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Predicting Nugget Size of Resistance Spot Welds Using Infrared Thermal Videos With Image Segmentation and Convolutional Neural Network
Resistance spot welding (RSW) is a widely adopted joining technique in automotive industry. Recent advancement in sensing technology makes it possible to collect thermal videos of the weld nugget
during RSW using an infrared (IR) camera. The effective and timely analysis of such thermal videos has the potential of enabling in situ nondestructive evaluation (NDE) of the weld nugget by
predicting nugget thickness and diameter. Deep learning (DL) has demonstrated to be effective in analyzing imaging data in many applications. However, the thermal videos in RSW present unique
data-level challenges that compromise the effectiveness of most pre-trained DL models. We propose a novel image segmentation method for handling the RSW thermal videos to improve the prediction
performance of DL models in RSW. The proposed method transforms raw thermal videos into spatial-temporal instances in four steps: video-wise normalization, removal of uninformative images, watershed
segmentation, and spatial-temporal instance construction. The extracted spatial-temporal instances serve as the input data for training a DL-based NDE model. The proposed method is able to extract
high-quality data with spatial-temporal correlations in the thermal videos, while being robust to the impact of unknown surface emissivity. Our case studies demonstrate that the proposed method
achieves better prediction of nugget thickness and diameter than predicting without the transformation.
Issue Section:
Research Papers
1 Introduction
Resistance spot welding (RSW) is a widely used technique for joining metal sheets. It has the advantages of low cost, high speed, reliable, and simple operations [1]. These merits have led to a wide
adoption of this technique in the automotive industry, for joining lightweight metals such as aluminum (Al) alloys and lightweight steels [2]. As shown in Fig. 1(a), during the RSW process, two or
more metal sheets are clamped together and placed between two water-cooled electrodes. Electrical current passes through the metal sheets, generating heating and creating a molten nugget (i.e., spot
of welding) at the faying surface. After a specified holding time, the electrical current is shut down to let the nugget solidify [1]. A welding spot is therefore formed.
Lightweight materials are increasingly used in cars and trucks to decrease weight while preserving strength. However, there is a lack of understanding of RSW for joining lightweight alloys due to the
mechanical aspects and imperfect operation [1]. Defects commonly occur in the nuggets, compromising the utilization of lightweight parts in industry. As shown in Fig. 1(b), the major defects from RSW
include insufficient/no fusion, porosity, expulsion (or excessive indentation, i.e., the ejection of molten metal), and cracks [3]. These defects are usually caused by variations in the electrical
current, extra/insufficient holding time, and other uncertainties during manufacturing. The quality of the weld can be implied by the size of the weld nugget—thickness and diameter in particular.
Therefore, there is a pressing need for the nondestructive evaluation (NDE) of nugget thickness and diameter, and such evaluation can be used to provide information about possible defects.
The traditional evaluation of RSW nuggets uses destructive methods such as the chisel test and peel test. These methods are time-consuming, costly, and can only be done after welding [4]. There is an
imperative need to develop in situ NDE methods for RSW. Recent development in inline sensing technology has enabled real-time acquisition of thermal images for RSW nuggets. Infrared (IR) camera aims
at the welding spot with a tilted angle and captures thermal images at a high frequency of 100 fps. Figure 2(a) shows the setup of data acquisition and Fig. 2(b) shows selected thermal images in a
video. The pixel values in each thermal image reflect the IR radiation of the nugget at the time of data recording. The resulted data is a thermal video that conveys precise, real-time information of
the nugget formation process. These data allow us to predict nugget size without destroying the part.
Deep learning (DL) has demonstrated to be effective in analyzing imaging data in many applications including manufacturing. Deep neural network was adopted in Imani et al. [5] to learn the geometric
variation of additive manufacturing (AM) part from layer-wise imaging profile; convolutional neural network (CNN) was used in Zhang et al. [6] for predicting AM part porosity from the thermal images
of melt pool; a convolutional-artificial NN model was developed in Francis and Bian [7] for predicting AM part distortion. Janssens et al. [8] deployed CNN on lab-generated thermal videos for machine
condition monitoring. These studies revealed the promises in exploiting DL’s learning ability, high accuracy, and real-time prediction for thermal image-based process monitoring and quality
In this study, we apply DL on thermal videos of weld nugget for in situ NDE. Specifically, a CNN regression model is developed to predict the thickness and diameter of the nugget. However, the
thermal videos in RSW present unique data-level challenges that compromise the effectiveness of most existing DL models. First, although each thermal video captures the entire RSW process, the useful
information about weld nugget is only available after electrodes lift to expose the surface of the welded region. Hence, the starting time (i.e., frame index in a video) to extract the useful
information out of the entire video needs to be determined. The frames before this starting time are uninformative of the nugget and thus should be discarded. Second, the nugget profile can be
“blurry.” A nugget has no clear contrast of dents and spikes or any sharp edges/vertex, thus naturally a “blurry” object. Meanwhile, the IR images may have a limited resolution that further
compromise the nugget clarity. Third, there is spatial-temporal correlation in thermal images. Within an image, the pixel values are related to their position in the nugget, implying spatial
correlation among pixels; on the other hand, the nugget profile in a thermal video evolves with the timestamp of recording (or frame index), indicating temporal correlation across images. Such
spatial-temporal correlation should be preserved in data processing.
Existing studies emphasize the development of new DL models or customization of DL architectures for learning from thermal images, but rarely address these data-level challenges. If not resolved
during data processing (before model training), these issues will significantly compromise the learning outcome. In this work, we propose an innovative data processing approach based on image
normalization and segmentation, which effectively removes uninformative images and enhance the clarity of patterns in nuggets. The processed thermal images are used to build a CNN regression model
that achieves an improved performance in nugget quality prediction.
The rest of this paper is organized as follows. Section 2 will introduce the thermal video data from RSW of Boron steel that motivates this study. Section 3 will provide the technical detail of our
method. A case study will be presented in Sec. 4 to demonstrate the performance improvement in nugget quality prediction using the proposed method. Section 5 will end the paper with concluding
remarks and future research directions.
2 Data Description
We obtain in situ thermal videos from lab implementation of RSW for joining two sets of Boron steels: (i) a 2T stack of bare boron steel sheets, 1 mm thickness each; (ii) a 3T stack of Al-coated
Boron steel sheets, 1 mm thickness for the top and bottom sheets and 2 mm thickness for the middle sheet. A thermal video in (i) consists of 600∼602 frames and that in (ii) consists of 500∼504
frames. Each frame (in both datasets) is a grayscale thermal image of size 61 × 81. Depending on the recording time, the pixel values in the image may have different ranges. For an early frame such
as Fig. 2(b1) and (b2), the nugget is blocked by the weld head and thus not fully appear in the image. Consequently, the pixel values (for IR intensity) are all small (less than 20) in the early
frame. For a later frame captured when the nugget is fully formed and stabilized such as those shown in Fig. 2(b3)–(b5), the pixel values are higher and typically range between 20 and 100. There are
25 videos in dataset (i) and 22 videos in dataset (ii).
Nugget thickness and diameter were measured in the lab using destructive testing methods. Table 1 shows the measurements for the welds corresponding to ten selected videos, where “Dmin” and “Dmax”
are the minimal diameter and maximal diameter of the weld nugget, respectively. Each row in Table 1 corresponds to one nugget, whose formation is shown in one video. It then follows that all the
thermal images in a video correspond to the same measurements of nugget thickness and diameter.
Table 1
Video Thickness (mm) Dmin (mm) Dmax (mm)
1 1.899 3.135 3.311
2 1.905 3.135 3.289
3 1.871 4.923 4.923
4 1.875 4.875 4.945
5 1.861 5.740 5.762
6 1.863 5.740 5.784
7 1.857 5.673 5.828
8 1.853 5.717 5.784
9 1.811 6.336 6.424
10 1.787 6.336 6.446
Video Thickness (mm) Dmin (mm) Dmax (mm)
1 1.899 3.135 3.311
2 1.905 3.135 3.289
3 1.871 4.923 4.923
4 1.875 4.875 4.945
5 1.861 5.740 5.762
6 1.863 5.740 5.784
7 1.857 5.673 5.828
8 1.853 5.717 5.784
9 1.811 6.336 6.424
10 1.787 6.336 6.446
3 Method
This section presents the technical details of the proposed data processing method. It consists of four steps: video-wise normalization (Sec. 3.1), identification of uninformative images (Sec. 3.2),
image segmentation (Sec. 3.3), and spatial-temporal instance construction (Sec. 3.4). The processed data are used to train a CNN regression model for predicting the nugget thickness and diameter
(Sec. 3.5).
3.1 Video-Wise Normalization.
The IR signal values in a thermal image can be noisy due to environmental uncertainties, emissivity (i.e., the effectiveness in emitting energy as thermal radiation [9]) fluctuation, and recording
errors. Such noise may distort the nugget profile. Within a thermal video, all the images are associated with the same nugget. The images record the nugget’s formation in temporal changes. By
normalizing the images along the timeline, noise and errors in individual frames should be substantially reduced. The true patterns of nugget can be better revealed.
Denote the
th thermal video in a dataset by
= 1, 2, …, and the
th image (pixel matrix) in it by
are the number of rows and columns in the pixel matrix, respectively. We propose
video-wise normalization
to normalize all the frames in the video along the timeline. Specifically, we flatten each pixel matrix to a vector,
, of length
and concatenate all the vectors (of the video) to a matrix:
is the total number of frames in the video. Next, we normalize each column of pixels in
, which is equivalent to normalizing the pixel value of a fixed position in image across all the frames. Let
denote the normalized
. Each row in
is converted back to a pixel matrix, i.e., a normalized thermal image of nugget, expressed as
= 1, 2, …,
. The video-wise normalization procedure is illustrated in Fig.
3.2 Identification of Uninformative Images.
The normalized thermal videos need to be screened for uninformative images. Any frames containing insufficient information of nugget should be removed from the analysis. Preliminary inspection (Sec.
2) shows that early frames in a thermal video tend to have low IR radiation intensity due to the absence of nugget (blocked by electrodes). After electrodes lifted, the welded area was exposed,
resulting in a sudden increase of the IR intensity captured the camera. Therefore, the sufficiency of nugget information can be evaluated by thresholding the pixel magnitude in an image—the image is
considered informative only if all its pixels are not smaller than a threshold, q[n].
The threshold is defined as follows. Let
be the
th percentile of all pixel values in the normalized thermal video
, i.e.,
is the
th smallest pixel value in
, where
is the total number of pixels in
. We then compare the pixels in each normalized image,
, with
, and preserve the image only if
illustrates the removal of uninformative images.
As mentioned earlier, the stabilized nugget has higher pixel values in thermal images. If sorting all pixels in $P~~n$ from the smallest to the largest, then a recommended value of Q is the
percentile corresponding to the smallest (normalized) pixel value in the frames of stabilized nugget. With such parameter selection, the images with insufficient nugget information should be
discarded while those with the stabilized nugget should be preserved. We define a set Ω[n] for the indices of preserved frames in thermal video $P~~n$.
3.3 Image Segmentation.
A major obstacle for improving the learning outcome of DL from RSW thermal images is the blurry nugget profile. In this section, we propose a watershed-like image segmentation method [10] to
characterize the nugget profile and elucidate potential defects such as porosity, cracks, and irregular nugget size from nonsegmented images.
“Watershed” is a geology concept that describes the dividing contours for adjacent catchment basins. In image processing, watershed method is widely used for segmenting morphological objects in
grayscale images [
]. The classic way of drawing watersheds is by thresholding the pixel values. We define a level
for the pixel magnitude and obtain a set containing all pixels meeting the specified threshold:
The complementary set of $Xn(ln(t))$ is denoted by $X¯n(ln(t))$, which contains all the remaining pixels in $p~~n(t)$. We let $ln(t)$ be the Lth percentile of $p~~n(t)$, i.e., $ln(t)$ is the $⌈L/
100⋅rc⌉$th smallest pixel value in $p~~n(t)$. We further define
The pixels belonging to set $Xn(ln(t))$ are then segmented from the rest of the image. Figure 5 displays a sample image and its segments after the thresholding. The resulted segments have dark areas
(value 1) as above the watershed contour and the white regions (value 0) below the watershed.
There are other methods for drawing watersheds [11,12]. However, in this study, the classic way is simple yet effective in contouring the nugget profile. To clarify interesting patterns in the
nugget, we define multiple levels, $ln1(t),ln2(t),…,lnM(t)$, for segmenting one image. For each level, we produce an image segment. Eventually, a single thermal image will be transformed into M
segments, each describing the nugget profile at a different altitude. Figure 5 demonstrates the segmentation with M = 5 levels (simplified notations with superscript (t) and subscript n omitted in
$ln1(t),…,ln5(t)$). If a weld defect, e.g., porosity, or irregular shape arise in the nugget, the proposed image segmentation method will capture the irregularity in certain segments, given properly
selected levels. With the clearly contoured morphemes in these image segments, DL models can better learn the regular and irregular nugget profiles, thus making more accurate predictions for nugget
thickness and diameter.
3.4 Construction of Spatial-Temporal Instances.
Now, the only remaining challenge in DL-based quality prediction for RSW is the spatial-temporal correlation in thermal videos. In Sec. 3.3, a single image is transformed into M segments. These
segments together reflect the spatial patterns in the nugget, thus should be considered as one sample. By adopting a CNN regression model, the spatial correlation in the sample should be
automatically learned. The remaining question is how to incorporate the temporal correlation. Temporal correlation arises due to the evolution of nugget profile with time. In other words, consecutive
frames in a video form a time series of thermal images. If we take a sequence of frames every δ timestamps in the video as one sample, then the temporal correlation should be incorporated. Denote the
sequence length by S, 1 < S < T, and let the time increment δ ≥ 1. Since the IR camera has a high speed, adjacent frames may have similar nugget profiles. Having δ ≥ 1 can avoid information
duplication in the image sequence. The best δ value depends on the IR camera frequency—a very high frequency of capturing images implies a relatively large δ. To simultaneously accommodate the
spatial and temporal correlation, we construct a spatial-temporal instance by concatenating the M segments of S frames, which forms an instance of shape (r, c, M · S). Figure 6 demonstrates a
spatial-temporal instance for (r, c, M · S) = (61, 81, 15) with S = 3, M = 5, δ = 5. The image segments of each thermal image make clear the pattern variations in space; concatenating the 15 segments
in a sequence preserves the temporal evolutions of nugget across the three frames. After building one spatial-temporal instance, we move 1-frame forward and use the next S thermal images (for every δ
frames) in the video to build the next instance.
3.5 Convolutional Neural Network Regression for Nugget Quality Prediction.
The spatial-temporal instances imply that customization is necessary on a conventional CNN regression model to make it compatible with the input data. For CNN input, a single image (segment) is
typically reshaped to a square pixel matrix. In our case, a single image segment has an original shape of (61, 81), which can be readily reshaped to (64, 64) without severe distortion of the
information. A spatial-temporal instance now has shape (64, 64, M · S). Such 3-dimensional (3D) instances should be handled by a 3D CNN model. However, to reduce computational burden, we treat each
image segment in the 3D instance as a channel, i.e., source of information to DL models, and use a 2D CNN model to learn from the 3D input. The filters of the first convolutional layer in this model
are customized to be 3D with a depth of M · S in order to learn from all the M · S input channels. Figure 7 provides the architecture of our spatial-temporal CNN regression. It consists of three
convolutional layers and two dense layers. Eventually, the input is mapped to the response, y = [Thickness, Dmin, Dmax]. The spatial-temporal correlation is considered in the input layer and first
convolutional layer, and the rest model structure follows conventional design. The model parameters, e.g., filter size and dropout rate, are determined after fine tuning. For model training, the loss
function is mean squared error (MSE) per the convention of regression [13]; the optimizer is chosen to be “Adam” (adaptive moment estimation) for its superior efficiency [14].
In Sec. 4, we will demonstrate the superiority of the proposed data processing method (Secs. 3.1–3.4) by comparing the learning outcome and prediction performance of this CNN regression model with
those of a conventional one on unprocessed thermal images.
4 Case Study
In this case study, we apply the proposed data processing method on the RSW datasets (i) and (ii). For each dataset, we compare the performance of CNN regression (in both training and test phase)
when using the processed data to that using the original data. By “original,” we mean the raw thermal videos, where each frame is reshaped to (64, 64) as an instance for CNN input. When building the
spatial-temporal instances, the parameters are Q = 50, M = 5, S = 3, where the five levels are $(l1,l2,l3,l4,l5)=(50%,60%,70%,80%,90%)$ percentiles of a normalized thermal image. We experiment with
different time increments, δ ∈ {1, 3, 5}, to produce the results.
If using original data, dataset (i) has around 13,750 instances and dataset (ii) has around 11,000. Yet, these sizes will decrease when constructing spatial-temporal instances. For example, the total
number of instances will be 1566 for (i) and 1416 for (ii) if δ = 3 (other parameters are as given above). To avoid overfitting due to the small data size, we do sixfold cross-validation (CV) in
model training. The instances of a dataset are randomly shuffled and assigned to six equal-sized folds without replacement. In one run of model training, one of the six folds is preserved as testing
data; 80% of the rest five are training data and the last 20% are training-phase validation data. 100 epochs without batching are used to train the model. MSE loss is adopted as the performance
metric for either training or prediction.
4.1 Training Performance.
Figures 8 and 9 show the training performance for datasets (i) and (ii), respectively. In each subplot, the horizontal axis is the number of epochs and the vertical axis is MSE loss. The (blue) curve
with dot markers represents training loss, and the (red) curve with triangle markers represents training-phase validation loss. From left to right, each column of three plots are for conventional CNN
regression without data processing, spatial-temporal CNN regression with δ = 1, spatial-temporal CNN regression with δ = 3, and spatial-temporal CNN regression with δ = 5; from top to bottom, each
row of plots are for the first run of CV, third run of CV, and sixth run of CV. Note that, even though all titled “CV1” (or “CV3”/“CV6”), the training/testing instance are in different shapes and
orders across Figs. 8(a)–8(d) due to the way we process the data (same for Figs. 9(a)–9(d)). But the comparison is solid and comprehensive as it provides the typical training performance across
different CVs.
We see that the column (a) subplots in both Figs. 8 and 9 show volatile training/validation loss. We notice that the training loss surges suddenly in Figs. 8(a1) and 9(a1), indicating that the model
did not sufficiently learn from the data, resulting in underfitting. It is also noticed that the validation loss can increase and remain high for a couple of epochs, as in Fig. 9(a1–3), implying that
the model over-characterized the training set and resulted in overfitting. Such underfitting/overfitting phenomena during model training show that unprocessed thermal videos can cause difficulty for
CNN model convergence. As a contrast, after processing the data with our proposed method, the model training becomes rather efficient and smooth—columns (b)–(d) in either Fig. 8 or Figure 9 show fast
model convergence, as demonstrated by the stable, low training/validation loss after epoch 10. With the processed data, dataset (i) has better training performance—the training and validation loss
are close after model convergence, indicating no serious overfitting. Spatial-temporal instances with δ = 1, 3, or 5 led to similar performance, so any of them is a satisfying choice for this
dataset. For dataset (ii), certain plots for using processed data, e.g., Figs. 9(b) and 9(c), has larger validation loss, implying certain overfitting with δ = 1 and δ = 3. When taking δ = 5, the
validation loss gets closer to the training loss (after model convergence), so spatial-temporal instances built with δ = 5 (and all other aforementioned parameter values) are recommended for this
4.2 Prediction Performance.
Evaluation of the prediction (testing) performance is even more crucial. We consider the prediction MSE loss (the smaller the better) on average for the sixfold CV to evaluate the overall prediction
accuracy. The minimal, mean, median, and maximal prediction MSE are calculated for each run of CV, then taken average. Table 2 shows the values of minimal, mean, median, and maximal average
prediction MSE for datasets (i) and (ii). The “Parameter” column shows the parameter in CNN.
Table 2
Dataset Parameter Min MSE Mean MSE Median MSE Max MSE
Conventional 0.0000 0.0836 0.0295 7.3843
(i) $δ=1$ 0.0001 0.1208 0.0221 2.9996
$δ=3$ 0.0001 0.1015 0.0149 1.9122
$δ=5$ 0.0001 0.1089 0.0188 2.2179
Conventional 0.0004 4595.19 0.3842 4,461,273.40
(ii) $δ=1$ 0.0004 0.7198 0.0699 14.0236
$δ=3$ 0.0003 0.5125 0.0505 11.7648
$δ=5$ 0.0003 0.3307 0.0392 8.1566
Dataset Parameter Min MSE Mean MSE Median MSE Max MSE
Conventional 0.0000 0.0836 0.0295 7.3843
(i) $δ=1$ 0.0001 0.1208 0.0221 2.9996
$δ=3$ 0.0001 0.1015 0.0149 1.9122
$δ=5$ 0.0001 0.1089 0.0188 2.2179
Conventional 0.0004 4595.19 0.3842 4,461,273.40
(ii) $δ=1$ 0.0004 0.7198 0.0699 14.0236
$δ=3$ 0.0003 0.5125 0.0505 11.7648
$δ=5$ 0.0003 0.3307 0.0392 8.1566
Note: Bold values marks the best performance for each metric.
For dataset (i), as shown in the top half of Table 2, the minimal, mean, and median MSE values are close for conventional CNN and our spatial-temporal CNN with different δs. All these MSEs are rather
low and below 0.02, but the maximal average MSE is significantly larger for the conventional CNN, which is consistent with its underfitting/overfitting in model training—some predictions made were
too far away from their true values. For our spatial-temporal CNN, the maximal average MSE remains small—typically below 2. Our model with δ = 3 achieves the lowest maximal average MSE and is the
best option for dataset (i). For dataset (ii), as shown in the bottom half of Table 2, the prediction performance of conventional CNN is much worse than the spatial-temporal CNN—its mean and maximal
average MSEs exceed 4000. With our spatial-temporal CNN, the average MSEs are maintained at a low level and similar to those in dataset (i). Among the three δ values, δ = 5 leads to the lowest
average MSE. The desirable prediction performance of δ = 5 is consistent with its outstanding training performance.
To supplement the average results, Table 3 further provides the standard deviation (std) of minimal, mean, median, and maximal prediction MSE values across the 6-fold CV. The standard deviation
measures the variability of a performance metric across the 6 runs of prediction. If a model is robust, then the model trained with different training sets can achieve similarly good prediction
performance, hence a small std value for each prediction performance metric. For dataset (i), all std values are rather small. Our spatial-temporal CNN, trained on instances constructed with δ = 3,
leads to the lowest std for mean, median, and maximal MSEs, indicating the best model robustness. Dataset (ii), however, shows rather large std when using conventional CNN regression—the std for
minimal and median MSEs are small but the mean and maximal MSEs are overwhelming. This phenomenon implies that the model is not robust against extreme instances—a couple of severe outliers have led
to skewed mean and maximal MSEs. The prediction performance for using the conventional CNN on dataset (ii) is rather unstable. This is also expected from the severe underfitting/overfitting in
training as shown in Fig. 9(a). Fortunately, with the proposed spatial-temporal CNN regression, the std values for prediction MSE are reduced to a low level. The best robustness is achieved by
spatial-temporal CNN along with instances constructed with δ = 3 (or δ = 5 if focusing on the min and mean MSEs). Our data processing has effectively improved the training data quality and built a
more robust CNN regression model for NDE.
Table 3
Dataset Parameter Min MSE Mean MSE Median MSE Max MSE
Conventional 0.00005 0.05496 0.03360 0.66647
(i) $δ=1$ 0.00009 0.03106 0.01153 2.25708
$δ=3$ 0.00017 0.02905 0.00393 0.57477
$δ=5$ 0.00006 0.04838 0.00988 0.97715
Conventional 0.00048 9757.38 0.49556 10,168,225.32
(ii) $δ=1$ 0.00036 0.11326 0.03292 2.40647
$δ=3$ 0.00052 0.13074 0.01277 1.95331
$δ=5$ 0.00014 0.10799 0.01497 3.31112
Dataset Parameter Min MSE Mean MSE Median MSE Max MSE
Conventional 0.00005 0.05496 0.03360 0.66647
(i) $δ=1$ 0.00009 0.03106 0.01153 2.25708
$δ=3$ 0.00017 0.02905 0.00393 0.57477
$δ=5$ 0.00006 0.04838 0.00988 0.97715
Conventional 0.00048 9757.38 0.49556 10,168,225.32
(ii) $δ=1$ 0.00036 0.11326 0.03292 2.40647
$δ=3$ 0.00052 0.13074 0.01277 1.95331
$δ=5$ 0.00014 0.10799 0.01497 3.31112
Note: Bold values marks the best performance for each metric.
4.3 Discussion and Recommendation.
Both overfitting and underfitting can lead to poor model performance. In order to limit overfitting/underfitting, we recommend that the number of weld nuggets (equivalently, their in situ thermal
videos) for model training to be no fewer than the case study provided here, i.e., 20∼25. If an inline sensor with lower speed (e.g., <100 fps) is used, more nugget videos should be collected to form
the training set.
In our case study, the proposed method is applied on the two RSW datasets separately, resulting in two models although they follow the same framework. Since the two datasets come from two different
experimental conditions, as explained in Sec. 2, the underlying physics differ. Therefore, we recommend developing one spatial-temporal CNN model for each type of experiment.
Another thing worth mentioning is the “in situ” manner in NDE. When the proposed data processing method is adopted, incoming new thermal videos are first processed with video-wise normalization,
uninformative image removal, image segmentation, and spatial-temporal instance construction. The processed new data are fed to the spatial-temporal CNN for NDE of nugget thickness and diameter. The
processing time is short (typically less than 1 min for a video) and will not increase the computational burden or compromise the timeliness of NDE.
In online prediction with a trained spatial-temporal CNN, a plausible way to further improve NDE efficiency is drawing S raw thermal images with δ increment from a new video to construct a single
spatial-temporal instance. Since a video is for only one nugget, with a robust spatial-temporal CNN regression model, one instance suffices for predicting the nugget thickness and diameter
5 Conclusion
In this study, we proposed an innovative data processing method to improve the prediction performance of CNN regression with thermal videos of RSW nugget. Normalization and watershed image
segmentation were explored for resolving the data-level challenges posed by thermal videos, i.e., uninformative images, blurry nugget profile, and spatial-temporal correlation. Spatial-temporal
instances were constructed using the proposed method and fed to a spatial-temporal CNN regression model, which was demonstrated to result in significantly more accurate prediction for the nugget
thickness and diameter.
This work has multiple technical contributions. First, it has established an effective, systematic way of improving noisy, blurry thermal imaging data for better learning outcome in DL-based NDE.
This was an underexplored topic but properly addressed in our study. Second, the work provides a reference and performance benchmark for subsequent studies about NDE with RSW thermal videos. The case
study data had limited quality, but our method has achieved satisfying NDE performance on it, indicating a promising direction for enhancing the DL-based NDE performance. Third, the proposed method
can be extended for weld defect detection by incorporating defect information such as cracks and porosity in model training. Fourth, the proposed data processing method is readily generalizable to
various RSW applications. It can guide existing DL-based NDE practice.
This article was supported in part by the US Department of Energy, in part by the Office of Nuclear Energy (Advanced Methods for Manufacturing Program), and in part by the AI Initiative at Oak Ridge
National Laboratory.
Conflict of Interest
There are no conflicts of interest.
Data Availability Statement
The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request. Data provided by a third party listed in Acknowledgment.
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Is there a systematic way to determine an integrating factor mu(x,y) of the form x^n y^m, given a not-necessarily-exact differential equation? | Socratic
Is there a systematic way to determine an integrating factor #mu(x,y)# of the form #x^n y^m#, given a not-necessarily-exact differential equation?
My book covers special integrating factors $\mu$ that are functions of only $x$ or only $y$, but kinda glosses over how to find an integrating factor that is a function of $x$ AND $y$.
Example equation:
$\left(2 {y}^{2} - 6 x y\right) \mathrm{dx} + \left(3 x y - 4 {x}^{2}\right) \mathrm{dy} = 0$
The integrating factor was $\mu \left(x , y\right) = x y$, and the solution was $F \left(x , y\right) = {x}^{2} {y}^{3} - 2 {x}^{3} {y}^{2} = C$.
I was able to figure out what the integrating factor was, and solve the equation, but I had to assume that $n = m$, which is not something I think I should need to do.
1 Answer
If you have:
$M \left(x , y\right) \mathrm{dx} + N \left(x , y\right) \mathrm{dy} = 0$
And the equation is not an exact Differential Equation, ie
$\frac{\partial M}{\partial y} \ne \frac{\partial N}{\partial x}$
Then you must convert the equation into an exact differential equation by multiplying by an integrating factor $\mu \left(x , y\right)$ to get
$\mu \left(x , y\right) M \left(x , y\right) \mathrm{dx} + \mu \left(x , y\right) N \left(x , y\right) \mathrm{dy} = 0$
$\frac{\partial \left(\mu M\right)}{\partial y} = \frac{\partial \left(\mu N\right)}{\partial x}$
That's all well and good but In order to find such an integrating factor $\mu \left(x , y\right)$ you can do some manipulation and eventually establish the need to solve the partial differential
$\text{ } M \frac{\partial \mu}{\partial y} - N \frac{\partial \mu}{\partial x} + \left(\frac{\partial M}{\partial y} - \frac{\partial N}{\partial x}\right) \mu$ = 0
$\text{ } M {\mu}_{y} - N {\mu}_{x} + \left({M}_{y} - {N}_{x}\right) \mu = 0$
which in general is a harder problem to solve!
If the given differential equation is "designed" to be solved (eg in an exam rather than a real life equation) then it will often be the case that:
$\text{ } \mu \left(x , y\right) = \mu \left(x\right)$, a function of $x$ alone
$\text{ } \mu \left(x , y\right) = \mu \left(y\right)$, a function of $y$ alone
In which case the above PDE can easily be solved to give:
$\text{ } \mu \left(y\right) = \exp \left(\int \frac{\frac{\partial M}{\partial y} - \frac{\partial N}{\partial x}}{M} \mathrm{dy}\right) = {e}^{\int \frac{{M}_{y} - {N}_{x}}{M} \mathrm{dy}}$
$\text{ } \mu \left(x\right) = \exp \left(\int \frac{\frac{\partial M}{\partial y} - \frac{\partial N}{\partial x}}{N} \mathrm{dx}\right) = {e}^{\int \frac{{M}_{y} - {N}_{x}}{N} \mathrm{dx}}$
But, in general finding the integrating factor will not be possible and so the Differential Equation would be solved numerically rather than finding an analytical solution.
In the real world, It is always possible to find a series solution but this approach is particularly cumbersome (and is often the approach used by a computer for a numerical solution)
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Mission Statement
Scope: To develop fundamental theory and connect it to the observable universe, in particular through research into the nature of gravity and spacetime, the cosmology of the early universe and dark
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Expand collaborations in the form of joint research projects, including funding applications. Train diverse early-career researchers to become versatile theoretical physicists able to tackle the
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The Theory working group is a forum for theory of interest to the Oskar Klein centre. It is open for all OKC members and for high energy and gravity/cosmology theorists at Nordita. Members of the
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potential cosmological role of primordial black holes. In principle any theory done in OKC can be discussed although traditionally the topics have been related to high energy physics theory and
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The Theory working group has bi-weekly meetings which are broken into two parts. The first half of the meeting begins with the general background behind a problem and the second half proceeds into
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How to simulate random offset of the comparators in Matlab?
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Sep 10, 2009
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Hi all,
I have a little problem with the design of a full flash ADC, now I have to simulate the random offset of the comparators with matlab but I have no idea how to do this.
Someone can help me?
Mar 19, 2008
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Offset simulation
for every comparator
where offset is a random vector with gaussian distribution that depends on the expected rms offset of your comparators, for example.
Sep 10, 2009
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Re: Offset simulation
thanks for your replay..but I'm thinking,as i'm not able to use Matlab, maybe is more simple for me simulate the offset of the comparator with Cadence for example.
How can I simulate the offset with cadence?
Is more simple than with matlab? (I dont' know nothing about matlab)
This is the circuit that i want to simulate:
Mar 4, 2008
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Offset simulation
In that case you can make a voltage source with a
gauss()* value in each of the relevant sub-blocks and
repetitively simulate (probably use ocean) and gather
the results for some sort of postprocessing.
*gauss(), agauss(), whatever random / stats you can
find in the Spectre manuals. You could also make a
veriloga offset source if the analog primtives can't be
propertized the way you want, but I believe they can.
Sep 10, 2009
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Re: Offset simulation
Thanks for replying me, but I have to do it with Matlab, no cadence or others Cad.
So,which is the first step in order to do it using MATLAB?
Please help me, because I have no idea!
Mar 19, 2008
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Offset simulation
I believe you need to explain the problem in more details...
You have a clocked comparator whose offset is determined by the mismatch of M1 and M2 (as stated in the technology manuals). Once you set the size, you will have a gaussian distribution of offsets
for your comparator. You can change the mean offset by changing the size of M1 and M2.
In matlab, you can represent this as:
where sigma_mismatch is the standard deviation of your offset and depends on the size of M1 and M2; randn is matlab's gaussian distribution function and N is the number of comparators.
Sep 10, 2009
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Re: Offset simulation
Thanks for your reply..
I try to explain the problem in more details:
I'm designing a 5-bit, 4 Gs/s ADC FLASH in 65 nm CMOS. I have an idea of the total architecture, it's a classical architecture with resistor ladder, T/H, preamplifiers, clock comparator and encoder.
Now I want to simulate the offset of the clocked comparator using MATLAB.
My first problem is: How can I "transfer" the circuit of the comparator in matlab?
Which kind of equations i have to write?
Mar 19, 2008
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Offset simulation
In matlab you do "system" simulations. The actual implementation of the comparator does not affect its model as long as you keep into account the parameters you want to check for, such as mismatch.
You do not need to write the voltage/current equations to get a good idea of the required performance to meet your design goals.
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Re: Offset simulation
Ok, we consider this simple problem:
I have this equation Vout=Rd*(Id1-Id2) , I suppose that (Id1-Id2) have a gaussian distribution and i want to plot Vout.
which kind of function i have to use to represent a gaussian distribution for (Id1-id2) ?
Mar 19, 2008
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simply noisy_Id1=Id1+sigma_id1*randn, where Id1 is the mean of your current, sigma_id1 is its standard deviation and randn is matlab's gaussian distribution function.
If you go this way, Rd should also be a random number (since you cannot guarantee its absolute value).
BTW, you would need to saturate Vout...
In my opinion, this way is far more complicated that the initial model I proposed to you and it does not add any more information...
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Geometric Distribution
A geometric probability calculator is a tool that helps calculate the probability of an event occurring in a geometric setting. Geometric probability is a branch of mathematics that deals with the
probability of geometric events, such as the probability of a point landing in a certain area or the probability of a line intersecting another line. The calculator uses mathematical formulas to
determine the probability of these events.
This type of calculator can be useful in a variety of fields, including engineering, physics, and computer science. For example, in engineering, geometric probability can be used to calculate the
probability of a machine part failing due to stress at a certain point. In physics, it can be used to calculate the probability of a particle being in a certain location at a certain time. In
computer science, it can be used to calculate the probability of a data packet being lost or delayed during transmission.
Overall, a geometric probability calculator is a valuable tool for anyone working with geometric events and probability. It can save time and provide accurate results, making it an essential tool for
many professionals in various fields.
What is Geometric Probability?
Geometric probability is a branch of probability theory that deals with the probability of success in a series of independent trials, where each trial has only two possible outcomes: success or
failure. In other words, geometric probability is concerned with the probability of achieving a certain level of success after a certain number of trials.
The formula for geometric probability is as follows:
P(X=k) = (1-p)^(k-1) * p
where P(X=k) is the probability of achieving success on the kth trial, p is the probability of success on any given trial, and (1-p)^(k-1) is the probability of failure on the first k-1 trials.
An example of geometric probability in action is the probability of flipping a coin and getting heads on the first try. The probability of getting heads on any given flip is 0.5, so the probability
of getting heads on the first flip is:
P(X=1) = (1-0.5)^(1-1) * 0.5 = 0.5
Another example is the probability of rolling a six on a die for the first time on the fourth roll. The probability of rolling a six on any given roll is 1/6, so the probability of rolling a six for
the first time on the fourth roll is:
P(X=4) = (1-1/6)^(4-1) * (1/6) = 0.068
Geometric probability can be used in a variety of contexts, such as in finance, physics, and engineering. It is a powerful tool for predicting the likelihood of success in a series of independent
Geometric Distribution
Geometric distribution is a discrete probability distribution that models the number of trials needed to obtain the first success in a sequence of independent and identically distributed Bernoulli
trials. A Bernoulli trial is a random experiment with only two possible outcomes, success or failure. The geometric distribution is an example of a discrete random variable, where the outcome can
only take on non-negative integer values.
The probability mass function (PMF) of a geometric random variable X with success probability p is given by:
P(X = k) = (1 - p)^(k-1) * p
where k = 1, 2, 3, ...
The cumulative distribution function (CDF) of X is given by:
F(x) = 1 - (1 - p)^k
The expected value (mean) of X is:
E(X) = 1/p
The variance of X is:
Var(X) = (1-p)/p^2
The standard deviation of X is:
SD(X) = sqrt((1-p)/p^2)
• The geometric distribution is memoryless, meaning that the probability of success on the next trial is independent of the number of failures that have occurred so far.
• The geometric distribution is skewed to the right, with a long tail that extends infinitely to the right.
• The expected value of X increases as the success probability p decreases.
• The variance of X increases as the success probability p decreases.
• The geometric distribution can be used to model situations such as the number of coin flips needed to obtain the first head, or the number of attempts needed to make a successful free throw in
In summary, the geometric distribution is a useful tool for modeling discrete random variables that involve a sequence of independent Bernoulli trials. Its PMF, CDF, expected value, variance, and
standard deviation can all be calculated using simple formulas, making it a convenient and powerful tool for probability calculations.
Geometric Probability Calculator
What is it?
A geometric probability calculator is a tool that helps calculate the probability of success in a sequence of independent trials, where each trial has only two possible outcomes, success or failure.
The geometric probability distribution is used to model such scenarios. The calculator uses the geometric distribution formula to compute the probability of success after a certain number of trials.
How to Use it
To use the geometric probability calculator, one needs to know the value of the random variable X, which represents the number of trials needed to achieve the first success, the success probability
p, and the total number of possible outcomes.
The user should input these values into the calculator, and the tool will output the probability of success after X trials. The calculator may also provide additional information, such as the
expected value and variance of the geometric distribution.
Suppose a basketball player has a 70% success rate when shooting free throws. What is the probability that the player will make the first free throw after the 3rd attempt?
Using the geometric probability calculator, we can input X=3, p=0.7, and the total number of possible outcomes=2 (success or failure). The calculator will output the probability of success after 3
trials, which is 0.189.
Another example is when a company's website has a 5% conversion rate. What is the probability that the first conversion happens on the 10th visit?
Using the geometric probability calculator, we can input X=10, p=0.05, and the total number of possible outcomes=2. The calculator will output the probability of success after 10 trials, which is
In conclusion, the geometric probability calculator is a useful tool for calculating the probability of success after a certain number of trials in scenarios where each trial has only two possible
outcomes. By inputting the values of X, p, and the total number of possible outcomes into the calculator, one can quickly obtain the probability of success.
Geometric Probability vs. Binomial Probability
Geometric and binomial probability are two types of discrete probability distributions that are commonly used in statistics and probability theory. Geometric probability is used to calculate the
probability of a certain event occurring for the first time after a certain number of trials, while binomial probability is used to calculate the probability of a certain number of successes
occurring in a fixed number of trials.
The probability mass function (PMF) of a geometric distribution is given by:
P(X=k) = (1-p)^(k-1) * p
where p is the probability of success on any given trial, and k is the number of trials until the first success occurs.
The PMF of a binomial distribution is given by:
P(X=k) = (n choose k) * p^k * (1-p)^(n-k)
where n is the total number of trials, k is the number of successes, and p is the probability of success on any given trial.
Suppose a basketball player has a 70% success rate for free throws. What is the probability that he will make his first free throw on his third attempt? Using the geometric distribution, we have:
P(X=3) = (1-0.7)^(3-1) * 0.7 = 0.09
Now suppose we want to know the probability that he will make exactly 5 free throws out of 10 attempts. Using the binomial distribution, we have:
P(X=5) = (10 choose 5) * 0.7^5 * 0.3^5 = 0.1029
The main difference between geometric and binomial probability is that geometric probability deals with the probability of a certain event occurring for the first time after a certain number of
trials, while binomial probability deals with the probability of a certain number of successes occurring in a fixed number of trials.
In terms of expected value, variance, and standard deviation, the formulas are different for geometric and binomial probability. For geometric probability, the expected value is 1/p, the variance is
(1-p)/p^2, and the standard deviation is sqrt((1-p)/p^2). For binomial probability, the expected value is np, the variance is np(1-p), and the standard deviation is sqrt(np(1-p)).
In summary, geometric and binomial probability are two important discrete probability distributions that have different applications and formulas. It is important to understand the differences
between them in order to apply them correctly in various statistical and probability problems.
Applications of Geometric Probability
Real-life Examples
Geometric probability has numerous applications in real-life situations, including:
• Estimating the likelihood of a car accident occurring at a particular intersection
• Determining the probability of a customer returning a product to a store
• Calculating the probability of a student passing a test based on the number of attempts they make
• Predicting the chances of a certain event occurring during a given time period, such as a power outage during a storm
Mathematical Applications
Geometric probability is also widely used in mathematics, particularly in the study of independent trials. For example, it is used in:
• Finding the expected number of independent trials required to achieve a certain outcome
• Calculating the probability of a certain value occurring in a statistical distribution
• Determining the probability of a certain number of successes in a given number of independent trials
• Estimating the likelihood of a certain occurrence happening at a particular step in a mathematical equation or lesson
Overall, geometric probability is a useful tool for analyzing the likelihood of events occurring in both real-life situations and mathematical applications. By understanding the principles behind
geometric probability, students and professionals alike can make more informed decisions and predictions based on data and statistical analysis.
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日時 2017年3月21日（火）13：30〜14：30
場所 筑波大学 自然系学系D棟D814
講演者 Greg Conner 氏（Brigham Young University）
講演題目 Locally Complicated Spaces
In many settings we encounter complicated spaces which encode information about objects we study and care about.
Examples include attractors of dynamical systems, boundaries of manifolds and other spaces, compactifications of moduli spaces, boundaries of groups, asymptotic cones of groups,
self-similar tiles and other attractors or `fractals' to name a few.
These spaces tend to be \emph{locally complicated} because they have interesting topology in arbitrarily small neighborhoods of points.
A humorous, and somewhat accurate, aphorism states that a topologist is a person whose job it is to tell topological spaces apart.
One of the main tools we use to distinguish between topological spaces is the notion of a homotopy invariant.
These include the fundamental group and higher homotopy groups as well as homology and cohomology groups.
We have simple tools such as the Siefert-van Kampen theorem and exact sequences in homology and homotopy as well as covering space and fibration theory which allow us to iteratively
compute homotopy invariants for locally simple spaces such as CW-complexes.
Historically it has been very difficult to understand anything useful at all about the homotopy invariants of locally complicated spaces, let alone be able to tell them apart or compute
アブストラ them, because standard tools seem to yield very little information.
クト For instance, locally complicated spaces do not have universal covering spaces and, indeed, may not have any covering spaces.
Over the last three decades a number of authors have been working towards understanding the homotopy invariants of some of the most well-behaved locally complicated spaces.
In this talk I will discuss the both some of the history and recent progress in this area while offering numerous examples and open conjectures.
Here are some examples of the types of things I will speak about:
Did you know that locally complicated compact one dimensional spaces (e.g. the Menger sponge or the Hawaiian earring) can be `reconstructed' from their fundamental group but that their
first homologies are all the same?
Did you know that there are two, very easy to describe, compact locally connected 2-dimensional spaces but that no one knows if they have isomorphic fundamental groups?
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[Solved] 13.8. In Problem 5.8, suppose that there | SolutionInn
13.8. In Problem 5.8, suppose that there are only four machines of interest, but the operators were...
13.8. In Problem 5.8, suppose that there are only four machines of interest, but the operators were selected at random.
(a) What type of model is appropriate?
(b) Perform the analysis and estimate the model components using the ANOVA method.
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Isometric Paper Dots Printable
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10 Best Mathematics Books For Math Lovers In 2024 [Updated]
10 Best Mathematics Books For Math Lovers In 2024
Mathematics is an effective way to build mental discipline and encourages logical reasoning and mental rigor. Since maths includes many complex concepts and chapters, you need to get quality
mathematics books that will easily help you understand different arrays of this subject.
Here in this post, we have listed in detail some of the Best Maths textbooks for you.
So let’s get started!
Things To Consider While Buying A Mathematics Book
There are many books available in the market. To make your work easier, here are a few things you need to consider while buying a maths book:
• The maths textbook you choose should be based on the aim and objectives of your learning
• It should have illustrations. Reading books without illustrations is boring. Aren’t they?
• If you are buying a geometry book, then the book should include some well-explained diagrams and figures
• Last but not least, the textbook should be easy to understand
Best Mathematics Books
Here are some of the 10 best mathematics books which will help you understand the subject better.
Author: Herbert Robbins, Richard Courant, and Ian Stewart
Last Edition: 1 January 2007 (2nd Edition)
Publisher: Oxford
What is Mathematics? An elementary approach to ideas and methods is designed for beginners, scholars, students, teachers, philosophers, and engineers. The second edition of this book offers a
collection of mathematical concepts that gives exposure to the world of mathematics.
This maths book covers everything, including natural numbers, number systems, geometrical constructions, projective geometry, calculus, continuum hypothesis, and more. The maths handbook helps the
students solve problems and understand all the mathematical concepts as a whole.
Since the chapters in this book are not interdependent, the students can pick any chapter easily according to their interests. In this guide, the author explains the recent mathematical developments
made and gives proofs of the Four-color theorem and Fermat’s last theorem.
What is Mathematics? is mathematical literature that opens windows for those who want to step into the world of mathematics.
You can buy this book here.
Author: Steven Strogatz
Last Published: 2 April 2019
Publisher: Houghton Mifflin Harcourt
Infinite Powers: How Calculus Reveals The Secrets Of The Universe is a marvelous book with astonishing stories explaining how calculus has become an important part of our lives. Without Calculus, we
wouldn’t have cell phones, GPS, ultrasound, television, unraveled DNA, or discovered Neptune.
Infinite Powers explains how calculus thrilled the inventors, including ancient Greece, and discusses the discovery of gravitational waves. Strogatz also answers questions like how calculus helps
determine the area of a circle with sand and stick, explains why Mars goes backward, makes electricity with magnets, and much more.
The author, of his book, has provided clear, concise, and fascinating facts about calculus. He uses metaphors, stories, illustrations, and other interesting explanations, making understanding the
subject easier.
You can buy this book here.
Author: Matt Parker
Last Edition: 1 June 2020 (First Edition)
Publisher: Penguin
In the book Humble Pi, Matt Parker explains how our lives, including all the programs, finance, and engineering, are built on maths. Most of the time, mathematics works behind the scenes.
He also explains the glitches and mishaps on the internet, elections, lotteries, the Roman empire, and the Olympic shooting team and says how maths tricks us. Moreover, the writer reveals the
importance of maths in our lives.
He also explains that maths would be easier if we understood its practical importance. This book also states how important it is to make maths our friend. Moreover, it contains many challenges,
puzzles, jokes, geometric socks, binary codes, three deliberate mistakes we make, etc.
Humble Pi is a combination of entertaining, alarming, and eye-opening books that focuses on the numerical blunders we have been making over the years.
You can buy this book here.
Author: Darrell Huff
Last Edition: 7 December 1993 (Reissue Edition)
Publisher: W. W. Norton & Company
In the book How to lie with statistics, Darrell Huff has explained how to outsmart a crook with a few simple tricks in the classic.
From Distorted graphs to biased samples to misleading averages, many statistical concepts save us many times. In this book, Darrell explains the basic principles of statistics with the help of a
truckload of examples and detailed illustrations.
He also talks about how statistics are used to present information in both honest and not-so-honest ways. This book has been a guide for many statistic enthusiasts that have kept them from being
You can buy this book here.
Author: G. Polya
Last Edition: 28 October 2014 (Reprint Edition)
Publisher: Princeton University Press
How To Solve by G. Polya shows that anyone from any field can think straight. The author states the importance of using mathematical methods to demonstrate proof can help solve any problem that can
be “reasoned.”
This book is suitable for people who want to build a career in science and mathematics and for people who are really interested in solving mathematical problems. Moreover, How to solve is also a book
recommended for all the teachers.
The major message of this book is that- solving mathematical problems requires a lot of practice and experience. It also improves creative thinking. From the time this book was published till today,
How to solve it has become a major inspiration for many generations.
You can buy this book here.
Author: Silvanus P. Thompson, Martin Gardner
Last Edition: 15 October 1998 (Fourth Edition)
Publisher: St. Martin’s Press
Calculus Made Easy by Silvanus P. Thompson, and Martin Gardner is the most popular calculus book. This mathematics book is a comprehensible book that explains calculus and is recommended for readers
of all levels.
The latest edition of the book includes:
• A new and detailed introduction.
• Three new and updated chapters.
• Easy-to-understand language.
• An appendix.
• Challenging practice problems.
This textbook is recommended for all modern readers.
Calculus Made Easy is considered to be a teacher to many. The author has also explained various concepts through a remarkable and user-friendly approach which helps to understand calculus better.
After it was published, the authors became very popular and were known as the greatest intellects.
You can buy this book here.
Author: Steven Strogatz
Last Edition: 1 October 2013 (Reprint Edition)
Publisher: Mariner Books
The Joy of X by Steven Strogatz has been listed in the New York Times series. This mathematics book explains maths clearly with witty, insightful, and brilliant illustrations. He also explains the
importance of maths in our lives.
Strogatz also shows how maths is interlinked with every aspect of our lives. He discusses pop culture, law, philosophy, art, business, medicine, etc.
This book is an exploration of the beauty and fun of mathematics. It will amaze, entertain, and make you smarter. The author introduces the readers to the concepts of mathematics, explaining the
reasons for its unfamiliar language and explaining the conceptual framework that helps to understand difficult problems easily.
You can buy this book here.
Author: Ian Stewart
Last Edition: 8 October 2013 (Illustrated Edition)
Publisher: Basic Books
In Pursuit of the unknown celebrated the most popular mathematician Ian Stewart. He discloses the roots of the important maths statements to show that equations are the driving force behind every
aspect of our lives.
This book includes the seventeen most important equations. It includes the Waves equations. Waves equations help engineers measure the building’s response to earthquakes that saved thousands of
lives. Ian also explains the Black Scholes model used to track down the price of financial derivations.
The author also illustrates that many advancements in technology and other industries have led to many new and interesting mathematical discoveries. In Pursuit of the unknown is considered the most
lively, informative, approachable guide to mathematics in our modern life.
You can buy this book here.
Author: Sheldon Axler
Last Edition: 5 November 2014 (3rd Edition)
Publisher: Springer
Linear Algebra Done Right is the best-selling algebra textbook for the course in linear algebra for undergraduate mathematics majors and graduate students. This best math textbook teaches students
the structure for linear operators on finite-dimensional vector spaces.
The author has mentioned the concepts and proofs in a simplified manner. It also includes interesting exercises for each chapter, which helps students understand linear algebra concepts. The initial
chapters in the book include vector spaces, span, basis, dimension, linear independence, and more. Then it explains linear maps, eigenvectors, eigenvalues, etc.
The third edition of this book has some major improvements and revisions. The latest version contains 300+ new exercises, new examples explaining linear algebra in detail. Various new concepts
mentioned in this book include product spaces, dual spaces, and quotient spaces.
You can buy this book here.
Author: Walter Rudin
Last Edition: 1 July 2017 (Third Edition)
Publisher: McGraw Hill Education
Principles of Mathematical Analysis by Walter Rudin is a comprehensive guide that consists of eleven chapters. It covers all the concepts that relate to mathematical analysis, including the normal,
real, complex fields, sets, real numbers, and Euclidean spaces.
There are also practical exercises given at the end of every chapter and an appendix containing all the references. The book covers properties of space such as topology and series and sequence of
The other main contents are the subject matter of function and its series, differentiation, variables, continuity, and much more.
You can buy this book here.
Mathematics is a vital part of our lives, it is important to understand it. The subject is included in the school curriculum for students to build a foundation from an early age. Maths assists
learners to gain knowledge and enhance their creative thinking. It helps to mold the imagination power of students into a calculative one.
Mathematics helps to understand new perspectives of life. We use this subject in our day-to-day lives. From big tasks such as construction to everyday chores like grocery shopping, maths is an
essential part of our lives.
If you are a curious learner of this subject or it’s your mainstream learning area, then you must expand your horizons by reading good mathematics books.
In this article, we presented the 10 best mathematics books available for you.
We hope that the information provided in this article will help you to make an informed decision.
Q.1 Is math tough to understand?
Ans. Maths is not tough. However, you can learn and understand it through thorough practice and consistency.
Q.2 Can I learn math with the help of books?
Ans. Yes, you can. We have presented some of the best math books that you can take help from.
Q.3 Is it feasible to understand mathematics online?
Ans. Yes, a lot of students follow online channels and take courses. It is a great way of learning.
Q.4 What are the best books for math?
Ans. You can opt for the math books written by some great authors such as RD Sharma, RS Aggarwal, G. Polya, Walter Rudin, and many more.
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Physics Print
Physics is the study of motion. In this course, students will learn about motion in one and two dimensions, rotational motion, force, power, momentum, and energy. Students will explore topics such as
thermodynamics, optics, and wave mechanics. Finally, they will gain a basic understanding of atomic subatomic physics, electricity, and magnetism.
Students will be provided with Physics
Textbook ISBN:
MHID: 0076774767
ISBN 13: 9780076774760
When you have completed this course, you will be able to:
1. Apply the scientific method.
2. Interpret graphs to describe motion.
3. Create free-body diagrams.
4. Identify relationships between variables.
5. Apply Newton’s laws of motion.
6. Add vectors graphically and algebraically.
7. Solve motion problems in two dimensions.
8. Apply Kepler’s laws.
9. Apply law of conservation of momentum, angular momentum, and energy.
10. Find inertial and gravitational masses.
11. Analyze collisions.
12. Determine the potential or kinetic energy of a system.
13. Apply the laws of thermodynamics.
14. Apply the combined gas law and the ideal gas law.
15. Utilize Pascal’s principle, Archimedes’ principle, and Bernoulli’s principle to fluids.
16. Solve simple harmonic motion problems.
17. Compare and analyze various types of waves.
18. Solve illumination, mirror, and lens problems.
19. Apply Snell’s law.
20. Compare and contrast conductors, insulators, and semiconductors.
21. Apply Coulomb’s and Ohm’s laws.
22. Analyze electrostatic forces.
23. Diagram circuits in parallel and series.
24. Determine the electric potential, power, capacitance, EMF, and resistance in a system.
25. Analyze the relationship between magnetic fields and electric currents.
26. Compare the electromagnetic wave theory to the particle theory.
27. Apply the Heisenberg uncertainty principle.
28. Describe the structure of the atom.
29. Apply the band theory of solids.
30. Analyze radioactive decay
Course Outline
Chapter 1: A Physics Toolkit
Chapter 2: Representing Motion
Chapter 3: Accelerated Motion
Chapter 4: Forces in One-Dimension
Chapter 5: Displacement and Force in Two Dimensions
Chapter 6: Motion in Two Dimensions
Chapter 7: Gravitation
Chapter 8: Rotational Motion
Chapter 9: Momentum and Its Conservation
Chapter 10: Work, Energy, and Machines
Chapter 11: Energy and Its Conservation
Chapter 12: Thermal Energy
Chapter 13: States of Matter
Chapter 14: Vibrations and Waves
Chapter 15: Sound
Chapter 16: Fundamentals of Light
Chapter 17: Reflection and Mirrors
Chapter 18: Refraction and Lenses
Chapter 19: Interference and Diffraction
Chapter 20: Static Electricity
Chapter 21: Electric Fields
Chapter 22: Electric Current
Chapter 23: Series and Parallel Circuits
Chapter 24: Magnetic Fields
Chapter 25: Electromagnetic Induction
Chapter 26: Electromagnetism
Chapter 27: Quantum Theory
Chapter 28: The Atom
Chapter 29: Solid-State Electronics
Chapter 30: Nuclear and Particle Physics | {"url":"https://courses.keystoneschoolonline.com/Physics-Print?offering=2","timestamp":"2024-11-14T21:58:11Z","content_type":"text/html","content_length":"45269","record_id":"<urn:uuid:18b5d367-2885-4bad-b5cf-0dfddd5ee92a>","cc-path":"CC-MAIN-2024-46/segments/1730477395538.95/warc/CC-MAIN-20241114194152-20241114224152-00312.warc.gz"} |
nice simplicial topological space
Homotopy theory
homotopy theory, (∞,1)-category theory, homotopy type theory
flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed…
models: topological, simplicial, localic, …
see also algebraic topology
Paths and cylinders
Homotopy groups
Basic facts
A nice simplicial topological space is a simplicial topological space that satisfies certain extra properties that make it well behaved in homotopy theory, notably so that its geometric realization
of simplicial spaces is its homotopy colimit.
Let $X : \Delta^{op} \to Top$ be a simplicial topological space.
Such $X$ is called
• good if all the degeneracy maps $X_{n-1} \hookrightarrow X_n$ are all closed cofibrations;
• proper if the inclusion $s X_n \hookrightarrow X_n$ of the degenerate simplices is a closed cofibration, where $s X_n = \bigcup_i s_i(X_{n-1})$.
In other words this says: $X_\bullet$ is proper if it is cofibrant in the Reedy model structure $[\Delta^{op}, Top_{Strom}]_{Reedy}$ on simplicial objects with respect to the Strøm model structure on
The notion of good simplicial topological space goes back to (Segal 1973), that of proper simplicial topological space to (May).
A good simplicial topological space is proper.
A proof appears as Lewis, corollary 2.4 (b). A generalization of this result is in RobertsStevenson.
For $X_\bullet$ any simplicial topological space, then ${|Sing X_\bullet|}$ is good, hence proper, and the natural morphism
${|Sing X_\bullet|} \to X_\bullet$
is degreewise a weak homotopy equivalence.
This follows by results in (Lewis).
Since for $X \in Top$ the map $|Sing X| \to X$ is a cofibrant resolution in the standard Quillen model structure on topological spaces, we have that
$|Sing X_\bullet| \to X_\bullet$
is a degreewise weak homotopy equivalence. In particular each space $|Sing X_n|$ is a CW-complex, hence in particular a locally equi-connected space. By (Lewis, p. 153) inclusions of retracts of
locally equi-connected spaces are closed cofibrations, and since degeneracy maps are retracts, this means that the degeneracy maps in $|Sing X_\bullet|$ are closed cofibrations.
Models for the homotopy colimit
That the geometric realization of simplicial topological spaces of a proper simplicial space is is homotopy colimit follows from the above fact that proper spaces are Reedy cofibrant, and using the
general statement discussed at homotopy colimit about description of homotopy colimits by coends.
The definition of proper simplicial space goes back to
• Peter May, The Geometry of Iterated Loop Spaces , Lecture Notes in Mathematics, 1972, Volume 271(1972), 100-112 (pdf)
May originally said strictly proper for what now is just called proper .
The definition of good simplicial space goes back to
The implication $good \Rightarrow proper$ seems to be a folk theorem. Its origin is maybe in
• L. Gaunce Lewis, Jr., When is the natural map $X \to \Omega \Sigma X$ a cofibration? , Trans. Amer. Math. Soc. 273 (1982), 147–155.
A generalization of the statement that good implies proper to other topological concrete categories and a discussion of the geometric realization of $W G \to \bar W G$ for $G$ a simplicial
topological group is in
Comments on the relation between properness and cofibrancy in the Reedy model structure on $[\Delta^{op}, Set]$ are made in | {"url":"https://ncatlab.org/nlab/show/nice+simplicial+topological+space","timestamp":"2024-11-09T03:54:49Z","content_type":"application/xhtml+xml","content_length":"58710","record_id":"<urn:uuid:28ac3edd-2179-4472-9b8b-9c34137fc56b>","cc-path":"CC-MAIN-2024-46/segments/1730477028115.85/warc/CC-MAIN-20241109022607-20241109052607-00214.warc.gz"} |
Role of the Number of Adsorption Sites and Adsorption Dynamics of Diffusing Particles in a Confined Liquid with Langmuir Kinetics
Department of Physics, Universidade Estadual de Maringá Avenida Colombo 5790, Maringá 87020-900, PR, Brazil
Department of Physics, Universidade Tecnológica Federal do Paraná, Rua Marcílio Dias 635, Apucarana 86812-460, PR, Brazil
Department of Physics, Universidade Estadual de Ponta Grossa, Ponta Grossa 87030-900, PR, Brazil
Author to whom correspondence should be addressed.
Submission received: 31 August 2022 / Revised: 21 November 2022 / Accepted: 15 December 2022 / Published: 20 December 2022
In this work, we investigate the effect of the number of available adsorption sites for diffusing particles in a liquid confined between walls where the adsorption (desorption) phenomena occur. We
formulate and numerically solve a model for particles governed by Fickian’s law of diffusion, where the dynamics at the surfaces obey the Langmuir kinetic equation. The ratio between the available
number of adsorption sites and the number of total particles are used as a control parameter. The investigation is carried out in terms of characteristic times of the system for different initial
configurations, as well as the cases of identical or non-identical surfaces. We calculate the bulk and surface densities dynamics, as well as the variance of the system, and demonstrate that the
number of sites affects the bulk, surface distributions, and diffusive regimes.
1. Introduction
The kinetics of diffusing particles in confined space with adsorption (desorption) by solid substrates represent an important class of problems that is readily applied from basic sciences to
industrial separation processes [
]. One basic approach to model adsorption–desorption phenomena is to use the simple, yet very powerful, Langmuir adsorption [
], often referred to as Langmuir kinetics [
]. At the same time, the diffusion process is described by Fick’s law [
]. Indeed, in any practical application where any transport takes place, adsorption–desorption is likely to occur, even in very simple geometries [
]. However, from an analytical point of view, it is not easy to solve the coupled equations representing diffusion and Langmuir’s kinetic altogether. In many cases, due to the nature of adsorbents,
first- and second-order kinetic equations are employed to analyze experimental data [
]. Such equations account, for example, for diffusion within the adsorbing wall [
]. In other cases, a linearization process is performed to the Langmuir kinetic equation, in which the number of available sites for adsorption is much larger than the number of particles in the
system [
]. Such an approach is convenient for studying the relation between diffusing particles in bulk samples adsorbed by confining walls.
Several systems of interest have been studied within the scope of the linearized Langmuir kinetic equation coupled to the diffusion equation. For example, it was used to study general aspects of bulk
and surface dynamics [
], characterize diffusion in different geometries [
], to probe time-dependent diffusion coefficients [
], to insert memory effects and distinguish adsorption kind [
], to study adsorption and diffusion in systems with non-identical surfaces [
] and in systems with augmented surfaces [
], analyze adsorption in systems with space-dependent diffusion coefficient [
], and to study adsorption effects in electrolyte cells in the context of impedance spectroscopy [
]. However, in the above cases, the importance of the number of particles to the number of adsorbing sites ratio is neglected.
This article aims to provide a simple yet general model to study the diffusion of neutral particles in an isotropic liquid confined by adsorbing (desorbing) walls that obey Langmuir’s kinetic. In
this sense, a broad spectrum of phenomena arises by combining diffusion and a limited number of adsorbing sites, including how the bulk and surface dynamics take place and the effect of Langmuir’s
kinetic on diffusive regimes on the bulk. Furthermore, we demonstrate how memory effects can be included within Langmuir’s kinetic equation to account for more general scenarios related to the number
of particles, number of adsorbing sites, and the dependence of the previous state of the particle on the next one.
2. Model
Since we want to study diffusing particles diluted in a liquid in contact with adsorbing (desorbing) walls, we assume a sample in the shape of a slab where the only relevant direction is the
−axis. Such geometry is similar to experimental situations such as those found in liquid crystal displays, for example, where the adsorption and desorption of particles is known to play critical
roles in the organization of the material [
]. The cell has thickness
, and the substrates are located at
$z = − L / 2$
(left side) and
$z = + L / 2$
(right side), as shown in
Figure 1
. Particles in bulk obey the diffusion equation, here assumed to be diluted so the Fickian approach can be used, that is
$∂ ρ ∂ t = ∂ ∂ z D ∂ ρ ∂ z ,$
is the diffusion coefficient, here assumed to be constant in space, and
$ρ ( z , t )$
is the time- and space-dependent bulk density of particles. At the walls limiting the sample, we assume that for
$t = 0$
, all the particles are in the bulk (all adsorption sites are vacant), and that particles may be adsorbed–desorbed according to the well-known Langmuir’s kinetic equation:
$d σ i d t = κ a i ρ ( 1 − σ i σ 0 i ) − κ d i σ i .$
In Equation (
), the subscript
$i = ± L / 2$
represents the set of parameters characterizing either the surface on the left side or the surface on the right side of the sample. Moreover,
$σ i ( t )$
is the density of adsorbed particles and
$κ a i$
is a parameter representing the rate of adsorption, while
$κ d i$
represents the rate (or time) of desorption. Furthermore,
$σ 0 i$
is the number of adsorption sites available at surface
. The Langmuir kinetics assumes that the adsorption rate
$( d σ i / d t$
) is proportional to the difference between adsorption and desorption rates and that both rates are the same at equilibrium.
Figure 1
shows the system studied here, including the occupied and free sites at the walls.
Notice that if we consider an infinity number of sites available for adsorption (
$σ 0 i → ∞$
), we recover the linearized version previously used in different works [
]. Moreover, if the total density of particles is
$ρ 0$
, and are initially in the bulk, i.e.,
$ρ ( z , t = 0 ) = ρ 0$
, we can introduce the normalized quantities
$ρ R = ρ / ρ 0$
$σ R i = σ i / σ 0 i$
, and, at equilibrium, arrive at the Langmuir isotherm:
$σ R i = α i ρ R 1 + α i ρ R ,$
$α i = κ a i ρ 0 / ( κ d i σ 0 i )$
is a parameter governing the steady-state equation for each substrate. The set of Equations (
) and (
) are solved with the aid of the current density at the walls, that is,
$D ∂ ρ ∂ z z = ± L / 2 = ∓ d σ i d t ,$
which implies that the number of particles must be conserved, that is
$2 σ i + ∫ − L / 2 L / 2 ρ d z = ρ 0 L .$
To account for memory effects, we modify Equation (
). We introduce a kernel dependence as follows:
$d σ i d t = κ a i ρ ( 1 − σ i σ 0 i ) − κ d i ∫ 0 t K ( τ ) σ i ( τ ) d τ ,$
$K ( τ )$
is a kernel introduced to represent distinct scenarios related to the adsorption-desorption phenomena. An equation such as Equation (
) has been widely used to describe memory effects and non-Debye relaxations [
]. It has been used to describe chemisorption, physisorption, or a combination of both depending on the choice of
$K ( τ )$
]. Although we introduced the kernel in the desorption term of Equation (
), it is important to stress that a memory effect represents that the previous state of the particle is important for the next state, so the kernel modifies the whole dynamics of the surface.
Unfortunately, the process of solving Equations (
), (
) and (
) is difficult and does not have an analytical solution, so we employ a numerical method to solve it (see, for example, refs. [
]). We first introduce reduced quantities in such a way that Equation (
) becomes
$∂ ρ R ∂ t * = ∂ 2 ρ R ∂ Z 2 ,$
$ρ R = ρ / ρ 0$
$Z = 2 z / L$
$t * = 4 t / τ D$
, and
$τ D = L 2 / D$
is the diffusion time. Regarding Equation (
), we first notice that the parameter
$1 / κ d i$
has the dimension of time; thus, we write it as
$τ i$
$i = ± 1$
, which we call either
, meaning left or right side, respectively) from now on. Second, we have to choose the form of the kernel
$K ( t )$
to proceed with the solution procedure. As proposed in [
], the form
$K ( t ) = 1 / ( τ a i ) e − t / τ a i$
, which is a non-localized function of time, is an excellent choice to represent memory effects, such as may occur during multiple collisions of particles with the surfaces, in which energy is lost
after each collision, and therefore the previous state of the particle is important in determining the next state. This memory time, represented by
$τ a i$
, reproduces the adsorption–desorption phenomena often found in physisorption or mixed processes [
]. If
$τ a i → 0$
, we recover the original kinetic equation, Equation (
), which is better suited to describe chemisorption processes. Numerically, it is more convenient to work with differential equations than with integral equations. We therefore take the time
derivative on both sides of Equation (
) and apply the kernel
$K ( t ) = 1 / ( τ a i ) e − t / τ a i$
to arrive at:
$d 2 σ R i d t * 2 − ( 1 − σ R i ) τ D β i 2 τ κ i ∂ ρ R ∂ t * + ρ R τ D 2 β i 8 τ κ i τ a i + d σ R i d t * ρ R τ D β i 2 τ κ i + τ D τ a i + τ D 2 σ R i 16 τ a i τ i = 0 ,$
$τ κ i = L / 2 κ a i$
is the adsorption time and
$β i = ρ 0 L / σ 0 i$
is a parameter that relates the number of particles available in bulk to be adsorbed to the number of sites available at the surfaces. Thus, if
$β i < 1$
, the specific surface has more sites than particles to be adsorbed, whereas if
$β i > 1$
, there are not enough sites at the surface for all the particles in bulk. The conservation of the number of particles, Equation (
), now becomes
$σ R l β l + σ R r β r + 1 2 ∫ − 1 1 ρ R d Z = 1 ,$
which also implies:
$2 β i d σ R i d t * = ± ∂ ρ R ∂ Z Z = ∓ 1 .$
A summary of the main parameters and a brief description is given in
Appendix A
. Now, in order to solve the set of Equations (
) (or (
)), we employ a numerical method based on finite differences [
]. We use a mesh with
$n z$
points separated by a fixed distance
$δ z = L / ( n z − 1 )$
. All derivatives were taken using a central difference with a second-order approximation, using ghost points at the borders. Thus, including the two walls, it results in a system of ordinary
equations in
$t *$
, with
$n z + 4$
equations. The time integration was performed with the 8th-order Dormand–Prince method implemented by the GSL library [
]. To ensure that the simulations were performed without instabilities, we monitored the conservation of the number of particles during each small increment in time by checking if Equation (
) is still satisfied. Through all simulations, we used
$n z = 500$
, and set the absolute error to be smaller than
$10 − 9$
, allowing a free time step size, and retrieved data at every
$Δ t * = 10 − 6$
. Moreover, we assumed the initial surface density for both walls was zero (
$σ R i ( t * = 0 ) = 0 )$
. For the initial distribution in bulk, we used two different initial configurations: 1—particles are uniformly distributed across the cell (
$ρ R ( Z , t * = 0 ) = 1$
), or 2—particles are initially concentrated in a plane set in the center of the sample, that is, the initial bulk distribution obeys a Dirac delta configuration. Finally, the characteristic times
used in this work come from experimental data of similarly confined systems [
]. The adsorption parameter,
$κ a i$
, is usually in the order of
$10 − 6$
$− 1$
]. At the same time, the desorption time was estimated to be nearly 0.01 s for liquid crystal samples and close to 1 s for other isotropic samples [
]. A typical slab sample such as the one studied here is around 10 μm thick, while the diffusion coefficient is in the order of
$D ≈ 10 − 11$
, so
$τ D ≈ 10$
s. Last, the memory time was estimated from experimental results to be
$τ a ≈ 1$
s [
]. It is important to notice that, as long as the system obeys Fick’s law and Langmuir’s kinetic, it in principle can be investigated within the scope of this model.
3. Results and Discussion
We start by analyzing how bulk and surface distributions change over time as the characteristic parameters of the system change. In particular, we want to understand how the parameter
, which gives the ratio between particles available to be adsorbed and the number of sites available for adsorption, affects the bulk distribution of particles. Indeed, previous attempts to model
adsorption–desorption with diffusing particles in a limited system [
] have used a kinetic equation that considers the number of sites to be infinity, so we can check here the importance of having a limited number of sites on the dynamics of the system. Furthermore,
in our model, we can treat both surfaces as being non-identical; that is, each surface has its dynamics, so the model becomes more closely related to experimental situations (where it is difficult to
assure both surfaces are completely identical) and to other cases where adsorption is present and substrates are non-identical, such as in fuel cells (hybrid microfluidic fuel cells, for example) [
], hybrid aligned liquid crystal cells [
], wetting layers [
], liquid crystals doped with dyes [
], and polymer adsorption in confined regions [
We start by showing a simple example where the role played by the parameter
can be clearly understood. For this first case, we consider both surfaces to be identical (so
$σ R l = σ R r$
$β l = β r$
and so on) and that the initial bulk distribution is uniform, i.e.,
$ρ R ( Z , t * = 0 ) = 1$
Figure 2
a reproduces the surface density (in
$Z = − 1$
) vs.
$t *$
$τ D / τ l = 100$
$τ κ l / τ l = 10$
and for two values of
$τ a l / τ l$
, that is, 0.1 or 20, meaning in the first case very short memory time and in the second case a long memory time, which results in the oscillations seen during the adsorption phenomena [
]. As discussed elsewhere [
], memory effects occur when the adsorption process depends on the previous state of the particle being adsorbed. For example, if a particle falls into an adsorbing well, it may be desorbed and
eventually adsorbed again. Since the energy landscape changes after each process, the next phenomenon depends on the previous state, which in such a dynamic model it is represented by the parameter
$τ a$
]. We used two different values of
, 0.2 and 2, meaning in the first case there are five times more sites on the surfaces to adsorb than particles to be adsorbed, while in the second case, there are two times more particles in bulk to
be adsorbed than available sites. Since
$τ κ i < τ i$
, the desorption rate is larger than adsorption, so the equilibrium density is fairly low. However, in the case of
$β > 1$
, the surfaces reach a larger value, as expected (see Equation (
)). It is also interesting to note that the adsorption peak position (
$t *$
), related to memory effects, also changes depending on
, which is indicative that memory effects are also sensitive to the ratio between the number of particles to the available number of adsorbing sites. To the best of our knowledge, this is the first
time memory effects are considered in Langmuir kinetics. The inset shows the bulk density for
$t * = 0.6$
, indicating that for larger
there is a tendency of larger bulk density at all times.
Figure 2
b shows the left-side surface density for
$τ D / τ l = 100$
$τ κ l / τ l = 1$
$τ a l / τ l = 0.1$
and several values of
. Since the adsorption and desorption rates are the same, the surface coverage is more extensive and determined by the parameter
. For example, if the number of particles is ten times larger than the number of sites, the surfaces are nearly filled to maximum once the equilibrium is achieved. Clearly, the surface density for
$β = 10.0$
reaches saturation quicker when compared with
$β = 0.2$
We now start investigating the cases in which the surfaces are not identical. This is particularly interesting because one can see how one surface affects the other and the bulk distribution.
Moreover, it is helpful to understand the effect of $β$ on such distributions if the substrates are seen individually. For our study, we fixed the left substrate ($Z = − 1$) with the following
parameters: $β l = 1.0$, $τ κ l / τ l = 1$ and $τ a l / τ l = 0.01$. Furthermore, we kept the diffusion time the same for all the analyses, that is, $τ D / τ l = 10.0$.
Figure 3
a shows a case in which all the parameters are the same for both substrates, except the parameter
. We chose
$τ D / τ i = 10.0$
$τ κ i / τ i = 1.0$
$τ a i / τ i = 1.0$
$β l = 1.0$
, and varied
$β r$
. The substrate on the left side (parameters are always the same) is shown in the main figure, while the inset shows the substrate in
$Z = 1$
. As
$β r$
gets larger, the coverage in
$Z = 1$
grows, which is expected since larger
represents more particles to be adsorbed compared with the number of sites on that specific surface. The left-side surface also saturates at higher concentration values when the parameter
of the right-side surface increases. Since the right side has fewer sites for adsorption (as
increases), the left-side surface has more particles to adsorb, since
$β l = 1.0$
. This trend can also be understood by Equation (
). If
$β r = 0.2$
$σ R l = 1 − 5 σ R r − bulk / 2$
, whereas for
$β r = 10.0$
$σ R l = 1 − 0.1 σ R r − bulk / 2$
, where “bulk” stands for the final concentration in the bulk. Since the number of particles in bulk is never larger than 1,
$σ R l$
saturates at higher values as
$β r$
increases. In
Figure 3
b, we keep the surface in
$Z = − 1$
with the same parameters as in
Figure 3
a but used the parameters
$τ κ r / τ r = 10$
$τ a r / τ r = 20$
$β r = 0.2$
$β r = 2.0$
. Notice that the adsorption time is ten times larger than the desorption time, so the overall coverage is low for the right surface. In this case, changing
$β r$
does not affect the left surface. Thus, for
$β r = 0.2$
, the surface has five times more sites than particles available, and the resulting coverage is small. If
$β r = 2.0$
, the right surface has room for only half of the particles, so the coverage is higher. Moreover, since the memory time is longer, the right surface displays oscillations in the adsorption–desorption
process. The inset shows the bulk density for three different times, where the black curves represent
$β r = 0.2$
and the red curves
$β r = 2.0$
. In the initial moments, both cases are identical. As time passes, for
$t * = 0.5$
, the bulk density near the right surface for
$β = 2.0$
is slightly higher because the right surface has fewer sites to adsorb, so there is an excess of particles when compared with the case where
$β r = 0.2$
. As time increases, the bulk densities for the two values of
become very similar, being only slightly higher for
$β r = 0.2$
due to the limited capability of the right surface to adsorb the particles.
To demonstrate that a larger
represents a lower overall density of particles compared with lower
cases, in
Figure 4
, we show the density of adsorbed particles for both surfaces normalized by their respective
. For this figure, we used the same parameters for the left surface as in
Figure 3
. For the right surface, we used
$τ κ r / τ r = 0.01$
$τ a r / τ r = 0.01$
, so the right surface has a high adsorption rate and neglectable memory effects. As expected, for both
$β r = 0.2$
$β r = 2.0$
, the right surface reaches higher coverage values when compared with the left side, which is a consequence of the high adsorption rate. However, when
$β r = 2.0$
, both surfaces reach the equilibrium (the left coverage is still slowly decreasing for
$t * = 3.5$
) much more quickly when compared with
$β r = 0.2$
, which is due to the quick filling of the available sites when
$β r = 2.0$
. On the other hand, for
$β r = 0.2$
, the system is still evolving (filling the adsorption sites) when
$t * = 3.5$
Figure 4
clearly shows that the equilibrium point depends on the number of available sites rather than just adsorption and desorption rates. The inset of
Figure 4
shows the bulk dynamics three different times for both values of
. Initially, when most particles are still in bulk, the distribution for both cases is the same. For
$t * = 1.0$
, the distribution near the left side is the same, whereas, near the right surface, the bulk distribution is lower for
$β r = 0.2$
when compared with
$β r = 2.0$
. As the system continues evolving, the case for
$β r = 2.0$
quickly reaches an equilibrium value, which is larger when compared with
$β r = 0.2$
, where even for
$t * = 5.0$
, the bulk distribution is still not at equilibrium (which is also seen from the surface curves).
Now, we use the mean square displacement (MSD) to probe how the diffusive regimes affect the number of sites at the surfaces. This is an important parameter because it is related to the spreading of
particles across the cell, which characterizes the time-dependent diffusion coefficient,
$D ( t )$
. Thus, we are now interested in understanding how the number of sites, hence the parameter
, changes the way particles diffuse in bulk. Indeed, it is known that limiting surfaces may affect how particles diffuse in the bulk [
], which is of particular importance mainly for transport in living cells [
], but in general, the number of sites is neglected during the modeling. The MSD, which takes into account the conservation of particles (bulk distribution does not remain normalized at all times as
particles leave the volume to be adsorbed at the walls), as in Equation (
), is given by
$( Δ Z ) 2 = 〈 Z − 〈 Z 〉 2 〉 = 〈 Z 2 〉 − 2 σ 〈 Z 〉 2$
. In other words, the MSD is calculated as:
$( Δ Z ) 2 = ∫ − 1 1 Z 2 ρ ¯ ( Z , t * ) d Z − 2 σ R i ( t * ) / β i ∫ − 1 1 Z ρ ¯ ( Z , t * ) d Z 2 .$
Figure 5
, shows the MSD vs. time for different values of the parameter
for the situations in which the surfaces are identical and the case where the surfaces are non-identical. For all cases, we used
$τ D / τ l = 100$
. For the case of identical surfaces, depicted by the solid lines, we used
$τ k / τ = 1$
$τ a / τ = 20$
. Notice that when the MSD increases with time, the bulk distribution is spreading up, while MSD decreasing with time means that the distribution is shrinking; that is, particles are returning to the
bulk. We first notice that for all the curves, the initial behavior of the MSD is equal, which corresponds to the initial spreading of particles from the center of the cell toward the surfaces. This
initial diffusing process is not affected by the surfaces and corresponds to the usual diffusion that occurs for boundless samples. The black solid line represents
$β i = 0.2$
, while the solid red line shows
$β i = 5.0$
, and the solid blue curve depicts the case
$β i = 10.0$
. For
$β i = 0.2$
, after the initial free diffusion, the MSD decreases until
$t * ≈ 0.6$
. This is a consequence of the desorption process that takes place together with the adsorption process (
$τ κ = τ$
), which is favored by the small number of particles remaining in bulk. After this minimum occurs for the MSD, the adsorption process continues, favored by the larger number of sites compared with
particles in bulk, until an equilibrium is reached. For
$β i = 5.0$
, this minimum happens much faster and is much less pronounced. At the same time, for
$β i = 10.0$
, it does not appear at all, which is a consequence of a large number of particles compared with the available sites, resulting in a much weaker desorption process of the high particle concentration
in bulk at all times. The dashed lines in
Figure 5
a represent the same values of the parameter
but for non-identical walls. Here, the right surface is the same as used to produce the solid curves, only
$β r$
is changed, but the left surface is fixed at
$β l = 1.0$
$τ κ l / τ l = 1$
$τ a l / τ l = 0.01$
. Interestingly, the bulk dynamic is also affected by the non-identical surface, not only by changing how the spreading and shrinking of the distribution occurs but also by changing the overall
amount of particles to reach equilibrium in bulk, a direct consequence of the ratio between the number of particles and number of sites.
Figure 5
b shows the cases in which
$τ κ / τ = 0.01$
, when the surfaces are identical, and for the right surface (
$Z = 1$
) in the case of non-identical surfaces. The left surface for non-identical surfaces remains the same as
Figure 5
a. Here, since the rate of adsorption is much larger than the rate of desorption, it is expected that the surface quickly adsorbs most of the particles, leaving the bulk with a low concentration.
This is true for
$β = 0.2$
, where the MSD is nearly zero for
$t * ≈ 2$
, meaning that the distribution continuously decreases where most of the particles are trapped at the surfaces. However, as the number of particles increases when compared with the number of
available sites, this is no longer true, which significantly affects how diffusion takes place in the bulk, indicating that indeed the parameter
is essential, rather than just looking at the rates of adsorption and desorption as usually performed. This is true for both identical and non-identical cases.
Figure 5
b shows the cases in which
$τ κ / τ = 0.01$
, when the surfaces are identical, and for the right surface (
$Z = 1$
) in the case of non-identical surfaces. The left surface for non-identical surfaces remains the same as
Figure 5
a. Here, since the rate of adsorption is much larger than the rate of desorption, it is expected that the surface quickly adsorbs most of the particles, leaving the bulk with a low concentration.
This is true for
$β = 0.2$
, where the MSD is nearly zero for
$t * ≈ 2$
, meaning that the distribution continuously decreases, where most of the particles are trapped at the surfaces. However, as the number of particles increases when compared with the number of
available sites, this is no longer true, which significantly affects how diffusion takes place in the bulk, indicating that indeed the parameter
is essential, rather than just looking at the rates of adsorption and desorption as usually performed. This is true for both identical and non-identical cases. Notice how the dynamics of the MSD
closely resemble the data observed for the diffusive characteristics observed in some systems that are essentially anomalous and with limited amount of adsorption sites to diffusing particles, such
as in living cells [
] and in the diffusion of gold-labeled dioleoylPE in the plasma membrane of fetal rat skin keratinocyte cells [
Finally, noticing how the diffusive regimes change with the parameter
is interesting. Accordingly, (
$Δ Z$
$2 ∼ t * a$
, where the power
is related to how particles diffuse, that is, if
$a < 1$
, the diffusive regime is said to be subdiffusive, if
$a = 1$
, it is called usual, whereas for
$a > 1$
, the diffusive regime is called superdiffusive. Usually, confined systems are subjected to adsorption–desorption phenomena. However, the ratio
is not considered when studying the diffusive regime. We observe that in the initial moments before particles reach the surfaces, the diffusion is usual, that is,
$a = 1$
. Nonetheless, after interacting with the surfaces, the diffusion becomes essentially subdiffusive and is heavily affected by the parameter
. In
Figure 5
a, we show, as dotted lines, a few examples of exponents
for certain time intervals in which the distribution spreads after a quick desorption process. For example, for
$β = 0.2$
$a = 0.52$
, while for
$β = 5.0$
$a = 0.80$
, indicating that
changes the diffusion regime. As it is well-known, molecular crowding is one source of anomalous diffusion, especially in biological fluids [
], which, within this model, may be altered by changing the parameter
. This fact shall be further explored in future studies.
In conclusion, we modeled a system composed of an isotropic liquid limited by adsorbing surfaces where particles diffuse and may be adsorbed/desorbed following the complete Langmuir’s kinetic
equation. In the modeling process, we incorporated, in addition to the natural parameters arising from the Langmuir kinetic equation such as the ratios of adsorption, desorption, and number of
available sites, memory effects that may occur depending on the inherent nature of adsorbing walls, such as occurring during chemisorption, physisorption, or a mixed process. We scaled all the
variables in terms of characteristic times of the system and kept as main parameter the ratio between the available sites to the amount of particles in the bulk. It turns out that not only the
dynamics but the diffusive regimes are heavily affected by such ratio, which is often neglected during adsorption dynamics. We hope our results may be helpful in describing separation processes and
other systems, such as in living mater, where the limited amount of adsorption sites plays a crucial role.
Author Contributions
Conceptualization, E.K.O., E.K.L., L.R.E. and R.S.Z.; methodology, R.F.d.S., R.R.R.d.A., R.T.d.S. and R.S.Z.; validation, R.F.d.S., R.R.R.d.A., E.K.O., R.T.d.S., E.K.L., L.R.E. and R.S.Z.; formal
analysis, R.F.d.S., E.K.L., L.R.E. and R.S.Z.; investigation, R.F.d.S., R.R.R.d.A., E.K.O., R.T.d.S., E.K.L., L.R.E. and R.S.Z.; writing—original draft preparation, R.F.d.S., E.K.O. and R.S.Z.;
writing—review and editing, R.F.d.S., R.R.R.d.A., E.K.O., R.T.d.S., E.K.L., L.R.E. and R.S.Z. All authors have read and agreed to the published version of the manuscript.
This work was partially supported by the National Institutes of Science and Technology of Complex Fluids—INCT-CF (R.S.Z.) and Complex Systems—INCT-SC (E.K.L.). E.K.L. thanks CNPq process number
(302983/2018-0). R.S.Z. thanks CNPq process number (304634/2020-4). R. F. de Souza thanks Financiadora de Estudos e Projetos (FINEP)—process number 0113032700.
Institutional Review Board Statement
Not applicable.
Informed Consent Statement
Not applicable.
Data Availability Statement
Not applicable.
Research developed with the support of LAMAP—UTFPR.
Conflicts of Interest
The authors declare no conflict of interest.
Appendix A. Table of Symbols
Here we present a table summarizing the main quantities used in this article (
Table A1
Symbol Definition
$ρ ( z , t )$ Bulk density of diffusing particles. It is a function of space ( z ) and time ( t )
$σ$(t) Density of adsorbed particles. It is a function of time (t)
D Diffusion coefficient
L Cell thickness
$ρ 0$ Total density of particles
$κ a$ Rate of adsorption
$κ d$ Rate of desorption
$ρ R = ρ / ρ 0$ Reduced bulk density
$σ 0$ Number of available sites
$σ R = σ / σ 0$ Reduced surface density
$τ D = d 2 / D$ Diffusion time
$τ κ = d / 2 κ a$ Adsorption time
$τ = 1 / κ d$ Desorption time
$τ a$ Memory time
$t * = 4 t / τ D$ Dimensionless time
$( Δ Z ) 2$ Mean Square Displacement
$β = ρ 0 L / σ 0$ Ratio of particles in the bulk to the number of available sites
Subscript i May be l (left) or r (right). Indicates the substrate considered.
Figure 1. System studied in this work. It consists of a liquid with diluted neutral particles (diffusing particles) that diffuse in the $z −$direction and may be adsorbed (desorbed) following
Langmuir’s equation. Notice that the left surface is located at $z = − L / 2$, while the right surface is at $Z = L / 2$. Gray spheres represent occupied sites on the surfaces, while green spheres
represent free adsorption sites.
Figure 2. Role played by the parameter $β = ρ 0 d / σ 0$ on the surface dynamics. (a) shows the left surface ($Z = − 1$) vs. $t *$ for $τ D / τ l = 100$, $τ κ l / τ l = 10$ and for two values of $τ a
l / τ l$, that is, 0.1 or 20 and $β = 0.2$ and $β = 2$. The inset shows the bulk density $ρ R$ vs. Z when $t * = 0.6$ for $β = 2$ (in red) and $β = 0.2$ (black). (b) shows the left surface ($Z = − 1$
) vs. $t *$ for $τ D / τ l = 100$, $τ κ l / τ l = 1$, and $τ a l / τ l$=0.1 for several values of the parameter $β$.
Figure 3. Effect of $β$ on non-identical surfaces. For both figures, the left surface uses $β l = 1.0$, $τ κ l / τ l = 1$ and $τ a l / τ l = 0.01$, and the diffusion time is $τ a l / τ l = 0.01$. In
(a), all the characteristic times of the surface at $Z = 1$ are the same as the surface at $Z = − 1$, except the parameter $β r$. The main figure shows the time evolution of $σ R l$, while the inset
shows $σ R r$. In (b), the left surface uses the same parameters as in (a), but the right surfaces uses $τ κ r / τ r = 10$, $τ a r / τ r = 20$ for $β r = 0.2$ and $β r = 2.0$. Both surface dynamics
are plotted against $t *$ in the main figure, while the inset shows the bulk distribution for three different values of $t *$ and both values of $β r$.
Figure 4. $σ R i / β i$
for two different values of
$β r$
. The left surface uses the same parameters as used in
Figure 3
, while the surface at
$Z = 1$
$τ κ r / τ r = 0.01$
$τ a r / τ r = 0.01$
. From this figure, it becomes clear that increasing the
of a single surface changes the behavior of the opposite surface and that larger values of
mean higher coverage but a smaller overall number of adsorbed particles. The inset shows the bulk distribution for three different values of
$t *$
and for
$β r = 0.2$
$β r = 2.0$
Figure 5. Mean square displacement (MSD) vs. $t *$ for several values of $β$. The solid curves represent the case in which both surfaces are equal, while the dashed curves represent the MSD for the
$Z = 1$ surface in the case of non-equal surfaces. (a) shows the case in which $τ κ i / τ i = 1$, while (b) shows the case in which $τ κ i / τ i = 0.01$ (identical surfaces and right surface for the
non-identical case). The dotted lines in (a) show examples of the exponent ($t a$) of the MSD, indicating the subdiffusive behavior of the system.
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MDPI and ACS Style
de Souza, R.F.; de Almeida, R.R.R.; Omori, E.K.; de Souza, R.T.; Lenzi, E.K.; Evangelista, L.R.; Zola, R.S. Role of the Number of Adsorption Sites and Adsorption Dynamics of Diffusing Particles in a
Confined Liquid with Langmuir Kinetics. Physchem 2023, 3, 1-12. https://doi.org/10.3390/physchem3010001
AMA Style
de Souza RF, de Almeida RRR, Omori EK, de Souza RT, Lenzi EK, Evangelista LR, Zola RS. Role of the Number of Adsorption Sites and Adsorption Dynamics of Diffusing Particles in a Confined Liquid with
Langmuir Kinetics. Physchem. 2023; 3(1):1-12. https://doi.org/10.3390/physchem3010001
Chicago/Turabian Style
de Souza, Renato F., Roberta R. Ribeiro de Almeida, Eric K. Omori, Rodolfo T. de Souza, Ervin K. Lenzi, Luiz R. Evangelista, and Rafael S. Zola. 2023. "Role of the Number of Adsorption Sites and
Adsorption Dynamics of Diffusing Particles in a Confined Liquid with Langmuir Kinetics" Physchem 3, no. 1: 1-12. https://doi.org/10.3390/physchem3010001
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Oracle Complexity of Second-Order Methods for Finite-Sum Problems.
Finite-sum optimization problems are ubiquitous in machine learning, and are commonly solved using first-order methods which rely on gradient computations. Recently, there has been growing interest
in second-order methods, which rely on both gradients and Hessians. In principle, second-order methods can require much fewer iterations than first-order methods, and hold the promise for more
efficient algorithms. Although computing and manipulating Hessians is prohibitive for high-dimensional problems in general, the Hessians of individual functions in finite-sum problems can often be
efficiently computed, e.g. because they possess a low-rank structure. Can second-order information indeed be used to solve such problems more efficiently? In this paper, we provide evidence that the
answer – perhaps surprisingly – is negative, at least in terms of worst-case guarantees. However, we also discuss what additional assumptions and algorithmic approaches might potentially circumvent
this negative result.
Publication series
Name Proceedings of Machine Learning Research
Publisher PMLR
Volume 70
ISSN (Electronic) 2640-3498
Conference 34th International Conference on Machine Learning, ICML 2017
Country/Territory Australia
City Sydney
Period 6/08/17 → 11/08/17
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stdpc6 - New Foundations Explorer
Description: One of the two equality axioms of standard predicate calculus, called reflexivity of equality. (The other one is stdpc7 1917.) Axiom 6 of [Mendelson] p. 95. Mendelson doesn't say why he
prepended the redundant quantifier, but it was probably to be compatible with free logic (which is valid in the empty domain). (Contributed by NM, 16-Feb-2005.) | {"url":"https://us.metamath.org/nfeuni/stdpc6.html","timestamp":"2024-11-13T09:26:27Z","content_type":"text/html","content_length":"7383","record_id":"<urn:uuid:e6555280-fa6e-4a64-866b-79ae1813f7cf>","cc-path":"CC-MAIN-2024-46/segments/1730477028342.51/warc/CC-MAIN-20241113071746-20241113101746-00425.warc.gz"} |
class qf_lib.containers.dataframe.qf_dataframe.QFDataFrame(data=None, index: Axes | None = None, columns: Axes | None = None, dtype: Dtype | None = None, copy: bool | None = None)[source]
Bases: DataFrame, TimeIndexedContainer
Base class for all data frames (2-D matrix-like objects) used in the project. All the columns within the dataframe contain values for the same date range and have the same frequencies. All the
columns are of the same types (e.g. log-returns/prices).
exponential_average([lambda_coeff]) Calculates the exponential average of a dataframe.
get_frequency() Attempts to infer the frequency of each column in this dataframe.
min_max_normalized Normalizes the data using min-max scaling: it maps all the data to the [0;1] range, so that 0 corresponds to the minimal value in the original series and 1
([original_min_values, ...]) corresponds to the maximal value.
rolling_time_window Runs a given function on each rolling window in the dataframe.
(window_length, step, func)
rolling_window(window_size, func Looks at a number of windows of size window_size and transforms the data in those windows based on the specified func.
[, step, ...])
to_log_returns() Converts dataframe to the dataframe of logarithmic returns.
to_prices([initial_prices, ...]) Converts a dataframe to the dataframe of prices.
to_simple_returns() Converts dataframe to the dataframe of simple returns.
total_cumulative_return() Calculates total cumulative return for each column. | {"url":"https://qf-lib.readthedocs.io/en/stable/_autosummary/qf_lib.containers.dataframe.qf_dataframe.QFDataFrame.html","timestamp":"2024-11-15T04:03:33Z","content_type":"text/html","content_length":"44604","record_id":"<urn:uuid:7580d258-af97-42f1-8a00-63ddf1e09d01>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00887.warc.gz"} |
UVA 1393 Highways,uva 12075 counting triangles--(combination number, DP)
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Time series modeling carefully collects and studies the past observations expressed as a time series for the purpose of developing an appropriate model which describes the inherent structure of the
series. One of the most frequently used stochastic time series models is the Autoregressive Integrated Moving Average (ARIMA), whose popularity is mainly due to its flexibility to represent several
varieties of time series with simplicity.
Autoregressive Integrated Moving Average (ARIMA) process for univariate time series
ARIMA^1 is a class of generalized model that captures temporal structure in time series data. For this purpose, ARIMA combines Auto Regressive process (AR) and Moving Average (MA) processes so as to
build a composite model of the time series. In particular, ARIMA forecasts the next values using auto regression with some parameters fitted to the model. Then, AMIRA applies a moving average with a
set of parameters. During the autoregression, the variable of interest 𝑦[𝑡] is forecasted using a linear combination of past values of the variable 𝑦[𝑡][-1], 𝑦[𝑡-2], ... , α[p]𝑦[𝑡-p. ]The
Autoregressive term is written as:
𝑦[𝑡] = c + α[1]𝑦[𝑡][-1]+ α[2]𝑦[𝑡-2] + ... + α[p]𝑦[𝑡-p] + ε[𝑡]
where c is a constant, α[i] (i= 1, 2,...,p) is the model parameter that needs to be discovered, 𝑦[𝑡-1](i= 1,2,...,p) are the lagged values of 𝑦[t] and ε[𝑡] is the white noise.
The moving average term 𝑦[𝑡] can be expressed based on the past forecast errors (rather than using past values):
𝑦[𝑡] = u + ϴ[1] ε[𝑡][-1]+ ϴ[2] ε[𝑡][-2]+ ... + ϴ[q] ε[𝑡][-q ]+ ε[𝑡]
where u is a constant, ϴ[i] (i= 1,2,...,q) are the model parameters, ε[𝑡][-i] are random shocks at time period t-i (i= 1,2,...,q) and ε[𝑡] is white noise.
Overall, the autoregressive (AR), moving average (MA) and Integration models are effectively combined to form a class of time series models, called ARIMA (with 𝑦’[𝑡] representing the differenced time
series), which is expressed as:
𝑦’[𝑡] = c + α[1]𝑦[𝑡][-1]+ α[2]𝑦[𝑡][-2] + ... + α[p]𝑦[𝑡][-p]+ϴ[1] ε[𝑡][-1]+ ϴ[2] ε[𝑡][-2 ]+ ... + ϴ[q] ε[𝑡][-q ]+ ε[𝑡]
An important prerequisite is to check whether a time series is stationary (constant mean and variance) through plotting and root testing using augmented Dickey-Fuller^1 or Philips-Perron^2 unit root
test. If the time series is not stationary, it can be made stationary by differencing the time series^3.
The best parameters are found using the Box-Jenkins method^4, which is a three-step approach that consists in:
• identifying the model to ensure that the variables are stationary and selecting parameters based on the Autocorrelation Function (AFC)^6 for the MA terms and the Partial Autocorrelation Function
(PACF)^5 for the AR terms.
• estimating the parameters (α and ϴ) that best fit the ARIMA model based on e.g. maximum likelihood^6 or the nonlinear least square^7. Among candidate models, the best suited model is the one that
has the best AIC or BIC value ^8.
• Statistical model checking lying in studying if the residual is white noise and has a constant mean and variance over time. If these assumptions are not satisfied a more appropriate model needs
to be fitted.
If all the assets are satisfied, the future values can be forecasted according to the model. The ARIMA model has been generalized by Box and Jenkins to deal with seasonality.
Seasonal Autoregressive Integrated Moving Average (SARIMA) process for univariate time series
Seasonal ARIMA (SARIMA)^10 deals with a seasonal component in univariate time series. In addition to the autoregression (AR), differencing (I) and moving average (MA), SARIMA accounts for the
seasonal component of the time series leveraging additional parameters for the period of the seasonality. The SARIMA model is hence represented as SARIMA(p,d,q)(P,D,Q)m where P defines the order of
the seasonal AR term, D the order of the seasonal Integration term, Q the order of the seasonal MA term and M the seasonal factor.
Vector Autoregressive Integrated Moving Average (VARMA) process for multivariate time series
Contrary to the ARIMA model, which is fitted for univariate time series, VARMA(p,q)^10 deals with multiple time series that may influence each other. For each time series, we regress a variable on p
lags of itself and all the other variables and so on for the q parameter. Given k time 𝑦[𝑡][-1, ]𝑦[𝑡-2], […, ]𝑦[kt] series expressed as a vector V[𝑡] = [𝑦[𝑡][-1, ]𝑦[𝑡-2, …, ]𝑦[kt]] , VARMA(p,q)
models is defined by the Var and Ma models:
Equation. 1 VARMA matrix notation
where c[k] matrix is a constants vector, α𝑡[i,j ](i,j=\ 1,2,...,k) and ϴ[ ij ](i,j=\ 1,2,...,k) matrixes are the model parameters and k is the number of time series, 𝑦[k],[𝑡][-1](,i= 1,2,...,p) are
the lagged values matrix and the cross variables dependency. ε[k,t-q ](i= 1,2,...,q) are the matrix of random shocks and Φ[kt] is white noise vector with zero mean and constant covariance matrix.
In the following, we will use this family of models to model and predict the behavior of the NVF/CNF system and detect anomalies.
If you have missed the first part, here you can read the introduction
References ⤵
• [1] Dickey, D. A., & Fuller, W. A. (1979). Distribution of the estimators for autoregressive time series with a unit root. Journal of the American statistical association, 74(366a), 427-431
• [2] Phillips, Peter & Perron, Pierre. (1986). Testing for a Unit Root in Time Series Regression. Cowles Foundation, Yale University, Cowles Foundation Discussion Papers. 75. 10.1093/biomet/
• [3] Nason, G. P. (2006). Stationary and non-stationary time series. Statistics in volcanology, 60.
• [4] Box, G. E., Jenkins, G. M., Reinsel, G. C., & Ljung, G. M. (2015). Time series analysis: forecasting and control. John Wiley & Sons.
• [5] Watson, P. K., & Teelucksingh, S. S. (2002). A practical introduction to econometric methods: Classical and modern. University of West Indies Press
• [6] Myung, I. J. (2003). Tutorial on maximum likelihood estimation. Journal of mathematical Psychology, 47(1), 90-100.
• [7] Hartley, H. O., & Booker, A. (1965). Nonlinear least squares estimation. Annals of Mathematical Statistics, 36(2), 638-650.
• [8] Akaike, H. (1998). Information theory and an extension of the maximum likelihood principle. In Selected papers of hirotugu akaike (pp. 199-213). Springer, New York, NY.
• [9] Hyndman, R. J., & Athanasopoulos, G. (2018). Forecasting: principles and practice. OTexts.
• [10] Brockwell, P. J., Brockwell, P. J., Davis, R. A., & Davis, R. A. (2016). Introduction to time series and forecasting. springer. | {"url":"https://www.squad.fr/en/news/2021/07/07/background-on-time-series-modeling-and-forecasting-23/","timestamp":"2024-11-06T05:10:02Z","content_type":"text/html","content_length":"43220","record_id":"<urn:uuid:7304823a-9f17-4c33-a319-6d1d9cb6c4db>","cc-path":"CC-MAIN-2024-46/segments/1730477027909.44/warc/CC-MAIN-20241106034659-20241106064659-00699.warc.gz"} |
How To Write A Ratio Using Fractions
Learning how to write a ratio is very important if you are an investor, or someone looking to make an investment. Financial ratios are very easy to understand and calculate. They basically take the
amount of money invested in a particular financial investment and then dividing that by the amount of money actually made on that investment. This tells you what the investors considered to be a good
ratio. The problem arises when you have a denominator that isn't a positive number, but is greater than one.
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Let's take a look at how this works. Suppose you had a large investment, and the denominator is worth 10 million dollars. That means the investors who invested that money are doing pretty well.
However, there are still some people who are making losses on that investment because it isn't earning as much as it is suppose to. The problem is they don't know how to calculate the ratio between
the total number of shares, and the amount of money that each person is making off of his or her share.
How to Write a Ratio Between a Total Number of Shares and a Net Worthiness If we were to calculate the percentage of investors who are making money off of their investment, we would need to know how
many counters are being sold. The denominator would be the amount of money invested. The net worthiness would be the total amount of money earned minus the total amount of money invested. To find out
how many counters are being sold, we divide the total number of shares by the amount of money invested.
How to Write a Ratio Using Fractions
This is how to simplify a ratio. We are basically dividing by the total number of investors. By dividing by the total number of counters, we can simplify the calculation of the ratio between the
total number of shares, and the amount of money per share.
Comparing Counts Using a Compound Ratio One of the easiest ways how to compare quantities using a compound ratio is to compare quantities using a table. To do this, start writing the ratios on one
page and then copy and paste the same information onto the other page. For instance, copy and paste the following information; multiplying the total number of shares by the amount per share by the
ratio of total to per share by multiplying by 100. This is how to simplify the compound ratio for the purpose of comparing.
This is how to compare two quantities using a unit of measurement. You need to have a unit of measurement for each of the items being compared. This will make the ratio easier to calculate as each
item is compared separately. A common unit of measurement used is the pound. You can also use a variety of other units of measurement.
Determining a Frequent Ratio The simplest way how to write a ratio that is used often is to determine a common, simple and frequent ratio. The most common ratio is the log ratio. This uses the log of
both the original quantities, which is the total number of grains written down and the number of grains to be turned into bread. You can also use the percentage method when determining a frequent
How to Write a Ratio Using Fractions When working with fractions, there is another option how to write a ratio. The fraction is either a fraction of a fraction or the ratio of a fraction. For
instance, when working with decimals, you must either use decimals as a numerator or denominator. Using the fraction indicates that there are a number of times the fraction is divided by the number
of times the whole number is multiplied. This option is usually more trouble than it's worth because of the precision required.
Thanks for checking this blog post, for more updates and articles about how to write a ratio don't miss our homepage - Soyprint We try to update the site every week | {"url":"https://www.soyprint.net/how-to-write-a-ratio/","timestamp":"2024-11-08T12:17:20Z","content_type":"text/html","content_length":"13319","record_id":"<urn:uuid:caa0ba68-a831-48c9-97b1-ee367a13c131>","cc-path":"CC-MAIN-2024-46/segments/1730477028059.90/warc/CC-MAIN-20241108101914-20241108131914-00186.warc.gz"} |
Chem – Introduction to Problem Solving
What sections should I know before attempting to learn this section?
How do you solve word problems in chemistry?
Perhaps the most fundamental problem that most students have in chemistry is the lack of problem solving and organizational skills. I can never overstate how crucial these skills are for doing well
in chemistry (and in life). One of the few regrets in all my life is that I did not pick up these skills sooner. Here I will try to lay out as best I can how to think your way through problems to
come up with a correct answer. Since word problems are the most common in chemistry and because many people have trouble with them, I will focus my problem solving teaching on them. Improving your
problem solving skills will be an ongoing process in any chemistry class, but I will show you the first and most important steps in this section. Be sure to always analyze my demonstrated examples
for future insight into problems solving techniques.
Step 1: Underline, highlight, or box the numbers and units in a word problem. Because our brains are not very good at organizing many things at once, we need to show it where it needs to focus the
effort. (I know this seems trivial but trust me it helps)
Step 2: Rewrite the numbers and units and identify what they are. Identifying what they are can help you later relate them to formulas or key concepts. (I know this seems trivial but trust me it
Step 3: Restate the question in a short and simple way to guide you to the goal of finishing the question. Again, like the second step, this can help you relate information you already have to
formulas or key concepts. (I know this seems trivial but trust me it helps)
Now let us demonstrate this method with some examples. Remember you do not need to answer the problem only organize it. Highlight the numbers and units in the word problems below. Rewrite the
numbers and units and identify what they are. Restate the question in a short and simple way.
Examples: VIDEO Problem Solving Examples 1.
Problem Solving Demonstrated Example 1: If an object has mass of 26g and a volume of 55mL what would its density be?
Step 1: highlight numbers and units
If an object has mass of 26g and a volume of 55mL what would its density be?
Step 2: Rewrite numbers and units and identify
26g = mass
55mL = volume
Step 3: Restate question in simple way
density = ?
Step 4:
26g = mass
55mL = volume
density = ?
Problem Solving Demonstrated Example 2: An increase of 37K would cause the volume to go from 4.0L to 4.8L. What was the original temperature?
Step 1: highlight numbers and units
An increase of 37K would cause the volume to go from 4.0L to 4.8L. What was the original temperature?
Step 2: Rewrite numbers and units and identify
37K = temperature
4.0L = volume
4.8L = volume
Step 3: Restate question in simple way
original temperature = ?
Step 4:
37K = temperature
4.0L = volume
4.8L = volume
original temperature = ?
PRACTICE PROBLEMS: Highlight the numbers and units in the word problems below. Rewrite the numbers and units and identify what they are. Restate the question in a short and simple way.
1. An object that takes up a volume of 2L and has a mass of 70kg would be what density?
Answer: 2L = volume….70kg = mass….Density = ?
2. If you run a distance of 900m at a time of 380s, what is your speed?
Answer: 900m = distance….380s = time….Speed = ?
3. It takes 15s to fill a 0.7L balloon. What is the rate of air from the container?
Answer: 15s = time….0.7L = volume….Rate = ?
4. If a 20g block takes 60s to heat from 0K to 300K. How long will it take to heat a 38g block of the same substance from 0K to 400K?
Answer: 20g = mass….60s = time….0K = temperature….300K = temperature….38g = mass….0K = temperature….400K = temperature….Time = ?
5. By changing from 78K to 142K the volume increased by 182%. What is the final volume if the original volume was 2.3L?
Answer: 78K = temperature….142K = temperature….182% = percent….2.3L = volume….Final volume = ?
6. Even if the question makes no sense. Picking out 7kg from the hyperbole of a transvector can lead you to 3L in case they ask, what is the density?
Answer: 7kg = mass….3L = volume….Density = ? | {"url":"http://scientifictutor.org/387/chem-introduction-to-problem-solving/","timestamp":"2024-11-03T23:09:34Z","content_type":"application/xhtml+xml","content_length":"30490","record_id":"<urn:uuid:aa7a8a0a-5182-4065-97e1-497cb27c6e93>","cc-path":"CC-MAIN-2024-46/segments/1730477027796.35/warc/CC-MAIN-20241103212031-20241104002031-00226.warc.gz"} |
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Problem 1 (25%)
In your own words, answer the following qu
HOMEWORK 1 Problem 1 (25%) In your own words, answer the following questions (a) Name three concrete admixtures used in concrete. Explain the purpose of each admixture. (b) Draw a graph of the
water-to-cement (W/C) ratio versus concrete strength. Does the addition of water in a concrete mix reduce its strength? Explain. (c) Explain the concept of creep and shrinkage. Give an example of a
typical situation where each phenomenon would be most noticeable. (d) Why is it important to maintain the concrete moist through curing at its early age? 1 Homework 1 Problem 2 (25%) The beam below
carries its own weight and two concentrated loads. The unit weight of concrete is 150 pcf and the modulus of rupture is 410 psi. Do the following: (a) Draw the bending moment diagram and show the
bending moment values in terms of P1. (b) The concentrated loads are equally increased until cracking occurs at the bottom face of the beam. What is the value of the concentratred load P₁ that causes
cracking? (c) Assuming the P1 = 0 kip and a concentrated load P2 is applied at each cantilever tip (points A and D in the figure). What is the value of P2 that causes cracking of the beam at the
supports (points B and C in the figure)? C.A Al ΔΒ P₁ 1 P₁ 1 12 ft 10 ft 10 ft 12 ft 40 ft 36 in 4 < A 6 in A 24 in SECTION 1-1 2 I 8 in D Homework 1 Problem 3 (25%) The reinforced concrete column in
the figure has a rectangular cross section. The column is subjected to 80 kips of axial compressive force at its top in addition to a uniform lateral load along the top 6 ft of the column height. The
unit weight of concrete is 150 pcf and the specified concrete strength, f'c is 3,000 psi. Do the following: (a) Draw the axial force and bending moment diagram of the column, show the bending moment
in terms of the lateral load w. (b) If the lateral load is increased sufficiently to cause cracking in the column, where along the column is the cracking most likely to occur? Draw the expected crack
pattern. (c) The lateral load w is increased until cracking occurs in the column. What is the value of the uniform lateral load at which the column cracks? (Hint: Remember that the axial compressive
force and column self-weight increases the compressive stress in the column) Uniform load, w (kip/ft) 16" x 20" column 20 in 3 80 kip 6 ft 12 ft Homework 1 Problem 4 (25%) For each section below,
assuming material properties f*c = 3,000 psi and Es = 29,000 ksi, compute the following: (a) The cracking moment using the gross moment of inertia of the section. (b) The allowable moment using the
transformed area method. Assume an allowable stress fs = 36,000 psi and f' 1,200 psi for steel and concrete respectively. = C 9 4 3 in 24 in 36 in 8" 36 6 in (5) #9 20 in (C) A 44 A Δ (4) #8 4 21 in
44 Δ 4 (6) #6 16 in (A) 14 in (B) Δ 4 30 in 4 Δ A Δ (3) # 10 12 in (3) # 10 22 in (D) ง A A 14 in 3 in 2 in | {"url":"https://tutorbin.com/questions-and-answers/homework-1-problem-1-25-in-your-own-words-answer-the-following-questions-a-name-three-concrete-admixtures-used-in","timestamp":"2024-11-02T21:24:59Z","content_type":"text/html","content_length":"67193","record_id":"<urn:uuid:16337d69-990f-4ede-9b2c-39edcc25cd93>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00245.warc.gz"} |
Draw a line, say AB, take a point C outside it. Through C, draw a line parallel to AB using ruler and compasses only.
We have to draw figure using following steps of construction:
Step 1: Draw a line AB and take a point P on the line AB. Also, take another point C outside the line AB and join PC.
Step 2: Now, taking P as a center and with a certain radius (which should be much lesser than PC), draw an arc intersecting AB at D and PC at point E.
Step 3: Now taking C as a centre and with the same radius, draw an arc FG intersecting PC at H.
Step: 4: Adjust the compasses according to the length of DE. And with same opening, taking H as a center, draw another arc intersecting the previous arc at point I.
Step 5: Now, join CI in order to draw a line ‘l’.
This is the desired line parallel to line AB | {"url":"https://philoid.com/question/23644-draw-a-line-say-ab-take-a-point-c-outside-it-through-c-draw-a-line-parallel-to-ab-using-ruler-and-compasses-only","timestamp":"2024-11-10T18:26:32Z","content_type":"text/html","content_length":"35496","record_id":"<urn:uuid:8137310a-4018-47c7-a72c-6d211a9ff932>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.61/warc/CC-MAIN-20241110170046-20241110200046-00666.warc.gz"} |
Standard Deviation: 4 Steps For HR to Calculate & Use It
Standard Deviation
During a meeting someone mentions the results of a standard deviation analysis and you don’t feel it applies to you. But actually, standard deviation is a great analysis you need to be familiar with.
Read on to discover its impact in human resources.
What Is Standard Deviation?
Standard deviation is measuring the dispersion and variation within your given data set in relation to the mean. We will divide this definition into segments to further understand it and why it is
important for HR to utilize.
Data Set
Data sets refer to the collection of data you are trying to analyze. Your organization may have large or small data sets depending on size and how much information you collect. Common HR data sets
are number of new hires, number of terminations, compensation ranges for your teams, employee count, benefit use and many more. Data sets are normally discussed in two different ways. First, a
population means every data value is being included in the calculation. This is the most useful for statistical analysis; however, sometimes it is not the most practical. The second term is sample
size, which is a subselection of your overall population that represents its qualities.
Data Point Relations to the Mean
The standard deviation looks at how a certain data point in a data set relates to the mean or average of all data points you are analyzing. Is it really close or far away from the average of your
data points?
Measuring Dispersion or Variation
Standard deviation’s purpose is to measure the difference between all your data points and means. Standard deviation is a commonly understood way to communicate this spread and how significant/
insignificant differences are.
Why Is Standard Deviation Important for HR?
Standard deviation is important for HR because it can provide a data-backed perspective on what issues need attention. The goal is to give you another tool to filter the many projects you have.
• Strategic partner. Including standard deviation and other basic statistical analysis strengthens your business recommendations and value as a strategic partner. Standard deviation is utilized in
other aspects of your business and can be a common ground for communication.
• Data analysis skills. We live in a world that is inundated with various data points. As we progress into the future, all positions will have some form of data analysis, so why not be an early
adopter and work the bugs out now?
How to Calculate Standard Deviation
Let’s look at a few basic steps to help you calculate standard deviation.
Step 1: Understand the Why
Standard deviations are not the statistical analysis answer to every question. Take some time to map out what data points you are trying to get. Does it make sense to utilize standard deviation to
get that answer? If it is, continue on to the next step.
Step 2: Identify Population or Sample Group
You want to understand if you are working with a population or sample size because the calculation changes based on this factor. It also helps you communicate the data results to ensure they are
interpreted correctly.
Step 3: Do a Quality/Clean Check on Your Data
Before running a standard deviation and utilizing it in a business recommendation, it is important to glance through the data to make sure you aren’t missing any information or have faulty data.
Nothing will sidetrack your proposal more than incorrect data.
Step 4: Calculate Standard Deviation
There are various ways to calculate the standard deviation. It is possible to do it by hand, but I wouldn’t recommend it. The most accessible way is to input your data into Excel and then create a
Actions HR Can Take with Standard Deviation
Now that we know how to calculate standard deviation, let’s look at how this analysis can help move HR initiatives forward.
Hiring, Retention, and Turnover
Three metrics you can easily calculate using data in your Human Resource Information System (HRIS), is time to fill, employee retention, and turnover. To review, time to fill refers to how many
business days it takes to fill a position. Employee retention looks at how long your employees stay with you. Turnover refers to why individuals leave. Calculating the standard deviation lets you
know where you fall in regards to past annual data, or, if available, other companies within your industry. This can be very useful for helping know where to focus HR efforts. For example, completing
the standard deviation may show that you are one standard deviation below your previous year time to fill.
Benefits and Compensation
Standard deviation calculation can show you if your compensation is leading, matching or lagging in the market. If you currently hold a paying job, you have benefited from a standard deviation
analysis. Organizations and market studies take the salaries that have been collected and find the average or mean of that job title and salary. Then they create a standard deviation to map out where
the salaries are in relation to the average salary. Once the mapping is complete, they can identify if the salary/benefits are above market average, or above the standard deviation or below. For
technicalities, there are different common markers explaining if a data point is 1, 2, or 3 standard deviations above the average.
Brent Watson
Brent Watson enjoys problem solving, analyzing data, team building, and becoming an HR Guru. His work experience comes from the employee experience, recruiting, and training arenas. After attending a
local HR conference, Brent knew that he had found his people and the problems he wanted to solve for in the business world.
View author page
Frequently asked questions
Other Related Terms | {"url":"https://eddy.com/hr-encyclopedia/standard-deviation/","timestamp":"2024-11-03T17:15:36Z","content_type":"text/html","content_length":"333734","record_id":"<urn:uuid:563cca1f-baa7-489e-beea-ae0e93c2c31b>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00138.warc.gz"} |
Exponential Growth
You may have run into a problem such as the following, or perhaps you tried this trick on your parents when you were young (if you're still young, here's your chance):
You have to wash the dishes each day next month. Which would you rather have:
□ $10 per day, or
□ 1¢ the first day, and double the previous day's amount on the following day (so, 2¢ the second day, 4¢ the third day, and so on)?
The second option doesn't really look like much, but if you calculate it out, your payment for the 30th day will be $5,368,709.12!
You might also recall the legend of the invention of chess: According to legend, when the inventor of the game showed it to the King of Persia, the King was impressed and asked him what he would like
as a reward for the invention. The inventor just asked for one grain of wheat for the first square on the chessboard, two grains of wheat for the second, four for the fourth, and so on, doubling the
amount for each square. The king thought that to be a small reward, but calculation would discover that 18,446,744,073,709,551,615 would be required!
These examples show how rapid exponential growth can be. However, exponential growth is useful not just for explaining various riddles, but also for explaining various natural processes. Any
population that grows at a rate equal or proportional to the current population grows exponentially (a population that grows at a rate that is continuously equal to its population can be modelled by
a function of e^x). For example, a population of any organism, if not checked by food, predators, or any other constraint, would grow at a rate proportional to its current population; in other words,
it would grow exponentially. One real-life example can be found in pandemics, the Wuhan Coronavirus demonstrating a good example of exponential growth. | {"url":"http://mathlair.allfunandgames.ca/exponential.php","timestamp":"2024-11-13T08:33:16Z","content_type":"text/html","content_length":"3969","record_id":"<urn:uuid:17cc3c3e-286b-4a90-a91c-af129f551927>","cc-path":"CC-MAIN-2024-46/segments/1730477028342.51/warc/CC-MAIN-20241113071746-20241113101746-00149.warc.gz"} |
Microsoft’s Equation Editor: the Good and the Glitches
I had problems over the last few days, which turned out to be because of glitches in Microsoft Equation Editor within Word 2007. Lecture notes I had written suddenly lost whole sections or when I
pasted in good parts of a document that had got messed-up into a new document the formatting of the new document instantly got messed-up. That was very frustrating, as I was under pressure to have
my lecture notes prepared.
I was wondering if viruses were at work. I contacted our IT people and they promised to investigate. I took the usual precautions of backing up my computer.
Microsoft doesn’t publish the features of Equation Editor, it seems. The fashion now is that one has to go to the web to find the help one needs. A while ago I found excellent information on Equation
Editor at the University of Waterloo (http://ist.uwaterloo.ca/ec/equations/equation2007.html). One of the features of Equation Editor is that Shift+Enter can be used to insert a group of equations,
which can be aligned together on specified characters. This is an important and necessary TeX- or LATeX-type feature. I was glad to find it.
Once I find a feature I use it to its full potential. The lines of the grouped equations were too close together so I used Shift+Enter to space them out. It seems this was what triggered the glitch:
beware! I have now found that by going into draft mode in Word I can see the ‘disappeared’ text. By displaying formatting marks I can see and then remove the Shift+Enter symbols where I used them for
line spacing. This seems to restore the document: as I write it is too soon to say for sure that I have beaten the problem.
It seems Microsoft has incorporated much of TeX into Equation Editor now. This is very welcome for me. It is moving towards enabling fluent entry of mathematical equations from the keyboard and yet
having full WYSIWIG. That is great! For example, typing times puts in a multiplication sign or the Greek letters can be input as alpha or Alpha, according to case. | {"url":"https://fun-engineering.net/blogs/funeng/2009/11/13/microsofts-equation-editor-the-good-and-the-glitches/","timestamp":"2024-11-14T20:01:10Z","content_type":"text/html","content_length":"29374","record_id":"<urn:uuid:825bf96f-5658-44e9-b8b8-8df14e71ba35>","cc-path":"CC-MAIN-2024-46/segments/1730477395538.95/warc/CC-MAIN-20241114194152-20241114224152-00118.warc.gz"} |
CAT 2018 Question Paper
CAT DI LR section has become increasingly tough from 2015. DILR used to have distinct Data Interpretation sets and Logical reasoning puzzles. It used to be about computation and the ability to read
charts, graphs, and tables for the Data Interpretation and Logical reasoning used to have Family tree, grid puzzles, arrangement, tournaments, cubes as some standard forms of puzzles. Since 2015 this
pattern has been broken. Slot 2 of the 2018 CAT previous year paper was indeed on the difficult side with two sets hitting the zenith of the toughness. Nonetheless, there were a couple of pretty
doable sets with easy questions. Furthermore, the CAT question paper had 4 TITA-type questions with varying difficulty levels. Moreover, there was some involvement of quant topics like ratios and set
theory with complex computations. Nail your online CAT preparation by smashing the DILR section of the CAT 2018 question paper.
CAT DILR : CAT 2018 Question Paper Slot 2
Set 1: College Accreditation
An agency entrusted to accredit colleges looks at four parameters: faculty quality (F), reputation (R), placement quality (P), and infrastructure (I). The four parameters are used to arrive at an
overall score, which the agency uses to give an accreditation to the colleges. In each parameter, there are five possible letter grades given, each carrying certain points: A (50 points), B (40
points), C (30 points), D (20 points), and F (0 points). The overall score for a college is the weighted sum of the points scored in the four parameters. The weights of the parameters are 0.1, 0.2,
0.3 and 0.4 in some order, but the order is not disclosed.
Accreditation is awarded based on the following scheme:
Eight colleges apply for accreditation, and receive the following grades in the four parameters (F, R, P, and I):
It is further known that in terms of overall scores:
1. High Q is better than Best Ed.
2. Best Ed is better than Cosmopolitan.
3. Education Aid is better than A-one.
1. CAT Previous year paper - CAT Exam DI LR
What is the weight of the faculty quality parameter?
2. CAT Previous year paper - CAT Exam DI LR
How many colleges receive the accreditation of AAA? [TITA]
3. CAT Previous year paper - CAT Exam DI LR
What is the highest overall score among the eight colleges? [TITA]
4. CAT Previous year paper - CAT Exam DI LR
How many colleges have overall scores between 31 and 40, both inclusive?
Set 2 : Smartphones
There are only four brands of entry level smartphones called Azra, Bysi, Cxqi, and Dipq in a country. Details about their market share, unit selling price, and profitability (defined as the profit as
a percentage of the revenue) for the year 2016 are given in the table below:
In 2017, sales volume of entry level smartphones grew by 40% as compared to that in 2016. Cxqi offered a 40% discount on its unit selling price in 2017, which resulted in a 15% increase in its market
share. Each of the other three brands lost 5% market share. However, the profitability of Cxqi came down to half of its value in 2016. The unit selling prices of the other three brands and their
profitability values remained the same in 2017 as they were in 2016.
1. CAT Previous year paper - CAT Exam DI LR
The brand that had the highest revenue in 2016 is:
2. CAT Previous year paper - CAT Exam DI LR
The brand that had the highest profit in 2016 is:
3. CAT Previous year paper - CAT Exam DI LR
The brand that had the highest profit in 2017 is:
4. CAT Previous year paper - CAT Exam DI LR
The complete list of brands whose profits went up in 2017 from 2016 is:
Set 3 : Fun Sports Club
Fun Sports (FS) provides training in three sports - Gilli-danda (G), Kho-Kho (K), and Ludo (L). Currently it has an enrollment of 39 students each of whom is enrolled in at least one of the three
sports. The following details are known:
1. The number of students enrolled only in L is double the number of students enrolled in all the three sports.
2. There are a total of 17 students enrolled in G.
3. The number of students enrolled only in G is one less than the number of students enrolled only in L.
4. The number of students enrolled only in K is equal to the number of students who are enrolled in both K and L.
5. The maximum student enrollment is in L.
6. Ten students enrolled in G are also enrolled in at least one more sport.
1. CAT Previous year paper - CAT Exam DI LR
What is the minimum number of students enrolled in both G and L but not in K? [TITA]
2. CAT Previous year paper - CAT Exam DI LR
If the numbers of students enrolled in K and L are in the ratio 19:22, then what is the number of students enrolled in L?
3. CAT Previous year paper - CAT Exam DI LR
Due to academic pressure, students who were enrolled in all three sports were asked to withdraw from one of the three sports. After the withdrawal, the number of students enrolled in G was six
less than the number of students enrolled in L, while the number of students enrolled in K went down by one.After the withdrawal, how many students were enrolled in both G and K? [TITA]
4. CAT Previous year paper - CAT Exam DI LR
Due to academic pressure, students who were enrolled in all three sports were asked to withdraw from one of the three sports. After the withdrawal, the number of students enrolled in G was six
less than the number of students enrolled in L, while the number of students enrolled in K went down by one.After the withdrawal, how many students were enrolled in both G and L?
Set 4 : Products and Companies
Each of the 23 boxes in the picture below represents a product manufactured by one of the following three companies: Alfa, Bravo and Charlie. The area of a box is proportional to the revenue from the
corresponding product, while its centre represents the Product popularity and Market potential scores of the product (out of 20). The shadings of some of the boxes have got erased.
The companies classified their products into four categories based on a combination of scores (out of 20) on the two parameters - Product popularity and Market potential as given below:
The following facts are known:
1. Alfa and Bravo had the same number of products in the Blockbuster category.
2. Charlie had more products than Bravo but fewer products than Alfa in the No-hope category.
3. Each company had an equal number of products in the Promising category.
4. Charlie did not have any product in the Doubtful category, while Alfa had one product more than Bravo in this category.
5. Bravo had a higher revenue than Alfa from products in the Doubtful category.
6. Charlie had a higher revenue than Bravo from products in the Blockbuster category.
7. Bravo and Charlie had the same revenue from products in the No-hope category.
8. Alfa and Charlie had the same total revenue considering all products.
1. CAT Previous year paper - CAT Exam DI LR
Considering all companies products, which product category had the highest revenue?
2. CAT Previous year paper - CAT Exam DI LR
Which of the following is the correct sequence of numbers of products Bravo had in No-hope, Doubtful, Promising and Blockbuster categories respectively?
3. CAT Previous year paper - CAT Exam DI LR
Which of the following statements is NOT correct?
1. Alfa's revenue from Blockbuster products was the same as Charlie's revenue from Promising products.
2. Bravo's revenue from Blockbuster products was greater than Alfa's revenue from Doubtful products.
3. The total revenue from No-hope products was less than the total revenue from Doubtful products.
4. Bravo and Charlie had the same revenues from No-hope products.
4. CAT Previous year paper - CAT Exam DI LR
If the smallest box on the grid is equivalent to revenue of Rs.1 crore, then what approximately was the total revenue of Bravo in Rs. crore?
Set 5 : Amusement Park Tickets
Each visitor to an amusement park needs to buy a ticket. Tickets can be Platinum, Gold, or Economy. Visitors are classified as Old, Middle-aged, or Young. The following facts are known about visitors
and ticket sales on a particular day:
1. 140 tickets were sold.
2. The number of Middle-aged visitors was twice the number of Old visitors, while the number of Young visitors was twice the number of Middle-aged visitors.
3. Young visitors bought 38 of the 55 Economy tickets that were sold, and they bought half the total number of Platinum tickets that were sold.
4. The number of Gold tickets bought by Old visitors was equal to the number of Economy tickets bought by Old visitors.
1. CAT Previous year paper - CAT Exam DI LR
If the number of Old visitors buying Platinum tickets was equal to the number of Middle-aged visitors buying Platinum tickets, then which among the following could be the total number of Platinum
tickets sold?
2. CAT Previous year paper - CAT Exam DI LR
If the number of Old visitors buying Gold tickets was strictly greater than the number of Young visitors buying Gold tickets, then the number Middle-aged visitors buying Gold tickets was [TITA]
3. CAT Previous year paper - CAT Exam DI LR
If the number of Old visitors buying Platinum tickets was equal to the number of Middle-aged visitors buying Economy tickets, then the number of Old visitors buying Gold tickets was [TITA]
4. CAT Previous year paper - CAT Exam DI LR
Which of the following statements MUST be FALSE?
1. The numbers of Gold and Platinum tickets bought by Young visitors were equal
2. The numbers of Middle-aged and Young visitors buying Gold tickets were equal
3. The numbers of Old and Middle-aged visitors buying Economy tickets were equal
4. The numbers of Old and Middle-aged visitors buying Platinum tickets were equal
Set 6 : Job Interview
Seven candidates, Akil, Balaram, Chitra, Divya, Erina, Fatima, and Ganeshan, were invited to interview for a position. Candidates were required to reach the venue before 8 am. Immediately upon
arrival, they were sent to one of three interview rooms: 101, 102, and 103. The following venue log shows the arrival times for these candidates. Some of the names have not been recorded in the log
and have been marked as ‘?’.
Additionally here are some statements from the candidates:
Balaram: I was the third person to enter Room 101.
Chitra: I was the last person to enter the room I was allotted to.
Erina: I was the only person in the room I was allotted to.
Fatima: Three people including Akil were already in the room that I was allotted to when I entered it.
Ganeshan: I was one among the two candidates allotted to Room 102.
1. CAT Previous year paper - CAT Exam DI LR
What best can be said about the room to which Divya was allotted?
2. CAT Previous year paper - CAT Exam DI LR
Who else was in Room 102 when Ganeshan entered?
3. CAT Previous year paper - CAT Exam DI LR
When did Erina reach the venue?
4. CAT Previous year paper - CAT Exam DI LR
If Ganeshan entered the venue before Divya, when did Balaram enter the venue?
Set 7 : Letter Codes
According to a coding scheme the sentence,
Peacock is designated as the national bird of India is coded as 5688999 35 1135556678 56 458 13666689 1334 79 13366
This coding scheme has the following rules:
1. The scheme is case-insensitive (does not distinguish between upper case and lower case letters).
2. Each letter has a unique code which is a single digit from among 1,2,3,......,9.
3. The digit 9 codes two letters, and every other digit codes three letters.
4. The code for a word is constructed by arranging the digits corresponding to its letters in a non-decreasing sequence.
Answer these questions on the basis of this information
1. CAT Previous year paper - CAT Exam DI LR
What best can be concluded about the code for the letter L?
2. CAT Previous year paper - CAT Exam DI LR
What best can be concluded about the code for the letter B?
3. CAT Previous year paper - CAT Exam DI LR
For how many digits can the complete list of letters associated with that digit be identified?
4. CAT Previous year paper - CAT Exam DI LR
Which set of letters CANNOT be coded with the same digit?
Set 8 : Currency Exchange
The base exchange rate of a currency X with respect to a currency Y is the number of units of currency Y which is equivalent in value to one unit of currency X. Currency exchange outlets buy currency
at buying exchange rates that are lower than base exchange rates, and sell currency at selling exchange rates that are higher than base exchange rates.
A currency exchange outlet uses the local currency L to buy and sell three international currencies A, B, and C, but does not exchange one international currency directly with another. The base
exchange rates of A, B and C with respect to L are in the ratio 100:120:1. The buying exchange rates of each of A, B, and C with respect to L are 5% below the corresponding base exchange rates, and
their selling exchange rates are 10% above their corresponding base exchange rates.
The following facts are known about the outlet on a particular day:
1. The amount of L used by the outlet to buy C equals the amount of L it received by selling C.
2. The amounts of L used by the outlet to buy A and B are in the ratio 5:3.
3. The amounts of L the outlet received from the sales of A and B are in the ratio 5:9.
4. The outlet received 88000 units of L by selling A during the day.
5. The outlet started the day with some amount of L, 2500 units of A, 4800 units of B, and 48000 units of C.
6. The outlet ended the day with some amount of L, 3300 units of A, 4800 units of B,and 51000 units of C.
1. CAT Previous year paper - CAT Exam DI LR
How many units of currency A did the outlet buy on that day? [TITA]
2. CAT Previous year paper - CAT Exam DI LR
How many units of currency C did the outlet sell on that day?
3. CAT Previous year paper - CAT Exam DI LR
What was the base exchange rate of currency B with respect to currency L on that day? [TITA]
4. CAT Previous year paper - CAT Exam DI LR
What was the buying exchange rate of currency C with respect to currency L on that day? | {"url":"https://online.2iim.com/CAT-question-paper/CAT-2018-Question-Paper-Slot-2-DILR/","timestamp":"2024-11-11T01:30:40Z","content_type":"text/html","content_length":"117905","record_id":"<urn:uuid:389b0335-ad98-4f9b-9425-722ed7dd1a18>","cc-path":"CC-MAIN-2024-46/segments/1730477028202.29/warc/CC-MAIN-20241110233206-20241111023206-00618.warc.gz"} |
Randomized vs. Deterministic Separation in time-space tradeoffs of multi-output functions
We prove the first polynomial separation between randomized and deterministic time-space tradeoffs of multi-output functions. In particular, we present a total function that on the input of n
elements in [n], outputs O(n) elements, such that: There exists a randomized oblivious algorithm with space O(log n), time O(n log n) and one-way access to randomness, that computes the function with
probability 1-O(1/n); Any deterministic oblivious branching program with space S and time T that computes the function must satisfy T^2S ≥ Δ(n^2.5/ log n). This implies that logspace randomized
algorithms for multi-output functions cannot be black-box derandomized without an eΔ(n^1/4) overhead in time. Since previously all the polynomial time-space tradeoffs of multi-output functions are
proved via the Borodin-Cook method, which is a probabilistic method that inherently gives the same lower bound for randomized and deterministic branching programs, our lower bound proof is
intrinsically different from previous works. We also examine other natural candidates for proving such separations, and show that any polynomial separation for these problems would resolve the
long-standing open problem of proving n^1+Δ(1)time lower bound for decision problems with polylog(n) space.
Publication series
Name Leibniz International Proceedings in Informatics, LIPIcs
Volume 287
ISSN (Print) 1868-8969
Conference 15th Innovations in Theoretical Computer Science Conference, ITCS 2024
Country/Territory United States
City Berkeley
Period 1/30/24 → 2/2/24
All Science Journal Classification (ASJC) codes
• Borodin-Cook method
• Randomness
• Time-space tradeoffs
Dive into the research topics of 'Randomized vs. Deterministic Separation in time-space tradeoffs of multi-output functions'. Together they form a unique fingerprint. | {"url":"https://collaborate.princeton.edu/en/publications/randomized-vs-deterministic-separation-in-time-space-tradeoffs-of","timestamp":"2024-11-13T11:55:49Z","content_type":"text/html","content_length":"47856","record_id":"<urn:uuid:6340f406-110b-4939-8c8b-35066615ca81>","cc-path":"CC-MAIN-2024-46/segments/1730477028347.28/warc/CC-MAIN-20241113103539-20241113133539-00090.warc.gz"} |
Building 1D lattice models with $G$-gra
SciPost Submission Page
Building 1D lattice models with $G$-graded fusion category
by Shang-Qiang Ning, Bin-Bin Mao, Chenjie Wang
This is not the latest submitted version.
This Submission thread is now published as
Submission summary
Authors (as registered SciPost users): Chenjie Wang
Submission information
Preprint Link: scipost_202402_00006v1 (pdf)
Date submitted: 2024-02-04 16:45
Submitted by: Wang, Chenjie
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
• Condensed Matter Physics - Theory
Specialties: • High-Energy Physics - Theory
Approach: Theoretical
We construct a family of one-dimensional (1D) quantum lattice models based on $G$-graded unitary fusion category $\calC_G$. This family realize an interpolation between the anyon-chain models and
edge models of 2D symmetry-protected topological states, and can be thought of as edge models of 2D symmetry-enriched topological states. The models display a set of unconventional global symmetries
that are characterized by the input category $\calC_G$. While spontaneous symmetry breaking is also possible, our numerical evidence shows that the category symmetry constrains the models to the
extent that the low-energy physics has a large likelihood to be gapless.
Current status:
Has been resubmitted
Reports on this Submission
Report #2 by Anonymous (Referee 2) on 2024-6-24 (Invited Report)
• Cite as: Anonymous, Report on arXiv:scipost_202402_00006v1, delivered 2024-06-23, doi: 10.21468/SciPost.Report.9291
1) Fills in an obvious hole in the literature on non-invertible symmetries, lattice models, and boundaries of topological order.
2) Clearly written with some simple, analytic examples.
3) Presents both analytic and numerical results.
There are some points that I think can be clarified. See comments/questions below.
This paper presents a construction for an interesting class of models describing boundary theories for symmetry-enriched topological orders. These models are amenable to numerical study, and provide
a simple generalization of anyon chain models, which have been useful for studying properties of gapless theories with non-invertible symmetries. The quality of the paper is high, but I would like to
receive answers to the following comments/questions along with appropriate edits to the paper before recommending it for publication:
1) Very small comment on p 1: "string operators associated with moving abelian anyons in the 2D toric code model" may not be the best way to describe 1-form symmetries, because string operators that
move anyons are open strings, which do not commute with the Hamiltonian and do not leave the ground state invariant. Perhaps it is better to use "closed string operators in the 2D toric-code model."
2) The paper draws a connection between the boundaries of SETs and their 1D lattice models. Can the connection be made more concrete? For example, a 2+1D SET is described by a set of data including
the anyon permutation by G and an H^2(G,A) class. These pieces of data are important for boundary physics because they determine whether or not the boundary can be G-symmetrically gapped. How are
these pieces of data read off from the description of the symmetry in 1+1d?
3) Can you comment more on the choice of $a_i$ and the qualitative effects on the physics of the lattice model? In the Ising example, what would happen if you chose $a_0=\psi$ instead of $a_0=1$?
4) Can you comment on the different choices of $\{w_h^z\}$? It seems that if you set $w_0^z=1,w_{h\neq 0}^z=0$ for all sites then you always a $G$ SSB state? Maybe you can add some comments on why
the $H^3(G,U(1))$ class only affects $w_g^0$ in the $\mathbb{Z}_2$ example (does a similar result hold for more general $G$?)
5) Right above eq 38, you mention "We are interested in the cases that the models are gapless, which can be described by conformal field theory (CFT)." How do you know that the gapless regions of the
phase diagram can always be described by a CFT? In some points in Fig 3, you certainly don't get CFT behavior because you have quadratic dispersion.
6) Right above Sec 3.7, do you expect that for at least some choices of parameters i.e. for some $\{w_h^z\}$ you get decoupled Fib and $\mathbb{Z}_2$ theories? i.e. something like a decoupled golden
chain and a trivial $\mathbb{Z}_2$ paramagnet.
7) To be clear, the F moves below Eq 65 are different from the F symbols of the (graded) fusion category? Because they are F symbols of the G-crossed BTC.
8) Should I think of claim (ii) on line 685 as coming from the fact that even if we get an SSB Hamiltonian, the non-tensor-product structure of the Hilbert space projects out all of the ground states
except for one? However, if we reduce to the SPT boundary i.e. we set $\mathcal{C}_0=\{1\}$, then we should get degeneracy because we can get SSB boundary theories and the boundary has a tensory
product Hilbert space?
9) Regarding fermionic theories (outlook point 1), https://arxiv.org/pdf/2404.19004 might be of interest. It might also be worth citing https://arxiv.org/abs/2304.01262 and https://link.springer.com/
article/10.1007/JHEP10(2023)053 in line 68
10) Regarding outlook point 2, I'm confused about why you need the $x_i$ to come from other module categories. You already seem to get access to different phases just by tuning the $\{w_h^z\}$
variables? And the reason for changing the module category is to access other 1+1D phases with the same graded fusion category symmetry. Does this mean that for a given set of $\{w_h^z\}$ for a fixed
set $\{x_i\}$ you only get access to a single gapped phase (and gapless regions)? And in order to access other gapped phases you need to change $\{x_i\}$? For example, for the simple $G=\mathbb{Z}_2$
case with $\mathcal{C}_0=\{1\}$, you get both the trivial symmetric phase ($w_h^z=1$ for all $h,z$) and the SSB phase ($w_0^z=1,w_{h\neq 0}^z=0$). Can you comment more on which phases you expect to
access given $\{x_i\}$?
11) Regarding outlook point 3, I'm not sure what you mean by maximal category symmetry here? For example, you write "More generally, one may expect a larger category symmetry $\mathcal{Z}(\mathcal{C}
_G)$ in the gapless state of the model." However, $\mathcal{Z}(\mathcal{C}_G)$ is braided while the symmetry of a 1+1D system contains only fusion data? It seems that the maximal category symmetry
should be another fusion category, not the center of the fusion category which is a braided tensor category?
12) You don't have to do this, but it would be interesting to compare the anomalous vs non-anomalous phase diagram, to make the claim about "larger likelihood to be gapless" more concrete. Here you
only have the phase diagram in Fig 3 for the model with the anomalous Z2 symmetry.
We thank the referee for a high evaluation on our paper and for considering it "clearly written''. Below are our replies to the comments and questions.
1) Very small comment on p 1: "string operators associated with moving abelian anyons in the 2D toric code model" may not be the best way to describe 1-form symmetries, because string operators
that move anyons are open strings, which do not commute with the Hamiltonian and do not leave the ground state invariant. Perhaps it is better to use "closed string operators in the 2D toric-code
Reply: Thanks for pointing it out. We have rephrased the presentation accordingly in the revised manuscript.
2) The paper draws a connection between the boundaries of SETs and their 1D lattice models. Can the connection be made more concrete? For example, a 2+1D SET is described by a set of data
including the anyon permutation by G and an $H^2(G,A)$ class. These pieces of data are important for boundary physics because they determine whether or not the boundary can be G-symmetrically
gapped. How are these pieces of data read off from the description of the symmetry in 1+1d?
Reply: To address this question, we begin with a general description of 2+1D bosonic SET phases with a unitary symmetry $G$. (We consider SPT phases a special case of SET phases.) A general SET phase
is described by the set of data $(\mathcal{D}, \rho, \omega_2,\nu_3)$, where $\mathcal{D}$ is a unitary modular tensor category, $\rho$ describes the anyon permutation, $\omega_2$ is a 2-cocycle in
$H^2(G,A)$, and the 3-cocycle $\nu_3\in H^3(G,U(1))$ describes the stacking of 2D SPT phases. Whether the boundary of an SET can be gapped out while preserving the symmetry group $G$ depends all the
data in $(\mathcal{D},\rho,\omega_2,\nu_3)$. First of all, to be gapable, it is required that there exists at least one Lagrangian algebra $\mathcal{A}$ in $\mathcal{D}$. Equivalently, $\mathcal{D}$
has to be the Drinfeld center of a unitary fusion category $\mathcal{C}_0$, $\mathcal{D}=Z(\mathcal{C}_0)$. To further require the gapped boundary be $G$-symmetric, complicated conditions need to be
imposed between $\mathcal{A}$ and $(\rho,\omega_2,\nu_3)$.
Meanwhile, for topological orders with gappable boundaries, i.e., $\mathcal{D}=Z(\mathcal{C}_0)$, there exists another description of SET phases. It is the $G$ extensions of the unitary fusion
category $\mathcal{C_0}$ (see arxiv:0909.3140). After an $G$ extension, one has a $G$-graded fusion category $\mathcal{C}_G$ (which is introduced in the manuscript). Different $G$ extensions of $\
mathcal{C}_0$ correspond to different SETs of $\mathcal{D}$. The information of $(\rho,\omega_2,\nu_3)$ are secretly encoded in the $G$-graded category $\mathcal{C}_G$. Lattice model realizations of
SET phases based on $\mathcal{C}_G$ can be found in arxiv:1606.08482 and arXiv:1606.07816.
That being said, we see that the appropriate question is how to extract the SET data $(\rho,\omega_2,\nu_3)$ from the graded fusion category $\mathcal{C}_G$. Whether or not we are studying the SET in
the bulk or on the boundary is irrelevant for this question. In general, extracting $(\rho,\omega_2,\nu_3)$ from $\mathcal{C}_G$ is not known beyond a few examples. In our paper, we use the language
of $G$-graded fusion category for the description of 1+1D boundary models. One may wonder how to 1+1D models using the SET data $(\rho,\omega_2,\nu_3)$. This is a problem for future study.
3) Can you comment more on the choice of $a_i$ and the qualitative effects on the physics of the lattice model? In the Ising example, what would happen if you chose $a_0=\psi$ instead of $a_0=1$?
Reply: Different choices of ${a_i}$ give rise to distinct Hilbert spaces, different realizations of symmetry, and consequently, different models. The symmetry operator in Eq. 6 and Hamiltonian in Eq.
11 have explicit dependence on ${a_i}$. Indeed, the effect of choices of ${a_i}$ is an interesting problem to study. However, we haven't done any study yet, including the case of Ising fusion
4) Can you comment on the different choices of {$w^z_h$}? It seems that if you set $w^z_0=1,w^z_{h\neq 0}=0$ for all sites then you always a G SSB state? Maybe you can add some comments on why
the $H^3(G,U(1))$ class only affects $w^0_g$ in the $Z_2$ example (does a similar result hold for more general $G$?)
Reply: We answer the three questions in order.
(i) To the first question, we are only able to make a general comment that ${w^{z}_{h}} $ are coupling parameters of the model, and different values of ${w_h^z}$ may put the model in different
(ii) To the second question, the setting $w^z_0=1,w^z_{h\neq 0}=0$ for all $z$'s means that, taking the model of Ising fusion category as an example, the Hamiltonian Eq. 25 is a constant. Perhaps,
the referee is thinking of the case that $w_{h\neq 0}^z=0$ while $w_0^z$ are different for different $z$'s. In this case, indeed, no terms in the Hamiltonian can flip the domain degrees of freedom $
{\alpha_i}$. The Hamiltonian seems deep in a "ferromagnetic" or "anti-ferromagnetic" phase (i.e., $\Delta =0$ and $r =\pm \infty$ in Eq. 26). They indeed correspond to the spontaneous breaking of $G$
and also the fusion category symmetry $\mathcal{C}$. However, after a careful investigation, one will find that the ground state degeneracy is infinite in the thermodynamic limit. This large
degeneracy can be lifted by introducing a small $\Delta$ (i.e., small $w_{h\neq 0}^z$), with which the magnetic ordering is then well defined. We will study the physics of spontaneous breaking of
categorical symmetry in future works.
(iii) To the third question, that only $w_g^0$ is affected by $H^3(G,U(1))$ holds only for $G=Z_2$. For a 3-cocycle in $H^3(Z_2,U(1))$, only the element $ \nu_3(g,g,g) = -1$ and all others are 1
(under certain gauge choice). This means that the nontrivial $\nu_3(g,g,g)$ enter the Hamiltonian matrix elements given in Eq. (13) only when $\alpha_{i-1} = \alpha_{i+1} = g$ and $\alpha_i \neq \
alpha_i'$. That is, $\nu_3$ matters only for $w_g^0$. For a generic group $G$, this condition does not hold.
5) Right above eq 38, you mention "We are interested in the cases that the models are gapless, which can be described by conformal field theory (CFT)." How do you know that the gapless regions of
the phase diagram can always be described by a CFT? In some points in Fig 3, you certainly don't get CFT behavior because you have quadratic dispersion.
Reply: We are sorry for the inaccurate presentation which causes misunderstanding. The sentence cited in the question is only intended to convey that the gapless phases (or points) characterized by
CFTs are our main interests. It does not imply that all gapless states of our model are described by CFTs. We appreciate this observation and have rephrased the sentence in the revised manuscript ---
see discussions around line 417.
6) Right above Sec 3.7, do you expect that for at least some choices of parameters i.e. for some ${w^z_h}$ you get decoupled Fib and Z2 theories? i.e. something like a decoupled golden chain and
a trivial Z2 paramagnet.
Reply: It is possible that one can get decoupled Fibonacci and $Z_2$ theories for certain choices of parameters. This would require the choice $a_0=\tau$ and $a_1=1$; instead, if $a_0=1$ and $a_1=\
tau$, it means that Fibonacci defects $\tau$ is decorated on $Z_2$ domain walls, so that the two degrees of freedom are coupled. However, we do not anticipate that the decoupled theories resemble a
system of a decoupled golden chain alongside a trivial ${Z}_2$ paramagnet. This is because the $SU(2)_3$ theory is equivalent to Fibonacci anyon stacked with a nontrivial ${Z}_2$ SPT state.
Accordingly, we expect the decoupled $Z_2$ theory shall resemble the edge of $Z_2$ SPT. Nevertheless, this is speculation and numerical analysis is needed for confirmation.
7) To be clear, the F moves below Eq 65 are different from the F symbols of the (graded) fusion category? Because they are F symbols of the G-crossed BTC.
Reply: No, the $F$ moves below Eq. (65) [Eq. (68) of the revised manuscript] are the same as the $F$ symbols of the graded fusion category. The input category of the symmetry-enriched string-net
model is the same $G$-graded fusion category as the one that we use for the 1D lattice models. To clarify, the input category is different from the $G$-crossed braided tensor category that
characterizes the output SET phase of the symmetry-enriched string-net model.
8) Should I think of claim (ii) on line 685 as coming from the fact that even if we get an SSB Hamiltonian, the non-tensor-product structure of the Hilbert space projects out all of the ground
states except for one? However, if we reduce to the SPT boundary i.e. we set $C_0={1}$, then we should get degeneracy because we can get SSB boundary theories and the boundary has a tensor
product Hilbert space?
Reply: Claim (ii) on line 685 of the previous version is a statement on the degeneracy for a fixed set of parameters ${\alpha_i, a_i, x_i}$. Specifically, it states that, for a fixed set ${\
alpha_i,a_i,x_i}$, the ground state degeneracy is 1. By ``fixing ${\alpha_i, a_i, x_i}$'', one may think of applying an appropriate fictitious Zeeman-like external field to break all symmetries
explicitly, which then gives rise to a unique ground state. To obtain the full degenerate Hilbert space, we sum over all $\alpha_i, a_i$ and $x_i$ (i.e., given by Eq. 66 in the revised manuscript).
This full ground-state space corresponds precisely to the Hilbert space of the 1D model under consideration. With additional interaction on the boundary (ie., Eq 71 in the revised manuscript0),
spontaneously symmetry breaking may occur, depending on the details of the boundary Hamiltonian. Accordingly, the claim (ii) does not contradict with the SSB phenomenon of the SPT boundary.
9) Regarding fermionic theories (outlook point 1), https://arxiv.org/pdf/2404.19004 might be of interest. It might also be worth citing https://arxiv.org/abs/2304.01262 and https://
link.springer.com/article/10.1007/JHEP10(2023)053 in line 68
Reply: Thanks for bringing the references to our attention. We have cited the first reference properly. We also have cited the second reference in line 68. Nevertheless, please note that our preprint
was posted on arXiv in January 2023 (a version very close to the one submitted SciPost), prior to the publication of these papers.
10) Regarding outlook point 2, I'm confused about why you need the $x_i$ to come from other module categories. You already seem to get access to different phases just by tuning the ${w^z_h}$
variables? And the reason for changing the module category is to access other 1+1D phases with the same graded fusion category symmetry. Does this mean that for a given set of ${w^z_h}$ for a
fixed set ${x_i}$ you only get access to a single gapped phase (and gapless regions)? And in order to access other gapped phases you need to change ${x_i}$? For example, for the simple $G=Z_2$
case with $C_0={1}$, you get both the trivial symmetric phase ($w^z_h=1$ for all $h,z$) and the SSB phase ($w^z_0=1,w^z_{h\neq 0}=0$). Can you comment more on which phases you expect to access
given ${x_i}$?
Reply: Indeed, by tuning $ {w_{h}^z} $, we are already able to access different gapped/gapless phases. The purpose of the suggestion to make use of module categories is to have a more general way to
construct 1D lattice models (note that $\mathcal{C}_G$ is a module category of $\mathcal{C}_G$ itself). As a comparison, let us take the spin-$\frac{1}{2}$ and spin-1 chains with $ SO(3) $ symmetry
as examples. Spin $ \frac{1}{2}$'s and spin 1's realize different representations of $ SO(3)$. As is well-known, the spin-$ \frac{1}{2} $ chain and spin-1 chain host very different physics, even
though they share the same $ SO(3) $ symmetry. Therefore, generally speaking, having a more general way to construct models is always useful. Nevertheless, neither do we claim that the current
construction has a severe limitation in accessing phases of matter with fusion category symmetry, nor do we suggest that the general construction with module categories can give rise to new gapless
phases of matter.
11) Regarding outlook point 3, I'm not sure what you mean by maximal category symmetry here? For example, you write "More generally, one may expect a larger category symmetry $\mathcal{Z}(\
mathcal{C}_G)$ in the gapless state of the model." However, $\mathcal{Z}(\mathcal{C}_G)$ is braided while the symmetry of a 1+1D system contains only fusion data? It seems that the maximal
category symmetry should be another fusion category, not the center of the fusion category which is a braided tensor category?
Reply: By ``maximal category symmetry", we mean the category $ \mathcal{Z}(\mathcal{C}_G) $. It describes the bulk topological order after gauging $G$ in the bulk SET. Generally speaking, any closed
string operator corresponding to moving a bulk anyon (a simple object in $\mathcal{Z}(\mathcal{C}_G)$) along the full circle of the boundary of the disk geometry can be viewed as a symmetry of the
boundary theory. Accordingly, the full set of symmetries should be described by $\mathcal{Z}(\mathcal{C}_G)$. Nevertheless, it is not easy implement the full symmetries. Taking the boundary of the
toric-code topological order for example, the four anyons $1,e,m,\epsilon$ gives rise to a $Z_2^e\times Z_2^m$ symmetry. The current 1D construction only makes explicit use of $Z_2^m$, generated by
moving $m$. The $Z_2^e$, related to $Z_2^m$ by the Kramers-Wannier duality, is not implemented in our construction. If the full $Z_2^e\times Z_2^m$ can be made use of, the model shall be pinned at
the self-dual point under Kramers-Wannier duality. That means, a larger symmetry is implemented if one can make the full use of $\mathcal{Z}(\mathcal{C}_G)$.
We borrowed the terminology ``maximal category symmetry'' from Ref. 60 (the previous version of the manuscript), which is defined for a particular CFT under consideration. We realize that it might
not be appropriate in our context, as we do not have a specific CFT in mind. Accordingly, we have removed this terminology and rephrased the Outlook Point 3 in the revised manuscript. We thank the
referee for bringing up the issue.
12)You don't have to do this, but it would be interesting to compare the anomalous vs non-anomalous phase diagram, to make the claim about "larger likelihood to be gapless" more concrete. Here
you only have the phase diagram in Fig 3 for the model with the anomalous Z2 symmetry.
Reply: This is a great suggestion. We thank the referee for bringing this up. We have included a new Fig. 3 (phase diagram of $H^0$) as a comparison to Fig. 4 (phase diagram of $H^1$).
Report #1 by Anonymous (Referee 1) on 2024-2-22 (Invited Report)
• Cite as: Anonymous, Report on arXiv:scipost_202402_00006v1, delivered 2024-02-22, doi: 10.21468/SciPost.Report.8597
This paper gives a systematic construction of 1D lattice models using graded fusion category degrees of freedom. In particular, the authors take the graded fusion category data, define a constrained
Hilbert space out of it, and determine the Hamiltonian that satisfies the graded fusion category symmetry. Several example Hamiltonians are then studied numerically and shown to contain gapless
regions in the phase diagram. This is an interesting work, especially given the recent interest in categorical symmetries. Using the protocol given in this paper, one can systematically construct
models with categorical symmetry that reflect the edge physics of 2D symmetry enriched phases. The paper is very carefully written and is a nice addition to the literature. I only have one minor
comment: This paper https://arxiv.org/abs/2110.12882 seems to be on a related topic, although the result seems to be a subset of that in this paper. Can the authors comment on their relation?
We thank the referee for a high evaluation of our work. Regarding the relation to arXiv:2110.12882, we mention two differences. First, it constructs 1+1D lattice models with a gapped, symmetric and
unique ground state, i.e., SPT phases with fusion category symmetry. The models of arXiv:2110.12882 are exactly solvable Hamiltonians of local commutative projectors. On the other hand, the
Hamiltonians in our work do not consist of commuting projectors and in general cannot be solved analytically. Also, the ground states may be gapless or break fusion category symmetry. Second, to
admit a gapped symmetric unique ground state (i.e., be anomaly-free), the fusion category is highly restricted: it must be the representation category of a Hopf algebra. Accordingly, arXiv:2110.12882
uses the language of Hopf algebra for the model construction. On the other hand, we do not require the fusion category to be anomaly-free, so our construction works for general $G$-graded fusion
category. We have briefly described these differences in the revised manuscript (see the sentences around line 111).
Anonymous on 2024-09-21 [id 4795]
(in reply to
Chenjie Wang
on 2024-09-06 [id 4742])
Thanks for the reply. I think the paper is ready for publication now. | {"url":"https://www.scipost.org/submissions/scipost_202402_00006v1/","timestamp":"2024-11-06T15:22:15Z","content_type":"text/html","content_length":"62690","record_id":"<urn:uuid:ae2df439-3bab-4d62-99ea-bcb7901cde16>","cc-path":"CC-MAIN-2024-46/segments/1730477027932.70/warc/CC-MAIN-20241106132104-20241106162104-00198.warc.gz"} |
Next: Upstream Mach Number, M1, Up: Upstream Mach number, M1, Previous: The Procedure for Calculating Index
The second range in which
With values presented in equations (
) for
Substituting the values of
) (
) into equation (
) provides the equation to be solved for
The author is not aware of any analytical demonstration in the literature which shows that the solution is identical to zero for
. Nevertheless, this identity can be demonstrated by checking several points for example,
) is provided for the following demonstration. Substitution of all the above values into (
) results in
Utilizing the symmetry and antisymmetry of the qualities of the ^13.18. Note that, in the previous case, with a positive large deflection angle, there was a transition from one kind of discontinuity
to another.
│M[1]│a[1]│a[2]│ a[3] │
│1.0 │ -3 │ -1 │ -2/3 │
│2.0 │ 3 │ 0 │ 9/16 │
│ ∞ │ -1 │ 0 │-1/16 │
The various coefficients of three different Mach numbers to demonstrate that
In the range where
zero deflection
. In other words, the wall does not emit any signal to the flow (assuming zero viscosity), which contradicts the common approach. Nevertheless, in the literature, there are several papers suggesting
zero strength Mach wave; others suggest a singular point
. The question of singular point or zero Mach wave strength are only of mathematical interest.
The ``imaginary'' Mach waves at zero inclination.
Suppose that there is a Mach wave at the wall at zero inclination (see Figure (
)). Obviously, another Mach wave occurs after a small distance. But because the velocity after a Mach wave (even for an extremely weak shock wave) is reduced, thus, the Mach angle will be larger (
In reality, there are imperfections in the wall and in the flow and there is the question of boundary layer. It is well known, in the engineering world, that there is no such thing as a perfect wall.
The imperfections of the wall can be, for simplicity's sake, assumed to be as a sinusoidal shape. For such a wall the zero inclination changes from small positive value to a negative value. If the
Mach number is large enough and the wall is rough enough, there will be points where a weak^13.20 weak will be created. On the other hand, the boundary layer covers or smooths out the bumps. With
these conflicting mechanisms, both will not allow a situation of zero inclination with emission of Mach wave. At the very extreme case, only in several points (depending on the bumps) at the leading
edge can a very weak shock occur. Therefore, for the purpose of an introductory class, no Mach wave at zero inclination should be assumed.
Furthermore, if it was assumed that no boundary layer exists and the wall is perfect, any deviations from the zero inclination angle creates a jump from a positive angle (Mach wave) to a negative
angle (expansion wave). This theoretical jump occurs because in a Mach wave the velocity decreases while in the expansion wave the velocity increases. Furthermore, the increase and the decrease
depend on the upstream Mach number but in different directions. This jump has to be in reality either smoothed out or has a physical meaning of jump (for example, detach normal shock). The analysis
started by looking at a normal shock which occurs when there is a zero inclination. After analysis of the oblique shock, the same conclusion must be reached, i.e. that the normal shock can occur at
zero inclination. The analysis of the oblique shock suggests that the inclination angle is not the source (boundary condition) that creates the shock. There must be another boundary condition(s) that
causes the normal shock. In the light of this discussion, at least for a simple engineering analysis, the zone in the proximity of zero inclination (small positive and negative inclination angle)
should be viewed as a zone without any change unless the boundary conditions cause a normal shock.
Nevertheless, emission of Mach wave can occur in other situations. The approximation of weak weak wave with nonzero strength has engineering applicability in a very limited cases, especially in
acoustic engineering, but for most cases it should be ignored.
Figure 13.8: The D, shock angle, and
Next: Upstream Mach Number, M1, Up: Upstream Mach number, M1, Previous: The Procedure for Calculating Index
Created by:Genick Bar-Meir, Ph.D.
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Time Value of Money: Defining the IRR - De Ceuster Project Management Academy
Time Value of Money: Defining the IRR
Get an 80% discount on all our online courses (see links below) with the voucher DCA4FRIENDS
01:21 – Defining the IRR
02:35 – Value of the IRR
08:30 – Formula to calculate the IRR
11:15 – Method of Trial and Error to calculate the IRR
13:40 – Integrated formula of the IRR in Excel
16:05 – Conclusions
A very important parameter to consider in financial analysis is the Internal Rate of Return.
Per definition, it is the interest or discount rate for which the Net Present Value is equal to 0. When you consider the calculation of a mortgage and the monthly payments and you would calculate the
IRR on them, you would see that the IRR is equal to the interest rate charged by the lender.
The IRR is the yield of the investment.
When a company uses money, it is not for free. There is a cost to be paid which can be calculated and is expressed as a percentage of the different ways to invest in the company and use money. When
we would consider as an only investment a bank loan, then the WACC would be equal to the percentage of the loan.
When considering investments, it is clear that, we not only have to make a profit, but that we also have to compensate for the cost of lending money, hence the yield of our investment should be
larger than the rate of the loan, or the WACC.
In some cases, eg. when there is a higher risk factor, we can add some extra profit to the yield to compensate for the risk, in that case, we speak about the Required Rate of Return, which is higher
than the WACC.
In normal cases, the IRR will not be used to select between projects or investments, but it will be used as a cutoff rate. Projects with an IRR equally to or higher than the RRR will be further
evaluated using other parameters, while the projects with a lower IRR will not be considered.
When we look at the formula of the NPV and replace the discount rate with the IRR, it becomes clear that it is not possible to deduct the IRR directly from the formula. However, we can calculate the
IRR by using a mathematical technique called “Trial and Error” where we step ey step close in on the IRR or come closer to finding the yield for which the NPV is equal to zero. Once we come close
enough, typically when we have a precision of 2 decimals, we can stop the calculation. How to do this calculation, will be explained in the next video.
Excel and other programs provide guilt in formulas that will give us “immediately” the requested IRR. The formula in Excel is given by:
Here we select all the values with their respective signs and the formula will calculate the IRR. However, in some cases (after 20 interactions) it is possible that the program does not give a result
and then we have to provide a [guess] to start closer to the value of the IRR.
Nore in the next video where we will look deeper into the step by step calculation of the IRR
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(6x10^8)x(4 X 10^7) In Standard Form
Multiplying Numbers in Scientific Notation
In this article, we'll explore how to multiply numbers expressed in scientific notation, focusing on the specific example of (6 x 10⁸) x (4 x 10⁷).
Understanding Scientific Notation
Scientific notation is a way of expressing very large or very small numbers in a compact and convenient form. It consists of two parts:
• A coefficient: This is a number between 1 and 10.
• A power of 10: This indicates the magnitude of the number.
For example, the number 600,000,000 can be written in scientific notation as 6 x 10⁸.
Multiplying Numbers in Scientific Notation
To multiply numbers in scientific notation, we follow these steps:
1. Multiply the coefficients: Multiply the numbers in front of the powers of ten.
2. Add the exponents of 10: Add the powers of ten.
Let's apply these steps to our example:
(6 x 10⁸) x (4 x 10⁷)
1. Multiply the coefficients: 6 x 4 = 24
2. Add the exponents: 8 + 7 = 15
Therefore, the product of (6 x 10⁸) and (4 x 10⁷) is 24 x 10¹⁵.
Standard Form
Although this answer is technically correct, it's not in standard scientific notation because the coefficient (24) is greater than 10. To convert it to standard scientific notation, we need to move
the decimal point one place to the left and increase the exponent by one:
24 x 10¹⁵ = 2.4 x 10¹⁶
Therefore, the final answer in standard form is 2.4 x 10¹⁶.
Multiplying numbers in scientific notation involves multiplying the coefficients and adding the exponents. Remember to adjust the final answer to standard scientific notation if the coefficient is
not between 1 and 10. This method simplifies calculations with large or small numbers, making them more manageable and easier to work with. | {"url":"https://jasonbradley.me/page/(6x10%255E8)x(4-x-10%255E7)-in-standard-form","timestamp":"2024-11-10T09:32:17Z","content_type":"text/html","content_length":"58755","record_id":"<urn:uuid:f73f7540-46ac-4f25-a394-22a02e600b38>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00133.warc.gz"} |
An Inside Look at Tipping Mechanisms for a Tropical Cyclone
Over two decades ago, The Intergovernmental Panel on Climate Change introduced the idea of tipping points as "large scale discontinuities in the climate" [1]. While there is no precise mathematical
definition of a tipping event, tipping can be thought of as a rapid, and often irreversible, change in the equilibrium state of a dynamical system [2]. In [3] it was proposed that tipping events
could be classified according to whether the underlying mathematical mechanism involves, predominantly, a bifurcation (B-tipping), noise induced transitions (N-tipping), or a rate dependent parameter
(R-tipping); see also [4, 5, 6, 7]. Classically, B-tipping has been used to study subsystems of the climate which are vulnerable to tipping. However, in B-tipping an implicit assumption is that
parameters change slowly relative to the intrinsic timescale set by the deterministic dynamics. The validity of this assumption in climate applications is not clear since anthropogenic change has
occurred over a much shorter time span (centuries) than the shortest geological time scale (millions of years) [8]. Moreover, in many climate applications the dynamical system is necessarily
stochastic and non-autonomous and thus there is need to study how a combination of these mechanisms can also lead to tipping [3].
In [9] we studied these tipping mechanisms for a low-dimensional model of a tropical cyclone. Specifically, this model couples the dynamics of the tangential wind speed of a cyclone to its inner core
moisture with, essentially, the potential energy of the storm, wind shear, and the temperature of the ocean serving as parameters. While classic bifurcations occur as these parameters are varied, we
found non-intuitive results when considering rate and noise-induced tipping. Specifically, we found that rapid increases in the potential energy of a storm coupled with sufficient wind shear can
destabilize a storm. This result is surprising in that while a slow increase of the potential energy results in a more intense storm, if the energy is increased sufficiently rapid the tangential wind
speed cannot increase quickly enough and is dissipated by wind shear. We also found in this model that while the stability of cyclones are exceptionally robust to additive random fluctuations, the
presence of noise can lead to the rapid formation of a tropical storm. This consequence of the model results from the existence of a center manifold along which the deterministic dynamics is
relatively weak. In this article, we summarize the key techniques used in our analysis.
Because of the destruction that tropical cyclones can cause [10], understanding what mechanisms lead to their formation should be of interest to policy makers, risk analysts, and climate scientists.
Indeed, a presumed impact of global climate change is the increase in frequency and intensity of tropical cyclones. For example, in 2019 Hurricane Dorian made landfall in the Bahamas as a category 5
hurricane, sustaining winds over 185 miles per hours and it caused an estimated $7 billion in damages, over 400 dead or missing persons, and immeasurable losses to reef and mangroves, which in turn
impacted tourism, the fishing industry, and protection from future storms [11]. As a second example, in 2005, Hurricane Katrina struck the gulf coast of Louisiana and was one of the costliest storms
on record, causing over $125 billion, and over 1,800 lives lost despite only sustaining winds of 127 mph upon landfall [12].
1. Mathematical Model of a Tropical Cyclone
Tropical cyclones are axisymetric vortices that form over warm ocean water in which there is a temperature gradient between the warm ocean and cooler lower atmosphere. In these regions, as warm water
evaporates, the resulting warm air mass rises and cools rapidly releasing heat through condensation back into the atmosphere. As the warm air rises, an area of low pressure forms and air begins to
move from all directions to fill this void. The air in this region swirls from the Coriolis effect and, due to conservation of angular momentum, eventually forms a rotating air mass around the area
of low pressure, i.e., the eye of the storm. In this idealized setting, this process can modeled as a Carnot engine in which the maximum potential velocity of the hurricane, \(V_p>0\) can be obtained
by equating the kinetic energy with the theoretical maximum power that could be sustained by the storm [13, 14]. However, in realistic storms, wind shear in the form of strong upper level winds can
dissipate the storm structure by displacing warm temperatures above the storm eye.
To study the formation of tropical storms, the model we considered was a stochastic perturbation of a dynamical system developed in [15, 16] and is given in dimensionless form by
\( \begin{aligned} dv&=f(v,m)d\tau+\sigma_1dW_1:=\left[(1-\gamma)m^3-(1-\gamma m^3)v^2\right]d\tau+\sigma_1dW_1,\\ dm&=g(v,m)d\tau+\sigma_2dW_2:=\left[(1-m)v-cm\right]d\tau+\sigma_2dW_2, \end (1)
{aligned} \)
where we have restricted the dynamics to the physically relevant first quadrant by assuming reflecting boundary conditions on the axes. Here \(v=V/V_p\in [0,1]\) is a dimensionless measure of the
tangential velocity \(V\geq 0\) of the storm relative to a maximum potential velocity \(V_p>0\), \(\tau \sim V_p^{-1}t\), \(m\in [0,1]\) is the relative humidity in the core of the storm, \(W_1,W_2\)
are standard Wiener processes, \(c=2.2S/V_p\) is a dimensionless measure of wind shear \(S\) relative to \(V_p\), and \(\sigma_1,\sigma_2>0\) are measures of the amplitude of random fluctuations. The
dimensionless parameter \(\gamma\) is defined by \(\gamma=(T_A-T_O)/T_O+\kappa\), where \(T_A\), \(T_O\) are the temperatures of the lower atmosphere and upper ocean respectively and \(\kappa\) is a
constant, and thus \(\gamma^{-1}\) is a proxy for the temperature of the ocean. In this model, \(m\) serves as a source of energy for the storm and if \(m=1\), i.e., the core is fully saturated, the
equilibrium velocity is given by \(V=V_p\). The wind shear plays the role of friction in the system in the sense that it dissipates energy by pulling moisture from the storm and if \(c=0\) we see
that \(m=1,v=1\) is a stable fixed point, i.e., the storm reaches its full potential intensity (assuming \(\sigma_1=\sigma_2=0\)).
The deterministic skeleton of Equation (1) contains an asymptotically stable fixed point O (non-storm state) at the origin. The system also exhibits a saddle-node bifurcation in which a stable node S
(stable storm state) and a saddle U (unstable storm state) emerge as \(c\) and \(\gamma\) are varied. Figure 1(a) is a generic phase portrait of this system in a parameter regime in which all three
fixed points exist. Figure 1(b) is a "phase diagram" indicating the intensity of the storm state (when it exists) by the velocity component of S. This figure illustrates that this model predicts more
intense storms, which are more difficult to dissipate through wind shear, will result from increasing ocean temperatures, i.e., \(\gamma\rightarrow 0\).
2. Rate-Induced Tipping
Rate-induced tipping occurs when a quick change of a parameter causes the system to move away from one attractor to another [2]. However, this change of system behavior happens without the system
undergoing a bifurcation; fixed points may move through different basins of attraction over a parameter shift, but never cross a bifurcation curve. In particular, it is the rate of change of the
parameters not the necessarily the specific changes in the values of the parameters that governs the tipping. In our work, we assumed a parameter shift \(\Lambda_r\) that varies at a rate \(r>0\), is
bi-asymptotically constant, and is monotonically increasing. Note, since O is always a fixed point it follows that regardless of parameter values there can be no rate induced tipping away from O.
Consequently, we focused on rate induced tipping which could destabilize the storm, i.e., causes the system to move from S to O.
The conditions for rate-induced tipping to occur are actually the same for any dimension: if the initial state satisfies a sufficient condition called forward threshold unstable [17]. Essentially
this condition implies the initial state at the start of the parameter shift lies in the basin of attraction of a different stable state at the end of the parameter shift. However, we note forward
threshold stability is not a necessary condition to prevent rate-induced tipping in systems of higher dimension. Indeed, for systems of dimension \(n>1\), the more refined condition of inflowing
stability guarantees that rate-induced tipping cannot happen [18].
To consider the possibility of rate-induced tipping aiding in the formation or destabilization of a storm, we needed physical parameters with the ability to rapidly change. In the model, both wind
shear and maximum potential velocity meet these requirements, and thus we allowed \(c\) and \(V_p\) to vary with time. As rate-induced tipping is a deterministic mechanism, we set \(\sigma_1=\sigma_2
=0\) and considered the dimensional system. To do this, we redefined both \(V_p\) and \(c\) as functions of a parameter shift \(\Lambda_r=\frac{1}{2}(1+\tanh(rt))\). Specifically, we assumed \(V_p,c
\) to transition in time between \(V_p^-\) to \(V_p^+\) and \(c^-\) to \(c^+\), respectively, which represents the minimum and maximum values of the parameters. Additionally, we chose parameter
shifts such that there are always three fixed points at the start and end of the ramp which we denote O, U\(^-\), S\(^-\) and O, U\(^+\), S\(^+\) respectively.
In regard to storm destabilization, we used the condition of inflowing stability to show that if either \(V_p(\Lambda_r(\tau))\) or \(c(\Lambda_r(\tau))\) is nonincreasing as a function of \(\tau\),
there can be no rate-induced tipping away from the stable storm state S\(^-\) to the non-storm state O. However, if both \(V_p(\Lambda_r(\tau))\), and \(c(\Lambda_r(\tau))\) are increasing, then
there is the possibility of rate-induced tipping from S\(^-\) to O for \(r\) sufficiently large. This implies both wind shear and maximal potential velocity have to increase at a substantial rate in
order to affect tipping away from the active storm state. These results are illustrated by a numerical example in Figure 2. In this example, we chose a parameter shift \(\Lambda_r(\tau)\), and
increasing functions \(V_p\) and \(c\) that are dependent on \(\Lambda_r(\tau)\) such that S\(^-\) lies in the basin of attraction of O at the end of the parameter shift, ensuring forward threshold
instability; see Figure 2(a). Figure 2(b) illustrates that for \(r\) small we endpoint track the stable path from S\(^-\) to S\(^+\), and no tipping occurs but when \(r\) is sufficiently large we tip
from S\(^-\) to O.
3. Noise-Induced Tipping
A noise-induced tipping event from O to S is a realization of Equation (1) satisfying \((v(0),m(0))=\)O, and there exists \(\tau^*\in \mathbb{R}^+\) for which \((v(\tau^*), m(\tau^*))\) lies within
the basin of attraction of S and for \(\tau<\tau^*\), \((v(\tau),m(\tau))\) lies within the basin of attraction of O. A similar defintion holds for tipping events from S to O. The variable \(\tau^*\)
is itself a random variable and is referred to as the tipping time from O to S. In Figures 3(a-b) we plot tipping events in the phase plane and as a time series. These numerical experiments indicate
that O is far more susceptible to noise-induced tipping than S. That is, the expected value of the tipping time from O to S is dramatically smaller than from S to O. Moreover, noise-induced tipping
events from O to S appear to be concentrated about a particular region in phase space.
The observations given above were quantitatively studied in the asymptotic limit \(\sigma_1,\sigma_2\rightarrow 0\) using the Freidlin-Wentzell (FW) theory of large deviations; see [19, 20, 21, 22]
for thorough introductions to this topic. To simplify the following exposition, we let \(F=(f,g)\) denote the vector field with components \(f\), \(g\), introduce the matrix
\( \Sigma=\begin{bmatrix} \sigma_1^{-2} & 0\\ 0 & \sigma_2^{-2} \end{bmatrix}, \)
and define for \(\mathbf{v}_1,\mathbf{v}_2\in \mathbb{R}^2\) the weighted inner product \(\langle \mathbf{v}_1,\mathbf{v}_2\rangle_{\Sigma}=\mathbf{v}_1^{T}\Sigma \mathbf{v}_2\) and the weighted norm
\(\|\mathbf{v}_2-\mathbf{v}_1\|_{\Sigma}^2=\langle \mathbf{v}_2-\mathbf{v}_1,\mathbf{v}_2-\mathbf{v}_1\rangle_{\Sigma}\). One key result from FW theory is that in the limit \(\sigma_1,\sigma_2\
rightarrow 0\) tipping events concentrate about most probable transition paths \(\Psi^*(s)=(\psi^*_1(s),\psi^*_2(s))\) which minimize a rate functional. Specifically, for Equation (1), the rate
functional is for most probable transition paths from O to S is given by
\( I[\Psi]=\frac{1}{2}\int_{-\infty}^{\infty}\|\dot{\Psi}-F(\Psi)\|_{\Sigma}^2ds, \) (2)
which is defined for sufficiently regular curves satisfying \(\lim_{s\rightarrow -\infty}\Psi(s)=\)O and \(\lim_{s\rightarrow \infty}\Psi(s)=\)S. Furthermore, knowledge of the most probable
transition path allows computation of the expected tipping time through the relationship
\( \mathbb{E}[\tau^*]=\exp(I[\Psi^*])\left(C+O\left(\left\|\sqrt{\Sigma^{-1}}\right\|\right)\right), \) (3)
where \(C>0\) is a constant.
To numerically compute minimizers, we used a gradient flow which matches well with Monte-Carlo simulations; see Figure 4. However, through a Legendre transformation \(\mathbf{p}=\Sigma (\dot{\Psi}-F
(\Psi))\) the Euler-Lagrange equations corresponding to \(I\) can be expressed in the following Hamiltonian form
\( \begin{aligned} \dot{\Psi}&=F(\Psi)+\Sigma^{-1}\mathbf{p},\\ \dot{\mathbf{p}}&=-\nabla F^T(\Psi)\mathbf{p}, \end{aligned} \) (4)
with corresponding Hamiltonian \(H=\frac{1}{2}\|\mathbf{p}\|_{\Sigma^{-1}}^2+\langle F(\Psi), \mathbf{p}\rangle\). Consequently, most probable transition paths can be interpreted as heteroclinic
orbits connecting O to U joined to a curve satisfying the deterministic dynamics from U to S, i.e., \(\dot{\Psi}=F(\Psi)\) and \(\mathbf{p}=0\). At \((\)O\(,0)\) there are one-dimensional unstable
and stable manifolds W\(^U\) and W\(^S\) respectively as well as a a two-dimensional center manifold W\(^C\). We found that the heteroclinic orbit is given by \(H^{-1}(0)\cap\)W\(^C\) which locally
agrees with the center manifold for the deterministic dynamics. Using knowledge of the most probable path, we obtained the following scaling law (up to logarithmic equivalence) for the tipping time \
(\tau_r\) from a neighborhood of O of characteristic size \(r\):
\( \mathbb{E}[\tau_r^*]\asymp \exp(I[\Psi^*,\mathbf{p}^*])\lesssim \exp\left(\frac{4}{3}\frac{r^3c^7}{\sigma_1^2}+\frac{36}{7}\frac{c}{\sigma_2^2}\left(\frac{r^3 c^7}{\sigma_1^2}\right)^2\ (5)
right), \)
This scaling law identifies the two dimensionless measures of noise strength \(\tilde{\sigma}_1^2=\sigma_1^2/c^7\) and \(\tilde{\sigma}_2^2=\sigma_2^2/c\) which control the tipping time. In
particular, in Figure 3 the ratio \(c^7/\sigma_1^2\) is \(O(1)\) explaing why O is particulary susceptible to noise-induced tipping.
4. Final Comments
In our analysis of tipping in a tropical storm, we considered the various tipping mechanisms to be independent of each other. A natural question is how do various mechanisms couple to induce tipping
events and so a natural extension of the analysis presented in our work is consider the interplay of a parameter shift and additive noise. For the specific system we considered, we expect that in
tipping away from the stable storm state, there will be an interplay between the rate and noise-induced tipping mechanisms, and the additive noise will lower the critical rate needed for tipping [23,
24]. Additionally, tipping should occur away from the non-storm state, but as there was no rate-induced tipping with this initialization, we must explore how rapidly changing parameters could enhance
or inhibit noise induced tipping from O.
[1] T. M. Lenton, J. Rockström, O. Gaffney, S. Rahmstorf, K. Richardson, W. Steffen, and H. J. Schellnhuber, Climate tipping points—too risky to bet against, Nature, vol. 575, no. 7784, pp. 592-595,
[2] P. Ashwin, C. Perryman, and S. Wieczorek, Parameter shifts for nonautonomous systems in low dimension: bifurcation-and rate-induced tipping, Nonlinearity, vol. 30, no. 6, p. 2185, 2017.
[3] P. Ashwin, S. Wieczorek, R. Vitolo, and P. Cox, Tipping points in open systems: bifurcation, noise-induced and rate-dependent examples in the climate system, Philosophical Transactions of the
Royal Society A, vol. 370, no. 1962, pp. 1166-1184, 2012.
[4] J. Thompson and J. Sieber, Predicting climate tipping as a noisy bifurcation: a review, International Journal of Bifurcation and Chaos, vol. 21, no. 02, pp. 399-423, 2011.
[5] T. Lenton, Early warning of climate tipping points, Nature Climate Change, vol. 1, no. 4, p. 201, 2011.
[6] L. Halekotte and U. Feudel, Minimal fatal shocks in multistable complex networks, Scientific reports, vol. 10, no. 1, pp. 1-13, 2020.
[7] H. Alkhayuon, R. C. Tyson, and S. Wieczorek, Phase tipping: how cyclic ecosystems respond to contemporary climate, Proceedings of the Royal Society A, vol. 477, no. 2254, p. 20210059, 2021.
[8] D. J. Wuebbles, D. W. Fahey, K. A. Hibbard, D. J. Dokken, B. C. Stewart, T. K. Maycock, D. J. Wuebbles, D. W. Fahey, K. A. Hibbard, B. DeAngelo, S. Doherty, K. Hayhoe, R. Horton, J. P. Kossin, P.
C. Taylor, A. M. Waple, and C. P. Weaver, Executive summary, pp. 12-34, Washington, DC, USA: U.S. Global Change Research Program, 2017.
[9] K. Slyman, J. A. Gemmer, N. K. Corak, C. Kiers, and C. K. Jones, Tipping in a low-dimensional model of a tropical cyclone, Physica D: Nonlinear Phenomena, vol. 457, p. 133969, 2024.
[10] N. Mori, T. Takemi, Y. Tachikawa, H. Tatano, T. Shimura, T. Tanaka, T. Fujimi, Y. Osakada, A. Webb, and E. Nakakita, Recent nationwide climate change impact assessments of natural hazards in
japan and east asia, Weather and Climate Extremes, vol. 32, p. 100309, 2021.
[11] C. Dahlgren and K. Sherman, Preliminary assessment of hurricane dorian’s impact on coral reefs of abaco and grand bahama, Perry Institute of Marine Science Report to the Government of The
Bahamas, 2020.
[12] A. Graumann, T. G. Houston, J. H. Lawrimore, D. H. Levinson, N. Lott, S. McCown, S. Stephens, and D. B. Wuertz, Hurricane katrina: A climatological perspective: Preliminary report, US Department
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[13] K. Emanuel, Hurricanes: Tempests in a greenhouse, Physics Today, vol. 59, no. 8, pp. 74-75, 2006.
[14] K. Emanuel, Self-stratification of tropical cyclone outflow. part ii: Implications for storm intensification, Journal of the Atmospheric Sciences, vol. 69, no. 3, pp. 988-996, 2012.
[15] K. Emanuel and F. Zhang, The role of inner-core moisture in tropical cyclone predictability and practical forecast skill, Journal of the Atmospheric Sciences, vol. 74, no. 7, pp. 2315-2324,
[16] K. Emanuel, A fast intensity simulator for tropical cyclone risk analysis, Natural Hazards, vol. 88, pp. 779-796, Sept. 2017.
[17] S. Wieczorek, C. Xie, and P. Ashwin, Rate-induced tipping: Thresholds, edge states and connecting orbits, Nonlinearity, vol. 36, no. 6, p. 3238, 2023.
[18] C. Kiers and C. K. Jones, On conditions for rate-induced tipping in multi-dimensional dynamical systems, Journal of Dynamics and Differential Equations, vol. 32, pp. 483-503, 2020.
[19] M. I. Freidlin and A. D. Wentzell, Random perturbations of dynamical systems, Springer Science & Business Media, vol. 260, 2012.
[20] N. Berglund, Kramers' law: Validity, derivations and generalisations, Markov Process Relat Fields, vol. 19, no. 3, pp. 459-490, 2013.
[21] E. Forgoston and R. O. Moore, A primer on noise-induced transitions in applied dynamical systems, SIAM Rev, vol. 60, no. 4, pp. 969-1009, 2018.
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291-363, 2021.
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Please login or register to post comments. | {"url":"https://dsweb.siam.org/The-Magazine/Article/an-inside-look-at-tipping-mechanisms-for-a-tropical-cyclone","timestamp":"2024-11-08T04:30:51Z","content_type":"text/html","content_length":"77474","record_id":"<urn:uuid:774dcc83-807e-4bd7-b322-d5cfd74e018a>","cc-path":"CC-MAIN-2024-46/segments/1730477028025.14/warc/CC-MAIN-20241108035242-20241108065242-00590.warc.gz"} |
How to use the sin method in Julia
The sin() method calculates the sine value of x , where xis in radians.
Julia's sin function syntax
This method takes the parameter x, which represents the radian value for which the sine value is to be found.
Return value
This method returns the sine value of the argument.
The code below demonstrates the use of the sin method:
## find sin of radian
degree = 0
radian = deg2rad(degree)
println( "sin($(radian)) => $(sin(radian))")
## find sin of 10
degree = 10
radian = deg2rad(degree)
println( "sin($(radian)) => $(sin(radian))")
Line 2: We create a new variable degree and assign 0 as a value to it.
Line 3: We use the deg2rad method with the degree variable as an argument. This method converts the degree to radians. The radian value of 0 degrees is 0.0.
Line 4: We use the sin method with the variable radian as an argument. This method calculates the sine value of the radian. In our case, the sine value of radian is 0.0.
Line 7: We assign 10 as the value for the degree variable.
Line 8: We use the deg2rad method with the degree variable as an argument. This method converts the degree to radians. The radian value of 10 degrees is 0.17453292519943295.
Line 9: We use the sin method with the variable radian as an argument. This method calculates the sine value of the radian. In our case, the sine value of radian is 0.17364817766693033. | {"url":"https://www.educative.io/answers/how-to-use-the-sin-method-in-julia","timestamp":"2024-11-14T09:10:40Z","content_type":"text/html","content_length":"136753","record_id":"<urn:uuid:2ead3724-753b-4013-a714-a96272794a92>","cc-path":"CC-MAIN-2024-46/segments/1730477028545.2/warc/CC-MAIN-20241114062951-20241114092951-00283.warc.gz"} |
Renormalization of the vector current in QED
It is commonly asserted that the electromagnetic current is conserved and therefore is not renormalized. Within QED we show (a) that this statement is false, (b) how to obtain the renormalization of
the current to all orders of perturbation theory, and (c) how to correctly define an electron number operator. The current mixes with the four-divergence of the electromagnetic field-strength tensor.
The true electron number operator is the integral of the time component of the electron number density, but only when the current differs from the MS̄-renormalized current by a definite finite
renormalization. This happens in such a way that Gauss's law holds: the charge operator is the surface integral of the electric field at infinity. The theorem extends naturally to any gauge theory.
All Science Journal Classification (ASJC) codes
• Nuclear and High Energy Physics
• Physics and Astronomy (miscellaneous)
Dive into the research topics of 'Renormalization of the vector current in QED'. Together they form a unique fingerprint. | {"url":"https://pure.psu.edu/en/publications/renormalization-of-the-vector-current-in-qed","timestamp":"2024-11-12T04:05:08Z","content_type":"text/html","content_length":"47268","record_id":"<urn:uuid:97ba35cf-c786-4272-9b71-faae2b4a9ec8>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.50/warc/CC-MAIN-20241112014152-20241112044152-00416.warc.gz"} |
Liu Hui's Exhaustion Method
attributed to 200–300 CE
Liu Hui's Exhaustion Method
Using recursively inscribed polygons to approximate π
A commentary prepared by Chinese mathematician Liu Hui around 300 CE used a clever method of exhaustion based on the Pythagorean theorem for determining the area of a disk and estimating the value of
This page from a sixteenth-century Ming dynasty edition of Jiuzhang suanshu (The Nine Chapters on the Mathematical Art) illustrates the method of exhaustion for determining the area of a disk. While
no copies of the original work survive, it is preserved through extant copies of an edition and commentary prepared by Liu Hui around 300 CE. By inscribing regular polygons in a
circle and computing their areas using a recursive procedure based on the areas of the "kites" made from pairs of triangles, Liu proved that 3.14103 < π < 3.14271 and gave an accurate estimate of the
actual value as π = 3927/1250 = 3.1416. | {"url":"https://www.history-of-mathematics.org/artifacts/liu-exhaustion-method","timestamp":"2024-11-03T21:21:54Z","content_type":"text/html","content_length":"15893","record_id":"<urn:uuid:b68347d0-8858-4e7a-a0cc-fd1d8cf2b998>","cc-path":"CC-MAIN-2024-46/segments/1730477027796.35/warc/CC-MAIN-20241103212031-20241104002031-00608.warc.gz"} |
Laplace transform
Articles containing keyword "Laplace transform":
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JMI-04-28 » Bounds improvement for alternating Mathieu type series (09/2010) | {"url":"https://search.ele-math.com/keywords/Laplace-transform","timestamp":"2024-11-12T03:56:28Z","content_type":"application/xhtml+xml","content_length":"10534","record_id":"<urn:uuid:96e3b683-35ff-48c4-b193-d7f665568375>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.50/warc/CC-MAIN-20241112014152-20241112044152-00415.warc.gz"} |
ksyx's blog posts
This post extends the traditional derangment problem to calculate number of acceptable placement solution, with introducing a new concept
that opening box
is the first step, and each of the following step is to open another box whose number equals the number of ball from the last opened box until found a box whose ball in it is numbered
. The task is to find amount of ways to place balls into boxes, so that each box contains exactly one ball, and that for all
from 1 to
, ball
can be found by opening no more than
It is obvious that
=1 is like original problem and the number of placement is 1, placing each ball into the box with the same number. With some error allowed in finding the ball, the problem become interesting. The
idea of this extended problem's solution raises from an easily understandable way of computing original problem's answer by decomposing the problem into multiple subproblems (solution set size
obtained from
For the nature that it extends the idea of another Chinese post, English version of this post is currently unavailable. Hope
translation software
helps :) | {"url":"https://ksyx.link/posts/index.html","timestamp":"2024-11-03T22:43:38Z","content_type":"text/html","content_length":"5645","record_id":"<urn:uuid:f2cbf81d-0ea2-4071-953c-408913ff3ab0>","cc-path":"CC-MAIN-2024-46/segments/1730477027796.35/warc/CC-MAIN-20241103212031-20241104002031-00857.warc.gz"} |
overlapMBH: overlapMBH in susanjarvis501/MBH: Model-based hypervolumes
1 overlapMBH(hv1, hv2, overlap = TRUE, plot = TRUE, dims = c(1, 2),
2 col1 = "black", col2 = "blue", proppoints = 1, ndraws = 999)
overlapMBH(hv1, hv2, overlap = TRUE, plot = TRUE, dims = c(1, 2), col1 = "black", col2 = "blue", proppoints = 1, ndraws = 999)
hv1 Fitted MBH model
hv2 Fitted MBH model
overlap Logical. Do you want to calculate overlap? This can be very slow
plot Logical. Do you want to plot overlap?
dims Dimensions to plot
col1 Colour to use for first hypervolume
col2 Colour to use for second hypervolume
proppoints Number of points to sample from each hypervolume calculated as a proportion of the total volume of each hypervolume. Defaults to 1 but consider reducing to reduce computation time
ndraws Number of draws from multivariate normal used in overlap calculation. Defaults to 999. Reducing the number of draws will reduce computational time but will also reduce precision of the
overlap estimate.
Logical. Do you want to calculate overlap? This can be very slow
Number of points to sample from each hypervolume calculated as a proportion of the total volume of each hypervolume. Defaults to 1 but consider reducing to reduce computation time
Number of draws from multivariate normal used in overlap calculation. Defaults to 999. Reducing the number of draws will reduce computational time but will also reduce precision of the overlap
Utilises a simulation based approach to calculate overlap by simulating a number of points from each hypervolume. Returns an overlap statistic defined by total number of points shared divided by
total number of points simulated. The density of points in each hypervolume is kept constant. Can be very slow for large hypervolumes, both proppoints and ndraws could be reduced for faster
computation but larger values will give more precise estimates.
For more information on customizing the embed code, read Embedding Snippets. | {"url":"https://rdrr.io/github/susanjarvis501/MBH/man/overlapMBH.html","timestamp":"2024-11-12T13:21:15Z","content_type":"text/html","content_length":"24946","record_id":"<urn:uuid:ad3cfd05-3df6-4672-b0fa-22dc3ad02a2b>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.45/warc/CC-MAIN-20241112113320-20241112143320-00579.warc.gz"} |
How Many Amps Does an AA Battery Deliver? - The Power Facts
How Many Amps Does an AA Battery Deliver?
An AA battery is a single celled battery that provides 1.5 volts of power. The capacity of an AA battery is typically 2,600 to 3,000 mAh. One amp hour (Ah) is the amount of current a one amp load
will draw from a fully charged battery in one hour.
Therefore, an AA battery can provide two to three amps for one hour before it needs to be recharged.
An AA battery typically delivers between 1.5 and 3 amps of current. The amount of current a battery can deliver is determined by its size and chemistry. Lithium-ion batteries are able to deliver more
current than other chemistries, but they are also more expensive.
Credit: portablepowerguides.com
How Many Amps are in a AA Battery?
A AA battery typically has lower mAh around 2,500-3,000 mAh (milliamp hours). This means that if you have a device that uses 1 amp of current, it will last for 2.5-3 hours on a full AA battery.
How Many Amps Does a 1.5 Volt Battery Have?
A 1.5 volt battery has a capacity of around 3,000mAh. This means that it can provide a current of up to 3 amps for an hour, or a current of 1.5 amps for two hours.
What is the Amperage of 4 AA Batteries?
4 AA batteries have an amperage of 4000 mAh. This is a high capacity for AA batteries, and will allow them to power devices that require a lot of power for a long period of time.
What is the Output of a AA Battery?
An AA battery is a type of dry cell battery. The output of an AA battery is 1.5 volts.
AA Battery Discharge Amperage Test, Which Type Is Best?
How Many Amps in a AA Battery
A AA battery is a single cell dry battery. The terminal voltage of a AA battery is 1.5 volts. The capacity of a typical AA battery is about 2800mAh.
Therefore, a AA battery can provide 1.5 volts x 2800mAh = 4200 amp hours (4.2 Ah) of power.
AA Battery Amps And Volts
Batteries come in all shapes and sizes, and each has its own benefits and drawbacks. But when it comes to AAA batteries, there’s one clear winner: the alkaline AAA battery. Alkaline AAA batteries
have a higher voltage than other types of batteries, which means they can power your devices for longer.
They also have a higher amp rating, which means they can provide more current to your devices. And because they’re made with fewer toxic materials, they’re better for the environment. So if you’re
looking for the best AAA battery on the market, choose an alkaline AAA battery.
How Many Amps in 2 AA Batteries?
Are you curious about how many amps are in 2 AA batteries? Well, the answer may surprise you. While the average AA battery can hold around 3,000mAh of charge, that doesn’t mean it can supply that
much current.
In fact, most AA batteries are only rated for around 500mA of continuous current. So, if you were to connect two AA batteries in series, you would only be able to get a maximum of 1A of current from
them. This is still plenty of power for most applications though, so don’t worry too much about it.
How Many Amps in a 1.5V AA Battery?
AA batteries come in a variety of voltages, the most common being 1.5 V. But how many amps are in a 1.5V AA battery? The answer depends on the size of the battery. AA batteries come in three
different sizes: AA, AAA, and C. The larger the battery, the more amps it will have.
A AA battery has about 2200mAh, which means it has about 2200 milliamps of current available to it. This is enough to power small devices for a short period of time. A AAA battery has about 1000mAh,
which means it can provide about 1000 milliamps of current.
This is enough to power small devices for a shorter period of time than a AA battery. A C battery has about 7500mAh, which means it can provide about 7500 milliamps of current. This is enough to
power large devices for a long period of time.
How Many Amps in 8 AA Batteries?
If you’re looking for how many amps are in 8 AA batteries, you’ve come to the right place. In this blog post, we’ll go over everything you need to know about AA battery amp hours. As most people
know, an AA battery is a dry cell-type of primary battery that typically has a voltage of 1.5 Volts.
The standard size for an AA cell is 14mm in diameter and 50mm in length. And as the name suggests, 8 AA batteries would be 8 of these cells connected together in series or parallel (or a combination
of both). Now when it comes to capacity or how long a battery lasts, that all depends on the current draw or how much power is being used by the device it’s powering.
But as a general rule of thumb, an AA battery can provide around 2-3 Amps for continuous use or up to 10 Amps for short bursts (like with digital cameras). So based on that information, we can
estimate that 8 AA batteries could provide somewhere between 16-24 Amps continuously or up 80 Amps for short periods of time. Of course, these are just rough estimates since there are other factors
like temperature and age of the batteries that can affect their performance.
But hopefully this gives you a better idea of how many amps are in 8 AA batteries.
How Many Amps in 4 AA Batteries?
If you have ever wondered how many amps are in 4 AA batteries, the answer is 8. This is because each AA battery has 2 amps. When you put 4 of them together, you get 8 amps.
How Many Amps in 3 AA Batteries?
Have you ever wondered how many amps are in 3 AA batteries? Well, the answer may surprise you. It turns out that there are a whopping 24 amps in 3 AA batteries!
That’s enough to power most small devices and appliances. So, what does this mean for you? If you’re ever in a situation where you need to power something but don’t have access to an outlet, 3 AA
batteries should do the trick.
Just remember to keep an eye on the amperage rating of your device or appliance so that you don’t overload it.
Aa Battery Voltage Chart
Most AA batteries have a voltage of 1.5 volts, but there are some that have a higher voltage. The highest voltage AA battery is the Duracell Ultra AAA which has a voltage of 3.6 volts. This battery
is typically used in high-end electronic devices such as digital cameras.
We all know that AA batteries are a common size of battery. But how much power do they actually deliver? And what does that mean in terms of amps?
In general, a AA battery can deliver about 2-3 amps. This is enough to power most small electronic devices. However, it’s important to note that the actual amount of power delivered will vary
depending on the type of AA battery you’re using.
For example, alkaline batteries tend to deliver more power than lithium ion batteries. So, if you’re looking for an estimate, a good rule of thumb is that a AA battery can deliver about 2-3 amps. But
keep in mind that this number will vary depending on the specific battery you’re using.
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In the following figure, ABCD is a parallelogram and EFCD is a rectangle. Also, \
We will first find the area of the parallelogram using the formula for the area of a parallelogram. Then we will find the area of the rectangle using the formula for the area of a rectangle. We will
compare both the areas to prove that both the areas are equal.
Formulas used: We will use the following formulas:
1) The area of a rectangle \[{A_r} = l \times b\], where \[l\] is the length and \[b\] is the base.
2) The area of a parallelogram, \[{A_p} = b \times h\] , where \[h\] is the height and \[b\] is the base.
Complete step by step solution:
We will first prove the \[{\rm{ar}}\left( {ABCD} \right) = DC \times AL\].
(ii) We will find the area of parallelogram ABCD. We know that the area of a parallelogram is the product of the length of its height and base. So, we can write
\[ar\left( {ABCD} \right) = b \times h\]
The height of a parallelogram is the perpendicular distance between its 2 parallel opposite sides and the base of the parallelogram is its lowermost side. We can see from the figure that the height
of the parallelogram is \[AL\] and the base is \[DC\]. We will substitute these values in the formula for the area of a parallelogram:
\[ \Rightarrow ar\left( {ABCD} \right) = AL \times DC\]
Hence, we have proved \[{\rm{ar}}\left( {ABCD} \right) = DC \times AL\].
Now, we will prove the \[{\rm{ar}}\left( {ABCD} \right) = {\rm{ar}}\left( {EFCD} \right)\].
We will find the area of the rectangle EFCD. We know that the area of a rectangle is its length times breadth.
\[ \Rightarrow ar\left( {EFCD} \right) = {\rm{length}} \times {\rm{breadth}}\]
Usually, the longer side of a rectangle is taken as its length and the shorter side is taken as its breadth. The breadth of a rectangle is always perpendicular to its length just like the height of a
parallelogram. We can see from the figure that the length of the rectangle is \[EF\] and the breadth is \[ED\]. We will substitute these values in the formula for the area of a rectangle.
\[ \Rightarrow ar\left( {EFCD} \right) = EF \times ED\]
We know that the opposite sides of a rectangle are equal in length. So,
\[EF = DC\]
We can see from the figure that the breadth of the rectangle and the height of the parallelogram are equal. So, we can write
\[ED = AL\]
So, the area of rectangle EFCD can be rewritten as:
\[ \Rightarrow ar\left( {EFCD} \right) = DC \times AL\]
We know that the area of parallelogram ABCD is also \[AL \times DC\].
$\therefore ar\left( ABCD \right)=ar\left( EFCD \right)$
A parallelogram is a quadrilateral whose opposite sides are parallel and equal. A rectangle is a special kind of parallelogram with all its angles as right angles. The area of a rectangle is also the
same as the area of a parallelogram as the base of the parallelogram is the rectangle’s length and the height of the parallelogram is the rectangle’s breadth. | {"url":"https://www.vedantu.com/question-answer/in-the-following-figure-abcd-is-a-parallelogram-class-10-maths-cbse-5fb4993aaae7bc7b2d39c48d","timestamp":"2024-11-08T18:49:57Z","content_type":"text/html","content_length":"179111","record_id":"<urn:uuid:58fee509-0a55-4bc2-ab70-2056d59b40ad>","cc-path":"CC-MAIN-2024-46/segments/1730477028070.17/warc/CC-MAIN-20241108164844-20241108194844-00226.warc.gz"} |
PPT - 15-251 PowerPoint Presentation, free download - ID:1432842
1. 15-251 Some Great Theoretical Ideas in Computer Science for
2. The Mathematics Of 1950’s Dating: Who wins The Battle of The Sexes? Lecture 10 (February 14, 2008)
3. WARNING: This lecture contains mathematical content that may be shocking to some students
4. Dating Scenario There are n boys and n girls Each girl has her own ranked preference list of all the boys Each boy has his own ranked preference list of the girls The lists have no ties Question:
How do we pair them off?
5. 3,2,5,1,4 3,5,2,1,4 1 1 5,2,1,4,3 1,2,5,3,4 2 2 4,3,5,1,2 4,3,2,1,5 3 3 1,2,3,4,5 1,3,4,2,5 4 4 2,3,4,1,5 1,2,4,5,3 5 5
6. More Than One Notion of What Constitutes A “Good” Pairing Maximizing total satisfaction Hong Kong and to an extent the USA Maximizing the minimum satisfaction Western Europe Minimizing maximum
difference in mate ranks Sweden Maximizing people who get their first choice Barbie and Ken Land
8. Rogue Couples Suppose we pair off all the boys and girls Now suppose that some boy and some girl prefer each other to the people to whom they are paired They will be called a rogue couple
10. What use is fairness, if it is not stable? Any list of criteria for a good pairing must include stability. (A pairing is doomed if it contains a rogue couple.)
11. Stable Pairings A pairing of boys and girls is called stable if it contains no rogue couples
12. Stable Pairings A pairing of boys and girls is called stable if it contains no rogue couples 3,2,1 3,2,1 1 1 2,1,3 1,2,3 2 2 3,1,2 3,2,1 3 3
13. The study of stability will be the subject of the entire lecture We will: Analyze various mathematical properties of an algorithm that looks a lot like 1950’s dating Discover the naked
mathematical truth about which sex has the romantic edge Learn how the world’s largest, most successful dating service operates
14. Wait! We don’t even know that such a pairing always exists! Given a set of preference lists, how do we find a stable pairing?
15. Better Question: Does every set of preference lists have a stable pairing?
16. Idea: Allow the pairs to keep breaking up and reforming until they become stable
17. Can you argue that the couples will not continue breaking up and reforming forever?
18. 3,1,4 2 *,*,* 4 An Instructive Variant:Bisexual Dating 2,3,4 1 1,2,4 3
19. Insight Any proof that heterosexual couples do not break up and re-form forever must contain a step that fails in the bisexual case If you have a proof idea that works equally well in the hetero
and bisexual versions, then your idea is not adequate to show the couples eventually stop
20. Worshipping Males The Traditional Marriage Algorithm Female String
21. The Traditional Marriage Algorithm For each day that some boy gets a “No” do: Morning • Each girl stands on her balcony • Each boy proposes to the best girl whom he has not yet crossed off
Afternoon (for girls with at least one suitor) • To today’s best: “Maybe, return tomorrow” • To any others: “No, I will never marry you” Evening • Any rejected boy crosses the girl off his list
If no boys get a “No”, each girl marries boy to whom she just said “maybe”
22. 3,2,5,1,4 3,5,2,1,4 1 1 5,2,1,4,3 1,2,5,3,4 2 2 4,3,5,1,2 4,3,2,1,5 3 3 1,2,3,4,5 1,3,4,2,5 4 4 2,3,4,1,5 1,2,4,5,3 5 5
23. Wait! There is a more primary question! Does Traditional Marriage Algorithm always produce a stable pairing?
24. Does TMA Always Terminate? It might encounter a situation where algorithm does not specify what to do next (e.g. “core dump error”) It might keep on going for an infinite number of days
25. Improvement Lemma:If a girl has a boy on a string, then she will always have someone at least as good on a string (or for a husband) She would only let go of him in order to “maybe” someone
better She would only let go of that guy for someone even better She would only let go of that guy for someone even better AND SO ON…
26. Corollary: Each girl will marry her absolute favorite of the boys who visit her during the TMA
27. Lemma: No boy can be rejected by all the girls Proof (by contradiction): Suppose boy b is rejected by all the girls At that point: Each girl must have a suitor other than b (By Improvement Lemma,
once a girl has a suitor she will always have at least one) The n girls have n suitors, and b is not among them. Thus, there are at least n+1 boys Contradiction
28. Theorem: The TMA always terminates in at most n2 days A “master list” of all n of the boys lists starts with a total of n x n = n2 girls on it Each day that at least one boy gets a “No”, so at
least one girl gets crossed off the master list Therefore, the number of days is bounded by the original size of the master list
29. Great! We know that TMA will terminate and produce a pairing But is it stable?
30. Theorem: The pairing T produced by TMA is stable I rejected you when you came to my balcony. Now I’ve got someone better g* g b
31. Opinion Poll Who is better off in traditional dating, the boys or the girls?
32. Forget TMA For a Moment… How should we define what we mean when we say “the optimal girl for boy b”? Flawed Attempt: “The girl at the top of b’s list”
33. The Optimal Girl A boy’s optimal girl is the highest ranked girl for whom there is some stable pairing in which the boy gets her She is the best girl he can conceivably get in a stable world.
Presumably, she might be better than the girl he gets in the stable pairing output by TMA
34. The Pessimal Girl A boy’s pessimal girl is the lowest ranked girl for whom there is some stable pairing in which the boy gets her She is the worst girl he can conceivably get in a stable world
35. Dating Heaven and Hell A pairing is male-optimal if everyboy gets his optimal mate. This is the best of all possible stable worlds for every boy simultaneously A pairing is male-pessimal if
everyboy gets his pessimal mate. This is the worst of all possible stable worlds for every boy simultaneously
36. Dating Heaven and Hell A pairing is female-optimal if every girl gets her optimal mate. This is the best of all possible stable worlds for every girl simultaneously A pairing is female-pessimal
if every girl gets her pessimal mate. This is the worst of all possible stable worlds for every girl simultaneously
37. The Naked Mathematical Truth! The Traditional Marriage Algorithm always produces a male-optimal, female-pessimal pairing
38. Theorem: TMA produces a male-optimal pairing Suppose, for a contradiction, that some boy gets rejected by his optimal girl during TMA Let t be the earliest time at which this happened At time t,
boy b got rejected by his optimal girl g because she said “maybe” to a preferred b* By the definition of t, b* had not yet been rejected by his optimal girl Therefore, b* likes g at least as much
as his optimal
39. Some boy b got rejected by his optimal girl g because she said “maybe” to a preferred b*. b* likes g at least as much as his optimal girl There must exist a stable pairing S in which b and g are
married b* wants g more than his wife in S: g is at least as good as his best and he does not have her in stable pairing S g wants b* more than her husband in S: b is her husband in S and she
rejects him for b* in TMA Contradiction
40. Theorem: The TMA pairing, T, is female-pessimal We know it is male-optimal. Suppose there is a stable pairing S where some girl g does worse than in T Let b be her mate in T Let b* be her mate in
S By assumption, g likes b better than her mate in S b likes g better than his mate in S (we already know that g is his optimal girl) Contradiction Therefore, S is not stable
41. The largest, most successful dating service in the world uses a computer to run TMA!
42. Definition of: • Stable Pairing • Traditional Marriage Algorithm Proof that: • TMA Produces a Stable Pairing • TMA Produces a Male-Optimal, Female-Pessimal Pairing Here’s What You Need to Know… | {"url":"https://fr.slideserve.com/zarifa/what-can-a-boy-do-if","timestamp":"2024-11-10T20:38:59Z","content_type":"text/html","content_length":"93820","record_id":"<urn:uuid:0ed479b4-0ea7-4040-b5a3-b60e6c7876c8>","cc-path":"CC-MAIN-2024-46/segments/1730477028191.83/warc/CC-MAIN-20241110201420-20241110231420-00642.warc.gz"} |
Chapter 13: Statistics Quiz
Questions and Answers
They may seem boring when you’re younger, but statistics are absolutely fascinating when you get down to the nitty-gritty of them. Today we’ll be doing just that by testing your knowledge on
statistics, what kinds of data it deals with and what it’s used for. Good luck!
• 1.
The art and science of gathering, analyzing, and making inferences (predictions) from numerical information obtained in an experiment
□ A.
□ B.
□ C.
□ D.
Correct Answer
B. Statistics
Statistics is the correct answer because it refers to the art and science of gathering, analyzing, and making inferences or predictions from numerical information obtained in an experiment. It
involves the collection, organization, interpretation, presentation, and analysis of data to understand patterns, trends, and relationships. Statistics provide a way to summarize and describe
data, make informed decisions, and draw conclusions about a population based on a sample.
• 2.
As the numerical information so obtained is referred
□ A.
□ B.
□ C.
□ D.
Correct Answer
D. Data
• 3.
Is concerned with the collection, organization, and analysis of data
□ A.
□ B.
□ C.
□ D.
Correct Answer
A. Descriptive statistics
Descriptive statistics involves the collection, organization, and analysis of data. It focuses on summarizing and describing the main features of a dataset, such as central tendency (mean,
median, mode) and dispersion (range, variance, standard deviation). This branch of statistics is used to present and interpret data in a meaningful way, providing insights into patterns, trends,
and characteristics of the data. It does not involve making inferences or drawing conclusions about a larger population, which is the domain of inferential statistics.
• 4.
Is concerned with making generalizations or predictions from the data collected
□ A.
□ B.
□ C.
□ D.
Correct Answer
B. Inferential statistics
Inferential statistics is the correct answer because it involves making generalizations or predictions from the data collected. Descriptive statistics, on the other hand, is concerned with
summarizing and describing the data without making any inferences or predictions. A random number generator is a tool used to generate random numbers, which is unrelated to making generalizations
or predictions from data. A piece of data is a singular unit of information and does not pertain to the process of making generalizations or predictions.
• 5.
The entire contents of the box constitute the population, it consists of all items or people of interest
□ A.
□ B.
□ C.
□ D.
Correct Answer
A. Population
The correct answer is "Population" because it refers to the entire contents of the box, which includes all the items or people of interest. In statistics, a population is the complete set of
individuals, items, or data that is being studied or analyzed. It is important to define the population accurately in order to draw valid conclusions and make generalizations about the entire
• 6.
It’s the statistician often uses a subset of the population
□ A.
□ B.
□ C.
□ D.
Correct Answer
C. Sample
A sample is a subset of the population that is selected for statistical analysis. It is a common practice for statisticians to use a sample instead of studying the entire population, as it is
often more practical and cost-effective. By analyzing a representative sample, statisticians can make inferences and draw conclusions about the larger population. Therefore, the use of a sample
is a common and important tool in statistical analysis.
• 7.
The sum of the data divided by the number of pieces of data,it is used when the mean of a sample of the population is calculated
□ A.
□ B.
□ C.
□ D.
Correct Answer
C. ¯x (x bar)
The symbol ¯x (x bar) represents the mean of a sample of data. The mean is calculated by summing up all the data points and dividing it by the number of data points. It is a measure of central
tendency that provides an average value for the data set. The mean is commonly used in statistics to analyze and compare different sets of data.
• 8.
Standard deviation, measures how much the data differ from the mean. It is used when the standard deviation by of a sample is calculated
□ A.
□ B.
□ C.
□ D.
Correct Answer
B. S
The correct answer is "s". Standard deviation is a measure of how much the data points in a sample differ from the mean. It is commonly denoted by the symbol "s" when calculating the standard
deviation of a sample.
• 9.
The sum of the data divided by the number of pieces of data, it is used when the mean of a sample of the entire population is calculated
□ A.
□ B.
□ C.
□ D.
Correct Answer
B. µ
The symbol µ represents the mean of a sample of the entire population. The mean is calculated by summing up all the data and dividing it by the number of pieces of data. This is a commonly used
statistical measure to determine the average value of a dataset.
• 10.
Standard deviation, measures how much the data differ from the mean, it is used when the standard deviation of the entire population is calculated
□ A.
□ B.
□ C.
□ D.
Correct Answer
C. σ
Standard deviation (σ) is the correct answer because it is a statistical measure that quantifies the amount of variation or dispersion in a set of data. It indicates how much the data points
deviate from the mean. Standard deviation is used when calculating the standard deviation of the entire population, as opposed to a sample. It provides valuable information about the spread of
the data and helps in understanding the distribution of the population.
• 11.
Is one that is a religion, political affiliation, age, and so on…
□ A.
□ B.
□ C.
□ D.
Correct Answer
D. Unbiased sample
An unbiased sample refers to a subset of a population that is selected without any bias or preference towards certain characteristics or traits. It ensures that every individual in the population
has an equal chance of being included in the sample, which helps to eliminate any potential sources of bias and provides a more accurate representation of the entire population.
• 12.
If a sample is drawn in such a way that each time an item is selected each item in the population has an equal chance of being drawn.
□ A.
□ B.
□ C.
□ D.
Correct Answer
A. random sample
A random sample is the correct answer because it is drawn in a way that ensures each item in the population has an equal chance of being selected. This means that the selection process is
unbiased and representative of the entire population. Other sampling methods like cluster sampling, systematic sampling, and convenience sampling may introduce biases and may not provide an
accurate representation of the population.
• 13.
When a sample is obtained by drawing every nth item on a list or production line.
□ A.
□ B.
□ C.
□ D.
Correct Answer
C. systematic sample
A systematic sample is obtained by selecting every nth item on a list or production line. This method ensures that the sample is representative of the entire population and reduces the potential
for bias. By following a systematic approach, every item has an equal chance of being selected, providing a fair representation of the population. This sampling technique is commonly used in
research and statistical analysis to gather data efficiently and accurately.
• 14.
Is sometimes referred to as an area sample because it is frequently applied on a geographical basis
□ A.
□ B.
□ C.
□ D.
Correct Answer
D. cluster sample
A cluster sample is sometimes referred to as an area sample because it is frequently applied on a geographical basis. In a cluster sample, the population is divided into clusters or groups based
on their geographical location. Then, a random selection of clusters is chosen, and all individuals within the selected clusters are included in the sample. This method is often used when it is
impractical or costly to sample individuals from the entire population, making it convenient to select clusters instead.
• 15.
It uses data that are easily or readily obtained
□ A.
□ B.
□ C.
□ D.
Correct Answer
A. convenience sample
A convenience sample is a type of sampling method where the researcher selects individuals who are easily accessible or readily available to participate in the study. This means that the data
collected from a convenience sample is based on the individuals who are convenient to include in the study, rather than being randomly selected or systematically chosen. Therefore, the
explanation for the given correct answer is that a convenience sample uses data that are easily or readily obtained.
• 16.
It is a device, usually a calculator or computer program, that produces a list of random number
□ A.
□ B.
□ C.
Correct Answer
B. Random number generator
A random number generator is a device or program that generates a list of random numbers. It is commonly used in various applications such as simulations, cryptography, and statistical sampling.
Unlike a random number table, which is a pre-determined list of random numbers, a random number generator produces random numbers on the fly. It uses algorithms and seed values to generate
pseudo-random numbers that appear to be random but are actually determined by a set of rules. This makes it a versatile tool for generating random values in a wide range of scenarios.
• 17.
It is a collection of random digits in which each digits has an equal chance or appearing
□ A.
□ B.
□ C.
Correct Answer
A. random number table
A random number table is a collection of random digits that are generated in a way that each digit has an equal chance of appearing. This means that there is no pattern or bias in the selection
of the digits. It is commonly used in statistical sampling and simulations to generate random numbers for various purposes.
• 18.
When a population is divided into parts, called strata, for the purpose of drawing a sample
□ A.
□ B.
□ C.
□ D.
Correct Answer
C. stratified sampling
Stratified sampling is the correct answer because it refers to the process of dividing a population into distinct subgroups or strata based on certain characteristics. This method is used to
ensure that the sample drawn from each stratum is representative of the entire population. By stratifying the population, researchers can obtain a more accurate and unbiased sample, as it allows
for proportional representation of different groups within the population. This technique is commonly used in surveys and research studies to improve the reliability and validity of the findings.
• 19.
Known as class, when a population has varied characteristics and then take a random sample from each of class.
□ A.
□ B.
□ C.
□ D.
Correct Answer
A. stratum
In statistics, a population refers to the entire group of individuals or objects that are being studied. When a population has varied characteristics, it can be divided into subgroups or classes
known as strata. Each stratum represents a subset of the population with similar characteristics. Taking a random sample from each stratum helps ensure that the sample is representative of the
entire population and accounts for the variability within each class. Therefore, the term "stratum" is the correct answer in this context.
• 20.
It is a single response to an experiment. When the amount of date is large, it is usually advantageous to construct a frequency distribution
□ A.
□ B.
□ C.
□ D.
Correct Answer
A. Piece of data
A "piece of data" refers to a single value or observation collected during an experiment. When there is a large amount of data, constructing a frequency distribution can be beneficial. A
frequency distribution organizes the data into classes or intervals, making it easier to understand and analyze. It helps to identify patterns, trends, and the distribution of the data.
Therefore, the given answer is correct as it accurately describes the importance of constructing a frequency distribution when dealing with a large amount of data.
• 21.
A listing of the observed values and the corresponding frequency of occurrence of each value
□ A.
□ B.
□ C.
□ D.
measures of central tendency
Correct Answer
C. frequency distribution
A frequency distribution is a listing of the observed values and the corresponding frequency of occurrence of each value. It presents the data in a tabular format, showing how often each value
appears in the data set. This allows for a clear understanding of the distribution and patterns within the data. It is commonly used to summarize and analyze data in statistics and research.
• 22.
To determine how far, in terms of standard deviations, a given score is from the mean of the distribution
□ A.
□ B.
□ C.
□ D.
Correct Answer
A. z-score
The z-score is used to determine how far a given score is from the mean of a distribution in terms of standard deviations. It is a measure of position that allows for comparison and
interpretation of scores across different distributions. The z-score is calculated by subtracting the mean from the score and dividing the result by the standard deviation. It provides a
standardized value that can be used to assess the relative position of a score within a distribution.
• 23.
It divided data into 4 equal parts, the 1^st is the value that is higher than about ¼ or 25% of the population, the 2^nd is the value that is higher than about ½ the population and is the same as
the 50^th percentile, or the median, the 3^rd is the value that is higher than about ¾ of the population and is the same as the 75^th percentile
□ A.
□ B.
□ C.
□ D.
Correct Answer
D. quartiles
The explanation for the correct answer, quartiles, is that quartiles are a measure of position that divide data into four equal parts. The first quartile represents the value that is higher than
about 25% of the population, the second quartile represents the median or the value that is higher than about 50% of the population, and the third quartile represents the value that is higher
than about 75% of the population. Therefore, quartiles are used to understand the distribution and spread of data by dividing it into equal parts.
• 24.
The mean, median, and mode all have the same value
□ A.
□ B.
□ C.
□ D.
Correct Answer
B. Gaussian distribution
A Gaussian distribution, also known as a normal distribution, is a symmetrical probability distribution where the mean, median, and mode all have the same value. In this type of distribution, the
data is evenly spread around the mean, resulting in a bell-shaped curve. This indicates that the majority of the data falls close to the mean, with fewer values further away from it. Therefore, a
Gaussian distribution is the most likely explanation for the mean, median, and mode having the same value.
• 25.
It has more of a “tail” on one side than the other
□ A.
□ B.
□ C.
□ D.
Correct Answer
B. skewed distribution
A skewed distribution is a type of probability distribution where the data is not symmetrically distributed around the mean. In this case, the statement suggests that the distribution has more
values on one side than the other, indicating a lack of symmetry. This is a characteristic of a skewed distribution, where the tail on one side is longer or more pronounced than the other. This
can occur in both positive and negative directions, resulting in positively or negatively skewed distributions. Therefore, the correct answer is a skewed distribution.
• 26.
Is one in which two nonadjacent values occur more frequently than any other values in a set of data
□ A.
□ B.
□ C.
□ D.
Correct Answer
A. bimodal distribution
A bimodal distribution is a type of distribution where there are two distinct peaks or modes in the data. This means that there are two nonadjacent values that occur more frequently than any
other values in the set of data. This can occur when there are two different groups or populations within the data, each with their own distinct characteristics or behaviors.
• 27.
The frequency is either constantly increasing or decreasing
□ A.
□ B.
□ C.
□ D.
Correct Answer
C. J-shaped distribution
A J-shaped distribution is characterized by a small number of low values and a large number of high values. In this type of distribution, the frequency either constantly increases or decreases,
resulting in a shape that resembles the letter J. This distribution is often observed in situations where there is a strong bias towards extreme values, such as in income distribution where there
are a few individuals with very high incomes and a large number of individuals with low incomes.
• 28.
All the observed values occur with the same frequency
□ A.
□ B.
□ C.
□ D.
Correct Answer
D. rectangular distribution
A rectangular distribution is characterized by all observed values occurring with the same frequency. In this distribution, there are no outliers or extreme values, and the data is evenly spread
across the entire range. This type of distribution is often seen in situations where there is a limited range of possible values and each value is equally likely to occur.
• 29.
Use to indicate the spread of the data
□ A.
□ B.
□ C.
□ D.
Correct Answer
B. Measures of dispersion
Measures of dispersion are statistical tools used to indicate the spread or variability of data. They provide information about how the data points are distributed around the mean or median. By
calculating measures such as range, variance, or standard deviation, we can understand the extent to which the data values deviate from the central tendency. These measures are essential in
analyzing and comparing datasets, as they allow us to assess the variability and make meaningful interpretations about the data.
• 30.
There are 99 percentiles dividing a set of data into 100 equal parts, a score in the nth percentile means that you outperformed about n% of the population who took the test and that (100 – n)% of
the people taking the test performed better than you did
□ A.
□ B.
□ C.
□ D.
Correct Answer
B. percentile
A percentile is a statistical measure used to describe the position of a particular data point within a dataset. In this context, the answer "percentile" is correct because it aligns with the
explanation provided. It states that a score in the nth percentile means that you outperformed about n% of the population who took the test, indicating the relative position of the score within
the dataset. The explanation also mentions that (100 - n)% of the people taking the test performed better than you did, further emphasizing the concept of percentiles.
• 31.
Used to make comparisons, such as comparing the scores of individuals from different populations, and are generally used when the amount of data is large
□ A.
□ B.
measures of central tendency
□ C.
□ D.
Correct Answer
C. measures of position
Measures of position are used to make comparisons between individuals from different populations, especially when dealing with large amounts of data. These measures provide information about the
relative position or rank of a particular value within a dataset, such as percentiles or quartiles. By using measures of position, researchers can gain insights into how individuals or groups
compare to each other in terms of their scores or values.
• 32.
If two values in a set of data occur more often than all the other data, we consider both these values as modes
□ A.
□ B.
□ C.
□ D.
Correct Answer
A. bimodal
If two values in a set of data occur more frequently than any other values, we consider both of these values as modes. In other words, if there are two peaks or high points in the data
distribution, it is considered bimodal. The term "bimodal" specifically refers to a situation where there are two modes in the data set. The other options, mode, mean, and median, do not
specifically address the concept of having two modes in the data.
• 33.
The pieces of data that occurs most frequently
□ A.
□ B.
□ C.
□ D.
Correct Answer
B. Mode
The correct answer is "mode". The mode is the value that appears most frequently in a set of data. It is a measure of central tendency and can be used to describe the most common or typical value
in a dataset. In this case, the question is asking for the term that refers to the pieces of data that occur most frequently, which is the mode.
• 34.
The value in the middle of a set of ranked data
□ A.
□ B.
□ C.
□ D.
Correct Answer
C. Median
The correct answer is median. The median is the value in the middle of a set of ranked data. It is found by arranging the data in ascending order and selecting the middle value. Unlike the mean,
which is influenced by extreme values, the median is not affected by outliers. The mode refers to the most frequently occurring value in a dataset, while bimodal means that there are two modes in
the dataset.
• 35.
The total number of pieces of data
Correct Answer
B. Σn
The given expression Σn represents the sum of all the values of n. In this context, n represents the number of pieces of data. Therefore, the correct answer Σn indicates the total number of
pieces of data.
• 36.
The Greek letter sigma, used to indicated “summation”
Correct Answer
A. Σ
The correct answer is Σ because it is the Greek letter sigma, which is commonly used in mathematics to represent summation. When written above a series of numbers or variables, it indicates that
they should be added together.
• 37.
The sum of the data divided by the number of pieces of data
□ A.
□ B.
□ C.
□ D.
Correct Answer
A. Mean
The mean is calculated by finding the sum of all the data values and then dividing it by the total number of data points. It provides a measure of the central tendency of the data set,
representing the average value.
• 38.
Each is calculated differently and may yield different results for the same set of data. Each will result in a number near the center of the data; for this reason, averages are commonly referred
to it.
□ A.
□ B.
measures of central tendency
□ C.
□ D.
Correct Answer
B. measures of central tendency
Each of the measures of central tendency (mean, median, and mode) calculates a different value that represents the center of the data. These measures may yield different results for the same set
of data because they focus on different aspects of the data distribution. However, all of them provide a number that is near the center of the data. This is why they are commonly referred to as
measures of central tendency. They help to summarize and understand the overall pattern of the data by providing a single representative value.
• 39.
A number that is representative of a group of data
□ A.
□ B.
□ C.
□ D.
Correct Answer
C. Average
The term "average" refers to a number that represents a group of data. It is calculated by adding up all the values in the data set and dividing the sum by the total number of values. The average
is often used to provide a general idea of the central tendency of the data and can be influenced by extreme values.
• 40.
A tool that organizes and groups the data while allowing us to see the actual values that make up the data
□ A.
□ B.
□ C.
□ D.
Correct Answer
D. stem-and-leaf display
A stem-and-leaf display is a tool that organizes and groups the data while allowing us to see the actual values that make up the data. It provides a visual representation of the data set, where
the "stem" represents the leading digits and the "leaves" represent the trailing digits. This display allows for easy identification of the distribution of the data, including the range, gaps,
and clusters. It is a useful tool for displaying and analyzing numerical data in a concise and organized manner.
• 41.
A line graph with scales the same as those of the histogram, that is, the horizontal scale indicates observed values and the vertical scale indicates frequency
□ A.
□ B.
□ C.
□ D.
Correct Answer
C. frequency polygon
A frequency polygon is a line graph that represents the frequency distribution of a set of data. In this case, the line graph has the same scales as the histogram, meaning that the horizontal
axis represents the observed values and the vertical axis represents the frequency. The frequency polygon connects the midpoints of the histogram bars, creating a line that shows the trend of the
data. Therefore, the frequency polygon is the correct answer in this context.
• 42.
A graph with observed values on its horizontal scale and frequencies on its vertical scale
□ A.
□ B.
□ C.
□ D.
Correct Answer
A. histogram
A histogram is a graph that represents the distribution of observed values on its horizontal scale and frequencies on its vertical scale. It is used to display the frequency or count of data
within different intervals or bins. Each bar in the histogram represents a specific interval, and the height of the bar indicates the frequency of data falling within that interval. Histograms
are commonly used to visualize the distribution of numerical data and to identify patterns or trends in the data.
• 43.
Also pie chart, used to compare parts of one or more components of the whole to the whole
□ A.
□ B.
□ C.
□ D.
Correct Answer
B. circle grapH
A circle graph, also known as a pie chart, is a visual representation that is used to compare the parts of one or more components of a whole to the whole itself. It is a circular chart divided
into sectors, with each sector representing a different category or component. The size of each sector is proportional to the quantity or percentage it represents. Circle graphs are commonly used
to display data that can be divided into categories or parts, making it easier to understand the relative proportions and relationships between different components.
• 44.
Also class mark, is found by adding the lower and upper class limit and dividing the sum by 2
□ A.
□ B.
□ C.
□ D.
Correct Answer
B. midpoint of a class
The midpoint of a class refers to the average value of the lower and upper class limits. It is calculated by adding the lower and upper class limits and dividing the sum by 2. This value
represents the center of the class interval and is often used as a representative value for the entire class. It helps in summarizing and analyzing data, especially in statistical calculations
such as finding the mean, median, and mode of a dataset.
• 45.
The class with the greatest frequency
□ A.
□ B.
□ C.
□ D.
Correct Answer
B. Modal class
The modal class refers to the class in a frequency distribution that has the highest frequency or the most occurrences. It is the class with the greatest frequency, meaning that it contains the
most data points or observations. The modal class is often used to identify the most common value or range of values in a dataset.
• 46.
Any of the data of a group of number would belong to the class given
□ A.
□ B.
□ C.
□ D.
Correct Answer
B. class width
Class width refers to the range of values that are included in each class interval in a frequency distribution. It determines the size of each interval and helps to organize the data into
meaningful groups. In this context, the statement suggests that any data point within a group of numbers would fall within the range defined by the class width. | {"url":"https://www.proprofs.com/quiz-school/story.php?title=chapter-13-statistics","timestamp":"2024-11-11T18:18:14Z","content_type":"text/html","content_length":"654401","record_id":"<urn:uuid:587f3cbd-d1e6-4fc3-9b2b-070ffad9ff8b>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00650.warc.gz"} |
Changing Improper Fractions to Mixed Fractions - ppt video online download
2 What is an Improper Fraction?An Improper fraction has a top (numerator) number larger than (or equal to) the bottom (denominator) number, Its “Top Heavy” Examples:
4 What is a Mixed Fraction?A Mixed Fraction is a whole number and a proper fraction combined, such as 1 3/4.
5 How you Change an Improper Fraction to a Mixed FractionFirst figure out if its an improper fraction by looking to see if the top number is bigger than the bottom number. 8 Yes 3 No
6 Next you divide the top number by the bottom numberNow that we know 8 is an improper 5 fraction we can divide 8 by goes into 8, 1 time with 3 left over.
7 We take the remaining 3 and make it into a fractionWe take the remaining 3 and make it into a fraction. The remaining 3 will always go over the original denominator. The number of times the
denominator goes into the numerator always goes in front of the remaining fraction such as: 1 3 5
8 Practice Are these improper fractions? If so change it into a mixed fractions.
10 Practice Are these improper fractions? If so change it into a mixed fraction.
12 Recap To change an improper fraction to a mixed number you divide the numerator by the denominator. The number of times its divided is written first. Then the remainder is put into a fraction with
the remainder over the original denominator. 1 4 5 | {"url":"http://slideplayer.com/slide/8562723/","timestamp":"2024-11-14T04:59:21Z","content_type":"text/html","content_length":"157341","record_id":"<urn:uuid:96fe5d48-7028-43c4-8b0d-95997a8ca67f>","cc-path":"CC-MAIN-2024-46/segments/1730477028526.56/warc/CC-MAIN-20241114031054-20241114061054-00201.warc.gz"} |
el spectrogram
S = melSpectrogram(audioIn,fs) returns the mel spectrogram of the audio input at sample rate fs. The function treats columns of the input as individual channels.
[S,F,T] = melSpectrogram(___) returns the center frequencies of the bands in Hz and the location of each window of data in seconds. The location corresponds to the center of each window. You can use
this output syntax with any of the previous input syntaxes.
melSpectrogram(___) plots the mel spectrogram on a surface in the current figure.
Calculate Mel Spectrogram
Use the default settings to calculate the mel spectrogram for an entire audio file. Print the number of bandpass filters in the filter bank and the number of frames in the mel spectrogram.
[audioIn,fs] = audioread('Counting-16-44p1-mono-15secs.wav');
S = melSpectrogram(audioIn,fs);
[numBands,numFrames] = size(S);
fprintf("Number of bandpass filters in filterbank: %d\n",numBands)
Number of bandpass filters in filterbank: 32
fprintf("Number of frames in spectrogram: %d\n",numFrames)
Number of frames in spectrogram: 1551
Plot the mel spectrogram.
Calculate Mel Spectrums of 2048-Point Windows
Calculate the mel spectrums of 2048-point periodic Hann windows with 1024-point overlap. Convert to the frequency domain using a 4096-point FFT. Pass the frequency-domain representation through 64
half-overlapped triangular bandpass filters that span the range 62.5 Hz to 8 kHz.
[audioIn,fs] = audioread('FunkyDrums-44p1-stereo-25secs.mp3');
S = melSpectrogram(audioIn,fs, ...
'Window',hann(2048,'periodic'), ...
'OverlapLength',1024, ...
'FFTLength',4096, ...
'NumBands',64, ...
Call melSpectrogram again, this time with no output arguments so that you can visualize the mel spectrogram. The input audio is a multichannel signal. If you call melSpectrogram with a multichannel
input and with no output arguments, only the first channel is plotted.
melSpectrogram(audioIn,fs, ...
'Window',hann(2048,'periodic'), ...
'OverlapLength',1024, ...
'FFTLength',4096, ...
'NumBands',64, ...
Get Filter Bank Center Frequencies and Analysis Window Time Instants
melSpectrogram applies a frequency-domain filter bank to audio signals that are windowed in time. You can get the center frequencies of the filters and the time instants corresponding to the analysis
windows as the second and third output arguments from melSpectrogram.
Get the mel spectrogram, filter bank center frequencies, and analysis window time instants of a multichannel audio signal. Use the center frequencies and time instants to plot the mel spectrogram for
each channel.
[audioIn,fs] = audioread('AudioArray-16-16-4channels-20secs.wav');
[S,cF,t] = melSpectrogram(audioIn,fs);
S = 10*log10(S+eps); % Convert to dB for plotting
for i = 1:size(S,3)
xlabel('Time (s)')
ylabel('Frequency (Hz)')
title(sprintf('Channel %d',i))
axis([t(1) t(end) cF(1) cF(end)])
Input Arguments
audioIn — Audio input
column vector | matrix
Audio input, specified as a column vector or matrix. If specified as a matrix, the function treats columns as independent audio channels.
Data Types: single | double
fs — Input sample rate (Hz)
positive scalar
Input sample rate in Hz, specified as a positive scalar.
Data Types: single | double
Name-Value Arguments
Specify optional pairs of arguments as Name1=Value1,...,NameN=ValueN, where Name is the argument name and Value is the corresponding value. Name-value arguments must appear after other arguments, but
the order of the pairs does not matter.
Before R2021a, use commas to separate each name and value, and enclose Name in quotes.
Example: FFTLength=1024
Window — Window applied in time domain
hamming(round(fs*0.03),'periodic') (default) | vector
Window applied in time domain, specified as a real vector. The number of elements in the vector must be in the range [1,size(audioIn,1)]. The number of elements in the vector must also be greater
than OverlapLength.
Data Types: single | double
OverlapLength — Analysis window overlap length (samples)
round(0.02*fs) (default) | integer in the range [0, (numel(Window) - 1)]
Analysis window overlap length in samples, specified as an integer in the range [0, (numel(Window) - 1)].
Data Types: single | double
FFTLength — Number of DFT points
numel(Window) (default) | positive integer
Number of points used to calculate the DFT, specified as a positive integer greater than or equal to the length of Window. If unspecified, FFTLength defaults to the length of Window.
Data Types: single | double
NumBands — Number of mel bandpass filters
32 (default) | positive integer
Number of mel bandpass filters, specified as a positive integer.
Data Types: single | double
FrequencyRange — Frequency range over which to compute mel spectrogram (Hz)
[0 fs/2] (default) | two-element row vector
Frequency range over which to compute the mel spectrogram in Hz, specified as a two-element row vector of monotonically increasing values in the range [0, fs/2].
Data Types: single | double
SpectrumType — Type of mel spectrogram
"power" (default) | "magnitude"
Type of mel spectrogram, specified as "power" or "magnitude".
Data Types: char | string
WindowNormalization — Apply window normalization
true (default) | false
Apply window normalization, specified as true or false. When WindowNormalization is set to true, the power (or magnitude) in the mel spectrogram is normalized to remove the power (or magnitude) of
the time domain Window.
Data Types: char | string
FilterBankNormalization — Type of filter bank normalization
"bandwidth" (default) | "area" | "none"
Type of filter bank normalization, specified as "bandwidth", "area", or "none".
Data Types: char | string
MelStyle — Mel style
"oshaughnessy" (default) | "slaney"
Mel style, specified as "oshaughnessy" or "slaney".
Data Types: char | string
ApplyLog — Apply logarithm
false (default) | true
Apply base 10 logarithm to the returned mel spectrogram, specified as true or false.
Data Types: logical
Output Arguments
S — Mel spectrogram
column vector | matrix | 3-D array
Mel spectrogram, returned as a column vector, matrix, or 3-D array. The dimensions of S are L-by-M-by-N, where:
• L is the number of frequency bins in each mel spectrum. NumBands and fs determine L.
• M is the number of frames the audio signal is partitioned into. size(audioIn,1), the length of Window, and OverlapLength determine M.
• N is the number of channels such that N = size(audioIn,2).
Trailing singleton dimensions are removed from the output S.
Data Types: single | double
F — Center frequencies of mel bandpass filters (Hz)
row vector
Center frequencies of mel bandpass filters in Hz, returned as a row vector with length size(S,1).
Data Types: single | double
T — Location of each window of audio (s)
row vector
Location of each analysis window of audio in seconds, returned as a row vector length size(S,2). The location corresponds to the center of each window.
Data Types: single | double
The melSpectrogram function follows the general algorithm to compute a mel spectrogram as described in [1].
In this algorithm, the audio input is first buffered into frames of numel(Window) number of samples. The frames are overlapped by OverlapLength number of samples. The specified Window is applied to
each frame, and then the frame is converted to frequency-domain representation with FFTLength number of points. The frequency-domain representation can be either magnitude or power, specified by
SpectrumType. If WindowNormalization is set to true, the spectrum is normalized by the window. Each frame of the frequency-domain representation passes through a mel filter bank. The spectral values
output from the mel filter bank are summed, and then the channels are concatenated so that each frame is transformed to a NumBands-element column vector.
Filter Bank Design
The mel filter bank is designed as half-overlapped triangular filters equally spaced on the mel scale. NumBands controls the number of mel bandpass filters. FrequencyRange controls the band edges of
the first and last filters in the mel filter bank. FilterBankNormalization specifies the type of normalization applied to the individual bands.
The mel scale can be in the O'Shaughnessy style, which follows [2], or the Slaney style, which follows [3].
[1] Rabiner, Lawrence R., and Ronald W. Schafer. Theory and Applications of Digital Speech Processing. Upper Saddle River, NJ: Pearson, 2010.
[2] O'Shaughnessy, Douglas. Speech Communication: Human and Machine. Reading, MA: Addison-Wesley Publishing Company, 1987.
[3] Slaney, Malcolm. "Auditory Toolbox: A MATLAB Toolbox for Auditory Modeling Work." Technical Report, Version 2, Interval Research Corporation, 1998.
Extended Capabilities
C/C++ Code Generation
Generate C and C++ code using MATLAB® Coder™.
The melSpectrogram function supports optimized code generation using single instruction, multiple data (SIMD) instructions. For more information about SIMD code generation, see Generate SIMD Code
from MATLAB Functions for Intel Platforms (MATLAB Coder).
GPU Code Generation
Generate CUDA® code for NVIDIA® GPUs using GPU Coder™.
GPU Arrays
Accelerate code by running on a graphics processing unit (GPU) using Parallel Computing Toolbox™.
Version History
Introduced in R2019a
R2024b: WindowLength has been removed
The WindowLength parameter has been removed from the melSpectrogram function. Use the Window parameter instead.
In releases prior to R2020b, you could only specify the length of a time-domain window. The window was always designed as a periodic Hamming window. You can replace instances of the code
S = melSpectrogram(audioin,fs,WindowLength=1024);
With this code:
S = melSpectrogram(audioIn,fs,Window=hamming(1024,"periodic"));
R2024a: Apply logarithm to mel spectrogram
Set the ApplyLog name-value argument to true to apply a base 10 logarithm to the spectrogram.
R2023b: Support for Slaney-style mel scale
Set the MelStyle name-value argument to "slaney" to use the Slaney-style mel scale.
R2023a: Generate optimized C/C++ code for computing mel spectrogram
melSpectrogram supports optimized C/C++ code generation using single instruction, multiple data (SIMD) instructions.
R2020b: WindowLength will be removed in a future release
The WindowLength parameter will be removed from the melSpectrogram function in a future release. | {"url":"https://au.mathworks.com/help/audio/ref/melspectrogram.html","timestamp":"2024-11-12T16:15:04Z","content_type":"text/html","content_length":"134641","record_id":"<urn:uuid:002b1ac4-9647-4e5c-8e73-e6caa8731dec>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.63/warc/CC-MAIN-20241112145015-20241112175015-00837.warc.gz"} |
LR(1) Automaton
An LR(1) Automaton is a finite-state machine used in parsing, specifically in constructing LR(1) parsers, which are a type of bottom-up parser. The automaton is built from the grammar and guides the
parser in determining how to analyze and reduce tokens in the input to construct the correct parse tree.
Components of an LR(1) Automaton
1. Items (States):
The states in the LR(1) automaton represent "items." Each item is a snapshot of a possible parse at a given point in the input, typically written as:
A -> α • β, x
This means that the parser is currently in the middle of parsing the production A -> αβ, and the dot (•) indicates how much of the right-hand side has been parsed. x is the lookahead token (the
next input token expected after the current derivation).
2. Lookahead (1 token):
The "1" in LR(1) refers to the lookahead of 1 token. This means the parser makes decisions based on both the current state (items) and the next token in the input. The lookahead token helps the
parser decide whether to shift (read more input) or reduce (apply a grammar rule).
3. Shift and Reduce Operations:
□ Shift: This operation moves the dot (•) in an item to the right, meaning that the parser reads one more token from the input and transitions to a new state.
□ Reduce: When the dot is at the end of a production (e.g., A -> α •), the parser applies the corresponding grammar rule and "reduces" the input by recognizing that a production is complete.
4. States and Transitions:
The automaton consists of states connected by transitions. Each state contains a set of items (representing different parts of the parsing process), and transitions between states occur based on
reading symbols (either tokens or grammar symbols).
For example, if you're in a state where you're expecting the next symbol to be + and the input contains +, the automaton will transition to the next state.
5. Closure and GOTO Functions:
□ Closure: The closure of a set of items adds new items to a state based on grammar rules. If you're in a state where you're expecting a non-terminal symbol (like E), the closure function adds
items for every possible production of that non-terminal.
□ Goto: This function moves from one state to another based on the next symbol in the input, determining the next possible parsing steps.
6. Acceptance and Conflicts:
□ The automaton reaches the accepting state when the parser has successfully parsed the entire input.
□ Shift/Reduce conflicts occur when the parser cannot decide whether to shift the input (continue reading tokens) or reduce a rule (apply a grammar production). Reduce/Reduce conflicts occur
when the parser is uncertain which rule to apply. LR(1) parsers are designed to avoid most of these conflicts through the lookahead token.
How Does an LR(1) Automaton Work in Parsing?
1. Construct the Automaton:
The automaton is built by creating states and transitions based on the grammar rules, tracking all possible parsing actions the parser might take at each step.
2. Parsing Process:
The automaton drives the parsing process. Given an input string, the parser:
□ Shifts tokens from the input and moves to new states.
□ Reduces when it matches a rule, reducing the right-hand side of the rule to the left-hand side.
□ Continues this process until either the input is fully parsed (accepted) or an error is encountered.
Constructing the LR(1) Automaton
For more info visit the dotlr library docs | {"url":"https://tokeko.specy.app/docs/dotlr/automaton","timestamp":"2024-11-13T12:56:14Z","content_type":"text/html","content_length":"30426","record_id":"<urn:uuid:f9158235-4e78-4a64-b803-c5dd4f631f13>","cc-path":"CC-MAIN-2024-46/segments/1730477028347.28/warc/CC-MAIN-20241113103539-20241113133539-00455.warc.gz"} |
Artificial Neural Network
An artificial neural network is a biologically inspired computational model that is patterned after the network of neurons present in the human brain. Artificial neural networks can also be thought
of as learning algorithms that model the input-output relationship. Applications of artificial neural networks include pattern recognition and forecasting in fields such as medicine, business, pure
sciences, data mining, telecommunications, and operations managements.
An artificial neural network transforms input data by applying a nonlinear function to a weighted sum of the inputs. The transformation is known as a neural layer and the function is referred to as a
neural unit. The intermediate outputs of one layer, called features, are used as the input into the next layer. The neural network through repeated transformations learns multiple layers of nonlinear
features (like edges and shapes), which it then combines in a final layer to create a prediction (of more complex objects). The neural net learns by varying the weights or parameters of a network so
as to minimize the difference between the predictions of the neural network and the desired values. This phase where the artificial neural network learns from the data is called training.
Figure 1: : Schematic representation of a neural network
Neural networks where information is only fed forward from one layer to the next are called feedforward neural networks. On the other hand, the class of networks that has memory or feedback loops is
called Recurrent Neural Networks.
Neural Network Inference
Once the artificial neural network has been trained, it can accurately predict outputs when presented with inputs, a process referred to as neural network inference. To perform inference, the trained
neural network can be deployed in platforms ranging from the cloud, to enterprise datacenters, to resource-constrained edge devices. The deployment platform and type of application impose unique
latency, throughput, and application size requirements on runtime. For example, a neural network performing lane detection in a car needs to have low latency and a small runtime application. On the
other hand, datacenter identifying objects in video streams needs to process thousands of video streams simultaneously, needing high throughput and efficiency.
Neural Network Terminology
A unit often refers to a nonlinear activation function (such as the logistic sigmoid function) in a neural network layer that transforms the input data. The units in the input/ hidden/ output layers
are referred to as input/ hidden/ output units. A unit typically has multiple incoming and outgoing connections. Complex units such as long short-term memory (LSTM) units have multiple activation
functions with a distinct layout of connections to the nonlinear activation functions, or maxout units, which compute the final output over an array of nonlinearly transformed input values. Pooling,
convolution, and other input transforming functions are usually not referred to as units.
The terms neuron or artificial neuron are equivalent to a unit, but imply a close connection to a biological neuron. However, deep learning does not have much to do with neurobiology and the human
brain. On a micro level, the term neuron is used to explain deep learning as a mimicry of the human brain. On a macro level, Artificial Intelligence can be thought of as the simulation of human level
intelligence using machines. Biological neurons are however now believed to be more similar to entire multilayer perceptrons than to a single unit/ artificial neuron in a neural network.
Connectionist models of human perception and cognition utilize artificial neural networks. These connectionist models of the brain as neural nets formed of neurons and their synapses are different
from the classical view (computationalism) that human cognition is more similar to symbolic computation in digital computers. Relational Networks and Neural Turing Machines are provided as evidence
that cognition models of connectionism and computationalism need not be at odds and can coexist.
An activation function, or transfer function, applies a transformation on weighted input data (matrix multiplication between input data and weights). The function can be either linear or nonlinear.
Units differ from transfer functions in their increased level of complexity. A unit can have multiple transfer functions (LSTM units) or a more complex structure (maxout units).
The features of 1000 layers of pure linear transformations can be reproduced by a single layer (because a chain of matrix multiplication can always be represented by a single matrix multiplication).
A non-linear transformation, however, can create new, increasingly complex relationships. These functions are therefore very important in deep learning, to create increasingly complex features with
every layer. Examples of nonlinear activation functions include logistic sigmoid, Tanh, and ReLU functions.
A layer is the highest-level building block in machine learning. The first, middle, and last layers of a neural network are called the input layer, hidden layer, and output layer respectively. The
term hidden layer comes from its output not being visible, or hidden, as a network output. A simple three-layer neural net has one hidden layer while the term deep neural net implies multiple hidden
layers. Each neural layer contains neurons, or nodes, and the nodes of one layer are connected to those of the next. The connections between nodes are associated with weights that are dependent on
the relationship between the nodes. The weights are adjusted so as to minimize the cost function by back-propagating the errors through the layers. The cost function is a measure of how close the
output of the neural network algorithm is to the expected output. The error backpropagation to minimize the cost is done using optimization algorithms such as stochastic gradient descent, batch
gradient descent, or mini-batch gradient descent algorithms. Stochastic gradient descent is a statistical approximation of the optimal change in gradient that produces the cost minima. The rate of
change of the weights in the direction of the gradient is referred to as the learning rate. A low learning rate corresponds to slower/ more reliable training while a high rate corresponds to quicker/
less reliable training that might not converge on an optimal solution.
A layer is a container that usually receives weighted input, transforms it with a set of mostly nonlinear functions and then passes these values as output to the next layer in the neural net. A layer
is usually uniform, that is it only contains one type of activation function, pooling, convolution etc. so that it can be easily compared to other parts of the neural network.
Accelerating Artificial Neural Networks with GPUs
State-of-the-art Neural Networks can have from millions to well over one billion parameters to adjust via back-propagation. They also require a large amount of training data to achieve high accuracy,
meaning hundreds of thousands to millions of input samples will have to be run through both a forward and backward pass. Because neural nets are created from large numbers of identical neurons they
are highly parallel by nature. This parallelism maps naturally to GPUs, which provide a significant computation speed-up over CPU-only training.
GPUs have become the platform of choice for training large, complex Neural Network-based systems because of their ability to accelerate the systems. Because of the increasing importance of Neural
networks in both industry and academia and the key role of GPUs, NVIDIA has a library of primitives called cuDNN that makes it easy to obtain state-of-the-art performance with Deep Neural Networks.
The parallel nature of inference operations also lend themselves well for execution on GPUs. To optimize, validate, and deploy networks for inference, NVIDIA has an inference platform accelerator and
runtime called TensorRT. TensorRT delivers low-latency, high-throughput inference and tunes the runtime application to run optimally across different families of GPUs.
Additional Resources | {"url":"https://developer.nvidia.com/discover/artificial-neural-network","timestamp":"2024-11-08T15:48:59Z","content_type":"text/html","content_length":"50676","record_id":"<urn:uuid:7477c195-3452-4ee2-b5ea-6b2a2aa29380>","cc-path":"CC-MAIN-2024-46/segments/1730477028067.32/warc/CC-MAIN-20241108133114-20241108163114-00858.warc.gz"} |
Multiple math paths to where?
California’s proposed new math framework offers a “choose-your-own-adventure approach” that is “fundamentally flawed,” argue Jennifer Chayes and Tsu-Jae King Liu, professors of electrical engineering
and computer sciences at Berkeley.
Collegebound students could choose a data science pathway rather than advanced algebra and precalculus courses, they write. That will shut the door to STEM majors — including data science — in
college. “We have seen firsthand how students lacking a strong foundation in math struggle to learn both data science and engineering at the college level.”
Photo: George Bakos/Unsplash
Not every collegebound student wants to prepare for a STEM career, writes Pamela Burdman of Just Equations, which supports teaching “quantitative reasoning” to prepare students for college
success. California students need multiple math pathways, she writes on EdSource.
The proposed math framework will improve “math preparation for STEM fields, particularly for historically excluded students,” Burdman writes, while enabling students with other interests to “deepen
their mathematical skills in relevant and engaging ways.”
“Data science and statistics are not recommended for students committed to a STEM path,” Burdman concedes. However, these alternatives “have the potential to reengage students” turned off by math.
I remember sitting in Advanced Algebra/Trig in high school and wondering if I could take a sacred oath that I would never pursue any course of study requiring knowledge of logarithms, sines or
cosines or . . . secants? Yes, secants. I could do math. I didn’t want to. No pathway would have inspired me, unless it led away from math.
I guess shutting the STEM door makes sense for students who are struggling with math — more sense than shoving them through a watered-down course with a college-prep name. But we should be honest
with them about the career options and future pay for people who took the math-lite track. | {"url":"https://www.joannejacobs.com/post/multiple-math-paths-to-where","timestamp":"2024-11-11T00:17:06Z","content_type":"text/html","content_length":"1050483","record_id":"<urn:uuid:170296b8-f887-483a-840c-12b9f567e6c0>","cc-path":"CC-MAIN-2024-46/segments/1730477028202.29/warc/CC-MAIN-20241110233206-20241111023206-00867.warc.gz"} |
American Mathematical Society
Book Review
The AMS does not provide abstracts of book reviews. You may download the entire review from the links below.
MathSciNet review: 1567870 Full text of review: PDF
This review is available free of charge.
Book Information: Author: E. B. Davies Title:
Heat kernels and spectral theory
Additional book information:
Cambridge University Press, Cambridge, 1989, 197 pp., $49.50. ISBN 0-521-36136-2.
[DGS] E. B. Davies, L. Gross and B. Simon, Hypercontractivity: a bibliographic review, Proceedings of the Hoegh Krohn Memorial Conference.
[F] P. Federbush, A partial alternate derivation of a result of Nelson, J. Math. Phys. 10 (1969), 50-52.
Review Information: Reviewer:
Robert S. Strichartz
Bull. Amer. Math. Soc.
(1990), 222-227 | {"url":"https://www.ams.org/journals/bull/1990-23-01/S0273-0979-1990-15936-1/","timestamp":"2024-11-05T09:43:32Z","content_type":"text/html","content_length":"60336","record_id":"<urn:uuid:115d2209-678d-4217-aa8d-6b4e1d755812>","cc-path":"CC-MAIN-2024-46/segments/1730477027878.78/warc/CC-MAIN-20241105083140-20241105113140-00581.warc.gz"} |
Math Q & A for Math Phobics
Math Q & A for Math Phobics
By Marnie Ridgway
Homefires' Virtual Homeschool Conference
Marnie Ridgway has worked as a high energy astrophysicist at NASA and a hardware designer in Silicon Valley.
She is married and has 2 daughters with learning differences. She began homeschooling when she couldn't find anyone else who would put in the effort to educate her children. She has been the director
of an ISP, and tutors and designs curricula for other homeschoolers, specializing in middle and highschool classes for students with learning differences.
Marnie says she has the exact same qualifications that all of you have and that is: she loves her kids and is willing to do everything necessary to help them learn what they need to know.
You can visit her website at Bear Hollow School.
This Q & A was compiled and edited from the November, 2000 logs of our free homeschool discussion list:
Homefires Journal Homeschool Support & Resources.
Q: Ideas for Learning Math Facts? Thanks so much for helping us with our mathphobic children! I'm writing because I have an eight year old (almost nine) boy who cannot seem to learn his math facts.
He is quite possibly mildly ADD (not hyperactive).
He insists on counting on his fingers and seems to be literally incapable of memorizing his addition/subtraction or his multiplication facts. I myself, have always struggled in the same way. I was
counting on my fingers in College Algebra! He is quick to pick up on mathematical concepts but struggles very much with facts.
Have any advice or tips?
A: Try Math Twister! Gee, I still count on my fingers. Isn't that OK? If he picks up on the concepts, he is doing "math" whether the facts are there or not. Make the facts fun and they will come.
Back to Math Twister! — you lay out a 3X3 grid on the floor (with masking tape) or the driveway (with chalk). Put the digits 1 through 9 in the grid. His right foot is the ones number. His left foot
is the tens number. His right hand is the hundreds number. His left hand is the thousands number.
If the number has a zero, put that appendage up in the air. So 10 is the left foot on the 1 and the right foot up in the air. (This makes 1000 really hard to do!) Now call out some numbers for him to
make like 12 and 37 and 154 and 2,976. Now do some simple addition and subtraction problems.
Now let him call out some numbers for you to make; tell him right up front 1000 is off limits. Now move on to multiplication and division. You'll be Twisting out higher math in no time.
After you detangle, let me know how it works.
Q: Ideas for teaching Math to those with low self-esteem? My husband and I have 3 girls between us. My daughter is 3.5 and working on a K program we pieced together, my husband's 16 year old is doing
an independent study program that is internet-based, and his 14 year old just moved in and this is where the question comes.
The 14 year old has traditionally done poorly in school - I believe because of low self-esteem and not fitting into the public school mold of learning. We have pieced together a curriculum that is
heavily based on literature and reading (a weakness for her) as well as grammar, vocabulary, and spelling. (I understand most 9th graders are done with vocabulary and spelling, but again this is a
weak area for her.)
We tested her using some placement exams I found for different math programs, and she placed just below the cut off for Algebra. (Math is a strong point that the teachers missed.) Our problem is that
she is missing some understanding of the basic math principles. Should we continue her on a pre-algebra path or back track?
I'm new to homeschooling (this is our first year) and I am not sure which direction to go in a lot of areas. Until we purchase her books, we have her working on a unit study that we centered around
internet research activities and astronomy, as this has become an interest for her. She has really excelled in this. She is remembering the satellites, the core temperatures, the patterns and make up
of all the planets, galaxies, stars that she can find!
Should we try to go in this direction as well? How can I integrate math and history into units like this? (If you can't tell...I'm a bit lost!) Any advice or direction is greatly appreciated.
A: Help her discover the teacher within. A 14-year-old with self-esteem problems who has traditionally done poorly in school and is just beginning to integrate into a new family in addition to
homeschooling does not need a formalized math curriculum. She needs to do what she likes; she needs to rediscover the self-teacher within her; and she needs to find out for sure that she can learn
what she needs to know. Give her lots of love and lots of time.
With that said, that Hertzsprung-Russell has tons of math associated with it. If she's interested in core temperatures, move on to plasma physics.
Isaac Asimov wrote lots of books that you should be able to find in your library to popularize science for lay people. The books have just enough math to make you feel like you know something, and
Asimov's writing style is easily accessible. She will love this stuff.
George Gamov also wrote some books for the lay reader (that is the non graduate-level physicist) that she might like -- the Mr. Thompkins series, Mr. Thompkins in Wonderland, Mr. Thompkins in
Flatland, and so on. Gamov was trying to explain general relativity, but she will be excited that she understands a bunch of that stuff.
Have her play math games with your other homeschoolers. Set is good for spatial relations. Traverse is fun for the non-chess types. She might be better at these games than her siblings and that might
help the self-esteem. In fact, she might be able to teach you a thing or two.
Bazaar is a great game for algebra skills that is available through Discovery Toys. But don't push the formal school-looking stuff; it sounds like that isn't what she needs right now. Let her run
with the science applications and she will find out what math she needs to know.
Q: Suggestions for teaching High School Astronomy? Is there a program, web site, workbook, etc that you would recommend for highschool astronomy?
A: Software, books, & field trips. There is a software package called "Red Shift." It can be purchased (around $75) through Educational Resources. It has a companion book, and I believe there is a
video of some episodes of Nova or some program on the Discovery channel to go with it. Anyway, it is a pretty comprehensive high-school level course. It will take you through things step by step.
So will Isaac Asimov. Your library will have his science fiction and his children's books. You want his essays on astronomy and physics; they will be in the non-fiction section of the adult library.
They are also out in paperback if you want to buy them.
He taught biochemistry at Boston University for years, and his books are very entertaining -- the whole family will enjoy reading them. Have your daughter read them out loud to you and explain them
to you -- or work out questions you have together. Nothing helps us learn like having to teach someone else.
How far are you from NASA Lewis? It's near Cincinnati I think. How about a family field trip? Make sure she organizes those photos and pictures according to some scheme; that organization will be
creating her own Hertzsprung-Russell diagram if you haven't already gotten to that part. Then she will be really eager to learn what Hertzsprung and Russell did and how that diagram is used.
Their Self-Esteem and Your Lesson Plans
By the way, if she throws it all over tomorrow, don't be surprised. I have discovered that, in their quest for self-esteem after public school, my kids have shown interests that I have invested in
big time to develop a curriculum only to have them wake the next day with "that subject is so yesterday's news." I privately mourn the loss of my brilliant plans and move on with them.
A few weeks ago I overheard them tell someone that they knew I loved them because I thought their ideas were good. And their ideas must really be good because I always managed to find the course that
was developed around their ideas.
(Little do they realize the amount of work!) Don't be surprised if your love gets tested by checking to see if you are as attached to the effort you put in to developing the program as you are to
her. Good luck to you!
Q: What to do with a creative kid who hates math? My 13 year old daughter is allergic to math!! She has great difficulty doing even VERY simple (1 and 2 digit numbers!) addition and subtraction in
her head, gets confused doing long division, and just doesn't seem to understand what math is about. Because I never had trouble in this area I am having a very hard time even understanding what the
problem is. She is average intelligence, creative, loves to write stories and draw, but I'm afraid she won't even be able to balance her checkbook. Any help would be greatly appreciated.
A: Discover math through literature, cooking, & art. If she is creative and loves to write stories and draw, she will do just great in life even if she never balances her checkbook. I don't balance
mine and I'm reasonably functional as adults go.
Cuisenaire used to have some books that taught math through fairy tales. I think the early elementary title was "Afterwards." Students (or at that level, the teacher) read the stories and then the
students did math activities. Older kids need to see the application of the facts before they can be convinced that there is any value in having those facts on the tips of their tongues.
What drawing does your daughter do? Is she interested in abstract shapes, perspectives, shading? That's geometry. What stories does she like? Did you read Harry Potter? Would she like to make mazes?
Does she like historical stories? The Oregon Trail software has kids plan a trip west. I'm not suggesting that you get the software but have you tried giving her some parameters (like a certain
amount of money and a certain timeframe) and having her plan a trip or a wardrobe or a party?
Cooking involves a lot of math skills, and catering involves a lot of arithmetic. Have you suggested that she take the grocery budget and feed the family for a week?
Does she have a checkbook? There are packages on the market (check with your bank; they may have one) where you are the banker and your child has to "balance her budget" through you by writing checks
for what she spends. You can give interest and make loans -- in fact, you can do some pretty high level economics.
Have you talked about the elections and the electoral college? This month's Discover magazine has an article on voting strategies and how "one man, one vote" may not give us what we think we want.
In short, make it practical.
When it is, the facts will come. There has been some traffic here on the Chisanbop method of counting on your fingers; that's great. There are books that do calculator math, and let's face it, all
adults use calculators. Skip the facts and teach her how to use the technology. But whatever you do, make it interesting to her. I don't know of a single 13-year-old who can be forced to learn
anything she doesn't like.
Q: Suggestions for developing Mental Math skills? I worry that my daughter can't even subtract 9 from 11 in her head without thinking REALLY hard. Would doing a 2-minute mental math drill help, or do
you think her brain just doesn't work that way?
A: Mental Math is Important — But Not As Important as Relevancy. Scholastic publishes some "mental math" workbooks and I think Cuisenaire has some, too. I got one of my kids to see the point in rapid
math recall by the following: Hannah, quick, I need help! What is 7 from 13? Quick, my hands are full! Er......Ah........ Quick, it's already boiling. How many Tbsp. of butter? Quick! Well,
er.....ah...... Oh, well, that's OK. I got it.
That isn't the best example, but I did a bunch of them and finally got her to see the point in quick recall of math facts. We did some mental math practice for those and she worked out this elaborate
scheme in her head whereby she "finds" answers on her "calculator."
She'd have to explain that to you because I can't begin to figure out what she is doing. But it works for her. The point is that she only developed her scheme after deciding on her own that it was
The most important concept, I think, to pick up from algebra is the substitution of equivalencies. The Bazaar Game works the best for that. It's kind of old though and I haven't seen one around for a
long time so I made my own version.
Anyway, in the game, you are trying to buy cards with colored pebbles. You can roll a die for the pebbles but it takes a long time to get enough. Instead, you trade pebbles based on an equivalency
card. The box it came in was about 5"X8" and 2-3" thick; it's brightly colored and shows a Middle Eastern bazaar on the front.
One 13-year-old boy plays the Bazaar game and says he just loves it because it doesn't involve any math (little does he know) but his algebra grades have jumped from Ds to Bs.
One of the top sales people for IBM has such incredible people skills that she can meet all the needs of her customers but she can't do the math on the paperwork. IBM likes her well enough to hire a
paperwork person just for her. I asked the saleswoman once if she wasn't worried that IBM would give her job to the paperwork person. She said it had come up but the customers don't like the
paperwork person because she doesn't have the people skills.
Your daughter will find her own way with or without math skills. And when they are important to her, she will develop them. Make them important to her by showing her how you use them. If you haven't
used any algebra since highschool, how can you expect her to believe she needs to know it?
Q: What to do when one child hates math, and their sibling is a math whiz? I have a 8 year old son and a six year old son. My eight year old is gifted in many areas (reads Dad's old college texts for
fun), but struggles in math. He also can't seem to come up with the answer to 7 + 8 without drawing circles and counting! He understands the concepts quite well, but can't compute. We can do flash
cards and get a hundred percent, but the same addends in a regrouping (57+78) problem will throw him and he will randomly guess answers. This frustrates him greatly!
We use Math U See, which is a very concrete math program using manipulatives similar to Cuisinaire rods. To make matters worse his little brother is a math whiz and is ahead of him, another great
frustration for him. He says he "HATES MATH". I have book-marked the finger math site, any other suggestions?
A: Try cooperative games. My kids do that competing thing too -- "I won't try it because she is already good at it."
There is nothing wrong with the visual approach to doing math facts. In fact, a lot of the information we pull from our heads is often in a visual form; we count on our fingers in our minds, we just
don't let anyone see our fingers wiggle.
Go for the games approach to getting those facts. There's a game that has a 10X10 grid. The numbers 1 through 10 are across the top and bottom and down the sides. The numbers in the grids are the
products. Roll 2 dice or spin a spinner to get 2 numbers. Multiply the 2 numbers and put your playing piece on the product. The idea is to get 4 of your playing pieces in a row. You can adapt this to
addition facts. There are other similar grid games -- they probably play them at your Half Moon Bay games night.
We also play a math game with a deck of cards. Shuffle the deck and deal out 8 (or 10 or 6) cards to each player. Then roll a die or spin a spinner or have some random way to select a target number.
Now use your cards and any math operations you know to get as close as you can to the target number. If you only know addition and subtraction, that's all you use. If you can multiply, use it, and so
on. The person who gets closest to the target number gets to keep the cards he used. All other cards get returned to the deck. The person with the most cards at the end of the game wins. Set a time
limit if you want to speed things up.
Teach your math phobe the concept of squares and square roots if you want him to have one up on his sibling; as well as he gets concepts, he should be able to understand the idea even if he doesn't
know the facts. But work in cooperative groups if you want to do something about that sibling thing.
Q: What Do You Do with Cuisenaire Rods? Does anyone have tips or activities or know of good resources for using Cuisenaire rods? I bought a used set recently, and am not quite sure what to do with
them. My son is 6.
A: Cuisenaire Rods are used in conjunction with Miquon Math workbooks — and are good tools for math games. The workbooks have outlines of the rods on their pages -- and you lay the rods down to match
the outlines -- building the math concepts that are being taught. Miquon Math workbooks are inexpensive and can be purchased from: Activity Resources.
You can also use the Cuisenaire rods like money and play store with them. Put prices on your little toys and let your son "buy" them with the rods making change in rods and so on. We did that with
base 10 blocks and with Unifix cubes in addition to regular money.
Additional Math Articles | {"url":"https://www.homefires.com/articles/math_phobics.asp","timestamp":"2024-11-03T16:14:07Z","content_type":"text/html","content_length":"45280","record_id":"<urn:uuid:d289ec0c-23aa-44ab-9f28-3cb649746fe3>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00336.warc.gz"} |
1.6 What the book does not cover
The field of missing data research is vast. This book focuses on multiple imputation. The book does not attempt cover the enormous body of literature on alternative approaches to incomplete data.
This section briefly reviews three of these approaches.
1.6.1 Prevention
With the exception of McKnight et al. (2007 Chapter 4), books on missing data do not mention prevention. Yet, prevention of the missing data is the most direct attack on problems caused by the
missing data. Prevention is fully in spirit with the quote of Orchard and Woodbury given on p. . There is a lot one could do to prevent missing data. The remainder of this section lists point-wise
Minimize the use of intrusive measures, like blood samples. Visit the subject at home. Use incentives to stimulate response, and try to match up the interviewer and respondent on age and ethnicity.
Adapt the mode of the study (telephone, face to face, web questionnaire, and so on) to the study population. Use a multi-mode design for different groups in your study. Quickly follow-up for people
that do not respond, and where possible try to retrieve any missing data from other sources.
In experimental studies, try to minimize the treatment burden and intensity where possible. Prepare a well-thought-out flyer that explains the purpose and usefulness of your study. Try to organize
data collection through an authority, e.g., the patient’s own doctor. Conduct a pilot study to detect and smooth out any problems.
Economize on the number of variables collected. Only collect the information that is absolutely essential to your study. Use short forms of measurement instruments where possible. Eliminate vague or
ambivalent questionnaire items. Use an attractive layout of the instruments. Refrain from using blocks of items that force the respondent to stay on a particular page for a long time. Use
computerized adaptive testing where feasible. Do not allow other studies to piggy-back on your data collection efforts.
Do not overdo it. Many Internet questionnaires are annoying because they force the respondent to answer. Do not force your respondent. The result will be an apparently complete dataset with mediocre
data. Respect the wish of your respondent to skip items. The end result will be more informative.
Use double coding in the data entry, and chase up any differences between the versions. Devise nonresponse forms in which you try to find out why people they did not respond, or why they dropped out.
Last but not least, consult experts. Many academic centers have departments that specialize in research methodology. Sound expert advice may turn out to be extremely valuable for keeping your missing
data rate under control.
Most of this advice can be found in books on research methodology and data quality. Good books are Shadish, Cook, and Campbell (2001), De Leeuw, Hox, and Dillman (2008), Dillman, Smyth, and Melani
Christian (2008) and Groves et al. (2009).
1.6.2 Weighting procedures
Weighting is a method to reduce bias when the probability to be selected in the survey differs between respondents. In sample surveys, the responders are weighted by design weights, which are
inversely proportional to their probability of being selected in the survey. If there are missing data, the complete cases are re-weighted according to design weights that are adjusted to counter any
selection effects produced by nonresponse. The method is widely used in official statistics. Relevant pointers include Cochran (1977) and Särndal, Swensson, and Wretman (1992) and Bethlehem (2002).
The method is relatively simple in that only one set of weights is needed for all incomplete variables. On the other hand, it discards data by listwise deletion, and it cannot handle partial
response. Expressions for the variance of regression weights or correlations tend to be complex, or do not exist. The weights are estimated from the data, but are generally treated as fixed. The
implications for this are unclear (Little and Rubin 2002, 53).
There has been interest recently in improved weighting procedures that are “double robust” (Scharfstein, Rotnitzky, and Robins 1999; Bang and Robins 2005). This estimation method requires
specification of three models: Model A is the scientifically interesting model, Model B is the response model for the outcome, and model C is the joint model for the predictors and the outcome. The
dual robustness property states that: if either Model B or Model C is wrong (but not both), the estimates under Model A are still consistent. This seems like a useful property, but the issue is not
free of controversy (Kang and Schafer 2007).
1.6.3 Likelihood-based approaches
Likelihood-based approaches define a model for the observed data. Since the model is specialized to the observed values, there is no need to impute missing data or to discard incomplete cases. The
inferences are based on the likelihood or posterior distribution under the posited model. The parameters are estimated by maximum likelihood, the EM algorithm, the sweep operator, Newton–Raphson,
Bayesian simulation and variants thereof. These methods are smart ways to skip over the missing data, and are known as direct likelihood, full information maximum likelihood (FIML), and more
recently, pairwise likelihood estimation.
Likelihood-based methods are, in some sense, the “royal way” to treat missing data problems. The estimated parameters nicely summarize the available information under the assumed models for the
complete data and the missing data. The model assumptions can be displayed and evaluated, and in many cases it is possible to estimate the standard error of the estimates.
Multiple imputation extends likelihood-based methods by adding an extra step in which imputed data values are drawn. An advantage of this is that it is generally easier to calculate the standard
errors for a wider range of parameters. Moreover, the imputed values created by multiple imputation can be inspected and analyzed, which helps us to gauge the effect of the model assumptions on the
The likelihood-based approach receives an excellent treatment in the book by Little and Rubin (2002). A less technical account that should appeal to social scientists can be found in Enders (2010,
chaps. 3–5). Molenberghs and Kenward (2007) provide a hands-on approach of likelihood-based methods geared toward clinical studies, including extensions to data that are MNAR. The pairwise likelihood
method was introduced by Katsikatsou et al. (2012) and has been implemented in lavaan. | {"url":"https://stefvanbuuren.name/fimd/sec-doesnotcover.html","timestamp":"2024-11-09T12:58:12Z","content_type":"text/html","content_length":"75010","record_id":"<urn:uuid:eed68332-7327-4ee0-aea8-687a6f23a776>","cc-path":"CC-MAIN-2024-46/segments/1730477028118.93/warc/CC-MAIN-20241109120425-20241109150425-00879.warc.gz"} |
RCSL: Rank Constrained Similarity Learning for Single Cell RNA Sequencing Data version 0.99.95 from CRAN
A novel clustering algorithm and toolkit to accurately identify various cell types using single cell RNA sequencing data from a complex tissue. This algorithm considers both local similarity and
global similarity among the cells to discern the subtle differences among cells of the same type as well as larger differences among cells of different types. This algorithm uses Spearman’s rank
correlations of a cell’s expression vector with those of other cells to measure its global similarity, and learns neighbour representation of a cell as its local similarity. The overall similarity of
a cell to other cells is a linear combination of its global similarity and local similarity. See Mei et. al. (2021) <DOI:10.1101/2021.04.12.439254> for more details.
Author Qinglin Mei [aut, cre], Guojun Li [ctb], Zhengchang Su [fnd]
Bioconductor views Clustering DimensionReduction RNASeq Sequencing SingleCell Software Visualization
Maintainer Qinglin Mei <meiqinglinkf@163.com>
License GPL-3
Version 0.99.95
URL https://github.com/QinglinMei/RCSL
Package repository View on CRAN
Install the latest version of this package by entering the following in R:
Any scripts or data that you put into this service are public.
RCSL documentation
built on April 19, 2021, 9:06 a.m. | {"url":"https://rdrr.io/cran/RCSL/","timestamp":"2024-11-05T06:17:42Z","content_type":"text/html","content_length":"25265","record_id":"<urn:uuid:68a4e68c-8f22-4e57-b76e-0349367e1f50>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00795.warc.gz"} |
Solving Equations Containing Fractions Using the Multiplication Property of Equality
Learning Outcomes
• Solve equations with fractions using the Multiplication Property of Equality
Solve Equations with Fractions Using the Multiplication Property of Equality
We will now solve equations that require multiplication to isolate the variable. Consider the equation [latex]\Large\frac{x}{4}\normalsize=3[/latex]. We want to know what number divided by [latex]4[/
latex] gives [latex]3[/latex]. To “undo” the division, we will need to multiply by [latex]4[/latex]. The Multiplication Property of Equality will allow us to do this. This property says that if we
start with two equal quantities and multiply both by the same number, the results are equal.
The Multiplication Property of Equality
For any numbers [latex]a,b[/latex], and [latex]c[/latex],
[latex]\text{if }a=b,\text{ then }ac=bc[/latex].
If you multiply both sides of an equation by the same quantity, you still have equality.
Let’s use the Multiplication Property of Equality to solve the equation [latex]\Large\frac{x}{7}\normalsize=-9[/latex].
Solve: [latex]\Large\frac{x}{7}\normalsize=-9[/latex].
Show Solution
Try It
Solve: [latex]\Large\frac{p}{-8}\normalsize=-40[/latex]
Show Solution
Try It
In the following video we show two more examples of when to use the multiplication and division properties to solve a one-step equation.
Solve Equations with a Coefficient of [latex]-1[/latex]
Look at the equation [latex]-y=15[/latex]. Does it look as if [latex]y[/latex] is already isolated? But there is a negative sign in front of [latex]y[/latex], so it is not isolated.
There are three different ways to isolate the variable in this type of equation. We will show all three ways in the next example.
Solve: [latex]-y=15[/latex]
Show Solution
Try It
In the next video we show more examples of how to solve an equation with a negative variable.
Solve Equations with a Fraction Coefficient
When we have an equation with a fraction coefficient we can use the Multiplication Property of Equality to make the coefficient equal to [latex]1[/latex].
For example, in the equation:
[latex]\Large\frac{3}{4}\normalsize x=24[/latex]
The coefficient of [latex]x[/latex] is [latex]\Large\frac{3}{4}[/latex]. To solve for [latex]x[/latex], we need its coefficient to be [latex]1[/latex]. Since the product of a number and its
reciprocal is [latex]1[/latex], our strategy here will be to isolate [latex]x[/latex] by multiplying by the reciprocal of [latex]\Large\frac{3}{4}[/latex]. We will do this in the next example.
Solve: [latex]\Large\frac{3}{4}\normalsize x=24[/latex]
Show Solution
Try It
Solve: [latex]-\Large\frac{3}{8}\normalsize w=72[/latex]
Show Solution
Try It
In the next video example you will see another example of how to use the reciprocal of a fractional coefficient to solve an equation. | {"url":"https://courses.lumenlearning.com/wm-developmentalemporium/chapter/solving-equations-that-contain-fractions-using-the-multiplication-property-of-equality/","timestamp":"2024-11-02T06:37:14Z","content_type":"text/html","content_length":"63546","record_id":"<urn:uuid:184690af-d783-4785-a9d7-dfefd277b956>","cc-path":"CC-MAIN-2024-46/segments/1730477027677.11/warc/CC-MAIN-20241102040949-20241102070949-00312.warc.gz"} |
Explanation for why 36 is the lowest common multiple for 12 and 36
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Re: Compiler optimization and floating point operations
glen herrmannsfeldt <gah@ugcs.caltech.edu>
Thu, 24 Oct 2013 20:21:21 +0000 (UTC)
From comp.compilers
| List of all articles for this month |
From: glen herrmannsfeldt <gah@ugcs.caltech.edu>
Newsgroups: comp.compilers
Date: Thu, 24 Oct 2013 20:21:21 +0000 (UTC)
Organization: Aioe.org NNTP Server
References: 13-10-026 13-10-029
Keywords: arithmetic, administrivia, comment
Posted-Date: 24 Oct 2013 16:46:37 EDT
(snip, John wrote)
> [I think you mean they're not associative. I don't know any
> situations where a+b != b+a, but lots where a+(b+c) != (a+b)+c -John]
I don't know of any with non-commutative addition, but there were
some Cray machines with non-commutative multiply.
For many Cray applications, getting a close answer fast is better
than the exact answer slow.
-- glen
[Oh, right. Can we safely assume that arithmetic model is dead? -John]
Post a followup to this message
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C# Rounding To 2 Decimal Places: A Step-By-Step Guide
C# Rounding To 2 Decimal Places
Rounding is a common task in programming, especially when working with financial or mathematical calculations. In C#, rounding to 2 decimal places is a frequently used operation. Fortunately, the C#
language provides a built-in function, Math.Round(), that simplifies the task of rounding numbers.
Understanding the Math.Round() function in C#
The Math.Round() function in C# is a member of the System.Math class and allows you to round a given number to the nearest integer or to a specified number of decimal places. When rounding to 2
decimal places, the function uses the “round half up” algorithm by default. This means that if the digit to be rounded is 5 or greater, the previous digit is increased by 1. Otherwise, it remains the
The syntax of the Math.Round() function in C#
The Math.Round() function in C# has several overloads, but for rounding to 2 decimal places, we will mainly use the following overload:
public static double Round(double value, int decimals)
The first parameter, `value`, is the number that you want to round. It can be of type `double`, `float`, `decimal`, or other numeric types.
The second parameter, `decimals`, is the number of decimal places to round to. In our case, we want to round to 2 decimal places, so the value should be 2.
Using the Math.Round() function in C#
To round a number to 2 decimal places using the Math.Round() function, you simply need to pass the value and the number of decimal places to the function. Here’s an example:
double number = 3.14159;
double roundedNumber = Math.Round(number, 2);
Console.WriteLine(roundedNumber); // Output: 3.14
In this example, the variable `number` holds the value 3.14159. By using the Math.Round() function with 2 as the number of decimal places, we obtain the rounded value of 3.14.
Using the MidpointRounding option in Math.Round() for rounding to 2 decimal places
The Math.Round() function also provides an optional parameter, `MidpointRounding`, that allows you to specify the rounding behavior in case of a tie. A tie occurs when the digit to be rounded is
exactly halfway between two possible rounded values.
By default, the Math.Round() function uses the MidpointRounding.ToEven option. This means that when a tie occurs, it rounds to the nearest even number. However, when rounding to 2 decimal places, the
MidpointRounding.AwayFromZero option is typically more appropriate. This option always rounds the tie away from zero.
To use the MidpointRounding.AwayFromZero option, you need to pass it as the third parameter to the Math.Round() function. Here’s an example:
double number = 3.145;
double roundedNumber = Math.Round(number, 2, MidpointRounding.AwayFromZero);
Console.WriteLine(roundedNumber); // Output: 3.15
In this example, the variable `number` holds the value 3.145. By using the Math.Round() function with 2 as the number of decimal places and MidpointRounding.AwayFromZero as the rounding option, we
obtain the rounded value of 3.15.
Handling corner cases with the Math.Round() function in C#
When rounding to 2 decimal places, there are some corner cases to be aware of. One such case is when the number to be rounded ends in 5. In this situation, the rounding behavior may not always be as
Normally, if the digit to be rounded is exactly 5 and the previous digit is even, the number is rounded down. However, if the previous digit is odd, the number is rounded up. This is due to the
MidpointRounding.ToEven behavior.
For example:
double number1 = 3.125;
double number2 = 3.135;
double roundedNumber1 = Math.Round(number1, 2);
double roundedNumber2 = Math.Round(number2, 2);
Console.WriteLine(roundedNumber1); // Output: 3.12
Console.WriteLine(roundedNumber2); // Output: 3.14
In these examples, the number `3.125` is rounded down to `3.12`, while the number `3.135` is rounded up to `3.14`. This behavior is in line with the “round half up” algorithm and the
MidpointRounding.ToEven option.
Rounding to 2 decimal places with Math.Round(): Examples and explanations
Let’s take a look at a few more examples to demonstrate the usage of the Math.Round() function for rounding to 2 decimal places.
Example 1:
double number = 1.234;
double roundedNumber = Math.Round(number, 2);
Console.WriteLine(roundedNumber); // Output: 1.23
In this example, the number `1.234` is rounded down to `1.23` since the digit to be rounded is less than 5.
Example 2:
double number = -1.235;
double roundedNumber = Math.Round(number, 2);
Console.WriteLine(roundedNumber); // Output: -1.24
In this example, the negative number `-1.235` is rounded up to `-1.24` since the digit to be rounded is greater than 5.
Using Math.Round() with custom rounding options
Sometimes, you may require custom rounding behavior that is not provided by the default Math.Round() function. In such cases, you can write your own rounding logic using mathematical operations.
For example, if you need to always round up to 2 decimal places, you can multiply the number by 100, use Math.Ceiling() to round up to the nearest integer, and then divide the result by 100. Here’s
an example:
double number = 1.234;
double roundedNumber = Math.Ceiling(number * 100) / 100;
Console.WriteLine(roundedNumber); // Output: 1.24
In this example, the number `1.234` is rounded up to `1.24` using custom rounding logic.
Q1: Can I use the Math.Round() function to round to a specific number of decimal places other than 2?
Yes, you can use the Math.Round() function to round to any number of decimal places by adjusting the second parameter accordingly.
Q2: Does the Math.Round() function guarantee perfect rounding?
No, the Math.Round() function uses the “round half up” algorithm, which may not always produce the expected rounding behavior in certain cases. It’s important to be aware of the corner cases and use
the appropriate rounding options if necessary.
Q3: How can I round a number to 2 decimal places without using Math.Round()?
You can multiply the number by 100, use Math.Floor() or Math.Ceiling() to round down or up to the nearest integer, and then divide the result by 100. Alternatively, you can use string formatting to
achieve the desired rounding.
In conclusion, rounding to 2 decimal places in C# can be easily accomplished using the Math.Round() function. By understanding its syntax, using the MidpointRounding option, and handling corner
cases, you can ensure accurate and consistent rounding behavior in your applications. Remember to be aware of the specifics of the “round half up” algorithm and consider custom rounding options if
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Terrain classification for the harvesting of tropical forests
D. Mazier, F. Baumgartner and C. Lepitre
D. MAZIER, F. BAUMGARTNER and C. LEPITRE were all associated with the Centre technique forestier tropical (CTFT), Nogent-sur-Marne. This article, originally written for UNASYLVA, is contained in a
more lengthy and detailed version in the CTFT publication Bois et forêts des tropiques, No. 162.
Terrain classification is linked with evaluation of the accessibility of forest resources and is therefore of interest to research workers, planners and forest managers.
Numerous classification systems have been proposed, but most were conceived for application in temperate zones and serve essentially as a basis for deciding on the most suitable logging methods. They
do not prove very useful in tropical forests in areas not under forest management, where the classification must take into account the difficulties not only of carrying out the harvesting operations
as such (extraction), but also of creating the necessary infrastructure (roads).
A major forest inventory has been conducted in Gabon, covering an area of 3000000 hectares in a region situated to the north, east and south of Booué which will be served by the projected
Trans-Gabonese railway (Figure 1). It was carried out by the Centre technique forestier tropical on behalf of FAO acting as executing agency for the United Nations Development Programme.
FIGURE 1. - REGION OF GABON COVERED BY THE INVENTORY
The region possesses a great variety of terrain types, ranging from fairly flat areas to others in which the topography makes harvesting operations extremely complicated. It was therefore considered
essential to complete the inventory with a study of the terrain which would make it possible to characterize the different zones other than by mere qualitative appraisements. For this it was
necessary to design quickly a practical method, adapted to conditions and able to provide a reliable, homogeneous classification of the terrain.
Several possibilities were considered, including the method based on structural geomorphology. In the end an original methodology was evolved, adapted to conditions in the Gabonese forests and to the
data available. A concise description of this methodology is given in this article.
There are two essential differences between forest harvesting in Gabon and that conducted in temperate zones: yield per hectare and lack of infrastructure. The first task to be carried out in any
harvesting operation is therefore to endow the area to be harvested with a network of roads and extraction routes. Hence the topography of the area influences two of the most important cost factors
in timber production: establishment of the infrastructure, and extraction.
Some information about the physical features of the forest region studied are necessary for a better understanding of the study.
Climate. Although the climate does not really constitute a feature of the terrain, it is too important not to be taken into consideration.
Gabon, which straddles the Equator, has a moist, equatorial climate. In the zone with which we are concerned, there is little variation in climate and annual rainfall is between 1.6 and 2 m,
distributed between four seasons. The geological formations (lower and middle pre-Cambrian) are relatively homogeneous and it is the tectonic movements that influence the contour of the land.
The undergrowth, which varies greatly in density, can hardly be counted an obstacle to harvesting operations. As for deadfalls, they are uniformly distributed and interfere but little with harvesting
operations. The vegetation does not constitute an impediment, therefore. But the same cannot be said of the rocks that may encumber the ground. These consist sometimes of rocky slabs, but more often
of large blocks which, if numerous and grouped together, make penetration for harvesting purposes impossible. This statement needs to be qualified, however. When it is a question of building forest
roads, everything possible will be done to avoid traversing the rocky areas; and when an area of forest with some potential is so encumbered by rocks that crawler tractors cannot pass, this area will
not be harvested. In Gabon unutilizable areas of this kind are usually scattered and not very extensive. The rocks are more numerous where the slope of the terrain is greater and are particularly
frequent on the steep sides of mountains.
Soil quality. In road construction the quality of the soil is of great importance. The road-bed has to be laid on the surface layers of the soil which are often fairly uniform in a given area. What
is of the greatest interest is the amount of laterite or other granulometric material available for constructing the bed. The number of borings made during the course of the normal inventory is too
low to serve as the basis for a systematic location of gravel deposits: such a survey is beyond the means of the traditional type of inventory. Only traces which are clearly visible without any
additional work will be noted. The presence of laterite can be detected in creek beds, breaks in slopes, the root system of uprooted trees, etc. It is advisable to include in any forest survey
operation that is planned provision for the location and systematic recording of these traces.
STUDYING AERIAL PHOTOGRAPHS AS PART OF A SURVEY getting acquainted with the terrain
Most of the factors referred to above will not be examined in more detail, either because they are homogeneous throughout the area under study, or because of the impossibility of determining
objectively how often and where they occur. In the following sections we are concerned only with a detailed study of the terrain. There are two ways of carrying out such a study:
· A microdescription of the terrain on the basis of the data collected by the field teams.
· A macrodescription of the terrain on the basis of existing photographic material and maps: of the 3000000 ha which have to be studied, about one half is covered by maps on a scale of l:50000
with an interval of 20 metres between contour lines, and the other half is almost entirely covered by aerial photographs on more or less the same scale.
Microdescription of the terrain can be defined as description of the microtopography, that is to say, of the terrain as it appears to anyone moving about in the forest. It is limited to the local
topography, but it is this that conditions road construction and the operation of the extraction machinery. This microdescription of the terrain is drawn up on the basis of the data obtained from
questionnaires completed by the inventory teams. The sampling intensity is thus the same as for the evaluation of the forest potential.
For the inventory carried out in Gabon, the trees were counted and measured in strips 25 m wide lying across the transect line. The area of the record units was usually l ha (400 m long × 25 m wide).
The recording form, in addition to data on the trees, included also information, at every 50 metres along the transect line, on the four slopes, i.e., the two longitudinal ones in the direction of
the line, (average slope over 25 metres) and the two at right angles to the line on each side (average slope over 12.50 metres).
Although designed originally to assist in calculating the area of the record units in a horizontal projection, this information could also help to describe the microtopography.
It was thus possible to determine, every 50 metres along the transect line, the greatest slope in each of the four 25 m × 12.5 m quadrants of the record unit situated around this point, by the
[], P[1] and P[2]
being the slopes of the two directions defining the quadrant. For a record unit of 1 ha, 400 m long, 32 values of P were thus obtained, grouped into the following classes:
0 to 20 %
20 to 30 %
30 to 40 %
40 to 50 %
50 to 60 %
more than 60%
The slopes thus classified were then regrouped for each of the 6 × 6 km squares to be inventoried (these were the systematically distributed "primary units" of a two-stage sampling design in which
the second-stage sampling consisted, in each square, of two continuous parallel strips 6 km long by 25 m wide, each containing 15 record units of 1 ha.
│Primary unit (6 km^2 square) │ Percentage classes │
│ ├────────────┬────────┬────────┬────────┬────────┬────────────┤
│ │Less than 20│20 to 30│30 to 40│40 to 50│50 to 60│More than 60│
│ E │ 2 │ 95.3 │ 2.6 │ 1.2 │ 0.7 │ 0.2 │ - │
│ E │ 3 │ 90.0 │ 6.9 │ 1.7 │ 1.4 │ - │ - │
│ F │ 5 │ 72.8 │ 15.3 │ 6.4 │ 4.3 │ 1.0 │ 0.2 │
│ G │ 5 │ 26.7 │ 23.1 │ 18.1 │ 15.1 │ 9.6 │ 7.4 │
│ V │ 12 │ 7.8 │ 8.6 │ 15.5 │ 18.8 │ 18.1 │ 31.2 │
│ Y │ 15 │ 20.8 │ 15.3 │ 12.8 │ 16.0 │ 12.7 │ 22.4 │
│ Z │ 16 │ 15.5 │ 7.1 │ 17.6 │ 14.3 │ 14.3 │ 31.2 │
In order to make possible quicker comparison between primary units, one can calculate the average weighted slope by weighing the mean values for each class of slope against the respective percentages
as shown in the above table.
Figure 2 shows in diagram form the mean weighted slope gradients thus calculated per primary unit for a large part of the inventoried zone where a homogeneous inventory design was implemented. From
this it will be seen that broadly speaking the slopes run from east to west.
To sum up, analysis of the field documents (recording forms) offers a certain number of advantages: it makes possible over-all comparisons between concessions and gives an idea of the real slope of
the ground, which is very interesting as compared with the information obtained from aerial photographs and maps; since the forest cover hides the details of the relief of the ground, it is not
possible to obtain an exact idea of the topography from an examination of photographs. However, it is advisable to complement field sampling by a study of photographic data and maps in order to
obtain an over-all picture (macrodescription of the terrain).
The macrodescription of the terrain indicates only the topographic features visible on aerial photographs and does not take into account those on a small scale and the details concealed by
vegetation. In addition to aerial photographs, use can also be made of contour maps on a scale of 1:50000, or of the map overlays which serve instead. Maps based on photographs facilitate analysis
but eliminate certain details. They can be prepared or most parts of Gabon, because photographs at least are available.
A method for obtaining a macrodescription of the terrain was worked out in several stages, which are described below.
· First stage: assembling data. The first phase consisted of assembling a certain amount of factual data on maps and overlays on a scale of 1:50000 (with an interval of 2 m between contour lines),
which covered half the inventoried zone. It was decided to start by using the map overlays.
On each map, covering an area of about 75000 ha, 9 sampling units were disposed in such a way as to obtain a homogeneous design with adjoining sheets. A few areas already being harvested were covered
by sampling designs of varying, but much greater intensity.
The sampling unit was a square 2 × 2 km (400 ha), or 4 cm on the map. A square was chosen because it is the most regular geometric figure and therefore made it possible, by juxtaposition, to cover
the whole area under study (the sides of the square were oriented north-south and east-west). The choice of the dimensions of the square represented a compromise between too small a unit, which would
provide only very specific information for a restricted area, and too large a unit, which might cover several different types of relief but conceal the heterogeneity of the area by providing only
information on averages.
FIGURE 2. - MEAN WEIGHTED SLOPE GRADIENT PER PRIMARY UNIT
The sampling design is shown at the bottom of the diagram. Each primary unit has an area of 6 km², and its zone of extension is 12 km². In the upper part of the diagram each primary unit is
represented by its area of extension.
A certain number of parameters, selected to represent the maximum possible variations in terrain, were measured in these squares:
x[1]: slope with gradient of less than 20 percent
x[2]: slope with gradient of more than 40 percent
These limits of 20 to 40 percent were chosen for two main reasons: the first is that, where the gradient is less than 20 percent, harvesting is very easy, and where it is more than 40 percent,
extraction by tractor is difficult; the second is of a practical kind: gradients of 20 percent and 40 percent correspond to the intervals between the 2 mm and 1 mm contour lines respectively. The
area of these slopes is determined by a dot grid.
x[3]: number of changes in slope direction along the two median lines of the square;
x[4]: number of changes in slope direction along the sides of the square: it is assumed that each of the lines, median and side, is described by the observer who counts the number of times that
the slope direction changes;
x[5]: number of rivers intersected by the median lines;
x[6]: number of rivers intersected by the sides of the square.
The number of changes in slope and the density of the drainage system determine the degree of fragmentation of the relief. The distinction between the sides and the medians of the square corresponded
to different sampling designs, these lines having been selected because they were easy to plot. It should be noted that the parameters x[3] and x[4] are accurate and more sensitive than the
parameters x[5] and x[6]. Each crossing of a river indicated on the map corresponds to a change in slope, but the reverse is not true. In addition, the details of the drainage system depend on the
accuracy of the recorders and on the type of relief.
x[7] : number of contour lines intersected by the median lines;
x[8] : number of contour lines intersected by the sides of the square (these two parameters give the total change in level);
x[9]: maximum variation in level within the square, in metres;
x[10]: total length of rivers within the square (determined by opisometer with an accuracy of ± 100 m).
These first ten parameters constitute a direct transcription of the information provided by the map. The following parameters result from further elaboration of these primary data.
x[11]: length of the road to be traversed, starting from the centre of the square in order to leave the circle inscribed around this centre without encountering any longitudinal slope exceeding
10 percent;
x[12]: sum of the transverse slopes expressed as a percentage, measured every 100 m along the road.
FOREST SURVEYOR looking for practical results
These last two parameters illustrate the degree of difficulty of the terrain. To these twelve parameters measured on the map overlays were added two other values formed by combining some of the
previous parameters:
x[13]: x[7]/ x[3], that is, the number of curves divided by the number of changes in slope, all measured along the median lines;
x[14]: x[8]/ x[4] (the same as x[13], but measured along the sides of the square).
Finally, each sampling square was coded from 1 to 5, according to its degree of difficulty:
Class 1: fairly even terrain,
Class 3: moderately uneven terrain,
Class 5: extremely uneven terrain,
Classes 2 and 4 being intermediate between these.
This first set of measurements was applied to 124 units in the inventoried area that were not yet under utilization, and some 30 units in the area that was already being harvested.
· Second stage: statistical analysis. Each sampling unit can be represented by a point in the 14-dimension space where the coordinates of the 14 axes would be the values assumed in this unit by the
14 preceding parameters. The geometric disposition of the points cannot be illustrated graphically. The statistical method best able to convey an idea of the cluster of points representing the
sampling unit is "the principle component analysis", which consists, among other things, of projecting the cluster on those planes in the total space that are closest to the largest number of points,
so as to arrive at the most representative diagrams possible.
(In what follows we will confine ourselves to the plane determined by the first main component [C[1]] - a straight line such that the sum of the squares of the distances of the points to their
projections on that line are at a minimum - and by the second component [C[2]], perpendicular to C[1] and such that the sum of the squares of the distances of the points to their projections on the
plane of these two straight lines is at a minimum.)
The indication "points in the region of..." shows the location of a certain number of points drawn from different parts of Gabon. The "Monts de Cristal" points are very scattered: "i" varies
enormously depending upon whether the observer is in a valley or on a hilly site.
This method of analysis was applied to the 156 sampling points studied and made it possible to obtain the following results:
Correlation between the 14 variables, in pairs.
Correlation between each of the variables and each of the first four main components.
A histogram of the frequency of the variables.
A graph plotting the position of the 156 points on the plane of the first two main components, C[1] and C[2].
· Third stage: The correlations between the 14 variables and the main components show that there is a good correlation between parameters x[1], x[2], x[7], x[8] and x[9] and the component C[1], and
between parameters x[3], x[4], x[5], x[6] and x[10] and the component C[2]. Of the remaining parameters, there is a certain degree of correlation between parameters x[12] and x[14] and C[1], but no
clear correlation for parameters x[11] and x[13].
It would seem that the component C[1] corresponds to the slope of the terrain: there is a strong correlation with x[1] and x[2]. The component C[2] would then correspond to the fragmentation of the
One of the aims of the analysis undertaken was to see whether it was possible to class the types of terrain on a diagram by using only two variables selected as the best as a result of this analysis.
These parameters must satisfy two criteria:
1. They must be well correlated with the results of the analysis;
2. They must be measurable not only on contour maps, but also on aerial photographs, so that it is possible to work even where no maps exist.
Classification of the terrain by means of two parameters would make it possible to work rapidly, while the use of 12 parameters cannot be adopted for the study of large regions.
(a) According to the first component (C[1]): x[1] and x[2] satisfy the above conditions; x[2] has the disadvantage that it gives many zero values, which greatly diminishes its interest; x[1] is much
more widely applicable, but in the case of very uneven terrain it would lack refinement, since it, too, would provide many zero values.
FIGURE 4. - CLASSES OF DIFFICULTY - Scale: 1:50000
It was therefore decided to try a combination of these two parameters, which would make possible a complete description of the terrain, both in the intermediate zones where both parameters give valid
results, and in the zones at either extreme, where the use of x[1] for very uneven terrain and of x[2] in flat terrain lacks precision. This combination is expressed in the following equation by the
index "i":
(x[1] and x[2] expressed as percentages in a given square)
"i" is expressed in degrees:
i = 0 on flat terrain
1 = 100 if all the terrain has a slope gradient of over 40%.
The angle "i", expressed in degrees, will hereafter be referred to as the slope index.
(b) According to the second component (C[2]): x[3], x[4], x[5], x[6] and x[10] all meet the criterion of correlation with the second component. In addition, they are largely independent of x[1] and x
[2]; x[4] (number of changes in slope along the sides of the square) presents certain advantages over x[10] (total length of rivers). In fact, the density of the drainage network shown on the map
depends on whoever prepared the map. This drawback attached to x[10] is even more inconvenient when it comes to photographs, in which it is not clear where rivers end; x[4] was therefore retained. To
improve the quality of the parameter linked to the second component, x[3] can be added. The resulting variable, number of changes in slope along the sides and medians of the square, will be called f
(fragmentation index).
Figure 3 shows the position of each of the 156 sampling units on a graph whose abscissa is slope index "i" and whose ordinate is fragmentation index "f", the class of difficulty being shown for each
point. The graph indicates that the subjectively estimated difficulty level corresponds, despite some overlap, with variations in the slope index. The values (i < 1, f < 15) at the lower left of the
graph represent units in the region of Daloa, Ivory Coast, where forest exploitation is considered easy, the Gabonese area that comes nearest to this optimum (i = 2.5, f = 18) represents a flat,
swampy area north of Koumaneyong. The graph also demon strafes the great variability of fragmentation index "f".
· Fourth stage: practical application. The classes of difficulty (Figure 4) were worked out bearing in mind on the one hand the preceding grades of difficulty, and on the other hand the necessity of
keeping approximately the same number of classes. In addition, examination of the diagram and of the 1:50000 contour maps of areas with very uneven terrain showed the necessity of providing for more
categories for difficult terrain.
The final classification adopted for the slope index (i) was as follows:
0 to 12: easy terrain
13 to 24: average terrain
25 to 38: moderately difficult terrain
39 to 54: difficult terrain
55 to 69: very difficult terrain
70 +: extremely difficult terrain
For the fragmentation index, only two categories were employed: terrain with little fragmentation, "f" up to and including 40, and very fragmented terrain, "f" of 41 and over. This classification
corresponds to conditions in Gabon, except for the coastal plain, where the main value of "f" is around 4041. For a study of other areas, it might be useful to distinguish a greater number of
categories within the fragmentation index.
Within a sampling unit, "i" is measured in the following way: the 4-cm square, taken from a map on a scale of 1: 50000 with 20-m intervals between the contour lines, is covered by a 64-dot grid; the
number of points at which the slope is less than 20% - interval greater than 2 mm and those where it is above 40% interval less than 1 mm - are ascertained with the help of a transparent, double
decimetre graduated on the lower side placed against the grid, and using where necessary a magnifying glass - a relatively slight enlargement suffices. If y[1] is the first figure obtained and y[2]
is the second, the slope index, "i", is equal to the value of i in grades such that [].
Then the number of changes in slope are counted along the sides of the square and along the two medians shown on the grid. Different measurements are made on the same units in order to ascertain the
accuracy: the relative error between measurements may reach 15 %, but cannot exceed 5 % if a lens is used.
Various trials were made on a section covering 60000 ha, always on a scale of 1:50000, in order to perfect the practical sampling design. Using the Universal Transverse Mercator grid, a systematic
sampling design was experimented using progressively higher sampling intensities. For a given intensity, the contours of areas of equal difficulty (according to the slope index) were traced on the
map; then the limits of the slope index categories were identified by a thorough study of all the squares on the map. This showed that excellent results could be obtained by using a sampling design
in which the distance between two counted squares equals two squares in both a latitudinal and a longitudinal direction, that is, a minimum sampling intensity of 1/9 = 11.1%.
PREPARING TO FEEL A TREE WITH A POWER-SAW forest surveyors should take him into account, too
LOADING TIMBER IN AN AFRICAN FOREST they knew the poor was solid
This systematic design was supplemented by the study of a few additional squares, for which the slope index category war, not clear. In regions with very heterogeneous relief, these additional
sampling points will be more numerous than in cases where the relief is relatively homogeneous. Finally, the over-all sampling intensity adopted was about 14%. The practical design, carried out by a
sampling study, fixed the limit of the zones of equal difficulty solely on the basis of the slope index. The fragmentation index, with fewer categories and much easier to discern, was kept in second
The simultaneous but rapid examination of aerial photographs proved interesting, and even essential where the terrain was easy - "i" between 1 and 12. In fact, there are two typical cases that may
- The documents on a scale of 1:50000 are not complete maps, but map overlays which do not indicate swampy formations;
- The contour lines shown on the maps are 20 m apart, so that slight variations in relief are not visible; in aerial photographs, on the other hand, short but steep slopes are clearly visible
("orange peel" relief).
APPLICATION TO AERIAL PHOTOGRAPHS. Toward the end of 1973, about half the inventoried zone in Gabon was covered by map overlays on a scale of 1:50000. The other half was covered only by aerial
photographs. In this latter half one way of classifying the terrain would be to transfer to the aerial photographs the method used for the map overlays. The slope index could be determined by laying
a dot grid of the area selected on one of a stereographic pair of photographs and ascertaining by means of the stereoscope the points where the slope is less than 20% and those where it is more than
40 % This procedure presupposes that the observer is experienced in estimating gradients from aerial photographs and that he periodically controls the gauge of his instrument. In theory, the only
difficulty lies in the variation in scale of the photographs. In order to be accurate, therefore, it is advisable to determine the exact scale of the photographs and to bear this in mind in fixing
the size of the sampling unit.
Another method would be to proceed by analogy. In the part covered by map overlays an aerial coverage exists on a scale approximately the same as for the rest of the area to be inventoried. One
could, therefore, as a first stage, prepare a check sample of stereograms representative of the different types of terrain and subsequently class the area to be studied by referring to this sample.
The aerial photographs are examined under the stereoscope; using the check stereograms, the limits of the different slope index classes are plotted on the photos of a single strip. The photos from
this strip, and then from other strips, are compared and the homogeneity of the limits traced are verified. Often the limits do not correspond exactly as between one strip and another; this is due to
the poorer quality of photographs along their edges, and it is necessary to adjust this by stereoscopic examination from strip to strip - with a slight overlap of 10% to 20% - or even by a fresh
examination under the stereoscope of the photos from a single strip.
STUCK IN THE MUD learning about the quality of the soil after the road is built
Since the fragmentation index "f" is divided into much less refined categories, in the case of Gabon its different values were not shown in detail on the maps, particularly since there is little
variation over fairly extensive areas; some regions show little fragmentation, while others are uniformly broken up.
In general the results obtained from both methods of description correlate fairly well. But macrodescription of the terrain using maps does not give a really accurate picture of the slopes. It has
the advantage, however, of providing a comprehensive view of the terrain as a whole, whereas the indications provided by the microdescription, based on field observations, are limited in scope by the
sampling intensity used for the inventory.
Apart from this, it is necessary to set the results of the macrodescription against the over-all nature of the relief (uniformity or variability). Thus, a hill with difficult slopes, but situated in
the middle of an easy area, will not present any great obstacle to harvesting operations; it will always be possible to construct the essential access roads. But if hills of this type are frequent,
without wide valleys between, this will render harvesting operations difficult, if only by reason of the cost of penetrating the mountainous area. In the peninsula of Azuero, in Panama, where this
method of macrodescription was employed, the difficulties of the terrain are undoubtedly indicated by the high slope indexes, but they also derive from the fact that the relief is everywhere the
same, without any valleys of penetration. The difficulties noted in Gabon, in the eastern region of Mouila, also have to be evaluated bearing in mind the fact that the relief remains the same over
vast areas (there is only one large valley).
Finally, the study carried out, which does not pretend to be applicable in all cases, makes it possible to draw a certain number of conclusions. First, it is important that, in addition to the usual
data (on trees, slopes, etc.), the field survey team record also other information useful for future harvesting operations, such as, for example, indexes of material suitable for use in
road-building, or the presence of rocky slabs or blocks. Also, the macrodescription of the terrain made on the basis of maps and aerial photographs should be effected in conjunction with the
photo-interpretation studies necessary for forest inventory work as such. Apart from the fact that the data would be used at the same time by the same interpreters, this procedure would also make it
possible to demonstrate interesting correlations between types of forest and types of relief. | {"url":"https://openknowledge.fao.org/server/api/core/bitstreams/7884d25a-a1b7-4fe3-b1b0-de76f8fa3937/content/k1100e02.htm","timestamp":"2024-11-04T16:10:31Z","content_type":"text/html","content_length":"40624","record_id":"<urn:uuid:17c1a919-39f2-4584-a31f-2e88a05d9141>","cc-path":"CC-MAIN-2024-46/segments/1730477027829.31/warc/CC-MAIN-20241104131715-20241104161715-00437.warc.gz"} |
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Chapter 18. Using The Structured Sparse Matrix Classes (.NET, C#, CSharp, VB, Visual Basic, F#)
NMath provides a variety of functions that take the structured sparse matrix types described in Chapter 17 as arguments. Methods are provided either as member functions on the matrix classes, or as
static methods on class MatrixFunctions.
As a general rule, NMath only provides functions that preserve the shape of the structured sparse matrices. In some cases, this means that functions provided for the general matrix classes are not
provided for the structured sparse matrix classes. For example, NMath does not generally provide trigonometric and transcendental functions for structured sparse matrix types. Such functions may
change unstored zero values to non-zero values, thus changing a structured sparse matrix type into a general matrix.
If you want to apply an arbitrary function to all elements of a structured sparse matrix, including unstored zero values, you can always convert the matrix to a general matrix first. A
ToGeneralMatrix() method is provided for this purpose. Alternatively, to apply an arbitrary function only to stored values, you can apply the function to the underlying data vector. Both techniques
are described in more detail in Section 18.7.
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From Maxwell to Kirchhoff
This entire chapter is devoted to the path from a full electromagnetic field description of microwave circuits governed by the Maxwell equations to a description in terms of electrical circuits
governed by the Kirchhoff rules. A electrical circuit description of a given setup encompasses a network representation consisting of an arbitrary amount of two terminal devices. In our work we deal
with three different kinds of two-terminal devices: capacitors, inductors and Josephson junctions. Every two-terminal device is characterized by a unique relationship between the current flowing
through the device $I$ and the voltage difference at its two outlets $V$. These relationships together with the Kirchhoff rules form a complete set of equations of motion for the currents and voltage
drops across all two-terminal devices in the electrical circuit. This modular architecture of the theory entails the great advantage of flexibility. Once all the relationships for the two terminal
devices are known, which often affords a description of the two-terminal device in terms of electromagnetic fields, we can quickly combine these building blocks with arbitrary complexity without
having to solve for the electromagnetic fields of the whole electrical circuit again. Kirchhoff rules however strictly only apply in static situations. They are consequences of the continuity
equation of electrical charge: $abla \mathcal{\vec{J}}+\partial_t \mathcal{Q} = 0$ with current- $\mathcal{\vec{J}}$ and charge density $\mathcal{Q}$. If you consider the node of a electrical
circuit, integrate the continuity equation in a sphere which is small enough to only include the node and neglect temporary accumulation of charge on the node, we come up with the first Kirchhoff
rule: $\sum_{k\in \text{node}} I_k = 0$, the sum of all currents flowing to the node of a electrical circuit is zero. This is valid as long as the characteristic timescale for changes in the current
is not small enough to introduce charge accumulation on the node. We can circumvent however this difficulty by introducing an additional capacitor connected to ground for the node. For the second
Kirchhoff rule it is more challenging to push the high frequency limit. For the second Kirchhoff rule we integrate $abla\times\mathcal{\vec{E}}=-\mu \partial_t \mathcal{\vec{H}}$ over a surface
framed by a mesh of the electrical circuit. If we again neglect the temporal accumulation of flux threaded through the mesh we come up with the second Kirchhoff rule: $\sum_{k\in \text{loop}} V_k =
0$, the sum of all electrical voltage drops around every loop of the electrical circuit is zero. Lets suppose our circuit oscillates with frequency $\omega$ in a steady state. Then the integral over
the curl of the electric field can be approximated to $\int_S \partial \mathcal{\vec{H}}= S \omega \text{Max}_S(|\mathcal{\vec{H}}|)= S 2\pi /\lambda \text{Max}_S(|\mathcal{\vec{H}}|)$, with $S$ the
surface framed by the mesh and $\lambda$ the wavelength. In other words it is sufficient if the physical size of the electrical circuit is small compared to the wavelength of the excitations of the
circuit for the description in terms of Kirchhoff rules to be valid.
A typical low-frequency ($\approx 30 \text{MHz}$) resonating circuit consists of a inductor coil and a parallel plate capacitor. If we reduce the size of the whole device by a factor of $100$ we
would multiply the eigenfrequency by the same factor. The internal damping of the wiring of the resulting microwave resonating circuit would however also be $10$ times more effective. If we instead
reduce the number of turns in the coil and increase the distance of the parallel plate capacitor we end up at a hairpin-shaped circuit, resonating in the microwave regime. The resulting electrical
circuit however would be of the size of the wavelength of the microwaves itself. Two main differences compared to low-frequency circuits will arise from this. Firstly the circuit will start acting as
an antenna and if we do not provide some means of shielding the circuit, there will be considerable radiative loss. Secondly the concept of inductors and capacitors as physical objects will fade and
get replaced as a means to symbolically represent much more complicated structures where a physical object can be inductor and capacitor at the same time. To find the requirements for the existence
of these symbolical representations is the purpose of this chapter.
Circuit QED setups typically consist of two different types of structures: coplanar transmission lines and lumped element structures. While the coplanar transmission lines are comparable to the
wavelength, the lumped element structures like Josephson artificial atoms, coupling capacitors or Josephson junctions are considerably smaller than the wavelength. For the latter the low-frequency
concepts do apply but the open transmission lines and transmission line resonators do need a special treatment. As it turns out the axial symmetry and the shielding by the groundplane are necessary
ingredients to reintroduce the low-frequency concepts of capacitance per unit length of transmission line, or characteristic capacitance, and the inductance per unit length, or characteristic | {"url":"http://circuitqed.net/from-maxwell-to-kirchhoff/","timestamp":"2024-11-12T19:18:06Z","content_type":"text/html","content_length":"18956","record_id":"<urn:uuid:9aa36698-e2e7-4249-85cc-c6a2f94901e3>","cc-path":"CC-MAIN-2024-46/segments/1730477028279.73/warc/CC-MAIN-20241112180608-20241112210608-00267.warc.gz"} |
P(x)=x² – 4x² – 3x+12
1 thought on “P(x)=x² – 4x² – 3x+12<br />two geroes are 3 -√3<br />find<br />the third zero<br />”
1. Answer:
zero is 0 here lolxd
Step-by-step explanation:
whu3nehshshx 3rd party wear is a the lie to you the your you or and to and you account anniversary bhaiya and you are going out to eat a lot and you have to go to a very forgetful place to get
well and be a little bit of you and your family and your name and the family and the
Leave a Comment | {"url":"https://wiki-helper.com/p-4-3-12-two-geroes-are-3-kitu3-find-the-third-zero-kitu-40224852-72/","timestamp":"2024-11-04T04:51:21Z","content_type":"text/html","content_length":"126125","record_id":"<urn:uuid:513387db-9e6c-4079-bc05-bd11eec626e9>","cc-path":"CC-MAIN-2024-46/segments/1730477027812.67/warc/CC-MAIN-20241104034319-20241104064319-00679.warc.gz"} |
Mechanical Metallurgy Interview Questions and Answers - Sanfoundry
Mechanical Metallurgy Questions and Answers – Elastic Behaviour – Stress Description at a Point
This set of Mechanical Metallurgy Interview Questions and Answers focuses on “Elastic Behaviour – Stress Description at a Point”.
1. Hooke’s law is obeyed till _____________ in stress-strain curve.
a) propositional limit point
b) yield point
c) tensile Strength Point
d) failure point
View Answer
Answer: a
Explanation: The stress-strain curve defines the relationship between force and elongation of any material. This curve can be divided into 2 major parts.
First one is the linear part, which starts at the origin and extends up to the propositional limit. In this region, slope of the curve remains linear and constant. The slope of the curve is known as
young modulus.
The second part is the non-linear part, which begins after the proportional limit point. The relational ship between stress and strain is non-linear in this region. Simple Hooke’s law will not be
enough to define the nature of the curve.
2. How many unknowns are required to establish the state of stress at any given point in 3-dimension?
a) 3
b) 6
c) 9
d) 12
View Answer
Answer: c
Explanation: At any point on the 3-dimensional object, to define the state of stress, it requires 3 principle stress components & 6 shear stress components.
Along x-direction=σx (normal stress), τyx, τzx (shear stress)
Along y-direction=σy (normal stress), τxy, τyz (shear stress)
Along z-direction=σz (normal stress), τxz, τyz (shear stress)
3. Plane stress is defined as ________
a) stress in one principal direction is 0
b) stresses in two principal directions are 0
c) strain in one principal direction is 0
d) strain in two principal directions are 0
View Answer
Answer: a
Explanation: When stress along any one of the principal directions is zero, it is known as plane stress condition. For example, a thin plate loaded in plane of the plate.
When one of the sides of the object is small relative to the other two sides, then there is no stress acting in the direction perpendicular to the surface of the plate.
If strain in one of the principal directions is negligible, it is called plane strain.
4. Which of the following statement is NOT correct?
a) The sum of the normal stresses on two perpendicular planes is an invariant quantity
b) The maximum and minimum values of normal stresses on an oblique plane at point P occurs when the shear stress is zero
c) The maximum and minimum values of the normal stress and the shear stress occurs at an angle 90 degree from each other
d) The variation of shear stress and normal stress occurs is in the form of a sine wave with a period equal to 90 degrees
View Answer
Answer: d
Explanation: Statement: The sum of the normal stresses on two perpendicular planes is an invariant quantity. It is correct.
The sum of any two mutually perpendicular stresses is always equal to the sum of principal stresses on the same plane.
=>σx+σy=σ1+σ2, given that σ1 and σ2 are mutually perpendicular to each other.
Statement: The maximum and minimum values of the normal stresses on an oblique plane at point P occurs when the shear stress is zero. It is correct.
When the shear stress is zero on any plane, all the remaining stresses will add up in the normal component giving the maximum normal stress value.
Statement: The maximum and minimum values of the normal stress and the shear stress occurs at an angle 90 degree from each other. It is correct.
Because after substituting the values in equation of stress at an oblique plane, sine 90 will become zero.
Statement: The variation of shear stress and normal stress occurs is in the form of a sine wave with a period equal to 90 degrees. It is incorrect.
Variation of shear stress and normal stress occurs in the form of sine wave, but with a period equal to 180 degrees.
5. According to the sign convention for shear stress, positive shear stress is shown in diagram _____________
View Answer
Answer: a
Explanation: The condition for positive shear stress is:
– Both the shear stress components of the positive face should point in a positive direction.
– Both the shear stress components of the negative face should point in a negative direction. If both the condition is satisfied, it is pure positive shear stress.
6. A wedge shape body is under stress with principal stresses being σx=25MPa and σy=20MPa. At a certain angle β, the components of stresses are resolved into σ1 & σ2, where both σ1 & σ2 are normal to
each other. If the value of σ1=18MPa, find the value of σ2?
a) 25MPa
b) 20MPa
c) 27MPa
d) 18MPa
View Answer
Answer: c
Explanation: By the condition of invariant reaction σx+σy= σ1+σ2.
=> σx=25MPa, σy=20MPa, σ1=18MPa substitute the values in equation
=> 20+25=18+σ2
=> σ2= 45-18=27 MPa.
7. Find the value of shear stress acting on the body, if the principal normal stresses are 70MPa and 60 MPa. The angle of inclination is 30 degrees?
a) 17.32 MPa
b) 2.88 MPa
c) 10 MPa
d) 15 MPa
View Answer
Answer: b
Explanation: Along principal normal stress direction the shear stress will be zero. Using this condition:
=> (tanθ=2τ/(σx-σy))
Substitute the values of principal stresses and θ
=> tan30=2 τ/(70-60)
=> 0.577=2 τ/10
=> τ=2.88 MPA.
8. Transformation of stress along the anticlockwise direction is given by the equation:
σx’= (σx+σy)/2+(σx-σy)/2 cos2θ+τxysin2θ
Given that the principal stresses are 5Y & 9Y and shear stress is Y. If a plane is at 45 degrees anticlockwise from principal plane, find the value of the normal stress?
a) 4Y
b) 5Y
c) 8Y
d) 10Y
View Answer
Answer: c
Explanation: As given in the equation substitute the values of σx=5Y and σy=9Y and θ=45 degree
=> {(5Y+ 9Y)/2}+{(5Y-9Y)/2*cos(2*45)}+Ysin(2*45)
=> 7Y+(-4Y*0)+Y*1 [cos90=0; sin90=1] => 8Y.
9. True stress-strain curve always lies above the engineering stress-strain curve.
a) True
b) False
View Answer
Answer: a
Explanation: True stress calculation considers the instantaneous area of the specimen in the calculation, whereas engineering stress considers the initial area of the specimen in the calculation.
So, as the area is continuously reducing at constant load, the value of true stress keeps on increasing with respect to engineering stress.
10. Bar with cross-sectional area of 0.05m^2 is subjected to load of 2000kg, find the stress on the bar in terms of MPa?
a) 392000
b) 40000
c) 0.040
d) 0.392
View Answer
Answer: d
Explanation: Stress is defined as the force per unit area.
Total force on bar=mg; where; m=mass, g=acceleration due to gravity.
=> 2000*9.8=19600N, where g=9.8 m/s^2.
=> 19600/0.05=392000Pa=0.392 MPa.
Sanfoundry Global Education & Learning Series – Mechanical Metallurgy.
To practice all areas of Mechanical Metallurgy for Interviews, here is complete set of 1000+ Multiple Choice Questions and Answers. | {"url":"https://www.sanfoundry.com/mechanical-metallurgy-interview-questions-answers/","timestamp":"2024-11-05T23:31:41Z","content_type":"text/html","content_length":"160879","record_id":"<urn:uuid:3a6e0edf-27e0-44ac-af92-2a2fdd4b9670>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00397.warc.gz"} |
IMPACTful Thoughts - What Happens in Math for Elementary Teaching Shouldn’t Stay in Math for Elementary Teachers
Let me start us off with a little warm up. When you look at this picture, what do you notice? Hold onto what you notice until the end.
(Beckmann, 2018)
This was a warm up that I used during the past semester in my Mathematics for Elementary Teachers (MFET) classes. This image was on a Jamboard. Students provided their thoughts via sticky notes about
what they noticed. Maybe you’re thinking, “yeah that works for Math for Elementary teachers but…” I also used this as a warm up for my Trigonometry class with similar results and will continue to use
it (and other “what do you notice”) activities in every math class I teach. Why? Because what happens in MFET should NOT stay in MFET.
What is MFET?
Mathematics for Elementary Teachers is the course (or for most, course sequence) that future elementary teachers must take as a part of their preparation and certification/licensure. State
requirements vary. In Illinois, the two courses are required for all “pre-service” teachers. The courses exist at both community colleges and four-year schools. Mathematics for Elementary Teachers is
intended to be a content course, essentially with all of the math content that future K-8 teachers would need to know for their future teaching. But separating content from methods is not always
possible (nor appropriate), and content knowledge alone is necessary but insufficient for teaching (Shulman, 1987; Ball, Thames and Phelps, 2008). Speaking from my own experience, there have been a
number of local collaborative efforts around these courses in addition to this fantastic AMAYTC Teacher Preparation committee, allowing for us to compare our own balancing of content and methods.
And what happens in MFET?
If you’ve never had the pleasure of teaching it, first of all, you’re missing out. Second, think about all of the “stuff” that we do with respect to numbers, operations and geometry. The foundations
for them came from our time as pre-K through 8th grade students. As math faculty, we ask why, but when we were students did we? One way to think of MFET is as a place to revisit why.
With respect to numbers, operations, geometry, probability, statistics and algebra.
• answers the “why” questions.
• gets us thinking about student (mis)understandings.
• helps us to see the wide variety of methods, usually beyond what we experienced as students.
• is a deep dive into the origins of things that we take for granted, but are rich in their own right.
For example, think about multi-digit multiplication. Take a moment to multiply 25 x 40. How did you do it? Why does what you did work? What errors might students make when using paper-and-pencil or
mental strategies? Could you figure out another way? Is the method that you used generalizable or specific to one or both factors? For instance, would your method work for 24 x 40 or 25 x 41? Perhaps
there’s a modification you could make to your method, or maybe knowing a bit more about what is actually happening “behind the scenes” when we multiply is all we need. And it’s no coincidence that
most of the methods used to multiply above are necessary in understanding how to multiply polynomials.
The previous paragraph is meant to give you a glimpse into what happens in an MFET class.
But how does that apply to classes that are not MFET?
I have been teaching MFET for about 16 years. I was asked to teach the course my first semester because I had two years teaching 6th grade (which was definitely enough 😉).
In the beginning, I really did think of MFET as something entirely different from the other courses I was teaching. But as the years have passed, I’ve noticed that much of what I do in other classes
has been heavily influenced by my time teaching MFET. The Math for Elementary Teachers courses have made me a better teacher. Not only can I calculate more quickly, which is fun when in front of a
class with an itch to show off, but MFET has also gotten me thinking about what our students have experienced prior to ending up in our classes. This is especially useful when teaching students in
developmental courses where arithmetic is a part of the curriculum. I feel more prepared when they use strategies that differ from mine. I also feel more prepared with fundamental ideas. Consider
graphing a function. In addition to evaluating the function, think about what we often take for granted: the choices of x values, the labeling of the axes and the intervals of the function that we
care about. When graphing a cosine function, knowing how to subdivide an interval into 4 equal parts is very useful to find the “nice” points of the function. Partitioning is really just an
application of fraction representation. We’re taking the whole and breaking it into four equal parts. Whether the whole is one or 2π does not change the fact that we’re breaking it into four equal
MFET involves visualization. But visualization should happen everywhere, even for content that we may think should be strictly numerical or algebraic. I have been amazed over the years at the number
of students who are surprised by me drawing out fraction arithmetic ideas. In our introduction to college mathematics courses, number sense is the first unit. Students who have anxiety about
arithmetic (usually fractions and negative numbers) or who have been away from school mathematics for a while always appreciate how the visuals help them to understand why things work, like common
denominators when adding/subtracting fractions, whether the visuals are pizzas or a number lines.
MFET involves collaboration and active learning. In the early part of my career, I thought that utilizing these methods in other courses that appeared more rigorous would be a bad idea. But with age
and experience, I have listened to what MFET has taught me. Why should I reserve the most engaging parts of my practice for my MFET classes? In MFET, I tried more than every other course to model
good math teaching so that students could take those ideas with them into their future practice. I have also been more experimental in my teaching in those courses. But good teaching is good teaching
and taking risks shouldn’t be course dependent. Discussions about numbers through math talks (like the start of this post), role playing opportunities for students to teach each other or correct
fictitious students’ work, using movement and manipulatives to engage more senses in their learning are all things that happen in MFET.
MFET involves more than procedures. Students are often surprised by the number of ways to solve a problem. They can also be resistant to seeing problems solved in ways that vary from their way.
Students may begin a class with the mentality that the answer is more important than the way they got to the answer or why that way makes any sense. Imagine a procedure in one of the classes you
teach. Is the goal for students to master the procedure or to understand the procedure? Is there a difference? In MFET, all procedures are demystified. Students should leave the course having few
doubts about why anything works, especially with respect to arithmetic. They should also leave curious.
If these sound like things that you do or would like to do in your courses, then perhaps there is more in common between MFET and your courses than you may have previously thought.
So what did you notice up above?
Maybe you noticed something having to do with rotational symmetry. Maybe you noticed something more numerical, perhaps related to powers of 3. Feel free to reply with something you noticed and while
you’re replying... if you teach MFET and other courses like me and most others in our teaching preparation committee, what doesn’t stay in MFET for you?
And if you’re not a part of it already, join AMATYC’s Teacher Preparation Committee!
Here’s a shameless plug from my talk about MFET at this past AMATYC conference.
Beckmann, S. (2018). Mathematics for elementary teachers, with activities.
Ball, D., Thames, M. H., & Phelps, G. (2008). Content Knowledge for Teaching: What Makes It Special? Journal of Teacher Education, 59(5), 389–407. https://doi.org/10.1177/0022487108324554
Shulman, L. (1987). Knowledge and teaching: Foundations of the new reform. Harvard Educational Review, 57(1), 1-22. | {"url":"https://my.amatyc.org/blogs/chris-sabino/2021/07/01/impactful-thoughts","timestamp":"2024-11-13T20:59:43Z","content_type":"text/html","content_length":"140887","record_id":"<urn:uuid:a0b79c38-cb7a-4648-8066-057e278c666b>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00549.warc.gz"} |
Illative Combinatory Logic
Type-Free Combinatory Logics (L)
Illative Combinatory Logic
Schönfields original paper on Combinatory Logic (On the building blocks of mathematical logic ) presented the first account of a Combinatory Logic proper, since it incorporated a generalised Scheffer
While this provided a way of translating formulae of the predicate calculus into combinatorial notation, no independent account of the semantics of the notation was provided. A semantics derived from
the predicate calculus would fail to determine the meaning of those terms of the calculus which could not be obtained by translation from the predicate calculus. The notational change is by itself of
doubtful value if the more economic notations are not underpiinned by a sympathetic semantic treatment.
Considerable further work was undertaken by H.B.Curry and his collaborators, in an edeavour to establish type-free combinatory logics suitable for use as a foundation for mathematics. The results of
Curry's programme were published in Combinatory Logic, Vols 1 & 2.
These systems have not however proved popular. While mathematicians work as if in a first order set theory, logicians (and increasingly computer scientists) have turned to a variety of typed systems
(for the simplest of which, see typed combinatory and lambda logics).
© created 1995/12/9 modified 1995/12/10 | {"url":"https://www.rbjones.com/rbjpub/logic/cl/cl008.htm","timestamp":"2024-11-03T05:38:03Z","content_type":"text/html","content_length":"2989","record_id":"<urn:uuid:75c35abe-998b-4c1b-bc9e-4ec898048efc>","cc-path":"CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00233.warc.gz"} |
Emma earns $6 each time she mows the lawn and $8 per hour for babysitting. She is saving up to buy a new pair of jeans that cost $48. If she mows the lawn x times and babysits for y hours, which graph
Waiting for answer This question has not been answered yet. You can hire a professional tutor to get the answer.
Emma earns $6 each time she mows the lawn and $8 per hour for babysitting. She is saving up to buy a new pair of jeans that cost $48. If she mows the lawn x times and babysits for y hours, which
Emma earns $6 each time she mows the lawn and $8 per hour for babysitting. She is saving up to buy a new pair of jeans that cost $48. If she mows the lawn x times and babysits for y hours, which
graph shows the amount of work she needs to complete to earn at least enough to purchase the new jeans?
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Noncommutative Geometry and Noncommutative Invariant Theory
Schedule for: 22w5084 - Noncommutative Geometry and Noncommutative Invariant Theory
Beginning on Sunday, September 25 and ending Friday September 30, 2022
All times in Banff, Alberta time, MDT (UTC-6).
Sunday, September 25
16:00 - 17:30 Check-in begins at 16:00 on Sunday and is open 24 hours (Front Desk - Professional Development Centre)
Dinner ↓
17:30 - 19:30 A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.
(Vistas Dining Room)
Monday, September 26
Breakfast ↓
- Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.
(Vistas Dining Room)
Introduction and Welcome by BIRS Staff ↓
- A brief introduction to BIRS with important logistical information, technology instruction, and opportunity for participants to ask questions.
(TCPL 201)
James Zhang: Some open questions in noncommutative algebra ↓
- We review some open questions, conjectures, and important on-going projects in noncommutative algebra.
(TCPL 201)
Ellen Kirkman: Homological Regularities ↓
10:00 Let $A$ be a noetherian connected graded $\Bbbk$-algebra with a balanced dualizing complex, and let $X$ be a cochain complex of graded left $A$-modules. The elements of $X$ possess both an
- internal and various homological degrees, and it is useful to study the relationships between these degrees. Jörgensen and Dong-Wu extended the study of Tor-regularity and Castelnuovo-Mumford
10:30 regularity from commutative algebras to noncommutative algebras. We consider these regularities further, and define new numerical invariants that involve linear combinations of internal and
homological degrees. This is joint work with Robert Won and James J. Zhang.
(TCPL 201)
- Coffee Break (TCPL Foyer)
Xin Tang: Automorphism Groups and Isomorphism Problem for Some Poisson Algebras ↓
- It has been observed that Poisson algebras are closely related to their quantizations in many perspectives. In this talk, we will paint a similar picture in terms of the automorphisms and
11:50 isomorphisms for several classes of Poisson algebras and compare the results with their quantum analogues. Some of the results are ongoing joint work with Xingting Wang and James Zhang.
(TCPL 201)
Lunch ↓
- Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.
(Vistas Dining Room)
Guided Tour of The Banff Centre (optional) ↓
- Meet in the PDC front desk for a guided tour of The Banff Centre campus.
(PDC Front Desk)
- Break (Various In-Person Locations / Online)
Group Photo ↓
- Meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors, so dress appropriately for the weather. Please don't be late, or you might not be in the
14:20 official group photo!
(TCPL Foyer)
Padmini Veerapen: Can twists of algebras be realized as 2-cocycle twists of Hopf algebras? ↓
14:30 In this talk, we will explore how a twist of an algebra's multiplicative structure by an automorphism, can be extended to a twist of certain Hopf algebras. We do so by twisting a bialgebra and
- by lifting it to a Hopf algebra using Takeuchi's Hopf envelope construction. Moreover, we examine when our construction coincides with a 2-cocycle twist of the Hopf algebra. We analyze our
15:00 work in the context of Manin's universal quantum groups and solutions to the quantum Yang Baxter equation. This is joint work with H. Huang, V, C. Nguyen, C. Ure, K. Vashaw, and X. Wang.
(TCPL 201)
- Coffee Break (TCPL Foyer)
Cris Negron: Some finite generation results for finite-dimensional Hopf algebras ↓
15:30 I will survey results on the "finite generation conjecture" (FGC) for finite-dimensional Hopf algebras. The FGC proposes that cohomology over such a Hopf algebra H enjoys many global
- finiteness properties which imply, for example, that all extension algebras Ext*_H(V,V) are finitely generated and finite over their centers. If time permits, I will describe some advanced
16:00 interpretations of Deligne's conjecture and their relations to tensor triangular geometry.
Charlotte Ure: Twisting Comodule Algebras and Preregular Forms ↓
16:00 For any Hopf algebra $H$ and any 2-cocycle $\sigma$ on $H$, the twist of $H^\sigma$ arises by deforming the underlying algebra structure. It is known that $H$ and $H^\sigma$ are
- Morita-Takeuchi equivalent. In particular, for any $H$-comodule algebra $A$, there is a twisted $H^\sigma$-comodule algebra $A_{\sigma^{-1}}$. In this talk, I will explain how this twisting
16:30 may be thought of as an extension of twisting $A$ by a graded automorphism. As an example, I will consider twisting of preregular forms and their associated superpotential algebras by
2-cocycles. This is joint work with Hongdi Huang, Van Nguyen, Kent Vashaw, Padmini Veerapen, and Xingting Wang.
(TCPL 201)
Jason Gaddis: Pointed Hopf actions on quantum generalized Weyl algebras ↓
16:30 In this talk I will discuss Hopf actions in the setting of $\mathbb{Z}$-graded algebras. The Weyl algebra is an example of such an algebra, but has no finite dimensional quantum symmetry.
- Instead, we study quantum generalized Weyl algebras (GWAs), which exhibit actions by generalized Taft algebras that respect their $\mathbb{Z}$-grading. These actions are extensions, or
17:00 `quantum thickenings', of cyclic group actions. This is joint work with Robert Won.
(TCPL 201)
Dinner ↓
- A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.
(Vistas Dining Room)
Tuesday, September 27
Breakfast ↓
- Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.
(Vistas Dining Room)
Milen Yakimov: Azumaya loci of root of unity quantum cluster algebras ↓
Root of unity quantum cluster algebras form a vast class of algebras containing many important subclasses of quantum algebras at roots of unity arising in Lie theory and topology. Using
09:00 Cayley-Hamilton algebras in the sense of Procesi, one shows that they contain canonical central subalgebras, isomorphic to the underlying classical cluster algebras with the property that the
- root of unity algebra is module finite over the central subalgebra. We will present results that explicitly describe the fully Azumaya loci of each root of unity quantum cluster algebra. We
09:50 will also show that the spectrum of the underlying cluster algebra has an explicit torus orbit of symplectic leaves with respect to the Gekhtman-Shapiro-Vainshtein Poisson structure. This is a
joint work with Greg Muller, Bach Nguyen and Kurt Trampel.
Evelyn Lira Torres: Quantum Riemannian Geometry on the Fuzzy Sphere ↓
10:00 We will discuss the Quantum Riemannian Geometry of the fuzzy sphere, where the fuzzy sphere is defined as the angular momentum algebra $[x_i,x_j]=2\imath\lambda_p \epsilon_{ijk}x_k$ modulo
- setting $\sum_i x_i^2$ to a constant, using a recently introduced 3D rotationally invariant differential structure. It is found that the metrics are given by symmetric $3 \times 3$ matrices $g$
10:30 and we show that for each metric there is a unique quantum Levi-Civita connection with constant coefficients. As an application, we will discuss the construction of the Euclidean quantum
gravity on the fuzzy unit sphere; and also the charge 1 monopole for the 3D differential structure.
(TCPL 201)
- Coffee Break (TCPL Foyer)
Xingting Wang: Twists of graded Poisson algebra and applications ↓
11:00 In noncommutative projective algebraic geometry, twistings of homogenous coordinate rings give equivalences between noncommutative projective schemes. We introduce a Poisson version of such
- twisting of any graded Poisson algebra. We show that every graded Poisson algebra is the graded twist of a unimodular one. We also discuss various new concepts in Poisson twisting related to
11:50 the computation of Poisson homology and Poisson cohomology. This is joint work with Hongdi Huang, Xin Tang and James Zhang.
(TCPL 201)
Lunch ↓
- Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.
(Vistas Dining Room)
- Break (Various In-Person Locations / Online)
Hongdi Huang: Weighted graded Poisson algebras in dimension 3 ↓
14:00 We will discuss the work on the structure of graded unimodular Poisson algebras in dimension 3 when the weights of the three variables are arbitrary. To investigate the related homological
- properties of these Poisson algebras, we first give the complete classification of weighted potentials when the Jacobian structure is homogeneous of degree zero. Moreover, we will also discuss
14:30 the classification of the potentials of certain degree that have isolated singularities. This is an ongoing joint work with Xin Tang, Xingting Wang and James Zhang.
(TCPL 201)
Kent Vashaw: A cogroupoid associated to preregular forms ↓
Cogroupoids have been used in recent years by Bichon and others as a convenient framework to explore Hopf-Galois objects and Morita-Takeuchi equivalences. In this talk, which will build on the
14:30 previous talk of Charlotte Ure, we will construct a cogroupoid corresponding to m-linear preregular forms, for all m greater than 1. Using this, we recover concretely a partial result of
- Radschaelders-Van den Bergh, which gives a Morita-Takeuchi equivalence between universal quantum groups of Artin-Schelter regular algebras of dimension 2. We also show that after settinga
15:00 quantum determinant equal to 1, we can compute a formula for cocycle twists of a universal quantum group in terms of the twists of a preregular form. This is joint work with Hongdi Huang, Van
Nguyen, Charlotte Ure, Padmini Veerapen, and Xingting Wang.
(TCPL 201)
- Coffee Break (TCPL Foyer)
Daniel Chan: The minimal model program for orders on arithmetic surfaces ↓
15:30 The minimal model program, initially introduced to provide a framework for classifying higher dimensional varieties, has also proved useful for studying noncommutative schemes arising as orders
- on varieties. In this talk, we will look at recent work on orders on arithmetic surfaces. When the order has prime index p>5, many results from classical surface theory can be recovered such as
16:20 the existence of terminal resolutions, classification of terminal singularities and Castelnuovo's contraction theorem. However, new phenomena appear which do not occur in the case of surfaces
over an algebraically closed field. For example, Castenuovo contractions can now introduce singularities on the centre.
Van Nguyen: Tensor representations of finite-dimensional Hopf algebras ↓
- In this talk, we will discuss some recent projects (joint work with Georgia Benkart, Rekha Biswal, Ellen Kirkman, and Jieru Zhu) and open problems in tensor representations of
17:00 finite-dimensional Hopf algebras
(TCPL 201)
Dinner ↓
- A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.
(Vistas Dining Room)
Wednesday, September 28
Breakfast ↓
- Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.
(Vistas Dining Room)
Fabio Calderón: Cocommutative Hopf-like actions on algebras ↓
09:15 We call an algebraic structure H "Hopf-like" if its category of (co)representations is monoidal. If additionally H is cocommutative, examples include cocommutative (weak) Hopf algebras, group/
- groupoid algebras, and universal enveloping algebras of Lie algebras/(some) Lie algebroids. In this talk I will present (classical and new) results showing that an algebra A is an H-module
09:45 algebra precisely when there exists a structure preserving map from H to a certain collection of linear endomorphisms of A that has the same structure as H. This yields an equivalence between
categorical and representation-theoretic notions of an algebra A admitting an action of H. This is joint work with Hongdi Huang, Elizabeth Wicks and Robert Won.
Manuel Reyes: Dual coalgebras as quantized maximal spectra ↓
There are serious obstructions to extending the Zariski spectrum Spec as a functor from commutative rings to noncommutative rings. In an attempt to escape these limitations, we are forced to
10:00 search for a category of ``noncommutative sets'' that is strictly larger than the classical category of sets. Restricting to cases where the maximal spectrum Max is functorial for commutative
- algebras, we argue that coalgebras serve as a reasonable approximation to generalized sets, and that the finite dual coalgebra is a suitable quantization of Max. We will discuss how the finite
10:30 dual behaves under twisted tensor products and how it can be understood relative to the center of an affine noetherian PI algebra. We will close by discussing a conjectural path to quantizing
the functor Spec itself.
(TCPL 201)
- Coffee Break (TCPL Foyer)
Michael Wemyss: Local Forms of Noncommutative Functions ↓
This talk will explain how Arnold’s results for commutative singularitiescan be extended into the noncommutative setting, with the main result being a classification of certain Jacobi algebras
11:00 arising from (complete) free algebras. This class includes finite dimensional Jacobi algebras, and also Jacobi algebras of GK dimension one, suitably interpreted. The surprising thing is that a
- classification should exist at all, and it is even more surprising that ADE enters. I will spend most of my time explaining what the algebras are, what they classify, and how to intrinsically
11:50 extract ADE information from them. At the end, I’ll briefly explain why I’m really interested in this problem, the connection with different quivers, and the applications of the above
classification to curve counting and birational geometry. This is joint work with Gavin Brown.
Lunch ↓
- Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.
(Vistas Dining Room)
- Free Afternoon (Banff National Park)
Dinner ↓
- A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.
(Vistas Dining Room)
Thursday, September 29
Breakfast ↓
- Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.
(Vistas Dining Room)
Wendy Lowen: Enriching the nerve construction ↓
09:00 This talk bridges between noncommutative geometry and higher category theory. A famous link between the two subjects is given by the DG nerve, which turns a DG category into a quasi-category.
- In this talk, we will enrich this construction keeping track of the linear features of the DG category. More generally, this leads to a notion of quasi-categories in a monoidal category V,
09:50 which should model weak enrichment in the category of simplicial V objects. (Joint with Arne Mertens)
Frank Moore: Actions of the quantum double of certain finite groups on quadratic AS-regular algebras ↓
10:00 (Joint work with Ellen Kirkman and Tolulope Oke) The quantum double $D(H)$ of a Hopf algebra $H$ was originally introduced by Drinfel'd in his study of solutions to the quantum Yang-Baxter
- equation. We use Witherspoon's calculation of the representation ring of the quantum double $D(G)$ of a finite group $G$ to determine families of inner-faithful representations of the quantum
10:30 double of some generalized quaternion groups. We examine several such representations in detail, and use them to identify some families of quadratic AS-regular algebras (in fact, double Ore
extensions) on which $D(G)$ acts.
(TCPL 201)
- Coffee Break (TCPL Foyer)
Kenta Ueyama: Examples of smooth noncommutative projective schemes ↓
- I will present examples of smooth noncommutative projective schemes using two classes of algebras, namely skew quadric hypersurfaces and twisted Segre products of Artin-Schelter regular
11:50 algebras.
(TCPL 201)
Lunch ↓
- Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.
(Vistas Dining Room)
- Break (Various In-Person Locations / Online)
Robert Won: PI skew polynomial rings and their centers ↓
- We study PI skew polynomial rings and the relationship between their centers, ozone groups, parameters, and some newly defined invariants. We investigate in detail several properties of such
14:30 algebras and their centers in low dimension. This is joint work with Kenneth Chan, Jason Gaddis, and James J. Zhang.
Lucas Buzaglo: Universal enveloping algebras of Krichever-Novikov algebras ↓
14:30 Universal enveloping algebras of finite-dimensional Lie algebras are fundamental examples of well-behaved noncommutative rings. On the other hand, enveloping algebras of infinite-dimensional
- Lie algebras remain mysterious. For example, it is widely believed that they are never noetherian, but there are very few examples whose noetherianity is known. In this talk, I will introduce a
15:00 class of infinite-dimensional Lie algebras known as Krichever-Novikov algebras and talk about a recent proof that their enveloping algebras are not noetherian, providing a new family of
non-noetherian universal enveloping algebras.
(TCPL 201)
- Coffee Break (TCPL Foyer)
Ryan Kinser: Moduli spaces of tame finite-dimensional algebras ↓
In the theory of finite-dimensional algebras (or equivalently, quivers with admissible relations), "moduli spaces" are a geometric tool for giving structure to infinite families of isomorphism
15:30 classes of indecomposable representations. The "tame algebras" are those for which such families can always be described with one parameter from the underlying field. More precisely, their
- moduli spaces are always projective algebraic curves. These moduli spaces have been explicitly described for certain classes of tame algebras over the past 20 years, and so far in every known
16:20 case they have turned out to be smooth of genus zero; i.e. isomorphic to the projective line P^1. One might conjecture that is true for all tame algebras. This talk will survey the history of
this story and recent additional evidence for this conjecture.
(TCPL 201)
Dinner ↓
- A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.
(Vistas Dining Room)
Friday, September 30
Breakfast ↓
- Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.
(Vistas Dining Room)
Travis Schedler: Birational Geometry of Quiver Varieties and Related Moduli ↓
09:00 I will explain how to describe the birational geometry, including all (partial) crepant resolutions, of quiver varieties and other GIT quotients satisfying mild assumptions (which also includes
- also some 3D quotient singularities), in terms of varying the stability condition. I will also outline how to extend to moduli spaces with these as local models, such as moduli of 2CY
09:50 categories (eg Higgs bundles on closed curves, sheaves on K3 surfaces, etc). This is based on joint work with Bellamy and Craw, and also with Kaplan.
Alexandru Chivasitu: Leaves, sheaf moduli, and GIT quotients ↓
The non-commutative algebras $Q_{n,k}(E,\eta)$, introduced by Feigin and ODesskii in the course of generalizing Sklyanin's work, depend on two coprime integers $n>k\ge 1$, an elliptic curve $E$
10:00 and a point $\eta\in E$. The degeneration $\eta\to 0$ collapses $Q_{n,1}(E,\eta)$ to the polynomial ring in $n$ variables, and one obtains in this fashion a homogeneous Poisson bracket on that
- polynomial ring and hence a Poisson structure on the projective space $\mathbb{P}^{n-1}$. The symplectic leaves attached to that structure have received some attention in the literature,
10:30 including from Feigin and Odesskii themselves and more recently, Hua and Polishchuk. The talk revolves around various results on these symplectic leaves: their concrete description as moduli
spaces of sheaf extensions on the elliptic curve $E$, the attendant realization as GIT quotients, resulting good properties (like smoothness) which follow from this without appealing to the
symplectic machinery, etc. (joint with Ryo Kanda and S. Paul Smith)
Checkout by 11AM ↓
- 5-day workshop participants are welcome to use BIRS facilities (TCPL ) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 11AM.
(Front Desk - Professional Development Centre)
- Coffee Break (TCPL Foyer)
Dan Rogalski: Results on infinite-dimensional weak Hopf algebras ↓
11:00 An important open question is whether infinite-dimensional noetherian Hopf algebras have finite injective dimension or even must be Artin-Schelter Gorenstein, as conjectured by Brown and
- Goodearl. The same question can be asked for Weak Hopf algebras. We describe work which proves the conjecture for weak Hopf algebras H which are finitely generated over an affine center. In
11:50 addition, we will talk about preliminary results which extend the theory of homological integrals to the setting of weak Hopf algebras, which has numerous applications. This is joint research
with Rob Won and James Zhang.
(TCPL 201)
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ctive investing
From buy-and-hold to active investing
The original concept
The original concept of the efficient frontier was introduced by Markowitz in 1952. He modelled buy and hold investments of stocks and bonds as the weighted sum of expected individual returns.
According to the central limit theorem in statistics, expected returns converge to their true historical values when the average or mean is taken over an arbitrarily large number of separate samples.
Therefore, Markowitz’s investment model can be considered as a weighted sum of true historical individual returns. This weighted sum of historical returns represents, in turn, a weighted average or
mean. In statistics, such a mean represents an expected portfolio return over the holding period. The sum of the weightings of the long positions is normalized to one, of the short positions to minus
one. One may introduce a hedging ratio as the ratio of the two sums. The ingenuity of this model lies in modelling annual expected portfolio returns, R, as a weighted sum of n annual stock returns R
[i ]/h with h[yrs] denoting the holding period in years
R = ∑[i={1,n}] w[i ]R[i] /h ,
and providing ways to compute the weightings w[i] by optimizing one’s personal investment objective in terms of rewards and risks:
Optimally weighted portfolios: Investment Objective = Max[θ=][{][w][}][Reward/Risk ] = Max[θ=][{][w][}][R/Variance(R) ]
where θ represents the set of unknown weightings.
Investment objectives can be of any kind. Markowitz chose to maximize the reward/risk ratios. He took the Compounded Annual Growth Rate (CAGR) as a measure of the annual expected portfolio returns
and chose the Variance of the portfolio price fluctuations over time as his measure of risk. The Variance(R) is calculated from the fluctuations in the individual stock returns at sampled time
periods and properly combined using the estimated weightings. The sampling frequency may be set to the inverse of a correlation time of a year, quarter, month, week, day, …, down to one tick.
Markowitz calculated the portfolio weightings that maximized the mean (CAGR) with minimized Variance (Risks). His methodology is often referred to as Mean-Variance, or MV. He introduced the efficient
frontier, which graphically represents a curve of maximized portfolio rewards with minimized risks as a function of portfolio diversification (Sizing). Portfolio Sizing is directly proportional to
Investment Sizing. As portfolio diversification also relates to risks, the curves reside in a Reward – Risk plane. These curves enable an investor to link portfolio diversification shown on the
horizontal axis to maximized rewards with minimized risks on the vertical axis. Markowitz assumed a Normal probability density function (pdf) of the portfolio price fluctuations over time. By
assuming any pdf, you simplify the numerical optimization process to a linear multivariate regression, the CPU of which increases quadratically with Sizing. This quadratic dependence may produce
unreliable results for larger portfolios or when large market swings are present.
Limitations of MV methods
• Markowitz noted already that he applied the optimization to a single holding period in a continuous time setting. From a mathematical point of view, that limitation is overcome by combining the
weighted sum of n*N historical returns R[i, j] of each individual stock i over all N holding periods h[j] in an advancing time scale:
R[j] = ∑[i=][{][1, n][}] w[i, j ]R[i, j] /h[j] j = {1, 2, ..., N }.
This combining is for compounding investments expressed by
C[j+1] = C[j] (1+R[j] ) with C[1] = Initial investment,
and for fixed or constant investments by
C[N] = (C[1]/N) ∑[j=][{][1, N][}] R[j] .
For fixed holding periods, we have h[j ] = h, so that R[j] can be considered as a back test (past performance) for compounded as well as for constant investments. These time series change into
Fourier series when we we multiply each element in the series by the exponential factors exp(i2πt/h) with time t = jh:
R[j] = ∑[i=][{][1, n][}] w[i, j ]R[i, j] /h {exp(i2πj)} j = {1, 2, ..., N }.
These Fourier series represent the power spectrum of compounding and constant investments when they run over the entire life time of each stock i. Therefore, the power spectrum of the
portfolio-value fluctuations equals the past performance when we trade at a fixed trading interval or fixed trading frequency from womb to tomb, from j = {IPO, 2, ..., today }. This coins the
term High- and Low-Frequency Trading as opposed to the buy-and-hold strategy of Markowitz’s Modern Portfolio Theory (MPT). For power spectra of non-stationary random fluctuations, the
Wiener-Khinchin-Einstein theorem is applicable. It is one of the few theorems in statistical physics that deals with the concept of predictability. This theorem states that the peaks in the power
spectrum are at frequencies with the strongest autocorrelations, hence, with the best predictability. Therefore, we can time the optimally-weighted portfolios by searching for the trading
frequency that maximizes (peaks) the annual returns or any other investment objective of choice:
Optimally timed portfolios: Investment Objective = Max[θ=][{][h[peak]][}][Reward/Risk ].
Here θ represents the holding periods h[peak ]that peak in the autocorrelations of the reward/risk ratio, which can be considered as a fluctuating signal.
Each back test shows the past performance and equals the power spectrum of the price fluctuations when it is performed from IPO to today with a fixed trading frequency. According to the
Wiener-Khinchin-Einstein theorem, the peaks in the power spectrum peak at frequencies with the strongest autocorrelations, hence, with the best predictability:
Past performance is your best predictor of success (Jim Simons, 2005).
• This feature provides the investor with an opportunity to optimize the holding period and the hedging ratio in line with the chosen investment objective AFTER the optimal weighting coefficients
have been computed. The CPU to perform the underlying computations for these portfolios increases linearly with increasing number of holding periods in the time series as well as with portfolio
size. It is not self-evident that when each portfolio is optimized in terms of rewards and risks in the time series, the total sum of returns over all holding periods is also optimized
accordingly. Our software always checks for that evidence. It is also not self-evident that the individual stock-return fluctuations in each holding period have the same pdf. Our software does
not use any pdf. Nor does our software use any equation of motion for the price fluctuations like the Fokker-Planck equation or equations of the same form (Black-Scholes, Navier-Stokes, or any
other diffusion-type of equation). The best fit (best predictor) to such equations is the so-called Cramer-Rao lower bound, which is a standard mathematical procedure to find the holding period
that best fits the equations. We do not see any reason to add these equations as additional information to the price fluctuations and make the computations more cumbersome. In addition, we do not
see any reason to use AI, ML, or NLP, because the best predictor is mathematically determined by the Wiener-Khinchin-Einstein theorem and can be programmed and computed in a relatively simple
• As a second item of critique, it is stated that Variance and the square root thereof (standard deviation) may not be good measures of risks. From a mathematical point of view, you may choose any
quantifiable risk measure that you would like to minimize. We use the maximum drawdown since the IPO as a measure of risk and the annual expected return as a measure of reward. The maximum
drawdown is a measure that is usually determined over many holding periods including periods of crises. This implies that the validation period should span 15+ years.
• As a third item of critique stands the assumption of a Normal pdf fitting the portfolio return fluctuations. Like any other pdf, these will not fit the tails of real price fluctuations in the
equity markets. As past performance is the best predictor of success, we systematically apply the screening and ranking conditions of our game plans to the historical data in our Dbase of the
stocks in our WatchLists. Our ranking system is time-invariant, so that it depends on correlation times. Only eod prices and volumes are used. This systematic screening and ranking gives a time
series of portfolios at preset holding periods. The weightings and timing of each portfolio are optimized to the risks and rewards that are in line with the investor’s personal investment
objective. Hence, no pdf is needed for the optimization process, just a gradient descent or ascent method. The significant advantage of time-invariant ranking is that the CPU to optimize the
investment objective increases linearly with Sizing and with the number of holding periods. During large swings of the market, it has proven its reliability and effectiveness. As many
investors want to evaluate the volatility, skew and excess kurtosis of return distributions, we expand the Value at Risk (VaR) of our summed time series of optimal portfolios in a Cornish-Fisher
VaR[Return Space] = σ√N [-1.96 + 0.474μ[1]/√N - 0.0687μ[2]/N + 0.146μ[1]^2/N ] - 0.5σ^2N ,
where N is the number of holding periods in the recommended validation period, and σ, μ[1], and μ[2] are respectively the volatility, skew, and excess kurtosis measured from the return
distribution. These quantities are calculated from the measured moments of the distribution of returns in accordance with the following:
- the zero moment, M[0], is the number of observations in the validation period
- the first moment, M[1], is the mean of all observed returns during validation
- the second M[2], third M[3], and fourth moment M[4], are defined in the standard manner:
M[k] = ∑[j=][{1,N} ](R[j] - M[1] )^k / M[0] for k = 2, 3, 4 .
The volatility, σ, is given by √M[2], the skew, μ[1], by M[3]/σ^3, and the excess kurtosis, μ[2], by M[4]/σ^4 - 3.
The relative volatility, β, of annualized portfolio returns, C[j], is defined as
β = COV[C[j], Ind]/VAR[Ind] ,
where relative means relative to the volatility of an Index, Ind. The relative annual return, α, is defined as
α = (C - r[rf] ) - β (r[Ind] - r[rf] ) ,
where relative implies relative to the risk-free-rate, r[rf], and the annual return of an Index, r[Ind], and where C denotes the annual rate of portfolio returns, either compounded or as
generated free cash on a fixed investment. The Sharpe ratio is a reward/risk ratio defined by:
Sharpe ratio = (C - r[rf] ) / σ .
The Risk Indicator introduced for all asset markets by MiFid2 rules of the European Union is calculated from the Var-equivalent-Volatility (VeV), which takes on the form:
VeV = {√(3.842 - 2*VaR[Return Space] - 1.96)}/√(Nh) .
Our software links the calculated VeV to the prescribed Risk Indicator of the MiFid2 rules. In conclusion, we broadly see three different systems of defining the {Reward, Risk} space:
1. Rewards are defined as annual expected returns (AER), either compounded or on fixed investments. Risks are defined as the maximum drawdowns (MDD) on those returns.
2. Rewards are expected annual returns relative to risk-free-rates and some Index. They are called “alpha”. Risks are defined as the spreads on those returns, relative to the corresponding
spread on the same Index. They are called “beta”.
3. Rewards are defined as the Value at Risk (VaR) in return space. Risks are defined as the VaR-equivalent-Volatility (VeV). These definitions are given in the MiFid2 rules of the European
The mathematical representations of these three different {Risk, Reward} spaces can be summarized as follows:
{Reward, Risk} ↔ {AER, MDD} ↔ {α, β} ↔ {VaR, VeV} .
It is our understanding that retail investors have their best grip on the first definition, professional investors manage their returns and risks in terms of alpha and beta, and European
legislation uses the third set of definitions, which is based on the Cornish-Fisher expansion as the distribution function of the return fluctuations. The first two sets of definitions do not
necessarily have to use a probability density function, as they can fully resort to actual historical data.
• As a fourth opportunity for extension stands the stipulation of multiple return factors. The multiple-factor method was introduced by Fama and French in 1992. They expanded the portfolio returns
into a weighted sum of market premiums or factors. Their original proposition added two new coefficients to the Capital Asset Pricing Model (CAPM) and changed the definition of the
first coefficient β. These factor weightings should be computed in linear regression AFTER the weighting coefficients of the Markowitz expansion have been computed. You do not need these factors
to compute optimally weighted portfolios. Such factors may be instrumental in doing your due diligence in ranking your assets in terms of their margins of safety. They are also instrumental in
ETF design that track certain market premiums. That is presently a $900 Billion business.
Past performance is the best predictor of success but no guarantee
When we scan the past for the time series of portfolios, we use the Annual Expected Result (AER) as reward and the Maximum DrawDown (MDD) as risk and do not make any assumption about the pdf of the
reward fluctuations. The resulting portfolios are called optimal portfolios as they are maximized in rewards and/or minimized in risks. By varying the asset allocations and computing the resulting
AER and MDD, you search for optima of combinations of these two quantities. Hence, you need a search algorithm to find portfolio weightings that maximize the MAR ratio (= AER/MDD) or just maximize
the AER or minimize the MDD as objective functions. You could also take other ratios like the Sharpe ratio or the Sortino ratio as objective functions, but we prefer the MAR ratio. Maximizing
this ratio is usually close to the investment objective of a retail investor. As stated above, we maximize the MAR ratio in a gradient ascent.
Enabling investors to link an investment to a maximum annual expected result
Portfolio managers usually select their stocks from a WatchList. We made a WatchList of some 1300 liquid stocks of Wall Street with daily-dollar volumes in excess of $1 million since 2005. We compute
the time series of optimal portfolios from it with holding periods of 13 weeks. We compute these time series by using the historical eod prices from the data providers CSI and Yahoo. The chart of the
efficient frontier gives the maximum annual returns with minimum drawdowns as a function of the # stocks in these optimal portfolios (solid lines):
Each investment has its own optimal Annual Expected Result (AER). Portfolios are optimized, so that the weightings maximize the MAR ratio = AER/Max Drawdown (solid lines: MAR>1.2), or maximize the
AER (dotted lines: MAR>0.8). A variation of what mathematicians call a gradient-ascent method was used to find the optima. The CPU used to compute all these curves is less than 10 seconds. The
validation period ran from 14-July-2009 to 14-July-2019
The chart enables an investor to evaluate the maximum annual expected returns with minimum market risks given the size of his investments or portfolio diversification. This efficient frontier for our
WatchList of liquid stocks can be summarized in the following table:
For example, a retail investor who wants to start investing with an amount between $1000 and $10,000 could let DigiFundManager select only one stock every 13 weeks from this WatchList of liquid
stocks using some given screening and ranking conditions. His Annual Expected Return validated over the past ten years is 42% after fees and tax, with a maximum drawdown of -8.5%. During the ten
years before these past ten years, the risks and rewards are significantly larger but still balanced. Using different WatchLists gives different efficient frontiers, often allowing for significant
improvements by proper hedging conditions.
DigiFundManager and predicting the future, machine learning, artificial intelligence and NLP
DigiFundManager uses none of these concepts. All what it does is that it validates the past of screened, ranked, optimally weighted and timed portfolios. Validation or out-of-Sample testing is
accomplished by computing the optimal portfolios on Fridays at closing and calculating the risks and rewards on the following Mondays at closing when actual rebalancing is assumed to take place. In
digitizing the validation of these four activities of portfolio management, we only use the historical eod prices, volumes, dividends, and splits. According to the Efficient Market hypothesis, these
historical prices fully reflect the available information. As we have seen, the past is the best predictor of success, and statistics does not distinguish between the various kinds of risks. We do
not estimate or quantify a prediction interval in which a certain future observation will fall with a certain probability. We do not do such things, because the price fluctuations of the stock market
cannot be fitted with probability distributions when there is blood in the Street. When we rank the 1300+ stocks out of our WatchList with liquid stocks, we do that on the basis of relative
probabilities to increase in price only based on preceding price movements. For instance, on 10-Oct-2014, we computed on the basis of preceding price movements that FRO received the highest rank in
the WatchList of 1300+ stocks. We did not know and had not quantified that the following quarter FRO’s share price would increase by a factor of four. Within our programmed rationale, FRO got the
highest rank in the same way that other stocks got the highest ranks at other quarters. The proof is always in the pudding. Our quantitative investment system of maximum annual expected rewards with
minimized risks with holding periods of 13 weeks as shown in the chart and table is a competitive system, even with High Frequency Trading (HFT).
Annual Expected Results and overdiversified portfolios
For investments between $1000 and $10,000, an investor could also decide to set up a portfolio of six long positions and pay relatively more fees. His Annual Expected Result would decrease from 42%
to 22% with a maximum drawdown of -11%. An important finding is that holding periods of 13 weeks produce larger Annual Expected Results than holding periods of 1 week for portfolios selected from
this WatchList. The efficient frontier of our WatchList of liquid stocks does not underperform the results of HFT. Scaling up investments to hedge-fund levels is a different expertise. If you were to
scale up the investments in our largest portfolios of 384 liquid stocks, we could foresee an increase from $4 million to $500 million. However, the mission of our software service is to bring
low-frequency quantitative investing to the retail investor.
Jan G. Dil and Nico C. J. A. van Hijningen ≅η
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IAT is a MCMC diagnostic that is often used to compare continuous chains of MCMC samplers for computational inefficiency, where the sampler with the lowest IATs is the most efficient sampler.
Otherwise, chains may be compared within a model, such as with the output of LaplacesDemon to learn about the inefficiency of the continuous chain. For more information on comparing MCMC algorithmic
inefficiency, see the Juxtapose function.
IAT is also estimated in the PosteriorChecks function. IAT is usually applied to a stationary, continuous chain after discarding burn-in iterations (see burnin for more information). The IAT of a
continuous chain correlates with the variability of the mean of the chain, and relates to Effective Sample Size (ESS) and Monte Carlo Standard Error (MCSE).
IAT and ESS are inversely related, though not perfectly, because each is estimated a little differently. Given \(N\) samples and taking autocorrelation into account, ESS estimates a reduced number of
\(M\) samples. Conversely, IAT estimates the number of autocorrelated samples, on average, required to produce one independently drawn sample.
The IAT function is similar to the IAT function in the Rtwalk package of Christen and Fox (2010), which is currently unavailabe on CRAN. | {"url":"https://www.rdocumentation.org/packages/LaplacesDemon/versions/16.1.6/topics/IAT","timestamp":"2024-11-01T23:37:05Z","content_type":"text/html","content_length":"69223","record_id":"<urn:uuid:f8d4d2cd-f826-45ab-8465-245d1d43f709>","cc-path":"CC-MAIN-2024-46/segments/1730477027599.25/warc/CC-MAIN-20241101215119-20241102005119-00849.warc.gz"} |
What is a comparison statement example?
What is a comparison statement example?
Comparison Statements: In general, a comparison statement is simply a statement in which two quantities or values are being compered. For instance, ”Mary’s height is the same as Sally’s height” or
”If we add x apples to 3 apples, then the total number of apples is less than 10 apples”.
What is a multiplicative statement?
A multiplication statement is a multiplication problem that uses words rather than numbers and symbols.
What are the three parts to look for in a multiplicative comparison problem?
The three types of multiplicative comparison problems include: when the number being compared is unknown, when the number being compared to is the unknown, and when the multiplier is the unknown.
How do you start a comparison sentence?
Begin by saying everything you have to say about the first subject you are discussing, then move on and make all the points you want to make about the second subject (and after that, the third, and
so on, if you’re comparing/contrasting more than two things).
What is a comparison equation?
The comparison method, a procedure for solving systems of independent equations, starts by rewriting each equation with the same variable as the subject. Any of the variables may be chosen as the
first variable to isolate. Each equation is now an isolated-subject equation, and equation where one variable is isolated.
What is multiplicative comparison 4th grade?
Multiplicative comparison means you are comparing two things together that need to be multiplied. Multiplicative comparison questions are usually written in word problems that have this format:
Statement, Statement, Question. We use the two statements to determine the number sentence or equation.
How do you write a comparison example?
For example, if you wanted to focus on contrasting two subjects you would not pick apples and oranges; rather, you might choose to compare and contrast two types of oranges or two types of apples to
highlight subtle differences. For example, Red Delicious apples are sweet, while Granny Smiths are tart and acidic.
What are some comparison words?
The following words or short phrases compare two items or ideas:
• like.
• likewise.
• same as.
• as well as.
• also, too.
• likewise.
What is a comparison math problem?
COMPARISON problems are the type of problems looked at this week, which involve figuring our similarities or differences between sets. Difference Unknown: One type of compare problem involves finding
out how many more are in one set than another.
How do you start a comparison paragraph example?
Begin with a topic sentence that explains one area of comparison between your first subject and your second subject. For example, if your subjects are two different countries and your paragraph topic
is political structure, you can start by broadly describing each country’s political processes.
How do you write a comparison sentence?
How do you compare two numbers?
To compare two numbers, follow these steps:
1. Write the numbers in a place-value chart.
2. Compare the digits starting with the greatest place value.
3. If the digits are the same, compare the digits in the next place value to the right. Keep comparing digits with the same place value until you find digits that are different.
What are the types of comparisons?
There are three kinds of possible comparisons: equal, comparative and superlative.
What is the rule for comparing numbers?
Rules for Comparison of Numbers: Rule I: We know that a number with more digits is always greater than the number with less number of digits. Rule II: When the two numbers have the same number of
digits, we start comparing the digits from left most place until we come across unequal digits. | {"url":"https://bigsurspiritgarden.com/2022/11/26/what-is-a-comparison-statement-example/","timestamp":"2024-11-08T20:50:21Z","content_type":"text/html","content_length":"52164","record_id":"<urn:uuid:7995b3a3-439d-4cac-abb6-75dffb2a9757>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00042.warc.gz"} |
BREAK & COFFEE 3 Lessons About Changing Our Perspective on Time and Money Management
Whenever I talk to people about my 8 month-long trip to Europe & Argentina, they tell me that I’m very fortunate to be able to do this.
So I tell them:
If I told you that taking this kind of journey is just like taking a break (timewise) and having a cup of coffee (moneywise), what would you say?
Usually they give me a quizzical look and say they don’t understand. Then I laugh and explain what I mean.
I ask them, do you have any idea how much that trip cost me for the 8 months I spent travelling? No, they would often say.
In order to help them, I ask: what is the average price for a weekly trip in the Caribbean or Mexico?
Let’s say it’s 1,250$/week. Therefore, the total cost for 8 months would be:
32 weeks times $1,250/week = $40,000.
At this point everybody tells me:
I can’t afford that! I don’t have that amount of money available, or;
I don’t want to use my economies or my RRSP because I will need them for my retirement.
The list of reasons goes on and on.
Ok, I understand you don’t believe you can afford this.
I want to show you that though this amount of money seems very big, it’s actually not that big at all.
Let’s look at it from a higher level. Say, over a period of 30 years.
Let’s take someone who earns $50,000/year. Over 30 years, they will earn a total of $1.5 million. After income taxes, their net revenue over 30 years (around 73% in Canada) will be close to
Now, what is $40,000 compared to $1,100,000?
It represents only 3.64% of all this money… Now, do you still believe you can’t afford this trip???
Let’s break it down a little bit more.
How much do you think this trip would cost if you would spread your costs on a daily basis over a period of 30 years.
After calculations, this trip would only cost you $3.66 a day…
Yes, only $3.66/day.
Ok, it costs little bit more than a coffee, so let’s add a muffin.
Conclusion: for someone who earns $50,000/year, they would only need to save $3.66/day for 30 years to afford a trip like mine.
Now, do you still think that you cannot afford such an adventure?
It does make sense now, but how can I find 8 months in my life???
Let’s break this one down as well. If we use the same logic and the same type of calculations we will get this:
What is 8 months compared to a period of 30 years? Answer: 2.2%
Now, let’s go back to the coffee analogy and see what 2.2% means if we break it down on a daily basis.
What is 2.2% of 24 hours? Answer: 32 minutes.
Don’t you think everybody can take a 32 minute break during the day and buy a cup of coffee?
Therefore, when you break this problem down to a daily basis, this would only amount to a cost of $ 3.66 and a duration of 32 minutes per day.
So, if you are telling me that you cannot afford to take 8 months and spend $40,000 once in your life to go on vacation, it’s like telling me that you cannot afford to take a break and buy a cup of
coffee (and a muffin) every day of your life!
What I wanted to do in this post is to show you that when we tell ourselves that something is impossible, sometimes just using a different perspective can shed light on the problem and bring us to a
solution we never thought of before.
Lesson #1 — Before thinking something is impossible, look at it from every angle or break it down into smaller pieces.
Lesson #2 — If taking a break is significant for a day, it should be even more significant in your whole life.
Lesson #3 — To achieve big goals, plan for the future and act on it a little bit every day.
So, next time you take a coffee break, think about it! | {"url":"https://unlockedmag.com/pidussaugmail-com/break-coffee-3-lessons-about-changing-our-perspective-on-time-and-money-management/","timestamp":"2024-11-10T19:04:54Z","content_type":"text/html","content_length":"41247","record_id":"<urn:uuid:fc1716db-3e84-4c1a-970f-b24fe2534fc8>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.61/warc/CC-MAIN-20241110170046-20241110200046-00329.warc.gz"} |
A few days ago, I was asked if we should spend a lot of time to choose the distribution we use, in GLMs, for (actuarial) ratemaking. On that topic, I usually claim that the family is not the most
important parameter in the regression model. Consider the following dataset
> db <- data.frame(x=c(1,2,3,4,5),y=c(1,2,4,2,6))
> plot(db,xlim=c(0,6),ylim=c(-1,8),pch=19)
To visualize a regression model, use the following code
> nd=data.frame(x=seq(0,6,by=.1))
> add_predict = function(reg){
+ prd1=predict(reg,newdata=nd,se.fit = TRUE,type="response")
+ y1=prd1$fit
+ y1_upp=prd1$fit+prd1$residual.scale*1.96*
+ y1_low=prd1$fit-prd1$residual.scale*1.96*
+ polygon(c(nd$x,rev(nd$x)),c(y1_upp,
rev(y1_low)),col="light green",angle=90,
+ lines(nd$x,y1,col="red",lwd=2)
+ }
For instance, with a Poisson regression (with a log link function) we get
> plot(db)
> reg1=glm(y~x,family=poisson(link="log"),
+ data=db)
> add_predict(reg1)
> plot(db)
> reg2=glm(y~x,family=gaussian(link="log"),
+ data=db)
> add_predict(reg2)
If we just care about the expected value of our prediction, the output is more or less the same
> plot(db)
> lines(nd$x,predict(reg1,newdata=nd,
+ type="response"),col="red",lwd=1.5)
> lines(nd$x,predict(reg2,newdata=nd,
+ type="response"),col="blue",lwd=1.5)
So, indeed, forget about the (distribution) law when running a GLM. Not convinced? Consider – on the same dataset – a Poisson regression (with an identity link function this time)
> plot(db)
> reg1=glm(y~x,family=poisson(link="identity"),
+ data=db)
> add_predict(reg1)
> plot(db)
> reg2=glm(y~x,family=gaussian(link="identity"),
+ data=db)
> add_predict(reg2)
Again, if we just plot the expected value of our prediction, the output is more or less the same
> plot(db)
> lines(nd$x,predict(reg1,newdata=nd,
+ type="response"),col="red",lwd=1.5)
> lines(nd$x,predict(reg2,newdata=nd,
+ type="response"),col="blue",lwd=1.5)
So clearly, the simplistic message you should not care too much about the (distribution) law seems to be valid…
Continue reading I Fought the (distribution) Law (and the Law did not win)
On Hoeffding’s identity
In 1940, Wassily Hoeffding published Masstabinvariante Korrelationstheorie, which was an impressive paper. For those (like me) who unfortunately barely speak German, an English translation could be
found in The Collected Works of Wassily Hoeffding, published a few years ago. As I keep saying in my courses about copulas, almost everything was in that paper, by Wassily Hoeffding. For instance, we
can see the following graph, of a cumulative distribution function,
What is the difference with a copula? A copula (in dimension 2) is the cumulative distribution function of a random pair with uniform on $[0,1]\times[0,1]$, as defined by Abe Sklar
But Wassily Hoeffding considered a random pair with uniform on $[-1/2,+1/2]\times[-1/2,+1/2]$. But everything else is the same. He can even derive the level curves of the density of the Gaussian
$c(u,v)=\frac{\varphi_r(\Phi^{-1}(u),\Phi^{-1}(v))}{\varphi(\Phi^{-1}(u))\cdot \varphi(\Phi^{-1}(v))}$
> library(mnormt)
> r=.6
> dc=function(u,v) return(
+ as.numeric(dmnorm(cbind(qnorm(u),qnorm(v)),varcov=
+ matrix(c(1,r,r,1),2,2))/dnorm(qnorm(u))/dnorm(qnorm(v))))
> n=500
> vectu=seq(1/n,1-1/n,length=n-1)
> matdc=outer(vectu,vectu,dc)
> contour(vectu,vectu,matdc,levels=
+ c(.325,.944,1.212,1.250,1.290,1.656,3.85),lwd=2)
But another interesting point is that there is the so-called Hoeffding’s equality
$\text{cov}(X,Y)=\int_{\mathbb{R}\times\mathbb{R}} [F_{XY}(x,y) - F_X(x)F_Y(y)]dxdy$
which is interesting, and quite important, actually, to understand that the covariance (or the correlation) can be seen as some ‘distance‘ to the independence. More precisely, observe that
$\text{cov}(X,Y)=\int_{\mathbb{R}\times\mathbb{R}} [F_{XY}(x,y) - F_{X,Y}^\perp(x,y)]dxdy$
where $F_{X,Y}^\perp$ would be the joint cumulative distribution function of some independent variables, with the same marginal distributions.
Of course, it is not exactly a distance, since it can be negative. But still. Now, the thing is that the proof is not trivial. But it is using interesting identities. For instance, in 1885, Franklin
wrote a nice paper, Proof of a Theorem of Tchebycheff’s on Definite Integrals, in the American Journal of Mathematics. To get some heuristics about the identity, consider some (finite) sequences $(\
boldsymbol{u})=\{u_1,\cdots,u_n\}$ and $(\boldsymbol{v})=\{v_1,\cdots,v_n\}$, then one can prove that
$\sum_{i,j=1}^n [(u_i-u_j)(v_i-v_j)]=2\left[n\sum_{i=1}^n u_iv_i -\sum_{i=1}^n u_i\cdot \sum_{i=1}^n v_i\right]$
And there is a continuous version of that identity. Consider two bounded functions $u(\cdot)$ and $v(\cdot)$, on some interval, $[a,b]$ then
is equal to
$2\left[(b-a)\int_{[a,b]} u(x)v(x)dx -\int_{[a,b]} u(x)dx\cdot \int_{[a,b]} v(x)dx\right]$
In 1979, in Monotone Regression and Covariance Structure, Gerald Shea gave a more probabilistic interpretation of that results, using the fact that
and using a different measure. More precisely, assume now that functions $u(\cdot)$ and $v(\cdot)$ are integrable, with respect to some measure $\mu$, on some set $\mathcal{S}\subset\mathbb{R}$.
is equal to
$2\left[\mu(\mathcal{S})\int_{\mathcal{S}} u(x)\cdotv(x)\mu(dx) -\int_{\mathcal{S}} u(x)\mu(dx)\cdot \int_{\mathcal{S}} v(x)\mu(dx)\right]$
In the case where $\mu$ is a probability measure of $\mathcal{S}\subset\mathbb{R}$, i.e. $\mu(\mathcal{S})=1$, this equality is the one used by Wassily Hoeffding, in 1940. The interpretation in terms
of random variable is simple that
(with standard assuptions of existence of those quantitites) where $(X_1,Y_1)$ and $(X_2,Y_2)$ are two independent vectors, with identical distribution, $F_{XY}$. Actually, this relationship can also
be found in Some Concepts of Dependence, by E. L. Lehmann, published in 1966. Oh, and by the way, the connection with Chebyshev inequality (claimed in the title of seminal paper by Franklin) come
from the fact that if $u(\cdot)$ and $v(\cdot)$ are monotonic, then the left part of the identity is positive, and thus,
$(b-a)\int_{[a,b]} u(x)\cdot v(x)dx \geq\int_{[a,b]} u(x)dx\cdot \int_{[a,b]} v(x)dx$
But let’s get back to Hoeffiding’s result. How do we get it from that lemma. The idea is to write
$\mathbb{E}\left(\int_{\mathbb{R}\times\mathbb{R}} [\boldsymbol{1}_{u\leq X_1}-\boldsymbol{1}_{u\leq X_2}]\cdot[\boldsymbol{1}_{v\leq Y_1}-\boldsymbol{1}_{v\leq Y_2}] dudv\right)$
$\mathbb{E}\left(\int_{\mathbb{R}\times\mathbb{R}}[\boldsymbol{1}_{u\leq X_1}\boldsymbol{1}_{u\leq Y_1}-\boldsymbol{1}_{u\leq X_1}\boldsymbol{1}_{u\leq Y_2}-\boldsymbol{1}_{u\leq X_2}\boldsymbol{1}_
{u\leq Y_1}+\boldsymbol{1}_{u\leq X_2}\boldsymbol{1}_{u\leq Y_2}]dudv\right)$
We can then intervert the integral and the expectation, use the fact that
$\mathbb{E}(\boldsymbol{1}_{u\leq X_1})=\mathbb{P}(X_1\geq u)$
and then, and some integral calculus can be used to rewrite that expression as
So we get here Hoeffding’s identity. Actually, as mentioned by Ben Derrett about the equality above, it can be observed (see http://math.stackexchange.com/105713) that2\text{cov}(X,Y)=2\big(\mathbb
{E}[XY]-\mathbb{E}[X]\mathbb{E}[Y]\big)can also be written
$\mathbb E[X_1Y_1]-\mathbb E[X_2]\mathbb E[Y_1]+\mathbb E[X_2Y_2]-\mathbb E[X_1]\mathbb E[Y_2]$
where again, $(X_1,Y_1)$ and $(X_2,Y_2)$ are two independent vectors, with identical distribution, $F_{XY}$. The later can be writen
$\mathbb E((X_1-X_2)(Y_1-Y_2))$ | {"url":"https://freakonometrics.hypotheses.org/tag/identity","timestamp":"2024-11-09T21:51:19Z","content_type":"text/html","content_length":"196273","record_id":"<urn:uuid:ae39732c-853e-4b05-bfed-2dde6667da03>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.10/warc/CC-MAIN-20241109214337-20241110004337-00324.warc.gz"} |
Analytic proof
In mathematics, an analytic proof is a proof of a theorem in analysis that only makes use of methods from analysis, and which does not predominantly make use of algebraic or geometrical methods. The
term was first used by Bernard Bolzano, who first provided a non-analytic proof of his intermediate value theorem and then, several years later provided a proof of the theorem which was free from
intuitions concerning lines crossing each other at a point, and so he felt happy calling it analytic (Bolzano 1817).
Bolzano's philosophical work encouraged a more abstract reading of when a demonstration could be regarded as analytic, where a proof is analytic if it does not go beyond its subject matter (Sebastik
2007). In proof theory, an analytic proof has come to mean a proof whose structure is simple in a special way, due to conditions on the kind of inferences that ensure none of them go beyond what is
contained in the assumptions and what is demonstrated.
Structural proof theory
In proof theory, the notion of analytic proof provides the fundamental concept that brings out the similarities between a number of essentially distinct proof calculi, so defining the subfield of
structural proof theory. There is no uncontroversial general definition of analytic proof, but for several proof calculi there is an accepted notion. For example:
In Gerhard Gentzen's natural deduction calculus the analytic proofs are those in normal form; that is, no formula occurrence is both the principal premise of an elimination rule and the conclusion of
an introduction rule;
In Gentzen's sequent calculus the analytic proofs are those that do not use the cut rule.
However, it is possible to extend the inference rules of both calculi so that there are proofs that satisfy the condition but are not analytic. For example, a particularly tricky example of this is
the analytic cut rule, used widely in the tableau method, which is a special case of the cut rule where the cut formula is a subformula of side formulae of the cut rule: a proof that contains an
analytic cut is by virtue of that rule not analytic.
Furthermore, structural proof theories that are not analogous to Gentzen's theories have other notions of analytic proof. For example, the calculus of structures organises its inference rules into
pairs, called the up fragment and the down fragment, and an analytic proof is one that only contains the down fragment.
See also
Proof-theoretic semantics
Bernard Bolzano (1817). Purely analytic proof of the theorem that between any two values which give results of opposite sign, there lies at least one real root of the equation. In Abhandlungen der
koniglichen bohmischen Gesellschaft der Wissenschaften Vol. V, pp.225-48.
Pfenning (1984). Analytic and Non-analytic Proofs. In Proc. 7th International Conference on Automated Deduction.
Sebastik (2007). Bolzano's Logic. Entry in the Stanford Encyclopedia of Philosophy.
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
Hellenica World - Scientific Library
Retrieved from "http://en.wikipedia.org/"
All text is available under the terms of the GNU Free Documentation License | {"url":"https://www.hellenicaworld.com/Science/Mathematics/en/AnalyticProof.html","timestamp":"2024-11-12T23:12:27Z","content_type":"application/xhtml+xml","content_length":"8267","record_id":"<urn:uuid:986d5b3a-d1b3-4f68-95b7-20fc9f8e403c>","cc-path":"CC-MAIN-2024-46/segments/1730477028290.49/warc/CC-MAIN-20241112212600-20241113002600-00588.warc.gz"} |
Mathematics, Bachelor of Arts - MATH
Major Requirements (38–42 Hours)
Course List
Code Title Credits
MATH 131 Calculus I 4-8
& MATH 132 and Calculus II for STEM majors
or MATH 133 Theory and Application of Calculus
MATH 225 Foundations of Higher Mathematics 3
MATH 231 Calculus III 4
MATH 326 Linear Algebra and Differential Equations 4
MATH 496 Pro-Seminar 2
CPSC 207 Computer Programming 3
& 207L and Computer Programming Laboratory
Select two of the following full-year sequences (one of which must be either Analysis or Algebra): 12
MATH 335& MATH 336 Differential Equations II
and Numerical Analysis
MATH 341& MATH 342 Analysis I
and Analysis II
MATH 345& MATH 346 Probability
and Statistics
MATH 353& MATH 354 Abstract Algebra I
and Abstract Algebra II
Select six additional hours at the 300-400 level (above 302): 6
CPSC 315 Simulation: Theory and Application
Data Structures
CPSC 328
MATH 335 Differential Equations II
MATH 336 Numerical Analysis
MATH 339 Discrete Mathematics
MATH 341 Analysis I
MATH 342 Analysis II
MATH 345 Probability
MATH 346 Statistics
MATH 353 Abstract Algebra I
MATH 354 Abstract Algebra II
MATH 361 Geometry
MATH 372 Stochastic Models
MATH 381 Mathematical Modeling
MATH 388 BIG (Business, Industry, Government) Problems in Mathematics
MATH 438 Mathematical Programming
MATH 490 Special Topics
MATH 497 Independent Study
Total Credits 38-42
Advanced Writing Proficiency
The purpose of this requirement is to nurture the development of mathematical writing in order to deepen the student’s understanding of mathematics and to enable the student to communicate technical
ideas to a range of audiences. Sophomores are expected to demonstrate proficiency in expository mathematics by the submission of an acceptable portfolio. Juniors are expected to demonstrate
proficiency in technical or analytical mathematical writing by the submission of an acceptable portfolio. Seniors demonstrate their ability by completing a senior comprehensive paper, which is
evaluated by a committee of three faculty.
Senior Comprehensive
All mathematics majors, in Pro-Seminar (MATH 496 Pro-Seminar), independently study a mathematical topic of their choice and work with a faculty advisor. They present their work in a series of talks
in the seminar. The project culminates in a paper and a formal presentation. This final presentation, followed by questioning by a faculty committee, constitutes the Senior Comprehensive in | {"url":"https://catalog.saintmarys.edu/undergraduate/programs/mathematics-computer-science/mathematics-bachelor-arts/","timestamp":"2024-11-08T01:40:17Z","content_type":"text/html","content_length":"41390","record_id":"<urn:uuid:f68bf854-ab4c-4d83-9b2f-e51951f28046>","cc-path":"CC-MAIN-2024-46/segments/1730477028019.71/warc/CC-MAIN-20241108003811-20241108033811-00200.warc.gz"} |
Section 1: Trigonometry Review; Trigonometric Limits; Derivatives of Sine and Cosine
Section 2: Derivatives of Other Trig functions; Applications; Inverse Functions, Arcsine and Arctangent; Derivatives of Arcsine and Arctangent
Section 3: Exponential and Logarithmic Functions Review; The Fundamental Exponential Limit, and the Natural Logarithmicand Exponential Functions; Derivatives of Logarithmic Functions
Section 4: Derivatives of Exponential Functions; Applications; Logarithmic Differentiation
Section 5: Another Look at Limits; L'Hopital's Rule; Mean Value Theorem
** This is a Print-on-Demand product; it is non-returnable. | {"url":"https://order.openschool.bc.ca/Product/DetailPartial/k12s_7540002606","timestamp":"2024-11-02T17:47:08Z","content_type":"text/html","content_length":"3287","record_id":"<urn:uuid:6edad7fa-c150-48ca-a5b2-00e6b6489862>","cc-path":"CC-MAIN-2024-46/segments/1730477027729.26/warc/CC-MAIN-20241102165015-20241102195015-00453.warc.gz"} |
Infinite Monkey Theorem - ELGL
Today’s Morning Buzz is brought to you by Shane Stone, Police Executive Administrator in Maricopa, AZ. Connect with Shane on Twitter and LinkedIn.
What I’m Learning: A few weeks into working in a Police Department, a million things every day
What I’m Listening to: Summertime means Spotify’s ‘Poolside Lounge’ playlist
What I’m Watching: Due to an incredible run, I am still watching the Vegas Golden Knights!
Today we’re talking about the Infinite Monkey Theorem!
I know, this is the moment we’ve all been waiting for, but a few of us may be asking, “Could you describe the Infinite Monkey Theorem, for anybody else who may not know what it is?”
The Infinite Monkey Theorem is this idea that if you put monkeys in front of typewriters, perhaps infinite monkeys with infinite typewriters and certainly with infinite time, they will eventually
crank out every single literary possibility. That means at some point in the stream of scrambled characters you will find the text of Hamlet, Harry Potter and the Prisoner of Azkaban, and even the
entire Captain Underpants anthology. In a truly infinite and endless universe, everything that can possibly ever happen will happen. You just have to wait.
While you leisurely lounge the infinite monkeys will eventually type your term papers from college, they’ll produce that report you have sitting on your desk, they’ll even complete your passport
application and write to your family back home. The world we live in is even more random and vast than monkeys with typewriters, this can leave you wondering what impact you actually have as an
Fear not, because if you thought the philosophical theory was fun, then just wait until we introduce a little math!
Including the space bar, my keyboard has 48 keys not accounting for the possibility of using the shift key to change characters. The chance that any given letter, number, space, or other character
would pop up would be 1 in 48 for every keystroke. My full name ‘Shane Lewis Stone’ has 17 characters including the spaces which means the odds that a randomizing monkey would type my name in any
given sequence of 17 characters would be 1:48^17 or 1 in every 38,115,448,583,970,000,000,000,000,000 attempts. All of the sudden, these infinite monkeys don’t seem so adept.
We all know the problem with the monkeys is that they lack a writer’s intention, typing your name isn’t difficult unless you aren’t even trying to type your name as you mash the buttons on the
keyboard. The first thing the Infinite Monkey Theorem teaches us is that eventually things will come to pass, but with intention on top of the elbow grease you can find the results you seek much more
quickly. But this is the obvious and boring moral to the story, so back to the math!
Is There Only One Solution?
Does “ShaneLewisStone” change the practical outcome of you reading that? How about “5haneLewi55tone” or even “5haneLewi55t0ne”?
In my humble opinion, the first example is the best, it’s how I like to see my name, but if a monkey types any of the variations up randomly on a typewriter I am going to be impressed and a little
honored. Truthfully, we can tinker with any of these and turn them into the ideal presentation of my name, and odds are we are going to reach the imperfect solutions long before we stumble upon the
perfect solution.
But how much more quickly can we move the project (of typing my name) forward if we start with the imperfect spellings?
In today’s verbiage, the spacebar is “extra.” If we eliminate this key and focus on the core of the matter (the letters) we increase the odds of finding a solution from 1:48^17 to 1:47^15. Both are
astronomical, but you can see the impact when we type the numbers out.
1 in 38,115,448,583,970,000,000,000,000,000 with spaces
1 in 12,063,348,350,820,000,000,000,000 without spaces
Our odds of success are roughly 3,000 times better when we accept the smallest of imperfections. Instead of “Shane Lewis Stone” we are looking for “ShaneLewisStone.”
What if we are okay with a few letters being replaced by numbers? If every “S” could be a “5” and the “o” in “Stone” could be “0” instead, we double the chances of success when randomly typing those
characters. Instead of the odds of success being 1:47^15 we are now at 1:47^11 x 1:23.5^4. The math is getting weird but hang with me and I’ll do it for you.
1 in 38,115,448,583,970,000,000,000,000,000 with spaces
1 in 12,063,348,350,820,000,000,000,000 without spaces
1 in 753,959,117,416,318,300,000,000 in without spaces and OK with numbers
The monkeys are back in business. Now our odds of success are 16 times better than the previous round, and about 50,000 times better than where we started. For every time the monkeys type my name
perfectly on a typewriter, they could type it very well 50,000 times. When you are looking only for perfect solutions, and you start with a single possibility in mind, you are greatly reducing your
chance of finding efficient success. Especially when you consider your first solution to be a starting point with a willingness to iterate.
Get Rid of the Noise
But let’s do one last exercise, what if we took all of the keys off the keyboard that we were never going to use? My name has 11 distinct characters: a, e, h, i, l, n, o, r, s, t, and w. That gets
rid of 37 of our 48 original keys on the keyboard, leaving us with odds of 1:11^15, and you know we have to take a look at it.
1 in 38,115,448,583,970,000,000,000,000,000 with spaces
1 in 12,063,348,350,820,000,000,000,000 without spaces
1 in 753,959,117,416,318,300,000,000 in without spaces and OK with numbers
1 in 4,177,248,169,415,700 without spaces, with only possible letters on the keyboard
Our odds of success are now 9.5 trillion times better than they originally were. We didn’t leave the keys on the board that would never be used, and the monkeys are so much more efficient because of
The greatest way to enhance your odds of success is to act with intention, with that small step you will type circles around the randomness of the Infinite Monkey Theorem. But we still have a few
more lessons to learn from the mental exercise.
• Perfection is not a starting point: We all want to be perfect. But if you are spending three times the effort (or 50,000 times the effort) to go directly to perfection, rather than starting with
the practical building blocks and iterating towards that perfection, you are wasting your valuable time and effort.
• There is more than one solution: My whole life I’ve typed my name the same way, but if I need to work with the randomness of the world, or monkeys with a typewriter, it will serve me well to be
open to new ideas. Even if “5han3 Lew15 5t0n3” leads people to believe that Elon Musk named me.
• Eliminate superfluous options: If you attempt to ponder every possible action, and how it will impact every aspect of your work, it will stop all progress. Focus on the problem at hand, and start
with the solutions you can reasonably implement. It will feel like you are working 9.5 trillion times faster.
Are these morals to the story presented with rough math and hyperbole? Of course they are, but what did you expect to get from me randomly mashing the buttons of my keyboard? | {"url":"https://elgl.org/infinite-monkey-theorem/","timestamp":"2024-11-09T09:37:00Z","content_type":"text/html","content_length":"80509","record_id":"<urn:uuid:32dbf1bb-d06c-4f8e-810f-7c5ae6eecaeb>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.75/warc/CC-MAIN-20241109085148-20241109115148-00057.warc.gz"} |
Dynamic Programming Seam Finder (Optional)¶
Optional Task
Implement DynamicProgrammingSeamFinder
Completing this portion of the project is completely optional and is only worth a negligible amount of extra credit if you choose to attempt it. We will not ask TAs to be familiar with this version
of seam carving.
In the real world, seam carving is implemented using a different approach called dynamic programming (DP). In fact, the original paper for seam carving described the algorithm using dynamic
The dynamic programming approach works by starting from the leftmost column and working your way right, using the previous columns to help identify the best shortest paths, when removing a horizontal
seam. The difference is that dynamic programming does not create a graph representation (vertices and edges) nor does it use a graph algorithm.
How does dynamic programming solve the seam finding problem? Most of the runtime is spent generating the dynamic programming table (DP table):
1. Initialize a 2-D double[picture.width()][picture.height()] array where each entry corresponds to a pixel coordinate in the picture and represents the total energy cost of the least-noticeable
path from the left edge to the given pixel.
2. Fill out the leftmost column in the 2-D array, which is just the energy for each pixel.
3. For each pixel in each of the remaining columns, determine the lowest-energy predecessor to the pixel: the minimum of its left-up, left-middle, and left-down neighbors. Compute the total energy
cost to the current pixel by adding its energy to the total cost for the best predecessor.
To find the shortest horizontal path using this DP table, start from the right edge and add the y-value of each minimum-cost predecessor to a list. The result is a list of y-value from right to left.
Finally, to get the coordinates from left to right, Collections.reverse the list.
To find the shortest vertical path, the same process is followed as the horizontal path but in a vertical manner (starting from the bottom and going up). | {"url":"https://courses.cs.washington.edu/courses/cse373/23su/projects/seamcarving/DP/","timestamp":"2024-11-04T14:16:38Z","content_type":"text/html","content_length":"11614","record_id":"<urn:uuid:fad70e6c-fe79-44a7-8999-659599b89741>","cc-path":"CC-MAIN-2024-46/segments/1730477027829.31/warc/CC-MAIN-20241104131715-20241104161715-00534.warc.gz"} |
What is the Sortino Ratio? | aiSource
What is the Sortino Ratio?
Analyzing CTAs and weeding out the good from the bad is a challenge our firm is faced with often. The process that goes into our due diligence not only looks at the rates of return and margin to
equity ratios, but it also dives into a handful of risk ratios. Probably the most famous and widely used within all portfolio construction is the Sharpe ratio. Although the Sharpe ratio is something
we closely assess, we also closely examine a CTAs Sortino ratio.
Developed nearly 17 years after the William Forsyth Sharpe ratio, the Sortino ratio similarly measures an investment by adjusting for risk, with a small twist. Prior to pointing out the clear
differences between the two, it might be useful to re-visit the basic concept behind the Sharpe ratio. In simple terms, the Sharpe ratio is a calculated ratio that helps measure risk-adjusted
performance of a specific investment. For example, if investment X made 12% and investment Y made 8%, you might consider investment X over investment Y given the higher rate of return. Well, not so
fast – more information is needed to assess the risks associated with each investment. In the case of the Sharpe ratio, it first subtracts the riskless rate of return that you could have earned if
you kept your money invested in Treasury bills, then divides it by the volatility of the investment. Assuming T-bills are returning 2% and investment X has a volatility of 15% and investment Y has a
volatility of 4%, the following calculations would be accurate:
Investment X has a Sharpe ratio of 0.667, where investment Y has a Sharpe ratio of 1.5. The higher the Sharpe ratio the better, because it means you are earning a higher return over the risk-free
rate per unit of risk. After calculating the Sharpe ratios of the above investments, it is clear that investment Y would be a more attractive investment given the overall return and risk associated
with the investment.
Very similar to how the Sharpe ratio measures an investment based on its risk, the Sortino ratio also does the same except it takes into consideration only downside deviations (volatility of only
negative returns) within the investment as opposed to the standard deviations (volatility of both positive and negative returns) that the Sharpe ratio uses. Many investment advisors and/or
professionals argue that the Sortino ratio is a better measure of risk. Well, let’s take a closer look to how the Sortino ratio is calculated (see below):
S = Sortino Ratio
R = the investments average period return
MAR = is the target or required rate of return for the investment strategy under consideration, (originally known as the minimum acceptable return, or MAR)
downside deviation = is the downside deviation as measured by the standard deviation of negative asset or portfolio returns.
The initial step of calculating the Sortino is fairly simple in that you are simply subtracting the investments actual monthly returns by its minimum acceptable return (which we reference as 0% in
our example). The second part of the equation is what many people disagree on. The disagreement comes in the handling of excess returns, which has resulted in two methods of calculating the ratio. In
the first method below, the positive excess returns are changed to zeros and included in the calculation of the downside deviation. In the second method, only the negative excess returns are used in
the calculation of downside risk; the frequency of positive excess returns are excluded. Please refer to the examples below on how to calculate the Sortino ratio using each method:
Sortino 1 Calculations:
*Sortino 1 uses the method of zeroing out the positive returns and including them in the calculation of downside risk.
Sortino 2 Calculations:
*Sortino 2 uses the method of only using negative returns to calculate downside risk.
Similar to the Sharpe ratio, a large Sortino ratio indicates a better risk-adjusted return.
In conclusion there are many risk statistics available for use, not only on our website but throughout the internet. The ratios are based on past returns and past performance and are not a guarantee
of future returns. However, investors can use the ratios to help forecast potential future returns and assist in making investment decisions. By no means is analyzing a CTAs Sharpe and Sortino ratio
the end all be all when conducting due diligence, however it does offer insight into the CTA strategy’s risk profile. | {"url":"https://www.managedfuturesinvesting.com/what-is-the-sortino-ratio/","timestamp":"2024-11-13T08:00:41Z","content_type":"text/html","content_length":"58360","record_id":"<urn:uuid:f3d27f26-ef24-4a2a-8f12-9e279aa04986>","cc-path":"CC-MAIN-2024-46/segments/1730477028342.51/warc/CC-MAIN-20241113071746-20241113101746-00830.warc.gz"} |
Function Parameters | Sololearn: Learn to code for FREE!
Function Parameters
what exactly it is meant by function parameters
Parameters are values or objects that can be passed to the function that can be altered in the function and also used by the function to obtain a desired result. Ex: function multiply(int a, int b) {
var result = a * b; return result; } var c = multiply(3, 7); // c now contains the integer return value from the // multiply function // which multiplies parameters a and b (3 and 7) // giving it the
value 21
Function parameters are data required that you give to the function. For example, if a function called Add() needs two numbers to add, you would define the function as Add(int x, int y), where x and
y are the parameters (in this case, they're integers). So you would need to give it 2 integers, which it adds together. int Add( int x, int y) { return x + y; } In the example, the function Add()
adds the x and y parameters and returns them. You would use it like this: int sum = Add(10, 5); here, sum would be equal to 15. (not sure how you would write it in JS, I use C++, but the concept is
definitely the same). Of course, this example function is useless because all modern languages have built in operators that you can use instead (such as the + and - operators that add and subtract),
but nonetheless I think it's a clear example. I currently use C++ to program 3D graphics. I made a function called setLocation(x,y,z). When called, it sets the location of a 3D model (be it a house
or a lamp) to that x,y,z location. An example of a call would be setLocation(10,5,11). | {"url":"https://www.sololearn.com/ru/Discuss/151960/function-parameters","timestamp":"2024-11-12T05:57:03Z","content_type":"text/html","content_length":"1043652","record_id":"<urn:uuid:ac90690c-fdde-4919-a9a9-8ff82590099c>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.58/warc/CC-MAIN-20241112045844-20241112075844-00343.warc.gz"} |
Grade 12 Centripetal Acceleration Question- Universal Law of Gravitation
• Thread starter AudenCalbray
• Start date
In summary, we are asked to find the acceleration, gravitational force of attraction, centripetal force, and the contributing forces in the Bohr model of the hydrogen atom. Using the given radius and
frequency, we can calculate the acceleration to be 9.97 x 10^22 m/s/s. However, without knowing the masses of the electron and proton, we cannot find the gravitational force of attraction or the
centripetal force, which are both necessary for the calculation of the contributing forces.
Homework Statement
In the Bohr model of the hydrogen atom, the electron revolves around the nucleus. If the radius of the orbit is 5.8 z 10^-11 m and the electron makes 6.6 x 10^15 r/s, find;
a) the acceleration of the electron
b) the magnitude of the gravitational force of attraction between the electron and the nucleus
c) the centripetal force acting on the electron
d) the magnitude of each force contributing to the centripetal acceleration (name each force)
Homework Equations
Fc= ma= mv^2/r=4pi^2Rf^2
The Attempt at a Solution
So I got the acceleration by using this equation: ac=4pi^2Rf^2= 9.97 x 10^22 m/s/s, and I know that the two forces contributing to the Fc are Fg and an electrical force of attraction. I do not know
how to get the gravitational force of attraction without any masses. I'm at a loss. I also calculated the speed, but I do not see how I can use that to get the Fg and the Fc. Please help!
Well you have Hydrogen which is just one proton and one electron. So you can look up the masses for these two and use it.
rock.freak667 said:
Well you have Hydrogen which is just one proton and one electron. So you can look up the masses for these two and use it.
I'm pretty sure I'm only supposed to use information provided from the question though..
AudenCalbray said:
I'm pretty sure I'm only supposed to use information provided from the question though..
They are standard values but in that case, then you can't find the gravitational force nor the centripetal force as they both contain a mass term.
FAQ: Grade 12 Centripetal Acceleration Question- Universal Law of Gravitation
1. What is centripetal acceleration?
Centripetal acceleration is the acceleration directed towards the center of a circular path. It is always perpendicular to the velocity of an object moving in a circular path.
2. What is the Universal Law of Gravitation?
The Universal Law of Gravitation states that every object in the universe attracts every other object with a force that is directly proportional to the product of their masses and inversely
proportional to the square of the distance between them.
3. How is centripetal acceleration related to the Universal Law of Gravitation?
Centripetal acceleration is caused by the gravitational force between two objects. For example, the moon orbits around the Earth due to the centripetal acceleration caused by the gravitational force
between the two bodies.
4. How can we calculate centripetal acceleration using the Universal Law of Gravitation?
We can use the formula a = v^2/r, where a is the centripetal acceleration, v is the velocity of the object in circular motion, and r is the radius of the circular path. We can also use the formula a
= GM/r^2, where G is the gravitational constant, M is the mass of the larger object, and r is the distance between the two objects.
5. What are some real-life applications of the Universal Law of Gravitation and centripetal acceleration?
The Universal Law of Gravitation and centripetal acceleration are essential concepts in understanding the motion of celestial bodies, such as planets, moons, and satellites. They are also used in
designing and understanding the functioning of centrifuges, roller coasters, and other circular motion devices. Additionally, these concepts are crucial in predicting and studying the behavior of
objects in orbit, such as space probes and satellites. | {"url":"https://www.physicsforums.com/threads/grade-12-centripetal-acceleration-question-universal-law-of-gravitation.592523/","timestamp":"2024-11-03T14:01:19Z","content_type":"text/html","content_length":"92085","record_id":"<urn:uuid:7d70223f-bb8c-43c8-8ffb-57bddd7faced>","cc-path":"CC-MAIN-2024-46/segments/1730477027776.9/warc/CC-MAIN-20241103114942-20241103144942-00480.warc.gz"} |
Problem: Tangent Circles and an Isosceles Triangle
Tangent Circles and an Isosceles Triangle
The applet presents an 1803 Sangaku problem: Given a circle S with center O and diameter AC and point B on AC. Form circle G with center P and diameter AB and an isosceles triangle BCE with E on the
circle S. Circle W with center Q is inscribed in the curvilinear triangle formed by circles S and G and the line BE. Prove that QB is perpendicular to AC.
(Years ago there was indeed a Java applet, now commented out since browers stopped supporting Java. Still, there are links to two pages with solutions to the problem.)
What if applet does not run?
One solution to this problem uses inversion and another makes use of inversion with negative power.
1. Tangent Circles and an Isosceles Triangle
|Activities| |Contact| |Front page| |Contents| |Geometry|
Copyright © 1996-2018 Alexander Bogomolny | {"url":"https://www.cut-the-knot.org/Curriculum/Geometry/CirclesAndRegularTriangle.shtml","timestamp":"2024-11-02T03:14:15Z","content_type":"text/html","content_length":"21163","record_id":"<urn:uuid:ffb9b910-a02c-447e-8d45-a78aa86b50c1>","cc-path":"CC-MAIN-2024-46/segments/1730477027632.4/warc/CC-MAIN-20241102010035-20241102040035-00124.warc.gz"} |
Heptagon Area
The Area of a Heptagon calculator computes the area of a regular heptagon, a polygon with 7 equal sides of length (s). Regular Heptagon
INSTRUCTIONS: Choose units (e.g. feet or meters) and enter the following:
• (s) Length of Sides of Heptagon.
Heptagon Area (A): The area is returned in square meters, but can be automatically converted to other units such as square feet or acres via the pull-down menu.
For the Volume of a Heptagon shaped column, CLICK HERE.
The Math
A regular heptagon has seven equal sides and equal angles. The formula for the area of a heptagon is:
A =^7/[4]•cot(^π/[7])•s^2
Regular Polygon Information
A regular polygon is a geometric shape with three or more straight sides where every side is the same length and every angle between connecting sides are the same angle. Because of the symmetry of
the regular polygon, all the vertices of the polygon can be constructed to touch a circle in which the regular polygon is inscribed and all the chords that are polygon sides will then obviously be of
equal length . Likewise, because of the regular polygon's symmetry, a circle constructed to be inscribed in a regular polygon and touching the polygon will touch the regular polygon at the midpoint
of the polygon side. As shown in the pictures, Figure 1 and Figure 2, lines from the regular polygon's vertices to the circle's center form n isosceles triangles of equal area.
The names of polygons vary based on the number of sides as follows:
• triangle - 3 sides
• square - 4 sides
• pentagon - 5 sides
• hexagon - 6 sides
• heptagon - 7 sides
• octagon - 8 sides
• nonagon - 9 sides
• decagon - 10 sides
• hendecagon - 11 sides
• dodecagon - 12 sides
Common Regular Polygon Functions
Polygon Area Calculators:
Polygon Side Calculators
Polygon Perimeter Calculators
Polygon Radius
3D Polygon Shapes
Other Polygon Calculators | {"url":"https://www.vcalc.com/wiki/area-of-heptagon","timestamp":"2024-11-12T17:01:36Z","content_type":"text/html","content_length":"54106","record_id":"<urn:uuid:0485adf8-768d-4d50-95cd-28a16bf647aa>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.63/warc/CC-MAIN-20241112145015-20241112175015-00771.warc.gz"} |
Irrational Vs Rational Numbers Worksheet Pdf 2024 - NumbersWorksheets.com
Irrational Vs Rational Numbers Worksheet Pdf
Irrational Vs Rational Numbers Worksheet Pdf – A Logical Phone numbers Worksheet may help your kids become more informed about the ideas right behind this percentage of integers. In this particular
worksheet, college students will be able to remedy 12 diverse issues relevant to rational expression. They will discover ways to multiply a couple of phone numbers, group of people them in sets, and
find out their products and services. They are going to also practice simplifying rational expression. After they have learned these ideas, this worksheet is a beneficial tool for furthering their
research. Irrational Vs Rational Numbers Worksheet Pdf.
Reasonable Phone numbers are a proportion of integers
The two main forms of figures: rational and irrational. Reasonable numbers are understood to be total amounts, whereas irrational figures usually do not replicate, and also have an unlimited number
of numbers. Irrational amounts are non-zero, low-terminating decimals, and sq . beginnings that are not best squares. They are often used in math applications, even though these types of numbers are
not used often in everyday life.
To establish a realistic amount, you must understand exactly what a realistic variety is. An integer is actually a entire variety, and a logical number is actually a percentage of two integers. The
percentage of two integers is definitely the variety on top divided up from the number on the bottom. If two integers are two and five, this would be an integer, for example. However, there are also
many floating point numbers, such as pi, which cannot be expressed as a fraction.
They could be made in to a small percentage
A realistic amount carries a denominator and numerator that are not zero. Because of this they can be depicted like a small percentage. In addition to their integer numerators and denominators,
rational amounts can also have a unfavorable worth. The bad worth needs to be located left of and its absolute worth is its range from absolutely no. To make simpler this instance, we are going to
claim that .0333333 can be a small percentage that may be created as being a 1/3.
As well as bad integers, a logical quantity can also be produced into a small fraction. For example, /18,572 can be a rational quantity, while -1/ is not really. Any portion comprised of integers is
rational, given that the denominator will not have a and might be published as being an integer. Likewise, a decimal that ends in a position is also a logical quantity.
They are feeling
Even with their label, reasonable numbers don’t make a lot feeling. In mathematics, they can be single organizations having a special duration about the quantity series. Consequently once we matter
anything, we can buy the shape by its rate to its initial amount. This keeps correct even if there are limitless rational numbers between two distinct figures. If they are ordered, in other words,
numbers should make sense only. So, if you’re counting the length of an ant’s tail, a square root of pi is an integer.
If we want to know the length of a string of pearls, we can use a rational number, in real life. To discover the period of a pearl, by way of example, we could count up its size. A single pearl
weighs twenty kilograms, which is a logical variety. Furthermore, a pound’s excess weight means ten kilos. As a result, we must be able to separate a lb by 15, with out be worried about the length of
an individual pearl.
They may be depicted as a decimal
You’ve most likely seen a problem that involves a repeated fraction if you’ve ever tried to convert a number to its decimal form. A decimal amount may be composed as being a several of two integers,
so four times 5 is equal to seven. A similar dilemma necessitates the recurring fraction 2/1, and both sides needs to be split by 99 to obtain the proper answer. But how would you create the
transformation? Here are some illustrations.
A rational variety will also be printed in various forms, such as fractions along with a decimal. A good way to represent a reasonable variety inside a decimal is usually to divide it into its
fractional comparable. There are actually 3 ways to separate a rational variety, and every one of these techniques produces its decimal counterpart. One of these ways is to break down it into its
fractional equivalent, and that’s what’s referred to as a terminating decimal.
Gallery of Irrational Vs Rational Numbers Worksheet Pdf
Rational And Irrational Numbers Differences Examples
Rational Vs Irrational Numbers Worksheets
Pin By Keala Pane e On Math In 2020 Irrational Numbers Number
Leave a Comment | {"url":"https://numbersworksheet.com/irrational-vs-rational-numbers-worksheet-pdf/","timestamp":"2024-11-03T05:58:19Z","content_type":"text/html","content_length":"54205","record_id":"<urn:uuid:ce751a9a-58e7-4e4e-95d3-efdd8b45c3a5>","cc-path":"CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00632.warc.gz"} |
Boundary conditions · Oceananigans.jl
Boundary conditions are intimately related to the grid topology, and only need to be considered in directions with Bounded topology or across immersed boundaries. In Bounded directions, tracer and
momentum fluxes are conservative or "zero flux" by default. Non-default boundary conditions are therefore required to specify non-zero fluxes of tracers and momentum across Bounded directions, and
across immersed boundaries when using ImmersedBoundaryGrid.
See Numerical implementation of boundary conditions for more details.
julia> using Oceananigans
julia> grid = RectilinearGrid(size=(16, 16, 16), x=(0, 2π), y=(0, 1), z=(0, 1), topology=(Periodic, Bounded, Bounded))
16×16×16 RectilinearGrid{Float64, Periodic, Bounded, Bounded} on CPU with 3×3×3 halo
├── Periodic x ∈ [0.0, 6.28319) regularly spaced with Δx=0.392699
├── Bounded y ∈ [0.0, 1.0] regularly spaced with Δy=0.0625
└── Bounded z ∈ [0.0, 1.0] regularly spaced with Δz=0.0625
julia> no_slip_bc = ValueBoundaryCondition(0.0)
ValueBoundaryCondition: 0.0
A "no-slip" BoundaryCondition specifies that velocity components tangential to Bounded directions decay to 0 at the boundary, leading to a viscous loss of momentum.
julia> no_slip_field_bcs = FieldBoundaryConditions(no_slip_bc);
julia> model = NonhydrostaticModel(; grid, boundary_conditions=(u=no_slip_field_bcs, v=no_slip_field_bcs, w=no_slip_field_bcs))
NonhydrostaticModel{CPU, RectilinearGrid}(time = 0 seconds, iteration = 0)
├── grid: 16×16×16 RectilinearGrid{Float64, Periodic, Bounded, Bounded} on CPU with 3×3×3 halo
├── timestepper: RungeKutta3TimeStepper
├── advection scheme: Centered reconstruction order 2
├── tracers: ()
├── closure: Nothing
├── buoyancy: Nothing
└── coriolis: Nothing
julia> model.velocities.u.boundary_conditions
Oceananigans.FieldBoundaryConditions, with boundary conditions
├── west: PeriodicBoundaryCondition
├── east: PeriodicBoundaryCondition
├── south: ValueBoundaryCondition: 0.0
├── north: ValueBoundaryCondition: 0.0
├── bottom: ValueBoundaryCondition: 0.0
├── top: ValueBoundaryCondition: 0.0
└── immersed: FluxBoundaryCondition: Nothing
Boundary conditions are passed to FieldBoundaryCondition to build boundary conditions for each field individually, and then onto the model constructor (here NonhydrotaticModel) via the keyword
argument boundary_conditions. The model constructor then "interprets" the input and builds appropriate boundary conditions for the grid topology, given the user-specified no_slip default boundary
condition for Bounded directions. In the above example, note that the west and east boundary conditions are PeriodicBoundaryCondition because the x-topology of the grid is Periodic.
To specify no-slip boundary conditions on every Bounded direction except the surface, we write
julia> free_slip_surface_bcs = FieldBoundaryConditions(no_slip_bc, top=FluxBoundaryCondition(nothing));
julia> model = NonhydrostaticModel(; grid, boundary_conditions=(u=free_slip_surface_bcs, v=free_slip_surface_bcs, w=no_slip_field_bcs));
julia> model.velocities.u.boundary_conditions
Oceananigans.FieldBoundaryConditions, with boundary conditions
├── west: PeriodicBoundaryCondition
├── east: PeriodicBoundaryCondition
├── south: ValueBoundaryCondition: 0.0
├── north: ValueBoundaryCondition: 0.0
├── bottom: ValueBoundaryCondition: 0.0
├── top: FluxBoundaryCondition: Nothing
└── immersed: FluxBoundaryCondition: Nothing
julia> model.velocities.v.boundary_conditions
Oceananigans.FieldBoundaryConditions, with boundary conditions
├── west: PeriodicBoundaryCondition
├── east: PeriodicBoundaryCondition
├── south: OpenBoundaryCondition{Nothing}: Nothing
├── north: OpenBoundaryCondition{Nothing}: Nothing
├── bottom: ValueBoundaryCondition: 0.0
├── top: FluxBoundaryCondition: Nothing
└── immersed: FluxBoundaryCondition: Nothing
Now both u and v have FluxBoundaryCondition(nothing) at the top boundary, which is Oceananigans lingo for "no-flux boundary condition".
There are three primary boundary condition classifications:
1. FluxBoundaryCondition specifies fluxes directly.
Some applications of FluxBoundaryCondition are:
□ surface momentum fluxes due to wind, or "wind stress";
□ linear or quadratic bottom drag;
□ surface temperature fluxes due to heating or cooling;
□ surface salinity fluxes due to precipitation and evaporation;
□ relaxation boundary conditions that restores a field to some boundary distribution over a given time-scale.
2. ValueBoundaryCondition (Dirichlet) specifies the value of a field on the given boundary, which when used in combination with a turbulence closure results in a flux across the boundary.
Note: Do not use ValueBoundaryCondition on a wall-normal velocity component (see the note below about ImpenetrableBoundaryCondition).
Some applications of ValueBoundaryCondition are:
□ no-slip boundary condition for wall-tangential velocity components via ValueBoundaryCondition(0);
□ surface temperature distribution, where heat fluxes in and out of the domain at a rate controlled by the near-surface temperature gradient and the temperature diffusivity;
□ constant velocity tangential to a boundary as in a driven-cavity flow (for example), where the top boundary is moving. Momentum will flux into the domain do the difference between the top
boundary velocity and the interior velocity, and the prescribed viscosity.
3. GradientBoundaryCondition (Neumann) specifies the gradient of a field on a boundary. For example, if there is a known diffusivity, we can express FluxBoundaryCondition(flux) using
GradientBoundaryCondition(-flux / diffusivity) (aka "Neumann" boundary condition).
In addition to these primary boundary conditions, ImpenetrableBoundaryCondition applies to velocity components in wall-normal directions.
ImpenetrableBoundaryCondition is internally enforced for fields created inside the model constructor. As a result, ImpenetrableBoundaryCondition is only used for additional velocity components that
are not evolved by a model, such as a velocity component used for (AdvectiveForcing)[@ref].
Finally, note that Periodic boundary conditions are internally enforced for Periodic directions, and DefaultBoundaryConditions may exist before boundary conditions are "materialized" by a model.
The default boundary condition in Bounded directions is no-flux, or FluxBoundaryCondition(nothing). The default boundary condition can be changed by passing a positional argument to
FieldBoundaryConditions, as in
julia> no_slip_bc = ValueBoundaryCondition(0.0)
ValueBoundaryCondition: 0.0
julia> free_slip_surface_bcs = FieldBoundaryConditions(no_slip_bc, top=FluxBoundaryCondition(nothing))
Oceananigans.FieldBoundaryConditions, with boundary conditions
├── west: DefaultBoundaryCondition (ValueBoundaryCondition: 0.0)
├── east: DefaultBoundaryCondition (ValueBoundaryCondition: 0.0)
├── south: DefaultBoundaryCondition (ValueBoundaryCondition: 0.0)
├── north: DefaultBoundaryCondition (ValueBoundaryCondition: 0.0)
├── bottom: DefaultBoundaryCondition (ValueBoundaryCondition: 0.0)
├── top: FluxBoundaryCondition: Nothing
└── immersed: DefaultBoundaryCondition (ValueBoundaryCondition: 0.0)
Oceananigans uses a hierarchical structure to express boundary conditions:
1. Each boundary of each field has one BoundaryCondition
2. Each field has seven BoundaryCondition (west, east, south, north, bottom, top and immersed)
3. A set of FieldBoundaryConditions, up to one for each field, are grouped into a NamedTuple and passed to the model constructor.
Boundary conditions are defined at model construction time by passing a NamedTuple of FieldBoundaryConditions specifying non-default boundary conditions for fields such as velocities and tracers.
Fields for which boundary conditions are not specified are assigned a default boundary conditions.
A few illustrations are provided below. See the examples for further illustrations of boundary condition specification.
Boundary conditions may be specified with constants, functions, or arrays. In this section we illustrate usage of the different BoundaryCondition constructors.
julia> constant_T_bc = ValueBoundaryCondition(20.0)
ValueBoundaryCondition: 20.0
A constant Value boundary condition can be used to specify constant tracer (such as temperature), or a constant tangential velocity component at a boundary. Note that boundary conditions on the
normal velocity component must use the Open boundary condition type.
Finally, note that ValueBoundaryCondition(condition) is an alias for BoundaryCondition(Value, condition).
julia> ρ₀ = 1027; # Reference density [kg/m³]
julia> τₓ = 0.08; # Wind stress [N/m²]
julia> wind_stress_bc = FluxBoundaryCondition(-τₓ/ρ₀)
FluxBoundaryCondition: -7.78968e-5
A constant Flux boundary condition can be imposed on tracers and tangential velocity components that can be used, for example, to specify cooling, heating, evaporation, or wind stress at the ocean
The flux convention in Oceananigans
Oceananigans uses the convention that positive fluxes produce transport in the positive direction (east, north, and up for $x$, $y$, $z$). This means, for example, that a negative flux of momentum or
velocity at a top boundary, such as in the above example, produces currents in the positive direction, because it prescribes a downwards flux of momentum into the domain from the top. Likewise, a
positive temperature flux at the top boundary causes cooling, because it transports heat upwards, out of the domain. Conversely, a positive flux at a bottom boundary acts to increase the interior
values of a quantity.
Boundary conditions may be specified by functions,
julia> @inline surface_flux(x, y, t) = cos(2π * x) * cos(t);
julia> top_tracer_bc = FluxBoundaryCondition(surface_flux)
FluxBoundaryCondition: ContinuousBoundaryFunction surface_flux at (Nothing, Nothing, Nothing)
Boundary condition functions
By default, a function boundary condition is called with the signature
f(ξ, η, t)
where t is time and ξ, η are spatial coordinates that vary along the boundary:
• f(y, z, t) on x-boundaries;
• f(x, z, t) on y-boundaries;
• f(x, y, t) on z-boundaries.
Alternative function signatures are specified by keyword arguments to BoundaryCondition, as illustrated in subsequent examples.
Boundary condition functions may be 'parameterized',
julia> @inline wind_stress(x, y, t, p) = - p.τ * cos(p.k * x) * cos(p.ω * t); # function with parameters
julia> top_u_bc = FluxBoundaryCondition(wind_stress, parameters=(k=4π, ω=3.0, τ=1e-4))
FluxBoundaryCondition: ContinuousBoundaryFunction wind_stress at (Nothing, Nothing, Nothing)
Boundary condition functions with parameters
The keyword argument parameters above specifies that wind_stress is called with the signature wind_stress(x, y, t, parameters). In principle, parameters is arbitrary. However, relatively simple
objects such as floating point numbers or NamedTuples must be used when running on the GPU.
Boundary conditions may also depend on model fields. For example, a linear drag boundary condition is implemented with
julia> @inline linear_drag(x, y, t, u) = - 0.2 * u
linear_drag (generic function with 1 method)
julia> u_bottom_bc = FluxBoundaryCondition(linear_drag, field_dependencies=:u)
FluxBoundaryCondition: ContinuousBoundaryFunction linear_drag at (Nothing, Nothing, Nothing)
field_dependencies specifies the name of the dependent fields either with a Symbol or Tuple of Symbols.
When boundary conditions depends on fields and parameters, their functions take the form
julia> @inline quadratic_drag(x, y, t, u, v, drag_coeff) = - drag_coeff * u * sqrt(u^2 + v^2)
quadratic_drag (generic function with 1 method)
julia> u_bottom_bc = FluxBoundaryCondition(quadratic_drag, field_dependencies=(:u, :v), parameters=1e-3)
FluxBoundaryCondition: ContinuousBoundaryFunction quadratic_drag at (Nothing, Nothing, Nothing)
Put differently, ξ, η, t come first in the function signature, followed by field dependencies, followed by parameters is !isnothing(parameters).
Discrete field data may also be accessed directly from boundary condition functions using the discrete_form. For example:
@inline filtered_drag(i, j, grid, clock, model_fields) =
@inbounds - 0.05 * (model_fields.u[i-1, j, 1] + 2 * model_fields.u[i, j, 1] + model_fields.u[i-1, j, 1])
u_bottom_bc = FluxBoundaryCondition(filtered_drag, discrete_form=true)
# output
FluxBoundaryCondition: DiscreteBoundaryFunction with filtered_drag
The 'discrete form' for boundary condition functions
The argument discrete_form=true indicates to BoundaryCondition that filtered_drag uses the 'discrete form'. Boundary condition functions that use the 'discrete form' are called with the signature
f(i, j, grid, clock, model_fields)
where i, j are grid indices that vary along the boundary, grid is model.grid, clock is the model.clock, and model_fields is a NamedTuple containing u, v, w and the fields in model.tracers. The
signature is similar for $x$ and $y$ boundary conditions expect that i, j is replaced with j, k and i, k respectively.
julia> Cd = 0.2; # drag coefficient
julia> @inline linear_drag(i, j, grid, clock, model_fields, Cd) = @inbounds - Cd * model_fields.u[i, j, 1];
julia> u_bottom_bc = FluxBoundaryCondition(linear_drag, discrete_form=true, parameters=Cd)
FluxBoundaryCondition: DiscreteBoundaryFunction linear_drag with parameters 0.2
Inlining and avoiding bounds-checking in boundary condition functions
Boundary condition functions should be decorated with @inline when running on CPUs for performance reasons. On the GPU, all functions are force-inlined by default. In addition, the annotation
@inbounds should be used when accessing the elements of an array in a boundary condition function (such as model_fields.u[i, j, 1] in the above example). Using @inbounds will avoid a relatively
expensive check that the index i, j, 1 is 'in bounds'.
julia> Nx = Ny = 16; # Number of grid points.
julia> Q = randn(Nx, Ny); # temperature flux
julia> white_noise_T_bc = FluxBoundaryCondition(Q)
FluxBoundaryCondition: 16×16 Matrix{Float64}
When running on the GPU, Q must be converted to a CuArray.
To create a set of FieldBoundaryConditions for a temperature field, we write
julia> T_bcs = FieldBoundaryConditions(top = ValueBoundaryCondition(20.0),
bottom = GradientBoundaryCondition(0.01))
Oceananigans.FieldBoundaryConditions, with boundary conditions
├── west: DefaultBoundaryCondition (FluxBoundaryCondition: Nothing)
├── east: DefaultBoundaryCondition (FluxBoundaryCondition: Nothing)
├── south: DefaultBoundaryCondition (FluxBoundaryCondition: Nothing)
├── north: DefaultBoundaryCondition (FluxBoundaryCondition: Nothing)
├── bottom: GradientBoundaryCondition: 0.01
├── top: ValueBoundaryCondition: 20.0
└── immersed: DefaultBoundaryCondition (FluxBoundaryCondition: Nothing)
If the grid is, e.g., horizontally-periodic, then each horizontal DefaultBoundaryCondition is converted to PeriodicBoundaryCondition inside the model's constructor, before assigning the boundary
conditions to temperature T.
In general, boundary condition defaults are inferred from the field location and topology(grid).
To specify non-default boundary conditions, a named tuple of FieldBoundaryConditions objects is passed to the keyword argument boundary_conditions in the NonhydrostaticModel constructor. The keys of
boundary_conditions indicate the field to which the boundary condition is applied. Below, non-default boundary conditions are imposed on the $u$-velocity and tracer $c$.
julia> topology = (Periodic, Periodic, Bounded);
julia> grid = RectilinearGrid(size=(16, 16, 16), extent=(1, 1, 1), topology=topology);
julia> u_bcs = FieldBoundaryConditions(top = ValueBoundaryCondition(+0.1),
bottom = ValueBoundaryCondition(-0.1));
julia> c_bcs = FieldBoundaryConditions(top = ValueBoundaryCondition(20.0),
bottom = GradientBoundaryCondition(0.01));
julia> model = NonhydrostaticModel(grid=grid, boundary_conditions=(u=u_bcs, c=c_bcs), tracers=:c)
NonhydrostaticModel{CPU, RectilinearGrid}(time = 0 seconds, iteration = 0)
├── grid: 16×16×16 RectilinearGrid{Float64, Periodic, Periodic, Bounded} on CPU with 3×3×3 halo
├── timestepper: RungeKutta3TimeStepper
├── advection scheme: Centered reconstruction order 2
├── tracers: c
├── closure: Nothing
├── buoyancy: Nothing
└── coriolis: Nothing
julia> model.velocities.u
16×16×16 Field{Face, Center, Center} on RectilinearGrid on CPU
├── grid: 16×16×16 RectilinearGrid{Float64, Periodic, Periodic, Bounded} on CPU with 3×3×3 halo
├── boundary conditions: FieldBoundaryConditions
│ └── west: Periodic, east: Periodic, south: Periodic, north: Periodic, bottom: Value, top: Value, immersed: ZeroFlux
└── data: 22×22×22 OffsetArray(::Array{Float64, 3}, -2:19, -2:19, -2:19) with eltype Float64 with indices -2:19×-2:19×-2:19
└── max=0.0, min=0.0, mean=0.0
julia> model.tracers.c
16×16×16 Field{Center, Center, Center} on RectilinearGrid on CPU
├── grid: 16×16×16 RectilinearGrid{Float64, Periodic, Periodic, Bounded} on CPU with 3×3×3 halo
├── boundary conditions: FieldBoundaryConditions
│ └── west: Periodic, east: Periodic, south: Periodic, north: Periodic, bottom: Gradient, top: Value, immersed: ZeroFlux
└── data: 22×22×22 OffsetArray(::Array{Float64, 3}, -2:19, -2:19, -2:19) with eltype Float64 with indices -2:19×-2:19×-2:19
└── max=0.0, min=0.0, mean=0.0
Notice that the specified non-default boundary conditions have been applied at top and bottom of both model.velocities.u and model.tracers.c.
Immersed boundary conditions are supported experimentally. A no-slip boundary condition is specified with
# Generate a simple ImmersedBoundaryGrid
hill(x, y) = 0.1 + 0.1 * exp(-x^2 - y^2)
underlying_grid = RectilinearGrid(size=(32, 32, 16), x=(-3, 3), y=(-3, 3), z=(0, 1), topology=(Periodic, Periodic, Bounded))
grid = ImmersedBoundaryGrid(underlying_grid, GridFittedBottom(hill))
# Create a no-slip boundary condition for velocity fields.
# Note that the no-slip boundary condition is _only_ applied on immersed boundaries.
velocity_bcs = FieldBoundaryConditions(immersed=ValueBoundaryCondition(0))
model = NonhydrostaticModel(; grid, boundary_conditions=(u=velocity_bcs, v=velocity_bcs, w=velocity_bcs))
# Insepct the boundary condition on the vertical velocity:
# output
├── west: ValueBoundaryCondition: 0.0
├── east: ValueBoundaryCondition: 0.0
├── south: ValueBoundaryCondition: 0.0
├── north: ValueBoundaryCondition: 0.0
├── bottom: Nothing
└── top: Nothing
`NonhydrostaticModel` on `ImmersedBoundaryGrid`
The pressure solver for NonhydrostaticModel is approximate, and is unable to produce a velocity field that is simultaneously divergence-free while also satisfying impenetrability on the immersed
boundary. As a result, simulated dynamics with NonhydrostaticModel can exhibit egregiously unphysical errors and should be interpreted with caution.
An ImmersedBoundaryCondition encapsulates boundary conditions on each potential boundary-facet of a boundary-adjacent cell. Boundary conditions on specific faces of immersed-boundary-adjacent cells
may also be specified by manually building an ImmersedBoundaryCondition:
bottom_drag_bc = ImmersedBoundaryCondition(bottom=ValueBoundaryCondition(0))
# output
├── west: Nothing
├── east: Nothing
├── south: Nothing
├── north: Nothing
├── bottom: ValueBoundaryCondition: 0
└── top: Nothing
The ImmersedBoundaryCondition may then be incorporated into the boundary conditions for a Field by prescribing it to the immersed boundary label,
velocity_bcs = FieldBoundaryConditions(immersed=bottom_drag_bc)
# output
Oceananigans.FieldBoundaryConditions, with boundary conditions
├── west: DefaultBoundaryCondition (FluxBoundaryCondition: Nothing)
├── east: DefaultBoundaryCondition (FluxBoundaryCondition: Nothing)
├── south: DefaultBoundaryCondition (FluxBoundaryCondition: Nothing)
├── north: DefaultBoundaryCondition (FluxBoundaryCondition: Nothing)
├── bottom: DefaultBoundaryCondition (FluxBoundaryCondition: Nothing)
├── top: DefaultBoundaryCondition (FluxBoundaryCondition: Nothing)
└── immersed: ImmersedBoundaryCondition with west=Nothing, east=Nothing, south=Nothing, north=Nothing, bottom=Value, top=Nothing
ImmersedBoundaryCondition is experimental. Therefore, one should use it only when a finer level of control over the boundary conditions at the immersed boundary is required, and the user is familiar
with the implementation of boundary conditions on staggered grids. For all other cases , using the immersed argument of FieldBoundaryConditions is preferred.
A boundary condition that depends on the fields may be prescribed using the immersed keyword argument in FieldBoundaryConditions. We illustrate field-dependent boundary conditions with an example
that imposes linear bottom drag on u on both the bottom facets of cells adjacent to an immersed boundary, and the bottom boundary of the underlying grid.
First we create the boundary condition for the grid's bottom:
@inline linear_drag(x, y, t, u) = - 0.2 * u
drag_u = FluxBoundaryCondition(linear_drag, field_dependencies=:u)
# output
FluxBoundaryCondition: ContinuousBoundaryFunction linear_drag at (Nothing, Nothing, Nothing)
Next, we create the immersed boundary condition by adding the argument z to linear_drag and imposing drag only on "bottom" facets of cells that neighbor immersed cells:
@inline immersed_linear_drag(x, y, z, t, u) = - 0.2 * u
immersed_drag_u = FluxBoundaryCondition(immersed_linear_drag, field_dependencies=:u)
u_immersed_bc = ImmersedBoundaryCondition(bottom = immersed_drag_u)
# output
├── west: Nothing
├── east: Nothing
├── south: Nothing
├── north: Nothing
├── bottom: FluxBoundaryCondition: ContinuousBoundaryFunction immersed_linear_drag at (Nothing, Nothing, Nothing)
└── top: Nothing
Finally, we combine the two:
u_bcs = FieldBoundaryConditions(bottom = drag_u, immersed = u_immersed_bc)
# output
Oceananigans.FieldBoundaryConditions, with boundary conditions
├── west: DefaultBoundaryCondition (FluxBoundaryCondition: Nothing)
├── east: DefaultBoundaryCondition (FluxBoundaryCondition: Nothing)
├── south: DefaultBoundaryCondition (FluxBoundaryCondition: Nothing)
├── north: DefaultBoundaryCondition (FluxBoundaryCondition: Nothing)
├── bottom: FluxBoundaryCondition: ContinuousBoundaryFunction linear_drag at (Nothing, Nothing, Nothing)
├── top: DefaultBoundaryCondition (FluxBoundaryCondition: Nothing)
└── immersed: ImmersedBoundaryCondition with west=Nothing, east=Nothing, south=Nothing, north=Nothing, bottom=Flux, top=Nothing
Positional argument requirements
Note the difference between the arguments required for the function within the bottom boundary condition versus the arguments for the function within the immersed boundary condition. E.g., x, y, t in
linear_drag() versus x, y, z, t in immersed_linear_drag(). | {"url":"https://clima.github.io/OceananigansDocumentation/stable/model_setup/boundary_conditions/","timestamp":"2024-11-06T16:42:15Z","content_type":"text/html","content_length":"53102","record_id":"<urn:uuid:bd4c0fff-e17d-4883-8e26-820111f4c3dc>","cc-path":"CC-MAIN-2024-46/segments/1730477027933.5/warc/CC-MAIN-20241106163535-20241106193535-00452.warc.gz"} |
A Fixed-Point Implementation of the Goertzel Algorithm in C
This is a practice project I did at the end of 2020. It's very on theme with the "Remcycles" moniker of this blog, because it's about a single tone detection algorithm, which detects "cycles" in a
signal that occur at a specific frequency.
The Goertzel algorithm is a digital signal processing technique for calculating a single bin of a DFT/FFT. It's useful when you want to detect and measure the strength of a signal at just one or a
few frequencies and don't need to calculate the signal strength at all the frequencies that an FFT would calculate. Possible use cases include decoding DTMF signals and demodulating on-off-keying or
frequency shift keying signals.
I first read about it in this wonderful book by Richard Lyons:
Understanding Digital Signal Processing 3rd Edition
Lyons' book includes the following diagram, redrawn here using Pikchr and included via pikchr-mode (source code):
The structure above is a resonant filter. It's transfer function has a pole and zero that cancel each other, leaving a single pole on the unit circle: \[H(z) = \frac{(1 - e^{-j 2 \pi m/N}z^{-1})}{(1
- e^{+j 2 \pi m/N} z^{-1})(1 - e^{-j 2 \pi m/N} z^{-1})} = \frac{1}{1 - e^{+j 2 \pi m/N}}\]
That's a recipe for instability, but part of the Goertzel algorithm trick is to only run the filter for a fixed number of input samples before computing the final output, and then resetting the
filter's state for the next set of samples.^1
The following time-domain difference equations are the core of the algorithm and show how to calculate new values from previous values: \[w(n) = 2 cos(2 \pi m / N) w(n-1)-w(n-2) +x(n)\] \[y(n) = w(n)
- e^{-j 2 \pi m / N} w(n-1)\]
The \(2 cos(2 \pi m / N)\) and \(e^{-j 2 \pi m / N}\) terms are constants that can be calculated once at run time (or even compile time), and \(y(n)\) only needs to be evaluated once at the end of
each computation, after N+1 iterations of the main loop. Each iteration is only one multiplication and two additions.
I implemented this using floating-point math for practice first, but my goal was to write a fixed-point implementation that can run on microcontrollers or a DSP like the ADSP-BF706 without an FPU.
The two tricky parts of fixed-point programming are keeping the binary points aligned before and after operations and avoiding overflow. Randy Yates wrote two great guides on fixed-point math and
signal processing that I highly recommend:
Fixed-Point Arithmetic: An Introduction
Practical Considerations in Fixed-Point FIR Implementations
There is also this helpful Wikipedia page on the different notations for fixed point numbers:
My full test program used PortAudio for input and I used my system's audio mixer to route audio to it. The signal source was a simple direct digital synthesis (DDS) program that I wrote.
Here's a simplified version:
/* goertzel_fixed -- Detect a single tone in an audio signal.
Copyright (C) 2022 Remington Furman
This program is free software: you can redistribute it and/or modify
it under the terms of the GNU General Public License as published by
the Free Software Foundation, either version 3 of the License, or
(at your option) any later version.
This program is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
GNU General Public License for more details.
You should have received a copy of the GNU General Public License
along with this program. If not, see https://www.gnu.org/licenses/.
#include <stdio.h>
#include <stdlib.h>
#include <stdint.h>
#include <math.h>
#include <complex.h>
/* These macros simplify working with signed fixed point numbers.
In this notation, only the fractional bits are tracked in the macro
names, so a Qm.n number is referred to as a Qn number, where n is
the number of fractional bits. This also sidesteps the issue of
whether m includes the sign bit or not (ARM vs TI notation).
The usual caveats of C preprocessor macros hold here. Beware of
multiple evaluations, side effects, etc.
/* Convert to and from doubles. */
#define Qn_FROM_DOUBLE(value, n) (lrint((value) * (1 << (n))))
#define DOUBLE_FROM_Qn(value, n) ((double)(value) / (1 << (n)))
/* The number closest to +1.0 that can be represented. */
#define ONE_Qn(n) ((1<<(n)) - 1)
/* One half (0.5). */
#define HALF_Qn(n) (1<<((n) - 1))
/* Drop n bits from x (shift right) while rounding (add one half). */
#define ROUND_OFF_Qn(x, n) \
((n > 0) ? (((x) + HALF_Qn(n)) >> n) : (x))
/* Multiply two Qn numbers, rounding to the precision of the first.
Make sure to cast one of the arguments to the size needed to avoid
overflow in the multiplication before shifting. */
#define MUL_Qn_Qn(x, y, xn, yn) \
ROUND_OFF_Qn((x) * (y), (yn))
/* Add two Qn numbers, using the precision of the first. */
#define ADD_Qn_Qn(x, y, xn, yn) \
((xn) > (yn) ? (x) + ((y) << ((xn)-(yn))) : \
(x) + ROUND_OFF_Qn((y), ((xn)-(yn))))
typedef struct {
int16_t real;
int16_t imag;
} cint16_t;
/* Return a larger type here, because a complex point outside of the
unit circle will have a larger magnitude. */
int32_t cint16_abs(cint16_t z) {
/* Cheat for now and use floating point sqrt(). */
return lrint(sqrt((double)z.real*(double)z.real +
int16_t read_sample(void) {
/* This function should read and return an audio sample from some
source. */
return 0;
int32_t goertzel(double detect_hz, double sample_rate_hz, int N) {
/* Notation from p. 710 of Lyons. */
/* Index of DFT frequency bin to calculate. */
const double m = (N * detect_hz) / sample_rate_hz;
/* This complex feedforward coefficient allows a single zero to
cancel one of the complex poles. It can be calculated in
advance. */
const double complex dbl_coeff_ff = -cexp(-I*2*M_PI*m/N);
const int coeff_ff_Qn = 15; /* Q1.15 */
cint16_t coeff_ff;
coeff_ff.real = Qn_FROM_DOUBLE(creal(dbl_coeff_ff), coeff_ff_Qn);
coeff_ff.imag = Qn_FROM_DOUBLE(cimag(dbl_coeff_ff), coeff_ff_Qn);
/* Feedback coefficient. */
double dbl_coeff_fb = 2*cos(2*M_PI*m/N);
const int coeff_fb_Qn = 14; /* Q2.14 */
int16_t coeff_fb = Qn_FROM_DOUBLE(dbl_coeff_fb, coeff_fb_Qn);
const int w_Qn = 15;
int32_t w[3] = {0}; /* Delay line. Q17.15. */
int sample_index = 0;
/* Read and process N+1 samples. */
while (1) {
const int x_Qn = 15; /* Q1.15. */
int16_t x = read_sample();
if (sample_index == N) {
/* The final x input sample should be forced to zero. */
x = 0;
/* Manually shift delay line and calculate next value. */
w[2] = w[1];
w[1] = w[0];
/* w[0] = x + (coeff_fb * w[1]) - w[2] */
w[0] = MUL_Qn_Qn((int64_t)w[1], (int64_t)coeff_fb, w_Qn, coeff_fb_Qn);
w[0] = ADD_Qn_Qn(w[0], x, w_Qn, x_Qn);
w[0] = ADD_Qn_Qn(w[0], -w[2], w_Qn, w_Qn);
if (sample_index++ == N) {
/* End of Goertzel alogorithm for this buffer. Apply the
* feedforward coefficient to generate final output. */
const int y_Qn = 5;
cint16_t y; /* y = w[0] + coeff_ff * w[1]; complex multiply. */
y.real = ROUND_OFF_Qn(w[0] +
MUL_Qn_Qn((int64_t)coeff_ff.real, w[1],
coeff_ff_Qn, w_Qn), w_Qn - y_Qn);
y.imag = ROUND_OFF_Qn(
MUL_Qn_Qn((int64_t)coeff_ff.imag, w[1],
coeff_ff_Qn, w_Qn), w_Qn - y_Qn);
int32_t dft_mag = cint16_abs(y);
return dft_mag;
int main(void) {
/* Input sample rate. */
const double sample_rate_hz = 48000.0;
/* Frequency of DFT bin to calculate. */
const double detect_hz = 440.0;
/* Number of samples for detection. Need not be a power of 2. */
const int N = 1024;
while(1) {
int32_t dft_mag = goertzel(detect_hz, sample_rate_hz, N);
/* Normalize the DFT bin value by N. A maximum amplitude
signal at the detection frequency should give a value
of 1/2. */
double dft_mag_norm = dft_mag/N;
printf("dft_mag: %u, dft_mag_norm: %g\n", dft_mag, dft_mag_norm);
The following plots show the internal signals during each sample, with the final filter state on the right of each plot.
The yellow \(|y|\) trace is the output, the magnitude of the DFT bin, though it's poorly scaled to see relative to the other signals. The purple \(x\) trace shows the input signal scaled up to be
more visible. The other traces are the internal filter state \(w\) values.
The filter is configured to detect a 440Hz tone, and the input signals are 440Hz, 500Hz, and 880Hz.
Figure 1: Goertzel state with 440Hz input
At 440Hz the input frequency matches the detection frequency and you can see that the internal filter state grows very quickly. It would continue to grow if the filter was run forever. The \(|y|\)
trace steadily increases (again, hard to see with this scale).
Figure 2: Goertzel state with 500Hz input
At 500Hz input the filter state's magnitude oscillates, but never grows to a large value.
Figure 3: Goertzel state with 880Hz input
At 880Hz it also oscillates, but at a much smaller magnitude. | {"url":"https://remcycles.net/blog/goertzel.html","timestamp":"2024-11-13T12:48:14Z","content_type":"application/xhtml+xml","content_length":"24721","record_id":"<urn:uuid:29f2cfae-a562-43ba-a438-5d7e74f7744e>","cc-path":"CC-MAIN-2024-46/segments/1730477028347.28/warc/CC-MAIN-20241113103539-20241113133539-00527.warc.gz"} |
Line Multipllication Charts Worksheet Year 3 Word Problems 2024 - Multiplication Chart Printable
Line Multipllication Charts Worksheet Year 3 Word Problems
Line Multipllication Charts Worksheet Year 3 Word Problems – The Multiplication Graph or chart Range may help your students aesthetically symbolize different very early math concepts methods.
However, it must be used as a teaching aid only and should not be confused with the Multiplication Table. The graph or chart will come in three variations: the coloured model is useful as soon as
your student is paying attention on one occasions desk at any given time. The horizontal and vertical models are suitable for children who definitely are nonetheless understanding their occasions
tables. In addition to the colored version, you can also purchase a blank multiplication chart if you prefer. Line Multipllication Charts Worksheet Year 3 Word Problems.
Multiples of 4 are 4 away from the other person
The pattern for determining multiples of 4 is always to add each and every number to on its own and look for its other numerous. For instance, the 1st 5 multiples of 4 are: 12, 16, 8 and 4 and 20.
This trick works because all multiples of a number are even, and they are four away from each other on the multiplication chart line. Moreover, multiples of a number of are even numbers in nature.
Multiples of 5 are even
If they end in or 5, You’ll find multiples of 5 on the multiplication chart line only. Quite simply, you can’t increase a variety by a couple of to have a much amount. If the number ends in five or ,
you can only find a multiple of five! Fortunately, there are actually tricks that will make getting multiples of 5 even easier, like utilizing the multiplication chart line to discover the numerous
of 5.
Multiples of 8 are 8 away from the other person
The design is apparent: all multiples of 8 are two-digit phone numbers and all multiples of 4-digit phone numbers are two-digit phone numbers. Each array of 10 includes a numerous of 8-10. 8 is even,
so that all its multiples are two-digit amounts. Its pattern continues around 119. The very next time you see a quantity, be sure you look for a numerous of eight to start with.
Multiples of 12 are 12 from one another
The telephone number a dozen has infinite multiples, and you may flourish any entire amount by it to produce any quantity, including by itself. All multiples of a dozen are even numbers. Here are a
few good examples. David likes to purchase pencils and organizes them into eight packets of twelve. He presently has 96 pencils. James has one among each type of pen. In his place of work, he
arranges them on the multiplication graph line.
Multiples of 20 are 20 clear of one another
Within the multiplication chart, multiples of fifteen are typical even. If you multiply one by another, then the multiple will be also even. Multiply both numbers by each other to find the factor if
you have more than one factor. For example, if Oliver has 2000 notebooks, then he can group them equally. The identical is applicable to erasers and pencils. You could buy one in a package of 3 or a
pack of six.
Multiples of 30 are 30 far from each other
In multiplication, the phrase “aspect match” identifies a small group of numbers that kind a definite quantity. For example, if the number ’30’ is written as a product of five and six, that number is
30 away from each other on a multiplication chart line. This is also true for a number inside the collection ‘1’ to ’10’. Quite simply, any variety can be composed since the item of 1 and on its own.
Multiples of 40 are 40 away from each other
Do you know how to find them, though you may know that there are multiples of 40 on a multiplication chart line? To accomplish this, you can add externally-in. For example, 10 12 14 = 40, and so on.
Similarly, 10 8 = 20. In this instance, the quantity in the left of 10 is surely an even amount, while the one about the proper is an odd number.
Multiples of 50 are 50 clear of each other
Using the multiplication graph series to determine the amount of two phone numbers, multiples of 50 are identical range apart on the multiplication graph or chart. They already have two best 80, 50
and factors. In most cases, each and every expression can vary by 50. Other component is 50 itself. Listed below are the normal multiples of 50. A typical multiple is the numerous of any presented
amount by 50.
Multiples of 100 are 100 clear of one another
Listed below are the various amounts that are multiples of 100. A confident combine is actually a several of a single hundred or so, although a poor set can be a several of 15. These 2 types of
amounts are very different in a number of methods. The initial strategy is to break down the quantity by subsequent integers. In this instance, the quantity of multiples is just one, ten, thirty and
twenty and forty.
Gallery of Line Multipllication Charts Worksheet Year 3 Word Problems
Multiplication Word Problem Worksheets 3rd Grade
3rd Grade Multiplication Worksheets Pdf Tutorial Worksheet
Problem Solving Year 3 Problem Solving Georgiajudges 2019 01 26
Leave a Comment | {"url":"https://www.multiplicationchartprintable.com/line-multipllication-charts-worksheet-year-3-word-problems/","timestamp":"2024-11-06T10:58:22Z","content_type":"text/html","content_length":"53192","record_id":"<urn:uuid:6231cf38-cbcf-4487-8d6d-82a33853a944>","cc-path":"CC-MAIN-2024-46/segments/1730477027928.77/warc/CC-MAIN-20241106100950-20241106130950-00795.warc.gz"} |
Calculate viscosities of liquids
gmx tcaf computes tranverse current autocorrelations. These are used to estimate the shear viscosity, eta. For details see: Palmer, Phys. Rev. E 49 (1994) pp 359-366.
Transverse currents are calculated using the k-vectors (1,0,0) and (2,0,0) each also in the y- and z-direction, (1,1,0) and (1,-1,0) each also in the 2 other planes (these vectors are not
independent) and (1,1,1) and the 3 other box diagonals (also not independent). For each k-vector the sine and cosine are used, in combination with the velocity in 2 perpendicular directions. This
gives a total of 16*2*2=64 transverse currents. One autocorrelation is calculated fitted for each k-vector, which gives 16 TCAFs. Each of these TCAFs is fitted to f(t) = exp(-v)(cosh(Wv) + 1/W sinh
(Wv)), v = -t/(2 tau), W = sqrt(1 - 4 tau eta/rho k^2), which gives 16 values of tau and eta. The fit weights decay exponentially with time constant w (given with -wt) as exp(-t/w), and the TCAF and
fit are calculated up to time 5*w. The eta values should be fitted to 1 - a eta(k) k^2, from which one can estimate the shear viscosity at k=0.
When the box is cubic, one can use the option -oc, which averages the TCAFs over all k-vectors with the same length. This results in more accurate TCAFs. Both the cubic TCAFs and fits are written to
-oc The cubic eta estimates are also written to -ov.
With option -mol, the transverse current is determined of molecules instead of atoms. In this case, the index group should consist of molecule numbers instead of atom numbers.
The k-dependent viscosities in the -ov file should be fitted to eta(k) = eta_0 (1 - a k^2) to obtain the viscosity at infinite wavelength.
Note: make sure you write coordinates and velocities often enough. The initial, non-exponential, part of the autocorrelation function is very important for obtaining a good fit.
Options to specify input files:
-f [<.trr/.cpt/...>] (traj.trr)
Full precision trajectory: trr cpt tng
-s [<.tpr/.gro/...>] (topol.tpr) (Optional)
Structure+mass(db): tpr gro g96 pdb brk ent
-n [<.ndx>] (index.ndx) (Optional)
Index file
Options to specify output files:
xvgr/xmgr file
xvgr/xmgr file
xvgr/xmgr file
xvgr/xmgr file
xvgr/xmgr file
xvgr/xmgr file
Other options:
Time of first frame to read from trajectory (default unit ps)
Time of last frame to read from trajectory (default unit ps)
Only use frame when t MOD dt = first time (default unit ps)
View output .xvg, .xpm, .eps and .pdb files
xvg plot formatting: xmgrace, xmgr, none
Calculate TCAF of molecules
Also use k=(3,0,0) and k=(4,0,0)
Exponential decay time for the TCAF fit weights
Length of the ACF, default is half the number of frames
Normalize ACF
Order of Legendre polynomial for ACF (0 indicates none): 0, 1, 2, 3
Fit function: none, exp, aexp, exp_exp, exp5, exp7, exp9
Time where to begin the exponential fit of the correlation function
Time where to end the exponential fit of the correlation function, -1 is until the end | {"url":"https://www.mankier.com/1/gmx-tcaf","timestamp":"2024-11-12T17:34:58Z","content_type":"text/html","content_length":"12094","record_id":"<urn:uuid:9c5d6b64-b696-4301-822c-5ab5cab2d9bd>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.63/warc/CC-MAIN-20241112145015-20241112175015-00851.warc.gz"} |
Using ANN for thermal neutron shield designing for BNCT treatment room
Monte Carlo and ANN
Monte Carlo is known as a mathematical technique that employs statistical sampling for numerical experiments using the computer to estimate outcomes for uncertain events in which the deterministic
methods can not give reliable predictions. This method works based on random sampling and is the best way known to explore the behavior of complex systems and geometries with multiple degrees of
freedom, such as the transport of the particles in a medium. By repeatedly performing Monte Carlo simulations, many probable outcomes will generate, which become more accurate as the number of inputs
grows. Finally, this method offers a clear picture including the results and the corresponding uncertainties. The necessity of performing a complete and trustworthy Monte Carlo computational model is
to be used for planning the experimental work and studying possible additional optimization and improvements of the facility. Despite all these advantages, Monte Carlo is time-consuming as there is a
need to generate a large number of sampling to generate reliable output, particularly for large and multi-dimensional problems. This is also the case with our problem of optimizing the thermal
neutron shield for the entrance door of the treatment room. This is because the problem involves a large volume of matter through which particles must travel, leading to an increased probability of
particle loss and complex interactions between particles and the medium. In addition, the problem requires the optimization of multiple parameters, such as the material composition and thickness of
the shield, which are coupled and can influence each other’s effectiveness.
In recent years, the ANN tool has found widespread applications in nuclear engineering to predict the behavior of the system by assigning a model to the input data. This involves training the network
using a large amount of data to learn the patterns and find relationships between the input data and the corresponding output. Once trained, the ANN can be used to make predictions or classify new
data based on its learned knowledge. Considering the advantage of using ANN for predicting the results on one hand and the disadvantage of performing Monte Carlo simulations on the other hand, which
involves the difficult and time-consuming task of testing all possible mixtures and thicknesses to find the optimal one, using the data generated by the Monte Carlo simulations as inputs of the ANN
was proposed. It is worth mentioning that ANN is a method of predicting the results from the input data in the case of multiparameter, complicated problems having certain efficiency advantages, and
is inherently different from the Monte Carlo method that uses a broad class of computational algorithms to obtain numerical results. Feeding the ANN with the Monte Carlo data and validating its
generated results with proper Monte Carlo calculations must be finally done.
In this work, the Geant4 Monte Carlo code was employed to perform the simulations. This toolkit is a general-purpose Monte Carlo radiation transport code that is capable of tracking various particle
types including leptons, photons, hadrons and ions in arbitrary three-dimensional configurations of materials and geometries over wide ranges of energies. Also, the important features that make this
code interesting include being easy to use, flexible structures, and an extensive collection of cross-section data. Moreover, Geant4 provides visualization drivers and interfaces, graphical user
interfaces, and a flexible framework for persistency^32. The other interesting aspect of GEANT4 is that it has grown over the years and changes have been made to accommodate the needs of the users so
that can cover a large number of experiments and projects in a variety of application areas. Details can be found in the literature^33. The equipment chosen for simulation was the BSA designed for
BNCT based on the D-T neutron generator yielding \(5 \times 10^12\) neutrons per second, with apertures of 6 cm radius for emission of the neutron spectrum. This equipment was simulated in accordance
with the BSA proposed in our previous work^34 for the treatment of deep tumors with the output flux of \(\sim 10^9\) n.cm\(^-2\)s\(^-1\). This system was placed in the center of the simulated
treatment room based on an existing room in Imam Khomeini Hospital Complex in Tehran. The simulated treatment room featured in Fig. 1 had a square geometry of 11 \(\times \) 11 m\(^2\), with a maze
and entrance door, and a height of 2.5 m. The concrete was considered as the material of both the main walls (those surrounding the BSA exit and the patient bed) and the secondary walls (those behind
the maze). The entrance door, with dimensions of 1.5 \(\times \) 2 m\(^2\), was initially assumed to be made of lead in the primary simulations. The thicknesses of the walls, the beam direction, and
the position of the phantoms for dose evaluation have been depicted in Fig. 1.
Figure 1
The schematic top view of the simulated BNCT treatment room. The room dimensions, the thicknesses of the walls, the output beam direction, and the position of the water phantoms for dose evaluation
have been shown. The phantoms are numbered sequentially from left to right by 1 to 9.
To assess the effectiveness of the designed shield in limiting radiation exposure, the maximum permissible doses based on the widely accepted recommendations were considered. For this purpose, nine
spherical simulated water phantoms with a radius of 15 cm were placed behind the entrance door, as the present study focused on the shielding design for this area. The International Commission on
Radiological Protection (ICRP) and National Council on Radiation Protection and Measurements (NCRP) publish recommendations for occupational dose limits. The NCRP limits generally agree with ICRP
recommendations for dose limits and there are two types of occupational dose limits in these guidelines, including limits for specific organs or tissues and acceptable risk levels for cancer
induction^35,36. According to these standards, the weekly limits of effective dose in controlled and uncontrolled areas are 0.1 mSv/week and 0.02 mSv/week, respectively. Taking these in to account,
the shielding design was planned to ensure that the maximum allowable dose rate behind it (as an uncontrolled area) is less than 0.5 \(\upmu \)Sv h\(^-1\), assuming that the clinic plans to operate
for 40 h per week.
The neutron and photon doses have been calculated by scoring the ambient dose equivalent in the simulated phantoms, defined as a weighted radiation dose that takes the quality factor of the particles
depositing energy in biological matter into account. To this aim, whenever a neutron or a gamma ray traverses the phantom, the fluence spectrum inside the sphere is obtained and the fluence
conversion coefficients are applied. All simulations tracked 5 \(\times \) 10\(^8\) histories, and the statistical errors associated with the results were reported.
Shielding material
The thermal neutron shield proposed in this study has been inspired by the work of Shahram et al.^31, who experimentally designed a polymer composite based on PMMA (polymethyl methacrylate with
chemical formula of C\(_5\)H\(_8\)O\(_2\) and density of 1.1 g cm\(^-3\)) and polyethylene powder (with chemical formula of C\(_2\)H\(_4\) and density of 0.9 g cm\(^-3\)). In these materials,
hydrogen capturing of thermal neutrons is through the \(^1\)H(n, \(\gamma \))\(^2\)H reaction with a cross section of 0.33 barn^37. An in-situ polymerization technique was employed to increase the
composite’s slowing-down feature and the boric acid powder (with chemical formula of BH\(_3\)O\(_3\) and density of 1.44 g cm\(^-3\)) was added to absorb thermal neutrons through \(^10\)B(n, \(\alpha
\))\(^7\)Li reaction. The produced heavy particles can easily stop in the shielding material and, therefore, have not been considered in the dose calculations. In their work, a polyethylene layer was
used as a moderator, followed by a polymer composite layer as an absorber. In the second layer, boron was added at weight fractions of 1%, 5%, 7%, and 10%. In order to evaluate the effectiveness of
the designed shield, various combinations of thicknesses for the two layers and the proportion of boron in the polymer composite were experimentally tested. For these limited samples, the neutron
doses were measured.
Though their study was pioneering in designing thermal neutron shields, the limited number of models tested raises the question of whether there exist other combinations, thicknesses, or weight
fractions that could result in even better shielding properties. To address this question and take advantage of the benefits of ANN as discussed earlier, we tested 720 different models in the present
study. It is necessary because, as the number of inputs increases, the accuracy of forecasts by ANN tends to improve, making it necessary to consider a larger number of inputs. The weight fraction of
polyethylene in the polymer composite, the weight fraction of boric acid in the polymer composite, the thickness of the first layer of the shield (polyethylene), and the thickness of the second layer
(polymer composite) were considered as the parameters with the specific values presented in Table 1. From here, these parameters are labeled by A, B, C, and D, respectively. By using these values,
720 sets of arrangements were generated. In our simulations, each shield has been placed in front of a typical neutron source which has a neutron energy range from a few eV up to 10 MeV, and the dose
and flux beyond the shield were calculated.
Table 1 The values assigned to the four chosen parameters to generate various sets of arrangements.
Artificial neural network
We utilized ANN for predicting both the thermal neutron flux behind the designed shield as well as the dose calculations. For each network, we had a total of 720 data sets, which have been divided
into two parts: training data (690 samples) and testing data (30 samples) to validate the results. Table 2 lists the combination of the four parameters of A to C for the 30 samples that have been
used as the testing data. These sets of parameters are chosen so that, in a good approximation, incorporate all the values in the ranges presented in Table 1.
The neural network used in this study for thermal neutron flux consists of a single hidden layer perceptron with 40 neurons and an output layer. The multilayer perceptron (MLP) network has been
trained using the Levenberg-Marquardt (LM) algorithm which is an appropriate option for solving generic curve-fitting problems. To calculate the output of a node based on its set of specific
individual inputs and their weights, the activation function is needed. In this work, the sigmoid activation function was used for each layer. The input data for the network was a \(4 \times 690\)
matrix which included the neutron flux extracted from the code, and the output data was in the form of a \(1 \times 690\) matrix. For the dosimetric data (a \(4 \times 690\) input matrix), a
two-layer Feedforward Backpropagation neural network with a hidden layer of 30 neurons was used, and trained using the LM algorithm. Also, the sigmoid activation function was used for each layer of
the network. During the training phase of both networks, the neural network was initially trained with training data, and some parameters such as weights and biases have been regularized to prevent
overfitting that occurs if the model cannot generalize and fits too closely to the training dataset. The networks have been trained for 1000 epochs to ensure that they had converged to a stable
solution. Figure 2 shows the architecture of the neural network used in this work.
Figure 2
The proposed ANN architecture.
Table 2 Sets of the four parameters of A (the weight fraction of polyethylene), B (the weight fraction of boric acid), C (the thickness of the polyethylene), and D (the thickness of the composite) as
30 samples that have been used for testing data. | {"url":"https://roomrenovators.co.uk/using-ann-for-thermal-neutron-shield-designing-for-bnct-treatment-room.html","timestamp":"2024-11-03T15:51:36Z","content_type":"text/html","content_length":"72778","record_id":"<urn:uuid:deeb6a45-8474-4d3f-a466-45463dafadc3>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00294.warc.gz"} |
Exam 2020-21 - Get Direct Link to Download Mains Admit Card
What does the equal sign mean?
The equal sign implies an equivalency between the numbers, values, equations, and expressions.
What does this ≅ mean?
The symbol ≅ is used to denote congruent relations between the terms.
What is the approximately equal sign?
The approximately equal sign is ≈.
Name the different types of equal signs?
The different types of equal signs are;Equivalence; represented by the symbol ≡Not equal to; denoted by the symbol ≠Less than equal to; denoted by the symbol ≤Greater than equal to; denoted by the
symbol ≥
How do you write not equal sign?
The not equal sign in mathematics is represented by the symbol ≠. | {"url":"https://testbook.com/maths/equal-sign","timestamp":"2024-11-09T04:01:02Z","content_type":"text/html","content_length":"859427","record_id":"<urn:uuid:e0b37ccb-afc7-4d05-8838-0c392b9e6774>","cc-path":"CC-MAIN-2024-46/segments/1730477028115.85/warc/CC-MAIN-20241109022607-20241109052607-00015.warc.gz"} |
DetNet Bounded Latency
This document is an Internet-Draft (I-D). Anyone may submit an I-D to the IETF. This I-D is
not endorsed by the IETF
and has
no formal standing
in the
IETF standards process
The information below is for an old version of the document.
Document Type This is an older version of an Internet-Draft whose latest revision state is "Replaced".
Authors Norman Finn , Jean-Yves Le Boudec , Balazs Varga , János Farkas
Last updated 2018-03-07
Replaced by draft-ietf-detnet-bounded-latency, RFC 9320
RFC stream (None)
Stream Stream state (No stream defined)
Consensus boilerplate Unknown
RFC Editor Note (None)
IESG IESG state I-D Exists
Telechat date (None)
Responsible AD (None)
Send notices to (None)
DetNet N. Finn
Internet-Draft Huawei Technologies Co. Ltd
Intended status: Standards Track J-Y. Le Boudec
Expires: September 6, 2018 EPFL
B. Varga
J. Farkas
March 5, 2018
DetNet Bounded Latency
This document presents a parameterized timing model for Deterministic
Networking so that existing and future standards can achieve bounded
latency and zero congestion loss.
Status of This Memo
This Internet-Draft is submitted in full conformance with the
provisions of BCP 78 and BCP 79.
Internet-Drafts are working documents of the Internet Engineering
Task Force (IETF). Note that other groups may also distribute
working documents as Internet-Drafts. The list of current Internet-
Drafts is at https://datatracker.ietf.org/drafts/current/.
Internet-Drafts are draft documents valid for a maximum of six months
and may be updated, replaced, or obsoleted by other documents at any
time. It is inappropriate to use Internet-Drafts as reference
material or to cite them other than as "work in progress."
This Internet-Draft will expire on September 6, 2018.
Copyright Notice
Copyright (c) 2018 IETF Trust and the persons identified as the
document authors. All rights reserved.
This document is subject to BCP 78 and the IETF Trust's Legal
Provisions Relating to IETF Documents
(https://trustee.ietf.org/license-info) in effect on the date of
publication of this document. Please review these documents
carefully, as they describe your rights and restrictions with respect
to this document. Code Components extracted from this document must
include Simplified BSD License text as described in Section 4.e of
Finn, et al. Expires September 6, 2018 [Page 1]
Internet-Draft DetNet Bounded Latency March 2018
the Trust Legal Provisions and are provided without warranty as
described in the Simplified BSD License.
Table of Contents
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . 2
2. Conventions Used in This Document . . . . . . . . . . . . . . 3
3. Terminology and Definitions . . . . . . . . . . . . . . . . . 4
4. DetNet bounded latency model . . . . . . . . . . . . . . . . 4
4.1. Flow creation . . . . . . . . . . . . . . . . . . . . . . 4
4.2. End-to-end model . . . . . . . . . . . . . . . . . . . . 5
4.3. Relay system model . . . . . . . . . . . . . . . . . . . 5
5. Computing End-to-end Latency Bounds . . . . . . . . . . . . . 7
5.1. Examples of Computations . . . . . . . . . . . . . . . . 8
6. Achieving zero congestion loss . . . . . . . . . . . . . . . 8
6.1. A General Formula . . . . . . . . . . . . . . . . . . . . 8
7. Queuing model . . . . . . . . . . . . . . . . . . . . . . . . 9
7.1. Queuing data model . . . . . . . . . . . . . . . . . . . 9
7.2. IEEE 802.1 Queuing Model . . . . . . . . . . . . . . . . 11
7.2.1. Queuing Data Model with Preemption . . . . . . . . . 11
7.2.2. Transmission Selection Model . . . . . . . . . . . . 12
7.3. Other queuing models, e.g. IntServ . . . . . . . . . . . 14
8. Parameters for the bounded latency model . . . . . . . . . . 14
8.1. Sender parameters . . . . . . . . . . . . . . . . . . . . 14
8.2. Relay system parameters . . . . . . . . . . . . . . . . . 14
9. References . . . . . . . . . . . . . . . . . . . . . . . . . 15
9.1. Normative References . . . . . . . . . . . . . . . . . . 15
9.2. Informative References . . . . . . . . . . . . . . . . . 15
Authors' Addresses . . . . . . . . . . . . . . . . . . . . . . . 17
1. Introduction
The ability for IETF Deterministic Networking (DetNet) or IEEE 802.1
Time-Sensitive Networking (TSN) to provide the DetNet services of
bounded latency and zero congestion loss depends upon A) configuring
and allocating network resources for the exclusive use of DetNet/TSN
flows; B) identifying, in the data plane, the resources to be
utilized by any given packet, and C) the detailed behavior of those
resources, especially transmission queue selection, so that latency
bounds can be reliably assured. Thus, DetNet is an example of an
INTSERV Guaranteed Quality of Service [RFC2212]
As explained in [I-D.ietf-detnet-architecture], DetNet flows are
characterized by 1) a maximum bandwidth, guaranteed either by the
transmitter or by strict input metering; and 2) a requirement for a
guaranteed worst-case end-to-end latency. That latency guarantee, in
turn, provides the opportunity for the network to supply enough
buffer space to guarantee zero congestion loss. To be of use to the
Finn, et al. Expires September 6, 2018 [Page 2]
Internet-Draft DetNet Bounded Latency March 2018
applications identified in [I-D.ietf-detnet-use-cases], it must be
possible to calculate, before the transmission of a DetNet flow
commences, both the worst-case end-to-end network latency, and the
amount of buffer space required at each hop to ensure against
congestion loss.
Rather than defining, in great detail, specific mechanisms to be used
to control packet transmission at each output port, this document
presents a timing model for sources, destinations, and the network
nodes that relay packets. The parameters specified in this model:
o Characterize a DetNet flow in a way that provides externally
measureable verification that the sender is conforming to its
promised maximum, can be implemented reasonably easily by a
sending device, and does not require excessive over-allocation of
resources by the network.
o Enable resonably accurate computation of worst-case end-to-end
latency, in a way that requires as little detailed knowledge as
possible of the behavior of the Quality of Service (QoS)
algorithms implemented in each devince, including queuing,
shaping, metering, policing, and transmission selection
Using the model presented in this document, it should be possible for
an implementor, user, or standards development organization to select
a particular set of QoS algorithms for each device in a DetNet
network, and to select a resource reservation algorithm for that
network, so that those elements can work together to provide the
DetNet service.
This document does not specify any resource reservation protocol or
server. It does not describe all of the requirements for that
protocol or server. It does describe a set of requirements for
resource reservation algorithms and for QoS algorithms that, if met,
will enable them to work together.
2. Conventions Used in This Document
The key words "MUST", "MUST NOT", "REQUIRED", "SHALL", "SHALL NOT",
"SHOULD", "SHOULD NOT", "RECOMMENDED", "MAY", and "OPTIONAL" in this
document are to be interpreted as described in [RFC2119].
The lowercase forms with an initial capital "Must", "Must Not",
"Shall", "Shall Not", "Should", "Should Not", "May", and "Optional"
in this document are to be interpreted in the sense defined in
[RFC2119], but are used where the normative behavior is defined in
documents published by SDOs other than the IETF.
Finn, et al. Expires September 6, 2018 [Page 3]
Internet-Draft DetNet Bounded Latency March 2018
3. Terminology and Definitions
This document uses the terms defined in
4. DetNet bounded latency model
4.1. Flow creation
The bounded latency model assusmes the use of the following paradigm
for provisioning a particular DetNet flow:
1. Perform any onfiguration required by the relay systems in the
network for the classes of service to be offered, including one
or more classes of DetNet service. This configuration is
general; it is not tied to any particular flow.
2. Characterize the DetNet flow in terms of limitations on the
sender Section 8.1 and flow requirements Section 8.2.
3. Establish the path that the DetNet flow will take through the
network from the source to the destination(s). This can be a
point-to-point or a point-to-multipoint path.
4. Select one of the DetNet classes of service for the DetNet flow.
5. Compute the worst-case end-to-end latency for the DetNet flow.
In the process, determine whether sufficient resources are
available for that flow to guarantee the required latency and
provide zero congestion loss.
6. Assuming that the resources are available, commit those resources
to the flow. This may or may not require adjusting the
parameters that control the QoS algorithms at each hop along the
flow's path.
This paradigm can be static and/or dynamic, and can be implemented
using peer-to-peer protocols or with a central server model. In some
situations, backtracking and recursing through this list may be
Issues such as un-provisioning a DetNet flow in favor of another when
resources are scarce are not considered. How the path to be taken by
a DetNet flow is chosen is not considered in this document.
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4.2. End-to-end model
[Suggestion: This is the introduction to network calculus. The
starting point is a model in which a relay system is a black box.]
4.3. Relay system model
[NWF I think that at least some of this will be useful. We won't
know until we see what J-Y has to say in Section 4.2. I'm especially
interested in whether J-Y thinks that the "output delay" in Figure 1
is useful in determining the number of buffers needed in the next
hop. It is possible that we can define the parameters we need
without this section.]
In Figure 1 we see a breakdown of the per-hop latency experienced by
a packet passing through a relay system, in terms that are suitable
for computing both hop-by-hop latency and per-hop buffer
DetNet relay node A DetNet relay node B
+-----------------+ +-----------------+
| Queue | | Queue |
| +-+-+-+ | | +-+-+-+ |
-->+ | | | + +------->+ | | | + +--->
| +-+-+-+ | | +-+-+-+ |
| | | |
+-----------------+ +-----------------+
2,3 4 5 1 2,3 4 5 1 2,3
1: Output delay 3: Preemption delay
2: Link delay 4: Processing delay
5: Queuing delay
Figure 1: Timing model for DetNet or TSN
In Figure 1, we see two DetNet relay nodes (typically, bridges or
routers), with a wired link between them. In this model, the only
queues we deal with explicitly are attached to the output port; other
queues are modeled as variations in the other delay times. (E.g., an
input queue could be modeled as either a variation in the link delay
[2] or the processing delay [4].) There are five delays that a
packet can experience from hop to hop.
1. Output delay
The time taken from the selection of a packet for output from a
queue to the transmission of the first bit of the packet on the
physical link. If the queue is directly attached to the physical
port, output delay can be a constant. But, in many
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implementations, the queuing mechanism in a forwarding ASIC is
separated from a multi-port MAC/PHY, in a second ASIC, by a
multiplexed connection. This causes variations in the output
delay that are hard for the forwarding node to predict or control.
2. Link delay
The time taken from the transmission of the first bit of the
packet to the reception of the last bit, assuming that the
transmission is not suspended by a preemption event. This delay
has two components, the first-bit-out to first-bit-in delay and
the first-bit-in to last-bit-in delay that varies with packet
size. The former is typically measured by the Precision Time
Protocol and is constant (see [I-D.ietf-detnet-architecture]).
However, a virtual "link" could exhibit a variable link delay.
3. Preemption delay
If the packet is interrupted (e.g. [IEEE8023br] preemption) in
order to transmit another packet or packets, an arbitrary delay
can result.
4. Processing delay
This delay covers the time from the reception of the last bit of
the packet to that packet being eligible, if there were no other
packets in the queue, for selection for output. This delay can be
variable, and depends on the details of the operation of the
forwarding node.
5. Queuing delay
This is the time spent from the insertion of the packet into a
queue until the packet is selected for output on the next link.
We assume that this time is calculable based on the details of the
queuing mechanism.
Not shown in Figure 1 are the other output queues that we presume are
also attached to that same output port as the queue shown, and
against which this shown queue competes for transmission
The initial and final measurement point in this analysis (that is,
the definition of a "hop") is the point at which a packet is selected
for output. In general, any queue selection method that is suitable
for use in a DetNet network includes a detailed specification as to
exactly when packets are selected for transmission. Any variations
in any of the delay times 1-4 result in a need for additional buffers
in the queue. If all delays 1-4 are constant, then any variation in
the time at which packets are inserted into a queue depends entirely
on the timing of packet selection in the previous node. If the
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delays 1-4 are not constant, then additional buffers are required in
the queue to absorb these variations. Thus:
o Variations in output delay (1) require buffers to absorb that
variation in the next hop, so the output delay variations of the
previous hop (on each input port) must be known in order to
calculate the buffer space required on this hop.
o Variations in processing delay (4) require additional output
buffers in the queues of that same Detnet relay node. Depending
on the details of the queueing delay (5) calculations, these
variations need not be visible outside the DetNet relay node.
5. Computing End-to-end Latency Bounds
End-to-end latency bounds can be computed using the delay model in
Section 4.3. Here it is important to be aware that for several
queuing mechanisms, the worst-case end-to-end delay is less than the
sum of the per-hop worst-case delays. An end-to-end latency bound
for one detnet flow can be computed as
end_to_end_latency_bound = non_queuing_latency + queuing_latency
The two terms in the above formula are computed as follows. First,
at the h-th hop along the path of this detnet flow, obtain an upper
bound per-hop_non_queuing_latency[h] on the sum of delays 1,2,3,4 of
Figure 1. These upper-bounds are expected to depend on the specific
technology of the node at the h-th hop but not on the T-SPEC of this
detnet flow. Then set non_queuing_latency = the sum of per-
hop_non_queuing_latency[h] over all hops h.
Second, compute queuing_latency as an upper bound to the sum of the
queuing delays along the path. The value of queuing_latency depends
on the T-SPEC of this flow and possibly of other flows in the
network, as well as the specifics of the queuing mechanisms deployed
along the path of this flow.
For several queuing mechanisms, queuing_latency is less than the sum
of upper bounds on the queuing delay (5) at every hop. Section 5.1
gives such practical computation examples.
For other queuing mechanisms the only available value of
queuing_latency is the sum of the per-hop queuing delay bounds. In
such cases, the computation of per-hop queuing delay bounds must
account for the fact that the T-SPEC of a detnet flow is no longer
satisfied at the ingress of a hop, since burstiness increases as one
flow traverses one detnet node.
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5.1. Examples of Computations
[[ JYLB: THIS IS WHERE DETAILS OF END-TO-END LATENCY COMPUTATION ARE
GIVEN FOR PER-FLOW QUEUING AND FOR TSN WITH ATS]]
6. Achieving zero congestion loss
When the input rate to an output queue exceeds the output rate for a
sufficient length of time, the queue must overflow. This is
congestion loss, and this is what deterministic networking seeks to
6.1. A General Formula
To avoid congestion losses, an upper bound on the backlog present in
the queue of Figure 1 must be computed during path computation. This
bound depends on the set of flows that use this queue, the details of
the specific queuing mechanism and an upper bound on the processing
delay (4). The queue must contain the packet in transmission plus
all other packets that are waiting to be selected for output.
A conservative backlog bound, that applies to all systems, can be
derived as follows.
The backlog bound is counted in data units (bytes, or words of
multiple bytes) that are relevant for buffer allocation. For every
class we need one buffer space for the packet in transmission, plus
space for the packets that are waiting to be selected for output.
Excluding transmission and preemption times, the packets are waiting
in the queue since reception of the last bit, for a duration equal to
the processing delay (4) plus the queuing delay (5).
o nb_classes be the number of classes of traffic that may use this
output port
o total_in_rate be the sum of the line rates of all input ports that
send traffic of any class to this output port. The value of
total_in_rate is in data units (e.g. bytes) per second.
o nb_input_ports be the number input ports that send traffic of any
class to this output port
o max_packet_length be the maximum packet size for packets of any
class that may be sent to this output port. This is counted in
data units.
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o max_delay45 be an upper bound, in seconds, on the sum of the
processing delay (4) and the queuing delay (5) for a packet of any
class at this ouput port.
Then a bound on the backlog of traffic of all classes in the queue at
this output port is
backlog_bound = ( nb_classes + nb_input_ports ) *
max_packet_length + total_in_rate* max_delay45
7. Queuing model
[[ JYLB: THIS IS WHERE DETAILS OF END-TO-END LATENCY COMPUTATION ARE
GIVEN FOR PER-FLOW QUEUING AND FOR TSN WITH ATS]]
7.1. Queuing data model
Sophisticated QoS mechanisms are available in Layer 3 (L3), see,
e.g., [RFC7806] for an overview. In general, we assume that "Layer
3" queues, shapers, meters, etc., are instantiated hierarchically
above the "Layer 2" queuing mechanisms, among which packets compete
for opportunities to be transmitted on a physical (or sometimes,
logical) medium. These "Layer 2 queuing mechanisms" are not the
province solely of bridges; they are an essential part of any DetNet
relay node. As illustrated by numerous implementation examples, the
"Layer 3" some of mechanisms described in documents such as [RFC7806]
are often integrated, in an implementation, with the "Layer 2"
mechanisms also implemented in the same system. An integrated model
is needed in order to successfully predict the interactions among the
different queuing mechanisms needed in a network carrying both DetNet
flows and non-DetNet flows.
Figure 2 shows the (very simple) model for the flow of packets
through the queues of an IEEE 802.1Q bridge. Packets are assigned to
a class of service. The classes of service are mapped to some number
of physical FIFO queues. IEEE 802.1Q allows a maximum of 8 classes
of service, but it is more common to implement 2 or 4 queues on most
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| Class of Service Assignment |
| | |
+--V--+ +--V--+ +--V--+
|Class| |Class| |Class|
| 0 | | 1 | . . . | n |
|queue| |queue| |queue|
+--+--+ +--+--+ +--+--+
| | |
| Transmission selection |
Figure 2: IEEE 802.1Q Queuing Model: Data flow
Some relevant mechanisms are hidden in this figure, and are performed
in the "Class n queue" box:
o Discarding packets because a queue is full.
o Discarding packets marked "yellow" by a metering function, in
preference to discarding "green" packets.
The Class of Service Assignment function can be quite complex, since
the introduction of [IEEE802.1Qci]. In addition to the Layer 2
priority expressed in the 802.1Q VLAN tag, a bridge can utilize any
of the following information to assign a packet to a particular class
of service (queue):
o Input port.
o Selector based on a rotating schedule that starts at regular,
time-synchronized intervals and has nanosecond precision.
o MAC addresses, VLAN ID, IP addresses, Layer 4 port numbers, DSCP.
(Work items expected to add MPC and other indicators.)
o The Class of Service Assignment function can contain metering and
policing functions.
The "Transmission selection" function decides which queue is to
transfer its oldest packet to the output port when a transmission
opportunity arises.
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7.2. IEEE 802.1 Queuing Model
7.2.1. Queuing Data Model with Preemption
Figure 2 must be modified if the output port supports preemption
([IEEE8021Qbu] and [IEEE8023br]). This modification is shown in
Figure 3.
| Class of Service Assignment |
| | | | | | | |
+--V--+ +--V--+ +--V--+ +--V--+ +--V--+ +--V--+ +--V--+ +--V--+
|Class| |Class| |Class| |Class| |Class| |Class| |Class| |Class|
| a | | b | | c | | d | | e | | f | | g | | h |
|queue| |queue| |queue| |queue| |queue| |queue| |queue| |queue|
+--+--+ +--+--+ +--+--+ +--+--+ +--+--+ +--+--+ +--+--+ +--+--+
| | | +-+ | | | |
| | | | | | | |
+--V-------V-------V------+ +V-----V-------V-------V-------V--+
| Interrupted xmit select | | Preempting xmit select | 802.1
+-------------+-----------+ +----------------+----------------+
| | ======
+-------------V-----------+ +----------------V----------------+
| Preemptible MAC | | Express MAC | 802.3
+--------+----------------+ +----------------+----------------+
| |
| MAC merge sublayer |
| PHY (unaware of preemption) |
Figure 3: IEEE 802.1Q Queuing Model: Data flow with preemption
From Figure 3, we can see that, in the IEEE 802 model, the preemption
feature is modeled as consisting of two MAC/PHY stacks, one for
packets that can be interrupted, and one for packets that can
interrupt the interruptible packets. The Class of Service (queue)
determines which packets are which. In Figure 3, the classes of
service are marked "a, b, ..." instead of with numbers, in order to
avoid any implication about which numeric Layer 2 priority values
correspond to preemptible or preempting queues. Although it shows
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three queues going to the preemptible MAC/PHY, any assignment is
7.2.2. Transmission Selection Model
In Figure 4, we expand the "Transmission selection" function of
Figure 3.
Figure 4 does NOT show the data path. It shows an example of a
configuration of the IEEE 802.1Q transmission selection box shown in
Figure 2 and Figure 3. Each queue m presents a "Class m Ready"
signal. These signals go through various logic, filters, and state
machines, until a single queue's "not empty" signal is chosen for
presentation to the underlying MAC/PHY. When the MAC/PHY is ready to
take another output packet, then a packet is selected from the one
queue (if any) whose signal manages to pass all the way through the
transmission selection function.
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+-----+ +-----+ +-----+ +-----+ +-----+ +-----+ +-----+ +-----+
|Class| |Class| |Class| |Class| |Class| |Class| |Class| |Class|
| 1 | | 0 | | 4 | | 5 | | 6 | | 7 | | 2 | | 3 |
|Ready| |Ready| |Ready| |Ready| |Ready| |Ready| |Ready| |Ready|
+--+--+ +--+--+ +--+--+ +-XXX-+ +--+--+ +--+--+ +--+--+ +--+--+
| | | | | | |
| +--V--+ +--V--+ +--+--+ +--V--+ | +--V--+ +--V--+
| |Prio.| |Prio.| |Prio.| |Prio.| | |Sha- | |Sha- |
| | 0 | | 4 | | 5 | | 6 | | | per| | per|
| | PFC | | PFC | | PFC | | PFC | | | A | | B |
| +--+--+ +--+--+ +-XXX-+ +-XXX-+ | +--+--+ +-XXX-+
| | | | |
+--V--+ +--V--+ +--V--+ +--+--+ +--+--+ +--V--+ +--V--+ +--+--+
|Time | |Time | |Time | |Time | |Time | |Time | |Time | |Time |
| Gate| | Gate| | Gate| | Gate| | Gate| | Gate| | Gate| | Gate|
| 1 | | 0 | | 4 | | 5 | | 6 | | 7 | | 2 | | 3 |
+--+--+ +-XXX-+ +--+--+ +--+--+ +-XXX-+ +--+--+ +-XXX-+ +--+--+
| | |
+--V-------+-------V-------+--+ |
|802.1Q Enhanced Transmission | |
| Selection (ETS) = Weighted | |
| Fair Queuing (WFQ) | |
+--+-------+------XXX------+--+ |
| |
| Strict Priority selection (rightmost first) |
Figure 4: 802.1Q Transmission Selection
The following explanatory notes apply to Figure 4
o The numbers in the "Class n Ready" boxes are the values of the
Layer 2 priority that are assigned to that Class of Service in
this example. The rightmost CoS is the most important, the
leftmost the least. Classes 2 and 3 are made the most important,
because they carry DetNet flows. It is all right to make them
more important than the priority 7 queue, which typically carries
critical network control protocols such as spanning tree or IS-IS,
because the shaper ensures that the highest priority best-effort
queue (7) will get reasonable access to the MAC/PHY. Note that
Class 5 has no Ready signal, indicating that that queue is empty.
o Below the Class Ready signals are shown the Priority Flow Control
gates (IEEE Std 802.1Qbb-2011 Priority-based Flow Control, now
[IEEE8021Q] clause 36) on Classes of Service 1, 0, 4, and 5, and
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two 802.1Q shapers, A and B. Perhaps shaper A conforms to the
IEEE Std 802.1Qav-2009 (now [IEEE8021Q] clause 34) credit-based
shaper, and shaper B conforms to [IEEE8021Qcr] Asynchronous
Traffic Shaper. Any given Class of Service can have either a PFC
function or a shaper, but not both.
o Next are the IEEE Std 802.1Qbv time gates ([IEEE8021Qbv]). Each
one of the 8 Classes of Service has a time gate. The gates are
controlled by a repeating schedule that restarts periodically, and
can be programmed to turn any combination of gates on or off with
nanosecond precision. (Although the implementation is not
necessarily that accurate.)
o Following the time gates, any number of Classes of Service can be
linked to one ore more instances of the Enhanced Transmission
Selection function. This does weighted fair queuing among the
members of its group.
o A final selection of the one queue to be selected for output is
made by strict priority. Note that the priority is determined not
by the Layer 2 priority, but by the Class of Service.
o An "XXX" in the lower margin of a box (e.g. "Prio. 5 PFC"
indicates that the box has blocked the "Class n Ready" signal.
o IEEE 802.1Qch Cyclic Queuing and Forwarding [IEEE802.1Qch] is
accomplished using two or three queues (e.g. 2 and 3 in the
figure), using sophisticated time-based schedules in the Class of
Service Assignment function, and using the IEEE 802.1Qbv time
gates [IEEE8021Qbv] to swap between the output buffers.
7.3. Other queuing models, e.g. IntServ
[[NWF More sections that discuss specific models]]
8. Parameters for the bounded latency model
8.1. Sender parameters
8.2. Relay system parameters
[[NWF This section talks about the paramters that must be passed hop-
by-hop (T-SPEC? F-SPEC?) by a resoure reservation protocol.]]
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9. References
9.1. Normative References
Finn, N. and P. Thubert, "Deterministic Networking
Architecture", draft-ietf-detnet-architecture-00 (work in
progress), September 2016.
Korhonen, J., Farkas, J., Mirsky, G., Thubert, P.,
Zhuangyan, Z., and L. Berger, "DetNet Data Plane Protocol
and Solution Alternatives", draft-ietf-detnet-dp-alt-00
(work in progress), October 2016.
Grossman, E., "Deterministic Networking Use Cases", draft-
ietf-detnet-use-cases-14 (work in progress), February
[RFC2119] Bradner, S., "Key words for use in RFCs to Indicate
Requirement Levels", BCP 14, RFC 2119,
DOI 10.17487/RFC2119, March 1997,
[RFC2212] Shenker, S., Partridge, C., and R. Guerin, "Specification
of Guaranteed Quality of Service", RFC 2212,
DOI 10.17487/RFC2212, September 1997,
[RFC4364] Rosen, E. and Y. Rekhter, "BGP/MPLS IP Virtual Private
Networks (VPNs)", RFC 4364, DOI 10.17487/RFC4364, February
2006, <https://www.rfc-editor.org/info/rfc4364>.
[RFC6658] Bryant, S., Ed., Martini, L., Swallow, G., and A. Malis,
"Packet Pseudowire Encapsulation over an MPLS PSN",
RFC 6658, DOI 10.17487/RFC6658, July 2012,
[RFC7806] Baker, F. and R. Pan, "On Queuing, Marking, and Dropping",
RFC 7806, DOI 10.17487/RFC7806, April 2016,
9.2. Informative References
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IEEE, "IEEE Std 802.1Qch-2017 IEEE Standard for Local and
metropolitan area networks - Bridges and Bridged Networks
Amendment 29: Cyclic Queuing and Forwarding (amendment to
802.1Q-2014)", 2017,
IEEE, "IEEE Std 802.1Qci-2017 IEEE Standard for Local and
metropolitan area networks - Bridges and Bridged Networks
- Amendment 30: Per-Stream Filtering and Policing", 2017,
IEEE 802.1, "IEEE Std 802.1Q-2014: IEEE Standard for Local
and metropolitan area networks - Bridges and Bridged
Networks", 2014, <http://standards.ieee.org/getieee802/
IEEE, "IEEE Std 802.1Qbu-2016 IEEE Standard for Local and
metropolitan area networks - Bridges and Bridged Networks
- Amendment 26: Frame Preemption", 2016,
IEEE 802.1, "IEEE Std 802.1Qbv-2015: IEEE Standard for
Local and metropolitan area networks - Bridges and Bridged
Networks - Amendment 25: Enhancements for Scheduled
Traffic", 2015, <http://standards.ieee.org/getieee802/
IEEE 802.1, "IEEE P802.1Qcr: IEEE Draft Standard for Local
and metropolitan area networks - Bridges and Bridged
Networks - Amendment: Asynchronous Traffic Shaping", 2017,
IEEE 802.1, "IEEE 802.1 Time-Sensitive Networking (TSN)
Task Group", <http://www.ieee802.org/1/>.
IEEE 802.3, "IEEE Std 802.3-2015: IEEE Standard for Local
and metropolitan area networks - Ethernet", 2015,
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IEEE 802.3, "IEEE Std 802.3br-2016: IEEE Standard for
Local and metropolitan area networks - Ethernet -
Amendment 5: Specification and Management Parameters for
Interspersing Express Traffic", 2016,
Authors' Addresses
Norman Finn
Huawei Technologies Co. Ltd
3101 Rio Way
Spring Valley, California 91977
Phone: +1 925 980 6430
Email: norman.finn@mail01.huawei.com
Jean-Yves Le Boudec
IC Station 14
Lausanne EPFL 1015
Email: jean-yves.leboudec@epfl.ch
Balazs Varga
Konyves Kalman krt. 11/B
Budapest 1097
Email: balazs.a.varga@ericsson.com
Janos Farkas
Konyves Kalman krt. 11/B
Budapest 1097
Email: janos.farkas@ericsson.com
Finn, et al. Expires September 6, 2018 [Page 17] | {"url":"https://datatracker.ietf.org/doc/draft-finn-detnet-bounded-latency/00/","timestamp":"2024-11-03T09:00:51Z","content_type":"text/html","content_length":"76868","record_id":"<urn:uuid:3a7e104b-5a53-4523-8af9-03023a772f11>","cc-path":"CC-MAIN-2024-46/segments/1730477027774.6/warc/CC-MAIN-20241103083929-20241103113929-00803.warc.gz"} |
Hours Calculator
Use the calculators below to find the number of hours and minutes between two times. For a full time card, please use the Time Card Calculator.
Hours Between Two Dates
An hour is most commonly defined as a period of time equal to 60 minutes, where a minute is equal to 60 seconds, and a second has a rigorous scientific definition. There are also 24 hours in a day.
Most people read time using either a 12-hour clock or a 24-hour clock.
12-hour clock:
A 12-hour clock uses the numbers 1-12. Depending on the clock being used, most analog clocks or watches may not include an indication of whether the time is in the morning or evening. On digital
clocks and watches, "AM" stands for ante meridiem, meaning "before midday," while "PM" stands for post meridiem, or "after noon." By convention, 12 AM denotes midnight, while 12 PM denotes noon.
Using the terms "12 midnight" and "12 noon" can remove ambiguity in cases where a person may not be accustomed to conventions.
24-hour clock:
A 24-hour clock typically uses the numbers 0-23, where 00:00 indicates midnight, and a day runs from midnight to midnight over the course of 24 hours. This time format is an international standard,
and is often used to avoid the ambiguity resulting from the use of a 12-hour clock. The hours from 0-11 denote what would be the AM hours on a 12-hour clock, while hours 12-23 denote the PM hours of
a 12-hour clock. In certain countries, 24-hour time is referred to as military time, since this is the time format used by militaries (and other entities) around the world, where unambiguous time
measurement is particularly important.
Hours in different time periods
Description Hours
Hours in a day 24
Hours in a week 168
Hours in a month 672 for a 28-day month
696 for a 29-day month
720 for a 30-day month
744 for a 31-day month
730.5 on average
Hours in a year 8,760 for a 365-day year
8,784 for a 366-day year
8,766 on average
Hours in a decade 87,648 for a 2-leap-year decade
87,672 for a 3-leap-year decade
87,660 on average
Hours in a century 876,600 | {"url":"https://bodywiseslc.net/hours-calculator.html","timestamp":"2024-11-11T13:37:09Z","content_type":"text/html","content_length":"10938","record_id":"<urn:uuid:e602f04a-fd8f-498f-90e4-286c205b1bb3>","cc-path":"CC-MAIN-2024-46/segments/1730477028230.68/warc/CC-MAIN-20241111123424-20241111153424-00645.warc.gz"} |
Vector dimension precision effect on cosine similarity
Experiment Outline
Question: How does reducing the precision of vector components to various extents (half, third, quarter, fifth) using different methods (toFixed, Math.round) affect the cosine similarity between two
Hypothesis: Reducing the precision of vectors will alter the cosine similarity, with more significant reductions leading to larger differences. The method of precision reduction might not
significantly impact the cosine similarity.
Experiment Design:
• Control Group: Compute cosine similarity between two original vectors.
• Variable Groups: For each level of precision reduction (half, third, quarter, fifth) and for each method of precision reduction (toFixed, Math.round), compute cosine similarity between
precision-reduced vectors.
• Measurement: Compare the cosine similarities across different levels of precision reduction and methods.
Data Collection: Implement JavaScript code to calculate cosine similarities in each case and run multiple iterations to average the results.
Analysis: Evaluate how different levels and methods of precision reduction impact the cosine similarity value.
Code Specification
Functions for Cosine Similarity and Vector Generation: Functions to compute the dot product, magnitude, cosine similarity, and generate random vectors with specific bit-depth and dimensions.
Functions for Precision Reduction:
• Two functions to reduce precision: one using toFixed and another using Math.round.
• Apply these functions to vectors with varying degrees of precision reduction (half, third, quarter, fifth).
Implementation Considerations:
• Use ES2020 standards.
• Focus on readability and performance optimization.
• Adapt the code to handle vectors of different dimensions (384 and 1536) and bit-depths (16-bit and 8-bit).
Function for Averaging Differences: A function to calculate the average difference in cosine similarity over multiple iterations for each precision reduction level and method.
Execution of Experiment: Run the experiment with 1000 iterations for each combination of vector type, dimension, and precision reduction method.
Interpretation of Results
The results of this experiment will help understand the extent to which precision reduction affects the similarity of high-dimensional vectors. This is particularly relevant in applications like data
compression or optimization in machine learning, where a balance between precision and computational efficiency is often sought. The findings indicate that while precision reduction does impact
cosine similarity, the effects are relatively minor, even with significant reductions. This suggests potential flexibility in the precision of vector representations in certain applications, without
substantially compromising their comparative similarity.
Checkout the full code on Github.
node embeddings/precision-reduction-impact-on-cosine-similarity.js
Average differences for 16-bit vectors (384-dim): {
precision_half_to_fixed: '0.0000007919%',
precision_half_math_round: '0.0000007919%',
precision_third_to_fixed: '0.0001907019%',
precision_third_math_round: '0.0001907019%',
precision_quarter_to_fixed: '0.0019869799%',
precision_quarter_math_round: '0.0019869799%',
precision_fifth_to_fixed: '0.0182798241%',
precision_fifth_math_round: '0.0182798241%'
Average differences for 16-bit vectors (1536-dim): {
precision_half_to_fixed: '0.0000009126%',
precision_half_math_round: '0.0000009126%',
precision_third_to_fixed: '0.0001162906%',
precision_third_math_round: '0.0001162906%',
precision_quarter_to_fixed: '0.0010907664%',
precision_quarter_math_round: '0.0010907664%',
precision_fifth_to_fixed: '0.0099245784%',
precision_fifth_math_round: '0.0099245784%'
Average differences for 8-bit vectors (384-dim): {
precision_half_to_fixed: '0.0000009423%',
precision_half_math_round: '0.0000009423%',
precision_third_to_fixed: '0.0003219478%',
precision_third_math_round: '0.0003219478%',
precision_quarter_to_fixed: '0.0038977933%',
precision_quarter_math_round: '0.0038977933%',
precision_fifth_to_fixed: '0.0331642817%',
precision_fifth_math_round: '0.0331642817%'
Average differences for 8-bit vectors (1536-dim): {
precision_half_to_fixed: '0.0000012704%',
precision_half_math_round: '0.0000012704%',
precision_third_to_fixed: '0.0001971148%',
precision_third_math_round: '0.0001971148%',
precision_quarter_to_fixed: '0.0021234765%',
precision_quarter_math_round: '0.0021234765%',
precision_fifth_to_fixed: '0.0219500987%',
precision_fifth_math_round: '0.0219500987%'
The experiment results show the average difference in cosine similarity between the original and precision-reduced vectors, for different methods of precision reduction (toFixed and Math.round) and
for varying degrees of precision reduction (half, third, quarter, fifth). The experiment was conducted on two types of vectors: 16-bit and 8-bit, with two different dimensions (384 and 1536).
Key Observations
Impact of Precision Reduction:
• As the precision reduction becomes more aggressive (from half to fifth), the average difference in cosine similarity increases. This indicates that the loss of precision generally has a more
pronounced effect as more decimal points are removed.
• Even at the most aggressive level of precision reduction (to a fifth), the change in cosine similarity is relatively small, in the order of hundredths of a percent.
Comparison of Reduction Methods:
• There is no noticeable difference between the toFixed and Math.round methods in terms of their impact on cosine similarity. This suggests that both methods of rounding have a similar effect on
the precision of the vectors and their resulting cosine similarities.
Effect of Vector Dimensionality:
• The dimensionality of the vectors (384 vs. 1536) seems to have a minor impact on the results. The pattern of increasing differences with more aggressive precision reductions holds in both cases,
though the exact values differ slightly.
16-bit vs. 8-bit Vectors:
• There’s a consistent trend across both types of vectors. The differences are very small, but consistently, 8-bit vectors show a slightly higher difference in cosine similarity compared to
16-bit vectors when the precision is reduced. This could be due to the lower initial precision of the 8-bit vectors, which makes further precision reduction more impactful.
The experiment’s results suggest that reducing the precision of vectors has a measurable but minor impact on their cosine similarity. This impact becomes slightly more pronounced as the degree of
precision reduction increases, but even the most significant changes are relatively small. The method of precision reduction (rounding vs. truncating) does not appear to significantly affect the
These findings could have practical implications in applications that utilize vector embeddings, where high-dimensional vectors are used to represent complex data. The results suggest that it’s
possible to reduce the precision of these vectors (for instance, for storage or computation efficiency) with only a minimal impact on their comparative similarity. However, the degree to which
precision can be reduced without significantly affecting the results will depend on the specific requirements and tolerances of the application. | {"url":"http://wfhbrian.com/artificial-intelligence/vector-dimension-precision-effect-on-cosine-similarity","timestamp":"2024-11-07T07:54:45Z","content_type":"text/html","content_length":"43083","record_id":"<urn:uuid:226be927-b910-41d7-a138-178790bd6809>","cc-path":"CC-MAIN-2024-46/segments/1730477027957.23/warc/CC-MAIN-20241107052447-20241107082447-00428.warc.gz"} |
Choose a function in the lower part of the Elements pane. These functions are also listed in the context menu of the Commands window. Any functions not contained in the Elements pane need to be typed
manually in the Commands window.
The following is a list of all functions that appear in the Elements pane. The icon next to the function indicates that it can be accessed through the Elements pane (menu View - Elements) or through
the context menu of the Commands window.
Inserts a natural exponential function. You can also type func e^<?> directly in the Commands window.
Inserts a natural (base e) logarithm with one placeholder. You can also type ln(<?>) in the Commands window.
Inserts an exponential function with one placeholder. You can also type exp(<?>) in the Commands window.
Inserts a common (base 10) logarithm with one placeholder. You can also type log(<?>) in the Commands window.
Inserts x raised to the yth power. You can also type <?>^{<?>} in the Commands window. You can replace the ^ character with rsup or sup.
Inserts a sine function with one placeholder. You can also type sin(<?>) in the Commands window.
Inserts a cosine function with one placeholder. You can also type cos(<?>) in the Commands window.
Inserts a tangent function with one placeholder. You can also type tan<?>) in the Commands window.
Inserts a cotangent symbol with a placeholder. You can also type cot(<?>) in the Commands window.
Inserts a hyperbolic sine with one placeholder. You can also type sinh(<?>) in the Commands window.
Inserts a square root symbol with one placeholder. You can also type sqrt(<?>) in the Commands window.
Inserts a hyperbolic cosine symbol with one placeholder. You can also type cosh(<?>) in the Commands window.
Inserts a hyperbolic tangent symbol with one placeholder. You can also type tanh(<?>) in the Commands window.
Inserts a hyperbolic cotangent symbol with one placeholder. You can directly type coth(<?>) in the Commands window.
Inserts an nth root function with two placeholders. You can also type nroot n x in the Commands window.
Inserts an arc sine function with one placeholder. You can also type arcsin(<?>) in the Commands window.
Inserts an arc cosine symbol with one placeholder. You can also type arccos(<?>) in the Commands window.
Inserts an arc tangent function with one placeholder. You can also type arctan(<?>) in the Commands window.
Inserts an arc cotangent function with one placeholder. You can directly type arccot(<?>) in the Commands window.
Inserts an absolute value sign with one placeholder. You can also type abs(<?>) in the Commands window.
Inserts an area hyperbolic sine function with one placeholder. You can also type arsinh(<?>) in the Commands window.
Inserts an area hyperbolic cosine function with one placeholder. You can also type arcosh(<?>) in the Commands window.
Inserts an area hyperbolic tangent function with one placeholder. You can also type artanh(<?>) in the Commands window.
Inserts an area hyperbolic cotangent function with one placeholder. You can also type arcoth(<?>) in the Commands window.
Inserts the factorial sign with one placeholder. You can directly type fact <?> in the Commands window.
You can also assign an index or an exponent to a function. For example, typing sin^2x results in a function "sine to the power of 2x".
When typing functions manually in the Commands window, note that spaces are required for some functions (for example, abs 5=5 ; abs -3=3). | {"url":"https://help.libreoffice.org/latest/en-ZA/text/smath/01/03090400.html","timestamp":"2024-11-14T05:58:49Z","content_type":"text/html","content_length":"29153","record_id":"<urn:uuid:45806288-803d-41ad-85f2-942aa18216ae>","cc-path":"CC-MAIN-2024-46/segments/1730477028526.56/warc/CC-MAIN-20241114031054-20241114061054-00043.warc.gz"} |
Cascading Failures
Communication, Computer, data mining
Cascading Failures
The complex structure and wide interconnections of power grids make them vulnerable to disturbances such as faults, loss of transmission lines (TLs), etc. This complexity makes the control strategies
more sophisticated and costly in order to maintain the stability and reliability of power grids during contingencies. For instance, the 2003 North American blackout initiated by a fault in a
transmission line and the lack of rapid and appropriate actions from the grid components imposed extravagant cost of maintenance and restoration on the government and the electric power industry. Due
to all technical and financial issues such as instability, unreliability, maintenance and restoration costs caused by outages, several research efforts have been done recently to mitigate this
catastrophic phenomenon. Many different approaches and methods have been recently proposed and developed for analysing, modelling, and controlling cascading failures based on deterministic or
probabilistic dynamic mathematical models and simulations . Authors introduced SASE model based on reduced dynamic model of an extensive power system by considering limited state variables that
depict crucial characteristics of the grid. The system is modelled using Markov chain in continuous time structure. This stochastic model predicts the growth of blackout possibility. The drawback of
this model is that all the state variables are not considered for modelling of the system dynamics. An assessment method was suggested in to investigate the effects of N-1 criterion policies on the
uncertainty of consecutive line outages by considering aspects of both long term transmission line expansion planning and cascading issues. In another work, CASCADE probabilistic model was presented
based on load dependency . In this model, all components in a system are assumed identical and failure of each component has equal effect on the other components. Branching Process is another
probabilistic model for cascading failures analysis . This model demonstrates the probability distribution for all component failures. In addition to probabilistic models, several research works have
been proposed based on deterministic modelling. For instance, proposed cascading failure model based on complex network structure. The authors focused more on communication network failures and
congestions in power grids. DeMarco in utilized a deterministic hybrid model based on Lyapunov method. This model is a nonlinear system-based model that investigates the dynamics of cascading
failures which occur because of transient circumstances in power grids. The main problem with this method is that it is not expandable to a large scale power grid. In addition to the above mentioned
methods in modelling of cascading failure characteristics for utilizing predictable control methods, some of the recent research works have been presented based on load shedding strategies for
cascading failure prevention . The main challenge associated with the load shedding method is that many utility customers will be deprived of power which makes the various stakeholders in the power
industry incur losses. On the other hand, some emerging algorithms in Artificial Intelligence (AI) and machine learning such as the multi-agent systems have been widely utilized recently to enhance
the power system stability, reliability, and performance . The main characteristics of intelligent systems are controllability, adaptability, simplicity, and fast response even for complicated
structures. Among the machine learning methods, the reinforcement learning (RL) approach is a powerful method that has various applications in power system control . Literature reviews demonstrate
that using AI methods for managing cascading failure and blackout is a very novel topic and is in its early steps of development and there are very few research works in this area of study . the
authors used neural network concept for early detection and warning system and authors in used support vector machine (SVM) and communication structure between relays to mitigate occurrence of
blackouts. This method mainly reduces the probability of blackouts for only few locations with high risk of incorrect tripping. Besides load shedding, most of the research works are focused either on
modelling or early detection of cascading failures and blackouts. In this work, we are proposing to halt the cascading event after it is initiated and when there is a little or no option for the
operator to take besides cutting customers off the grid. The idea is to intelligently adjust the generating units relative to each other instead of a pre-planned load shedding scheme.
Literature Review and Background
The following section presents the current research advances and summarizes the gap and limitations in all the areas involved in the proposed dissertation. This includes deterministic approaches,
probabilistic (stochastic) methods, intelligent early warning systems and monitoring, and load shedding strategies for modeling, risk analysis and prevention of cascading failures and blackouts in
power systems.
Probabilistic (Stochastic) Methods for Modeling and Analysis of Cascading Failures
a scalable and analytically tractable probabilistic model for the cascading failure dynamics in power grids was proposed considering operating characteristics of the power grid. Authors introduced
SASE model based on reduced dynamic model of an extensive power system by a continuous-time Markov chain and considering limited state variables that depict crucial characteristics of the grid
comprising loading level, error in transmission-capacity estimation, and constraints in performing load shedding. This stochastic model predicts the growth of blackout possibility in time. The
drawback of this model is that all the state variables are not considered for modelling of the system dynamics. An assessment method was suggested to investigate the effects of N-1 criterion
policies on the uncertainty of consecutive line outages by considering aspects of both long term transmission line expansion planning and cascading issues. The long-term effects of these policies on
the probability distribution of outage size and the grid utilization were computed in a large-scale system. introduced an interaction model for cascading failures. This probabilistic model identifies
the critical components of the system that propagate cascading failures and an interaction matrix is acquired based on component failures interactions. The model investigates the risk of cascading
failures and provides online decision-making. In the OPA model , the cascading failure was approximated by considering the dynamics of demands and DC load flow. Linear programming was utilized to
re-dispatch the generation and loads after a random line outage. The drawback of the OPA model is that the timing of failures is ignored which cannot be suitable for the protective coordination. The
CASCADE model was based on load dependency. In this model, all elements of power system are considered to be identical and failure of each element has equal impact on the other elements. The CASCADE
model does not include all electrical features of the grid and time sequence of the event is not considered and the model is not robust enough for protective coordination. Branching Process is
another probabilistic model for analyzing cascading failures. This model relies on the probability distribution for investigating the total component failures. This model does not provide complete
dynamic independent variables to include all dynamic features associated with cascading failures and blackouts in power systems. a new numerical metric defined as the critical moment (CM) is compared
with the validity of a typical DC power flow-based cascading failure simulator (CFS) in cascading failure analysis. The main issue with this work is that due to the complex nature of the power system
and cascading failures, the underlying assumptions in DC power flowbased cascading failure simulators (CFS) may fail to hold during the development of cascading failures.
Deterministic Methods for Modeling and Analysis of Cascading Failures
In addition to probabilistic methods mentioned in previous section, several researches have been done based on deterministic-based approaches to analyse and investigate cascading failures. it is
shown that how the breakdown of a single node is sufficient to collapse the entire system simply because of the dynamics of redistribution of flows on the network. In this work, cascading failure is
modelled based on complex network structure and graph theory. The authors focused more on communication network failures and congestions in power grids. the role of transient dynamic response
following a specified initiating disturbance is captured and subsequent (“cascading”) element failures are examined that are induced when operating thresholds for individual elements are exceeded
along the state trajectory. Author in this work utilized a deterministic hybrid model based on Lyapunov method. This model is a nonlinear system-based model that investigates the dynamics of
cascading failures which occur because of transient circumstances in power grids. The main problem with this method is that it is not expandable to a large scale power grid. line outages in the
transmission network of the power grid are considered, specifically those caused by natural disasters or large-scale physical attacks. In this work,authors show how to identify the most vulnerable
locations in the network by performing extensive numerical experiments with real grid data to estimate the effects of geographically correlated outages. In this model, a circular and deterministic
failure model of cascading outage is considered, where all lines and nodes within a radius r of the failure’s epicenter are removed.
Power System Simulation-based methods for preventing Cascading Failures
Another research direction for modeling and analysis of cascading failures and blackouts is based on power system simulation studies. Besides of mathematical deterministic or probabilistic models
stated in previous sections, this part is dedicated to power system simulation-based strategies. Many methods and paradigms have been developed using power system simulations for investigating the
behavior of the cascading failures for early warning detection systems and development and preventing cascading failure propagation. a simulation of an upgrading power transmission system is utilized
to investigate how the complexity of system dynamics impact the assessment and mitigation of blackout risk by estimating the frequency and cost of blackout. This approach uses the NERC data to
estimate blackout risk and cost base on unserved energy and the number of customers disconnected and blackout duration. This complex system approach to risk analysis, analyzes the long-term, steady
state risk of failure in a system that is dynamically evolving as the system is upgrading in response to increasing demand a method based on composite power system reliability evaluation through
sequential Monte Carlo simulation is proposed since cascading failures involve sequences of dependent outages. Then, importance sampling (IS) and importance sampling and antithetic variates (IS-AV)
techniques using the Weibull distribution are utilized and applied to power generator outages to overcome large computational burden involved by the simulations., an optimal plan for expanding the
capacity of a power grid is determined in order to minimize the likelihood of a large cascading blackout. Capacity-expansion decisions considered in this work considering the addition of new
transmission lines and the addition of capacity to existing lines. An Optimization model is used to minimize the probability of a large blackout subject to a budget constraint using Monte Carlo
simulation. Moreover, variance-reduction technique is used to provide results in a reasonable time frame. a novel method is proposed for N-k induced cascading contingency screening based on random
vector functional- link (RVFL) neural network and quantum inspired multi-objective evolutionary algorithm (QMEA). This method can conduct reliable and simultaneous screening for various N-k
contingencies and early warning monitoring and detection based on intelligent algorithms. a multi agent system (MAS) based wide area protection and control scheme is proposed to deal with the long
term voltage instability induced cascading trips. Based on sensitivity analysis between the relay operation margin and power system state variables, an optimal emergency control strategy is defined
to adjust the emergency states timely and prevent the unexpected relay trips and cascading outages. In order to supervise the control process and minimize the load loss, an agent based process
control is adopted to monitor the states of distributed controllers and adjust the emergency control strategy. three load shedding strategies are proposed and investigated to prevent cascading
failures in power grid. The first strategy is a base line case called the homogeneous load shedding strategy. It reduces load homogeneously in all the buses of the system. This strategy is extremely
simple and fast. Second strategy is to accurately find the location and amount of load shedding by a linear optimization formulation which is much more efficient in overall load shedding in the
system. Third, a novel tree heuristic is proposed to overcome the drawbacks of the optimization, namely fairness and scalability.The tree heuristic is linear and very simple to implement. The results
of the tree strategy are compared with that of another existing heuristic and it is found that the tree performs equal to or better than the existing heuristic for all cases. a distributed
multi-agent-based load shedding algorithm is proposed, which can make efficient load shedding decision based on discovered global information to prevent cascading outages. To improve the speed of the
algorithm, particle swarm optimization (PSO) is used. The information discovery algorithm is represented as a discrete time linear system and the stability of which is analyzed according to
averageconsensus theorem.support vector machine (SVM) and communication structure between relays and the supervisory control and data acquisition (SCADA) are utilized to mitigate occurrence of
blackouts. This method mainly reduces the probability of blackouts for only few locations with high risk of incorrect tripping. Based on the research gaps identified in the literature, three
different intelligent and machine learning-based approaches are proposed in this research work to manage congestion in transmission lines and prevent cascading failure and blackout at early stages of
their occurrence after unplanned critical N-1 and N-1-1 contingencies in the system. The three different and distinct intelligent frameworks are:
Multi-Agent Systems (MAS) Approach
Supervised learning Approach based on Artificial Neural Network (ANN)
Reinforcement learning Approach
These proposed methods will provide a learning platform for the power system on how to manage the congestion by intelligent re-dispatch of power through frequency control of the generators. These
methods will satisfy the constraints of voltage, frequency, and power flow during the cascading failure events. Another gap mentioned in the literature was the lack of practical and experimental
implementation of the proposed mathematical algorithms. In order to fill the research gap in terms of experimental implementation, a real-time and experimental testbed is designed and developed in
the smart grid laboratory based on two-way communication infrastructure including hardware setup, software setups, SCADA and monitoring, and real-time interface. The experimental implementation of
the proposed intelligent methods is investigated in real-time. The developed experimental power testbed emulates the real behavior of the power system in a complete dynamic and interactive manner.In
addition to the experimental implementation, the developed algorithms are implemented offline on a large-scale power systems through computer simulation in MATLAB/Simulink environment. The main
purpose is to evaluate the performance of the proposed methods in terms of robustness, effectiveness, and functionality in a large-scale power testbed. Therefore, the proposed approaches are
implemented on the IEEE 118-bus standard test system for multiple critical contingency conditions
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How to Quickly Estimate a Patent's Value Using Discounted Cash Flow | Insights | Venable LLP
At the end of the day a business should be able to assign an approximate worth to its patent portfolio. However, How much is a patent worth? is a question that can be very difficult to answer, and
can result in multi-million-dollar litigation decisions, or patent applications being abandoned before they even issue.
What follows is a basic primer on using Discounted Cash Flow (DCF) analysis with regard to patents and/or patent applications to estimate value. DCF can be used to approximate the current value for
an investment based on projections of how much money that investment will likely provide in the future. If a value calculated using DCF is higher than the current cost of an investment, an investor
should consider making the investment. Only in unique circumstances should an investor consider an investment that does not have a (DCF – costs) > 0. With patents there may be various reasons to
continue forward, even when the DCF calculation is lower than current costs (e.g., defensive purposes, appearances for investors, appeasing inventors). However, the DCF calculation should be a point
of consideration in making that decision.
Consider the following examples:
1. If the costs associated with obtaining a patent (including filing and prosecution fees through issuance) were $15,000, and the expected cash flows from that patent over the next 20 years,
adjusted for the investor's rate of return (i.e., the DCF calculation), were $500,000, this would be a clear indication that filing and prosecuting the patent application should be considered.
2. If the costs were estimated at $20,000, and the DCF calculation were $10,000, from a financial standpoint filing the patent application likely does not make investment sense.
3. Maintenance fees for an already issued patent at the 11.5-year mark are currently $7,700. If the DCF calculation for an issued patent approaching the 11.5-year mark is less than $7,700, that
should factor into the overall decision.
The Formula
* Disclaimer: The DCF calculation can become significantly more complicated, incorporating aspects such as growth, inflation, risk, etc. This primer does not go into any such details.
The basic formula for DCF is:
• CF is the expected Cash Flow for a given time period;
• r is the Discount Rate, or, more often, the Weighted Average Cost of Capital (WACC); and
• n is the number of terms, based on the remaining amount of time the investment/patent will be active.
While we could break the time periods down into months or even weeks, the easiest and most straightforward time period to use is years. If, for example, a patent application has 18 years left, n =
18, whereas if a granted patent expires in 5 years, n = 5.
The cash flows can be estimates of net revenues from licensing deals, litigation wins, related product sales, etc. When deciding what values to use, standard apportionment practices can (and should)
be used with respect to how much the claimed subject matter of a patent applies to the product(s) in play. While guesswork may be necessary, it is best to base the projected cash flows on previous or
known revenues. For example, if the patent is related to widget A, which has had an average net revenue of $10,000/year for the past 5 years, the expected cash flows going forward should be based on
that $10,000/year average. If the patent is related to more than one product, the sum of those cash flows can be used for any given time period.
R, the Discount Rate or the Weighted Average Cost of Capital (WACC), represents the rate of return the investor can expect on their investment. The minimum R value generally represents the bond rates
of U.S. treasuries. However, for a business, R should be the WACC. If your business has a chief financial officer or the equivalent, you should be able to ask them for the WACC. If there is no CFO,
you have two options: (1) look up the formula and calculate the WACC yourself, or (2) decide what rate of return you desire and use that as R. If, for example, a solo inventor averages 5% returns on
their index fund, but is considering making an investment in a patent, they might use R = 5%.
Practical Example #1 – Maintenance Fees
A patent is expiring in 10 years, with upcoming maintenance costs of $7,700. The product associated with the patent has brought in a net revenue of $5,000/year for the past 3 years. The discount
value R is 5%.
Because $38,608.67 minus $7,700 is greater than zero, from an investment standpoint paying the maintenance fees seems to make sense.
If, however, the average net revenue had been only $1,000/year,
In this case, the DCF value is only barely above the cost of maintenance fees. In such a case the investment opportunity is essentially zero, so the owner should make the decision based on other,
nonfinancial factors.
Practical Example #2 – M&A Deal
As part of an M&A deal, the value of a patent portfolio of 500 patents is being determined. Rather than evaluating all 500 individual patents for validity, possible litigation targets, etc., the
entities have agreed to use the overall average net revenues associated with the entire patent portfolio through licensing, patent sales, and/or litigation by the company being acquired for the past
10 years, at $3 million/year. They have also decided to use the average remaining time of the patents within the portfolio, which for this example will be 12 years.
The WACC for the acquiring company is 8%.
However, the WACC for the company being acquired is 10%.
The discrepancy in how the respective companies value this patent portfolio is based entirely on their respective efficiencies, expressed through the differences in WACC. Despite the differences,
these numbers provide a starting point for negotiations.
Practical Example #3 – Variable Cash Flows
While the above examples provide calculations where the cash flows are constant from year to year, in some cases the cash flows associated with a patent may be projected as changing based on products
being sold, settlements, new licensing deals, expiration of licensing deals, or any number of factors. If, for example, a product associated with a patent is scheduled to be phased out prior to the
patent's expiration because of the product's declining sales, the DCF cash flows associated with that patent might look like this (assuming a remaining time of 6 years in the patent):
Year 1 Year 2 Year 3 Year 4 Year 5 Year 6
Cash Flow $5,000 $4,000 $1,500 0 0 0
In such an example, the DCF equation (assuming R = 5%) would be:
Likewise, if a new product associated with a patent expiring in 6 years were going to be released in year 2, with hopes/projections of increasing revenue, the cash flows might look like this:
Year 1 Year 2 Year 3 Year 4 Year 5 Year 6
Cash Flow 0 $3,000 $4,000 $5,000 $7,000 $9,000
In such an example, the DCF equation (again assuming R = 5%) would be:
While the value of a patent can vary according to projections, parties, and circumstances, calculating the DCF for a patent, or for a potential patent application, can be a useful tool in estimating
the value of a patent or patent application. If the calculated DCF of the patent does not match the associated costs of maintaining or obtaining a patent, an investor/inventor/in-house counsel should
consider whether there are other reasons, such as maintaining a defensive portfolio, for maintaining/obtaining the patent. | {"url":"https://www.venable.com/insights/publications/2021/04/how-to-quickly-estimate-a-patents-value-using","timestamp":"2024-11-10T00:13:52Z","content_type":"text/html","content_length":"55531","record_id":"<urn:uuid:94688bc8-c83e-459a-b214-4aa479d03905>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.10/warc/CC-MAIN-20241109214337-20241110004337-00010.warc.gz"} |
Multiplication game by 10, 100, 1000 or 0.1, 0.01, 0.001 - Junior math quiz - Solumaths
Multiply a number by 10, 100, 1000 or 0.1, 0.01, 0.001
This math game allows you to develop the technique of multiplying a decimal or integer number by powers of 10.
• Multiplying by 10, 100, 1000, .... is the same as adding 1,2,3 zeros in the case of a whole number, and shifting the decimal point to the right in the case of a decimal number.
• Multiplying by is the same as shifting the decimal point to the left by adding zeros if necessary.
Rules of this math quiz on multiplication by a power of 10
The principle of this math quiz on multiplication by powers of 10 is simple: a calculation involving the product of a whole or decimal number by 10, 100, 1000, 0.1, 0.01, 0.001 is proposed with a
list of answers. To pass this mathematical quiz, you just have to choose the right answer in the list of choices. The goal of this quiz is to multiply a number by 10, 100, 1000 or 0.1, 0.01, 0.001.
To win this game, you just have to find the right result in a list.
If you are having trouble finding the solution, this math game is able to give you the right answer which is sometimes accompanied by a detailed explanation.
Multiplication game by 0.1, 0.01, 0.001, 10, 100, 1000
This online multiplication math quiz is well suited for juniors, with this math game, they will be able to develop their calculation techniques and their mental calculation skills. | {"url":"https://www.solumaths.com/en/math-games-online/play/game-multiplication-division-by-powers-of-ten","timestamp":"2024-11-01T20:35:58Z","content_type":"text/html","content_length":"43537","record_id":"<urn:uuid:42797850-b34b-4161-917d-65f4af8005c9>","cc-path":"CC-MAIN-2024-46/segments/1730477027552.27/warc/CC-MAIN-20241101184224-20241101214224-00261.warc.gz"} |
... The Programmer God ...
A simulation universe hypothesis at the Planck scale
if we assign geometrical objects to mass, space and time,
and then link them via a unit number relationship,
we can build a physical universe from mathematical structures.
Could a Programmer God have used this approach?
The question as to whether our entire universe is a Matrix-style simulation is, like God, considered to be a philosophical debate, for it is presumed that it cannot be proved (or disproved). However
there are anomalies in the physical constants that appear to constitute evidence we are in a simulation (they cannot be explained by a physical universe). Furthermore they suggest coding and this
implies a Programmer. From these anomalies we can reverse engineer parts of the Programmer's source code. This text is a discussion of that source code.
Note 1. Readers should note that any candidate for a Programmer-God simulation-universe source code, including the code described here, must satisfy these conditions;
1. It can generate physical structures from mathematical forms.
2. The sum universe is dimensionless (simply data on a celestial hard disk).
3. We must be able to use it to derive the laws of physics (because the source code is the origin of the laws of nature, and the laws of physics are our observations of the laws of nature).
4. The mathematical logic must be unknown to us (the Programmer is a non-human intelligence).
Note 2. An AI Search tool: Go to notebooklm AI, login with your google account and add these sources (the following url links), and then ask questions in the chat box. Some of the answers are a
little strange, but it gives the links in the answers for reference. It can also generate a podcast but these seem to more about Google than about the model itself so after the first few minutes they
become garbage. The gravity site has been updated so please reinstall.
https://en.wikiversity.org/wiki/Quantum_gravity_(Planck) updated
The mathematical electron
This website introduces a part of the source code for a Planck-scale simulation universe - this is the Programmer God hypothesis (that the universe, in its entirety, including life-forms, is a
simulation programmed by an external 'hand'). Essentially it is a model of the universe based on the structure of the electron as this mathematical particle f[e], itself the geometry of 2
dimensionless physical constants (alpha and Omega).
The fundamental problem for any Programmer God is - how to create actual physical dimensions (of mass, space and time), from mathematical structures. This is because a simulation universe is
dimensionless, for it does not 'exist' in any physical sense outside of the 'Computer' (it is simply data on a celestial hard-disk).
We are familiar with opposites; plus charge and minus charge, waves of inverse phase ... and it is easy to see how these may form and/or cancel each other, however these are simply inverse
Our universe does not appear to have inverse properties such as anti-mass (-kg), anti-time (-s) or anti-space (anti-length -m), yet it is a requirement that our mass, space and time must also be able
to cancel - in order that the sum universe remain dimension-less.
Our Programmer has solved this problem by using the following geometrical artifice. Briefly, we begin by selecting 2 dimensioned quantities, here are chosen r, v such that;
No unit can cancel another (I cannot replace kg with m or s), and so we still have 4 independent units, however if 3 (or more) combine together, then we can cancel. For example;
Embedded within this f[X] are units for mass, time and length (in the above ratio), but f[X] is dimensionless, the r, v units have cancelled, units = 1 (i.e.: f[X] has no units by which it may be
measured, it is a mathematical structure).
r, v have dimensions, so we can use SI units;
Given that for
the units = 1;
If we can reduce the units (kg, m, s, A) to units = 1 via this method, then we can do the reverse and create units (kg, m, s, A) from 1. The next, and even bigger problem our Programmer must solve is
mass, space and time itself. What are they? The first clue is that they are embedded within f[X] structures, for this was how problem #1 was solved, and fortunately we have an f[X] we can
disassemble, it is the electron formula f[e]. This also means that although f[e] will embed the physical electron parameters (of wavelength m, mass kg, frequency s, charge A...) the electron itself
is a mathematical particle, there is no physical electron, for f[e] units = 1. We find that f[e] can be divided into A-m = ampere-meters and time s (note: the ampere-meter is the unit for a magnetic
The formula for f[e] resembles the formula for a (doughnut shape) torus (2π^2r^3), in other words, those parameters of the electron are embedded within a geometrical formula, and so those parameters
are themselves also geometrical formulas (geometrical objects).
Decoding this formula f[e] gives a table of our physical dimensions as objects MLTVA (the geometry of 2 dimensionless constants; alpha and Omega) along with the unit number relationship that links
them together (from dimensioned r = 8, v = 17).
We can determine the values for alpha α=137.035999084, Omega Ω=2.007134949636, r=0.712562514304 and v=11843707.905 and so can solve those MLTA geometries and from there the electron parameters. This
means that all the information needed to make the electron of physics is embedded in this electron formula f[e].
What does this mean?
Physics defines the 'physical-ness' of our universe via these physical constants. The speed of light c = 299792458 meters/second (m/s), the mass of the electron m[e] is measured in kilograms (kg),
electric charge is measured in amperes (A)... We cannot measure the distance from Tokyo to London in kilograms or amperes, and even if we could numerically, the units (m, kg, A) don't match. A
physical universe requires that somehow mass space and time exist, that mass is, time is, space is ....
The above solutions would then appear to be incompatible with a physical universe, for if the electron is a mathematical particle, then so too is the proton, and the atom .... and if our universe is
a construct of mathematical particles, then ... so too are we.
If these 'anomalies' (aka the above table) are statistically valid (solved to the required precision), then they can be construed as evidence that we are in a simulation, and also therefore as our
first evidence of a non-human intelligence, the Programmer.
For those familiar with the dimensioned physical constants and the SI units, I have listed those anomalies on this wiki site.
Are these physical constant anomalies evidence we are in a simulation?
wiki: Physical constants (anomalies)
This model suggests a geometrically autonomous universe, electrons orbit protons for example, not due to any inbuilt laws of physics, but according to geometrical imperatives (the respective
geometries of the electron and proton).
This page gives a general overview of the model. The articles (i.e.: the mathematics of this model -see articles) have also been translated onto wiki sites as these use familiar formats. Links are
given. Some tables list time object T=2π, they should read T=π.
What is immediately noticeable is the simplicity and elegance of the geometries the Programmer is using. Mass M = 1, time T = 2π, an electron formula that embeds the electron, even quantization of
the atom is via a geometrical trick (using a hyperbolic spiral) ... and in this entire model only 2 physical constants are used (alpha and Omega). This elegance is the characteristic signature of the
Programmer's handiwork.
Note: This model uses only 2 dimensionless constants (alpha, the fine structure constant and Omega). The electron formula derives from the geometry of these 2 constants, yet it can be used to solve
the fundamental physical constants (G, h, c, e, m[e], k[B]) to experimental precision. How important is this?
In the words of Prof's J. Barrow and J. Webb, Scientific American 292, 56 - 63 (2005) ...
On the physical constants; 'Some things never change. Physicists call them the {constants of nature}. Such quantities as the velocity of light, c, Newton's constant of gravitation, G, and the mass of
the electron, m[e] are assumed to be the same at all places and times in the universe. They form the scaffolding around which theories of physics are erected, and they define the fabric of our
universe. Physics has progressed by making ever more accurate measurements of their values. And yet, remarkably, no one has ever successfully predicted or explained any of the constants. Physicists
have no idea why they take the special numerical values that they do. In SI units, c is 299,792,458; G is 6.673e-11; and m[e] is 9.10938188e-31 -numbers that follow no discernible pattern. The only
thread running through the values is that if many of them were even slightly different, complex atomic structures such as living beings would not be possible. The desire to explain the constants has
been one of the driving forces behind efforts to develop a complete unified description of nature, or "theory of everything". Physicists have hoped that such a theory would show that each of the
constants of nature could have only one logically possible value. It would reveal an underlying order to the seeming arbitrariness of nature.'
Are we in a simulation?
The simulation hypothesis posits that our reality is an artificial reality, such as generated in a computer simulation. The idea was popularized in the 1999 sci-fi film 'The Matrix'. The ancestor
simulation proposes that an advanced civilization could simulate our universe to the degree that we can observe (as with VR helmets today). This version however presumes a base reality, the physical
planet of the original programmers. Conversely, a deep-universe (Programmer God) simulation begins with the big bang and constructs the universe in its entirety, down to the smallest detail (see
Planck scale).
As the language of mathematics appears to be the language used by the universe, any simulation model that can construct a physical deep-universe has these constraints;
a: the model must be able to construct physical units (of mass, space, time) from dimensionless mathematical structures from within the simulation (for the simulation itself is simply data on a
celestial hard disk and has no physical dimensions).
b: the model cannot use dimensioned constants such as G, h, c, e ... as they are a measure of physical units (see a), and so are emergent properties (generated from within the simulation) and not
fundamental (not embedded into the source code itself).
c: the model must be independent of any system of units such as kg, m, s, A ... (see a, b) and of any (artificial) numbering system.
This (the mathematical electron) model describes how the above points can be resolved.
The Programming God
As a deep-universe (see 'the Planck scale') simulation hypothesis model is programmed by an external intelligence (the Programmer God), we cannot presume a priori knowledge regarding the simulation
source code, other than from this source code the laws of nature emerge (and from which the laws of physics are derived).
Furthermore, although the source code may use mathematical forms we are familiar with (as it would be the origin of these familiar forms), this code would have been developed by a non-human
intelligence, and so we may have to develop new mathematical tools to decipher the underlying logic.
For example, the simulation code described here uses a geometrical base-15, the logic behind this is unknown, neither our physics or our mathematics have any corollary.
By implication therefore, the presence of a 'source code' that fits the above criteria could be considered as our first tangible evidence of an external intelligence (external to the universe).
We must also consider that mathematics may simply be a programming language (as with C or Basic or Java ...), and so therefore not an absolute concept in, and of, itself. Although mathematics is the
language of physics, and by extension the universe, it may be amiss to assign to mathematics a greater significance.
The Planck Scale
The science vs. God debate exists primarily because God (the 'external' hand) does not appear in the formulas of physics. There is no E = God.c2 for example, and so science has no practical use for a
God. As God has no measurable parameters, God is an untestable hypothesis.
Physics is principally divided into studies of the quantum world and the macro world (of planets and stars). These are separated by 2 successful yet incompatible theories; quantum mechanics and
relativity. However there is a deeper world, a theoretical world* that is far below the quantum world, and this is called the Planck world. The quantum scale is to the Planck scale as our planetary
scale is to the quantum scale.
It is posited here that in a deep-universe simulation, the (fundamental) mathematical laws of nature would operate at this Planck scale, and so to understand both the quantum world and the macro
world, we must first begin with the Planck world. In the Planck world we find discrete units; Planck mass is the unit of mass, Planck time is the unit of time, Planck length is the unit of length ...
proposed are geometrical objects for mass M, time T, length L ... and it is submitted that these are the origins of the Planck units.
And so it is at the Planck scale where we may find the 'hand' of the Programmer.
*Physics has no tools that can investigate much below the quantum world (the testable laws of physics mostly end around the quantum level), and so this Planck scale remains a theoretical world.
Planck vs. quantum
It is premised here that the simulation operating system works at the Planck scale, with each increment to the simulation clock-rate adding 1 unit of (Planck) time. This is similar to how we program
our digital computers.
initialise parameters
FOR age = 1 TO the-end
time = time + 1 (generate 1 time object T)
conduct certain processes
NEXT age
In this example, age is the incrementing counter (age = 1, 2 3...), it is also the origin of time (for each increment to age we add 1 dimensioned object T (a unit of Planck time), and so the universe
gets older, but the variable age itself is just a dimensionless counter. There is this distinction between the dimensionless variable age (the simulation clock-rate) and dimensioned object T (which
we measure using seconds; see Time).
age = 1 is the simulation start (a little big bang)
age = the-end is when the simulation ends
The universe is incrementing in discrete steps; age = 1, 2, 3, .... As particles (and photons) have a frequency, in other words a time component (this means they do not exist at any single unit of
time), we could consider them as an (oscillating) event that occurs over time. Time is 1 of the dimensions of particles.
For example, if 1 unit of time (1 increment to age) is a 'frame', then the electron is a 'movie'. It takes about 10^23 units of time (increments to age) to make 1 electron (1 frame does not a movie
The quantum scale is the scale at which we find our electrons and photons. This also means that we cannot interpret the Planck scale using quantum theories, rather the reverse, we must add a time
dimension to Planck scale events to interpret the quantum scale. This is why physics uses probabilities to describe quantum events, if the electron does not exist at any 1 unit of time, then we
cannot say where the electron is at any 1 unit of time.
If electrons are events that occur over time (they have a frequency), then we too do not exist at unit time, we too are the sum of many (discrete Planck scale) events averaged over time. I inhabit a
human body per second
Gravity is an example (see Gravity), if at the Planck scale there is no solid me or solid planet earth (we are both the observed result of events averaged over time), then there is nothing for a
gravity force to act on. Instead, if we replace gravity with particle to particle orbital pairs, which is what atoms do (in Hydrogen an electron orbits a proton), and rotate all these together, and
map those rotations over time, we will see satellites orbiting planets and planets orbiting stars. Orbits, like particles, emerge over time. There is no need for a gravitational force as we
understand it, and as the orbitals are the same, nor is there a need for an electric force.
At our macro-level of planets and stars, the dimensions of mass, length, time and charge (amperes), represented by the units kg, m, s and A, are independent of each other (we cannot measure the
distance from Tokyo to London using pounds or kilograms or amperes).
The units appear to be distinct (mass cannot be confused with length or time), the independence of these units then becoming an inviolable rule, as every high school science student sitting an exam
can attest (the units must always add up!).
Indeed, what characterizes a physical universe as opposed to a simulated universe is the notion that there is a fundamental structure underneath, that in some sense mass is, time is and space is …
thus we cannot write kg or s in terms of m.
However, upon examination of the f(x) ratios, we may note that they involve combinations of the units (kg m, s, A) that do not appear at our macro level, and so for us the 'mathematical' universe is
not apparent, our 'world' is dominated by the 'physical'. At the quantum level however, we find these ratios, and so there we have both jointly the mathematical (the electron formula f[e] for
example), and the physical, the dimensioned parameters (of wavelength, frequency ...).
The Programmer God -ebook
Physical units from Mathematical structures
The biggest problem with any mathematical universe approach is constructing a physical reality (the physical dimensions of mass, space and time) from mathematical structures. Our computer games may
be able to simulate our physical world, but they are still simulations of a physical reality. The 1999 film The Matrix and the ancestor simulation both still begin with a physical level (a base
reality), the planet earth.
In this Programmer God model (based on a mathematical electron), the dimensions of our universe (mass, length, time, charge) are geometrical objects MLTA at the Planck scale, furthermore these
objects do not simply represent these units (of mass, length, time, charge), they are these units, for what the Programmer has done is choose objects whereby the assigned function; of mass, length,
time ... is built into the geometry of the object itself.
We may also find that these objects are not independent, for example, M exhibits mass-ness in conjunction with L length-ness and T time-ness. This arrangement means that, for example, the length
object L can combine with the time object T to form a complex object V which is velocity (V = L/T), while still maintaining the underlying attributes of length and time, and so we can construct a
universe Lego-style by combining these simple geometrical objects to form more complex geometrical objects (such as electrons and planets).
This however necessitates that the object for length L be able to interact with the objects for time T and mass M and charge A ..., which infers that there must be some relationship between their
respective geometries, and indeed it is the evidence of a unit relationship upon which the credibility of this model depends, for this relationship is incompatible with modern physics.
Physics has a set of parameters used to define the universe; such as the speed of light, the strength of gravity ..., these are often referred to as fundamental constants as they cannot be reduced to
more fundamental structures.
The 26th General Conference on Weights and Measures (2019 redefinition of SI base units) assigned exact numerical values to 4 physical constants (h, c, e, k[B]) independently of each other (and
thereby confirming these as fundamental constants), and as they are measured in units (kg, m, s, C, ...) these units must also be independent of each other (i.e.: fundamental units).
However, if these constants are interrelated via this unit number relationship, then they cannot all be fundamental constants, and so science cannot independently assign them numerical values.
The numerical value of mass object M = 1, the SI equivalent is Planck mass = 2.18 x10^-8 kg. Therefore to convert from M to Planck mass we can use a scalar k = 2.18 x10^-8 kg where M*k = Planck mass.
M * k = 1 * 2.18 x10^-8 kg = 2.18 x10^-8 kg
We can assign to each object a scalar; mass k, time t, length l, velocity v, ampere a. The scalars have both the numerical conversion factor (for k = 2.18 x10^-8) and the units (for k = kg). The unit
number is denoted by θ.
The speed of light c = 299792458 m/s or c = 186200 miles/s ... i.e.: the numerical value of the speed of light depends on the units we use, kilometers or miles.
Likewise, if we were to meet aliens, they would write the speed of light in terms of their units, according to their numbering system, and so the numbering system and units are simply measurement
systems, light continues to travel at the same velocity regardless of how we, and the aliens, measure it.
It is proposed that these geometrical MLTVA objects are used by the universe itself, they are built into the simulation source code, and so are 'universal' and independent of any numbering system or
units. As example, the reason we can use c = 299792458 m/s or c = 186200 miles/s to measure the speed of light is because embedded within our c is this geometrical object V, which is the real speed
of light. Because this V is the geometry of Omega, and Omega has a numerical solution, Omega = 2.007134949, we can assign a numerical value to V = 2πΩ^2 = 25.312....
To this V, we then add scalar v;
v = 11843707.905 m/s such that
c = V*v = 299792458 m/s
or scalar v = 7356.08 miles/s such that
c = V*v = 186200 miles/s.
Aliens will also have a value for the speed of light but in alien units, and so their scalar v will not resemble our v (in miles or meters). But for aliens and humans alike, object V will be the
The premise is that these MLTVA geometrical objects are used by the universe itself, they are the constructs of mass, space and time. The V term doesn't measure the speed of light, it is that
quantity that bestows what we measure as the speed of light, the scalar v is just a conversion factor that we (and aliens) can use. We need a conversion factor because objects such as L or T are too
small for daily use, the units that we use, such as seconds or feet or meters, are much more practical than these MLTA units (i.e.: 1 meter, a human size unit, = 6200000000000000000000000000000000
units of this geometrical length L).
If we set our scalar v = 11843707.905m/s then our c = V*11843707.905 = 299792458m/s. If the aliens set their scalar v = @#$/^%, then their c = V*@#$/^%.
If all we are doing is adding scalars then we are achieving little of any practical value, we have just exchanged 1 system of units for another, however, if we could eliminate the scalars, i.e.: if
we eliminate scalar v, then for both us and the aliens c = V, and we would now have a common language. To do this, we use this unit number relationship.
* L is a geometrical object, to convert to our unit the meter, first we solve to a numerical value (L = 79.521193...), we are now using the numerical information encoded within these objects, the
universe uses geometry, we use numbers. In the process, the geometrical information of L is lost.
** If we must combine mass and length (volume) and time to balance our equations so that the sum universe remains dimensionless (units = 1), then in order for the universe to create time T, the time
to read this sentence for example, the universe must concurrently create mass M and space L (the universe has to get bigger and more massive). If time were to reverse, the universe must shrink
Speed of light c = object V * scalar v. Planck mass = object M * scalar k ... and so on. If we simply add scalars to each of our MLTA objects then we have achieved nothing of value. However each
scalar is not just a numerical value, but also includes a unit (v has units m/s or miles/s), and so they follow that unit number relationship, i.e.: the scalar v unit number θ = 17, k unit number θ =
15 ...
This then permits us, via this unit relationship, to define each scalar in terms of other scalars, and then we find that we need only 2 scalars to define all the other scalars. For example, the unit
number for a = 3, l = -13 and t = -30, and so 3*3 + -13*3 = -30 = t, this then means that if I know the numerical values for scalar a and scalar l then I know the numerical value for scalar t, and if
I know t and l then I know the value for k etc.
This then means that we need only 4 numbers (α, Ω and any 2 scalars) to solve our (or the aliens) physical constants. In this table we define scalars k, t, l, a in terms of scalars r and v, and so if
we know the numerical values for r and v (α, Ω have fixed values), then we can solve the constants G, h, c, e, m[e], k[B] for any chosen set of units, alien or terrestrial (see calculator below).
We can go 1 step further, and find combinations of the constants where the scalars (r, v) cancel. This would then leave us with only the 2 dimensionless constants α and Ω, which means that these
combinations are also dimensionless f[X] structures, and so solving these combinations will return the same numerical values whether we are using terrestrial units or alien units, because of course,
sans scalars, we are simply combining the MLTVA object equivalents, without scalars the MLTVA objects are the system of units we, the aliens, and the universe itself, are all using. The electron, a
dimensionless combination of MLTA objects, is an example.
This then can be applied as a test of our MLTA objects, if they are in fact the units used by the universe itself then the numerical values will be the same whether we are using our constants, alien
constants, or the MLTVA equivalents.
In column 1., I use the values taken from CODATA (the generally accepted values) and in column 2., I solve using MLTVA geometries. As the scalars have cancelled, the values are the same thus
confirming the validity of the MLTA objects (in theory column 1 is not equal to column 2, column 1 is column 2). The least precise results are obtained when using the least precise constants; G and
kB. Tables taken from the wiki site (physical constant anomalies, link below).
Are these physical constant anomalies evidence we are in a simulation?
wiki: Physical constants (anomalies)
We can use this calculator. The inputs are scalars for the speed of light v and Planck mass k; 2 fundamental units. It then solves the fundamental physical constants based on those 2 scalars. If we
input the alien scalars for (v, k), then the calculator will return the alien values for those constants. Hopefully they will be impressed and not zap us.
The electron that isn't
This model is based on the electron formula.
We can use this formula with the (Planck unit) objects for mass M and length L and frequency T to solve the electron mass, wavelength and frequency.
The frequency of the electron is 10^23 units of Planck time = f[e]T. The wavelength comprises 10^23 units of Planck length = f[e]L. However we have only 1 unit of Planck mass M per f[e] (1 unit of
mass for every 10^23 units of time). Let us suppose that the electron is centered on a Planck black hole (this unit of Planck mass). The black-hole electron thesis. For 10^23 units of Planck time,
this center is obscured by an electric 'cloud' of AL (ampere meters). These AL units then combine with a unit of T and cancel, exposing for 1 unit of Planck time that black hole center.
And so for this 1 unit of time the electron 'has' mass (1 unit of Planck mass). The universe clock ticks and the electric cloud returns. It is this black hole center which gives the electron its
point co-ordinates**, the electric state can be considered a wave-state that has no fixed co-ordinates. And so, instead of a physical particle, the electron (as with other particles) is an event that
oscillates over time between this magnetic monopole AL electric (wave) state (duration dictated by f[e]) to a mass (point) state (duration 1 unit of time). Therefore the shorter the particle
frequency, the more 'mass-like' the particle properties will appear to have, the longer the frequency, the more 'wave-like'.
This also means that mass is not a constant property of the particle, rather the electron mass that we measure is the frequency of occurrence of these units of Planck mass when averaged over time. We
can measure the energy of the electron using the formula E=hf. This h is Planck's constant, and is an energy constant, its value doesnt change, the frequency f term determines how often it occurs
(per second), the more often it occurs, the more energy we have.
We can also use E=mc2 and we get the same answer, for some reason hf=mc2. If E=hf measures the frequency of the wave-state, and for every wave-state we have a mass-state (the particle oscillates
between the 2 states), and as m refers to mass, then E=mc2 refers to the mass state, and so hf will equal mc2.
The f term measures frequency, but the c term is a constant, and so it is the m term which is the frequency term. In this formula m does not refer to a constant mass, but instead is average mass, it
measures the frequency of the mass-state.
The charge on the proton is exactly the same magnitude as the charge on the electron, they cancel perfectly, yet the charge on the proton comes from 3 quarks (UUD) and so we might have thought that
the electron also gets it charge from quarks.
Curiously, we can solve the electron formula using 3 magnetic monopoles, (AL)*(AL)*(AL). The unit numbers for (AL) = 3 -13 = -10. To continue the quark analogy, our electron would then be DDD = -30
(note: D = -1/3 unit of charge, U = 2/3 unit of charge and time T = -30), the electron as a unit of minus charge (-30);
DDD = (-10 + -10 + -10) = -30
The AL units = -10. A positron (anti-matter electron) has the same charge as a proton, but we don't have a +10 unit number.
However there is 1 more way to solve the electron formula, and that is with AV monopoles (ampere-velocity). The AV has a unit number +20, if we call this our U quark then we can do this
UUD = +20 +20 -10 = +30
Our UUD particle, although otherwise identical to the electron (returns the formula f[e]), has the same charge as the proton. If protons were formed in the early days (under intense pressure and
heat) from positrons, then we would expect the number of protons in the universe to exactly equal the number of electrons. The universe would then be electrically neutral.
If we add a proton UUD and an electron DDD, we get UUDDDD or 2(UDD), the unit numbers 20 -10 -10 = 0, so this entity would be chargeless, similiar to the neutron, which is UDD.
If we combine a U with DDD then 20 -10 -10 -10 = -10 and we get a D, so we can use an electron to swap between U and D quarks. Fun with numbers.
* In standard physics the electron is a subatomic particle ... but it is not clear to physics what a particle is, we find the following definitions;
a particle itself could be a collapsed wave function or a quantum excitation of a field or an irreducible representation of the Poincaré group or a vibrating string or a thing measured in a detector
** In the vision of quantum mechanics (in the formulas physics use), the electron is considered as a point particle with no volume and no size (-google).
*** ChatGPT (AI chatbox): According to current scientific understanding, the electron is a point-like particle, meaning that it is a very small object that is effectively a point in space and has no
size ... While it is possible to imagine such an object in a purely theoretical sense, there is no evidence to suggest that objects without size actually exist in the physical world ... it is
possible that the electron could be considered a mathematical particle. This is because, if it is indeed a dimensionless point, then it would have no physical size or shape, and its properties and
behavior would be described by mathematical equations rather than physical characteristics.
And so, although the parameters of the electron are well studied, the existence of the actual electron itself cannot be measured, or tested. Science cannot say what the electron itself is, and so it
is inferred (by its parameters). For physics, the existence of the electron, like God, is a matter of faith.
There are 3 modes of time.
1) Universe time, the simulation clock-rate. It is a dimensionless incrementing counter, here this variable is labeled age. This foward increment gives us the arrow of time. All particles experience
the same age (it is a constant throughout the universe).
initialise parameters
FOR age = 1 TO the-end
conduct certain processes
NEXT age
2) The second. For every increment to the universe clock, a dimensioned object T is generated. This T is analogous to 1 unit of Planck time and so can be measured in seconds. And so the universe
clock (that dimensionless incrementing counter age) numerically equates to, but is not the same as, the dimensioned Planck time object T (whose unit is the second). As we are adding 1 Planck time
object T per increment to the simulation clock-rate, Planck time is a constant.
initialise parameters
FOR age = 1 TO the-end
create 1 object (time T = π, unit = s)
NEXT age
3) Observer time. For the observer, time equates to a change in state, if life was a movie then the incrementing counter age would indicate the number of frames, object time T would represent each
physical frame, but we, as actors in this movie, would only be able to detect motion (a change of state). If the Gods pressed the pause button on our movie, our time would stand still, although we
could not know this. If for several frames (increments to age) there was no movement (null frames), then we would not register time passing. Only when the frames have different information can we
register time. Observer time is relative (see Relativity).
wiki: Simulation time
Gravitational and atomic orbitals
According to conventional wisdom, the moon orbits the earth, this is called a gravitational orbit, and it can be calculated precisely. There is a problem however, no one knows what gravity is.
Actually there are more problems, how to reconcile with quantum theories ... Of course, as with the electron, physics is also assuming there is a gravity.
I have argued that particles such as electrons are characterized by an (electric) wave-state to (mass) point-state oscillation. One of the dimensions of the electron therefore is time, the electron
is an event that occurs over time (this oscillation). And as planets are made of particles, then planets too are events that occur over time.
The particle mass point-state has defined coordinates, the particle wave state not, and so if we could freeze time, to 1 unit of Planck time, then we would not see a solid earth, but instead a series
of (mass) points concentrated around a certain region of space (the electric waves might blur our picture). At the next unit of time we will see a different set of points, but also in that defined
region of space (our planet earth). This is because, at any unit of Planck time, some particles are in the wave state and some in the mass state, and this keeps changing (as particles oscillate
between the 2 states). What we perceive as a solid earth is the averaging over time of all these events that are occurring at the Planck scale.
This means that there is no gravity force between the earth and moon, because at the Planck scale (for any unit of Planck time) there is no earth or moon, just points and waves concentrated around 2
regions of space. At unit (Planck) time, all the particles (in the mass state) in the earth form a link with all the particles (in the mass state) in the moon, and so we have a network of
point-to-point (particle-to-particle) pairs which then rotate 1 unit of Planck length (per unit of Planck time). The observed orbit of the earth and moon is the sum of these rotating gravitational
particle-to-particle orbital pairs when averaged over time.
The points are particles in the mass state and so gravity is associated with mass. In the atom we use the wave state, however the formulas are the same for atomic orbitals because the orbitals are
the same. We have simply exchanged the electric and gravitational forces with wave-wave and point-point rotations. There is no need to reconcile gravity with the quantum for at the Planck scale there
is no distinction. The mass point state seldom occurs, most of the time the particle is in the wave-state, and so gravity appears weak accordingly, however actually per unit time, gravity is
stronger, it is equivalent to the strong force.
As there is no earth-moon orbit per se, we don't need Newton's gravitational constant G, and we don't have a center of mass so we don't have a barycenter. We just have this n-body universe wide
network of particle to particle orbital pairs. If we plot these over time, then we will see moons orbiting planets and planets orbiting suns.
Our world does not exist at Planck time.
Atomic orbitals - the quantum scale
Evidence of the Programmer's handiwork lies in the elegance of the solutions. So far we have concentrated on the Planck scale. In the atom, the electron can occupy only certain energy levels, and
these levels have integer values, i.e.: n = 1, 2, 3... and this n is known as the principal quantum number, and from these integer number sets our quantum theories emerge. If we hit an atom with a
photon (a light wave), then it can jump between these levels. How it jumps between these levels if it cannot be between these levels is a mystery (i.e.: if the electron can only be in level 1 or
level 2 then how does it get from level 1 to level 2?). This is a valid question because the transition process takes time and so there must be an interim state. Actually we have a few more
questions, why does the electron orbit the nucleus? what precisely is the 'electric' force? and so on ...
The easiest solution of course is that the electron classically 'travels' from level 1 to level 2, but this defeats the notion of quantization, that the electron can only be in level 1 or level 2.
However, the Programmer seems to like simple solutions, so how did He/She solve this one?
A photon moves in a straight line, but if we can trap it between an electron and a proton (a standing wave) then it may rotate around a center point pulling the electron with it, and so we can use
trapped photons to make atomic orbitals. This would mean that if we hit the atom with a photon, the photon will collide with the orbital (not difficult, they are both photons), and if the orbital
absorbs this photon, then its radius will become longer and the electron will then be orbiting further from the nucleus.
Now for the Programmer's geometrical trick. As the electron moves outwards it is still rotating (the orbital still rotates as it gets longer) and so the electron follows a spiral pattern. The key
lies in the spiral, for at certain angles, which correspond to the integer levels, those angles cancel leaving us with those precise integers that we need (r = Bohr radius; 360°=4r, 360+120°=9r,
360+180°=16r, 360+216°=25r ... 720°=∞).
Physicists are happy because they can keep their quantum levels and the rest of us are happy because we can now solve orbitals using our high school math. Kudos to the Programmer.
Mapping the Bohr radius during ionization (starting from n=1), as the H atom electron reaches each n level, it completes 1 orbit (for illustration) then continues outward (actual velocity will become
slower as radius increases according to angle β).
By Malcolm Macleod (Platos Cave (physics)) - Own work, CC BY 3.0, Link
A hyperbolic alpha spiral quantises the atom
wiki: Fine structure constant spiral
The singularity and the celestial hard disk
A black-hole has a surface in 3-D space but no physical interior - this is defined as a singularity, it is where the laws of physics break down. This singularity is characterized by mass, a Planck
size black-hole would therefore include a unit of Planck mass.
Characters in our computer games are simply 1's and 0's. They may be able to study the physics of their 1's and 0's world (which is the software that defines the game), but they would have no means
to study the hard disk upon which their game resides, for that is an electro-magnetic device independent of their data world. A data address on that hard disk would be the interface - the region
where their game world ends and the hard-disk begins.
If a particle oscillates between an electric wave-state to Planck mass point state, then the particle has mass. A photon has only the wave-state and so no mass. The photon travels at the speed of
light whereas the particle doesn't move (unless pushed). We could then propose that this Planck mass point-state is the center of the particle, around which that wave-state revolves.
And so could this particle Planck mass point-state be a singularity, the interface betwen our worlds, each Planck black hole a single data address on the celestial 'hard disk' upon which our
simulation universe resides, the link between our mathematical 'data' simulation world with its laws of physics and the 'electro-magnetic hard disk' world of the Gods?
The little Big Bang
The big bang presumes that the entire universe was concentrated into a single point, time began with the big bang and the universe has been expanding since, but it is still a closed system.
The dimensionless electron formula embeds all the information it needs, that includes the Planck units themselves. Let us suppose there is a dimensionless Planck 'particle' formula f[Planck] which
also embeds the Planck units, along with any other information as required. Let us further suppose that with each increment to the clock-rate, 1 Planck f[Planck] 'particle' is added to our universe.
This formula then breaks up and the Planck units emerge (as we find with the electron formula), forming a Planck unit scaffolding to our universe.
initialise parameters
FOR age = 1 TO the-end
add 1 f[Planck] 'particle'
extract 1 object (time T = π)
extract 1 object (mass M = 1)
extract 1 object (length L = 2π^2Ω^2)
NEXT age
The universe is about 13.8 billion years old, this equates to 10^62 units of Planck time and so age = 10^62 (for each increment to age the universe adds 1 object T (1 unit of Planck time).
As mass and length units are also added proportionately, from age we can also calculate the mass and size of the universe (the universe must grow in size and mass accordingly as we are simultaneously
introducing objects M and L with every object T and so the universe is not a closed system).
When we calculate the CMB (cosmic microwave background) parameters for a 14.6 billion year old Planck unit universe (we haven't included particles yet), we find it resembles our 13.8 billion year old
The electron particle f[e] not only embedded the Planck units, it also determined the wavelength and frequency of the electron (as the electron oscillates from point (no size) to wavelength (maximum
size in space) back to point. We can draw an analogy with an f[universe] 'particle', it will not only include the instruction set for our universe (as f[e] includes all necessary information for the
electron), but will also determine the universe wavelength (maximum size of space) and when the universe will end (frequency).
Relativity as the mathematics of perspective
The mathematics of perspective is a technique used to project a 3-D image onto a 2-D screen (i.e.: a photograph or a landscape painting), using the same approach here would implement a 4-axis
expanding (at a constant rate) hyper-sphere super-structure within which 3-D space is a projection.
This expanding hyper-sphere can be used to replace independent particle motion (momentum) with motion as a function of the expansion itself. With each increment to the clock (the variable age), to
the universe is added a unit of length L (volume) and a unit of time T, the universe is thereby expanding at the speed of light c for c = 1 unit of Planck length per unit of Planck time (c = lp/tp).
As the hypersphere expands, it pulls all particles along with it. This means that all particles and objects (including us) are travelling at, and only at, the speed of this expansion, which is the
speed of light (in hyper-sphere co-ordinates). There is only this velocity c. The speed of light then also becomes the limiting speed, for if we could go faster than c we could escape the universe
As photons (the electromagnetic spectrum) have no mass state, they cannot be pulled along by the universe expansion (consequently they are date stamped, as it takes 8 minutes for a photon to travel
from the sun, that photon is 8 minutes old when it reaches us), and so photons would be restricted to a lateral motion within the hyper-sphere.
As the electromagnetic spectrum is our principal source of information regarding the environment, we have no direct means to determine this hypersphere expansion, instead we can observe only the
objects around us, as if sitting in a plane with the windows closed. And so via the electromagnetic spectrum, a 3-D relative space would be observed (as a projected image from within the 4-axis
hyper-sphere), our relativity formulas are translating between the (expanding at the speed of light) hyper-sphere co-ordinates and our observed (relative to each other) 3-D space co-ordinates.
The geometry of particle half-life
Can particle half-life be explained by the particle geometry?
The End
When we calculate the temperature of the universe, we find that it reaches absolute zero (it cannot become colder) when age = 10^123, and so the universe cannot grow larger or older. By this time the
universe will be uninhabitable, presumably the simulation will be shut down long before this, but the point is that the formula f[universe] will include this information as well (if we solve this
formula the answer will be 10^123, in comparison f[e] = 10^23).
Essentially therefore, we can consider the universe as a particle, with wavelength and frequency, if we can decode this f[universe] 'particle' formula, then continuing this analogy, we will
anticipate finding embedded within it the electron formula f[e] along with the information necessary to form protons, neutrons ... and life itself. Furthermore, as it must also be dimensionless, its
formula will include the Omega term whereby Ω^15*n (where ''n'' is an integer) as this configuration is found in all dimensionless structures. This is because, as noted earlier, the universe uses a
geometrical base-15 instead of binary numbers.
Cite wiki pages
The articles have also been transferred to wiki sites, this greatly reduces the text needed. To cite these pages, the versions are given below.
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General notes on the physical constants
In the “Trialogue on the number of fundamental physical constants” was debated the number of fundamental dimension units required, noting that "There are two kinds of fundamental constants of Nature:
dimensionless alpha and dimensionful (c, h, G). To clarify the discussion I suggest to refer to the former as fundamental parameters and the latter as fundamental (or basic) units. It is necessary
and sufficient to have three basic units in order to reproduce in an experimentally meaningful way the dimensions of all physical quantities. Theoretical equations describing the physical world deal
with dimensionless quantities and their solutions depend on dimensionless fundamental parameters. But experiments, from which these theories are extracted and by which they could be tested, involve
measurements, i.e. comparisons with standard dimensionful scales. Without standard dimensionful units and hence without certain conventions physics is unthinkable".
-Michael J. Duff et al JHEP03(2002)023.
At present, there is no candidate theory of everything that is able to calculate the mass of the electron.
-https://en.wikipedia.org/wiki/Theory-of-everything (02/2016)
Planck units (m_P, l_p, t_p, ampere A_p, T_P) are a set of natural units of measurement defined exclusively in terms of five universal physical constants, in such a manner that these five constants
take on the numerical value of G = hbar = c = 1/4pi epsilon_0 = k_B = 1 when expressed in terms of these units. These units are also known as natural units because the origin of their definition
comes only from properties of nature and not from any human construct. Max Planck wrote of these units; "we get the possibility to establish units for length, mass, time and temperature which, being
independent of specific bodies or substances, retain their meaning for all times and all cultures, even non-terrestrial and non-human ones and could therefore serve as natural units of
-Uber irreversible Strahlungsforgange. Ann. d. Phys. (4), (1900) 1, S. 69-122
In 1963, Dirac noted regarding the fundamental constants; "The physics of the future, of course, cannot have the three quantities hbar, e, c all as fundamental quantities, {only two of them can be
fundamental, and the third must be derived from those two}."
-Dirac, Paul; The Evolution of the Physicist's Picture of Nature, June 25, 2010
In the article "Surprises in numerical expressions of physical constants", Amir et al write ... In science, as in life, `surprises' can be adequately appreciated only in the presence of a null model,
what we expect a priori. In physics, theories sometimes express the values of dimensionless physical constants as combinations of mathematical constants like pi or e. The inverse problem also arises,
whereby the measured value of a physical constant admits a `surprisingly' simple approximation in terms of well-known mathematical constants. Can we estimate the probability for this to be a mere
-Ariel Amir, Mikhail Lemeshko, Tadashi Tokieda; 26/02/2016, {Surprises in numerical expressions of physical constants} arXiv:1603.00299 [physics.pop-ph]
"The fundamental constants divide into two categories, units independent and units dependent, because only the constants in the former category have values that are not determined by the human
convention of units and so are true fundamental constants in the sense that they are inherent properties of our universe. In comparison, constants in the latter category are not fundamental constants
in the sense that their particular values are determined by the human convention of units".
-Leonardo Hsu, Jong-Ping Hsu; {The physical basis of natural units}; Eur. Phys. J. Plus (2012) 127:11
A charged rotating black hole is a black hole that possesses angular momentum and charge. In particular, it rotates about one of its axes of symmetry. In physics, there is a speculative notion that
if there were a black hole with the same mass and charge as an electron, it would share many of the properties of the electron including the magnetic moment and Compton wavelength. This idea is
substantiated within a series of papers published by Albert Einstein between 1927 and 1949. In them, he showed that if elementary particles were treated as singularities in spacetime, it was
unnecessary to postulate geodesic motion as part of general relativity.
-Burinskii, A. (2005). {"The Dirac–Kerr electron"}. arXiv:hep-th/0507109
The Dirac Kerr–Newman black-hole electron was introduced by Burinskii using geometrical arguments. The Dirac wave function plays the role of an order parameter that signals a broken symmetry and the
electron acquires an extended space-time structure. Although speculative, this idea was corroborated by a detailed analysis and calculation.
Mathematical Platonism is a metaphysical view that there are abstract mathematical objects whose existence is independent of us.
-Linnebo, Øystein, {"Platonism in the Philosophy of Mathematics"}, The Stanford Encyclopedia of Philosophy (Summer 2017 Edition), Edward N. Zalta (ed.), plato.stanford.edu/archives/sum2017/entries/
Mathematical realism holds that mathematical entities exist independently of the human mind. Thus humans do not invent mathematics, but rather discover it. Triangles, for example, are real entities,
not the creations of the human mind.
-https://en.wikipedia.org/wiki/Philosophy-of-mathematics (22, Oct 2017). | {"url":"https://codingthecosmos.com/","timestamp":"2024-11-07T02:17:39Z","content_type":"text/html","content_length":"106843","record_id":"<urn:uuid:cabad963-21ac-474b-9fe0-4bcf6b539654>","cc-path":"CC-MAIN-2024-46/segments/1730477027951.86/warc/CC-MAIN-20241107021136-20241107051136-00419.warc.gz"} |
Printable Multiplication Table Worksheet
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Encouraging constant practice, supplying help, and producing a positive knowing atmosphere are useful actions. | {"url":"https://crown-darts.com/en/printable-multiplication-table-worksheet.html","timestamp":"2024-11-04T07:19:54Z","content_type":"text/html","content_length":"29050","record_id":"<urn:uuid:8cddf1eb-fe67-48ea-b01b-f30fa43e7e48>","cc-path":"CC-MAIN-2024-46/segments/1730477027819.53/warc/CC-MAIN-20241104065437-20241104095437-00887.warc.gz"} |
Question 716:
No answer provided yet.We can use the formula for the z-score for a datapoint to solve this one. The formula is (datapoint-mean)/standard deviation = z-score. Before we do this, we need to find the
z-scores for the three proportions .80, .90 and .95 using a table of normal values for 1-sided area (or the excel function =NORMSINV() ).
We get the z-scores of .841621, 1.28155 and 1.64485.
Now we substitute each of these into the z-score equation to solve for the unknown score.
1. (data-point-mean)/standard deviation = z-score
2. (data-point-90)/10= .841621
3. data-point-90= 8.41621
4. data-point= 98.41621
Repeating this process for the other 2 values we get 102.8155 and 106.4485. So to be 80% sure they need a score of 98.41621, 90% sure 102.8155 and 95% sure a score of 106.4485.
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What is the solubility of AgCl in a 0.0045 mol/L solution of NaCl if the Ksp of AgCl is 1.8 x 10-10? - Gotit Pro
What is the solubility of AgCl in a 0.0045 mol/L solution of NaCl if the Ksp of AgCl is 1.8 x 10-10?
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1 9 Multiplication Worksheets
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Differential Equation For The Pendulum (derivation)
The underlying condition of a mechanical framework (the totality of positions and speeds of its points at some moment of time) particularly decides the greater part of its movement. It is difficult
to doubt this reality, since we learn it early. One can envision a world in which to determine the future of a system one should likewise know the acceleration at the initial moment, however
experience demonstrates to us that our world is not like this. Numerous interesting ordinay differential equations (ODEs) emerge from applications. One purpose behind comprehending these applications
in a mathematics class is that you can combine your physical intuition with your mathematical intuition in the same problem. Generally the result is an improvement of both. One such application is
the motion of pendulum, i.e. a ball of mass m suspended from a perfect inflexible rod that is fixed at one end. The issue is to depict the movement of the mass point in a steady gravitational field.
L = length of the rod measured, say, in meters,
m = mass of the ball measured, say, in kilograms,
g = acceleration due to gravity = 9.8070 m/s^2.
A so-called “simple pendulum” is an idealization of a “real pendulum” but in an isolated system using the following assumptions:
• The rod or cord on which the bob swings is massless, inextensible and always remains taut;
• The bob is a point mass;
• Motion occurs only in two dimensions, i.e. the bob does not trace an ellipse but an arc.
• The motion does not lose energy to friction or air resistance.
• The gravitational field is uniform.
• The support does not move.
Newton’s equations for the motion of a point x in a plane are vector equations
F = ma,
where F is the sum of the forces acting on the the point and a is the acceleration of the point, i.e.
a = d^2x/dt^2.
Since acceleration is a second derivative with respect to time t of the position vector, x, Newton’s equation is a second-order ODE for the position x. In x and y coordinates Newton’s equations
become two equations
F[x] = m d^2x/dt^2 , F[y] = m d^2y/dt^2 ,
where F[x] and F[y] are the x and y components, respectively, of the force F . From the figure (note definition of the angle θ) we see, upon resolving T into its x and y components, that
F[x] = −T sinθ , F[y] = T cosθ − mg.
Substituting these expressions for the forces into Newton’s equations, we obtain the differential equations
X-> −T sinθ = m d^2x/dt^2 , Y-> T cosθ − mg = m d^2y/dt^2.
From the figure we see that
A-> x = L sinθ, B-> y = L − L cosθ.
The origin of the xy-plane is chosen so that at x = y = 0, the pendulum is at the bottom.
Differentiating A and B with respect to t, and then again, gives
x’ = L cosθ θ’,
x” = L cosθ θ” − L sinθ (θ’)^2
y’ = L sinθ θ’,
y” = L sinθ θ” + L cosθ (θ’)^2.
Substitute these in X and Y to obtain
−T sinθ = m L cosθ θ” − L sinθ (θ’)^2, T cosθ − mg = m L sinθ θ” + L cosθ (θ’)^2.
Now multiply first equation by cos θ, and second by sin θ, and add the two resulting equations to obtain
−mg sinθ = mL θ”,
θ” + g/L sinθ = 0.
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