that if we take an input signal \(u\), feed it into the system (1), compute the output \(y\), and feed this \(y\) into the system (2), we get the
original signal \(u\) back as the output of (2) (assuming zero initial conditions for both \(x\) and \(z\)).
Run computer simulations to verify that the inverse you constructed in part (a) indeed works as expected. Check what happens if you vary the initial conditions.
Problem 7
a. On Thursday before the break, we will discuss a lemma which says that we can go from a controllable pair \((A,b)\) to its corresponding controllable canonical form \((\bar A,\bar b)\) via a
coordinate transformation \(x=P\bar x\). In class we will derive \(P=\mathcal C(A,b) \mathcal C^{-1}(\bar A,\bar b)\) and verify \(\bar b=P^{-1}b\), but not that \(\bar A=P^{-1}AP\). Finish the
proof by verifying this last claim.
b. Prove that any two minimal realizations of a given transfer function \(g(s)\) can be obtained from each other by a coordinate transformation. (Hint: use the result of the previous problem.)
Problem 8
Consider the system
\[ \dot x=\begin{bmatrix} 0 & 1 & 0\\ -1 & 2 & 5\\ 1 & -1 & -3\end{bmatrix}x+ \begin{bmatrix} 1\\-1\\1\end{bmatrix}u \]
Find a state feedback law \(u=Kx\) such that the poles of the closed-loop system are \(-1\) and \(-2\pm i\). | {"url":"https://courses.grainger.illinois.edu/ece515/sp2024/homework/hw08.html","timestamp":"2024-11-13T22:26:36Z","content_type":"application/xhtml+xml","content_length":"36809","record_id":"<urn:uuid:30121855-9768-448f-80a7-faf789d3d8c7>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00169.warc.gz"} |
Write a C# Program to Find the Sum of First N Numbers [3 Methods] - AspDotnetHelp.com
Write a C# Program to Find the Sum of First N Numbers [3 Methods]
If you are looking to find the sum of first N numbers in C#, check out this complete tutorial. I have explained how to write a C# program to find the sum of first N numbers.
To calculate the sum of the first N numbers in C#, you can write a program that uses a for loop to iterate from 1 to N, adding each number to a cumulative sum. Start by prompting the user to enter
the value of N, then use a loop to compute the sum, and finally, display the result. This approach provides a simple and efficient way to perform the calculation in C#.
Write a C# Program to Find the Sum of First N Numbers
The sum of the first N numbers is the total of all numbers starting from 1 to N. For instance, if N is 5, the sum would be 1 + 2 + 3 + 4 + 5.
Below is a complete example of a C# program that calculates the sum of the first N numbers.
using System;
class Program
static void Main()
Console.WriteLine("Enter the value of N:");
int N = Convert.ToInt32(Console.ReadLine());
int sum = SumOfNumbers(N);
Console.WriteLine($"The sum of the first {N} numbers is: {sum}");
static int SumOfNumbers(int n)
int sum = 0;
for (int i = 1; i <= n; i++)
sum += i;
return sum;
• Reading Input: The program begins by asking the user to enter the value of N.
• Sum Calculation: It then calculates the sum using the SumOfNumbers function.
• For Loop: Inside this function, a for loop runs from 1 to N, adding each number to sum.
• Result Output: Finally, the program displays the total sum.
Here, you can see the output after I ran the code using a Visual Studio C# console application. I enter 10, and then it shows me the output in the screenshot below as 55, as the sum of the first 10
numbers (1 to 10) is 55.
Alternatively, you can also write the C# program like the below:
using System;
class Program {
static int SumOfFirstNNumbers(int n) {
int sum = 0;
for (int i = 1; i <= n; i++) {
sum += i;
return sum;
static void Main() {
Console.WriteLine("Enter a number:");
int number = Convert.ToInt32(Console.ReadLine());
Console.WriteLine($"The sum of first {number} natural numbers is: {SumOfFirstNNumbers(number)}");
Find the Sum of the First N Numbers in C# using the Arithmetic Formula
You can also use an arithmetic formula in C# to find the sum of the first N numbers. Here is a complete code:
using System;
class Program {
static int SumOfFirstNNumbers(int n) {
return n * (n + 1) / 2;
static void Main() {
Console.WriteLine("Enter a number:");
int number = Convert.ToInt32(Console.ReadLine());
Console.WriteLine($"The sum of first {number} natural numbers is: {SumOfFirstNNumbers(number)}");
This method is more efficient than the loop method, especially for large numbers, as it does not require iteration.
Once you run the code using Visual Studio, you can see the output in the below screenshot.
C# program to find sum of first n numbers using recursion
In C#, you can also use the recursion to find the sum of the first N numbers.
Here is a complete program. In this C# program, the function calls itself with n – 1 until n becomes 0. At this point, it starts unwinding and adds each number from 1 to N.
using System;
class Program {
static int SumOfFirstNNumbers(int n) {
if (n <= 0) {
return 0;
return n + SumOfFirstNNumbers(n - 1);
static void Main() {
Console.WriteLine("Enter a number:");
int number = Convert.ToInt32(Console.ReadLine());
Console.WriteLine($"The sum of first {number} natural numbers is: {SumOfFirstNNumbers(number)}");
Writing a C# program to find the sum of the first N numbers is straightforward, and I have shown here how to write a program to find the sum of first N numbers in C# using various methods like a for
loop, an arithmetic formula, and recursion.
You may also like:
Bijay Kumar is a renowned software engineer, accomplished author, and distinguished Microsoft Most Valuable Professional (MVP) specializing in SharePoint. With a rich professional background spanning
over 15 years, Bijay has established himself as an authority in the field of information technology. He possesses unparalleled expertise in multiple programming languages and technologies such as
ASP.NET, ASP.NET MVC, C#.NET, and SharePoint, which has enabled him to develop innovative and cutting-edge solutions for clients across the globe. Read more… | {"url":"https://aspdotnethelp.com/write-a-csharp-program-to-find-the-sum-of-first-n-numbers/","timestamp":"2024-11-05T19:26:31Z","content_type":"text/html","content_length":"150060","record_id":"<urn:uuid:b70e1c8f-379f-4e00-b007-466dc4a4be46>","cc-path":"CC-MAIN-2024-46/segments/1730477027889.1/warc/CC-MAIN-20241105180955-20241105210955-00235.warc.gz"} |
Single-Variable Calculus Review
Section 1 Single-Variable Calculus Review
Multivariable calculus is all about trying to generalize the ideas of single-variable calculus, so it's important to review your single-variable calculus knowledge. The problems below are not
remotely close to a complete review, but they highlight some of the key ideas we'll be using in the next few weeks. Try them on your own before checking your answers.
Subsection 1.1 Derivative Review
1. For a single variable function \(f\text{,}\) how do you picture the derivative \(f'(a)\) at a point \(a\) on the graph of \(f\text{?}\) Solution
\(f'(a)\) is the slope of the line tangent to the graph of \(f(x)\) at \(x = a\text{:}\)
2. What information do we get about a function from knowing whether its derivative \(f'(a)\) at a point \(a\) is positive or negative? Solution
Knowing whether the tangent line has positive or negative slope at \(a\) tells us whether the function \(f\) is increasing or decreasing at \(a\text{.}\)
3. The owner of a cupcake truck has found that the number of cupcakes she sells per day depends on the price she charges. Let \(C(p)\) be the number of cupcakes she sells when the price of a cupcake
is \(p\) cents. Interpret in words the statement that \(C'(300) = -2\text{;}\) be sure to explain the units involved in this statement. (Imagine you were trying to explain to someone who's never
taken calculus what the practical meaning of \(C'(300) = -2\) is in terms of cupcakes and price.) Solution
In words, the statement \(C'(300) = -2\) says that, when the price of a cupcake is 300 cents, the instantaneous rate of change in the number of cupcakes sold in a day (with respect to price) is \
(-2\) cupcakes per cent. In less technical language, for each cent the owner increases the price above 300 cents, she can expect to sell 2 fewer cupcakes (and similarly, for each cent she
decreases the price, she can expect to sell 2 more cupcakes). But these expectations are only reasonable for small price changes. So, for example, it would be reasonable to use this guideline to
estimate the number of cupcakes she'd sell if she changed the price to 299 or 302 cents, but it would not be reasonable to use this to estimate the number of cupcakes she'd sell if she raised the
price to 700 cents. (That's what being an “instantaneous” rate of change means.)
4. The actual definition of the derivative \(f'(x)\) of a function \(f(x)\) is expressed as a limit; fill in the boxes to finish the definition.
\begin{equation*} f'(x) = \lim_{\fcolorbox{blue}{white}{$\vphantom{h}\phantom{h \to 0}$}} \fcolorbox{red}{white}{$\vphantom{\dfrac{1}{1}}\phantom{\dfrac{f(x + h) - f(x)}{h}}$} \end{equation*}
Explain the meaning of this definition; what does the expression in the red box represent? How do you visualize it on a graph of \(f\text{?}\) Solution
The definition is
\begin{equation*} f'(x) = \lim_{\fcolorbox{blue}{white}{$h \to 0$}} \fcolorbox{red}{white}{$\dfrac{f(x + h) - f(x)}{h}$} \end{equation*}
The expression in the red box, \(\dfrac{f(x + h) - f(x)}{h}\text{,}\) is the slope of the secant line between the points \((x, f(x))\) and \((x + h, f(x + h))\text{:}\)
When we let \(h \to 0\text{,}\) the purple point approaches the green point, and the secant line approaches the tangent line at \(x\text{,}\) so the slope of the secant line, \(\dfrac{f(x + h) -
f(x)}{h}\text{,}\) approaches the slope of the tangent line, \(f'(x)\text{.}\)
Subsection 1.2 Integration Review
1. Here's the graph of a function \(f\text{.}\)
1. How do you visualize \(\displaystyle \int_1^7 f(x)\,dx\) on this graph? Solution
It's the signed area between \(y = f(x)\) and the \(x\)-axis, between \(x = 1\) and \(x = 7\text{.}\) That is, it's the area of the green region minus the area of the red region:
2. Like the derivative, the definite integral \(\displaystyle \int_1^7 f(x)\,dx\) is really defined as a limit. Explain what limit this is. Solution
If we slice the interval \([1, 7]\) into \(n\) pieces of equal width \(\Delta x\) and label the endpoints \(x_0 = 1, x_1, x_2, ..., x_n = 7\text{,}\) then here's one way to approximate the
signed area \(\displaystyle \int_1^7 f(x)\,dx\) using rectangles:
Here, the signed area of the \(k\)-th rectangle is \(f(x_k) \Delta x\text{,}\) so the total area in the rectangles is \(\displaystyle \sum_{k=1}^n f(x_k) \Delta x\text{.}\)
If we use more rectangles, so that each one is thinner, our approximation gets more and more accurate:
So, \(\displaystyle \int_1^7 f(x)\,dx\) is the limit of these approximations as the number of rectangles increases without bound, or \(\boxed{\lim_{n \to \infty} \sum_{k=1}^n f(x_k) \Delta x}
2. Below is a list of integrals. Any variables in the integral other than the variable of integration should be considered constants. For example, in the integral \(\displaystyle \int xyz\ dy\text
{,}\) the \(dy\) tells us that \(y\) is the variable of integration, so we'll treat \(x\) and \(z\) as constants.
One of the integrals below cannot be evaluated by hand; identify that one, and evaluate the rest. (Can you check your answers by differentiating them?)
If you need to review your techniques of integration, check out this review material.
1. \(\displaystyle \int x \sin(x)\,dx\) Solution
We'll integrate by parts. Let \(u = x\) and \(dv = \sin x\,dx\text{.}\) Then, \(du = dx\) and \(v = - \cos x\text{,}\) so \(\displaystyle \int x \sin x\,dx = - x \cos x - \int - \cos x\,dx =
\boxed{- x \cos x + \sin x + C}\text{.}\)
2. \(\displaystyle \int x \sin(x^2)\,dx\) Solution
You might be able to do this in your head, but if not, you can use substitution. If we substitute \(u = x^2\text{,}\) then \(du = 2x\,dx\text{,}\) so \(x\,dx = \frac{du}{2}\text{.}\) Then,
\begin{align*} \int x \sin(x^2)\,dx \amp = \int \sin(u) \frac{du}{2} \\ \amp = - \frac{1}{2} \cos u + C \\ \amp = \boxed{-\frac{1}{2} \cos (x^2) + C} \end{align*}
3. \(\displaystyle \int x \sin(xy)\,dy\) Solution
Again, you might be able to do this in your head, or you can use substitution. If we substitute \(u = xy\text{,}\) then (remembering that \(x\) is a constant and \(y\) is the variable), \(du
= x\,dy\text{,}\) so
\begin{align*} \int x \sin(xy)\,dy \amp = \int \sin(u)\,du \\ \amp = - \cos u + C \\ \amp = \boxed{-\cos(xy) + C} \end{align*}
4. \(\displaystyle \int e^{x^2 y}\,dx\) Solution
This one cannot be evaluated by hand. This isn't because \(e^{x^2 y}\) doesn't have an antiderivative (when we think of it as a function of \(x\text{,}\) with \(y\) a constant); after all,
the Fundamental Theorem of Calculus guarantees that every continuous function has a continuous antiderivative. Rather, mathematicians have proved that the antiderivatives of \(e^{x^2 y}\)
can't be expressed in terms of familiar functions like exponentials and trigonometric functions.
5. \(\displaystyle \int e^{x^2 y}\,dy\) Solution
We'll substitute \(u = x^2 y\text{.}\) Then (remembering that \(x\) is a constant and \(y\) is the variable), \(du = x^2\,dy\text{,}\) so \(dy = \frac{du}{x^2}\text{.}\) So,
\begin{align*} \int e^{x^2 y}\,dy \amp = \int e^u \, \frac{du}{x^2} \\ \amp = \frac{1}{x^2} \int e^u\,du \\ \amp = \frac{1}{x^2} e^u + C \\ \amp = \boxed{\frac{1}{x^2} e^{x^2 y} + C} \end | {"url":"https://people.math.harvard.edu/~jjchen/math21a/handouts/svc-review.html","timestamp":"2024-11-03T15:29:30Z","content_type":"text/html","content_length":"21359","record_id":"<urn:uuid:2fd79853-4fbe-47f5-a62a-3b186d8a5e29>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00829.warc.gz"} |
Math 4 Wisdom. "Mathematics for Wisdom" by Andrius Kulikauskas. | Research / Orientation
• Index
• Writings
Andrius Kulikauskas
• m a t h 4 w i s d o m - g m a i l
• +370 607 27 665
• My work is in the Public Domain for all to share freely.
• 读物书影片维基百科
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• A field has an intrinsic direction.
• A line gains direction when there is a basis.
• A line relates a field with a one-dimensional space.
• Orientation is related with a coordinate system.
• When you have two bases then you relate them with a linear transformation, and the determinant gives the orientation, whether it changes.
• What is the relationship between zero and the point at infinity? | {"url":"https://www.math4wisdom.com/wiki/Research/Orientation","timestamp":"2024-11-14T14:15:40Z","content_type":"application/xhtml+xml","content_length":"10538","record_id":"<urn:uuid:f42c1a43-caa0-472b-9d34-73f4d57d5383>","cc-path":"CC-MAIN-2024-46/segments/1730477028657.76/warc/CC-MAIN-20241114130448-20241114160448-00089.warc.gz"} |
This function is useful for testing whether or not initial values changed due to constraints when being passed through a Model specification function. If any initial value changes, then the
constrained values that are ouput in the fifth component of the Model specification are suitable as initial values, not the tested initial values.
A parameter may be constrained and this function may not discover the constraint, since the discovery depends on the initial values and whether or not they change as they are passed through the | {"url":"https://www.rdocumentation.org/packages/LaplacesDemon/versions/16.1.6/topics/is.constrained","timestamp":"2024-11-02T00:07:10Z","content_type":"text/html","content_length":"59571","record_id":"<urn:uuid:e9c08ffa-0948-42ef-9249-0645d8fc22ea>","cc-path":"CC-MAIN-2024-46/segments/1730477027599.25/warc/CC-MAIN-20241101215119-20241102005119-00009.warc.gz"} |
Making a case for Spacetime
Are space and time different things ?
How many dimensions are there?
$x$, $y$, and $z$ right?
How sure are you?
Is time a dimension?
Now when you think about it there is no way to conceptualize a four-dimensional object.
That should be it then, no fourth dimension, right? But what of the fact that you literally are further in time than when you started reading this. You’ve effectively moved forward in time.
So is time a dimension? Time is defined as, and I quote “The indefinite continued progress of existence and events in the past, present, and future regarded as a whole”
Here’s the thing. There’s a very interesting phenomenon called time dilation. The most interesting example of this is a thought experiment known as thetwin paradox. But more on that later; lets try
to understand first how motion can affect time.
It sounds crazy right?
The idea behind time dilation is a little complex. But it is central to time as well as space as I will soon prove to you.
If the video above confused you, don’t worry, We’ll get a little technical and prove it mathematically.
Here’s another way to look at it. Say you and a friend are standing next to each other.
Your friend is standing and holding a laser upwards towards a mirror hanging above. Don’t think about how the mirror is floating but just know that it is constantly floating above your friend.You’re
both trying to measure how much time [in seconds] it will take for the light to hit the mirror and then come back.
So you’re both standing still with your clocks. He shoots the laser up, it bounces off the mirror and comes back.
So for the example in the picture you’re seeing essentially would be the distance L divided by the speed of light [c]. For the sake of simplicity let’s take that value to be 10 seconds. So it takes
10 seconds for the light to bounce off of the mirror and come back. Now when you’re both not moving you will see the light take the same amount of time.
Now let’s say the other person started moving. You’re still standing still, yet now when you measure your times, you measure the time for the light to go up and then back down to be 10.4 seconds as
opposed to the 10 seconds before. Your friend tells you that his clock measured 10 seconds just like before.
“That’s strange. The distance between the other person and the mirror hasn’t changed, so why would the time increase?”
Think back to the video you watched before, and then look at the picture below. Imagine the person starts moving at point A, when the laser is fired. The light is now taking the blue path. It’s no
longer taking the path of the picture above because now the person is moving. The distance the light has to travel is now longer because the light is traveling diagonally. Similar to the previous
example, when neither of you were moving the light traveled length L and back. (2L) would be less than (2D/c)
So if you measured 10.4 seconds… what does that mean? Well, if you measured 10.4, and he measured 10 seconds, then it would seem that time was moving faster for him than it was for you. This unusual
phenomenon is called time dilation. Your time was 10.4… why is it that number specifically do you think? In fact, how would you know how much slower it would be? It turns out there is a way to
calculate this.
$$t’=\frac{t}{\sqrt{1-v^2/c^2}} $$
The time that you (the non-moving observer) measure, is denoted by t’. The way you would set the equation up for this specific example would be replacing t’ with 10.4 and t (the time measured by your
friend) with 10. The other values you can tell based on the picture.
So how does this happen in everyday life? I don’t think i’ve ever had a different time while i was driving in my car then my friends waiting for me somewhere else. Well, here’s why, the difference is
so incredibly small that it doesn’t actually create a problem. Think back to the Twin Paradox I told you about earlier, the idea is that if this principal is taken to the extreme, two twins could end
up meeting each other at very different ages after one of the twins gets on a space ship and orbits around the earth.
It’s all probably still a little crazy to wrap your head around the idea of time moving at different rates based on how fast your moving, believe me it’s weird. Here is another video that should help
out with how physicists think about this.
So what we’ve essentially unraveled here is how to travel in time! Not to the past (unfortunately), but to the future! In fact, you just did it when between now and when you started reading this
sentence. trippy.
P.S. Remember that in real life the mirror would have to be one light-second away($3 \times 10^8$ meters) away in order for the light to actually take that long to get to the other person. | {"url":"https://davidaw.ad/posts/08-31-making-a-case-for-spacetime/","timestamp":"2024-11-12T03:29:24Z","content_type":"text/html","content_length":"22267","record_id":"<urn:uuid:ea3ba9b3-43c2-4dfa-aad7-462f420fdbe0>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.50/warc/CC-MAIN-20241112014152-20241112044152-00897.warc.gz"} |
What is 0.2 as a percentage? (0.2 as a percent)
How to write 0.2 as a percentage? How to convert 0.2 decimal to percent? What is the decimal number 0.2 as a percentage? These are all different questions asking the same thing.
Here we will not only give you the answer to 0.2 as a percentage, but also explain what it means to convert the decimal number 0.2 to percent and show you how to do it.
Look at the decimal number 0.2 as 0.2 out of one. Furthermore, percent means per hundred. So the task is to convert 0.2 out of one to 0.2 out of one hundred.
The equation to solve the problem is displayed below where x is the 0.2 as a percentage.
We solve the equation for x above by first multiplying each side by one and then multiplying each side by one hundred. Thus, the answer to 0.2 as a percentage is:
On this page we showed you how to calculate 0.2 as a percentage using a formula to illustrate clearly what it actually means to convert 0.2 to a percentage.
Now that you know the underlying process and explanation behind it, note that the shortcut to converting 0.2 to percent is simply to multiply 0.2 by one hundred to get the answer. (0.2 x 100 = 20%)
Another shortcut that is also easy to remember to get a decimal number like 0.2 as a percentage, is to move the decimal point two places to the right and add the percent (%) sign.
Decimal to Percentage Calculator
Do you need to convert another decimal to percent? If so, please enter your decimal here!
What is 0.2001 as a percentage?
Here is the next decimal number on our list that we converted to percentage.
Privacy Policy | {"url":"https://decimal.info/decimal-to-percent/0/what-is-0.2-as-a-percentage.html","timestamp":"2024-11-10T04:25:03Z","content_type":"text/html","content_length":"6875","record_id":"<urn:uuid:64075dca-ea2e-4165-9600-e14684ea1f9a>","cc-path":"CC-MAIN-2024-46/segments/1730477028166.65/warc/CC-MAIN-20241110040813-20241110070813-00062.warc.gz"} |
From cppreference.com
(ranges TS)
The concept Incrementable<I> specifies the requirements on a type that can be incremented (with the pre- and post-increment operators). The increment operations (including those required by
WeaklyIncrementable) are required to be equality-preserving, and the type is required to be EqualityComparable.
Let a and b be incrementable objects of type I. Incrementable<I> is satisfied only if:
• If bool(a == b) then bool(a++ == b).
• If bool(a == b) then bool(void(a++), a) == ++b).
Equality preservation
An expression is equality preserving if it results in equal outputs given equal inputs.
• The inputs to an expression consist of its operands.
• The outputs of an expression consist of its result and all operands modified by the expression (if any).
Every expression required to be equality preserving is further required to be stable: two evaluations of such an expression with the same input objects must have equal outputs absent any explicit
intervening modification of those input objects.
Unless noted otherwise, every expression used in a requires-expression is required to be equality preserving and stable, and the evaluation of the expression may only modify its non-constant
operands. Operands that are constant must not be modified.
The requirement that a equals b implies ++a equals ++b allows the use of multi-pass algorithms with Incrementable types. | {"url":"https://omegaup.com/docs/cpp/en/cpp/experimental/ranges/iterator/Incrementable.html","timestamp":"2024-11-08T07:51:43Z","content_type":"text/html","content_length":"41735","record_id":"<urn:uuid:33ede6eb-99b2-4cdf-bdab-571faa4830d4>","cc-path":"CC-MAIN-2024-46/segments/1730477028032.87/warc/CC-MAIN-20241108070606-20241108100606-00569.warc.gz"} |
Stereographic projection - Alchetron, the free social encyclopedia
In geometry, the stereographic projection is a particular mapping (function) that projects a sphere onto a plane. The projection is defined on the entire sphere, except at one point: the projection
point. Where it is defined, the mapping is smooth and bijective. It is conformal, meaning that it preserves angles. It is neither isometric nor area-preserving: that is, it preserves neither
distances nor the areas of figures.
Intuitively, then, the stereographic projection is a way of picturing the sphere as the plane, with some inevitable compromises. Because the sphere and the plane appear in many areas of mathematics
and its applications, so does the stereographic projection; it finds use in diverse fields including complex analysis, cartography, geology, and photography. In practice, the projection is carried
out by computer or by hand using a special kind of graph paper called a stereographic net, shortened to stereonet, or Wulff net.
The stereographic projection was known to Hipparchus, Ptolemy and probably earlier to the Egyptians. It was originally known as the planisphere projection. Planisphaerium by Ptolemy is the oldest
surviving document that describes it. One of its most important uses was the representation of celestial charts. The term planisphere is still used to refer to such charts.
In the 16th and 17th century, the equatorial aspect of the stereographic projection was commonly used for maps of the Eastern and Western Hemispheres. It is believed that already the map created in
1507 by Gualterius Lud was in stereographic projection, as were later the maps of Jean Roze (1542), Rumold Mercator (1595), and many others. In star charts, even this equatorial aspect had been
utilised already by the ancient astronomers like Ptolemy.
François d'Aguilon gave the stereographic projection its current name in his 1613 work Opticorum libri sex philosophis juxta ac mathematicis utiles (Six Books of Optics, useful for philosophers and
mathematicians alike).
In 1695, Edmond Halley, motivated by his interest in star charts, published the first mathematical proof that this map is conformal. He used the recently established tools of calculus, invented by
his friend Isaac Newton.
This section focuses on the projection of the unit sphere from the north pole onto the plane through the equator. Other formulations are treated in later sections.
The unit sphere in three-dimensional space R^3 is the set of points (x, y, z) such that x^2 + y^2 + z^2 = 1. Let N = (0, 0, 1) be the "north pole", and let M be the rest of the sphere. The plane z =
0 runs through the center of the sphere; the "equator" is the intersection of the sphere with this plane.
For any point P on M, there is a unique line through N and P, and this line intersects the plane z = 0 in exactly one point P′. Define the stereographic projection of P to be this point P′ in the
In Cartesian coordinates (x, y, z) on the sphere and (X, Y) on the plane, the projection and its inverse are given by the formulas
( X , Y ) = ( x 1 − z , y 1 − z ) , ( x , y , z ) = ( 2 X 1 + X 2 + Y 2 , 2 Y 1 + X 2 + Y 2 , − 1 + X 2 + Y 2 1 + X 2 + Y 2 ) .
In spherical coordinates (φ, θ) on the sphere (with φ the zenith angle, 0 ≤ φ ≤ π, and θ the azimuth, 0 ≤ θ ≤ 2π) and polar coordinates (R, Θ) on the plane, the projection and its inverse are
( R , Θ ) = ( sin φ 1 − cos φ , θ ) = ( cot φ 2 , θ ) , ( φ , θ ) = ( 2 arctan 1 R , Θ ) .
Here, φ is understood to have value π when R = 0. Also, there are many ways to rewrite these formulas using trigonometric identities. In cylindrical coordinates (r, θ, z) on the sphere and polar
coordinates (R, Θ) on the plane, the projection and its inverse are
( R , Θ ) = ( r 1 − z , θ ) , ( r , θ , z ) = ( 2 R 1 + R 2 , Θ , R 2 − 1 R 2 + 1 ) .
The stereographic projection defined in the preceding section sends the "south pole" (0, 0, −1) of the unit sphere to (0, 0), the equator to the unit circle, the southern hemisphere to the region
inside the circle, and the northern hemisphere to the region outside the circle.
The projection is not defined at the projection point N = (0, 0, 1). Small neighborhoods of this point are sent to subsets of the plane far away from (0, 0). The closer P is to (0, 0, 1), the more
distant its image is from (0, 0) in the plane. For this reason it is common to speak of (0, 0, 1) as mapping to "infinity" in the plane, and of the sphere as completing the plane by adding a "point
at infinity". This notion finds utility in projective geometry and complex analysis. On a merely topological level, it illustrates how the sphere is homeomorphic to the one-point compactification of
the plane.
In Cartesian coordinates a point P(x, y, z) on the sphere and its image P′(X, Y) on the plane either both are rational points or none of them:
P ∈ Q 3 ⟺ P ′ ∈ Q 2
Stereographic projection is conformal, meaning that it preserves the angles at which curves cross each other (see figures). On the other hand, stereographic projection does not preserve area; in
general, the area of a region of the sphere does not equal the area of its projection onto the plane. The area element is given in (X, Y) coordinates by
d A = 4 ( 1 + X 2 + Y 2 ) 2 d X d Y .
Along the unit circle, where X^2 + Y^2 = 1, there is no inflation of area in the limit, giving a scale factor of 1. Near (0, 0) areas are inflated by a factor of 4, and near infinity areas are
inflated by arbitrarily small factors.
The metric is given in (X, Y) coordinates by
4 ( 1 + X 2 + Y 2 ) 2 ( d X 2 + d Y 2 ) ,
and is the unique formula found in Bernhard Riemann's Habilitationsschrift on the foundations of geometry, delivered at Göttingen in 1854, and entitled Über die Hypothesen welche der Geometrie zu
Grunde liegen.
No map from the sphere to the plane can be both conformal and area-preserving. If it were, then it would be a local isometry and would preserve Gaussian curvature. The sphere and the plane have
different Gaussian curvatures, so this is impossible.
The conformality of the stereographic projection implies a number of convenient geometric properties. Circles on the sphere that do not pass through the point of projection are projected to circles
on the plane. Circles on the sphere that do pass through the point of projection are projected to straight lines on the plane. These lines are sometimes thought of as circles through the point at
infinity, or circles of infinite radius.
All lines in the plane, when transformed to circles on the sphere by the inverse of stereographic projection, meet at the projection point. Parallel lines, which do not intersect in the plane, are
transformed to circles tangent at projection point. Intersecting lines are transformed to circles that intersect transversally at two points in the sphere, one of which is the projection point.
(Similar remarks hold about the real projective plane, but the intersection relationships are different there.)
The loxodromes of the sphere map to curves on the plane of the form
R = e Θ a ,
where the parameter a measures the "tightness" of the loxodrome. Thus loxodromes correspond to logarithmic spirals. These spirals intersect radial lines in the plane at equal angles, just as the
loxodromes intersect meridians on the sphere at equal angles.
The stereographic projection relates to the plane inversion in a simple way. Let P and Q be two points on the sphere with projections P′ and Q′ on the plane. Then P′ and Q′ are inversive images of
each other in the image of the equatorial circle if and only if P and Q are reflections of each other in the equatorial plane.
In other words, if:
P is a point on the sphere, but not a 'north pole' N and not its antipode, the 'south pole' S,
P′ is the image of P in a stereographic projection with the projection point N and
P″ is the image of P in a stereographic projection with the projection point S,
then P′ and P″ are inversive images of each other in the unit circle.
△ N O P ′ ∼ △ P ′ ′ O S ⟹ O P ′ : O N = O S : O P ′ ′ ⟹ O P ′ ⋅ O P ′ ′ = r 2
Wulff net
Stereographic projection plots can be carried out by a computer using the explicit formulas given above. However, for graphing by hand these formulas are unwieldy. Instead, it is common to use graph
paper designed specifically for the task. This special graph paper is called a stereonet or Wulff net, after the Russian mineralogist George (Yuri Viktorovich) Wulff.
To make a Wulff net, one places a grid of parallels and meridians on a hemisphere, and then stereographically projects these curves to the disk. Depending on the particular projection used, the
parallels and meridians may or may not match those usually encountered in geography. For example, the figure at left is constructed using the conventions of the Definition section above. Because the
projection point is (0, 0, 1), the Wulff net depicts the southern hemisphere z ≤ 0. The equator plots at the circular boundary of the Wulff net, and the south pole plots at the center of the Wulff
net. The parallels are chosen to be small circles about the y-axis, and all of the meridians pass through (0, 1, 0) and (0, −1, 0).
In the figure, the area-distorting property of the stereographic projection can be seen by comparing a grid sector near the center of the net with one at the far right of the net. The two sectors
have equal areas on the sphere. On the disk, the latter has nearly four times the area of the former. If one uses finer and finer grids on the sphere, then the ratio of the areas approaches exactly
On the Wulff net, the images of the parallels and meridians intersect at right angles. This orthogonality property is a consequence of the angle-preserving property of the stereoscopic projection.
(However, the angle-preserving property is stronger than this property. Not all projections that preserve the orthogonality of parallels and meridians are angle-preserving.)
For an example of the use of the Wulff net, imagine two copies of it on thin paper, one atop the other, aligned and tacked at their mutual center. Let P be the point on the lower unit hemisphere
whose spherical coordinates are (140°, 60°) and whose Cartesian coordinates are (0.321, 0.557, −0.766). This point lies on a line oriented 60° counterclockwise from the positive x-axis (or 30°
clockwise from the positive y-axis) and 50° below the horizontal plane z = 0. Once these angles are known, there are four steps to plotting P:
1. Using the grid lines, which are spaced 10° apart in the figures here, mark the point on the edge of the net that is 60° counterclockwise from the point (1, 0) (or 30° clockwise from the point (0,
2. Rotate the top net until this point is aligned with (1, 0) on the bottom net.
3. Using the grid lines on the bottom net, mark the point that is 50° toward the center from that point.
4. Rotate the top net oppositely to how it was oriented before, to bring it back into alignment with the bottom net. The point marked in step 3 is then the projection that we wanted.
To plot other points, whose angles are not such round numbers as 60° and 50°, one must visually interpolate between the nearest grid lines. It is helpful to have a net with finer spacing than 10°.
Spacings of 2° are common.
To find the central angle between two points on the sphere based on their stereographic plot, overlay the plot on a Wulff net and rotate the plot about the center until the two points lie on or near
a meridian. Then measure the angle between them by counting grid lines along that meridian.
Other formulations and generalizations
Some authors define stereographic projection from the north pole (0, 0, 1) onto the plane z = −1, which is tangent to the unit sphere at the south pole (0, 0, −1). The values X and Y produced by this
projection are exactly twice those produced by the equatorial projection described in the preceding section. For example, this projection sends the equator to the circle of radius 2 centered at the
origin. While the equatorial projection produces no infinitesimal area distortion along the equator, this pole-tangent projection instead produces no infinitesimal area distortion at the south pole.
Other authors use a sphere of radius 1/2 and the plane z = −1/2. In this case the formulae become
( x , y , z ) → ( ξ , η ) = ( x 1 2 − z , y 1 2 − z ) , ( ξ , η ) → ( x , y , z ) = ( ξ 1 + ξ 2 + η 2 , η 1 + ξ 2 + η 2 , − 1 + ξ 2 + η 2 2 + 2 ξ 2 + 2 η 2 ) .
In general, one can define a stereographic projection from any point Q on the sphere onto any plane E such that
E is perpendicular to the diameter through Q, and
E does not contain Q.
As long as E meets these conditions, then for any point P other than Q the line through P and Q meets E in exactly one point P′, which is defined to be the stereographic projection of P onto E.
All of the formulations of stereographic projection described thus far have the same essential properties. They are smooth bijections with smooth inverse (diffeomorphisms) defined everywhere except
at the projection point. They are conformal and not area-preserving.
More generally, stereographic projection may be applied to the n-sphere S^n in (n + 1)-dimensional Euclidean space E^n+1. If Q is a point of S^n and E a hyperplane in E^n+1, then the stereographic
projection of a point P ∈ S^n − {Q} is the point P′ of intersection of the line QP with E. In Cartesian coordinates (x[i], i from 0 to n) on the sphere and (X[i], i from 1 to n) on the plane, the
projection from Q = (1, 0, 0, ..., 0) is given by
X i = x i 1 − x 0 ( i from 1 to n )
S 2 = ∑ j = 1 n X j 2
the inverse is given by
x 0 = S 2 − 1 S 2 + 1 and x i = 2 X i S 2 + 1 ( i from 1 to n )
Still more generally, suppose that S is a (nonsingular) quadric hypersurface in the projective space P^n+1. In other words, S is the locus of zeros of a non-singular quadratic form f(x[0], ..., x[n
+1]) in the homogeneous coordinates x[i]. Fix any point Q on S and a hyperplane E in P^n+1 not containing Q. Then the stereographic projection of a point P in S − {Q} is the unique point of
intersection of QP with E. As before, the stereographic projection is conformal and invertible outside of a "small" set. The stereographic projection presents the quadric hypersurface as a rational
hypersurface. This construction plays a role in algebraic geometry and conformal geometry.
Complex analysis
Although any stereographic projection misses one point on the sphere (the projection point), the entire sphere can be mapped using two projections from distinct projection points. In other words, the
sphere can be covered by two stereographic parametrizations (the inverses of the projections) from the plane. The parametrizations can be chosen to induce the same orientation on the sphere.
Together, they describe the sphere as an oriented surface (or two-dimensional manifold).
This construction has special significance in complex analysis. The point (X, Y) in the real plane can be identified with the complex number ζ = X + iY. The stereographic projection from the north
pole onto the equatorial plane is then
ζ = x + i y 1 − z , ( x , y , z ) = ( 2 Re ζ 1 + ζ ¯ ζ , 2 Im ζ 1 + ζ ¯ ζ , − 1 + ζ ¯ ζ 1 + ζ ¯ ζ ) .
Similarly, letting ξ = X − iY be another complex coordinate, the functions
ξ = x − i y 1 + z , ( x , y , z ) = ( 2 Re ξ 1 + ξ ¯ ξ , − 2 Im ξ 1 + ξ ¯ ξ , 1 − ξ ¯ ξ 1 + ξ ¯ ξ ) .
define a stereographic projection from the south pole onto the equatorial plane. The transition maps between the ζ- and ξ-coordinates are then ζ = 1/ξ and ξ = 1/ζ, with ζ approaching 0 as ξ goes to
infinity, and vice versa. This facilitates an elegant and useful notion of infinity for the complex numbers and indeed an entire theory of meromorphic functions mapping to the Riemann sphere. The
standard metric on the unit sphere agrees with the Fubini–Study metric on the Riemann sphere.
Visualization of lines and planes
The set of all lines through the origin in three-dimensional space forms a space called the real projective plane. This space is difficult to visualize, because it cannot be embedded in
three-dimensional space.
However, one can "almost" visualize it as a disk, as follows. Any line through the origin intersects the southern hemisphere z ≤ 0 in a point, which can then be stereographically projected to a point
on a disk. Horizontal lines intersect the southern hemisphere in two antipodal points along the equator, either of which can be projected to the disk; it is understood that antipodal points on the
boundary of the disk represent a single line. (See quotient topology.) So any set of lines through the origin can be pictured, almost perfectly, as a set of points in a disk.
Also, every plane through the origin intersects the unit sphere in a great circle, called the trace of the plane. This circle maps to a circle under stereographic projection. So the projection lets
us visualize planes as circular arcs in the disk. Prior to the availability of computers, stereographic projections with great circles often involved drawing large-radius arcs that required use of a
beam compass. Computers now make this task much easier.
Further associated with each plane is a unique line, called the plane's pole, that passes through the origin and is perpendicular to the plane. This line can be plotted as a point on the disk just as
any line through the origin can. So the stereographic projection also lets us visualize planes as points in the disk. For plots involving many planes, plotting their poles produces a less-cluttered
picture than plotting their traces.
This construction is used to visualize directional data in crystallography and geology, as described below.
Other visualization
Stereographic projection is also applied to the visualization of polytopes. In a Schlegel diagram, an n-dimensional polytope in R^n+1 is projected onto an n-dimensional sphere, which is then
stereographically projected onto R^n. The reduction from R^n+1 to R^n can make the polytope easier to visualize and understand.
Arithmetic geometry
In elementary arithmetic geometry, stereographic projection from the unit circle provides a means to describe all primitive Pythagorean triples. Specifically, stereographic projection from the north
pole (0,1) onto the x-axis gives a one-to-one correspondence between the rational number points (x, y) on the unit circle (with y ≠ 1) and the rational points of the x-axis. If (m/n, 0) is a rational
point on the x-axis, then its inverse stereographic projection is the point
( 2 m n n 2 + m 2 , n 2 − m 2 n 2 + m 2 )
which gives Euclid's formula for a Pythagorean triple.
Tangent half-angle substitution
The pair of trigonometric functions (sin x, cos x) can be thought of as parametrizing the unit circle. The stereographic projection gives an alternative parametrization of the unit circle:
cos x = t 2 − 1 t 2 + 1 , sin x = 2 t t 2 + 1 .
Under this reparametrization, the length element dx of the unit circle goes over to
d x = 2 d t t 2 + 1 .
This substitution can sometimes simplify integrals involving trigonometric functions.
The fundamental problem of cartography is that no map from the sphere to the plane can accurately represent both angles and areas. In general, area-preserving map projections are preferred for
statistical applications, while angle-preserving (conformal) map projections are preferred for navigation.
Stereographic projection falls into the second category. When the projection is centered at the Earth's north or south pole, it has additional desirable properties: It sends meridians to rays
emanating from the origin and parallels to circles centered at the origin.
The stereographic is the only projection that maps all circles of a sphere to circles. This property is valuable in planetary mapping when craters are typical features. The set of circles passing
through the point of projection have unbounded radius, and therefore degenerate into lines.
In crystallography, the orientations of crystal axes and faces in three-dimensional space are a central geometric concern, for example in the interpretation of X-ray and electron diffraction
patterns. These orientations can be visualized as in the section Visualization of lines and planes above. That is, crystal axes and poles to crystal planes are intersected with the northern
hemisphere and then plotted using stereographic projection. A plot of poles is called a pole figure.
In electron diffraction, Kikuchi line pairs appear as bands decorating the intersection between lattice plane traces and the Ewald sphere thus providing experimental access to a crystal's
stereographic projection. Model Kikuchi maps in reciprocal space, and fringe visibility maps for use with bend contours in direct space, thus act as road maps for exploring orientation space with
crystals in the transmission electron microscope.
Researchers in structural geology are concerned with the orientations of planes and lines for a number of reasons. The foliation of a rock is a planar feature that often contains a linear feature
called lineation. Similarly, a fault plane is a planar feature that may contain linear features such as slickensides.
These orientations of lines and planes at various scales can be plotted using the methods of the Visualization of lines and planes section above. As in crystallography, planes are typically plotted
by their poles. Unlike crystallography, the southern hemisphere is used instead of the northern one (because the geological features in question lie below the Earth's surface). In this context the
stereographic projection is often referred to as the equal-angle lower-hemisphere projection. The equal-area lower-hemisphere projection defined by the Lambert azimuthal equal-area projection is also
used, especially when the plot is to be subjected to subsequent statistical analysis such as density contouring.
Some fisheye lenses use a stereographic projection to capture a wide-angle view. Compared to more traditional fisheye lenses which use an equal-area projection, areas close to the edge retain their
shape, and straight lines are less curved. However, stereographic fisheye lenses are typically more expensive to manufacture. Image remapping software, such as Panotools, allows the automatic
remapping of photos from an equal-area fisheye to a stereographic projection.
The stereographic projection has been used to map spherical panoramas. This results in effects known as a little planet (when the center of projection is the nadir) and a tube (when the center of
projection is the zenith).
The popularity of using stereographic projections to map panoramas over other azimuthal projections is attributed to the shape preservation that results from the conformality of the projection.
Stereographic projection Wikipedia
(Text) CC BY-SA | {"url":"https://alchetron.com/Stereographic-projection","timestamp":"2024-11-04T04:58:22Z","content_type":"text/html","content_length":"166680","record_id":"<urn:uuid:c48d4d23-980f-464b-8d25-32b8457993b5>","cc-path":"CC-MAIN-2024-46/segments/1730477027812.67/warc/CC-MAIN-20241104034319-20241104064319-00194.warc.gz"} |
CAPM Model Analysis and Sensitivity
This assignment delves into the Capital Asset Pricing Model (CAPM) and its application in analyzing stock returns. Students are tasked with conducting a CAPM analysis using regression tools,
identifying the model's limitations, particularly concerning market return estimations, and examining how research results are influenced by sector-specific characteristics. The focus is on
understanding the relationship between stock returns, market returns, and the impact of industry trends. | {"url":"https://desklib.com/document/test-of-camp-theory-report/","timestamp":"2024-11-14T13:49:49Z","content_type":"text/html","content_length":"825877","record_id":"<urn:uuid:fa614a40-3a77-4bed-b952-e8f6b2f102fc>","cc-path":"CC-MAIN-2024-46/segments/1730477028657.76/warc/CC-MAIN-20241114130448-20241114160448-00056.warc.gz"} |
Does Correlation Imply Causation? (3 Key Concepts) | jdmeducational
Does Correlation Imply Causation? (3 Key Concepts)
“Correlation does not imply causation.” This statement has been uttered so many times in stats classes, the statement has almost become a cliche! What does it really mean though?
In this article, we’ll learn about the correlation between two variables. We’ll also learn about causation – how can we determine if one event causes another?
Let’s get started!
What does correlation mean?
In the context of statistics, two variables are correlated if there is an association between the two variables.
If both variables are moving in the same direction, we have a positive correlation. If one variable increases as the other variable decreases, we have a negative correlation. In either case, the
variables change together; we can also say that they covary.
When there is no relationship between the two variables, there is a zero correlation.
This graph shows a positive correlation between time spent studying and exam grades.
This graph shows a negative correlation between temperature and hot chocolate sales.
This graph shows a zero correlation between the variables x and y.
What is a correlation coefficient?
In statistics, we can actually use techniques to measure the strength of the correlation between two variables. The most common tool used in statistics classes to measure the strength of correlation
is called the Pearson Correlation.
This coefficient is often used in linear regression and is represented by the variable r. It is a single number that represents the strength of the relationship between 2 variables. The value of r
lies in the interval:
-1 <= r <= 1
Negative values of r indicate a negative relationship between the variables; that is the variables move in the opposite direction. An r value of -0.8 shows a strong negative relationship whereas an r
value of -0.3 shows a weak negative relationship.
The closer the r value is to 0, the weaker the relationship.
Similarly, positive values of r indicate a positive relationship between the variables, so an r value of 0.9 is a very strong positive relationship. To compare the strength of the relationship
between two variables, we can compare the absolute value of the r values.
If you want to play a game to guess the correlation coefficient of scatter plots, try this. (Warning: this game is way more fun than it should be!)
Geogebra Guess the Correlation Coefficient
We’ve learned about correlation. It seems as though it would be easy to draw conclusions from correlated variables.
Often, our conclusions may be incorrect. Consider the relatively famous example of the relationship between ice cream sales and shark attacks. It turns out that the two variables are closely
The graphs of ice cream sales and shark attacks look like they are related somehow.
Source: statology.org
Notice how closely the two graphs align! The two variables of ice cream sales and shark attacks have a strong correlation.
But does this mean that eating ice cream makes you more likely to get attacked by a shark? Probably not! Also, I hope not because I eat a lot of ice cream!
So, what is happening here? Well, there may be another variable at work. People tend to eat more ice cream during the warmer months and they’re more likely to swim in the ocean in the warmer months,
hence the similarity in the two graphs.
Sometimes, two variables may appear to be related but perhaps the relationship is by random chance or by another variable.
Important note: correlation can verify the existence of a relationship between two variables but does not confirm that one variable causes the other.
Why doesn’t correlation imply causation?
There are a few reasons why correlation does not mean that one variable caused the other.
1. The presence of a third variable. There may be a third variable that affects both variables, making it seem as though there is a causal relationship when there isn’t. For example, in the ice
cream sales and shark attack example, the third variable, warmer temperatures, causes both ice cream sales to go up and shark attacks to increase. The third variable acts on each of the variables
• The apparent relationship between the two variables occurred by random chance, not because one variable caused the other. A famous example comes from the NFL. The outcome of the most recent
Washington Commanders (formerly the Washington Redskins) home game prior to a US presidential election correlated strongly. When Washington won, the incumbent US president won; when Washington
lost, the candidate from the opposing party won. This relationship was true from 1940 to 2000! (Source: wikipedia.org)
• There may be a sampling error in the study. If a study isn’t properly randomized, there can appear to be a correlation between two variables. This could be true of the sample but not of the
overall population.
What is causation?
Causation indicates that one event directly caused the other event. There is a cause and effect relationship between the two events. (Source: abs.gov)
Sometimes it is easy to discern the difference between correlation; other times, not so much. In theory, it seems as though it should be a straightforward task, determining causation but in practice
this can be challenging. Statisticians have developed methods that help understand whether two variables are correlated or whether one causes the other.
If you’re familiar with medical studies, you may be aware of the idea of randomizing samples and establishing control groups. In a controlled study, participants are divided into two groups.
Typically, one group receives the treatment, like a new medicine, and the other group receives a placebo.
If the treatment group shows a noticeably different outcome compared with the control group, there is a possible cause and effect relationship; that is, the new medicine may help alleviate the
condition being treated!
A control group allows researchers to find out if a treatment has a significant effect or not.
Strategies to help attain causation
It turns out there are 3 criteria that are necessary for establishing cause and effect between 2 variables. If we want to show that X causes Y, the following 3 conditions must be met:
3 criteria for establishing causation
1. Temporal sequencing – the cause X must precede the effect Y. This may seem like common sense, but it is important to know which event happened first.
2. Non-spurious relationship – The relationship between the 2 variables is not due to chance alone. (Our NFL example from above demonstrates a relationship that was due to chance.)
• Eliminate alternate causes – there is no other underlying 3rd variable that accounts for the relationship between the X and Y variables. The ice cream sales/shark attack example is an alternate
cause that accounts for the relationship between the two variables.
source: umks.edu
Now that we’ve learned a bit more about correlation and causation, you’ll have better tools analyzing data going forward. So, the next time you’re having a statistics related conversation with your
peers, impress them with your knowledge of correlation and causation!
You can learn how to find correlation coefficients in Excel here.
About the author:
Jean-Marie Gard is an independent math teacher and tutor based in Massachusetts. You can get in touch with Jean-Marie at https://testpreptoday.com/. | {"url":"https://jdmeducational.com/does-correlation-imply-causation-3-key-concepts/","timestamp":"2024-11-02T07:50:27Z","content_type":"text/html","content_length":"82068","record_id":"<urn:uuid:b9b0a611-fa30-471a-a3d7-405c3fa4ec0c>","cc-path":"CC-MAIN-2024-46/segments/1730477027709.8/warc/CC-MAIN-20241102071948-20241102101948-00201.warc.gz"} |
What is the area enclosed by r=thetacostheta-2sin(theta/2-pi) for theta in [pi/4,pi]? | HIX Tutor
What is the area enclosed by #r=thetacostheta-2sin(theta/2-pi) # for #theta in [pi/4,pi]#?
Answer 1
$A = {\int}_{\frac{\pi}{4}}^{\pi} \frac{1}{2} {\left(\theta \cos \theta + 2 \sin \left(\frac{\theta}{2}\right)\right)}^{2} d \theta$
Area formula is #A =int_a^b 1/2 r^2 d theta#
Since #sin (theta/2 -pi) = - sin (pi - theta/2)= -sin (theta/2)#
Applying this formula #A=int_(pi/4) ^pi 1/2 ( theta cos theta + 2 sin (theta/2))^2 d theta #
This cannot be solved manually .
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Answer 2
To find the area enclosed by the curve ( r = \theta \cos(\theta) - 2 \sin\left(\frac{\theta}{2} - \pi\right) ) for ( \theta ) in ( [\frac{\pi}{4}, \pi] ), you need to integrate ( \frac{1}{2} r^2 )
with respect to ( \theta ) over the given interval. The integral will give you the area of one loop of the curve. Then, you can subtract the area under the curve for ( \theta ) from ( \frac{\pi}{4} )
to ( \frac{\pi}{2} ) from the area under the curve for ( \theta ) from ( \frac{\pi}{2} ) to ( \pi ). This subtraction accounts for the loops in opposite directions.
The expression for the area ( A ) enclosed by the curve is:
[ A = \int_{\frac{\pi}{4}}^{\frac{\pi}{2}} \frac{1}{2} \left(\theta \cos(\theta) - 2 \sin\left(\frac{\theta}{2} - \pi\right)\right)^2 , d\theta - \int_{\frac{\pi}{2}}^{\pi} \frac{1}{2} \left(\theta \
cos(\theta) - 2 \sin\left(\frac{\theta}{2} - \pi\right)\right)^2 , d\theta ]
You can then calculate each integral separately using numerical methods or appropriate techniques like integration by parts or substitution.
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Answer from HIX Tutor
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
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Definite Integral to Limit of Riemann Sum Calculator - GEGCalculators
Definite Integral to Limit of Riemann Sum Calculator
Definite Integral Calculator
Q2: What is the definite integral as a limit of a sum?
A2: The definite integral as a limit of a sum, also known as the Riemann integral, represents the process of calculating the area under a curve by partitioning the area into infinitely many small
rectangles and taking the limit as the width of these rectangles approaches zero. It is defined by the limit of a Riemann sum, as mentioned in the previous answer.
Q3: Is there a definite integral as the limit of a Riemann sum module?
A3: There is no specific “module” in standard mathematical notation or terminology called the “definite integral as the limit of a Riemann sum module.” Instead, the concept of the definite integral
as a limit of a Riemann sum is a fundamental part of calculus and is typically explained as a mathematical concept or theory within calculus courses and textbooks.
To understand and work with definite integrals, you don’t need a specific module; you can use mathematical notation and concepts as described in the previous answers. Software and computational tools
often provide functions or methods for numerical approximation of definite integrals, but these are not referred to as “modules” in the context of calculus.
Q: What are some common techniques for evaluating definite integrals?
A: There are several techniques for evaluating definite integrals, including:
1. Direct Integration: This involves finding the antiderivative of the function and applying the fundamental theorem of calculus.
2. Integration by Substitution: Use a change of variables to simplify the integral.
3. Integration by Parts: Apply the integration analogue of the product rule for differentiation.
4. Trigonometric Integrals: Special techniques for trigonometric functions.
5. Partial Fraction Decomposition: Decompose a rational function into simpler fractions.
6. Improper Integrals: Handle integrals over unbounded intervals or functions with infinite discontinuities.
7. Numerical Methods: Use numerical techniques like the trapezoidal rule or Simpson’s rule for approximating definite integrals when analytical methods are impractical.
The choice of technique depends on the specific integral and its complexity.
GEG Calculators is a comprehensive online platform that offers a wide range of calculators to cater to various needs. With over 300 calculators covering finance, health, science, mathematics, and
more, GEG Calculators provides users with accurate and convenient tools for everyday calculations. The website’s user-friendly interface ensures easy navigation and accessibility, making it suitable
for people from all walks of life. Whether it’s financial planning, health assessments, or educational purposes, GEG Calculators has a calculator to suit every requirement. With its reliable and
up-to-date calculations, GEG Calculators has become a go-to resource for individuals, professionals, and students seeking quick and precise results for their calculations.
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A Couple of Beautiful Things I Learned Today
I just wanted to make this post to share a couple of beautiful moments from the workshop I attended today.
This one is something I managed to write myself, and I'm fairly proud of it. The problem was to write a function that would take a number and return true if it was even, and false if it wasn't.
function even(num){
return !(num%2);
This function relies on the fact that 0 evaluates falsey (that is to say, when it's type coerced to a boolean, it resolves as "false"). Since I know that the modulo operation will give a zero when a
number divides evenly, I just have to flip the value with a not (!) operator. After that, it'll give me true when num is even, and false when it isn't.
I felt proud of myself for finding this solution.
Next, we have some stuff that's obvious for anyone who's done any amount of formal logic, but felt like a real revelation/accomplishment for me.
The first part of the assignment was to reproduce "or" functionality, meaning the || operator, with only ! and && operators. The second was to reproduce "and" functionality (&&) with only ! and || to
work with.
function or(a, b){
return !(!a&&!b);
function and(a, b){
return !(!a||!b);
Previously I had written the "or" with if/else statements, accounting for !a&&b, a&&!b, and a&&b as potential cases that should return true. Finding a way to write it succinctly in a single line
without any of the accoutrements of JS control flow felt like a revelation.
The secret, of course is that the "or" operator will only return false when both operands are false. So we can represent that situation as !a&&!b. That will resolve to "true" when both a and b are
false, and it will resolve to "false" in all other scenarios. This is the opposite of the functionality we want from it, so the last step is to simply invert the resulting boolean.
Similarly, I had written a&&b as follows:
function and(a, b){
if (a) if (b) return true;
return false;
I couldn't understand how to constrain the || operator in an effective way, and honestly I needed a glimpse at someone else's solution in order to understand how to conceptualize my own.
Still, filling in the rest to reach this solution and cleaning it up in these ways felt beautiful. The solutions are elegant, perfect constructs. Not a single wasted character, all working in
concert. No flaws, no edge cases. Just logic.
Let's look at them again:
function or(a, b){
return !(!a&&!b);
function and(a, b){
return !(!a||!b);
ahhh, so serene.
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mole ratio
26 Aug 2024
Title: Understanding Mole Ratios: A Fundamental Concept in Chemistry
Abstract: Mole ratios are a crucial concept in chemistry, representing the quantitative relationship between the amounts of reactants and products in chemical reactions. In this article, we will
delve into the definition, significance, and mathematical representation of mole ratios.
Introduction: Chemical reactions involve the transformation of one or more substances (reactants) into new substances (products). The mole ratio is a measure of the quantitative relationship between
the amounts of reactants and products in these reactions. It is a fundamental concept that underlies many chemical calculations, including stoichiometry and limiting reagent problems.
Definition: The mole ratio is defined as the ratio of the number of moles of one substance to the number of moles of another substance. Mathematically, it can be represented as:
Mole Ratio = (moles of substance A) / (moles of substance B)
In ASCII format:
Significance: The mole ratio is significant because it allows chemists to predict the quantitative outcome of chemical reactions. By knowing the mole ratio, chemists can determine the amount of
product that will be formed from a given amount of reactant.
Mathematical Representation: The mole ratio can also be represented in terms of the coefficients of the balanced chemical equation. For example, consider the reaction:
2H2 + O2 → 2H2O
In this case, the mole ratio between hydrogen gas (H2) and oxygen gas (O2) is 2:1.
Mole Ratio = 2 / 1
Mole Ratio = n(H2) / n(O2)
Conclusion: The mole ratio is a fundamental concept in chemistry that represents the quantitative relationship between reactants and products in chemical reactions. Its significance lies in its
ability to predict the outcome of chemical reactions, making it an essential tool for chemists.
• Atkins, P., & de Paula, J. (2018). Physical Chemistry. Oxford University Press.
• Chang, R. (2015). Physical Chemistry for the Life Sciences. McGraw-Hill Education.
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Maths Selina Concise ICSE Solutions Class 10
Selina Concise Mathematics Class 10 ICSE Solutions Chapter 5 Quadratic Equations Ex 5C These Solutions are part of Selina Concise Mathematics Class 10 ICSE Solutions. Here we have given Selina
Concise Mathematics Class 10 ICSE Solutions Chapter 5 Quadratic Equations Ex 5C. Other Exercises Selina Concise Mathematics Class 10 ICSE Solutions Chapter 5 Quadratic Equations … [Read more...]
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James' Empty Blog
Honestly. What's the point in being on the other side of the world if not to be able to see interesting astronomical phenomenon that are not visible to the rest of you?
Apparently there's a lunar eclipse. But it's cloudy. Bah!
This is what it looked like to people up north in Hokkaido:
(Borrowed from
Jules cynically says it always looks like that through the smog anyway.
The Japanese Meteorological Agency is starting up a new early warming system for earthquakes at the start of October - of course earthquakes are not meteorological, but they are covered by the
"natural disaster" remit. Shame foreigners aren't, but that's another story.
According to what I've read, the basic idea seems to be that they hope to detect the initial "p-wave" tremors (the primary pressure wave which travels fast), and warn before the main s-wave
(secondary transverse wave which travels slower) hits. OTOH this page talks about a third type of "surface waves" which are slower still and cause the most damage. Anyway, with a speed of something
like 4km/sec, it will be challenging to get any warning out in the area close to the epicentre (where the damage will be focussed) early enough to matter. But even a few seconds may be enough for
people to duck under their desks. I just hope there won't be too many false alarms, or people will simply ignore them. We are regularly amused by the warnings of heavy rain over the loudspeakers in
the street, that we can barely hear over the noise of the rain that is already thundering down :-) To be fair they do sometimes beat the storm.
The Shinkansen has had an automated system like this in operation for some time. Even if the train doesn't have time to stop, any slowing down can only help. The one time there was a derailment, in
Niigata 2004, there still weren't any injuries.
An interesting example I spotted on Andrew Gelman's blog.
UK readers will remember a medical test where 6 people took a particular drug and all had an extreme life-threatening reaction ("cytokine storm", whatever that means). Apparently there were also 2
controls, who were not treated, and who (surprise) did not suffer the reaction.
But...with only 8 samples in total, the results are barely significant in frequentist terms. Perhaps the simplest way of analysing the result is to ask the following: since 6 people out of 8 fell
ill, and given the null hypothesis that the treatment and control outcomes are probabilistically identical, what is the probability that the 6 ill people would coincide with the 6 treated people?
This is a simple combinatorial question, the answer to which is 1/28 or 3.6% (there is some more detailed discussion at the link about the correct test to use). So it is just significant at the p
<0.05 threshold but not p<0.01. Given the number of medical trials taking place, we should expect such failures regularly.
But we don't, of course. The reason being, our prior expectation of someone naturally having such a life-threatening reaction, absent any provocation, is so low as to be virtually zero. Any plausible
Bayesian updating of the prior belief P(treatment is harmful) in the light of the observed data, is going to massively increase this probability, because the alternative hypothesis (that the
reactions occurred by chance) is even lower. And this is obviously what all the researchers and commentators have actually done in practice, even if not explicitly and precisely.
Eg let's model it as the test having two possibilities: either it is harmful (all subjects will suffer) or not (reaction has the background probability 0.0001, surely an overestimate). Given an
extremely complacent prior belief that the test is harmless with probability 0.999, the posterior after 6 test subjects have all reacted is given by:
P(test is harmful)=1*0.001/(1*0.001+0.0001^6*0.999) = 1, to as many significant digits as I can be bothered writing. That's a very trivial analysis of course, but real maths is hard to do in Blogger
(no LaTeX facility).
Great news. According to a new survey, almost 60% of Japanese actually think that foreigners are deserving of human rights.
Of course, that means that just over 40% don't.
Mind you, it does sometimes feel to us like we are living on an alien planet.
Interesting to see Nature jumping on the no job prospects for PhDs and postdocs bandwagon (via Pharyngula). (Disclaimer - I haven't yet read the Nature article - no access at home - but I'm assuming
that it draws the same obvious conclusion from the statistics.) Not so long ago their "jobs editor" Paul Smaglik was having a go at some anonymous blogging post-doc for daring to suggest that
anything was less than perfect in their work life. But it is of course blindingly obvious that if every tenured staff member mentors on average even a single PhD student at a time, them the
overwhelming majority of these PhDs will not subsequently go on to get tenured positions in academia.
Of course one can legitimately argue that it is fine for the vast majority of PhDs to not get academic jobs (and for a large majority of postdocs to never land a tenured position) - so long as they
are aware of the situation and walk into it with eyes open, that's OK. But that hardly justifies the sort of situation where people complain about (and are reported uncritically on) the "shortage" of
qualified staff simply because they "only" get 30 applicants per post rather than 75!
Just in case anyone was under the misapprehension that the USA had cornered the market in right-wing nutcases:
Japan activist posts finger to PM
If I was the PM, I'd proffer a finger in return - but keep it firmly attached. About the only thing that Abe has done right since taking office is to not go to openly worship war criminals.
I know that not owning a TV probably hurts my Japanese comprehension. But with adverts like this, (for rental of "24" season 6 DVDs) can you blame me?
I think it is clear that at least one of the following propositions is true: (1) the human species is very likely to go extinct before developing supernatural powers; (2) any civilization with
supernatural powers is extremely unlikely to create a significant number of "universes"; (3) we are almost certainly living in a universe designed and manufactured by a "Creator". It follows that the
belief that there is a significant chance that we will one day develop into a race with supernatural powers who create universes is false, unless we are currently living in a universe created by such
a being.
I don’t pretend to know which of these hypotheses is more likely, but think that none of them can be ruled out. My gut feeling, and it’s nothing more than that, is that there’s a 20 percent chance
we’re living in a created universe (maybe I really think it's much lower, but Pascal's wager and all that).
My argument is either (a) a quasi-religious triviality or (b) a major new scientific breakthrough, depending on your point of view. As for me, I'm staying out of it - I'm good enough at making
enemies in my day job without going out and actively looking for new ones elsewhere :-)
Sorry to lead on any Zaurus fans with the title - it's not truly a new Zaurus, but merely new to me. The Zaurus line is long since defunct (indeed even the last few models were little more than cash
cows to milk some more profit out of the design, with very little innovation), and my old SL-C860 is a little long in the tooth with the hinge now starting to misbehave a little. With this in mind, I
thought it was time to upgrade before it was too late. So a few weeks ago we headed off to Akihabara where I expected to pick up the newest (least old) SL-C3200 for about ¥50,000 or a little more.
You can learn more about the 3200 model here BTW.
Rather to our surprise, we managed to find a small Sofmap shop which sells 2nd hand stuff, which we had last visited several years ago (in fact I might have got my 860 there). Things often change
over that sort of time scale here, and our memories were hazy about its location anyway. But it is still there, and had the full range of Zaurus models (along with all sorts of PDAs and cameras). I
hadn't really gone looking for the older 3100 model, but I vaguely remembered that it was almost the same as the 3200 - just a slightly smaller disk, and less software for English learners, neither
of which I am bothered about. So for a further saving of about ¥8,000 compared to their 3200s (which were already well below the best "new" price I could find), I chose the former. Although nominally
2nd hand, I think they must be unsold shop returns as they are in near-immaculate condition.
So now I've got a new Zaurus to install all my favourite software, and an excuse to have another look at the full range of software available (which to be honest hasn't changed much). The machine is
clearly better than the 860 in many minor ways - better keyboard feel and layout, a nice Japanese-English dictionary and encyclopedia (especially now I can read it a bit), obviously massive disk
space (comparatively speaking) and a slightly more elegant base shape. There are some other minor updates to the bundled software which are moderately useful/entertaining, like a train timetable/
planner. It is also now very clear that my 860's battery was getting rather weak - the new machine lasts much better. Mostly it's the same however, which is what I wanted.
As well as gaining a shiny new SL-C3100, I have of course also gained a spare disposable SL-C860, which means I can play at being a Linux geek and install new distributions like pdaXrom on it - this
is a full X11 windows manager thing with huge number of applications available. Installing the basic package was straightforward, setting up things like the internet connection rather less so, but
I've even got (one method of) that working OK now (ok, jules fixed the last bit for me). I'm not sure that it is really that useful to me, especially since it means losing a lot of Sharp's inbuilt
Japanese language abilities (and/or losing a lot of hair trying to install enough bits and pieces to get back roughly to where I started). But it's something new to play with.
Via email, I hear that this paper from Stephen Schwartz is making a bit of a splash in the delusionosphere. In it, he purports to show that climate sensitivity is only about 1.1C, with rather small
uncertainty bounds of +-0.5C.
Usually, I am happy to let RealClimate debunk the septic dross that still infects the media. In fact, since I have teased them about their zeal in the past, it may seem slightly hypocritical of me to
bother with this. However, this specific paper is particularly close to my own field of research, and the author is also rather unusual in that he seems to be a respected atmospheric scientist with
generally rather mainstream views on climate science (although perhaps a bit critical of the IPCC here). However, his background is in aerosols, which suggests that he may have stumbled out of his
field without quite realising what he is getting himself into.
Anyway, without further ado, on to the mistakes:
Mistake number 1 is a rather trivial mathematical error. He estimates sensitivity (K per W/m^2) via the equation
where C is the effective heat capacity (mostly ocean) and t is the time constant of the system (more on this later).
His numerical values for t and C are 5+-1, and 16.7+-7 respectively (with the uncertainties at one standard deviation). It is not entirely clear what he really intends these distributions to mean
(itself a sign that he is a little out of his depth perhaps), but I'll interpret them in the only way I think reasonable in the context, as gaussian distributions for the parameters in question. He
claims these values gives S equal to 0.3+-0.09, although he also writes 0.3+-0.14 elsewhere. This latter value works out at 1.1C+-0.5C for a doubling of CO2. But the quotient of two gaussians is not
gaussian, or symmetric. I don't know how he did his calculation, but it's clearly not right.
In fact, the 16%-84% probability interval (the standard central 68% probability interval corresponding to +- 1sd of a gaussian, and the IPPC "likely") of this quotient distribution is really
0.18-0.52K/W/m^2 (0.7-1.9C per doubling) and the 2sd limit of 2.5% to 97.5% is 0.12-1.3K/W/m^2 (0.4-4.8C per doubling). While this range still focuses mostly on lower values than most analyses
support, it also reaches the upper range that I (and perhaps increasingly many others) consider credible anyway. His 68% estimate of 0.6-1.6C per doubling is wrong to start with, and doubly
misleading in the way that it conceals the long tail that naturally arises from his analysis.
Mistake number 2 is more to do with the physics. In fact this is the big error, but I worked out the maths one first.
He estimates a "time constant" which is supposed to characterise the response of the climate system to any perturbation. On the assumption that there is such a unique time constant, this value can
apparently be estimated by some straightforward time series analysis - I haven't checked this in any detail but the references he provides look solid enough. His estimate, based on observed 20th
century temperature changes, comes out at 5y. However, he also notes that the literature shows that different analyses of models give wildly different indications of characteristic time scale,
depending on what forcing is being considered - for example the response to volcanic perturbations has a dominant time scale of a couple of years, whereas the response to a steady increase in GHGs
take decades to reach equilibrium. Unfortunately he does not draw the obvious conclusion from this - that there is no single time scale that completely characterises the climate system - but presses
on regardless.
Schwartz is, to be fair, admirably frank about the possibility that he is wrong:
This situation invites a scrutiny of the each of these findings for possible sources of error of interpretation in the present study.
He also says::
It might also prove valuable to apply the present analysis approach to the output of global climate models to ascertain the fidelity with which these models reproduce "whole Earth" properties of
the climate system such as are empirically determined here.
Perhaps a better way of putting that would be to suggest applying the analysis to the output of computer models in order to test if the technique is capable of determining their (known) physical
properties. Indeed, given the screwy results that Schwartz obtained, I would have thought this should be the first step, prior to his bothering to write it up into a paper. I have done this, by using
his approach to estimate the "time scale" of a handful of GCMs based on their 20th century temperature time series. This took all of 5 minutes, and demonstrates unequivocally that the "time scale"
exhibited through this analysis (which also comes out at about 5 years for the models I tested) does not represent the (known) multidecadal time scale of their response to a long-term forcing. In
short, this method of analysis grossly underestimates the time scale of response of climate models to a long-term forcing change, so there is little reason to expect it to be valid when applied to
the real system.
In fact there is an elementary physical explanation for this: the models (and the real climate system) exhibit a range of time scales, with the atmosphere responding very rapidly, the upper ocean
taking substantially longer, and the deep ocean taking much longer still. When forced with rapid variations (such as volcanoes), the time series of atmospheric response will seem rapid, but in
response to a steady forcing change, the system will take a long time to reach its new equilibrium. An exponential fit to the first few years of such an experiment will look like there is a purely
rapid response, before the longer response of the deep ocean comes into play. This is trivial to demonstrate with simple 2-box models (upper and lower ocean) of the climate system.
Changing Schwartz' 5y time scale into a more representative 15y would put his results slap bang in the middle of the IPCC range, and confirm the well-known fact that the 20th century warming does not
by itself provide a very tight constraint on climate sensitivity. It's surprising that Schwartz didn't check his results with anyone working in the field, and disappointing that the editor in charge
at JGR apparently couldn't find any competent referees to look at it.
So, I was googling for some IDL code for statistical tests (why reinvent the wheel) and I came across this ugly documentation of a Kolmogorov-Smirnov test:
; OUTPUTS: Probability two populations are drawn from same
; underlying distribution.
That's from the documentation of the UKMO IDL library (which I don't actually have, although it seems freely available). Of course, it is dead wrong. A K-S test does not calculate this at all!
It's not just climate scientists, though - essentially the same error is contained in this equivalent routine from a German university astronomy and astrophysics department (a high google hit):
PROB gives the probability that the null hypothesis (DATA came from a
Gaussian distribution with unit variance) is correct.
This one is particularly amusing, because the immediately previous two lines are
IDL> data = randomn(seed, 50) ;create data array to be tested
IDL> ksone, abs(data), 'gauss_cdf', D, prob, /PLOT ;Use K-S test
thus indicating beyond any shadow of a doubt that in this example the data did come from a Gaussian distribution, irrespective of the value of "prob" that results from a single application of the
test (ignoring quibbles about pseudorandom versus "truly" random).
In case it's not clear enough, the error in both sets of documentation is that the K-S test actually reports the probability that a particular test statistic would be exceeded if the two
distributions were the same, in other words, a frequentist P(data=D|hypothesis=H) statement. The alternative P(H|D), which the documentation claims that the routine outputs, is an entirely different
beast which demands a Bayesian treatment - in particular, it depends critically on a prior P(H), for which there is (in general) no default, "objective", or "correct" choice.
Both these routines claim to be based directly on Numerical Recipes. It is notable that the text of NR (at least my edition) avoids making this particular error. However, the same (latter) routine,
with the same error in the documentation, turns up all over the place, with apparently no-one in NASA, Princeton, Washington Uni etc (to name but three) ever noticing...
I suppose I could be more sympathetic to the climate scientists who have apparently been seduced by such ubiquitous and intuitively appealing language, and who have as a result tried to reshape
probability theory so as to make such statements actually valid. OTOH, I still think they should still be prepared to accept that their theories are wrong when I point out the problems in words of
one syllable together with elementary examples :-)
Via Andy Ridgwell (whose latest paper was featured in Science's "best of the rest" recently [sub required]):
Thailand is dressing up errant policemen in "Hello Kitty" armbands to humiliate them. In order to take off the armband, they have to go up to a random member of the public, purring and meowing. If
the member of the public cannot say "poor pussy" three times while stroking the policeman, without smiling, then the policeman gets to take off the armband.
Japan, not to be outdone, has introduced a humiliation system for cats - dressing them up as Thai policemen:
Poor pussy indeed!
After the superhuman hibernation episode, and the supposed inability of Japanese to tell lambs from poodles, I'm a little unsure that this mundane story of a traffic accident is really bloggable:
A 54-year-old man continued to drive a large motorcycle about 2 km after hitting the center divider on a national highway and losing his right leg below the knee, police said Tuesday.
Kazuo Nagata, a salaried worker, was apparently unaware of the loss of his leg despite the acute pain he experienced from the impact, police said.
However, unlike the previous two stories, this one appears to be true.
This paper in Science has had a surprisingly muted reaction in the blogosphere. It's almost as if climate scientists aren't supposed to validate their methods and/or make falsifiable predictions.
In contrast to those rather underwhelmed posters, I think it's a really important step forwards, not just in terms of the actual prediction made (which, to be honest is not all that exciting) but
what it implies about how people are starting to think more quantitatively and rigorously about the science of prediction. Of course the Hadley Centre is well placed for this trend given their close
links to the UKMO. I could probably do the odd bit of trivial nit-picking about the paper if I felt like it, but that would be churlish in the absence of a better result. I am sure they are well on
the way to improving their system anyway (the paper was submitted way back in January).
A quick note about the forecast "plateau" in temperatures that was the focus of much of the news coverage: the central forecast may stay slightly below the 1998 observed peak until 2010, but the
spread around this forecast assigns significant probability to a higher value. If one assumes that the annual anomalies (relative to the forecast mean) are independent with each of 2008 and 2009
having a 30% chance of exceeding 1998 (just from eyeballing their plot), then that gives a 50% chance of a new record before 2010, and 75% including 2010, which is virtually the same as what I wrote
At first glance, I thought this was a sign that the CBI was actually considering putting their money where their mouths are (albeit in a feeble manner), by offering minuscule bursaries to science
students. But no, it's cheaper to lobby the Govt for it than to actually dip their hands into their own pockets.
As for "struggling to fill their posts", "struggling to fill their pockets on the backs of underpaid and exploited workers" would be more like it. "Only" 30 applicants per job? My heart bleeds. If
their research and development is only viable on the premise of a never-ending supply of lab fodder desperately scrabbling for the scraps on offer then maybe we wouldn't miss them so much. Just how
many people do they actually want the Govt to train (at vast expense) for each job on offer?
There's a simple solution in the free market that the CBI claim to believe in: PAY MORE MONEY, MORONS! I don't really mean purely "more money", rather a more general "better conditions" - but of
course advocates of Stern think that everything can be reduced to cash :-). Sorry to shout, but it really gets my goat to hear these fat cats, who are sitting pretty at the top of the capitalist
pyramid, desperately struggling to stick their snouts in the socialist trough of Govt subsidies with the intention of propping up their businesses with a never-ending supply of compliant and
desperate wage slaves.
I rather liked this comment (found via the Adam Smith Institute blog, who I see has linked to my previous post):
You’re a well compensated, shiny-suited male executive spending a week at a conference in Amsterdam. In the evenings you experience a “shortage” of women willing to sleep with you. How do you
solve this problem? Do you perhaps write to your MP demanding that the EU offer grants to nubile Ukranian girls to migrate to brothels in western Europe?
Note that the author is an ex-scientist following the abrupt closure of his lab, so may just possibly be even more bitter and twisted than me (note to self: must try harder).
Obviously Barbie was just anticipating the latest research :-)
However, NewScientist is still well behind the times, bleating recently that:
The most urgent problem for UK science is the shortage of enthusiastic new recruits. The proportion of teenagers choosing to study physics at ages 16 and 18 is in free fall. The situation in
engineering and maths is little better and in chemistry things are starting to decline too. Just about everyone bar the government accepts that the root cause is a shortage of schoolteachers
qualified in these subjects to inspire pupils. There will be no solution until this is officially accepted...
The "shortage of new recruits" is, I assert, merely the free market speaking: achieving a useful level of skill in scientific subjects is hard, and those who are capable of it can get much greater
rewards (certainly in financial terms) elsewhere. Note that even with the current supposedly "hard" science A-Levels, some universities have switched to 4 years rather than 3 for their degree
courses, at least for people who are considering research.
I find it disturbing that people can seriously propose that all we need is smooth-talking teachers to con pupils into a low-paid and insecure job with stringent intellectual demands, severe
competition for jobs and high failure rate, when they would be substantially better off elsewhere. As I've mentioned before, an average estate agent in the UK earns 50% more than a scientist, and if
you want to consider careers with perhaps more comparable intellectual demands, an average GP earns about 3 times as much, and has a secure job for life too. I'm not saying that these people aren't
worth their salaries, but for anyone who is considering becoming a scientist, and who thinks that they might want to buy a house (say) at some point in their adult lives, bear in mind that this is
the sort of financial competition you'll be up against.
Of course I should acknowledge that there are good things about being a scientist, especially for the eccentrics and independently-monied :-) But for normal people, it's a rather poor choice, and I'd
rather see people talking openly about the real problems than papering over the cracks. Yet more innocent post-doc fodder is most certainly not what we need.
Spotted in an advert in NewScientist:
"Women are therefore especially encouraged to apply. The Max Planck Society also wishes to employ more severely challenged persons..."
At least they didn't say "... even more severely challenged persons (if such a thing exists)..." :-)
Another summer, another week of walking along amazing mountain ridges...
Unlike Stoat I don't have any pictures of naked men to show for my time away. Just flowers and mountain views:
A fuller set of pictures will appear in due course on my web site (update: here). Please excuse me if I have a distant look in my eyes for the next week or two...
No, not a tale of turkey twizzlers, but dolphin meat in Japan. A couple of local politicians have dared to point out the bleeding obvious, that the dophins "traditionally" slaughtered off the coast
on Japan and then stuffed into schoolkids (no-one actually buys the stuff willingly) by politicians in the hope that they will be indoctrinated into this "traditional way of life" are actually not
fit for human consumption.
I look forward to the agriculture minister claiming that the Japanese intestines are adapted to mercury-rich food through their unique genetic heritage.
Perhaps not. Mercury poisoning has a long history in Japan - they basically invented the problem, and some people are justifiably touchy about the subject (which was covered up for decades, and
lawsuits from the infamous ~1950s pollution scandal continue today).
Actually Prime Minister Abe has just lost his 2nd agriculture minister in as many months, both due to embezzlement scandals (the last may actually have more to do with the ruling LDP's historic
defeat in the recent elections).
To be fair, the actual amount of dolphin eaten is probably small enough that the mercury isn't that big a health problem. But it all makes good knockabout politics.
There's a mildly interesting story in the papers about how a "betting market" is investigating funny dealings on a tennis match where someone lost to a substantially worse player, in suspicious
circumstances. Of course this is precisely the principle behind the sadly abandoned "Policy Analysis Market" - that for a price, even crooks will part with their information. Nevertheless. we should
not ignore the chicken and egg problem, that if it was not for the betting market, there would have been no incentive for anyone to throw the match. Similarly, trading futures on the life of a
specific politician gives people the chance to make two killings with one bullet.
One thing you can be sure is that no-one would ever pay a British tennis player to lose a match - why bother when they do it so reliably for free :-)
JQ says:
"And even a 10 per cent reduction in income, by 2050, would not actually be noticeable against the background noise of macroeconomic and individual income fluctuations."
10% reduction in income by 2050, or equivalently 20% by 2100, is of course the far (lunatic?) extreme of the worst case that Stern could put together, not a realistic estimate.
Before JQ has a sense of humour failure, I'd better point out to that the above quote was addressing the costs of mitigation, not the projected losses due to climate change. But of course in economic
terms, 10% is 10%. What's more, a figure an order of magnitude lower (for both mitigation costs, and climate change damage) would probably be more realistic. And note that the question isn't even
about changing the net economic growth rate over this period by as much as by 0.2% pa (realistically, 0.02% pa) but rather where to draw the balance between mitigation and adaptation so as to
minimise the total sum of these costs, which (assuming one believes the models at all) is very unlikely to be zero or less.
What no-one has yet explained to me is why I should be bothered about whether 3 generations down the line are "only" 8 times richer than me, rather than 10 (or more realistically, "only" 9.8 times
rather than 10). By all means let's hear the arguments for and against various policy decision, but don't dress it up in in the spuriously authoritative language of economic argument with the facts
carefuly concealed under claims of AGW-caused "global recession" on the side of the alarmists (including our recently departed Dear Leader Blair through the Stern report) opposed by equivalent
comments on mitigation costs on the other side. A plague o' both your houses! | {"url":"https://julesandjames.blogspot.com/2007/08/","timestamp":"2024-11-04T10:58:53Z","content_type":"application/xhtml+xml","content_length":"243890","record_id":"<urn:uuid:41fea386-3160-4bcd-ad25-14a75ca02194>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00143.warc.gz"} |
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Inequalities with
Absolute Value
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SOLVING LINEAR AND
Steepest Descent for
Solving Linear Equations
The Quadratic Equation
Linear equations in two
who invented the method of division of polynomial ?
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Author Message
nasteldavit Posted: Saturday 30th of Dec 10:09
Can someone help me with my homework problems ? They are based on who invented the method of division of polynomial ?. I have read a some sample questions on greatest
common factor and algebra formulas but that didn’t really help me in solving the questions on my assignment. I couldn’t sleep last night since I have a deadline to
meet . But the problem is no matter how much time I invest, I just don’t seem to be getting the hang of it. Every question poses a new problem , one which seems to be
tougher than conquering Mt.Everest! I need some help right away . Somebody please guide me.
kfir Posted: Sunday 31st of Dec 17:02
You can check out Algebrator. This software literally helps you solve questions in math very fast. You can plug in the questions and this program will go through it
with you step by step so you can understand better as you solve them. There are some demos available so you can also get to know how incredibly helpful the program is.
I am sure your who invented the method of division of polynomial ? can be solved faster here.
From: egypt
DoniilT Posted: Monday 01st of Jan 09:31
Algebrator truly is a masterpiece for us algebra students. As already said in the post above , not only does it solve questions but it also explains all the
intermediary steps involved in reaching that final solution . That way you don’t just get to know the final answer but also learn how to go about solving questions
from the scratch , and it helps a lot in preparing for exams .
Arjanic Ongen Aliheen Posted: Tuesday 02nd of Jan 08:26
Oh really! I’m interested this software right away. Can someone please post a link to the website where I can order this software?
From: Sunny Cal, USA
Paubaume Posted: Wednesday 03rd of Jan 12:59
You can get all the details about the software here https://rational-equations.com/distancecirclesand-quadratic-equations.html.
From: In the stars...
where you left me,
and where I will wait
for you... always...
Author Message
nasteldavit Posted: Saturday 30th of Dec 10:09
Can someone help me with my homework problems ? They are based on who invented the method of division of polynomial ?. I have read a some sample questions on greatest common
factor and algebra formulas but that didn’t really help me in solving the questions on my assignment. I couldn’t sleep last night since I have a deadline to meet . But the
problem is no matter how much time I invest, I just don’t seem to be getting the hang of it. Every question poses a new problem , one which seems to be tougher than conquering
Mt.Everest! I need some help right away . Somebody please guide me.
kfir Posted: Sunday 31st of Dec 17:02
You can check out Algebrator. This software literally helps you solve questions in math very fast. You can plug in the questions and this program will go through it with you
step by step so you can understand better as you solve them. There are some demos available so you can also get to know how incredibly helpful the program is. I am sure your who
invented the method of division of polynomial ? can be solved faster here.
From: egypt
DoniilT Posted: Monday 01st of Jan 09:31
Algebrator truly is a masterpiece for us algebra students. As already said in the post above , not only does it solve questions but it also explains all the intermediary steps
involved in reaching that final solution . That way you don’t just get to know the final answer but also learn how to go about solving questions from the scratch , and it helps
a lot in preparing for exams .
Arjanic Ongen Aliheen Posted: Tuesday 02nd of Jan 08:26
Oh really! I’m interested this software right away. Can someone please post a link to the website where I can order this software?
From: Sunny Cal, USA
Paubaume Posted: Wednesday 03rd of Jan 12:59
You can get all the details about the software here https://rational-equations.com/distancecirclesand-quadratic-equations.html.
From: In the stars...
where you left me,
and where I will wait
for you... always...
Posted: Saturday 30th of Dec 10:09
Can someone help me with my homework problems ? They are based on who invented the method of division of polynomial ?. I have read a some sample questions on greatest common factor and algebra
formulas but that didn’t really help me in solving the questions on my assignment. I couldn’t sleep last night since I have a deadline to meet . But the problem is no matter how much time I invest, I
just don’t seem to be getting the hang of it. Every question poses a new problem , one which seems to be tougher than conquering Mt.Everest! I need some help right away . Somebody please guide me.
Posted: Sunday 31st of Dec 17:02
You can check out Algebrator. This software literally helps you solve questions in math very fast. You can plug in the questions and this program will go through it with you step by step so you can
understand better as you solve them. There are some demos available so you can also get to know how incredibly helpful the program is. I am sure your who invented the method of division of polynomial
? can be solved faster here.
Posted: Monday 01st of Jan 09:31
Algebrator truly is a masterpiece for us algebra students. As already said in the post above , not only does it solve questions but it also explains all the intermediary steps involved in reaching
that final solution . That way you don’t just get to know the final answer but also learn how to go about solving questions from the scratch , and it helps a lot in preparing for exams .
Posted: Tuesday 02nd of Jan 08:26
Oh really! I’m interested this software right away. Can someone please post a link to the website where I can order this software?
Posted: Wednesday 03rd of Jan 12:59
You can get all the details about the software here https://rational-equations.com/distancecirclesand-quadratic-equations.html. | {"url":"https://rational-equations.com/in-rational-equations/y-intercept/who-invented-the-method-of.html","timestamp":"2024-11-06T04:32:33Z","content_type":"text/html","content_length":"99158","record_id":"<urn:uuid:a631e309-e0ed-4514-9b9a-00998beba616>","cc-path":"CC-MAIN-2024-46/segments/1730477027909.44/warc/CC-MAIN-20241106034659-20241106064659-00787.warc.gz"} |
Simple Tips to Learn Multiplication Table of 4 for Kids - Intelligent Advices - Go through wide range of topicsSimple Tips to Learn Multiplication Table of 4 for Kids
Simple Tips to Learn Multiplication Table of 4 for Kids
Multiplication tables are the building blocks of mathematics. It is important for children to learn tables in order to solve arithmetic problems accurately. Once kids understand the multiplication
table of 2, it becomes easier for them to learn table 4. Learning times tables will enable them to solve problems related to multiplication, division, fraction, decimals, percentages, etc. Apart from
this, it makes it easier for them to calculate problems in their day to day life. For example, helping you in counting money or calculating bills for the items that you have purchased.
Learning table 4 will make calculations easier and faster for kids. At the initial stage, kids need engaging activities to learn multiplication tables. You cannot expect them to memorize 4 times
table with just a glance. They need time to grasp the numbers that they are learning in their little minds. Therefore, conducting or incorporating fun activities can help them to learn tables.
Besides this, you can also provide printable multiplication table charts for them to learn at their own pace.
Times tables are used to solve simple and complex problems effortlessly. Kids must get acquainted with multiplication tables in order to develop mathematical skills and solve any complex equations
within a fraction of seconds. Encourage your child to learn and practice tables on a regular basis so that they retain the numbers in their memory. Check out interesting tips to learn multiplication
tables of 4 given below.
Tips To Learn Table Of 4 For Kids
Sometimes, kids may find it difficult to memorize a table of 4. Therefore, it is necessary for you to engage them with easy activities to learn tables efficiently. A few tips to learn multiplication
tables of 4 are mentioned below:
• Use real life examples to teach tables: Explore more examples and opportunities to teach multiplication tables for kids. Make sure you take easy examples so that kids can understand the questions
before answering them. For example, you can ask kids to calculate the items that you have purchased. The price of a yoghurt can is 4$. What is the total cost of 4 yoghurt cans? The answer is 4 x
4= 16. Therefore, the total cost of 4 yoghurt cans is 16$. Similarly, you can ask simple questions so that they can learn a table of 4.
• Use building blocks to teach tables: Use legos or building blocks to teach multiplication tables. Provide them with building blocks of numbers written on them. Ask them to place the correct
multiples in front of the numbers. For example, 4 x 2 = 8, kids can search 8 and place it in front of 4 x 2. Similarly, you can make them practice table 4 in this way.
Table Of 4 Chart
Here is a table of 4 charts up to 20 for kids mentioned below:
4 x 1= 4 4 x 11= 44
4 x 2= 8 4 x 12= 48
4 x 3= 12 4 x 13= 52
4 x 4= 16 4 x 14= 56
4 x 5= 20 4 x 15= 60
4 x 6= 24 4 x 16= 64
4 x 7= 28 4 x 17= 68
4 x 8= 32 4 x 18= 72
4 x 9= 36 4 x 19= 76
4 x 10= 40 4 x 20= 80
Benefits Of Learning Times Table
Learning multiplication tables helps kids to score well in their academics. These are not only useful for mathematics but also for other subjects to calculate problems. Once they are acquainted with
a 3 times table, you can make them learn table 4 with engaging activities. Some of the advantages of learning times tables are mentioned below:
• Helps children to solve mathematical problems easily.
• Enables children to acquire a strong foundation of mathematical knowledge.
• Helps in solving problems related to fraction, decimals, percentages, multiplication, division, etc.
• Develops mathematical skills among children. | {"url":"https://intelligentadvices.com/simple-tips-to-learn-multiplication-table-of-4-for-kids/","timestamp":"2024-11-07T13:36:21Z","content_type":"text/html","content_length":"146916","record_id":"<urn:uuid:68921df4-594d-4703-99af-70358043ebb6>","cc-path":"CC-MAIN-2024-46/segments/1730477027999.92/warc/CC-MAIN-20241107114930-20241107144930-00464.warc.gz"} |
Interface ForwardRuleInfGraphI
• Method Summary
Modifier and Type
Adds a new Backward rule as a rules of a forward rule process.
Add a new deduction to the deductions graph.
Deletes a new Backward rule as a rules of a forward rule process.
Search the combination of data and deductions graphs for the given triple pattern.
Return the Graph containing all the static deductions available so far.
Return the Graph containing all the static deductions available so far.
Logger a derivation record against the given triple.
Set to true to cause functor-valued literals to be dropped from rule output.
Return true if derivation logging is enabled.
Return true if tracing should be acted on - i.e. if traceOn is true and we are past the bootstrap phase.
Methods inherited from interface org.apache.jena.graph.Graph
add, add, clear, close, contains, contains, delete, delete, dependsOn, find, find, find, getCapabilities, getEventManager, getPrefixMapping, getTransactionHandler, isClosed, isEmpty,
isIsomorphicWith, remove, size, sizeLong, stream, stream
Methods inherited from interface org.apache.jena.reasoner.InfGraph
find, getDerivation, getGlobalProperty, getRawGraph, getReasoner, prepare, rebind, rebind, reset, setDerivationLogging, testGlobalProperty, validate
• Method Details
□ shouldTrace
boolean shouldTrace()
Return true if tracing should be acted on - i.e. if traceOn is true and we are past the bootstrap phase.
□ addBRule
void addBRule(Rule brule)
Adds a new Backward rule as a rules of a forward rule process. Only some infgraphs support this.
□ deleteBRule
void deleteBRule(Rule brule)
Deletes a new Backward rule as a rules of a forward rule process. Only some infgraphs support this.
□ getDeductionsGraph
Graph getDeductionsGraph
Return the Graph containing all the static deductions available so far. Triggers a prepare if the graph has not been prepared already.
Specified by:
getDeductionsGraph in interface InfGraph
the deductions graph, if relevant for this class of inference engine or null if not.
□ getCurrentDeductionsGraph
Graph getCurrentDeductionsGraph
Return the Graph containing all the static deductions available so far. Does not trigger a prepare action.
□ addDeduction
Add a new deduction to the deductions graph.
□ findDataMatches
Search the combination of data and deductions graphs for the given triple pattern. This may different from the normal find operation in the base of hybrid reasoners where we are side-stepping
the backward deduction step.
□ shouldLogDerivations
boolean shouldLogDerivations()
Return true if derivation logging is enabled.
□ logDerivation
Logger a derivation record against the given triple.
□ setFunctorFiltering
void setFunctorFiltering(boolean param)
Set to true to cause functor-valued literals to be dropped from rule output. Default is true. | {"url":"https://jena.apache.org/documentation/javadoc/jena/org.apache.jena.core/org/apache/jena/reasoner/rulesys/ForwardRuleInfGraphI.html","timestamp":"2024-11-05T19:25:33Z","content_type":"text/html","content_length":"24427","record_id":"<urn:uuid:20ee2c0c-f79e-469c-9645-8249139ccde5>","cc-path":"CC-MAIN-2024-46/segments/1730477027889.1/warc/CC-MAIN-20241105180955-20241105210955-00503.warc.gz"} |
How many grams of copper (II) chloride should be added to 1.50 Liters of water if a 2.235 M solution is desired? | HIX Tutor
How many grams of copper (II) chloride should be added to 1.50 Liters of water if a 2.235 M solution is desired?
Answer 1
$\text{Approx. 1/2 kilo of copper(II) sulfate}$.
#"Concentration"# #=# #"Moles of solute"/"Volume of solution"#.
#"Moles of solute"="Concentration"xx"Volume of solution"#
#"Mass of cupric chloride "=3.3525*cancel(mol)xx134.45 *g*cancel(mol^-1)=??*g#
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Answer 2
To determine how many grams of copper (II) chloride should be added to 1.50 liters of water to make a 2.235 M solution, you first need to calculate the number of moles of copper (II) chloride
Use the formula: moles = molarity × volume (in liters)
Given: Molarity (M) = 2.235 M Volume (in liters) = 1.50 L
Calculate moles of copper (II) chloride needed: moles = 2.235 M × 1.50 L
Once you have the moles, you can use the molar mass of copper (II) chloride to convert moles to grams.
The molar mass of copper (II) chloride (CuCl2) is: Copper (Cu): 1 atom × 63.55 g/mol = 63.55 g/mol Chlorine (Cl): 2 atoms × 35.45 g/mol = 70.90 g/mol Total molar mass = 63.55 g/mol + 70.90 g/mol =
134.45 g/mol
Now, multiply the number of moles by the molar mass: grams = moles × molar mass
Calculate the grams of copper (II) chloride needed.
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Answer from HIX Tutor
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
Not the question you need?
HIX Tutor
Solve ANY homework problem with a smart AI
• 98% accuracy study help
• Covers math, physics, chemistry, biology, and more
• Step-by-step, in-depth guides
• Readily available 24/7 | {"url":"https://tutor.hix.ai/question/how-many-grams-of-copper-ii-chloride-should-be-added-to-1-50-liters-of-water-if--8f9af8504f","timestamp":"2024-11-05T21:57:28Z","content_type":"text/html","content_length":"571523","record_id":"<urn:uuid:6a527a9e-9394-4910-83bd-bb0d21046333>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00180.warc.gz"} |
Thermodynamics and Statistical Mechanics of Small Systems
by A. Puglisi, A. Sarracino, A. Vulpiani (eds)
Publisher: MDPI AG 2018
ISBN-13: 9783038970576
Number of pages: 336
Applications of the thermodynamic and statistical mechanics of small systems range from molecular biology to micro-mechanics, including models of nano-transport, Brownian motors, and (living or
artificial) self-propelled organisms.
Download or read it online for free here:
Download link
(14MB, PDF)
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It had to be done...
If you take crystal radios seriously enough and build them often enough then eventually you get around to building a performance DX set. This page introduces what I hope is my entry into the club.
This is basically a significant upgrade to my Teflon Set which I used as a starting point. The base boards are all that remains from that set, the coils and capacitors have been swapped for better
components. I did retain the FO-215 diode but may eventually look into paralleling three or four HP5082-2835 schottkys, who knows.. I prefer the simple double-tuned design without additional circuit
decorations such as a hobbydyne/selectivity enhancements, taps etc. The set audio transformation does include a Benny but the resistor is variable and can be adjusted to very low resistance
essentially eliminating this as well. What is left is "essential radio", Antenna tuner, Tank tuner, and Audio transformer. I include a QRM wave trap but that is an optional extra and has no insertion
loss to the circuit itself. What results is a set with quality performance although not most certainly not the very best possible. I have used the best components but the best capacitors are scarse
and of course my construction technique is OK at best. Still, it works for me. The set takes up most of a tabletop to run and looks quite impressive when one walks into the "shack".
The circuit diagram at left shows the layout from antenna/ground through to the audio transformer and headphones. In the following discussions I will describe the components mainly by category
starting with the antenna/ground as a unit, then a look at the coils, capacitors, diode and audio transformer. Next I focus on each of the three tuned circuits and model their performance from the
coil/capacitor characteristics. I compare the modeled Q against measured Q on each board. Finally I test the set in full and look at the sensitivity, Q, and resonance. The final conclusion is that
the set performs OK but not as well as hoped. The variable capacitors, while the best I can currently find, are only mediocre in quality and impact the final outcome. Other design considerations may
also play a part leaving much room for improvement. Overall this project has been a learning exercise and I hope the reader may find some useful ideas and tips.
ANTENNA n GROUND:
The signal arriving at the antenna consists of an AC voltage source Va, in series with some radiation resistance Rr, antenna capacitance Ca, antenna inductance La, antenna resistance Rr, and finally
a ground resistance for the return path to complete the circuit. Coupled to the antenna, the ATU consists of a coupling capacitance C2, and a tank with inductance L1 and capacitance C1. The
capacitance and inductance of the ATU is used to tune out the reactance of the various components of the antenna for which we do not have data. What to do?
My personal antenna consists of about 75' of 14 awg wire averaging about 25-30' high and with another 20' hanging under the gutters and finally a 25" vertical drop to my "shack". For such an antenna
Kuhn models a 30m antenna 3m high which approximates my situation kinda sorta, my antenna is 30+ meters long and 9 meters high. Wire resistance is negligible as is the radiation resistance which he
shows to vary between some 0.1 to 1.5 ohms. Such an antenna is capacitive by nature and will have about 220 - 375pF along with some small 20uH inductance, not too far off a standard "dummy" antenna.
This pretty much leaves earth resistance Rg and the main unknown parameter.
Aris fine sandy loam and the following table gives the resistivity (converted) to ohm-meters at left:
For my purposes I can assume a generic 15 ohm earth for my setup, an excellent situation indeed. To cover myself, I have driven four 3-ft rods into the ground at about 1 meter spacing.
The ATU for my radio has a 180uH coil in the tank. From my discussions in my Laboratory section, "Antenna Tuning Unit Notes" page, I have modeled the needed capacitance versus frequency for a
Tuggle-type ATU. For 1.1 MHz I get the following parameters for my antenna/ ground / ATU.
15 ohm / 180 uH Low resistivity Earth Case sensitivities
Ra Ca La L atu Qu f C cpl C tank Za Impedance LC Rp
Ohm pF uH uH kHz pF pF Ohm kOhm
15 300 20 180 585 1100 49 73 15 -j 344 728
The following charts show the modeled impedance Za / parallel resistance Rp, and the tuning capacitance tracking for the tuggle tuner, capacitances track rather very well indeed.
Coils are basket-wound with Litz 660/46 rope and all have high Q's in the 1500 more or less range at 1MHz, (Q vs F charts below). To measure Q I have utilized a "ring-down" technique as outlined in
"Coil Q" in my radio Laboratory area. The data is a bit messy, but generally shows overall very high Q over the broadcast frequency spectrum.
Coil Specifications on the set as follows:
L1 34 turns Litz 660/46 over 1 under 1, 5.2" dia av, 1.8" length, 181 uH and 819 Q
L2 40 turns Litz 660/46 over 1 under 1, 5.2" dia av, 2.2" length, 230 uH and 1137 Q
L3 46 turns Litz 660/46 over 1 under 1, 5.2" dia av, 2.5" length, 270 uH and 1066 Q
Inductance measurements at RF, Q for the case of F = 1 MHz
Capacitors are quite another matter. Caps are generally bought not made, and one is at the mercy of availability and quality for what is on offer. I have been singulary unable to obtain "holy grail"
caps with silver plated vanes, ceramic insulation, and effective brushes all in one package. Caps with ceramic insulation are available but I have found that this is no guarentee for quality. I have
tested ceramic insulated, aluminum-bladed capacitors with Q's (@ 1MHz) ranging from 5441 down to 487 (worse than the cheaps phenolic caps one sees on low-end sets). The caps in this set are small NOS
Russian 2-gang jobs that I found on eBay. The one on the tuning tank is very short, only 1.5 inches (under 4cm) and tested Moderate-Q, (Q @ 1 MHz = 1861). The two caps on the ATU and QRM are the same
style but slightly longer at 2.2 inches (5.8cm). They have trims with each gang and overall they are medium Q, (Q's @ 1MHz around 1100 to 1200). They are clearly superior than the original caps and
they are all matched, but one day I may find better.
Note: Ben Tongue's holy grail cap (BT A250) clocks in at a whopping Q = 13026 @ 1 MHz while a lowly phenolic capacitor such as the ones sold by the Crystal Set Society have a measured Q = 732 at the
same 1 Mhz (BT A250 and 420 AnP XSS on the charts). The ceramic insulation on my caps do not offer significant Q improvement.
Capacitor Specifications on the set as follows:
C1a (403) 14 to 388 pF and 1245 Q
C1b 15 to 325 pF and 1149 Q
C2 (408) 13 to 383 pF and 1861 Q
C3 (390) 14 to 375 pF and 1102 Q
Capacitance measurements at RF, Q for the case of F = 1 MHz
Note that Cap Q versus frequency is dependant upon the inductor used while Cap Q versus C should be "native" to the capacitor independant of inductor. Additionally, in the capacitor quality charts
below I also include Ben Tongues' holy grail cap for comparison. Cap Q remains a serious quality issue for my set.
I have chosen to use a Germanium FO-215 "holy grail" diode to rectify this set. I also have the option to parallel three to four HP 5082-2835's but have not gone down this road.. yet. Diode junction
resistance (Ro) measurements on the FO-215 give 144 and 165 kohm which is well matched for a Litz-coil tank. The Schockly equation used to evaluate diodes used the parameters n and Is to determine a
diodes' characteristic where "n" is the Ideality Factor and Is is the Reverse Saturation Current. What is wanted in crystal raio design is a determination of the diode junction (or zero-bias)
resistance. This is the resistance needed to match the diode to the tank and the parameter Is is needed to find this as follows: Ro is calculated via the equation Ro = VT * n / Is where VT is the
thermal voltage = k * T / q = 0.0257V at room temperature and Is and n are from the measured diodes. k is Boltzmann's constant = 1.38E-23 J/K and q is the electron charge = 1.609E-19 coulombs.
Is and n themselves are determined by making I:V characteristic measurements at small current levels. I have used a spreadsheet provided by Mike Tuggle that takes two pared observations Id1 Vd1 and
Id2 Vd2 to calculate the slope n of the characteristic curve and calculate Is.
Id = Is * {exp [(qe / (n*k*T)) * (V - I*Rs)] - 1} .. (Shockley diode equation) .. where:
Is = reverse saturation current (measured)
k = boltzmann = 1.38E-23 J/K
T = temp K = 300
qe = electron charge = 1.609E-19 cmb
n = ideality factor (measured slope of the characteristic)
Rs = series resistance in the circuit
The measurements are made at very small currents (0.5 to 1.0 mA) where the effects of series resistance Rs are negligible. The Shockley equation thus simplifies to:
Id = Is*(exp(Vd/(0.0256789*n))-1) and
Diode Junction Resistance Ro = VT * n / Is
T = 300K
VT = k*T/qe = 0.0256789
SO: Ro = k * T / qe * n / Is
The following chart gives the results of my measurements on two different FO-215 diodes.
A great deal more concerning diode characterization may be found in my "Laboratory" section.
I have described in detail the audio transformer in another page. Here it is worth to say a few words about it as I intend this to be a principal part of my Litz set. The transformer is based on the
KPB-02 multi-tapped transformer made in China and available on eBay.
The circuit diagram below shows the full range of transformations between tank/diode Z 10k to 200k ohm and phone Z from 8 ohm up to 100k ohm. I have not yet measured the optimal load Z for the set
but my expectation is that 200k ohms will be a minimum. While 200k ohms is the maximum Z handeled by the transformer, that range may be extended to nearly 900k ohms at the -3dB level. One cannot hear
(in general) a difference in sound level until a change of at least 3 dB takes place.
Due to pin limitations, I divide the output into three groups: 1) 10k to 100k ohms for traditional magnetic and piezo element phones, 2) 300 to 1500 ohms for sound powered phones, and 3) 8 to 32 ohms
for modern magnetic phones. This provides a variety of phone options for use with any radio connected to the transformer.
As part of my characterization for this set, I evaluated the Q for each main module, ATU, Main Tank, and QRM separately. I include both theoretical calculations based on the individual cap and coil Q
measurements as well as actual measurements of the unit Q itself. On the left hand chart for each module I plot the measured Q vs frequency for the module capacitor (green) and coil (blue). From
those two measurements I calculated a predicted tank Q (orange) and finally I also plot the actual measured tank Q (brown square). Each plotted curve has its algebraic function and precision (R2) as
well. High precision clearly does not necessarily mean high accuracy as the measured tank Q is always lower than the calculated Q. Other tank losses in addition to the coil and capacitor exist that
are not taken into account in my calculations.
Finally, from the calculated Rp (Rp = 2p * f * L * Qt) at 1.1 MHz from the measured Q data, I can also simulate the resonance curve expected for each unit. In the following charts, I show the
component Q with modeled unit Q along with the measured unit Q on the left hand chart, and the simulated resonance curve from the measured Q at 1.1 MHz on the right hand chart.
ANTENNA TUNING UNIT, L1C1a:
This measurement is made on the tank L1C1a only, C1b does not play a role. For the ATU one sees that the calculated Q versus frequency is optimistic, Q at 1.1 MHz is 377 instead of a calculated 510.
The calculations assume that all losses are contained in the cap and coil only. This appears not to be the case. Q = 377 is getting pretty low when one considers that the Q will be halved again when
coupled to the tuning tank.
MAIN TUNING UNIT, L2C2:
The main tuning tank measurements give me most cause for concern. The tank Q measurements are significantly below the calculations (by half) based on the component Q values. I have swapped out the
original cap which was starting to short frequently with another cap of somewhat lower Q. In both full tank measurements I find the calculated Q to be significantly higher than measured. Note that I
am measuring the Tank Q with diode attached but otherwise no load. There may be additional losses from the close proximity of the hookup wires to the tank coil, a generally messy configuration
necessitated by my setup. Measuring the tank Q in this manner is unusual, most likely for obvious reasons. Still, this is a test between theory and measurement and for the tank, the measured Q
appears overly low when compared to the predicted value. If the measurements are suspect then maybe things are brighter than this. Overall it does indicate that my caps, once again, are really
letting me down on what I hoped to be an excellent performance set. Calculated Q for 1.1 MHz is 645, actual measured tank Q is only 318... bummer.
Coupling the Tuning Tank (Q = 318) with the ATU (Q = 377) suggests a coupled Q approaching 174 (average of 318 and 377 divided by 2). Adding the diode and audio circuit to the mix will further lower
things. I need better caps!
WAVE TRAP, L3C3:
This module has the simplest circuit of the three with no "dangling" diode or capacitor to trouble the results. The wave trap, like the antenna tuner, measures out with Q pretty close and slightly
below the calculated Q from the cap and coil. Calculated Q for 1.1 MHz is 492, actual measured tank Q equals 412.
Working to sort things out. The chart below summarizes the test results for the radio at three different coil coupling spacings, 14 inches, 8 inches, and over coupled at just 2 inches. I am having
some issues with the signal generator, usure it is giving me the correct voltage output. I have it set to give a 200 mVpp sine but when I measure the output on my scope I am getting something more
like 4000 to 4800 mVpp. This is impacting my measurements and calculations and I remain uncertain of the efficiency calculations.
The set loaded Q factors are also a dissappointment, I was expecting Q's approaching 175 or so, presumably at critical coupling, about 8 inches in this case. What I measured, over and over, was
something nearer to 63. Under-coupling the set with a 14-inch separation between coils gives a more satisfactory Q = 147. Thats what I'm talking about!
Tabulated parameters as follows:
Pin = Power into the set in microwatts
Pout = Power out the back end (into the phones) also in microwatts
Eff = Set efficiency or Sensitivity and is the ratio on Pout to Pin * 100
BW = -3dB Bandwidth of the set at 1100khz (f res)
QL = Loaded Q of the set = f res / BW
Rx = Input resistance of the set in ohms
Rl = Load resistance on the tank in kohms
Rd = Junction resistance of the diode in the set
M = Mutual inductance between the coils
k = Coupling coefficient between the coils
Resonance curves for the tank.
Measurements were all made at the center of the broadcast band at 1100kHz. I carefully peaked the set before the 2, 8 and 14 inch cases and then also measured the Vdc output over a range of load
resistances. With this data I am able to calculate the power output in uW and determine the optimal load (Rl) for the set. A chart of the power output versus Rl is given below. For maximum power
transfer, the ideal load resistance increases with increasing coil separations. Actual power to the audio circuit shows only a slight diminishement between a 2 and an 8 inch coil separation, but
begins to drop dramatically as you approach 14 inch separation. Separate determinations were made for each spacing. All the subsequent testing for each case was made at the optimum load for maximum
power transfer.
The measured resonance curves are overlaid on the chart below. For the 2 inch spacing case I determined Rl and Q for the lower resonance peak near 1050 kHz. I remain bothered by the low Q for the 8
inch case, the 14 inch case sacrifices sensitivity to achieve its tight bandwidth characteristics. Curiously, the 1050 kHz peak on the 2 inch case has a higher Q than the 8 inch case.
14 inches between coils is under-coupled with high Q but sacrificing sensitivity, 8 inches is presumably near critical coupling and 2 inch spacing is woefully over-coupled giving two separate
resonance peaks. The mutual inductance (M) and coupling coefficients (k) are determined via resonance modeling against the measured data.
It is also possible to take the resonance curves generated above in the testing section and compare them against theoretical curves. This yields an estimate of the mutal inductance and coupling
coefficient for the various tank separations tested. Additional parameters that can be looked at and evaluated include the ATU and Tuning Tank Q's and the Q of the coupled circuit. In my "Radio
Laboratory" page I develop an Excel spreadsheet (with help from Mike Tuggle) to model coupled resonance. Using Excel's Solver routine I seek the parameters that give the closest fit to the actual
measurements. The charts below give a first look at the results. Much more to do... I am still digesting this as I go to press, stay tuned!
This radio represents my best effort to produce a performance set with high Q and good sensitivity. Care was taken along the way to test components and use good construction and design practices. The
results have been disappointing with the set loaded Q remaining stubbornly below 100 for coil separation at 8 inches. The principal factors for this poor performance have been noted with my inability
to secure high-Q capacitors being the main shortcoming. A crystal set is composed of several components and the final Q can never be above the Q of the poorest component.
This weak-link situation is well illustrated above in my "Unit Testing" section: For each assembled unit, ATU, Tank, and QRM, the "Q vs f" charts all show that the theoretical unit Q will always be
less than either of the individual components. When assembled together, the presence of additional tank losses means the measured Q vs frequency is systematically lower than the theoretical Q. This
was most seriously apparent for the main tuning tank where, at 1.1 MHz, I measured Q = 318 rather than the modeled 645. I am unsure how or if the presence of the diode impacted this measurement (but
a look at the circuit diagram at the top of the page suggests this should not be an issue). Both the ATU and the QRM consist of a simple tank circuit with no other components present.
Improvements for each unit can possibly be made in the following areas: 1) the coils are tied off with thick cotton string which can absorb moisture and lower Q. 2) The coils are mounted on the base
with wood dowels which may also absorb moisture. 3) The coil mounts raise the coils only about a half-inch off the base and the capacitors are also mounted within 1 to 2 inches of the coils. This
places the coils, and their magnetic fields, in close proximity to the wood base and other components.
All the above factors will have their impact on the final Q of the set. Still, without superior capacitors I am loathe to start reconstruction. Taller coil standoffs can easily be obtained and this I
will consider, but more serious modifications must wait until I can secure appropriate capacitors. I have certainly learned a lot form this project and so am happy with the results. Cheers to all who
may have read this far!
Portraits of set:
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Uncertainty Analysis
Uncertainty Analysis#
This section demonstrates the use of the Monte Carlo sampling (MCS) method for forward propagation of uncertainty. In uncertainty analysis, it is of interest to quantify how uncertain design
variables or parameters of a problem affect the quantity of interest (QoI) for a given system. This is done by sampling the distribution of the uncertain variables or parameters using a sampling
method and calculating the QoI for a large number of samples. The required statistics (mean, standard deviation, reliability) can then be calculated from the QoI values that are generated using MCS.
In the context of optimization, the statistics of interest are typically the mean and standard deviation of the objective function and the reliability of the constraint.
To illustrate the ideas of uncertainty analysis, consider a problem with the following objective and constraint
\[Y = f(\textbf{x}) = x_1^2 + 2x_2^2 + 3x_3^2\]
\[g(\textbf{x}) = x_1 + x_2 + x_3 - 3.5 \leq 0\]
In this problem, \(x_1\) is deterministic and other two variables are normally distributed with standard deviations: \(\sigma_2 = 0.06\) and \(\sigma_3 = 0.2\). For the following analysis, the design
variable values used are \(x = [1,1,1]^T\). This means that \(x_1 = 1\), \(x_2 \sim \mathcal{N}(1,0.06)\), and \(x_3 \sim \mathcal{N}(1, 0.2)\).
The first block of code below imports the required packages to perform the uncertainty analysis.
import numpy as np
import seaborn as sns
import matplotlib.pyplot as plt
from scipy.stats import uniform, norm
from time import time
from smt.sampling_methods import LHS
from scipy.stats.qmc import Halton, scale
Calculation of statistics (mean and standard deviation) of objective function#
This subsection will discuss the calculation of the statistics of the objective function based on the uncertain design variables. At the start, the general idea of the MCS method will be demonstrated
where a large number of samples will be drawn from the distributions of the random variables using random sampling and Halton sequences. The objective function statistics and distribution will be
calculated based on the samples that are generated the value of the objective function for those samples. However, when using the MCS method it is also important to determine the minimum number of
samples that can accurately obtain the statistics of a model to reduce computational cost. This is done by varying the number of samples from small to large values and observing the change in the
mean and standard deviation. The minimum number of samples for which there is no significant change in the estimated mean and standard deviation from the previous number of samples used can be used
to obtain the statistics of the objective function. Typically, one would start the analysis by performing the convergence of the MCS method to find the minimum number of samples.
The next block of code defines the objective function and constraint of the problem.
def function(x1, x2, x3):
Function for computing objective values
value = x1**2 + 2*x2**2 + 3*x3**2
return value
def constraint(x1, x2, x3):
Function for computing constraint values
value = x1 + x2 + x3 - 3.5
return value
MCS method for a large number of samples#
The first demonstration of MCS is with random sampling. The block of code below defines random variables for \(x_2\) and \(x_3\), and performs MCS with 1,000,000 random samples for computing mean,
standard deviation, and the output distribution.
# Defining random variables
rv_x2 = norm(loc=1, scale=0.06)
rv_x3 = norm(loc=1, scale=0.2)
num_samples = 1000000
x1 = 1
x2 = rv_x2.rvs(size=num_samples)
x3 = rv_x3.rvs(size=num_samples)
f_random_ = function(x1, x2, x3)
sns.kdeplot(x=f_random_, fill=True)
plt.ylabel("Probability density")
plt.title("Number of samples: {}".format(num_samples))
print("Estimated mean (true): {}".format(np.mean(f_random_)))
print("Estimated standard deviation (true): {}".format(np.std(f_random_)))
Estimated mean (true): 6.12930134987835
Estimated standard deviation (true): 1.236068224197124
From the above MCS with random sampling, the \(\mu\) is 6.13 and \(\sigma\) is 1.23. This distribution is slightly asymmetric and has a longer right tail.
In the second demonstration of MCS, the samples from the distribution will be drawn according to Halton sequences. In the context of sampling distributions, Halton sequences generated on a unit
hypercube (values are generated between the bounds of 0 and 1) represent probabilities of the occurrence of a realization of a random design variable or parameter. These probabilities can then be
transformed into the realizations of the random design variable or parameter using the inverse CDF function of the distribution of the random design variable or parameter. In this way, samples can be
drawn from the distributions of the uncertain variables and parameters and MCS can be performed.
The block of code below demonstrates this approach of using Halton sequences to perform MCS. Like the previous example of random sampling, the mean, standard deviation and output distribution are
computed using 1,000,000 samples.
# Defining the Halton sampler
halton_sampler = Halton(d=2, scramble=False)
# scramble is set to false to avoid Owen scrambling which is used to create non-deterministic Halton sequences
num_samples = 1000000
# Generating halton sequence samples and transforming them according to the distributions of the random variables
x1 = 1
x_halton = halton_sampler.random(n = num_samples)
x_halton = x_halton[1:,:] # Dropping the first element [0,0] of the Halton sequence since inverse CDF of 0 will be inf
x2_halton = rv_x2.ppf(x_halton[:,0])
x3_halton = rv_x3.ppf(x_halton[:,1])
f_halton = function(x1, x2_halton, x3_halton)
sns.kdeplot(x=f_halton, fill=True)
plt.ylabel("Probability density")
plt.title("Number of samples: {}".format(num_samples))
print("Estimated mean (Halton): {}".format(np.mean(f_halton)))
print("Estimated standard deviation (Halton): {}".format(np.std(f_halton)))
Estimated mean (Halton): 6.12716999446928
Estimated standard deviation (Halton): 1.235459444685999
The \(\mu\) and \(\sigma\) obtained using Halton sequences closely matches the values obtained using random sampling. The output distribution is also a good match for the one obtained using random
Convergence of the MCS method#
Now, MCS will be performed using random sampling, Latin Hypercube sampling (LHS) and Halton sequences using a varied number of samples drawn from the distribution. The previous analysis used a fixed
number of samples to perform the MCS but in the following analysis, the number of samples used for the MCS will be varied and the statistics will be calculated at every number of samples. In this
case, LHS and Halton sequences are used to generate samples between the bounds of 0 and 1. As mentioned previously, these values are treated as probabilities and the inverse CDF of the random
variables is used to draw samples from the distribution based on the values of these probabilities.
# Defining different samples sizes for MCS
samples = np.logspace(1, 4, 20, dtype=int)
# Defining the LHS sampler
lhs_sampler = LHS(xlimits=np.array([[0.0,1.0], [0.0,1.0]]), criterion="ese")
# Storing statistics
mean_lhs = []
sigma_lhs = []
mean_halton = []
sigma_halton = []
mean_random = []
sigma_random = []
# Storing the function values
F_lhs = []
F_random = []
F_halton = []
for sample in samples:
# LHS sampling
x_lhs = lhs_sampler(sample)
x2_lhs = rv_x2.ppf(x_lhs[:,0])
x3_lhs = rv_x3.ppf(x_lhs[:,1])
x1 = 1
f_lhs = function(x1, x2_lhs, x3_lhs)
# Halton sampling
x_halton = halton_sampler.random(n = sample)
if sample == samples[0]:
x_halton = x_halton[1:,:] # Dropping the first element [0,0] of the Halton sequence since inverse CDF of 0 will be inf
x2_halton = rv_x2.ppf(x_halton[:,0])
x3_halton = rv_x3.ppf(x_halton[:,1])
f_halton = function(x1, x2_halton, x3_halton)
# Random sampling
x2 = rv_x2.rvs(size=sample)
x3 = rv_x3.rvs(size=sample)
f_random = function(x1, x2, x3)
The next block of code plots the convergence history of the MCS with different number of samples. The convergence history plots the convergence of the mean and standard deviation versus the number of
samples used in the MCS. The relative change in the mean and standard deviation versus the number of samples is also plotted. A tolerance for the relative change in the statistics can be used to
determine when the MCS has converged. This can be used to determine the number of MCS samples required to accurately determine the statistics of the objective function.
fig, ax = plt.subplots()
ax.plot(samples, mean_lhs, marker=".", label="LHS")
ax.plot(samples, mean_random, marker=".", label="Random")
ax.plot(samples, mean_halton, marker=".", label="Halton")
ax.set_xlabel("Number of samples")
fig, ax = plt.subplots()
ax.plot(samples, sigma_lhs, marker=".", label="LHS")
ax.plot(samples, sigma_random, marker=".", label="Random")
ax.plot(samples, sigma_halton, marker=".", label="Halton")
ax.set_xlabel("Number of samples")
conv_mean_lhs = np.zeros(len(samples)-1)
conv_mean_random = np.zeros(len(samples)-1)
conv_mean_halton = np.zeros(len(samples)-1)
conv_sigma_lhs = np.zeros(len(samples)-1)
conv_sigma_random = np.zeros(len(samples)-1)
conv_sigma_halton = np.zeros(len(samples)-1)
for i in range(len(samples)-1):
conv_mean_lhs[i] = np.abs(mean_lhs[i+1] - mean_lhs[i]) / np.abs(mean_lhs[0])
conv_mean_random[i] = np.abs(mean_random[i+1] - mean_random[i]) / np.abs(mean_random[0])
conv_mean_halton[i] = np.abs(mean_halton[i+1] - mean_halton[i]) / np.abs(mean_halton[0])
conv_sigma_lhs[i] = np.abs(sigma_lhs[i+1] - sigma_lhs[i]) / np.abs(sigma_lhs[0])
conv_sigma_random[i] = np.abs(sigma_random[i+1] - sigma_random[i]) / np.abs(sigma_random[0])
conv_sigma_halton[i] = np.abs(sigma_halton[i+1] - sigma_halton[i]) / np.abs(sigma_halton[0])
fig, ax = plt.subplots()
ax.plot(samples[1:], conv_mean_lhs, label="LHS", marker=".")
ax.plot(samples[1:], conv_mean_random, label="Random", marker=".")
ax.plot(samples[1:], conv_mean_halton, label="Halton", marker=".")
ax.set_ylabel(r"$ |\mu^{(i)} - \mu^{(i-1)}| / |\mu^{(0)}|$")
ax.set_xlabel("Number of samples")
fig, ax = plt.subplots()
ax.plot(samples[1:], conv_sigma_lhs, label="LHS", marker=".")
ax.plot(samples[1:], conv_sigma_random, label="Random", marker=".")
ax.plot(samples[1:], conv_sigma_halton, label="Halton", marker=".")
ax.set_ylabel(r"$ |\sigma^{(i)} - \sigma^{(i-1)}| / |\sigma^{(0)}|$")
ax.set_xlabel("Number of samples")
Notice that as the number of samples increase, the \(\mu\) and \(\sigma\) also converge to the value obtained from MCS in the previous block of code. The above convergence plots show the benefit of
using LHS and Halton sequences to sample the distribution as compared to using random sampling. MCS performed using LHS and Halton sequences tends to converge faster to the known values of \(\mu\)
and \(\sigma\). This can be deduced from the quick reduction in the relative change of the mean and standard deviation that is brought by using the LHS and Halton sequence sampling method. It is also
deduced from the variation of mean and standard deviation with the number of samples as the values converge to those obtained from performing MCS using a large number of samples. This means a lower
number of samples must be drawn from the distribution and evaluated to calculate the statistics. This reduces the computational cost of performing the uncertainty analysis.
A similar convergence history can also be plotted for the output distribution of the objective function. This is done in the next code block. This convergence history shows that the output
distribution closely matches the true distribution as the number of samples increases. Here, the distribution obtained using random sampling and \(10^6\) MCS samples in an earlier code block is
treated as the true distribution.
for i in range(0, len(samples), 2):
fig, ax = plt.subplots()
sns.kdeplot(x=f_random_, fill=True, color = "red",label = 'True')
sns.kdeplot(x=F_lhs[i], fill=True, label = 'LHS')
sns.kdeplot(x=F_random[i], fill=True, label = 'Random')
sns.kdeplot(x=F_halton[i], fill=True, label = 'Halton')
plt.ylabel("Probability density")
plt.title("Number of samples: {}".format(samples[i]))
Probability of feasibility of a constraint#
When working with uncertainty in design variables or parameters for an optimization problem, the probability of feasibility of a constraint is of interest to a designer. The probability of
feasibility of the constraint used as an example in this section can be calculated using MCS. The distributions of the uncertain variables can be sampled using the methods shown previously and for
each sample, the value of the constraint can be calculated. Once all the samples are evaluated, the required probability can be calculated by simply dividing the number of samples that satisfy the
constraint by the total number of samples.
\[\text{Pr}(g(\textbf{x}) \leq 0) \approx \frac{\text{number of feasible samples}}{\text{total number of samples}}\]
Similar to the previous section, the next block of code uses random sampling and a large number of MCS samples to calculate the true probability of feasibility of the constraint.
num_samples = 1000000
x1 = 1
x2 = rv_x2.rvs(size=num_samples)
x3 = rv_x3.rvs(size=num_samples)
g_random_ = constraint(x1, x2, x3)
print("Estimated probability of feasibility (true): {}".format(len(g_random_[g_random_<=0])/num_samples))
Estimated probability of feasibility (true): 0.991639
The next block of code uses the LHS and Halton sequence sampling methods along with random sampling to calculate the reliability of the constraint. The number of samples used for MCS is varied and
the variation of the reliability with number of samples is plotted to visualize the convergence.
# Defining different samples sizes for MCS
samples = np.logspace(1, 4, 20, dtype=int)
# Storing reliabilities
p_lhs = []
p_halton = []
p_random = []
for sample in samples:
# LHS sampling
x_lhs = lhs_sampler(sample)
x2_lhs = rv_x2.ppf(x_lhs[:,0])
x3_lhs = rv_x3.ppf(x_lhs[:,1])
x1 = 1
g_lhs = constraint(x1, x2_lhs, x3_lhs)
# Halton sampling
x_halton = halton_sampler.random(n = sample)
if sample == samples[0]:
x_halton = x_halton[1:,:] # Dropping the first element [0,0] of the Halton sequence since inverse CDF of 0 will be inf
x2_halton = rv_x2.ppf(x_halton[:,0])
x3_halton = rv_x3.ppf(x_halton[:,1])
g_halton = constraint(x1, x2_halton, x3_halton)
# Random sampling
x2 = rv_x2.rvs(size=sample)
x3 = rv_x3.rvs(size=sample)
g_random = constraint(x1, x2, x3)
fig, ax = plt.subplots()
ax.plot(samples, p_lhs, marker=".", label="LHS")
ax.plot(samples, p_random, marker=".", label="Random")
ax.plot(samples, p_halton, marker=".", label="Halton")
ax.set_ylabel("Probability of Feasibility")
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Default - Job Saarnee
LONG ANSWER IMPORTANT QUESTION OF Designing and Analysis of Algorithm form UNIT 1 Q1: What do you mean by algorithm? Write the characteristic of an algorithm? (ALTU 2013-14) Q2: What do you
understand by asymptotic notation? Describe the type of asymptotic notation in detail? (AKTU 2013-14) or Discuss asymptotic notations in brief ? (AKTU 2014-15) Q3: …
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2 Marks IMPORTANT QUESTION OF Designing and Analysis of Algorithm form UNIT 1
2 Marks IMPORTANT QUESTION OF Designing and Analysis of Algorithm Q1: Solve the given recurrence T(n) = 4 T(n/4) + n? AKTU 2015-16 Q2: Justify why Quick Sort is better than merge sort? AKTU 2015-16
Q3: What is priority Queue? AKTU 2015-16 …
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The PSF of a Pinhole Camera
After introducing the Airy pattern in The Perfect Camera, I will show in this article how the PSF of a pinhole camera looks. A camera with a classical lens focuses the image that would be at infinity
on the detector, which means that you actually get (approximately) the Airy pattern as the image of a point source there. This is not true for a pinhole camera, so the Airy pattern is not a good
approximation of the PSF in that case.
The figure below shows the geometry that is used in the formulas that follow. The pinhole (or aperture) is located in the plane \(z=0\). The equation in the next section then computes the electrical
field \(E\) at a point \((x,y,z)\). In practice, this equation is used to compute the PSF in a plane \(z=d\), where \(d\) is the distance to the (planar) detector.
As was already mentioned in The Perfect Camera, it is necessary to use (at least) Fresnel diffraction to compute the PSF of a pinhole camera. In this article, I’ll do better and use the full
Rayleigh–Sommerfeld diffraction integral.
The Rayleigh–Sommerfeld Diffraction Integral
A general solution of the wave equations, which we need to solve to compute the PSF, is the Rayleigh–Sommerfeld diffraction integral. It is given by
\[E(x,y,z)=\frac{kz}{2\pi i}\iint_{-\infty}^{+\infty}\!E(x_0,y_0,0)\frac{e^{ikr}}{r^2}(1-\frac{1}{ikr})\,dx_0\,dy_0,\]
where \(k=2\pi/\lambda\) with \(\lambda\) the frequency of the wave, and \(r=\sqrt{(x-x_0)^2+(y-y_0)^2+z^2}\). \(E(x,y,z)\) is then the electrical field at the point \((x,y,z)\). This integral can be
interpreted as a convolution. If we define
\[h(x,y,z)=\frac{kz}{2\pi i}\frac{e^{ik\rho}}{\rho^2}(1-\frac{1}{ik\rho}),\]
where I’ve set \(\rho=\sqrt{x^2+y^2+z^2}\) for convenience, we get that
where the integral is simply the definition of convolution. The result is exactly the Rayleigh–Sommerfeld diffraction integral again. However, this time a basic impulse response \(h(x,y,z)\) can be
combined with an arbitrarily shaped pinhole through convolution, which can be done efficiently using the Fast Fourier Transform (FFT) algorithm. In the current article, I only use a circular pinhole,
but I have another article with more crazily shaped pinholes.
Let’s see how this works out in practice.
In Practice
When the Rayleigh–Sommerfeld integral is used to compute the PSF of a circular pinhole, the result is as shown in the following figure.
On the left, the PSF at a large distance from the aperture (490 mm, for a pinhole with a diameter of 0.3 mm and a wavelength of 550 nm) is shown. This PSF is very close to the Airy disk that was
computed in The Perfect Camera, so it turns out that 490 mm is far enough to approximate the theoretical result. But, as I’ve mentioned in the introduction, the PSF is expected to look different at
the actual distance of the detector for a (DSLR) pinhole camera, since that is much smaller. For my Nikon, that distance is 49 mm. The image on the right shows the PSF at that distance, and it indeed
looks significantly different from the image on the left. It is clear that this difference cannot be ignored in practice.
In a follow-up article, I will use this PSF to compute some simulated pinhole photos. PSFs like this can be useful to determine a suitable pinhole size for a given camera geometry.
Sources: The following article gives a nice overview of several near-field diffraction models: G. D. Gillena and S. Guha, “Modeling and propagation of near-field diffraction patterns: A more complete
approach”, Am. J. Phys., vol. 72, no. 9, pp. 1195–1201, 2004, doi: 10.1119/1.1767102. The geometry image is from Wikimedia Commons
Excellent article. I am trying to model diffraction effects in matlab using the Rayleigh-Sommerfeld impulse response. Specifically, I am trying to use the FFT to create filters as you suggest. I keep
getting results which are not consistent with the Airy disk, and I was wondering if the circular convolution property of Fourier Filters was the problem.
Have you had any success using the FFT to get the same result?
Both PSFs in the above figure were computed using FFT convolution. The circular convolution should not be a problem, if you don’t use the border of the result. To create these images, I used a very
large (in pixels) impulse image (h(x,y,z)), and a much smaller pinhole image (a disk in this case), so that I could crop the convolved image and still have a large enough result. You also have to be
careful that you don’t get Moiré patterns in your impulse image. Have you checked the real and imaginary part of that separately?
That would make sense. My impulse image, h(x,y,z), was the same size as the image to be convolved. How do you suggest to check for Moire? Visual inspection?
My result is similar to yours, however there are many regularly spaced disks as opposed to a large central one.
Yes, I think that visual inspection will be sufficient to check this. The stucture of h(x,y,z) quickly becomes very fine when you go away from the center, and this can produce aliasing if your
resolution is too low. You have to look at the real and imaginary parts separately to see this, however, since the magnitude image is very smooth. But if you say that you get many regularly spaced
disks in your final result, then that clearly points to a problem in your impulse image for me.
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MCDLII in Hindu Arabic Numerals
MCDLII = 1452
M C X I
MM CC XX II
MMM CCC XXX III
CD XL IV
D L V
DC LX VI
DCC LXX VII
DCCC LXXX VIII
CM XC IX
MCDLII is valid Roman numeral. Here we will explain how to read, write and convert the Roman numeral MCDLII into the correct Arabic numeral format. Please have a look over the Roman numeral table
given below for better understanding of Roman numeral system. As you can see, each letter is associated with specific value.
Symbol Value
I 1
V 5
X 10
L 50
C 100
D 500
M 1000
How to write Roman Numeral MCDLII in Arabic Numeral?
The Arabic numeral representation of Roman numeral MCDLII is 1452.
How to convert Roman numeral MCDLII to Arabic numeral?
If you are aware of Roman numeral system, then converting MCDLII Roman numeral to Arabic numeral is very easy. Converting MCDLII to Arabic numeral representation involves splitting up the numeral
into place values as shown below.
M + CD + L + I + I
1000 + 500 - 100 + 50 + 1 + 1
1000 + 400 + 50 + 1 + 1
As per the rule highest numeral should always precede the lowest numeral to get correct representation. We need to add all converted roman numerals values to get our correct Arabic numeral. The Roman
numeral MCDLII should be used when you are representing an ordinal value. In any other case, you can use 1452 instead of MCDLII. For any numeral conversion, you can also use our roman to number
converter tool given above.
Current Date and Time in Roman Numerals
The current date and time written in roman numerals is given below. Romans used the word nulla to denote zero because the roman number system did not have a zero, so there is a possibility that you
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Confusion Matrix - An easy way to remember and use
If you have ever done logistics regression, then you must be aware of the confusion matrix. The name is so apt for this important machine-learning concept. I used to get so confused trying to make
sense of the matrix and the formulas.
I struggled with it for a while until I figured out a way to remember the formulas and their meanings. In this blog post, I attempt to share that with you and I hope you find that helpful. Before we
get to the remembering part, let's get into the definitions first.
What is the confusion matrix?
The confusion matrix is a matrix of numbers built on classification machine-learning models. It is one of the most important evaluation tools to assess if the classification model did well. While
this matrix can be applied to both binary and multi-class classification problems, for simplicity, let's stick to binary classification for this article.
In simple terms, the matrix tells us the numbers of true-positives, false-positives, true-negatives and false-negatives. To understand these, let's pull up a matrix chart.
On the X-Axis we have the actual values from a dataset and on the Y-Axis we have the values predicted by a machine-learning binary classifier model. And below are the broad meanings of the four
quadrant blocks from the chart above.
TRUE POSITIVES - these are the number of samples which the model predicted as positive, and in reality too they were indeed positive.
TRUE NEGATIVES - these are the number of samples which the model predicted as negative, and in reality too they were indeed negative.
FALSE POSITIVES - these are the number of samples which the model predicted as positive, and in reality, they were negative.
FALSE NEGATIVES - these are the number of samples which the model predicted as negative, and in reality, they were positive.
Are you with me so far? Ok, moving on.
Key metrics
Now, there are numbers in the four quadrants in the matrix, right? If we play around with them we could derive some interesting insights that could be useful in telling how the model is performing.
Let's start with analysing how the model is doing overall. That is, all things aside, let's look at how many samples did the model predict correctly.
To get this we take the count of correct predictions (true-positives and true-negatives) and divide this number by the total number of samples. This measure is simply called Accuracy.
What does this tell us - Accuracy tells us how the model is able to correctly predict the classes overall*.* It's like a generalised metric and a starting point to look at while evaluating a model.
Where to use this -
• Example: Assessing a spam email classifier.
• Use case: You want to determine the overall effectiveness of the model in classifying emails as spam or non-spam. Accuracy provides an overall measure of how many emails were correctly
Precision is this - among the positive predictions, how many were right?
To get this, we take the count of true-positives and divide by the sum of true-positives and false positives.
What does this tell us - Precision indicates how well the model avoids false positive predictions.
Where to use this -
• Example: Evaluating a medical model trained for detecting a rare disease.
• Use case: In cases where a false positive result could lead to unnecessary treatments or interventions, precision is crucial. High precision ensures that the model minimizes the number of false
positives, reducing the chances of unnecessary actions.
Recall a.k.a Sensitivity a.k.a True Positive Rate
Recall is this - among the combination of true positives and false negatives, how many of them were true positives?
To get this, we take the count of true-positives and divide it by the sum of true-positives and false negatives.
What does this tell us - Recall indicates how well the model avoids false negative predictions.
When to use this -
• Example: Assessing a cancer detection model.
• Use case: When dealing with potentially life-threatening conditions, such as cancer, it is essential to have a high recall. A high recall value indicates that the model is effectively identifying
positive cases, minimizing the chances of false negatives and ensuring that true positive cases are not missed.
Specificity a.k.a True Negative Rate
Specificity is this - among the combination of true negatives and false positives, how many of them were true negatives?
To get this, we take the count of true-negatives and divide it by the sum of true-negatices and false positives.
What does this tell us - Specificity indicates how well the model avoids false positive predictions for negative instances.
When to use this -
• Example: Analyzing a credit card fraud detection system.
• Use case: In fraud detection, specificity is crucial to avoid falsely flagging legitimate transactions as fraudulent.
When to use what
Here's a quick summary of the previous section
Metric When to Use
Accuracy Use when you need an overall measure of the model's correctness and the costs of false positives and false negatives are similar.
Precision Use when the cost of false positives is high, and you want to minimize the chances of falsely identifying negative cases as positive.
Recall Use when the cost of false negatives is high, and you want to minimize the chances of missing positive cases.
Specificity Use when the cost of false positives for negative cases is high, and you want to minimize false alarms or false positive detections.
How to remember
As I said, I used to find it confusing to remember and interpret the meaning of the above-discussed metrics. So I came up with a system.
Let's look at the confusion matrix chart once again.
Here's my system for remembering the metrics.
I have built an acronym APRS which stands for Accuracy, Precision, Recall, and Specificity. I remember this sequence and built formulas around it.
Let's look at each of the metrics now.
To remember the formula for Accuracy, I use the following notation.
In my head, it looks like this:
To remember the formula for Precision, I use the following notation. Here q1 is quadrant 1 (top left).
In my head, it looks like this:
To remember the formula for Precision, I use the following notation.
In my head, it looks like this:
To remember the formula for Precision, I use the following notation.
In my head, it looks like this:
So imagine a GIF image cycling through the above patterns and I'm sure you will find it easy to recollect during an exam or an interview. It definitely helped me.
Code Examples
The following is the code to build and visualise the confusion matrix of a binary classification scenario.
import numpy as np
import matplotlib.pyplot as plt
import seaborn as sns
from sklearn.metrics import confusion_matrix
# True labels
y_true = [0, 1, 1, 0, 1, 0, 0, 1, 1, 1]
# Predicted labels
y_pred = [0, 1, 0, 0, 1, 1, 0, 1, 0, 1]
# Compute confusion matrix
matrix = confusion_matrix(y_true, y_pred)
# Create a heatmap visualization
labels = ['Negative', 'Positive']
ax = plt.subplot()
sns.heatmap(matrix, annot=True, fmt="d", cmap="Blues", ax=ax)
# Set labels, title, and axis ticks
ax.set_xlabel('Predicted Labels')
ax.set_ylabel('True Labels')
ax.set_title('Confusion Matrix')
# Show the plot
Well, there you go. That's my trick to remember the confusion matrix.
It is also key to remember that the APRS metrics we discussed here aren't the only ones used in the evaluation of classification models. There are other metrics like the F1 score and the ROC Curve
which can be used in conjunction with the above metrics for an even better evaluation. However, it largely depends on the context of the problem you are trying to solve.
If you have used the confusion matrix in any of your model building, comment below on what was the use case and how the confusion matrix helped you.
Thank you for reading through this article. Please feel free to leave your comment and if you found this blog useful, do leave a like and I'd highly appreciate it.
Did you find this article valuable?
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Class 8 Maths Chapter 3 Understanding Quadrilaterals MCQs
Class 8 Maths Chapter 3 Understanding Quadrilaterals MCQs (Questions and Answers) are provided here, online. These objective questions are designed for students, as per the CBSE syllabus (2021-2022)
and NCERT guidelines. Solving the chapter-wise questions will help students understand each concept and help to score good marks in exams. Also, learn important questions for class 8 Maths here at
Practice more and test your skills on Class 8 Maths Chapter 3 Understanding Quadrilaterals MCQs with the given PDF here.
MCQs on Class 8 Understanding Quadrilaterals
Multiple Choice Questions (MCQs) are available for Class 8 Understanding Quadrilaterals chapter. Each problem consists of four multiple options, out of which one is the correct answer. Students have
to solve the problem and select the correct answer.
1. Which of the following is not a quadrilateral?
A. Square
B. Rectangle
C. Triangle
D. Parallelogram
Answer: C
Explanation: A quadrilateral is a four-sided polygon but triangle is a three-sided polygon.
2. Which of the following quadrilaterals has two pairs of adjacent sides equal and its diagonals intersect at 90 degrees?
A. Square
B. Kite
C. Rhombus
D. Rectangle
Answer: B
3. Which one of the following is a regular quadrilateral?
A. Square
B. Trapezium
C. Kite
D. Rectangle
Answer: A
Explanation: A square has all its sides equal and angles equal to 90 degrees.
4. If AB and CD are two parallel sides of a parallelogram, then:
A. AB>CD
B. AB<CD
C. AB=CD
D. None of the above
Answer: C
5. The perimeter of a parallelogram whose parallel sides have lengths equal to 12 cm and 7 cm is:
A. 21 cm
B. 42 cm
C. 19 cm
D. 38 cm
Answer: D
Explanation: Perimeter of parallelogram = 2 (Sum of Parallel sides)
P = 2 (12 + 7)
P = 2 (19)
P = 38 cm
6. If ∠A and ∠C are two opposite angles of a parallelogram, then:
A. ∠A > ∠C
B. ∠A = ∠C
C. ∠A < ∠C
D. None of the above
Answer: B
Explanation: Opposite angles of a parallelogram are always equal.
7. If ∠A and ∠B are two adjacent angles of a parallelogram. If ∠A = 70°, then ∠B = ?
A. 70°
B. 90°
C. 110°
D. 180°
Answer: C
Explanation: The adjacent angles of parallelogram are supplementary.
∠A + ∠B = 180°
70° + ∠B = 180°
∠B = 180 – 70° = 110°
8. ABCD is a rectangle and AC & BD are its diagonals. If AC = 10 cm, then BD is:
A. 10 cm
B. 5 cm
C. 15 cm
D. 20 cm
Answer: A
Explanation: The diagonals of a rectangle are always equal.
9. Each of the angles of a square is:
A. Acute angle
B. Right angle
C. Obtuse angle
D. 180 degrees
Answer: B
Explanation: All the angles of square is at right angle.
10. The quadrilateral whose diagonals are perpendicular to each other is:
A. Parallelogram
B. Rectangle
C. Trapezium
D. Rhombus
Answer: D
11. Which of the following is not a regular polygon?
A. Square
B. Equilateral triangle
C. Rectangle
D. Regular hexagon
Answer: C. Rectangle
Explanation: A regular polygon is both equiangular and equilateral. But all four sides of a rectangle are not equal, thus it is not a regular polygon.
12. If the two angles of a triangle are 80° and 50°, respectively. Find the measure of the third angle.
A. 50°
B. 60°
C. 70°
D. 80°
Answer: A. 50°
Explanation: By the angle sum property of triangle, we know that;
Sum of all the angles of a triangle = 180°
Let the unknown angle be x
80° + 50° + x = 180°
x = 180° – 130°
x = 50°
13. In a parallelogram ABCD, angle A and angle B are in the ratio 1:2. Find the angle A.
A. 30°
B. 45°
C. 60°
D. 90°
Answer: C.60°
Explanation: As we know, the sum of adjacent angles of a parallelogram is equal to 180° and opposite angles are equal to each other.
Thus, in parallelogram ABCD angle A and angle B are adjacent to each other
Let angle A = x and angle B = 2x.
So, x + 2x = 180°
3x = 180°
x = 60°
14. The angles of a quadrilateral are in ratio 1:2:3:4. Which angle has the largest measure?
A. 120°
B. 144°
C. 98°
D. 36°
Answer: B.144°
Explanation: Suppose, ABCD is a quadrilateral.
Let angle A is x
x + 2x + 3x + 4x = 360° [Angle sum property of quadrilateral] 10x = 360°
x = 36°
Hence, the greatest angle is 4x = 4 x 36 = 144°
15. The length and breadth of a rectangle is 4 cm and 2 cm respectively. Find the perimeter of the rectangle.
A. 12 cm
B. 6 cm
C. 8 cm
D. 16 cm
Answer: A. 12 cm
Explanation: Given, length of rectangle is 4 cm
Breadth of rectangle = 2cm
By the formula of perimeter of rectangle, we know that;
Perimeter = 2 (Length + Breadth)
P = 2(4+2)
P = 2 x 6
P = 12 cm
16. The diagonals of a rectangle are 2x + 1 and 3x – 1, respectively. Find the value of x.
A. 1
B. 2
C. 3
D. 4
Answer: B.2
Explanation: The diagonals of a rectangle are equal in length.
2x + 1 = 3x -1
1 + 1 = 3x – 2x
2 = x
Thus, the value of x is 2.
17. The diagonals of a kite:
A. Bisects each other
B. Are perpendicular to each other
C. Does not bisect each other
D. None of the above
Answer: B. Are perpendicular to each other
Explanation: The diagonals of a kite are perpendicular to each other. They intersect at 90 degrees but does not bisect.
18. A rhombus has a side length equal to 5 cm. Find its perimeter.
A. 25
B. 10
C. 20
D. 30
Answer: C. 20
Explanation: A rhombus is a parallelogram that has all its four sides equal. Thus, the perimeter of rhombus,
P = 4 x side-length
P = 4 x 5
P = 20 cm
19. ABCD is a parallelogram. If angle A is equal to 45°, then find the measure of its adjacent angle.
A. 135°
B. 120°
C. 115°
D. 180°
Answer: A.135°
Explanation: The adjacent angles of a parallelogram sums up to 180°.
45° + x = 180°
x = 180° – 45°
x = 135°
20. The kite has exactly two distinct consecutive pairs of sides of equal length.
A. True
B. False
Answer: A. True
Explanation: A kite is a quadrilateral that has exactly two distinct consecutive pairs of sides of equal length. | {"url":"https://mathlake.com/Class-8-Maths-Chapter-3-Understanding-Quadrilaterals-MCQs","timestamp":"2024-11-06T02:46:06Z","content_type":"text/html","content_length":"15147","record_id":"<urn:uuid:d4570c83-e21b-42b5-ab57-3bb7addb99bc>","cc-path":"CC-MAIN-2024-46/segments/1730477027906.34/warc/CC-MAIN-20241106003436-20241106033436-00771.warc.gz"} |
MathJax Support In Org2Blog
Use MathJax with Org2Blog.
Thank you Jon for showing how to set this up in this post and this post and this post.
The only addition is that these instructions uses a more actively maintained plugin.
• Test it out using these (and more) examples
□ The word LaTeX in Math Mode (notice the italics)
□ The word LaTeX in Text Mode (notice the lack of italics)
□ Inline
☆ \(\sum_{i=0}^n i^2 = \frac{(n^2+n)(2n+1)}{6}\)
□ Equation
☆ \[\sum_{i=0}^n i^2 = \frac{(n^2+n)(2n+1)}{6}\]
2 thoughts on “MathJax Support In Org2Blog”
1. cdn.mathjax.org shutdown at the end of April so the documentation should not point to it. The link gives advice as to the best places to get it going forward.
1. Thanks Jon. I set it to cdnjs.cloudflare. | {"url":"https://wisdomandwonder.com/emacs/10660/mathjax-support-in-org2blog","timestamp":"2024-11-03T16:08:01Z","content_type":"text/html","content_length":"45355","record_id":"<urn:uuid:894b21dd-7b6a-4669-9e18-ab92f444c582>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00215.warc.gz"} |