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Multi-contrast anatomical subcortical structures parcellation
The human subcortex is comprised of more than 450 individual nuclei which lie deep in the brain. Due to their small size and close proximity, up until now only 7% have been depicted in standard MRI
atlases. Thus, the human subcortex can largely be considered as terra incognita. Here, we present a new open-source parcellation algorithm to automatically map the subcortex. The new algorithm has
been tested on 17 prominent subcortical structures based on a large quantitative MRI dataset at 7 Tesla. It has been carefully validated against expert human raters and previous methods, and can
easily be extended to other subcortical structures and applied to any quantitative MRI dataset. In sum, we hope this novel parcellation algorithm will facilitate functional and structural
neuroimaging research into small subcortical nuclei and help to chart terra incognita.
Subcortical brain structures are often neglected in neuroimaging studies due to their small size, limited inter-regional contrast, and weak signal-to-noise ratio in functional imaging (Forstmann et
al., 2016; Johansen-Berg, 2013). Yet, these small and diverse structures are prominent nodes in functional networks (Marquand et al., 2017; Ji et al., 2019), and they undergo pathological alterations
already at early stages of neurodegenerative diseases (Andersen et al., 2014; Koshiyama et al., 2018). Deep brain stimulation surgery, originally performed to reduce motor symptoms in essential
tremors, is now a promising therapeutic option in later stages of Parkinson’s disease and movement disorders, as well as refractory psychiatric illnesses in obsessive-compulsive disorder, anorexia,
or depression (Forstmann et al., 2017; Mosley et al., 2018). Evolutionary genetics even uncovered that in modern humans, Neanderthal-inherited alleles were preferentially down-regulated in
subcortical and cerebellar regions compared to other brain regions (McCoy et al., 2017), suggesting these structures to be essential in making us specifically human.
Despite their importance, these areas are particularly difficult to image. Furthermore, the size, shape, and location of these brain regions changes with development and aging (Fjell et al., 2013;
Keuken et al., 2013; Yeatman et al., 2014; Herting et al., 2018). Experience-based plasticity continuously remodels myelin (Tardif et al., 2016; Hill et al., 2018; Turner, 2019), iron and other
magnetic substances accumulate with age or pathology (Andersen et al., 2014; Zhang et al., 2018), both bringing changes in the MRI appearance of subcortical regions with diverse tissue
characteristics (Draganski et al., 2011; Keuken et al., 2017).
Thus, mapping the structure and function of the subcortex is a major endeavor as well as a major challenge for human neuroscience. Extensive work available from animal brain models unfortunately does
not translate in a straightforward way to human subcortical anatomy nor does it shed much light on its involvement in human cognition (Steiner and Tseng, 2017). Besides serious difficulties in
obtaining adequate measures of subcortical neural activity in functional MRI (de Hollander et al., 2017; Miletić et al., 2020), atlases and techniques for labeling accurately and reliably individual
subcortical structures have also been scarce (Frazier et al., 2005; Chakravarty et al., 2006; Ahsan et al., 2007; Yelnik et al., 2007; Qiu et al., 2010; Patenaude et al., 2011), typically labeling
the thalamus, striatum (or its subdivision into caudate and putamen), and globus pallidus (internal and external segments combined), sometimes the amygdala. However, recent advances in anatomical
MRI, combining multiple contrasts and/or quantitative MRI mapping and utilizing the higher resolution achievable with 7 Tesla (7T) and above have started to reduce the gap, each mapping a few
additional structures or sub-structures, primarily the iron-rich substantia nigra, red nucleus and sub-thalamic nucleus (Keuken et al., 2013; Xiao et al., 2015; Visser et al., 2016a; Visser et al.,
2016b; Wang et al., 2016; Makowski et al., 2018; Ewert et al., 2018; Iglesias et al., 2018; Pauli et al., 2018; Sitek et al., 2019). While these efforts generated valuable atlases, they do not yet
enable to identify many subcortical structures in individual subjects. Manual delineation, on the other hand, requires extensive manual labor from highly trained experts which cannot be easily
applied to large cohorts or clinical settings.
Here, we propose a new automated parcellation technique to identify and label 17 individual subcortical structures of varying size and composition in individual subjects, based on a large
quantitative 7T MRI database (Alkemade et al., 2020), using quantitative maps of relaxation rates R1 and R2* (1/T1 and 1/T2*, respectively) and quantitative susceptibility maps (QSM) as anatomical
contrasts. The algorithm, named Multi-contrast Anatomical Subcortical Structure Parcellation (MASSP), follows a Bayesian multi-object approach similar in essence to previous efforts (Fischl et al.,
2002; Eugenio Iglesias et al., 2013; Visser et al., 2016a; Garzón et al., 2018), combining shape priors, intensity distribution models, spatial relationships, and global constraints. The main
innovation of our approach is to explicitly estimate interfaces between subcortical structures based on a joint model derived from signed distance functions. Modeling interfaces in addition to the
structure itself provides a rich basis to encode relationships and anatomical knowledge in shape and intensity priors. A voxel-wise Markovian diffusion regularizes the combined priors for each
defined interface, lowering the imaging noise. Finally, the voxel-wise posteriors for the different structures and interfaces are further combined into global anatomical parcels by topology
correction and region growing taking into account volumetric priors, which regularizes parcellation results further in smaller nuclei with low or heterogeneous contrast. To validate the results from
this new method, in a thorough comparison with expert manual labeling, we show that the proposed method provides results very close from manual raters in many structures and exhibit reasonable bias
across the adult lifespan. The method can easily be extended to new structures, can be applied to any quantitative MRI dataset and is available in Open Source as part of Nighres (Huntenburg et al.,
2018), a neuroimage analysis package aimed at high-resolution neuroimaging.
The MASSP parcellation method presented here has been trained to parcellate the following 17 structures: striatum (Str), thalamus (Tha), lateral, 3rd and 4th ventricles (LV, 3V, 4V), amygdala (Amg),
globus pallidus internal segment (GPi) and external segment (GPe), SN, STN, red nucleus (RN), ventral tegmental area (VTA), fornix (fx), internal capsule (ic), periaqueductal gray (PAG),
pedunculopontine nucleus (PPN), and claustrum (Cl), see Figure 1. These structures include the most commonly defined subcortical regions (Str, Tha, Amg, LV), the main iron-rich nuclei (GPi, GPe, RN,
SN, STN), as well as smaller, less studied areas (VTA, PAG, PPN, Cl), white matter structures (ic, fx), and the central ventricles (3V, 4V).
The 17 subcortical structures currently included in the parcellation algorithm in axial (A), sagittal (B), and coronal (C) views.
MASSP uses a data set of ten expert delineations as a basis for its modeling. From the delineations, an atlas of interfaces between structures, shape skeletons, and interface intensity histograms are
generated, and used as prior in a multiple-step non-iterative Bayesian algorithm, see Figure 2 and Materials and methods.
The MASSP parcellation pipeline.
Validation against manual delineations
In a leave-one-out validation study comparing performance with the manual delineations, MASSP performed above 95% of the level of quality of the raters for Str, Tha, 4V, GPe, SN, RN, VTA, ic in terms
of Dice overlap, the most stringent of the quality measures (see Figures 3 and 4 and Table 1). Several of the smaller structures have lower overlap ratios likely due to their smaller size (GPi, STN,
PAG, PPN). Structures with an elongated shape (fx, Cl) remain challenging, due to the fact that small differences in location can substantially reduce overlap (Bazin et al., 2016). Despite these
challenges, when comparing the dilated Dice scores, all structures were above 75% of overlap, with most reaching over 90% of the manual raters ability. Note that the Dice coefficient is very
sensitive to size, as smaller structures will have lower overlap ratios for the same number of misclassified voxels. The dilated Dice coefficient is more representative of the variability regardless
of size, as the smaller structures can reach high levels of overlap, both in manual and automated parcellations (see Table 1). The average surface distance confirms these results, showing values
generally between one and two voxels of distance at a resolution of 0.7 mm, except in the cases of Amg, LV, fx, PPN, and Cl. These structures are generally more variable (LV), elongated (fx, Cl), or
have a particularly low contrast with neighboring regions (Amg, PPN).
Leave-one-out validation of the structures parcellated by MASSP, compared to the human rater with most neuroanatomical expertise.
Inter-rater variability for the human expert raters.
Comparison to other automated methods
To provide a basis for comparison, we applied other freely available methods for subcortical structure parcellation to the same 10 subjects. MASSP performs similarly to or better than Freesurfer, FSL
FIRST and a multi-atlas registration using ANTs (see Table 2). Multi-atlas registration provides high accuracy in most structures as well, but is biased toward under-estimating the size of smaller
and elongated structures where overlap is systematically reduced across the individual atlas subjects. Multi-atlas registration is also quite computationally intensive when using multiple contrasts
at high resolution. Finally, MASSP provides many more structures than Freesurfer and FSL FIRST, and can be easily applied to new structures based on additional manual delineations.
Application to new MRI contrasts
Quantitative MRI has only become recently applicable in larger studies, thanks in part to the development of integrated multi-parameter sequences (Weiskopf et al., 2013; Caan et al., 2019). Many data
sets, including large-scale open databases, use more common T1- and T2-weighted MRI. In order to test the applicability of MASSP to such contrasts, we obtained the test-retest subset of the Human
Connectome Project (HCP, Van Essen et al., 2013) and applied MASSP to the 45 pre-processed and skull-stripped T1- and T2-weighted images from each of the two test and retest sessions. While
performing manual delineations on the new contrasts would be preferable, the model is already rich enough to provide stable parcellations. Test-retest reproducibility is similarly high for MASSP and
Freesurfer, and are generally in agreement, see Figure 5 and Table 3.
Parcellation with Freesurfer (top, on T1w image) and MASSP (bottom, on T2w image) on Human Connectome Project data.
A common concern of brain parcellation methods is the risk of biases, as they are typically built from a small number of manual delineations. Our data set is part of a large scale study of the
subcortex, for which we obtained manual delineations of the STN, SN, RN, GPe, and GPi on 105 subjects over the adult lifespan (18–80 year old, see Alkemade et al., 2020 for details). First, we
investigated the impact of atlas size. We randomly assigned half of the subjects from each decade to two groups, and built atlas priors from subsets of 3, 5, 8, 10, 12, 15, and 18 subjects from the
first group. The subjects used in the atlas were taken randomly from each decade (18-30, 31-40, 41-50,51-60, 61-70, 71-80), so as to maximize the age range represented in each atlas. Atlases of
increasing size were constructed by adding subjects to previous atlases, so that atlases of increasing complexity include all subjects from simpler atlases. Results applying these atlases to
parcellate the second group are given in Figure 6. As in previous studies (Eugenio Iglesias et al., 2013; Bazin and Pham, 2008), performance quickly stabilized with atlases of more than five subjects
(no significant difference in Welch’s t-tests between using 18 subjects or any subset of 8 or more for all structures and measures).
MASSP parcellation scores as a function of increasing number of subjects included in the atlas.
Biases due to age differences
To more specifically test the influence of age on parcellation accuracy, we defined again six age groups by decade and randomly selected 10 subjects from each group. Each set of subjects was used as
priors for the five structures above, and applied to the other age groups. Results are summarized in Figure 7. Examining this age bias, we can see a decrease in performance when parcellating subjects
in the range of 60 to 80 years of age. The choice of priors seem to have a limited impact, which varies across structures. In particular, using priors from a similar age group is not always
MASSP parcellation scores over the lifespan.
Bias on individual measures
Finally, we investigated the impact of this decrease in performance in the estimation of anatomical quantities, see Figure 8. The bias did affect the morphometric measures of structure volume and
thickness, but the effects on the local measure of thickness was reduced compared to the global measure of volume. Quantitative MRI averages were very stable even when age biases are present in the
Regression of volume (log scale), structure thickness, R1, R2*, and QSM MRI parameters estimated using manual delineations versus MASSP automated parcellations.
For reference, we report structure volumes, thickness, R1, R2* and QSM values estimated from the entire AHEAD cohort for different age groups, extending our previous work based on manual delineations
on a different data set (Keuken et al., 2017; Forstmann et al., 2014). Results are given in Table 4, describing average volumes, thickness, and quantitative MRI parameters for young, middle-aged, and
older subjects for the 17 subcortical structures.
Our goal with the MASSP algorithm was to provide a fully automated method to delineate as many subcortical structures as possible on high-resolution structural MRI now available on 7T scanners. We
modeled 17 distinct structures, taking into account location, shape, volume, and quantitative MRI contrasts to provide individual subject parcellations. Based on our results, we can be confident that
the automated parcellation technique performs comparably to human experts, providing delineations within one or two voxels of the structure boundaries (dilated Dice overlap over 75% for all
structures, including in aging groups). Results were nearly indistinguishable from expert delineations for eight major structures (Str, Tha, 4V, GPe, SN, RN, VTA, ic), and smaller structures retain
high levels of overlap, comparable to trained human raters. This parcellation includes the most commonly defined structures (Str, Tha, SN, RN, STN) with overlap scores comparable to those previously
reported (Garzón et al., 2018; Visser et al., 2016a; Eugenio Iglesias et al., 2013; Chakravarty et al., 2013; Patenaude et al., 2011). More importantly, it also includes structures seldom or never
before considered in MRI atlases and parcellation methods, such as GPe, GPi, VTA, 3V, 4V, ic, fx, PAG, PPN, Cl. The technique handles structures of varying sizes well, as indicated by dilated overlap
and boundary distance. Additional structures can be added, if they can be reliably delineated by expert raters on single-subject MRI at achievable resolutions. Some enhancement techniques such as
building a multi-subject template (Pauli et al., 2018) or adding a denoising step (Bazin et al., 2019) may be beneficial. Co-registration to a high-precision atlas as in Ewert et al., 2018 may also
improve the initial alignment over the MASSP group average template.
Age biases are present both in expert manual delineations and automated parcellation techniques. Age trajectories in volume and quantitative MR parameters indicate systematic shifts in contrast
intensities and an increasing variability with age, associated with changing myelination, iron deposition, and brain atrophy (Draganski et al., 2011; Daugherty and Raz, 2013; Fjell et al., 2013;
Keuken et al., 2017). These changes seem only to impact the parcellation accuracy for age groups beyond 60 years and age-matched priors did not provide specific improvements, thus indicating that an
explicit modeling of age effects may be required to further improve parcellation quality in elderly populations. These results also point to exercising caution when applying automated parcellation
methods to study morphometry in elderly or diseased populations, where measured differences may include biases. They also point out that while global volume and local thickness are indeed affected by
such biases, quantitative MRI measures are much more robust. Note that this bias is likely present is many automated methods, although they have not been systematically investigated due to the
extensive manual labor required. Interestingly, biases also exist in expert delineations: when the size or shape of a structure is refined in neuroanatomical studies, experts may become more or less
conservative in their delineations. Automated methods provide a more objective measure in such case, as the source of their bias is explicitly encoded in the atlas prior delineations and
computational model. Important applications of subcortical parcellation also include deep-brain stimulation surgery (Ewert et al., 2018), where the number of structures parcellated by MASSP can help
neurosurgeons orient themselves more easily, although precise targeting will still require manual refinements, especially in neurodegenerative diseases.
We observed that dilated overlap, that is, the overlap of structures up to one voxel, provided a measure of accuracy largely independent of size, for automated or manual delineations. Imprecision in
the range of one voxel in the boundary is to be expected due to partial voluming which impacts Dice overlap. The dilated overlap measure is a better representative of performance and indicates that
conservative or inclusive versions of the subcortical regions can be obtained by eroding or dilating the estimated boundary by a single voxel. Such masks may be useful when separating functional MRI
signals between neighboring nuclei or when locating smaller features inside a structure. Additionally, the Bayesian estimation framework provides voxel-wise probability values, which can also be used
to further weight the contribution of each voxel within a region in subsequent analyses.
In summary, our method provides fast and accurate parcellation for subcortical structures of varying size, taking advantage of the high resolution offered by 7T and the specificity of quantitative
MRI. The algorithm is based on an explicit model of structures given in a Bayesian framework and is free of tuning parameters. Given a different set of regions of interest or different populations,
new priors can be automatically generated and used as the basis for the algorithm. If more MRI contrasts are available, the method can also be augmented to take them into account. The main
requirement for the technique is a set of manual delineations of all the structures of interest in a small group of representative subjects. Performance may further improve with the number of
included structures, as the number of distinct interfaces increases, refining in particular the intensity priors. In future works, we plan to include more structures or sub-structures and model the
effects of age on the priors. We hope that the method, available in open source, will help neuroscience researchers to include more subcortical regions in their structural and functional imaging
Our parcellation method has been developed for the MP2RAGEME sequence (Caan et al., 2019). Briefly, the MP2RAGEME consists of two interleaved MPRAGEs with different inversions and four echoes in the
second inversion. Based on these images, one can estimate quantitative MR parameters of R1, R2* and QSM. In this work, we used the following sequence parameters: inversion times TI1,2 = 670 ms,
3675.4 ms; echo times TE1 = 3 ms, TE2,1–4 = 3, 11.5, 19, 28.5 ms; flip angles FA1,2 = 4°, 4°; TRGRE1,2 = 6.2 ms, 31 ms; bandwidth = 404.9 MHz; TRMP2RAGE = 6778 ms; SENSE acceleration factor = 2; FOV
= 205×205 x 164 mm; acquired voxel size = 0.70×0.7 x 0.7 mm; acquisition matrix was 292 × 290; reconstructed voxel size = 0.64×0.64 x 0.7 mm; turbo factor (TFE) = 150 resulting in 176 shots; total
acquisition time = 19.53 min.
T1-maps were computed using a look-up table (Marques et al., 2010). T2*-maps were computed by least-squares fitting of the exponential signal decay over the multi-echo images of the second inversion.
R1 and R2* maps were obtained as the inverse of T1 and T2*. For QSM, phase maps were pre-processed using iHARPERELLA (integrated phase unwrapping and background phase removal using the Laplacian) of
which the QSM images were computed using LSQR (Li et al., 2014). Skull information was removed through creation of a binary mask using FSL’s brain extraction tool on the reconstructed uniform
T1-weighted image and then applied to the quantitative contrasts (Smith, 2002). As all images were acquired as part of a single sequence, no co-registration of the quantitative maps was required (see
Figure 9).
MP2RAGEME maps and delineations: quantitative R1 (left), quantitative R2* (middle), QSM (right).
Anatomical structure delineations
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Manual delineations of subcortical structures were performed by two raters trained by an expert anatomist, according to protocols optimized to use the better contrast or combination of contrasts for
each structure and to ensure a consistent approach across raters. The following 17 structures were defined on a group of 10 subjects (average age 24.4, eight female): striatum (Str), thalamus (Tha),
lateral, 3rd and 4th ventricles (LV, 3V, 4V), amygdala (Amg), globus pallidus internal segment (GPi) and external segment (GPe), SN, STN, red nucleus (RN), ventral tegmental area (VTA), fornix (fx),
internal capsule (ic), periaqueductal gray (PAG), pedunculopontine nucleus (PPN), and claustrum (Cl). Separate masks for left and right hemisphere were delineated except for 3V, 4V, and fx. In the
following the algorithm treats each side separately, resulting in a total of 31 distinct structures (see Figure 1).
Anatomical interface priors
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In order to inform the algorithm, we built a series of priors derived from the manual delineations. Each subject was first co-registered to a MP2RAGEME anatomical template built from 105 subjects
co-aligned with the MNI2009b atlas (Fonov et al., 2011) with the SyN algorithm of ANTs (Avants et al., 2008) using successively rigid, affine, and non-linear transformations, high levels of
regularization as recommended for the subcortex (Ewert et al., 2019) and mutual information as cost function.
The first computed prior is a prior of anatomical interfaces, recording the most likely location of boundaries between the different structures, defined as follows. Given two delineated structures
$i,j$, let $φi,φj$ be the signed distance functions to their respective boundary, that is, $φi(x)$ is the Euclidean distance of any given voxel to the boundary of i, with a negative sign inside the
structure. Then we define the interface $Bi|j$ with the distance function $di|j$:
(1) ${d}_{i|j}\left(x\right)=\mathrm{min}\left({\phi }_{i}\left(x\right),{\phi }_{j}\left(x\right)-\delta ,0\right)$
where $δ$ is a scale parameter for the thickness of the interface. These interfaces functions are not symmetrical, as the intensity inside i next to j is generally different from the intensity inside
j next to i. Based on this definition, the prior for a given interface based on N manual delineations is given by:
(2) $\begin{array}{c}\hfill P\left(x\in {B}_{i|j}\right)\sim \frac{1}{\sqrt{2\pi {\sigma }_{i|j}^{2}\left(x\right)}}\mathrm{exp}-\frac{1}{2}\frac{{\mu }_{i|j}^{2}\left(x\right)}{{\sigma }_{i|j}^{2}\
left(x\right)}\hfill \\ \hfill {\mu }_{i|j}\left(x\right)=\frac{1}{N}\sum _{n\in N}{d}_{i|j,n}\left(x\right),{\sigma }_{i|j}\left(x\right)=\sqrt{\frac{1}{N}\sum _{n\in N}{\left({d}_{i|j,n}\left(x\
right)-{\mu }_{i|j}\left(x\right)\right)}^{2}}+\delta \hfill \end{array}$
These probability functions are calculated for all possible configurations including $i|i$, which represent the inside of each structure. We thus have a total of $N2$ functions, but only a few are
non-zero at a given voxel x, and we may keep only the 16 largest values to account for any number of interfaces in 3D (Bazin et al., 2007). Finally, we need to scale the prior to be globally
consistent with the priors below by assuming that the 95th percentile of the highest kept $P(x∈Bi|j)$ values have a probability of 0.95. The scale parameter $δ$ is set to one voxel, representing the
expected amount of partial voluming. The resulting interface prior is shown in Figure 10A.
Anatomical interface (A) and skeleton (B) priors derived from the 10 manually delineated subjects.
Anatomical skeleton priors
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Next, we defined priors for the skeleton of each structure, representing their essential shape regardless of exact boundaries (Blum, 1973). As we are mostly interested in the most likely components
of the skeleton or medial axis $Si$, we follow a simple method to estimate its location:
(3) ${S}_{i}=\left\{x,|abla {\phi }_{i}\left(x\right)|<\frac{1}{2}\right\}$
We define as $si(x)$ the signed distance function of this discrete skeleton, and define prior probabilities as above:
(4) $\begin{array}{c}\hfill P\left(x\in {S}_{i}\right)\sim \frac{1}{\sqrt{2\pi {\sigma }_{i}^{2}\left(x\right)}}\mathrm{exp}-\frac{1}{2}\frac{{\mu }_{i}^{2}\left(x\right)}{{\sigma }_{i}^{2}\left(x\
right)}\hfill \\ \hfill {\mu }_{i}\left(x\right)=\frac{1}{N}\sum _{n\in N}{s}_{i,n}\left(x\right){\sigma }_{i}\left(x\right)=\sqrt{\frac{1}{N}\sum _{n\in N}{\left({s}_{i,n}\left(x\right)-{\mu }_{i}\
left(x\right)\right)}^{2}}+\delta \hfill \end{array}$
The skeletons are defined inside each structure, which implies $P(x∈Si)≤P(x∈Bi|i)$. To respect this relationship, we scale $P(x∈Si)$ with the same factor as $P(x∈Bi|i)$ but use $P(x∈Si)$ when
combining probabilities during the estimation stage. The obtained anatomical skeleton priors are given on Figure 10B.
Interface intensity priors
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While anatomical priors already provide rich information, they are largely independent of the underlying MRI. From the co-aligned quantitative MRI maps and manual delineations, we defined intensity
priors for every interface $i|j$, in the form of intensity histograms to ensure a flexible representation of intensity distributions. Given a quantitative contrast $Rn(x)$, we built a histogram $Hi|
j,n$ for each subject n and interface $i|j$. Histograms have 200 bins covering the entire intensity range within a radius of 10 mm from any of the delineated structures. To obtain an average
histogram, we combine each histogram with a weighting function $wn(x)$ giving the likelihood of the subject’s intensity measurement compared to the group:
(5) ${w}_{i|j,n}\left(x\right)=P\left(x\in {B}_{i|j}\right)\frac{1}{\sqrt{2\pi {\sigma }_{R}^{2}}}\mathrm{exp}-\frac{1}{2}\frac{{\left({R}_{n}\left(x\right)-{\mu }_{R}\left(x\right)\right)}^{2}}{{\
sigma }_{R}{\left(x\right)}^{2}}$
where $μR(x)$ is the median of the $Rn(x)$ values at x, and $σR(x)$ is 1.349 times the inter-quartile range of $Rn(x)$. These are robust estimators of the mean and standard deviation, used here
to avoid biases by intensity outliers. To further combine the R1, R2*, and QSM contrasts we take the geometric mean of the histogram probabilities: $Hi|j(x)=∏RHi|j(R(x))1/3$.
The last type of priors extracted from manual delineations are volume priors for each of the structure. Here, we assume a log-normal distribution for the volumes $Vi$ and simply estimate the mean
$μV,i$ and standard deviation $σV,i$ of $logVi,n$ over the subjects.
Voxel-wise posterior probabilities
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When parcellating a new subject, we first co-register its R1, R2*, and QSM maps jointly to the template and use the inverse transformation to deform the anatomical priors into subject space. Then we
derive voxel-wise posteriors as follows:
(6) $\begin{array}{c}\hfill P\left(x\in {B}_{i|j}|R\left(x\right)\right)\sim P\left(x\in {B}_{i|j}\right){H}_{i|j}\left(x\right)\text{if}ie j\hfill \\ \hfill \text{and}\hfill \\ \hfill P\left(x\in
{B}_{i|i}|{S}_{i}\left(x\right),R\left(x\right)\right)\sim \mathrm{max}\left(P\left(x\in {B}_{i|i}\right),P{\left(x\in {S}_{i}\right)}^{1/2}\right){H}_{i|i}\left(x\right)\hfill \end{array}$
Once again we should compute all possible combinations, but due to the multiplication of the priors we can restrict ourselves to the 16 highest probabilities previously estimated. To balance the
contribution of the anatomical priors and the intensity histograms, we also need to normalize the intensity priors sampled on the subject’s intensities. We use the same approach, namely assuming that
the 95th percentile of the highest kept $Hi|j(x)$ values have a probability of 0.95, separately for each contrast. The voxel-wise parcellation and posteriors obtained are shown in Figure 11A.
Successive parcellation results: (A) voxel-wise posteriors and parcellation, (B) diffused posteriors and parcellation, (C) topology-corrected posteriors and final region-growing parcellation.
The voxel-wise posteriors are independent from each other and do not reflect the continuous nature of the structures. The next step is to combine information from neighboring voxels. We define a
sparse Markov Random Field model for the posteriors:
(7) $P\left(x\in {B}_{i|j}|R,S,C\right)=\sum _{y\in C\left(x\right)}P\left(y\sim x|R\right)P\left(y\in {B}_{i|j}|R,S,C\right)$
with $P(y∼x|R)=∏Rexp-(R(y)-R(x))2/2σR2$, where $σR$ is the median of the standard deviations $σi|j,R$ of the contrast histograms $Hi|j(R(x))$. The neighborhood $C(x)$ is defined as x itself and
the four 26-connected neighboring voxels with highest probability $P(y∼x|R)$, thus representing the neighbors most likely to be connected to x. The model is similar to a diffusion process and can be
estimated with an iterated conditional modes (ICM) approach, updating sequentially the probabilities (Bazin and Pham, 2007):
(8) $P\left(x\in {B}_{i|j}|R,S,C\right)←\sum _{y\in C\left(x\right)}P\left(y\sim x|R\right)P\left(y\in {B}_{i|j}|R,S,C\right)$
from the initial voxel-wise posteriors until the ratio of changed parcellation labels decreases below 0.1, typically within 50–80 iterations. The diffused probabilities and parcellation are shown in
Figure 11B.
The final step of the parcellation algorithm takes a global view of the individual structures, growing from the highest posterior values inside toward the boundaries. This region growing approach
makes the implicit assumption that posterior maps should be monotonically decreasing from inside to outside, which is not necessarily the case. Therefore, we perform first a topology correction step
on the individual structure posteriors $P(x∈i|R,B,S,C)=maxi|jP(x∈Bi|j|R,S,C)$ with a fast marching algorithm (Bazin and Pham, 2007). While the corrected posterior is very similar to the original one
(see Figure 11C), it ensures that all regions obtained by growing to a threshold have spherical object topology.
Last, we turn the posteriors into optimized parcellations, by growing them concurrently (to avoid overlaps) until the target volume for each structure is reached. Given the volume $Vi(R,B,S,C)$ of
the parcellation of the diffused and topology-corrected posteriors, we define the following target volume:
(9) ${\stackrel{^}{V}}_{i}=P\left({V}_{i}|{\mu }_{V,i},{\sigma }_{V,i}\right){V}_{i}+\left(1-P\left({V}_{i}|{\mu }_{V,i},{\sigma }_{V,i}\right)\right)\mathrm{exp}{\mu }_{V,i}$
taking a weighted average of the volume estimated from the data and the prior volume. This approach ensures that even in extreme cases where some structures have low posteriors, they are still able
to grow to a plausible size. The region growing algorithm is driven from the most likely voxels, defined as $P(x∈i|R,B,S,C)-maxj≠iP(x∈j|R,B,S,C)$, and further modulated to follow isocontours of the
skeleton prior:
(10) $\begin{array}{cc}\hfill P\left(x←y\right)\sim & P\left(y\in i|R,B,S,C\right)-{\text{max}}_{je i}P\left(y\in j|R,B,S,C\right)\hfill \\ & -|P\left(y\in {S}_{i}\right)-P\left(x\in {S}_{i}\right)|\
hfill \end{array}$
Directionality of internal structures is a useful tool for understanding mechanical function in bones (Maquer et al., 2015). Here, we adapt this concept by using the skeleton isocontours as a
representation of internal directionality, maintaining the intrinsic shape of structures. Thus, voxels with highest probability compared to the other structures and with similar distance to the
internal skeleton are preferentially selected. The final parcellation is given in Figure 11C.
To validate the method against manual expert delineations, we compared the MASSP results and the expert delineations with the following three measures:
1. The Dice overlap coefficient (Dice, 1945) $D(A,B)=2A∩BA+B$, which measures the strict overlap between voxels in both delineation;
2. The dilated overlap coefficient $dD(A,B)=A∪d(B)+B∪d(A)A+B$, where $d(.)$ is a dilation of the delineation by one voxel, which measures the overlap between delineations allowing for one voxel
of uncertainty;
3. The average surface distance $asd(A,B)$, measuring the average distance between voxels on the surface boundary of the first delineation to the other one and reciprocally, which measures the
distance between both delineations.
We computed all three measures for the manual delineations from the two independent raters, as well as the ratio of overlaps (automated over manual) and distances (manual over automated) to compare
both performances, as detailed in the Results section.
Comparisons with other automated methods
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To assess the performance of MASSP compared to existing parcellation tools, we ran Freesurfer (Fischl et al., 2002), FSL FIRST (Patenaude et al., 2011) and a multi-atlas registration approach
(co-registering 9 of the 10 manually delineated subjects on the remaining one with ANTs [Avants et al., 2008] and labeling each structure by majority voting, similarly to the MAGeT Brain approach of
Chakravarty et al., 2013). Freesurfer and FIRST were run on the skull-stripped R1 map, while the multi-atlas approach used all three R1, R2*, and QSM contrasts. All methods were compared in terms of
Dice overlap, dilated overlap and average surface distance. We also assessed the presence of a systematic volume bias, defined as the average of the signed difference of the estimated structure
volume to the manually delineated volume, normalized by the manually delineated volume.
Application to new MRI contrasts
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Before applying MASSP to unseen contrasts, we need to convert its intensity prior histograms $Hi|j,R$ to the new intensities. In order to perform this mapping, we first created a groupwise median of
the HCP subjects, by co-registering every subject to the MASSP template using ANTs with non-linear registration and both T1w, T2w contrasts matched to the template’s R1 and R2* maps. The histogram
bins are then updated as follows:
(11) ${H}_{\text{bin},i|j,R}\equiv \sum _{x|R\left(x\right)\in \text{bin}}P\left(x\in {B}_{i|j}\right){H}_{i|j,R1}{H}_{i|j,R2*}{H}_{i|j,QSM}$
adding the joint probability of the quantitative contrasts weighted by their importance for each interface to define the new intensity histograms. This model is essentially projecting the joint
likelihood of the MASSP contrasts onto the new contrasts, assuming that the co-registration between the two is accurate enough. With these new histograms, we compared the test-retest reliability and
overall agreement of MASSP with Freesurfer parcellations included in the HCP pre-processed data set.
Measurement of structure thickness
Request a detailed protocol
Finally, when comparing derived measures obtained over the lifespan with MASSP compared to manual delineations, we explored the utility of a shape thickness metric, based on the medial
representation. Given the signed distance function $φi$ of the structure boundary and s[i] of the structure skeleton, the thickness is given by:
(12) $t{h}_{i}\left(x\right)=2\left({s}_{i}\left(x\right)-{\phi }_{i}\left(x\right)\right)$
Like in cortical morphometry, thickness is a local measure, defined everywhere inside the structure, and expected to provide additional information about anatomical variations. Indeed, a similar
measure of shape thickness has recently been able to highlight subtle anatomical changes in depression (Ho et al., 2020).
The proposed method, Multi-contrast Anatomical Subcortical Structure Parcellation (MASSP), has been implemented as part of the Nighres toolbox (Huntenburg et al., 2018), using Python and Java for
optimized processing. The software is available in open source from (release-1.3.0) and . A complete parcellation pipeline is included with the Nighres examples. Computations take under 30 min per
subject on a modern workstation.
The tool presented in this article is available in open source on Github (https://github.com/nighres/nighres). The atlases necessary to run the algorithm have been deposited on the University of
Amsterdam FigShare (https://doi.org/10.21942/uva.12074175.v1 and https://doi.org/10.21942/uva.12301106.v2). A single sample subject data set has been deposited on the University of Amsterdam FigShare
(https://doi.org/10.21942/uva.12280316.v2). All the measurements used to generate the figures included in the article have been deposited on the University of Amsterdam FigShare (https://doi.org/
52. Book
Handbook of Basal Ganglia Structure and Function
In: Steiner H, Tseng K, editors. Handbook of Behavioral Neuroscience, 24. Elsevier. pp. 1–1036.
58. Conference
Generation and evaluation of an ultra-high-field atlas with applications in DBS planning
Proceedings of SPIE Medical Imaging.
Article and author information
Author details
Nederlandse Organisatie voor Wetenschappelijk Onderzoek (VICI)
Nederlandse Organisatie voor Wetenschappelijk Onderzoek (STW)
• Anneke Alkemade
• Birte U Forstmann
The funders had no role in study design, data collection and interpretation, or the decision to submit the work for publication.
We thank Josephine Groot, Nikita Berendonk, Nicky Lute for their help collecting the AHEAD database, and Wietske van der Zwaag and Matthan Caan for their help in setting up the MP2RAGEME sequence. We
also thank Steven Miletić and Dagmar Timmann for stimulating discussions around this topic, and all undergraduate students who contributed to the manual delineations. This work was supported by a NWO
Vici grant (BF) and a NWO STW grant (AA, BF). HCP data were provided by the Human Connectome Project, WU-Minn Consortium (Principal Investigators: David Van Essen and Kamil Ugurbil; 1U54MH091657)
funded by the 16 NIH Institutes and Centers that support the NIH Blueprint for Neuroscience Research; and by the McDonnell Center for Systems Neuroscience at Washington University.
Human subjects: Informed consent and consent to publish, including consent to publish anonymized imaging data, was obtained for all subjects. Ethical approval was obtained with the University of
Amsterdam Faculty of Social and Behavioral Sciences LAB Ethics Review Board, with ERB number 2016-DP-6897.
© 2020, Bazin et al.
This article is distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use and redistribution provided that the original author and source are credited.
Views, downloads and citations are aggregated across all versions of this paper published by eLife.
A two-part list of links to download the article, or parts of the article, in various formats.
Downloads (link to download the article as PDF)
Open citations (links to open the citations from this article in various online reference manager services)
Cite this article (links to download the citations from this article in formats compatible with various reference manager tools)
1. Pierre-Louis Bazin
2. Anneke Alkemade
3. Martijn J Mulder
4. Amanda G Henry
5. Birte U Forstmann
Multi-contrast anatomical subcortical structures parcellation
eLife 9:e59430.
Further reading
1. Developmental Biology
2. Neuroscience
Otolith organs in the inner ear and neuromasts in the fish lateral-line harbor two populations of hair cells oriented to detect stimuli in opposing directions. The underlying mechanism is highly
conserved: the transcription factor EMX2 is regionally expressed in just one hair cell population and acts through the receptor GPR156 to reverse cell orientation relative to the other
population. In mouse and zebrafish, loss of Emx2 results in sensory organs that harbor only one hair cell orientation and are not innervated properly. In zebrafish, Emx2 also confers hair cells
with reduced mechanosensory properties. Here, we leverage mouse and zebrafish models lacking GPR156 to determine how detecting stimuli of opposing directions serves vestibular function, and
whether GPR156 has other roles besides orienting hair cells. We find that otolith organs in Gpr156 mouse mutants have normal zonal organization and normal type I-II hair cell distribution and
mechano-electrical transduction properties. In contrast, gpr156 zebrafish mutants lack the smaller mechanically evoked signals that characterize Emx2-positive hair cells. Loss of GPR156 does not
affect orientation-selectivity of afferents in mouse utricle or zebrafish neuromasts. Consistent with normal otolith organ anatomy and afferent selectivity, Gpr156 mutant mice do not show overt
vestibular dysfunction. Instead, performance on two tests that engage otolith organs is significantly altered – swimming and off-vertical-axis rotation. We conclude that GPR156 relays hair cell
orientation and transduction information downstream of EMX2, but not selectivity for direction-specific afferents. These results clarify how molecular mechanisms that confer bi-directionality to
sensory organs contribute to function, from single hair cell physiology to animal behavior.
2. Pain is a private experience observable through various verbal and non-verbal behavioural manifestations, each of which may relate to different pain-related functions. Despite the importance of
understanding the cerebral mechanisms underlying those manifestations, there is currently limited knowledge of the neural correlates of the facial expression of pain. In this functional magnetic
resonance imaging (fMRI) study, noxious heat stimulation was applied in healthy volunteers and we tested if previously published brain signatures of pain were sensitive to pain expression. We
then applied a multivariate pattern analysis to the fMRI data to predict the facial expression of pain. Results revealed the inability of previously developed pain neurosignatures to predict the
facial expression of pain. We thus propose a facial expression of pain signature (FEPS) conveying distinctive information about the brain response to nociceptive stimulations with minimal or no
overlap with other pain-relevant brain signatures associated with nociception, pain ratings, thermal pain aversiveness, or pain valuation. The FEPS may provide a distinctive functional
characterization of the distributed cerebral response to nociceptive pain associated with the socio-communicative role of non-verbal pain expression. This underscores the complexity of pain
phenomenology by reinforcing the view that neurosignatures conceived as biomarkers must be interpreted in relation to the specific pain manifestation(s) predicted and their underlying function
(s). Future studies should explore other pain-relevant manifestations and assess the specificity of the FEPS against simulated pain expressions and other types of aversive or emotional states.
3. Animals navigate by learning the spatial layout of their environment. We investigated spatial learning of mice in an open maze where food was hidden in one of a hundred holes. Mice leaving from a
stable entrance learned to efficiently navigate to the food without the need for landmarks. We developed a quantitative framework to reveal how the mice estimate the food location based on
analyses of trajectories and active hole checks. After learning, the computed ‘target estimation vector’ (TEV) closely approximated the mice’s route and its hole check distribution. The TEV
required learning both the direction and distance of the start to food vector, and our data suggests that different learning dynamics underlie these estimates. We propose that the TEV can be
precisely connected to the properties of hippocampal place cells. Finally, we provide the first demonstration that, after learning the location of two food sites, the mice took a shortcut between
the sites, demonstrating that they had generated a cognitive map. | {"url":"https://elifesciences.org/articles/59430","timestamp":"2024-11-12T20:12:56Z","content_type":"text/html","content_length":"502429","record_id":"<urn:uuid:2c4686e1-1d3e-461a-a7af-e62f7bfb3455>","cc-path":"CC-MAIN-2024-46/segments/1730477028279.73/warc/CC-MAIN-20241112180608-20241112210608-00720.warc.gz"} |
[GAP Forum] solving equations
stefan at mcs.st-and.ac.uk stefan at mcs.st-and.ac.uk
Tue May 18 21:11:12 BST 2010
Dear Forum,
Jan Schneider asked:
> can anybody tell me how to solve multiple equations with GAP?
> I know, it's a pretty easy question, but I just started working with GAP.
> For example, what do I have to do to let GAP calculate 5*x+y = 17, x*y=18?
You can proceed as follows:
First define the variables you need:
gap> x := Indeterminate(Rationals,1);; SetName(x,"x");
gap> y := Indeterminate(Rationals,2);; SetName(y,"y");
Then compute a reduced Groebner basis for the ideal of C[x,y] defined by
your equations, for lex order:
gap> ReducedGroebnerBasis([5*x+y-17,x*y-18],MonomialLexOrdering());
[ y^2-17*y+90, x+1/5*y-17/5 ]
Then solve the first equation y^2-17*y+90 = 0 for y (note that it is
an equation in y, only). Finally, insert the solutions into the second
equation x+1/5*y-17/5 = 0 to compute the possible values of x.
This process works in a similar way general.
-- You can find details in standard textbooks like
Cox, Little, O'Shea: Ideals, Varieties, and Algorithms:
An Introduction to Computational Algebraic Geometry and Commutative Algebra.
If you encounter a univariate polynomial whose Galois group is solvable,
you can use the GAP Package RadiRoot
( see http://www.gap-system.org/Packages/radiroot.html ) by Andreas Distler
to compute representations of its roots in terms of radicals.
There is presently no code in GAP to compute solutions numerically.
Hope this helps,
Stefan Kohl
More information about the Forum mailing list | {"url":"https://www.gap-system.org/ForumArchive2/2010/002801.html","timestamp":"2024-11-14T10:25:39Z","content_type":"text/html","content_length":"4127","record_id":"<urn:uuid:ca6e58df-1c70-4996-842b-b712a4898402>","cc-path":"CC-MAIN-2024-46/segments/1730477028558.0/warc/CC-MAIN-20241114094851-20241114124851-00718.warc.gz"} |
SCMRING4: Relocability for { \bf SCM } over Ring
theorem Th6
for R being non
trivial Ring
for s1, s2 being
(SCM R)
= IC
s2 & ( for a being
of R holds s1
a ) holds
theorem Th7
for n being
for R being non
trivial Ring
for a, b being
of R
for s1, s2 being
(SCM R)
for P1, P2 being
(SCM R)
for q being
NAT -defined
(SCM b[2]) -valued finite
halt-free Function
for p being non
[9] -autonomic FinPartState
(SCM R)
st p
s1 & p
s2 & q
P1 & q
P2 &
(Comput (P1,s1,n))
b & a
in dom
p holds
(Comput (P1,s1,n)) .
= (Comput (P2,s2,n)) .
theorem Th8
for n being
for R being non
trivial Ring
for a, b being
of R
for s1, s2 being
(SCM R)
for P1, P2 being
(SCM R)
for q being
NAT -defined
(SCM b[2]) -valued finite
halt-free Function
for p being non
[9] -autonomic FinPartState
(SCM R)
st p
s1 & p
s2 & q
P1 & q
P2 &
(Comput (P1,s1,n))
= AddTo
(a,b) & a
in dom
p holds
((Comput (P1,s1,n)) . a) + ((Comput (P1,s1,n)) . b) = ((Comput (P2,s2,n)) . a) + ((Comput (P2,s2,n)) . b)
theorem Th9
for n being
for R being non
trivial Ring
for a, b being
of R
for s1, s2 being
(SCM R)
for P1, P2 being
(SCM R)
for q being
NAT -defined
(SCM b[2]) -valued finite
halt-free Function
for p being non
[9] -autonomic FinPartState
(SCM R)
st p
s1 & p
s2 & q
P1 & q
P2 &
(Comput (P1,s1,n))
= SubFrom
(a,b) & a
in dom
p holds
((Comput (P1,s1,n)) . a) - ((Comput (P1,s1,n)) . b) = ((Comput (P2,s2,n)) . a) - ((Comput (P2,s2,n)) . b)
theorem Th10
for n being
for R being non
trivial Ring
for a, b being
of R
for s1, s2 being
(SCM R)
for P1, P2 being
(SCM R)
for q being
NAT -defined
(SCM b[2]) -valued finite
halt-free Function
for p being non
[9] -autonomic FinPartState
(SCM R)
st p
s1 & p
s2 & q
P1 & q
P2 &
(Comput (P1,s1,n))
= MultBy
(a,b) & a
in dom
p holds
((Comput (P1,s1,n)) . a) * ((Comput (P1,s1,n)) . b) = ((Comput (P2,s2,n)) . a) * ((Comput (P2,s2,n)) . b)
theorem Th11
for n being
for R being non
trivial Ring
for a being
of R
for loc being
for s1, s2 being
(SCM R)
for P1, P2 being
(SCM R)
for q being
NAT -defined
(SCM b[2]) -valued finite
halt-free Function
for p being non
[9] -autonomic FinPartState
(SCM R)
st p
s1 & p
s2 & q
P1 & q
P2 &
(Comput (P1,s1,n))
loc & loc
<> (IC (Comput (P1,s1,n))) +
1 holds
(Comput (P1,s1,n)) .
= 0.
R iff
(Comput (P2,s2,n)) .
= 0.
R )
theorem Th12
for k being
for R being non
trivial Ring
for s1, s2 being
(SCM R)
for q being
NAT -defined
(SCM b[2]) -valued finite
halt-free Function
for p being non
[5] -autonomic FinPartState
(SCM R)
st p
s1 &
s2 holds
for P1, P2 being
(SCM R)
st q
P1 &
P2 holds
for i being
(IC (Comput (P1,s1,i))) +
= IC (Comput (P2,s2,i))
(CurInstr (P1,(Comput (P1,s1,i))))
= CurInstr
(Comput (P2,s2,i))
) &
(Comput (P1,s1,i)) | (dom (DataPart p)) = (Comput (P2,s2,i)) | (dom (DataPart p))
DataPart (Comput (P1,(s1 +* (DataPart s2)),i)) = DataPart (Comput (P2,s2,i)) | {"url":"https://mizar.uwb.edu.pl/version/current/html/scmring4.html","timestamp":"2024-11-12T08:52:06Z","content_type":"text/html","content_length":"62834","record_id":"<urn:uuid:441238da-953a-4890-a1cc-f62d3940713c>","cc-path":"CC-MAIN-2024-46/segments/1730477028249.89/warc/CC-MAIN-20241112081532-20241112111532-00393.warc.gz"} |
The equation 12−kx2+8−ky2=1 represents
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Question Text The equation represents
Updated On Dec 7, 2022
Topic Conic Sections
Subject Mathematics
Class Class 11
Answer Type Text solution:1 Video solution: 1
Upvotes 208
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What is Spray Foam Insulation - Definition
What is Spray Foam Insulation – Definition
Spray foam insulation is a type of insulation that is sprayed in place through a gun. Spray foam insulation can be blown into walls, onto concrete slabs, on attic surfaces. Thermal Engineering
Spray Foam Insulation
Spray foam insulation is a type of insulation that is sprayed in place through a gun. Spray foam insulation can be blown into walls, onto concrete slabs, on attic surfaces, or under floors to
insulate and reduce air leakage. Spray foam can fill even the smallest cavities, creating an effective air barrier. Foam usually expands up to 30-60 times its liquid volume after it is sprayed in
place. It provides excellent resistance to air infiltration (unlike batts and blankets, which can leave bypasses and air pockets, and superior to some types of loose-fill). On the other han, the cost
of spray foam insulation can be higher compared to traditional insulation and most foams, with the exception of cementitious foams, release toxic fumes when they burn.
There are two types of spray foam insulation:
• Closed-cell foam. Closed-cell foams are better insulators. Their high-density cells are closed and filled with a gas that helps the foam expand to fill the spaces around it. Closed-cell foam is
very strong, and structurally reinforces the insulated surface.
• Open-cell foam. Open-cell foam cells are not as dense and are filled with air, which gives the insulation a spongy texture. Open-cell foam is porous, allowing water vapor and liquid water to
penetrate the insulation. On the other hand, open-cell foams will allow structural wood to breathe and they are about twice effective as a sound barrier.
Available foam insulation materials include:
Most are typically made with polyurethane or isocyanate. Cementitious foams are similar and can be applied in a similar manner but do not expand. These foams have higher fire resistance in comparison
to polyurethane or isocyanate foams.
Attic Insulation – Roof Insulation
A very important source of heat loss from a house is through roof and attic. Attic insulation is a thermally insulated, protective interior cladding procedure involving the use of glass or rock
wool, polyurethane foam or phenolic foam. It must be noted, there is a difference between insulating a pitched roof and a flat roof, and there is a difference between cold or warm loft insulation. A
cold roof insulation requires insulation at joist level to stop heat escaping through the unused roof space. A warm roof is insulated between and under the rafters of the roof itself.
The purpose of roof insulation is to reduce the overall heat transfer coefficient by adding materials with low thermal conductivity. Roof and attic insulation in buildings is an important factor to
achieving thermal comfort for its occupants. Roof insulation as well as other types of insulation reduce unwanted heat loss and also reduce unwanted heat gain. They can significantly decrease the
energy demands of heating and cooling systems. It must be added, there is no material which can completely prevent heat losses, heat losses can be only minimized.
Example of Insulation – Polyurethane Foam
Polyurethane foam (PUR) is a closed cell thermoset polymer. Polyurethane polymers are traditionally and most commonly formed by reacting a di- or poly-isocyanate with a polyol. Polyurethane foam
insulation is available in closed-cell and open-cell formulas. Polyurethane foam can be used as cavity wall insulation or as roof insulation, floor insulation, pipe insulation, insulation of
industrial installations. Insulating panels made from PUR can be applied to all elements of the building envelope. Another important aspect is that PUR can also be injected into existing cavity
walls, by using the existing openings and some extra holes.
Example – Heat Loss through a Wall
A major source of heat loss from a house is through walls. Calculate the rate of heat flux through a wall 3 m x 10 m in area (A = 30 m^2). The wall is 15 cm thick (L[1]) and it is made of bricks
with the thermal conductivity of k[1] = 1.0 W/m.K (poor thermal insulator). Assume that, the indoor and the outdoor temperatures are 22°C and -8°C, and the convection heat transfer coefficients on
the inner and the outer sides are h[1] = 10 W/m^2K and h[2] = 30 W/m^2K, respectively. Note that, these convection coefficients strongly depend especially on ambient and interior conditions (wind,
humidity, etc.).
1. Calculate the heat flux (heat loss) through this non-insulated wall.
2. Now assume thermal insulation on the outer side of this wall. Use polyurethane foam insulation 10 cm thick (L[2]) with the thermal conductivity of k[2] = 0.028 W/m.K and calculate the heat flux (
heat loss) through this composite wall.
As was written, many of the heat transfer processes involve composite systems and even involve a combination of both conduction and convection. With these composite systems, it is often convenient to
work with an overall heat transfer coefficient, known as a U-factor. The U-factor is defined by an expression analogous to Newton’s law of cooling:
The overall heat transfer coefficient is related to the total thermal resistance and depends on the geometry of the problem.
1. bare wall
Assuming one-dimensional heat transfer through the plane wall and disregarding radiation, the overall heat transfer coefficient can be calculated as:
The overall heat transfer coefficient is then:
U = 1 / (1/10 + 0.15/1 + 1/30) = 3.53 W/m^2K
The heat flux can be then calculated simply as:
q = 3.53 [W/m^2K] x 30 [K] = 105.9 W/m^2
The total heat loss through this wall will be:
q[loss] = q . A = 105.9 [W/m^2] x 30 [m^2] = 3177W
2. composite wall with thermal insulation
Assuming one-dimensional heat transfer through the plane composite wall, no thermal contact resistance and disregarding radiation, the overall heat transfer coefficient can be calculated as:
The overall heat transfer coefficient is then:
U = 1 / (1/10 + 0.15/1 + 0.1/0.028 + 1/30) = 0.259 W/m^2K
The heat flux can be then calculated simply as:
q = 0.259 [W/m^2K] x 30 [K] = 7.78 W/m^2
The total heat loss through this wall will be:
q[loss] = q . A = 7.78 [W/m^2] x 30 [m^2] = 233 W
As can be seen, an addition of thermal insulator causes significant decrease in heat losses. It must be added, an addition of next layer of thermal insulator does not cause such high savings. This
can be better seen from the thermal resistance method, which can be used to calculate the heat transfer through composite walls. The rate of steady heat transfer between two surfaces is equal to the
temperature difference divided by the total thermal resistancebetween those two surfaces.
We hope, this article, Spray Foam Insulation, helps you. If so, give us a like in the sidebar. Main purpose of this website is to help the public to learn some interesting and important information
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Excel Formula in Python: Multiply Cell Based on Conditions
In this tutorial, we will learn how to write an Excel formula in Python that multiplies a cell based on certain conditions. This formula uses the IF function along with the AND function to perform
conditional calculations. By understanding this concept, you will be able to automate calculations in Excel using Python code.
The formula we will be using is as follows:
=IF(AND(E3>=1, E3<1.099), B3*0.25, IF(AND(E3>=1.1, E3<=1.2), B3*0.5, B3))
Let's break down the formula step-by-step:
1. The first condition checks if the value in cell E3 is equal to or greater than 100.00% (1) and less than 109.9% (1.099). If this condition is met, the formula multiplies the value in cell B3 by
2. If the first condition is not met, the formula moves to the next IF statement.
3. The second condition checks if the value in cell E3 is equal to or greater than 110% (1.1) and equal to or less than 120% (1.2). If this condition is met, the formula multiplies the value in cell
B3 by 0.5%.
4. If neither of the conditions in step 1 and step 3 are met, the formula returns the value in cell B3 without any multiplication.
To understand how this formula works, let's look at some examples:
Example 1:
B C D E
1 1.05
2 1.15
3 1.2
4 1.3
Using the formula, the results would be:
• For cell C2, the value would be 0.25, because 1.05 is within the range of 100.00% to 109.9%.
• For cell C3, the value would be 1, because 1.15 is within the range of 110% to 120%.
• For cell C4, the value would be 3, because 1.3 does not meet any of the conditions, so the formula returns the value in cell B3 without any multiplication.
By following this tutorial, you now know how to write an Excel formula in Python to multiply a cell based on certain conditions. This can be useful for automating calculations and data analysis in
Excel using Python code.
An Excel formula
=IF(AND(E3>=1, E3<1.099), B3*0.25, IF(AND(E3>=1.1, E3<=1.2), B3*0.5, B3))
Formula Explanation
This formula uses the IF function along with the AND function to perform conditional calculations based on the value in cell E3.
Step-by-step explanation
1. The AND function is used to check if the value in cell E3 is equal to or greater than 100.00% (1) and less than 109.9% (1.099). If this condition is met, the formula multiplies the value in cell
B3 by 0.25.
2. If the condition in step 1 is not met, the formula moves to the next IF statement.
3. The next IF statement checks if the value in cell E3 is equal to or greater than 110% (1.1) and equal to or less than 120% (1.2). If this condition is met, the formula multiplies the value in
cell B3 by 0.5%.
4. If neither of the conditions in step 1 and step 3 are met, the formula returns the value in cell B3 without any multiplication.
For example, if we have the following data in cells B3 and E3:
| B | C | D | E |
| | | | |
| 1 | | | 1.05 |
| 2 | | | 1.15 |
| 3 | | | 1.2 |
| 4 | | | 1.3 |
The formula =IF(AND(E3>=1, E3<1.099), B30.25, IF(AND(E3>=1.1, E3<=1.2), B30.5, B3)) would return the following results: - For cell C2, the value would be 0.25, because 1.05 is within the range of
100.00% to 109.9%. - For cell C3, the value would be 1, because 1.15 is within the range of 110% to 120%. - For cell C4, the value would be 3, because 1.3 does not meet any of the conditions, so the
formula returns the value in cell B3 without any multiplication. | {"url":"https://codepal.ai/excel-formula-generator/query/96fQ3srZ/excel-formula-python-multiply-cell","timestamp":"2024-11-07T13:42:19Z","content_type":"text/html","content_length":"95499","record_id":"<urn:uuid:be879bc4-4245-4c9e-92ea-2e2363d4bfab>","cc-path":"CC-MAIN-2024-46/segments/1730477027999.92/warc/CC-MAIN-20241107114930-20241107144930-00139.warc.gz"} |
Exploring DNSSEC
While playing around with the Ethereum Name Service (ENS) and the DNSSEC oracle, I wanted to explore the DNS records of juniperspring.xyz related to DNSSEC. An introductory understanding of DNSSEC,
elliptic curve cryptography, and ECDSA is recommended for this post; recommended reading is provided as references^1^2.
Running dig juniperspring.xyz dnskey returns two records:
;; ANSWER SECTION:
juniperspring.xyz. 3316 IN DNSKEY 256 3 13 SGjhjkfPUdkl+lMIVSh0m/VBULv8dzacEjLi8F9ykoCTDxFYPMQQpYv+ mOPkEMdbFuoS11uZn7gijI4d/BMMjw==
juniperspring.xyz. 3316 IN DNSKEY 257 3 13 T428PVvB2uqJg1NaUXEoh+9lt1jbwx1Dqqu1had5cp7R48NEiGcTZlg8 +wdzDtnQrJosM+2G8fCrxnKJxYNJoQ==
Each DNSKEY record has the “Zone Key” flag set, yielding 256. The DNSKEY record containing the KSK public key additionally has the “Secure Entry Point” flag set, yielding 257. Thus the above entries
(from top to bottom) represent the ZSK public key and KSK public key, respectively.
The number 13 in this context represents the signature algorithm, specifically ECDSA Curve P-256 with SHA-256 (ECDSAP256SHA256)^3. Curve P-256 is also known as secp256r1; P-256 is the name given by
NIST. In ECDSA, the public key represents a (two-dimensional) point; the private key represents a one-dimensional scalar.
The public key $$Q_A$$ is generated by multiplication of the scalar $$d_A$$ by the known base point $$G$$. In other words, $$Q_A=d_A\cdot G$$, where the operator $$\cdot$$ represents multiplication
over the elliptic field.
ECDSA Public Key Compression
The textual representation of ECDSA public keys depends partially on whether it is a compressed or uncompressed representation. Multiplication is performed over a finite field; the size of the public
key is dependent on the size of the field.
The field size of NIST P-256 is 256 bits, or 32 bytes. An uncompressed ECDSA public key consists of an x and y‑coordinate, and since each coordinate is a field element (i.e., for NIST P-256, is at
most 32 bytes in length), an ECDSA public key for Curve P-256 requires 64 bytes, not including a prefix. When the (well-known) prefix 0x04 is prepended, the total length is 65 bytes.
The prefix is used to quickly differentiate between uncompressed (0x04) and compressed (0x02, 0x03) forms. The compressed form takes advantage of the property that the y‑coordinate of an ECDSA public
key can be unambiguously derived given the curve equation, x‑coordinate, and an odd-even flag (prefix). Solving the curve equation for y given x yields two possible values for y. One solution is
always even, one solution is always odd. By prefixing the compressed form with a well-known byte (0x02 for even, 0x03 for odd), the public key can be unambiguously specified with a total of $$32+1=
33$$ bytes.
ECDSA Key Format for DNSKEY
When used in DNSKEY records, ECDSA public keys are given in uncompressed form. References to some of the relevant RFCs are given below. Practically, this means that the DNSSEC public keys for
juniperspring.xyz should each be $32*2=64$ bytes long when no prefix is used. These keys are always exactly 64 bytes long, probably because the RFCs strictly specify uncompressed form (i.e., a prefix
would be redundant) and because small DNS records are desirable.
The RFC on DNSSEC records (RFC4034^4) describes the DNSKEY Public Key Field briefly:
The Public Key Field holds the public key material. The format depends on the algorithm of the key being stored and is described in separate documents.
ECDSA key formats for DNSSEC are specified in RFC6605^5; it says:
ECDSA public keys consist of a single value, called “Q” in FIPS 186-3. In DNSSEC keys, Q is a simple bit string that represents the uncompressed form of a curve point, “x | y”. … For P-256, each
integer MUST be encoded as 32 octets
Looking at the base64 strings from the two DNSKEY records, we have:
SGjhjkfPUdkl+lMIVSh0m/VBULv8dzacEjLi8F9ykoCTDxFYPMQQpYv+ mOPkEMdbFuoS11uZn7gijI4d/BMMjw==
T428PVvB2uqJg1NaUXEoh+9lt1jbwx1Dqqu1had5cp7R48NEiGcTZlg8 +wdzDtnQrJosM+2G8fCrxnKJxYNJoQ==
I convert these strings to hex with JavaScript:
let zsk = 'SGjhjkfPUdkl+lMIVSh0m/VBULv8dzacEjLi8F9ykoCTDxFYPMQQpYv+ mOPkEMdbFuoS11uZn7gijI4d/BMMjw=='
let ksk = 'T428PVvB2uqJg1NaUXEoh+9lt1jbwx1Dqqu1had5cp7R48NEiGcTZlg8 +wdzDtnQrJosM+2G8fCrxnKJxYNJoQ=='
let b64ToHex = (b64) => [...atob(b64)].map(c=> c.charCodeAt(0).toString(16).padStart(2,0)).join('')
b64ToHex(zsk) // '4868e18e47cf51d925fa53085528749bf54150bbfc77369c1232e2f05f729280930f11583cc410a58bfe98e3e410c75b16ea12d75b999fb8228c8e1dfc130c8f'
b64ToHex(ksk) // '4f8dbc3d5bc1daea8983535a51712887ef65b758dbc31d43aaabb585a779729ed1e3c34488671366583cfb07730ed9d0ac9a2c33ed86f1f0abc67289c58349a1'
Each of the resulting hex strings is 128 chars (64 bytes) long and does not contain a 0x04 prefix byte, consistent with the format of uncompressed ECDSA P-256 public keys.
DNSSEC and Namecheap
At the time of writing, the juniperspring.xyz domain is provided by Namecheap. The Namecheap web interface provides a toggle for enabling DNSSEC for the domain (DNSSEC is disabled by default).
ECDSAP256SHA256 as the signature algorithm by default, generates the requisite key pairs, and (I assume) stores the private keys on Namecheap’s servers. You can check the resulting DNSSEC records
with an online tool provided by Verisign Labs^6.
On the Ethereum DNSSEC Oracle
Having a working knowledge of DNS and DNSSEC enables a better understanding of the Ethereum Name Service (ENS)^7 and the Ethereum DNS Oracle^8. Submitting a proof to the DNSSEC oracle is
significantly more expensive (multiple times the cost of purchasing juniperspring.eth for four years) than purchasing a .eth domain.
Purchasing the juniperspring.eth domain name was relatively inexpensive at the time of writing, relative to the expense of submitting a proof to the DNSSEC oracle. Understanding DNSSEC signatures and
validation helps explain the high gas cost of the latter.
The ENS docs^9 explain:
Submitting proof to DNSSEC oracle takes up a lot of gas because it is heavy computation work. It will take up even more gas if you submit the first domain under the specific TLD. This is because
it submits proof of both your domain and its parent domain(eg: matoken.live, as well as .live).
Understanding DNS and DNSSEC elucidates ENS; understanding ENS and the DNSSEC oracle elucidates DNS and DNSSEC. | {"url":"https://juniperspring.org/posts/dnssec/","timestamp":"2024-11-05T23:19:59Z","content_type":"text/html","content_length":"42565","record_id":"<urn:uuid:6b448dba-eac5-4617-827c-b25f174fd920>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00090.warc.gz"} |
Network Coding Cooperation Performance Analysis in Wireless Network over a Lossy Channel, M Users and a Destination Scenario
Network Coding Cooperation Performance Analysis in Wireless Network over a Lossy Channel, M Users and a Destination Scenario ()
1. Introduction
The design and analysis of wireless cooperative transmission protocols have gained significant recent attention to improve diversity over lossy and fading channels. The fundamental idea of
cooperation, rooted in the seminal paper of Cover and El Gamal [1] , is that users within a network transmit their own information, and serve as relays to help other users’ transmissions. The
achievable gains for relay channels are shown in [2] . Several energy-efficient cooperative transmission schemes have been investigated in [3] characterizing their outage behavior. In [4] , a
cooperative network coding (NC) system for a sensor cluster is proposed to save significant amount of the transmission power, by decreasing the Automatic Repeat Requests (ARQ) for a small number of
users over the physical layer over AWGN, unlike this work where NC is proposed for a large number of users over erasure channel. Cooperation via distributed channel codes across orthogonal user
channels became popular because of its simplicity, effectiveness, and practicality [5] and [6] . A survey of recent information-theoretic results and coding solutions for cooperative wireless
communications is given in [7] and [8] , which show how to improve the diversity. This paper contributes deterministic protocols that achieve the diversity and saves in ARQ significantly. In [9] ,
physical layer cooperative NC is proposed and the Bit Error Rate (BER) for Decode and forward (DF) relay shows that combining a large number of packets does not affect the channel bandwidth
significantly mainly over decode-and-forward channel. In [10] , a comprehensive performance analysis of node cooperation is shown over the physical layer which shows that NC is a practical system
with encouraging performance and it gives the capacity theoretical limits for some NC protocols.
Most existing cooperative protocols operate in a timesharing (or frequency-sharing) manner, such that each user sends its own messages and relays its partners’ messages in separate time slots (or
frequency bands). In order to improve system throughput, it is possible to combine the users’ messages. For example, in [11] , each user transmits a linear combination of its own waveform and the
noisy waveforms received from its partners resulting in significant bandwidth savings at the cost of increased decoding complexity over a Half-Duplex Cooperative Channel. In [12] and [13] ,
cooperation is achieved through NC, as applied in [14] and [15] , where each user transmits an algebraic superposition of its own and its partners’ information, and decoding at the destination is
carried out by iterating between codewords from the two users. These schemes provide substantial coding gain over cooperative diversity techniques without NC, motivating the use of cooperation via
NC. NC in [14] allows users to recombine several input packets at a relay. This is a novel paradigm which exploits the broadcast nature of wireless channels, without resorting to more traditional
unicast or multicast transmissions. With NC, the relay can perform certain functions on the input data instead of just forwarding it. Indeed, multicast protocols based on random linear NC have been
explored in lossy channels exploiting cooperation between forwarding nodes and multiple destination nodes, showing improved throughput [16] , however, this work shows that even over one destination,
NC is a practical system in terms of bandwidth savings and complexity, when using deterministic combination instead of random combinations.
In this paper, we design and analyze a complete system that enables cooperation among multiple users through the use of NC. In contrast to [7] [9] and [11] [12] [13] , where the physical layer NC is
investigated, we consider NC over equal-length packets implemented as a sublayer on top of the link/MAC layer, as proposed in [16] [17] .
Aiming for a low complexity, practical solution suitable for the proposed scenario with small header requirements, we resort to NC over a binary field (simple XOR-ing) in a deterministic manner which
deceases the Jordan Gaussian Elimination (JGE) decoding process significantly, beside to the fact that we have no worries to repeat the same combined packet more than once when it is randomly
combined resulting to more unique linear equations for the transmitted packets. We allow for packet losses in the channel and try to recover all users’ messages at a common destination and among the
M users. The destination and the M users receive a set of linear combinations of users’ packets in two stages, with an optional third stage in case of failure to decode some messages in the previous
two stages. A user operates in cooperative mode when it forwards a network coded packet, i.e., a linear combination of its partners’ packets. We apply both analytical and simulation tools to
investigate the probability of packet error at the destination for low complexity protocols with and without binary NC. The advantage of the system is; its simplicity and practicality while still
maintaining performance gains over non-network coding solutions.
The proposed deterministic NC protocols in this work decrease the number of the transmitted packets in the second and third stages significantly which improves the network traffic and the
transmission power compared to the cooperative network without applying NC. Moreover, these protocols provide the advantage of the ability to retrieve the lost packet using JGE rather than just
asking for ARQ, taking into consideration that the applied JGE in some scenarios is simple, because the linear equations resulted from combining the packets are made by a deterministic algorithm that
gives such high diversity (independently) as shown in our work [4] where two packets deterministically combined result in a simplified JGE process with several advantages.
The paper is organized as follows: In Section 2, we describe the system model, starting with the cooperative benchmark model, and then, how users’ packets are combined for the three stages when
applying NC. Section 3 presents the probability analysis for successful recovery of all users’ packets at the destination in an erasure channel. This is followed by some simulation results and
analysis of system performance in Section 4. Section 5 concludes the paper.
2. System Model
We consider a wireless network that consists of M users N[1], N[2], ∙∙∙, N[M] that transmit messages X[1], X[2], ∙∙∙, X[M] respectively to a common destination D and to each others, where users can
hear one another over orthogonal erasure channels. We assume the channels between any two users and between the users and D can be modelled as a random packet erasure channel with q, and p
probability of packet loss between the one user and another, and between the users and D respectively, as shown in Figure 1.
Communication is performed in two or three broadcasting stages of M time slots each, using TDMA. Note that the results hold for other multiple access techniques, including FDMA. User N[i] generates
at each stage a binary packet X[i] which is broadcast during the i^th time slot. All packets are of the same length. For simplicity, we assume no packet erasure coding. The destination informs the
users with one bit broadcast feedback message when it successfully decodes all M messages. We assume that this feedback message is always perfectly transmitted.
In the first transmission stage, each user N[i] broadcasts its own packet X[i] in one time slot. Each of the M users receives a packet with probability 1 − q from the other M − 1 users. The
destination D receives a packet from each of the M users with probability 1 − p. Stage 1 ends after M time slots.
In the second transmission stage, we propose deterministic and non-deterministic cooperative combination strategies (protocols) with their relative merits and disadvantage as discussed in Subsections
2.2 to 2.4 below. Our proposed combination strategies can be seen as simple NC operation over binary field. Indeed, as in NC, a node computes a linear combination of incoming packets. However, in
traditional NC [14] , each packet is multiplied by a random coefficient, and all multiplicative coefficients are sent in the header. In our setup, all multiplicative coefficients are 1, and hence
there is no need for additional header information.
Figure 1. System Model with M = 4 users and destination D.
If not all packets are received after the second stage, either the system asks for ARQ or it goes for the third stage. The same mechanism is performed if there are missing packets in the third stage,
so, either a further stages are required or ARQ to start the re- transmission for the missing packets.
In this work, we propose two or three stages with optional further stages as shown in Figure 2.
The block diagram in Figure 2 shows that the transmission can finish in one, two, or three stages with sending next transmission or asking for ARQ to re-transmit what is not decoded with the option
of further stages if needed, however, our proposed model is for a maximum of three stages.
Figure 2. Block diagram for three transmission stages with optional further stages.
In the first stage, a maximum of M single packets could be received at D, so, if fewer than M packets received, i.e., kM, the system optionally goes for the next stage. In the second stage, NC is
applied to generate other M unique packets (unlike the single packets), to be sent to D. Hence, D will be receiving a maximum of 2 M unique packets at the end of stage 2.
The system goes for rank check to ensure that the received unique packets at the end of stage 2, i.e., (k + l) will consists M independent linear equations (rank M) which is needed to solve JGE
decoding process.
If fewer than M unique linear independent equations received; (rank M), the system goes for the third stage. In this stage, NC is applied again to obtain more M unique packets (other than what is
sent by the second stage) to be transmitted to D.
At the end of this stage, D will be receiving a maximum of 3 M unique linear equations, so, if the exact received unique packets at D can make the rank M received linear equations, i.e., (k + l + n)
can obtain M unique independent linear equations; the system goes for the next transmission, otherwise, the system resorts to ARQ.
In this work, we used Matlab to test the received packets rank, simply, we put the pivot 1 for the packet received at D, either single or combined, and zero if neither received nor combined.
Equation (1) shows the first stage (single packet):
where CT[1], CT[2], ∙∙∙, CT[M ]are the received packets at D and M nodes. So, if D receives the packet from the second node; D will put in the second raw of the receiving matrix, but if the second
packet is not received; D will put in the second raw.
When filling the whole matrix at D; rank test is performed over the total matrix to confirm obtaining rank M matrix or not, i.e., if the total number of the received linear equations provides M
linearly independent equations.
The same for combined packets, which means if the D receives a combined packet consists of X[1], X[2], and X[5] from the third node; D will put in the third raw of the received matrix.
Based on above, D will be having M 2M and 3M at the end of stage one, two and three respectively.
D confirms full reception when the rank of its matrix gives rank M, i.e., collecting M unique independent linear equations from the M, 2M or 3M received packets.
2.1. Benchmark Cooperative System
Figure 3 shows the system under investigation for M = 3 and D as an example for the benchmark cooperative system without applying NC,
Figure 3. Three user’s benchmark cooperative system without NC, solid is for first stage and dots for cooperative benchmark second stage.
Where q, and p are the probability of packet loss between one user and another, and between the users and D respectively.
Based on Figure 3, the system transmits three packets in the first stage and nine packets in the second stage, ending up with sending 12 packets in total transmission, in general, M^2 + M = M(M + 1).
Equation (1) shows the full received matrix at D and M nodes.
2.2. Stage 2 Combinations
Choice of the best combination strategy depends on the losses encountered. The main advantage of deterministic combinations over their non-deterministic counterparts is that almost no header
information is needed to inform D and the M users about which packets were combined, in fact, in some combination strategies requires only 1 bit as a header. Only one combined packet (of the same
size as the individual packets X[i]) from each user is sent to D and M. Thus, stage 2 requires only M transmission time slots and only M transmitted packets from the M nodes compared to M (M − 1)
packets in the case of the benchmark system.
2.2.1. Stage 2 M − 1 Packet Deterministic Combination
If all M users successfully recover their M − 1 partners’ packets sent from Stage 1, the system switches to fully cooperative mode, whereby user N[i] combines packets received from all other users
except its own, as given in Equation (2):
Equation (3) gives the received matrix at D and M nodes:
However, if any user fails to receive any one of its partners’ packets from Stage 1, only that user behaves selfishly by retransmitting only its own packet without resorting to Equation (2),
moreover, the raw of that particular node in the receiving matrix will be a raw of single packet. As an example, if user two does not cooperate; the second raw of the receiving matrix will be. Thus,
the network will remain in fully selfish operation only when all M users have not received one or more packets from all of their partners. In situations where such losses on such a large scale are
likely to occur; we propose to combine only two packets either in deterministic or non-deterministic way as described next in Section 2.2.2 in order to avoid falling back to selfish network
Figure 4 shows three users communicate with each other’s and with a destination D in two stages via this protocol.
Figure 4 illustrates that the users N[1], N[2] and N[3] broadcast the packets of X[1], X[2] and X[3] to D through the erasure channels with loss probability of p and to each others in with loss
probability of p in the first stage. In the second stage; N[1], N[2] and N[3] broadcast the packets of X[2] X[3], X[1] X[3] and X[1] X[2] respectively to the D and each other over the same channels
as in the first stage. Accordingly, each user receives four packets, two single ones in the first stage (M − 1), and two combined packets in the second stage (M − 1),
Figure 4. Three user’s cooperative system is solid for first stage and dots are for cooperative second stage.
taking into consideration that each user knows its own packet even before the broadcasting, unlike the D that does not know any packet before the first stage. In general, each user and the D will be
collecting or able to obtain 2M packets in term of M single packets and M combined ones.
Only one bit header is required in this protocol to confirm whether the transmitted packet is single or combined.
As in traditional wireless NC, in order to recover all M X[i]’s, a minimum of M linearly independent equations are needed at the receiving side. Since operations are done in the binary field and all
equations either reveal one unknown or contain binary sums of M − 1 unknowns, it is enough for D to receive at least M + 1 unique packets from the M users (either in selfish or cooperative mode) to
obtain M linearly independent equations at D, and just M unique packets at any user because the user knows its own packet. Under such case, D and the M users can recover all sources even if it has
not received both packets (sent during both stages) from one user. In fact, in many cases, D can even recover all M X[i]s packets with M unique packets received under many cases as will be shown in
Section 3.
One of the most important advantage for this protocol that it is possible to have M linearly equations at D and M even if there is a dead channel between two users or one user and D where a user is
considered dead if it cannot be heard by D or another user in both stages.
2.2.2. Stage 2 2-Packet Deterministic Combination
This protocol is ideal when any user N[i] is very likely to receive from stage 1 its nearest neighbour’s packet but unlikely to hear from all other M − 2 users.
In this protocol, the user transmits cooperatively in stage 2 by combining its own packet X[i] with either X[i][+1] or X[i][−1]. Addressing is cyclic such that if i = M, then i + 1 = 1.
Of course, if any user N[i] does not receive its immediate neighbour’s packet, it operates in selfish mode transmitting only its own packet in stage 2.
Same header is needed in this protocol as the previous one, i.e., one bit to confirm whether the transmitted packet is combined or single.
In Figure 5; each node sends one combined packet to the other three users and D+ resulting to a total of four packets at each user and D in this stage. Accordingly, 2 M unique packets at the end of
this stage (M packets from the first stage, and other different M packets from the first stage).
Taking into consideration that each user combines its own packet to the next packet only.
The receiving matrix at D is shown in Equation (4) for the case of the example in Figure 5 when all nodes cooperate:
Figure 5. The cooperative second stage for M = 4 with-packet deterministic combination.
In [4] , this protocol was proposed for wireless sensor network resulting to significant power saving and less traffic communication.
2.2.3. Stage 2 2-Packet Non-Deterministic Combination
This protocol is a variation of the above deterministic protocol described above, but ensures a higher likelihood of more users transmitting cooperatively in stage 2. For this strategy, any user N[i]
combines its own packet X[i] with any one other user’s packet X[j] it has received. Simulation results show that instead of finding an X[j] randomly from the buffer of received packets from stage 1,
performance is improved if X[j] is chosen when the difference
Combination of two packets at each user in such a non-deterministic way requires a header with log[2]M bits, as D and the M users need to know j.
2.2.4. Stage 2 All-Received Non-Deterministic Combination
Whilst the previous two combination strategies ensure that more users operate in cooperative mode than in the strategy of Subsection 2.2, they limit the possibility of recovery of all M users’
messages at D and the M users with as little as possible number of received packets k. We thus propose another non-deterministic combination strategy where user N[i] combines its own packet with all
other packets received from other users, such that:
where [i]. If all users’ packets are received by N[i], C[i] is made up of the sum (in binary field) of all M users’ packets. This strategy also boils down to the strategy of Subsection 2.2.3 if only
one other packet is received by N[i].
Due to its non-deterministic nature, this strategy requires an M-bit header listing all j’s combined.
After Stage 2, a total of 2M unique equations are generated by the users. A maximum of 2M unique equations is received at D and the M users, when and only when all users operate in cooperative mode.
The more users operating in selfish mode, the less unique equations are obtained. Due to transmission over a lossy network, only k ≤ 2M unique packets (unique equations) will be received by D and the
M users.
We go to the third stage of transmission if D or any user fails to receive at least M linearly independent equations from M or M + 1 unique packets in the M − 1 protocol, because it is impossible for
D or the M users to recover all M source messages. This can happen when more than one user is not heard by D at both stages (dead users). The third stage should provide novel information to D in the
form of linear combinations not previously received at D. We propose deterministic and non deterministic combination strategies for stage 3 next. This stage also requires only M time slots for
transmission. D and the M users can recover the M users’ messages when M linearly independent equations are received from all three stages. At this point, only if there are dead sources in the
network will D not be able to recover all M users’ messages.
Accordingly, D received matrix will be a random mix of ones and zeroes, which makes the JGE decoding process to be complicated and consumes significant processing time delay. Moreover, the random
combination could result to repeat the same combined packets, i.e., same ones and zeroes in D the received matrix. In such case, the replica transmitted packet will have no any unique information
which means that it consumes transmission power for no any benefit. Unlike the proposed deterministic combination where the JGE decoding process is sampler [4] and there is no any possibility to send
the same packet twice because both of the single and the combined packets are all unique.
2.3. Stage 3
The third stage is needed when the system fails to collect M linearly independent equations, i.e., when (k + l) unique linear equation shown in Figure 2, does not provide rank M matrix at the
receiving side, which means that JGE will not be able to process the data. Moreover, the third stage solve the problem of two dead users, because it is possible to obtain the rank M matrix even with
two dead users, however, a special combination is needed to do so, which is proposed in the sub-sections bellow.
2.3.1. Stage 3 M/2 Odd-Even Deterministic Combination
In this protocol, M/2 packets are combined at each user, resulting in a simple, deterministic solution. This is shown in Equation (6), where odd numbered users N[i] combine only all received packets
from their odd-numbered partners excluding their own packet. The algorithm is symmetric for even-numbered users.
All operations above are in binary field, and % denotes modulo 2 residual. Again, if user N[i] does not have all packets required to generate
Equation (7) shows the received matrix for fully cooperative stage:
Equation (7) shows the combined odd pivots in the receiving matrix when all combined packets are received either at D or M nodes. The last column is either one or zero depending on the number of
node, i.e., 1 is the number of nodes are odd and zero when it is even. for 6 nodes as an example, the second raw has just two combined packets, which are X[4] and X[6] (even combination), hence
pivots are 1 and the rest are zeroes, and same raw 3, where the pivots are ones just at X[3] and X[5] (odd combination). The last raw and last column depends on the number of users and whether it is
even or odd.
For 1 < M < 6, each user N[i] combines X[i] with X[i][+1], with cyclic addressing as shown in Table 1.
2.3.2. Stage 3 2-Packet Deterministic Combination
To reduce the possible number of users in selfish mode that would occur in the above strategy, we propose another simple, deterministic odd-even strategy whereby only two packets are combined as
shown in Equation (8).
Odd numbered users combine only the odd neighbour’s received packet X[i][+2] to their own packet. The algorithm is symmetric for even-numbered users. Again addressing is circular as in Subsection
So, each raw in the receiving matrix will have either two pivots of ones and the rest
Table 1. Packets combination when M < 6 for two and three stages.
are zeroes in the cooperative mode, and just one pivot of number one when selfish as shown in Equation (9):
Equation (9) is for odd number of nodes, however, if the number of nodes are even, just the last raw will be changing to which is for the last even node with the first even node as the combination is
a circular as in Subsection 2.3.
2.3.3. Stage 3 2-Packet Non-Deterministic Combination
This strategy is a fusion of the strategies proposed in Subsections 2.3.1 and 2.3.2 where Odd-numbered user N[i] combines its own packet with any other odd-numbered N[j]’s packet, such that [j]’s at
N[i]. The scheme is symmetric for even-numbered users and boils down to the strategy of Subsection 2.3.2 if
2.3.4. Stage 3 All-Odd-Even Received Non-Deterministic Combination
This strategy is a fusion of Subsections 2.3.1 and 2.3.2. Odd-numbered users combine all received packets from other odd-numbered users. The same goes for even-numbered users. Thus, a maximum of M/2
packets can be combined (yielding Section 2.3.1 strategy) and a minimum of 2 (Section 2.3.2 strategy). A (log[2]M) − 1 bit header is needed.
3. Probability Analysis
In this section, we determine the overall probability of destination D being able to recover all M source messages given that it has received less than 2M packets during the first two stages from the
M nodes. We assume that all nodes destination are within transmission range of one another.
The probability for each user is better than the probability of D, because the user knows its own packet, hence, we work out the probability of D as it is worse than users, though it is possible
easily to repeat the work for each user, but we will just work for the probability of full reception for D.
After the first transmission stage, the destination will receive k packets from the M nodes, where k ≤ M different packets have been received by D at the end of stage 1 with probability of error
(PER) given in Equation (10):
If decoding is not finished after the first stage, the second stage of transmission will follow.
3.1. Benchmark Probability of Error for Stage 2
We are seeking the PER to finish the decoding after stage 2 for the benchmark protocol where each user sends its own packet twice.
So, the PER for the packet to not be received after stage 2 is p^2, so, the PER to be received is: (1 − p^2), accordingly, D will finish decoding all M packet after the second stage as shown in
Equation (11):
3.2. PER for M − 1 Deterministic Combination Stage 2
Accordingly, the probability of receiving k packets after stage 2 is shown in Equation (12):
And P[Symb] is shown in (13):
In this stage, each user will either switch to fully cooperative behaviour (if it has received and decoded all the M − 1 partners’ packets) with probability
After the second stage of transmission is finished, we are interested in the probability
As we have M different packets, D must receive M linearly independent equations, i.e., any number of different equations that gives rank M receiving matrix at D. Accordingly, D evaluates the received
rank matrix, and in the case of having M rank, D informs the M users that all packets have been recovered; otherwise, more transmission’s stage is needed or D will declare that more uses are dead.
As the combination in the second stage is deterministic, the M users generates maximum of 2M unique packets. However, D needs to receive just M + 1 unique packet to guarantee obtaining rank M
receiving matrix.
In the case when D cannot hear from two users (the case of two dead users), D cannot receive M linear independent packets, which means that the received matrix’s rank is less than M, though M + 1
unique linear equations have been received.
Accordingly, we can confirm that D successfully decodes all M users’ packet when M + 1 unique packets received at D gives rank M received matrix at D as shown in Figure 6(a) and Figure 6(b).
Figure 6. M + 1 received packets at D. (a) and (b) are the solvable cases.
Where 1 denotes successfully received packet, 0 denotes lost packet, S/C denotes a packet received in either selfish or cooperative mode, and X denotes any possibility of either received or lost
packet. However, D can guarantee to obtain the M rank receiving matrix when D receives the packet in cooperative mode for one user received in stage 1 as shown in Figure 1. This means that any M + 1
different packets give M linearly independent equations, taking into consideration that we are seeking the definite probability to obtain the rank M received matrix at D.
Equation (15) and Equation (16) show the PER for these cases.
The decoding will successfully finish if for all (M + 1) − k time slots in which transmission failed in the first time slot, the transmission is successful in the second time slot. The decoding will
succeed irrespective of the behaviour (selfish or cooperative as long M rank linear equations received) of users in these M − k time slots, this is illustrated in Figure 6(a),
Moreover, even if one of M − k users’ packets is not received in the first time slots, remains not received in the second time slots (dead case), the decoding will succeed if, in the remaining k time
slots, the destination receives at least two user’s packets in cooperative mode.
An example of this case is illustrated in Figure 6(b) where C[r] denotes to a packet received in cooperative mode.
So, for case 2 PEP is given by Equation (16):
With these two successful decoding outcomes, we obtain Equation (17):
where Pc is the probability to receive the packet in cooperative mode.
Though M + 1 unique equations always give M linear independent equations, which is enough to recover the M received packets, D can obtain the M linear independent equations from just M packets as
shown in the following cases:
The probability P[d][,2](k) can be calculated by analyzing the probability of the two scenarios that lead to successful decoding of all M nodes after stage 2, illustrated in Figure 7.
Before describing them, we note the following important fact; Let l ≤ M to be the number of combined equations received from l different nodes. We focus on the l × l subsystem containing these l
combined equations, restricted only on the corresponding l nodes. This subsystem has full rank l if l%2 = 0, i.e., the even number of combined packets for subsystem of l has full rank received
matrix, hence, its solvability.
Figure 7(a) illustrates the first solvable case when, after receiving k ≥ 0 selfish equations in the first stage, in the second stage, each of the remaining M − k nodes succeed in transmitting either
selfish or combined equation. If out of these M − k equations, l ≤ M − k are combined equations, the system is solvable, otherwise, it is not solvable. Note that whatever is received by k nodes
already received in the first stage is irrelevant. The probability of the first solvable case is given in Equation (18):
Figure 7(b) illustrates the second solvable case when, after receiving k ≥ 1 selfish equations in the first stage, in the second stage, M − k − 1 out of the remaining M − k nodes succeed in
transmitting either selfish or combined equation, and one node is not successful for the second time. Then, if during the second stage at least one combined equation is received from any of the k
nodes already received in the first stage, the system is solvable. Note that in this case, the number l ≤ M − k − 1 of combined equations among the M − k − 1 combined schemes for different M’s
equations received by nodes unsuccessful in the first stage but successful in the second stage, is irrelevant. The probability that the second solvable case occurs is shown in Equation (19):
If two or more nodes do not succeed in transmitting any equation to the destination node during both stages, the system can be given in Equation (20):
Replacing Equation (20) in Equation (14), the probability P[d][,2] is obtained.
If D fails to receive at least M linearly independent equations from M users, a third
Figure 7. M received packets at D. (a) and (b) are the solvable cases. (a) Solvable Case 1. (b) Solvable Case 2.
transmission stage is considered whereby novel information in the form of linear combinations not previously received at D is transmitted as proposed in Sections 2.3.1 to 2.3.4.
Then, probability Pd + 3 of successful decoding after three stages can be calculated similarly as in Equation (20), as D will declare the ability to decode all M packets when receiving rank M matrix.
Probabilistic analysis for other schemes is possible but in some cases tedious. We leave the complete probabilistic analysis for our future work, and explore the performances of all the schemes,
including the three stage transmission, using simulations in the following section.
Figure 8 shows the block diagram for the system design with NC over the erasure channel for two stages transmitting.
The block diagram shows that the receiver tends to check the packets reception from the first stage transmission, and if not received, it tends to retrieve the packet by the helps of the combined
packets received by the second cooperative stage.
If any packet neither received nor retrieved, is then asked to be retransmitted unlike the scenario where NC is not applied which is retransmitting the packet in the case when it is not received
4. Results
This section presents the results and observations of our simulation experiments to determine how our proposed cooperative protocols based on NC reduce the probability that all N[1+] N[2+], ∙∙∙ + N
[M] are not successfully decoded, P[e]+ at D node ,with no extra transmitted packets compared to the benchmark protocol, where q, and p are the probability of packet loss in the channels between one
user and another, and between the users and D respectively.
The benchmark protocol (Figure 3) shows that M (M + 1) packets are needed compared to 2M when applying NC (Figure 3) for two stages system. Accordingly, our aim is to find how NC helps improving PER
with no extra transmitted packets compared to benchmark protocol, beside to the benefits to compare all protocols with each others to understand their behaviour, taking into consideration the saving
in transmitted packets (2M or 3M) and in the need for ARQ, which justifies the saving in data rate.
Figure 8. System set-up Block diagram with NC. The Figure justifies the improvement in PEP.
It is important to confirm that these results have been obtained by Matlab when applying the M rank indicator for full reception, i.e., to confirm whether the system managed to decode the information
or ARQ is needed.
Figure 9 shows the behaviour of P[e] when applying all the proposed protocols as compared to the benchmark protocol for transmission in two stages. Results show how deterministic protocols are as
competitive to the non-deterministic (random) with respect to P[e], but with the advantage of the deterministic combining such as 1 bit header in the M − 1 protocol, the easier Gaussian elimination
decoding as the received packets are either single or combined in a deterministic way and the ability to solve one dead channel in the case of M − 1 protocol. Moreover, we can see that the M − 1
deterministic combination outperforms all received combined at high error rate i.e., p = 0.25, and this shows how the deterministic combination is stable and can stand dropping large number of
packets, taking into consideration that the channel is allowed to drop M − 1 packets in the way.
Combinations of 2-packets provide acceptable P[e] improvement though their performance is considerably weaker than both the M − 1 deterministic and all received combined protocols. In addition, the
deterministic degree 2 next neighbour deterministic protocol is as competitive as the closest neighbour protocol, but does not require header fields, this is justified by using the same erasure
probabilities for all distances. The P[e] behaviour of the M − 1 deterministic combination protocol for different number M of nodes is presented in Figure 10, showing that the M − 1 deterministic
protocol is as competitive to the non-deterministic all-received combined protocol even for a large number of nodes. It also shows that increasing the number of nodes M deteriorates performance of
both cooperative protocols and the benchmark non-cooperative protocols.
Moreover, Figure 10 shows the worsening performance of the M − 1 deterministic combined protocol when increasing the erasure probability q between nodes. The
Figure 9. P[e] for 2-stage schemes (M = 10 and q = 0).
Figure 10. P[e] for 2-stage M − 1 deterministic and all received combined schemes for different M’s.
threshold behaviour of q is notable, where the performance at low values of p is significantly improved if q ≤ 0.3, and otherwise, deteriorate significantly.
Figure 11 shows the M − 1 deterministic combination when changing the erasure probability between the M users. The clear threshold in Figure 8 is justified by obtaining the sufficient number of
unique linear equations, i.e., when more than 30% losses packets, it is becoming harder to obtain the unique M + 1 linear equations.
In the final part of results, we focus on the deterministic three stages to give more details through the results as the theoretical analysis is tedious.
Figure 12 shows the performance of the deterministic system that transmits odd-even protocol Section 2.3.1 in the third stage, instead of repeating fully combined (M − 1) protocol in Section 2.2.1.
As noted in Sections 2.2.1, fully combined equations are used to solve the problem of one dead channel. The motivation for odd-even equations is that reception of these equations enables recovery of
any two users that were unable to send data to the destination (two dead channels). However, the destination can make use of only one odd and even equation; the rest of possibly received equations of
this type are useless. Note that the number of users that will send odd-even cooperative equation in the third stage depends on q.
This explains the PER results in Figure 11 where there is an optimum value of q (q = 0.2) for which the system performs better than for fully cooperative case q = 0.
Figure 13 illustrates the interval of q values, q = [0.3, 0.6] for which it is better to send odd-even (Equation (2)) than to repeat the fully combined (Equation (3)) in the third stage.
From the user viewpoint, the decision to send odd-even or fully combined equation during the third stage may be decided based on the previous channel state information. For smaller values of q, the
strategies where a user randomly decides whether to send
Figure 11. P[e] for 3-stage M − 1 deterministic combination example, M = 10 and different q’s.
Figure 12. P[e] performance of deterministic combined/odd-even cooperative scheme for three transmission stages.
odd-even or fully combined equation with appropriate optimal probabilistic decision that depends on q have a potential to further improve the overall PER performance.
Moreover, Figure 13 shows the performance gained by the third transmission stage, when the channel between users are good (0.03), indeed, the P[e] improved from 5.3 e^−3 to 4e^−5 at p = 0.1, the cost
of this good improvement is sending M extra packets.
When comparing three stages strategies, we notice that three stages deterministic out performs the three stages random combination clearly after the threshold p = 0.3, which showed in Figure 6 and
Figure 9, though just one bit header file is required. We can notice that using random combination when p is more than 0.35 is slightly better than our proposed deterministic combination, but it
requires M header file.
Same observation is interestingly noticed at two stages, where our proposed deterministic out performs the two stages random combination just after p = 0.3.
Figure 13. P[e] performance of deterministic combined/odd-even cooperative scheme for three transmission stages.
Finally, Figure 14 shows that repeating odd-even in the third stage results to worsen performance than the random combination, so, it is suggested not to repeat odd-even combination in the third
Based on above Figures, we can see clearly the gain obtained in P[e] when applying NC compared with the Benchmark when NC is not applied, taking into consideration that our protocols use just the
simple XOR combination which is simple and easy to be retrieved at D using XOR again in the same manner. Moreover, the different protocols give different performance, so, the proper protocol depends
on the application it is aimed for.
5. Conclusions
Motivated by recent results from [11] [12] [13] on the diversity and coding gains obtained by using NC in cooperation protocols, we propose several practical transmission protocols for a typical data
gathering scenario that boasts simplicity and low-complexity while still maintaining performance gains over non-NC solution.
We designed simple practical protocols that apply the deterministic NC at M nodes to improve the system data rate and to reduce the traffic significantly, through combination among the M nodes and
with D.
Applying NC in a cooperative manner allowed the system to recover all the data at the M nodes and D even in the case of one dead channel in the two stages, and two dead channels in the three stages
transmission. Moreover, the ARQ improved significantly, in fact, the ARQ reduced from 10^−^1 to 10^−^4 for the whole system with better data rate and simplicity of decoding the data with Jordan
Gaussian Elimination.
We derive analytical expressions for the probability that the destination does not recover all users’ packets, after one and two transmission stages adopting our proposed
Figure 14. Comparison of three and two stages for M = 10 at good erasure probability (q = 0.03).
cooperative protocols based on deterministic NC. Simulation results demonstrate a perfect match with the analytic results and also show the performance benefits of using deterministic compared with
non deterministic protocols. Moreover, we show that deterministic combination is as competitive as the non deterministic but without the overhead of headers the advantage of using simple Gaussian
Elimination decoding process. We also prove that it is sufficient for D to recover M + 1 unique packets from any three stages to completely recover all M users’ messages when using M − 1 protocol.
Our future work is to show the good benefit in term of data security when applying the proposed protocols, besides extending the system for more than one destination.
Moreover, applying the proposed protocol over the physical layer is under investigation and will be published soon with Partial Unit Memory Turbo Code (PUMTC) as the forward error correction tool,
and to show what is the impact of the accumulative noise when combining the packets together in Amplify-and-Forward (AF) relay and with Decode-Re-encode-and-Forward (DF) relay with the channel
capacity limits in both scenarios.
Finally, applying the proposed protocols over real applications is among our future plan as well, such as applying these protocols over mobile networks, and wireless sensor networks
Philadelphia University deserves my acknowledgements for the good atmosphere they maintain for their researchers and for the financial support for this research.
Moreover, I am always thankful for Dr. Lina Stankovic and Dr. Vladimir Stakovic from the University of Strathclyde, Glasgow, UK, for their support with the technical issues, beside their proof
reading and technical modifications in my whole research work. They always add good values for my research work. | {"url":"https://scirp.org/journal/paperinformation?paperid=72397","timestamp":"2024-11-13T12:48:42Z","content_type":"application/xhtml+xml","content_length":"150181","record_id":"<urn:uuid:c13d2dc0-011f-45fc-872b-01be529292e6>","cc-path":"CC-MAIN-2024-46/segments/1730477028347.28/warc/CC-MAIN-20241113103539-20241113133539-00655.warc.gz"} |
How will you concatenate two strings in PHP?
Last updated 5 years, 7 months ago | 4511 views 75 5
PHP | Combine two strings
There are several ways to concatenate two strings in PHP.
Use the concatenation operator [.] and [.=]
There are two string operators that are used for concatenation.
[.]: Concatenation operator | Used to combine two string values to create a new string.
[.=]: Concatenation assignment operator | Appends the argument on the right side to the argument on the left side.
Concatenation operator
$str1="Hello World! ";
$str2="Welcome to StudyZone4U.com";
echo $str1 . $str2;
// output
//This will produce following result −
//Hello World! Welcome to StudyZone4U.com
Concatenation assignment operator
$str1 = "Hello World! ";
// Use og concatenation assignment operator
$str2 .= "Welcome to StudyZone4U.com";
echo $str2;
// Output:
// Hello World! Welcome to StudyZone4U.com
Double quotes strings
PHP variables also work inside double-quotes. So if you place the PHP variable inside double quotes and just put a space between both the PHP variable will work the same
$str1 = "Hello World!";
$str2 = "Welcome to StudyZone4U.com";
// Use of Double quotes
// The curly braces have used here to make the things clear.
echo "{$str1} {$str2}";
// Output:
// Hello World! Welcome to StudyZone4U.com
In the above example, you can directly use the PHP variable without curly braces. The curly braces have used in the above example to make things clear.
// without curly braces
echo "$str1 $str2";
// Output:
// Hello World! Welcome to StudyZone4U.com
Use a cmmma "," with echo() statement
This used only when you what to print the thing directly on the webpage. The echo statement can take multiple variables as a list by using a comma to separate the variables.
So, let's see who its work
$str1 = "Hello World!";
$str2 = "Welcome to StudyZone4U.com";
echo $str1, ' ', $str2;
// Output:
// Hello World! Welcome to StudyZone4U.com | {"url":"http://www.studyzone4u.com/post-details/how-will-you-concatenate-two-strings-in-php","timestamp":"2024-11-01T19:27:40Z","content_type":"text/html","content_length":"53066","record_id":"<urn:uuid:688c3147-219c-41ba-a57f-62d628d538e1>","cc-path":"CC-MAIN-2024-46/segments/1730477027552.27/warc/CC-MAIN-20241101184224-20241101214224-00392.warc.gz"} |
Fill in the gaps with correct answer choosing from the brackets... | Filo
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Fill in the gaps with correct answer choosing from the brackets. is maximum if theta = ____.
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Question Text Fill in the gaps with correct answer choosing from the brackets. is maximum if theta = ____.
Updated On Mar 4, 2024
Topic Trigonometry
Subject Mathematics
Class Class 11
Answer Type Text solution:1 | {"url":"https://askfilo.com/user-question-answers-mathematics/fill-in-the-gaps-with-correct-answer-choosing-from-the-37353139323939","timestamp":"2024-11-11T17:06:19Z","content_type":"text/html","content_length":"245393","record_id":"<urn:uuid:0163ca7e-ef9b-42f2-b7a0-0aa618f59d96>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00287.warc.gz"} |
This implements sparse arrays of arbitrary dimension on top of numpy and scipy.sparse. It generalizes the scipy.sparse.coo_matrix and scipy.sparse.dok_matrix layouts, but extends beyond just rows and
columns to an arbitrary number of dimensions.
Additionally, this project maintains compatibility with the numpy.ndarray interface rather than the numpy.matrix interface used in scipy.sparse
These differences make this project useful in certain situations where scipy.sparse matrices are not well suited, but it should not be considered a full replacement. The data structures in pydata/
sparse complement and can be used in conjunction with the fast linear algebra routines inside scipy.sparse. A format conversion or copy may be required.
Sparse arrays, or arrays that are mostly empty or filled with zeros, are common in many scientific applications. To save space we often avoid storing these arrays in traditional dense formats, and
instead choose different data structures. Our choice of data structure can significantly affect our storage and computational costs when working with these arrays.
The main data structure in this library follows the Coordinate List (COO) layout for sparse matrices, but extends it to multiple dimensions.
The COO layout, which stores the row index, column index, and value of every element:
row col data
It is straightforward to extend the COO layout to an arbitrary number of dimensions:
dim1 dim2 dim3 … data
0 0 0 . 10
0 0 3 . 13
0 2 2 . 9
3 1 4 . 21
This makes it easy to store a multidimensional sparse array, but we still need to reimplement all of the array operations like transpose, reshape, slicing, tensordot, reductions, etc., which can be
challenging in general.
This library also includes several other data structures. Similar to COO, the Dictionary of Keys (DOK) format for sparse matrices generalizes well to an arbitrary number of dimensions. DOK is
well-suited for writing and mutating. Most other operations are not supported for DOK. A common workflow may involve writing an array with DOK and then converting to another format for other
The Compressed Sparse Row/Column (CSR/CSC) formats are widely used in scientific computing are now supported by pydata/sparse. The CSR/CSC formats excel at compression and mathematical operations.
While these formats are restricted to two dimensions, pydata/sparse supports the GCXS sparse array format, based on GCRS/GCCS from which generalizes CSR/CSC to n-dimensional arrays. Like their
two-dimensional CSR/CSC counterparts, GCXS arrays compress well. Whereas the storage cost of COO depends heavily on the number of dimensions of the array, the number of dimensions only minimally
affects the storage cost of GCXS arrays, which results in favorable compression ratios across many use cases.
Together these formats cover a wide array of applications of sparsity. Additionally, with each format complying with the numpy.ndarray interface and following the appropriate dispatching protocols,
pydata/sparse arrays can interact with other array libraries and seamlessly take part in pydata-ecosystem-based workflows.
This library is licensed under BSD-3 | {"url":"https://sparse.pydata.org/en/stable/","timestamp":"2024-11-05T12:22:42Z","content_type":"text/html","content_length":"38237","record_id":"<urn:uuid:40362fb8-4e70-4cb0-89ab-adfba94fca22>","cc-path":"CC-MAIN-2024-46/segments/1730477027881.88/warc/CC-MAIN-20241105114407-20241105144407-00074.warc.gz"} |
RE: st: Advanced linear regression question (non constant random perturb
[Date Prev][Date Next][Thread Prev][Thread Next][Date index][Thread index]
RE: st: Advanced linear regression question (non constant random perturbation variance)
From "Nick Cox" <[email protected]>
To <[email protected]>
Subject RE: st: Advanced linear regression question (non constant random perturbation variance)
Date Wed, 28 Jun 2006 14:12:49 +0100
It sounds as if Guillermo has a known break- or change-point.
This sounds straightforward as a -ml- problem.
You just need to write your short driver program. That is,
more or less, what you need to do in R or S-Plus,
is it not?
[email protected]
[email protected]
> I've asked a similar question before. I can't remember what
> conclusion I
> came to. But having gone through the emails I think the closest Stata
> comes to dealing with heteroscedastic regression is a program called
> -regh- .
> I think these problems are better dealt with in R or Splus...
Guillermo Villa <[email protected]>
> I want to estimate a linear regression in which the variance of the
> random perturbation is not constant. I do not want this variance to
> depend on some explanatory variable, rather I have two types of
> observations and each of this types should have its own
> constant variance.
> My sample is divided in two parts (I = I1 + I2). Then, ei follows a
> normal distribution with mean 0 and variance sigma1 if i belongs to I1
> and ei follows a normal distribution with mean 0 and variance
> sigma2 if
> i belongs to I2.
> I suppose this model should be estimated using GLS, but I do not know
> how to tell Stata that here the random perturbation variance is not
> constant. Any idea?
* For searches and help try:
* http://www.stata.com/support/faqs/res/findit.html
* http://www.stata.com/support/statalist/faq
* http://www.ats.ucla.edu/stat/stata/ | {"url":"https://www.stata.com/statalist/archive/2006-06/msg00915.html","timestamp":"2024-11-12T09:14:29Z","content_type":"text/html","content_length":"9501","record_id":"<urn:uuid:da05f2b3-19ef-4011-b58e-3ca136a00ae9>","cc-path":"CC-MAIN-2024-46/segments/1730477028249.89/warc/CC-MAIN-20241112081532-20241112111532-00878.warc.gz"} |
1,721 research outputs found
This paper studies the set cover problem under the semi-streaming model. The underlying set system is formalized in terms of a hypergraph $G = (V, E)$ whose edges arrive one-by-one and the goal is to
construct an edge cover $F \subseteq E$ with the objective of minimizing the cardinality (or cost in the weighted case) of $F$. We consider a parameterized relaxation of this problem, where given
some $0 \leq \epsilon < 1$, the goal is to construct an edge $(1 - \epsilon)$-cover, namely, a subset of edges incident to all but an $\epsilon$-fraction of the vertices (or their benefit in the
weighted case). The key limitation imposed on the algorithm is that its space is limited to (poly)logarithmically many bits per vertex. Our main result is an asymptotically tight trade-off between $\
epsilon$ and the approximation ratio: We design a semi-streaming algorithm that on input graph $G$, constructs a succinct data structure $\mathcal{D}$ such that for every $0 \leq \epsilon < 1$, an
edge $(1 - \epsilon)$-cover that approximates the optimal edge \mbox{($1$-)cover} within a factor of $f(\epsilon, n)$ can be extracted from $\mathcal{D}$ (efficiently and with no additional space
requirements), where $f(\epsilon, n) = \left\{ \begin{array}{ll} O (1 / \epsilon), & \text{if } \epsilon > 1 / \sqrt{n} \\ O (\sqrt{n}), & \text{otherwise} \end{array} \right. \, .$ In particular for
the traditional set cover problem we obtain an $O(\sqrt{n})$-approximation. This algorithm is proved to be best possible by establishing a family (parameterized by $\epsilon$) of matching lower
bounds.Comment: Full version of the extended abstract that will appear in Proceedings of ICALP 2014 track
We investigate the power of randomized algorithms for the maximum cardinality matching (MCM) and the maximum weight matching (MWM) problems in the online preemptive model. In this model, the edges of
a graph are revealed one by one and the algorithm is required to always maintain a valid matching. On seeing an edge, the algorithm has to either accept or reject the edge. If accepted, then the
adjacent edges are discarded. The complexity of the problem is settled for deterministic algorithms. Almost nothing is known for randomized algorithms. A lower bound of $1.693$ is known for MCM with
a trivial upper bound of $2$. An upper bound of $5.356$ is known for MWM. We initiate a systematic study of the same in this paper with an aim to isolate and understand the difficulty. We begin with
a primal-dual analysis of the deterministic algorithm due to McGregor. All deterministic lower bounds are on instances which are trees at every step. For this class of (unweighted) graphs we present
a randomized algorithm which is $\frac{28}{15}$-competitive. The analysis is a considerable extension of the (simple) primal-dual analysis for the deterministic case. The key new technique is that
the distribution of primal charge to dual variables depends on the "neighborhood" and needs to be done after having seen the entire input. The assignment is asymmetric: in that edges may assign
different charges to the two end-points. Also the proof depends on a non-trivial structural statement on the performance of the algorithm on the input tree. The other main result of this paper is an
extension of the deterministic lower bound of Varadaraja to a natural class of randomized algorithms which decide whether to accept a new edge or not using independent random choices
In this paper, we consider time-space trade-offs for reporting a triangulation of points in the plane. The goal is to minimize the amount of working space while keeping the total running time small.
We present the first multi-pass algorithm on the problem that returns the edges of a triangulation with their adjacency information. This even improves the previously best known random-access
Test results are presented for a 24 cell, two sq ft (4kW) stack. This stack is a precursor to a 25kW stack that is a key milestone. Results are discussed in terms of cell performance, electrolyte
management, thermal management, and reactant gas manifolding. The results obtained in preliminary testing of a 50kW methanol processing subsystem are discussed. Subcontracting activities involving
application analysis for fuel cell on site integrated energy systems are updated
In this paper, we study linear programming based approaches to the maximum matching problem in the semi-streaming model. The semi-streaming model has gained attention as a model for processing
massive graphs as the importance of such graphs has increased. This is a model where edges are streamed-in in an adversarial order and we are allowed a space proportional to the number of vertices in
a graph. In recent years, there has been several new results in this semi-streaming model. However broad techniques such as linear programming have not been adapted to this model. We present several
techniques to adapt and optimize linear programming based approaches in the semi-streaming model with an application to the maximum matching problem. As a consequence, we improve (almost) all
previous results on this problem, and also prove new results on interesting variants
Scholars of presidential primaries have long posited a dynamic positive feedback loop between fundraising and electoral success. Yet existing work on both directions of this feedback remains
inconclusive and is often explicitly cross-sectional, ignoring the dynamic aspect of the hypothesis. Pairing high-frequency FEC data on contributions and expenditures with Iowa Electronic Markets
data on perceived probability of victory, we examine the bidirectional feedback between contributions and viability. We find robust, significant positive feedback in both directions. This might
suggest multiple equilibria: a candidate initially anointed as the front-runner able to sustain such status solely by the fundraising advantage conferred despite possessing no advantage in quality.
However, simulations suggest the feedback loop cannot, by itself, sustain advantage. Given the observed durability of front-runners, it would thus seem there is either some other feedback at work and
/or the process by which the initial front-runner is identified is informative of candidate quality
Previous work involving Born-regulated gravity theories in two dimensions is extended to four dimensions. The action we consider has two dimensionful parameters. Black hole solutions are studied for
typical values of these parameters. For masses above a critical value determined in terms of these parameters, the event horizon persists. For masses below this critical value, the event horizon
disappears, leaving a ``bare mass'', though of course no singularity.Comment: LaTeX, 15 pages, 2 figure
We introduce an analytical model based on birth-death clustering processes to help understanding the empirical log-periodic corrections to power-law scaling and the finite-time singularity as
reported in several domains including rupture, earthquakes, world population and financial systems. In our stochastics theory log-periodicities are a consequence of transient clusters induced by an
entropy-like term that may reflect the amount of cooperative information carried by the state of a large system of different species. The clustering completion rates for the system are assumed to be
given by a simple linear death process. The singularity at t_{o} is derived in terms of birth-death clustering coefficients.Comment: LaTeX, 1 ps figure - To appear J. Phys. A: Math & Ge
We give an AM protocol that allows the verifier to sample elements x from a probability distribution P, which is held by the prover. If the prover is honest, the verifier outputs (x, P(x)) with
probability close to P(x). In case the prover is dishonest, one may hope for the following guarantee: if the verifier outputs (x, p), then the probability that the verifier outputs x is close to p.
Simple examples show that this cannot be achieved. Instead, we show that the following weaker condition holds (in a well defined sense) on average: If (x, p) is output, then p is an upper bound on
the probability that x is output. Our protocol yields a new transformation to turn interactive proofs where the verifier uses private random coins into proofs with public coins. The verifier has
better running time compared to the well-known Goldwasser-Sipser transformation (STOC, 1986). For constant-round protocols, we only lose an arbitrarily small constant in soundness and completeness,
while our public-coin verifier calls the private-coin verifier only once
The algebra of observables of SO_{q}(3)-symmetric quantum mechanics is extended to include the inverse \frac{1}{R} of the radial coordinate and used to obtain eigenvalues and eigenfunctions of a \
q-deformed Coulomb Hamiltonian | {"url":"https://core.ac.uk/search/?q=author%3A(Feigenbaum%20J%20A)","timestamp":"2024-11-08T16:03:38Z","content_type":"text/html","content_length":"182560","record_id":"<urn:uuid:8801df1f-12fe-41a8-984f-403a9eb26503>","cc-path":"CC-MAIN-2024-46/segments/1730477028067.32/warc/CC-MAIN-20241108133114-20241108163114-00059.warc.gz"} |
Plotting with R (Part I)
Raymond event 2020-09-23 visibility 480
For data analyst, it is critical to use charts to tell data stories clearly. R has numerous libraries to create charts and graphs. This article summarizes the high-level R plotting APIs (incl.
graphical parameters) and provides examples about plotting Pie Chart, Bar Chart, BoxPlot, Histogram, Line and Scatterplot using R.
Device, screen and layout
Before plotting, it is important to understand R's
Category Functions
Graphical devices These APIs provide controls over multiple graphics devices:
dev.list(), dev.cur(), dev.set(number), dev.off()
These APIs can be used to define a number of regions within the current device which can, to some extent, be treated as separate graphics devices. It is useful for
generating multiple plots on a single device.
Screens split.screen, screen(n), erase.screen()
* cannot work with multiple graphic device
Layouts (not compatible with layout divides the device up into as many rows and columns as there are in matrix mat, with the column-widths and the row-heights specified in the respective
split.screen) arguments.
layout(matrix), layout.show(n)
The following are some code examples (script R26.GraphicDevices.R) using these APIs:
# list devices
# split screen
split.screen(c(1, 2))
# layout
layout(matrix(1:4, 2, 2))
For example, the following code snippet will split the screen into 4 regions:
layout(matrix(1:4,2,2), widths=c(1, 3),heights=c(3, 1))
Graphic functions
The following table summarizes R graphic functions that can be used in plotting:
Function Description
plot(x) lot of the values of x (on the y-axis) ordered on the x-axis
plot(x, y) bivariate plot of x (on the x-axis) and y (on the y-axis)
sunflowerplot(x, y) the points with similar coordinates are drawn as a flower which petal number represents the number of points
pie(x) circular pie-chart
boxplot(x) “box-and-whiskers” plot
stripchart(x) plot of the values of x on a line (an alternative to boxplot() for small sample sizes)
coplot(x~y | z) bivariate plot of x and y for each value (or interval of values) of z
interaction.plot (f1, (f1, f2, y)
f2, y)
if f1 and f2 are factors, plots the means of y (on the y-axis) with respect to the values of f1 (on the x-axis) and of f2 (different curves); the option fun allows to choose the
summary statistic of y (by default fun=mean)
matplot(x,y) bivariate plot of the first column of x vs. the first one of y, the second one of x vs. the second one of y, etc.
dotchart(x) if x is a data frame, plots a Cleveland dot plot (stacked plots line-by-line and column-by-column)
fourfoldplot(x) visualizes, with quarters of circles, the association between two dichotomous variables for different populations (x must be an array with dim=c(2, 2, k), or a matrix with dim=c
(2, 2) if k = 1)
assocplot(x) Cohen–Friendly graph showing the deviations from independence of rows and columns in a two dimensional contingency table
mosaicplot(x) ‘mosaic’ graph of the residuals from a log-linear regression of a contingency table
pairs(x) if x is a matrix or a data frame, draws all possible bivariate plots between the columns of x
plot.ts(x) if x is an object of class "ts", plot of x with respect to time, x may be multivariate but the series must have the same frequency and dates
ts.plot(x) Similar as above but if x is multivariate the series may have different dates and must have the same frequency
hist(x) histogram of the frequencies of x
barplot(x) histogram of the values of x
qqnorm(x) quantiles of x with respect to the values expected under a normal law
qqplot(x, y) quantiles of y with respect to the quantiles of x
contour(x, y, z) contour plot (data are interpolated to draw the curves), x and y must be vectors and z must be a matrix so that dim(z)=c(length(x), length(y)) (x and y may be omitted)
filled.contour (x, y, Similar as above but the areas between the contours are coloured, and a legend of the colours is drawn as well
image(x, y, z) Similar as above but the actual data are represented with colours
persp(x, y, z) Similar as above but in perspective
stars(x) if x is a matrix or a data frame, draws a graph with segments or a star where each row of x is represented by a star and the columns are the lengths of the segments
symbols(x, y, ...) draws, at the coordinates given by x and y, symbols (circles, squares, rectangles, stars, thermometers or “boxplots”) which sizes, colours, etc, are specified by supplementary
termplot(mod.obj) plot of the (partial) effects of a regression model (mod.obj)
Commonalities of graphic functions
There are some common shared parameters for these plotting functions:
• add=FALSE: if TRUE superposes the plot on the previous one (if it exists)
• axes=TRUE: if FALSE does not draw the axes and the box
• type="p":
"p": points
"l": lines
"b": points connected by lines
"o": Similar as above but the lines are over the points
"h": vertical lines
"s": steps, the data are represented by the top of the vertical lines
"S": Similar as above but the data are represented by the bottom of the vertical lines
• xlim=, ylim= specifies the lower and upper limits of the axes, for example with xlim=c(1, 10) or xlim=range(x)
• xlab=, ylab= annotates the axes (character vector)
• main= main title (character vector)
• sub= sub-title
Simple examples
The following code snippet shows some basic examples (script R27.GraphicalFunctions.R) using these common parameters:
# plot
x <- rnorm(30,20,10)
plot(x, type="p", main="Plot with Type p", )
plot(x, type="l", main="Plot with Type l", add=FALSE)
plot(x, type="b", main="Plot with Type b", add=FALSE)
plot(x, type="o", main="Plot with Type o", add=FALSE)
plot(x, type="h", main="Plot with Type h", add=FALSE)
plot(x, type="s", main="Plot with Type s", add=FALSE)
plot(x, type="S", main="Plot with Type S", add=FALSE)
Low level plotting commands
Low level plotting commands are used to affect an existing graph. They can be used to add these items to the graph:
• data labels
• lines and points
• legends
• title, sub title
• …
The following table summarizes all the low-level plotting commands:
Commands Description
points(x, y) adds points (the option type= can be used)
lines(x, y) Similar as above but with lines
text(x, y, labels, ...) adds text given by labels at coordinates (x,y); a typical use is: plot(x, y, type="n"); text(x, y, names)
mtext(text, side=3, line=0, ...) adds text given by text in the margin specified by side (see axis() below); line specifies the line from the plotting area
segments(x0, y0, x1, y1) draws lines from points (x0,y0) to points (x1,y1)
arrows(x0, y0, x1, y1, angle= 30, Same as above with arrows at points (x0,y0) if code=2, at points (x1,y1) if code=1, or both if code=3; angle controls the angle from the shaft of the arrow to the
code=2) edge of the arrow head
abline(a,b) draws a line of slope b and intercept a
abline(h=y) draws a horizontal line at ordinate y
abline(v=x) draws a vertical line at abcissa x
abline(lm.obj) draws the regression line given by lm.obj
rect(x1, y1, x2, y2) draws a rectangle which left, right, bottom, and top limits are x1, x2, y1, and y2, respectively
polygon(x, y) draws a polygon linking the points with coordinates given by x and y
legend(x, y, legend) adds the legend at the point (x,y) with the symbols given by legend
title() adds a title and optionally a sub-title
axis(side, vect) adds an axis at the bottom (side=1), on the left (2), at the top (3), or on the right (4); vect (optional) gives the abcissa (or ordinates) where tick-marks are
box() adds a box around the current plot
rug(x) draws the data x on the x-axis as small vertical lines
Graphic parameters
Graphs can be improved using graphical parameters. They can be used either as options of graphical functions or with function par.
For example, the following code snippet will set the device background color as green for all the following plots:
In next part, I will show plotting examples of different chart types.
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A complete educational program in algebra and geometry for a school course?
SOCKET-0072017-11-30 16:15:20
2017-11-30 16:15:20
A complete educational program in algebra and geometry for a school course?
We need a list of references to completely eliminate gaps from school mathematics (grade 11).
I would also really like to learn how to prove theorems, but I don’t know a suitable problem book ..
Please recommend literature (textbook + problem book), which also contains derivations of important formulas and I could learn to think like a real mathematician.
I plan to become a free student at NMU, but I don’t know how to prove at all ... It would be ideal if they threw off a textbook that provides examples of many proofs.
Answer the question
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2 answer(s)
I'm in the process of learning myself, so I could be wrong. But I would venture to suggest that in order to be able to prove theorems or complete assignments for proofs from leaflets, you need to
prove theorems from textbooks, read it - put the book aside - proved it, it didn’t work out - we read the book again.
The ability to prove theorems is the ability to think logically.
For example, they proved the Great Fermat theorem (the proof is a whole book !!), i.e. it just takes a few days to rewrite it. Fermat himself proved this simply by reading some text, and simply the
lack of space in the margins did not allow him to state the proof. But in any case, he did not mean such proof - he had some other and very simple ... or he was mistaken))
Didn't find what you were looking for?
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Infinities as numbers: purging the epsilons and deltas from proofs
Part 6 of a six-part series of adaptations from Terence Tao’s book “Structure and Randomness". Archives
[1.5] Ultrafilters, non-standard analysis, and epsilon management Categories
In the previous article we saw that fruitful analogies between finitary and infinitary mathematics can allow the techniques of one to shed light on the other. Here we borrow the power
of infinitary math—in particular, ultrafilters and non-standard analysis—to simplify proofs of finitary statements.
Arguments in hard analysis are notorious for their profusion of "epsilons and deltas", a more familiar example being the $(\epsilon, \delta)$-definition of convergence from high school
calculus. One may have to keep track of a whole army of epsilons, some of which are "small", "very small" (i.e. negligible as compared to even the "small" epsilons), "very very small"
and so on. This "epsilon management" is exacerbated with those unsightly quantifiers ("for every $\varepsilon$ there exists $N$ such that...") sprinkled within any statement. These
quantifers need care to weave together, and need careful untangling to comprehend. To borrow a rather mild example from the previous article,
Finite convergence principle. If $\varepsilon > 0$ and $F$ is a function from the positive integers to the positive integers, and $0 \leq x_1 \leq x_2 \leq \dotsb \leq x_M \leq 1$
is such that $M$ is sufficiently large depending on $F$ and $\varepsilon$, then there exists an integer $N$ with $1 \leq N < N + F(N) \leq M$ such that $|x_n - x_m| \leq \
varepsilon$ for all $N \leq n, m \leq N + F(N)$.
(Anyone knows of more convoluted examples?)
"Automating" epsilon management has progressed in "asymptotic notation" like the $O()$ family of notations, as well as the $ \ll$- and $\sim$-type symbols, which rigorously formulate
the respective qualitative ideas of "bounded by", "much smaller than" and "comparable in size to", without resorting to explicit quantities like $ \varepsilon$ and $ N$.
However, the absence of actual quantities inhibits detailed study; for instance, sums and products of "bounded numbers" (i.e. $O(1)$) are also bounded, but it's meaningless (obstructed
by an axiom of set theory) to say that the set of $O(1)$ is closed under addition and multiplication. Non-standard analysis solves this problem by adding new numbers into our number
system, including infinities and infinitesimals.
Such "new numbers" are defined using non-principal ultrafilters, which are method to find the $p$-limit of any sequence $(x_n) = x_1, x_2, \dotsc$ of real numbers. If $(x_n)$ converges
then the $p$-limit is the usual limit. However, the sequence $1, 2, 3, \dotsc$ has a $p$-limit of the ordinal $\omega$, which you can think of as "the smallest infinity". You can get
bigger infinities from the $p$-limits of sequences like $\omega, 2\omega, 3\omega, \dotsc$ (which understandably converges to $\omega^2$).
The standard real number system together with all possible $p$-limits forms the set of non-standard numbers, or hyperreal numbers. The analogue of an $O(1)$ number is then a hyperreal
number that is smaller than some standard real number. In fact, this set is a ring and we can readily apply every insight from ring theory to it. This is made possible from the
principle that non-standard numbers can be manipulated just like standard numbers, also known as:
Transfer principle. Every proposition valid over real numbers is also valid over the hyperreals.
This allows us to take reciprocals of infinities to get infinitesimals for use in calculus. In fact, allowing calculus to rigorously work with infinitesimals was a major motivation for
the development of non-standard analysis. For any infinitesimal $\varepsilon$, the $p$-limit of the sequence $1, \varepsilon, \varepsilon^2, \dotsc$ is much, much smaller than $\
varepsilon$. This process can be iterated to churn out a heirarchy of infinitesimals that shrink at a ridiculous pace, simplifying epsilon management. Tao says that:
"it lets one avoid having to explicitly write a lot of epsilon-management phrases such as 'Let $\eta_2$ be a small number (depending on $\eta_0$ and $\eta_1$) to be chosen later' and
'… assuming $ \eta_2$ was chosen sufficiently small depending on $\eta_0$ and $\eta_1$', which are very frequent in hard analysis literature..."
I guess ultrafilters do not change a proof in essence but greatly simplifies its language, freeing one's attention for the big picture, as opposed to wading in a swamp of $\forall\
varepsilon_1\exists\varepsilon_2\forall N_1 \exists N_2 \gg N_1/(\varepsilon1\varepsilon_2)$.
Further Discussion
The original blog post details several important limitations to the above properties. It also develops some interesting properties of ultrafilters, such as their connection to the
usual limits (and identities concerning them) and propositional logic. Many of these properties are explained with a wonderfully illustrative analogy of an ultrafilter as a "voting
system": In a sequence $(x_n)$, each integer $ i = 1, 2, \dotsc$ "votes" on some real number $x_i$, and the $p$-limit is the elected candidate. Different ultrafilters are distinguished
by how much influence each integer has on the final decision (voting is unfair!)
The connection with propositional logic comes with asking each integer a yes-no question and evaluating the $p$-limit, which would be either yes ($1$) or no ($0$). A property $P(n)$ of
integers (i.e. $n > 0$) is "$p$-true" (i.e. almost surely true) if the "decision" after asking the integers "do you satisfy $P$?" is "yes". $p$-truths satisfy the laws of logic, but
tautologically true statements are $p$-true!
[1.5] Terence Tao. Ultrafilters, non-standard analysis, and epsilon management. In Structure and Randomness: Pages from Year One of a Mathematical Blog. American Mathematical Society
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Finite convergence principle. If $\varepsilon > 0$ and $F$ is a function from the positive integers to the positive integers, and $0 \leq x_1 \leq x_2 \leq \dotsb \leq x_M \leq 1$ is such that $M$ is
sufficiently large depending on $F$ and $\varepsilon$, then there exists an integer $N$ with $1 \leq N < N + F(N) \leq M$ such that $|x_n - x_m| \leq \varepsilon$ for all $N \leq n, m \leq N + F(N)$.
Transfer principle. Every proposition valid over real numbers is also valid over the hyperreals. | {"url":"https://www.herngyi.com/blog/infinities-as-numbers-purging-the-epsilons-and-deltas-from-proofs","timestamp":"2024-11-11T04:06:29Z","content_type":"text/html","content_length":"48255","record_id":"<urn:uuid:68f6077e-cb00-418a-a202-60d1e6a9f52f>","cc-path":"CC-MAIN-2024-46/segments/1730477028216.19/warc/CC-MAIN-20241111024756-20241111054756-00622.warc.gz"} |
Election Reversals
Strange things can happen in elections. Some of that strangeness arises from factors related to candidates, cranky voting machines, skewed opinion polls, downright fraud, and other human and
electromechanical foibles. Some of it comes out of inevitable quirks in election procedures, especially when more than two candidates are involved.
Suppose there are three candidates, Heather, Angela, and Kathy, and the election procedure calls for ranking the candidates in order of preference. The voters understand this to mean listing their
top preference first, their middle preference second, and their bottom preference third.
Three voters put Heather first, Angela second, and Kathy third. Three voters have Kathy first, Angela second, and Heather third. Four voters have Heather first, Kathy second, and Angela third. Four
voters have Angela first, Kathy second, and Heather third. When the votes are tallied, Heather is the top choice for seven voters, Angela is top for four voters, and Kathy is top for three voters.
Heather wins.
However, because of a misunderstanding, the officials tallying the ballots actually treat a candidate listed first as the bottom preference and a candidate listed third as the top preference. The
surprise is that this “reverse” tally gives the same order of finish (Heather, Angela, and Kathy) as the original tally instead of the expected reverse ranking (Kathy first, Angela second, and
Heather third). Heather wins again. Moreover, it’s clear you can’t get the results of one tally simply by reversing the order of finish in the other tally.
Common sense suggests questioning the reliability of any election procedure if it produces the same result when preferences are reversed.
“Surprisingly, this seemingly perverse behavior can sincerely occur with most standard election procedures,” Donald G. Saari of the University of California, Irvine, and Steven Barney of the
University of Wisconsin Oshkosh write in the current Mathematical Intelligencer. An election like the one described above actually occurred in an academic department to which Saari once belonged.
Such a counterintuitive outcome–in which a candidate can come out on top for one set of voter preferences and for its reverse–are a troubling consequence of using election procedures that call simply
for a plurality vote and, in effect, bias the results.
“It should be a concern because, . . . rather than a rare and obscure phenomenon, we can expect some sort of reversal behavior about 25 percent of the time with the standard plurality vote,” Saari
and Barney note. These results come out of considerations of mathematical symmetry related to voting systems.
Interestingly, for three-candidate elections, only the system known as the Borda count never exhibits a reversal bias. In such an election, voters assign 2 and 1 points, respectively, to their top-
and second-ranked candidates. The candidate with the highest point total wins.
All other methods that involve some sort of ranking–including approval voting, where electors can vote for as many candidates as they wish, and the candidate with the most votes wins–can admit
counterintuitive outcomes.
So, shenanigans aren’t always to blame for unexpected results. Sometimes, it’s just the choice of election procedure. | {"url":"https://www.sciencenews.org/article/election-reversals","timestamp":"2024-11-05T17:05:08Z","content_type":"text/html","content_length":"290048","record_id":"<urn:uuid:c61d0deb-ad3d-4eaa-a0e9-0a7306c0d3ea>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00603.warc.gz"} |
Note the distinction between binary operation and binary relation, a binary operation is a function, which is a binary relation with some extra conditions.
Left fold formalizes the idea that we want to look at objects of the form $\left(\left(\left({x}_{0}\beta {x}_{1}\right)\beta {x}_{2}\right)\beta \beta ―\beta {x}_{k}\right)$, we also have a
right fold:
Since all bracketings yield the same value, our definition of fold to be the left fold was quite arbitrary. We can also now look at previous things under simpler lights, recall a definition of ${\
beta }_{i=0}^{k}{a}_{i}$ that you're familiar with, this is simply just $fold\left(+,\left({a}_{0},\mathrm{\beta ¦},{a}_{k}\right)\right)$
note: informally this says that you can do nothing, and you can also undo anything
A trivial group is a group that has one element
Note that at this point in time we can write things like $a+b$ and that makes sense, if we were to write something like $xa+b$ this should not make sense because at this point we only have one
operator, and not a multiplication operator yet, but we can think of $xa$ as "syntactic sugar" for $a+a+\mathrm{\beta ¦}+a$ (with $x$ repetitions) and it's alright.
It should be clear by now that $\beta $ is not a group since it cannot have an identity element because there are no elements as candidate for this position. At this point in time we know that in a
group there is at least one element $e$ with the above properties, we will now find out that there is exactly one such element.
Can the above be generalized for any extra group properties?
Given a group $\left(G,\beta \right)$ then since there is only one operation in question, instead of writing the operator between every two pairs of elements we may limit it to $ab$, to refer to $a
\beta b$ to reduce visual clutter
note: since $\mathrm{Β·}$ is also a simple thing to write, we may also use it in place of $\beta $
If the power of an element with finite order yields the identity then the order divides the power
Notice that the equation stated above is special, firstly given two elements in $a,b\beta H$ it combines $a$ and ${b}^{\beta 1}$, the special thing here is the ${b}^{\beta 1}$, originally we
don't know if this element is a member of $H$ because in the proof we verify if indeed it is a group, and we don't know that inverses exists until we verify them, but we do know when it's combined
with ${b}^{\beta 1}$ it produces an element of $H$.
Note that we need $H$ to be non-empty otherwise the empty set would satisfy the above, and we know that the empty set is not a group. It also helps us bootstrap the proof. | {"url":"https://www.openmath.net/algebra/groups/binary_operations_and_groups.html","timestamp":"2024-11-02T10:59:07Z","content_type":"text/html","content_length":"98746","record_id":"<urn:uuid:eed891b5-055d-4f22-8587-73c0b686ecc3>","cc-path":"CC-MAIN-2024-46/segments/1730477027710.33/warc/CC-MAIN-20241102102832-20241102132832-00367.warc.gz"} |
Data Preparation for Machine Learning (7-Day Mini-Course)
Author: Jason Brownlee
Data Preparation for Machine Learning Crash Course.
Get on top of data preparation with Python in 7 days.
Data preparation involves transforming raw data into a form that is more appropriate for modeling.
Preparing data may be the most important part of a predictive modeling project and the most time-consuming, although it seems to be the least discussed. Instead, the focus is on machine learning
algorithms, whose usage and parameterization has become quite routine.
Practical data preparation requires knowledge of data cleaning, feature selection data transforms, dimensionality reduction, and more.
In this crash course, you will discover how you can get started and confidently prepare data for a predictive modeling project with Python in seven days.
This is a big and important post. You might want to bookmark it.
Let’s get started.
Who Is This Crash-Course For?
Before we get started, let’s make sure you are in the right place.
This course is for developers who may know some applied machine learning. Maybe you know how to work through a predictive modeling problem end to end, or at least most of the main steps, with popular
The lessons in this course do assume a few things about you, such as:
• You know your way around basic Python for programming.
• You may know some basic NumPy for array manipulation.
• You may know some basic scikit-learn for modeling.
You do NOT need to be:
• A math wiz!
• A machine learning expert!
This crash course will take you from a developer who knows a little machine learning to a developer who can effectively and competently prepare data for a predictive modeling project.
Note: This crash course assumes you have a working Python 3 SciPy environment with at least NumPy installed. If you need help with your environment, you can follow the step-by-step tutorial here:
Crash-Course Overview
This crash course is broken down into seven lessons.
You could complete one lesson per day (recommended) or complete all of the lessons in one day (hardcore). It really depends on the time you have available and your level of enthusiasm.
Below is a list of the seven lessons that will get you started and productive with data preparation in Python:
• Lesson 01: Importance of Data Preparation
• Lesson 02: Fill Missing Values With Imputation
• Lesson 03: Select Features With RFE
• Lesson 04: Scale Data With Normalization
• Lesson 05: Transform Categories With One-Hot Encoding
• Lesson 06: Transform Numbers to Categories With kBins
• Lesson 07: Dimensionality Reduction with PCA
Each lesson could take you 60 seconds or up to 30 minutes. Take your time and complete the lessons at your own pace. Ask questions and even post results in the comments below.
The lessons might expect you to go off and find out how to do things. I will give you hints, but part of the point of each lesson is to force you to learn where to go to look for help with and about
the algorithms and the best-of-breed tools in Python. (Hint: I have all of the answers on this blog; use the search box.)
Post your results in the comments; I’ll cheer you on!
Hang in there; don’t give up.
Lesson 01: Importance of Data Preparation
In this lesson, you will discover the importance of data preparation in predictive modeling with machine learning.
Predictive modeling projects involve learning from data.
Data refers to examples or cases from the domain that characterize the problem you want to solve.
On a predictive modeling project, such as classification or regression, raw data typically cannot be used directly.
There are four main reasons why this is the case:
• Data Types: Machine learning algorithms require data to be numbers.
• Data Requirements: Some machine learning algorithms impose requirements on the data.
• Data Errors: Statistical noise and errors in the data may need to be corrected.
• Data Complexity: Complex nonlinear relationships may be teased out of the data.
The raw data must be pre-processed prior to being used to fit and evaluate a machine learning model. This step in a predictive modeling project is referred to as “data preparation.”
There are common or standard tasks that you may use or explore during the data preparation step in a machine learning project.
These tasks include:
• Data Cleaning: Identifying and correcting mistakes or errors in the data.
• Feature Selection: Identifying those input variables that are most relevant to the task.
• Data Transforms: Changing the scale or distribution of variables.
• Feature Engineering: Deriving new variables from available data.
• Dimensionality Reduction: Creating compact projections of the data.
Each of these tasks is a whole field of study with specialized algorithms.
Your Task
For this lesson, you must list three data preparation algorithms that you know of or may have used before and give a one-line summary for its purpose.
One example of a data preparation algorithm is data normalization that scales numerical variables to the range between zero and one.
Post your answer in the comments below. I would love to see what you come up with.
In the next lesson, you will discover how to fix data that has missing values, called data imputation.
Lesson 02: Fill Missing Values With Imputation
In this lesson, you will discover how to identify and fill missing values in data.
Real-world data often has missing values.
Data can have missing values for a number of reasons, such as observations that were not recorded and data corruption. Handling missing data is important as many machine learning algorithms do not
support data with missing values.
Filling missing values with data is called data imputation and a popular approach for data imputation is to calculate a statistical value for each column (such as a mean) and replace all missing
values for that column with the statistic.
The horse colic dataset describes medical characteristics of horses with colic and whether they lived or died. It has missing values marked with a question mark ‘?’. We can load the dataset with the
read_csv() function and ensure that question mark values are marked as NaN.
Once loaded, we can use the SimpleImputer class to transform all missing values marked with a NaN value with the mean of the column.
The complete example is listed below.
# statistical imputation transform for the horse colic dataset
from numpy import isnan
from pandas import read_csv
from sklearn.impute import SimpleImputer
# load dataset
url = 'https://raw.githubusercontent.com/jbrownlee/Datasets/master/horse-colic.csv'
dataframe = read_csv(url, header=None, na_values='?')
# split into input and output elements
data = dataframe.values
X, y = data[:, :-1], data[:, -1]
# print total missing
print('Missing: %d' % sum(isnan(X).flatten()))
# define imputer
imputer = SimpleImputer(strategy='mean')
# fit on the dataset
# transform the dataset
Xtrans = imputer.transform(X)
# print total missing
print('Missing: %d' % sum(isnan(Xtrans).flatten()))
Your Task
For this lesson, you must run the example and review the number of missing values in the dataset before and after the data imputation transform.
Post your answer in the comments below. I would love to see what you come up with.
In the next lesson, you will discover how to select the most important features in a dataset.
Lesson 03: Select Features With RFE
In this lesson, you will discover how to select the most important features in a dataset.
Feature selection is the process of reducing the number of input variables when developing a predictive model.
It is desirable to reduce the number of input variables to both reduce the computational cost of modeling and, in some cases, to improve the performance of the model.
Recursive Feature Elimination, or RFE for short, is a popular feature selection algorithm.
RFE is popular because it is easy to configure and use and because it is effective at selecting those features (columns) in a training dataset that are more or most relevant in predicting the target
The scikit-learn Python machine learning library provides an implementation of RFE for machine learning. RFE is a transform. To use it, first, the class is configured with the chosen algorithm
specified via the “estimator” argument and the number of features to select via the “n_features_to_select” argument.
The example below defines a synthetic classification dataset with five redundant input features. RFE is then used to select five features using the decision tree algorithm.
# report which features were selected by RFE
from sklearn.datasets import make_classification
from sklearn.feature_selection import RFE
from sklearn.tree import DecisionTreeClassifier
# define dataset
X, y = make_classification(n_samples=1000, n_features=10, n_informative=5, n_redundant=5, random_state=1)
# define RFE
rfe = RFE(estimator=DecisionTreeClassifier(), n_features_to_select=5)
# fit RFE
rfe.fit(X, y)
# summarize all features
for i in range(X.shape[1]):
print('Column: %d, Selected=%s, Rank: %d' % (i, rfe.support_[i], rfe.ranking_[i]))
Your Task
For this lesson, you must run the example and review which features were selected and the relative ranking that each input feature was assigned.
Post your answer in the comments below. I would love to see what you come up with.
In the next lesson, you will discover how to scale numerical data.
Lesson 04: Scale Data With Normalization
In this lesson, you will discover how to scale numerical data for machine learning.
Many machine learning algorithms perform better when numerical input variables are scaled to a standard range.
This includes algorithms that use a weighted sum of the input, like linear regression, and algorithms that use distance measures, like k-nearest neighbors.
One of the most popular techniques for scaling numerical data prior to modeling is normalization. Normalization scales each input variable separately to the range 0-1, which is the range for
floating-point values where we have the most precision. It requires that you know or are able to accurately estimate the minimum and maximum observable values for each variable. You may be able to
estimate these values from your available data.
You can normalize your dataset using the scikit-learn object MinMaxScaler.
The example below defines a synthetic classification dataset, then uses the MinMaxScaler to normalize the input variables.
# example of normalizing input data
from sklearn.datasets import make_classification
from sklearn.preprocessing import MinMaxScaler
# define dataset
X, y = make_classification(n_samples=1000, n_features=5, n_informative=5, n_redundant=0, random_state=1)
# summarize data before the transform
print(X[:3, :])
# define the scaler
trans = MinMaxScaler()
# transform the data
X_norm = trans.fit_transform(X)
# summarize data after the transform
print(X_norm[:3, :])
Your Task
For this lesson, you must run the example and report the scale of the input variables both prior to and then after the normalization transform.
For bonus points, calculate the minimum and maximum of each variable before and after the transform to confirm it was applied as expected.
Post your answer in the comments below. I would love to see what you come up with.
In the next lesson, you will discover how to transform categorical variables to numbers.
Lesson 05: Transform Categories With One-Hot Encoding
In this lesson, you will discover how to encode categorical input variables as numbers.
Machine learning models require all input and output variables to be numeric. This means that if your data contains categorical data, you must encode it to numbers before you can fit and evaluate a
One of the most popular techniques for transforming categorical variables into numbers is the one-hot encoding.
Categorical data are variables that contain label values rather than numeric values.
Each label for a categorical variable can be mapped to a unique integer, called an ordinal encoding. Then, a one-hot encoding can be applied to the ordinal representation. This is where one new
binary variable is added to the dataset for each unique integer value in the variable, and the original categorical variable is removed from the dataset.
For example, imagine we have a “color” variable with three categories (‘red‘, ‘green‘, and ‘blue‘). In this case, three binary variables are needed. A “1” value is placed in the binary variable for
the color and “0” values for the other colors.
For example:
red, green, blue
1, 0, 0
0, 1, 0
0, 0, 1
This one-hot encoding transform is available in the scikit-learn Python machine learning library via the OneHotEncoder class.
The breast cancer dataset contains only categorical input variables.
The example below loads the dataset and one hot encodes each of the categorical input variables.
# one-hot encode the breast cancer dataset
from pandas import read_csv
from sklearn.preprocessing import OneHotEncoder
# define the location of the dataset
url = "https://raw.githubusercontent.com/jbrownlee/Datasets/master/breast-cancer.csv"
# load the dataset
dataset = read_csv(url, header=None)
# retrieve the array of data
data = dataset.values
# separate into input and output columns
X = data[:, :-1].astype(str)
y = data[:, -1].astype(str)
# summarize the raw data
print(X[:3, :])
# define the one hot encoding transform
encoder = OneHotEncoder(sparse=False)
# fit and apply the transform to the input data
X_oe = encoder.fit_transform(X)
# summarize the transformed data
print(X_oe[:3, :])
Your Task
For this lesson, you must run the example and report on the raw data before the transform, and the impact on the data after the one-hot encoding was applied.
Post your answer in the comments below. I would love to see what you come up with.
In the next lesson, you will discover how to transform numerical variables into categories.
Lesson 06: Transform Numbers to Categories With kBins
In this lesson, you will discover how to transform numerical variables into categorical variables.
Some machine learning algorithms may prefer or require categorical or ordinal input variables, such as some decision tree and rule-based algorithms.
This could be caused by outliers in the data, multi-modal distributions, highly exponential distributions, and more.
Many machine learning algorithms prefer or perform better when numerical input variables with non-standard distributions are transformed to have a new distribution or an entirely new data type.
One approach is to use the transform of the numerical variable to have a discrete probability distribution where each numerical value is assigned a label and the labels have an ordered (ordinal)
This is called a discretization transform and can improve the performance of some machine learning models for datasets by making the probability distribution of numerical input variables discrete.
The discretization transform is available in the scikit-learn Python machine learning library via the KBinsDiscretizer class.
It allows you to specify the number of discrete bins to create (n_bins), whether the result of the transform will be an ordinal or one-hot encoding (encode), and the distribution used to divide up
the values of the variable (strategy), such as ‘uniform.’
The example below creates a synthetic input variable with 10 numerical input variables, then encodes each into 10 discrete bins with an ordinal encoding.
# discretize numeric input variables
from sklearn.datasets import make_classification
from sklearn.preprocessing import KBinsDiscretizer
# define dataset
X, y = make_classification(n_samples=1000, n_features=5, n_informative=5, n_redundant=0, random_state=1)
# summarize data before the transform
print(X[:3, :])
# define the transform
trans = KBinsDiscretizer(n_bins=10, encode='ordinal', strategy='uniform')
# transform the data
X_discrete = trans.fit_transform(X)
# summarize data after the transform
print(X_discrete[:3, :])
Your Task
For this lesson, you must run the example and report on the raw data before the transform, and then the effect the transform had on the data.
For bonus points, explore alternate configurations of the transform, such as different strategies and number of bins.
Post your answer in the comments below. I would love to see what you come up with.
In the next lesson, you will discover how to reduce the dimensionality of input data.
Lesson 07: Dimensionality Reduction With PCA
In this lesson, you will discover how to use dimensionality reduction to reduce the number of input variables in a dataset.
The number of input variables or features for a dataset is referred to as its dimensionality.
Dimensionality reduction refers to techniques that reduce the number of input variables in a dataset.
More input features often make a predictive modeling task more challenging to model, more generally referred to as the curse of dimensionality.
Although on high-dimensionality statistics, dimensionality reduction techniques are often used for data visualization, these techniques can be used in applied machine learning to simplify a
classification or regression dataset in order to better fit a predictive model.
Perhaps the most popular technique for dimensionality reduction in machine learning is Principal Component Analysis, or PCA for short. This is a technique that comes from the field of linear algebra
and can be used as a data preparation technique to create a projection of a dataset prior to fitting a model.
The resulting dataset, the projection, can then be used as input to train a machine learning model.
The scikit-learn library provides the PCA class that can be fit on a dataset and used to transform a training dataset and any additional datasets in the future.
The example below creates a synthetic binary classification dataset with 10 input variables then uses PCA to reduce the dimensionality of the dataset to the three most important components.
# example of pca for dimensionality reduction
from sklearn.datasets import make_classification
from sklearn.decomposition import PCA
# define dataset
X, y = make_classification(n_samples=1000, n_features=10, n_informative=3, n_redundant=7, random_state=1)
# summarize data before the transform
print(X[:3, :])
# define the transform
trans = PCA(n_components=3)
# transform the data
X_dim = trans.fit_transform(X)
# summarize data after the transform
print(X_dim[:3, :])
Your Task
For this lesson, you must run the example and report on the structure and form of the raw dataset and the dataset after the transform was applied.
For bonus points, explore transforms with different numbers of selected components.
Post your answer in the comments below. I would love to see what you come up with.
This was the final lesson in the mini-course.
The End!
(Look How Far You Have Come)
You made it. Well done!
Take a moment and look back at how far you have come.
You discovered:
• The importance of data preparation in a predictive modeling machine learning project.
• How to mark missing data and impute the missing values using statistical imputation.
• How to remove redundant input variables using recursive feature elimination.
• How to transform input variables with differing scales to a standard range called normalization.
• How to transform categorical input variables to be numbers called one-hot encoding.
• How to transform numerical variables into discrete categories called discretization.
• How to use PCA to create a projection of a dataset into a lower number of dimensions.
How did you do with the mini-course?
Did you enjoy this crash course?
Do you have any questions? Were there any sticking points?
Let me know. Leave a comment below.
The post Data Preparation for Machine Learning (7-Day Mini-Course) appeared first on Machine Learning Mastery. | {"url":"https://www.aiproblog.com/index.php/2020/06/28/data-preparation-for-machine-learning-7-day-mini-course/","timestamp":"2024-11-09T14:14:49Z","content_type":"text/html","content_length":"71537","record_id":"<urn:uuid:3772f8e6-96da-4107-9af9-7a98bf137d6e>","cc-path":"CC-MAIN-2024-46/segments/1730477028118.93/warc/CC-MAIN-20241109120425-20241109150425-00309.warc.gz"} |
pyunicorn (Unified Complex Network and RecurreNce analysis toolbox) is an object-oriented Python package for the advanced analysis and modeling of complex networks. Beyond the standard measures of
complex network theory (such as degree, betweenness and clustering coefficients), it provides some uncommon but interesting statistics like Newman’s random walk betweenness. pyunicorn also provides
novel node-weighted (node splitting invariant) network statistics, measures for analyzing networks of interacting/interdependent networks, and special tools to model spatially embedded complex
Moreover, pyunicorn allows one to easily construct networks from uni- and multivariate time series and event data (functional/climate networks and recurrence networks). This involves linear and
nonlinear measures of time series analysis for constructing functional networks from multivariate data (e.g., Pearson correlation, mutual information, event synchronization and event coincidence
analysis). pyunicorn also features modern techniques of nonlinear analysis of time series (or pairs thereof), such as recurrence quantification analysis (RQA), recurrence network analysis and
visibility graphs.
pyunicorn is fast, because all costly computations are performed in compiled C code. It can handle large networks through the use of sparse data structures. The package can be used interactively,
from any Python script, and even for parallel computations on large cluster architectures.
To generate a recurrence network with 1000 nodes from a sinusoidal signal and to compute its network transitivity, you can simply run:
import numpy as np
from pyunicorn.timeseries import RecurrenceNetwork
x = np.sin(np.linspace(0, 10 * np.pi, 1000))
net = RecurrenceNetwork(x, recurrence_rate=0.05) | {"url":"https://www.pik-potsdam.de/~donges/pyunicorn/index.html","timestamp":"2024-11-04T01:02:47Z","content_type":"text/html","content_length":"9760","record_id":"<urn:uuid:1221ecf1-474e-4d26-a497-5353f71b94ab>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00130.warc.gz"} |
Predict Score on the basis of Studied Hours - Study Trigger
Predict Score on the basis of Studied Hours
Predict Score on the basis of Studied Hours in Machine Learning
by Mahesh Verma
written by Mahesh Verma
Predict Score on the basis of Studied Hours
You have a DataFrame containing information about students, including their “Study Hours” and “Exam Scores.” How could you use linear regression to predict a student’s exam score based on the number
of hours they studied?
For the above questions, let’s divide our solution into some steps :
STEP 1 : Creating DataFrame:
import pandas as pd
# Create a DataFrame with student data
data = {'study_hours': [12, 21, 31, 44, 15, 25, 37, 42, 27, 17, 14, 23, 33, 46, 19, 35, 39, 40, 24, 18],
'exam_scores': [50, 70, 84, 97, 52, 73, 87, 95, 75, 58, 52, 72, 87, 99, 62, 85, 90, 92, 73, 59 ]}
df = pd.DataFrame(data)
You can also create dataframe taking random value using random() of numpy library like:
import pandas as pd
import numpy as np
# Generating random data
num_students = 100
study_hours = np.random.randint(1, 10, num_students) # Random study hours between 1 and 10
exam_scores = 50 + 10 * study_hours + np.random.normal(0, 5, num_students) # Exam scores based on study hours
# Creating the DataFrame
data = {'study_hours': study_hours, 'exam_scores': exam_scores}
df = pd.DataFrame(data)
# Displaying the first few rows of the DataFrame
This will create a DataFrame with two columns: “study_hours” and “exam_scores”. Now, let’s use linear regression to predict exam scores based on study hours. We can use the scikit-learn library for
this purpose:
Step 2 : Imports the necessary libraries :
Imports the matplotlib.pyplot module, which is used for data visualization.
Imports necessary functions and classes from scikit-learn for data splitting, linear regression modeling, and performance evaluation.
import matplotlib.pyplot as plt
from sklearn.model_selection import train_test_split
from sklearn.linear_model import LinearRegression
from sklearn.metrics import mean_squared_error, r2_score
Step 3 : Splits the DataFrame into input features X (study hours) and target variable y (exam scores).
X = df[['study_hours']]
y = df['exam_scores']
Step 4 : Splits the data into training and testing sets using 80% for training and 20% for testing. The random_state=0 ensures reproducibility.
X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2, random_state=0)
Step 5 : Initializes a linear regression model.
model = LinearRegression()
Step 6 : Trains the linear regression model using the training data.
model.fit(X_train, y_train)
Step 7 : Uses the trained model to make predictions on the test data and calculates the R-squared score (r2) and Mean Squared Error (mse) to evaluate the model’s performance.
y_pred = model.predict(X_test)
r2 = r2_score(y_test, y_pred)
mse = mean_squared_error(y_test, y_pred)
Step 8 : Plots the test data points and the fitted line obtained from the linear regression model. Labels the axes, provides a title, and shows the plot.
plt.scatter(X_test, y_test, label='Test Data')
plt.plot(X_test, y_pred, color='red', label='Fitted Line')
plt.xlabel('Study Hours')
plt.ylabel('Exams Score')
plt.title('Linear Regression')
Step 9 : Prints the Mean Squared Error and R-squared Error calculated earlier, providing insights into the model’s accuracy and fit to the data.
print(f"Mean Squared Error: {mse:.2f}")
print(f"R-Squared Error : {r2:.2f}")
Let’s combine all these steps and the code :
import pandas as pd
import numpy as np
import matplotlib.pyplot as plt
from sklearn.model_selection import train_test_split
from sklearn.linear_model import LinearRegression
from sklearn.metrics import mean_squared_error,r2_score
# Create a DataFrame with employee data
data = {'study_hours': [12, 21, 31, 44, 15, 25, 37, 42, 27, 17, 14, 23, 33, 46, 19, 35, 39, 40, 24, 18],
'exam_scores': [50, 70, 84, 97, 52, 73, 87, 95, 75, 58, 52, 72, 87, 99, 62, 85, 90, 92, 73, 59 ]}
df = pd.DataFrame(data)
# Split the data into features (Study Hours) and target (Exam Scores)
X = df[['study_hours']]
y = df['exam_scores']
# Split the data into training and testing sets
X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2, random_state=42)
# Create a Linear Regression model
model = LinearRegression()
# Train the model on the training data
model.fit(X_train, y_train)
# Make predictions on the test data
y_pred = model.predict(X_test)
r2 = r2_score(y_test, y_pred)
# Calculate the Mean Squared Error (MSE)
mse = mean_squared_error(y_test, y_pred)
# Plot the data points and the fitted line
plt.scatter(X_test, y_test, label='Test Data')
plt.plot(X_test, y_pred, color='red', label='Fitted Line')
plt.xlabel('Study Hours')
plt.ylabel('Exams Score')
plt.title('Linear Regression')
print(f"Mean Squared Error: {mse:.2f}")
print(f"R-Squared Error : {r2:.2f}")
You will see the below chart after executing the above code :
The code provided generates a scatter plot (plt.scatter()) representing the test data points and overlays a red line plot (plt.plot()) depicting the fitted line obtained from the linear regression
model. Let’s break down what this visualization represents:
1. Scatter Plot (Blue Points):
• X-Axis: Study Hours
• Y-Axis: Exam Scores
• Each blue point on the scatter plot represents a data point from the test dataset. The x-coordinate represents the study hours, and the y-coordinate represents the corresponding exam scores.
2. Fitted Line (Red Line):
• The red line is the fitted line generated by the linear regression model based on the input features (study hours) and the predicted exam scores.
• The slope and intercept of this line are determined by the regression model during training.
• This line represents the model’s best attempt to capture the underlying relationship between study hours and exam scores in the test data.
• By visually comparing the scatter plot of the test data points with the fitted red line, you can observe how well the linear regression model fits the given data.
• If the fitted line closely follows the trend of the data points, it indicates that the linear regression model has captured the underlying pattern in the relationship between study hours and exam
• Any deviations between the data points and the fitted line might suggest areas where the model does not perform well. It’s essential to consider factors like outliers, noise in the data, or
non-linear relationships when interpreting the fit.
• The visualization provides a clear representation of how the linear regression model predicts exam scores based on study hours, making it easier to explain the model’s behavior to stakeholders or
Time to Predict the Score on the basis of Studied Hours
# Given study hours for which you want to predict marks
study_hours = int(input("Enter Studied Hours"))
# Reshape the new_experience to match the input shape the model expects
study_hours = np.array(study_hours).reshape(-1, 1)
# Use the trained model to predict the salary for the given experience
predicted_marks = model.predict(study_hours)
print(f"Predicted Score for {study_hours[0][0]} hours are : {predicted_marks[0]:.2f}")
FAQ’s on Linear Regression Problem
What is Linear Regression, and how is it applied in predicting scores based on marks?
Linear Regression is a statistical method used to model the relationship between a dependent variable (such as exam scores) and one or more independent variables (such as marks). In predicting scores
based on marks, linear regression helps establish a linear equation that best fits the relationship between these variables.
What are the key steps involved in performing a Linear Regression for score prediction?
The key steps include data collection, data preprocessing, splitting the data into training and testing sets, creating a linear regression model, training the model, making predictions, and
evaluating the model’s performance using metrics like Mean Squared Error (MSE) or R-squared.
Why is it essential to split the data into training and testing sets when working with Linear Regression?
Splitting the data helps in training the model on one subset and testing its performance on another. This ensures that the model does not simply memorize the data but generalizes well to unseen data,
providing a reliable evaluation of its predictive ability.
What role do coefficients play in a Linear Regression equation for score prediction?
Coefficients in a Linear Regression equation represent the relationship between independent and dependent variables. In the context of predicting scores based on marks, coefficients indicate how a
change in marks influences the predicted scores.
How can Linear Regression be affected by outliers in the data when predicting scores from marks?
Outliers can significantly impact Linear Regression by skewing the regression line. They can disproportionately influence the slope and intercept of the line, leading to inaccurate predictions.
Identifying and handling outliers is crucial to maintain the model’s accuracy.
Is Linear Regression the only method for predicting scores based on marks?
No, while Linear Regression is commonly used, there are other machine learning techniques like Decision Trees, Random Forest, and Neural Networks that can also be applied for score prediction based
on marks. The choice of method depends on the complexity of the relationship and the dataset.
How can one interpret the results obtained from a Linear Regression model predicting scores from marks?
Interpretation involves understanding the coefficients to see how much a one-unit change in marks affects the predicted score. Additionally, model evaluation metrics like MSE or R-squared provide
insights into the accuracy and goodness of fit of the model.
Are there any specific preprocessing techniques applied to marks data before performing Linear Regression?
Yes, preprocessing techniques like normalization or standardization of marks data can be applied to ensure consistency in scale, which aids in the accurate interpretation of coefficients and model
performance evaluation.
Can Linear Regression predict scores accurately for various subjects, or does it require customization for each subject?
Linear Regression can predict scores for different subjects if there is a linear relationship between marks and scores across subjects. However, customization might be necessary if the relationships
significantly vary between subjects.
How can one improve the accuracy of a Linear Regression model predicting scores based on marks?
Improving accuracy can be achieved through feature engineering, handling outliers, using advanced regression techniques, and ensuring a representative dataset. Regular evaluation and refinement of
the model also contribute to enhanced accuracy.
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Mahesh Verma
I have been working for 10 years in software developing field. I have designed and developed applications using C#, SQL Server, Web API, AngularJS, Angular, React, Python etc. I love to do work with
Python, Machine Learning and to learn all the new technologies also with the expertise to grasp new concepts quickly and utilize the same in a productive manner.
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UMBC CMSC202, Computer Science II, Spring 1998, Sections 0101, 0102, 0103, 0104 and Honors
Thursday February 5, 1998
Assigned Reading:
• Programming Abstractions in C: 5.1-5.2
Handouts (available on-line): none
Topics Covered:
• The Fibonacci numbers are defined recursively, so we can easily write a program with a recursive function that computes the n^th Fibonacci number. ( Sample run.)
□ Note the use of argc and argv in the parameter list of the main function. These parameters allow us to access the command line arguments that were typed in at the UNIX prompt.
□ This program is really slow. It took 3 minutes and 22 seconds to compute the 42nd Fibonacci number. The reason is that the fib() function is called over and over again for the same values.
For example, to compute fib(n), the value of fib(1) is computed fib(n-1) times. Since the Fibonacci numbers grow very quickly, this makes our program very slow.
□ One way to fix the problem of repeatedly computing the same values of fib(i) is to use a memoization table. (Note: the word is not "memorization".) In this approach, after we compute the
value of fib(i) for the first time, we record it in the memoization table. If the function fib() is called with parameter i again, the value from the table is retrieved and returned. This
approach reduces the number of recursive calls dramatically and gives us a much faster program. The sample run shows that the new running time to compute fib(42) is less than 0.1 seconds.
• The greatest common divisor (gcd) of two numbers m and n is the largest integer that divides both m and n. We show how recursion is used to find a fast algorithm to compute gcd.
□ First, we implement a naive algorithm to compute the gcd of two numbers. This program simply tries every number between 2 and the smaller of m and n. The largest divisor is returned. The
sample run shows that this program is quite slow. It took 22 seconds in one case to compute the gcd of two large-ish numbers.
□ Euclid, an ancient Greek mathematician, showed that
gcd(m,n) = gcd(n, m % n).
Using Euclid's recursive algorithm, we can produce a faster program to compute the gcd of two numbers. The sample run shows a running time of less than 0.1 seconds for the same input that
took 22 seconds previously. Note that the worst case input to Euclid's gcd algorithm is two consecutive Fibonacci numbers.
□ We can also implement Euclid's algorithm using a while loop instead of recursion. (Program and sample run.) With modern optimizing compilers, this program isn't necessarily faster than the
recursive version. One disadvantage of not using recursion is that Euclid's original formulation is lost.
• Another classic example of problem solving by recursion is the Towers of Hanoi problem. Program and sample run.
• Finally, we looked at an example of mutually recursion where two functions call each other. In our program, the function count_alpha() does not call itself directly, but does call the function
count_non_alphas(). Since count_non_alphas() also calls count_alpha(), the execution of count_alpha() does not return before another call to count_alpha() is issued. Hence we must take great care
to ensure that the recursion terminates properly. In this program, we rely on the null character at the end of the string to trigger the base case of the recursion. See sample run.
Last Modified: 22 Jul 2024 11:27:46 EDT by Richard Chang
Back up to Spring 1998 CMSC 202 Section Homepage | {"url":"https://userpages.cs.umbc.edu/chang/cs202.s98/lectures/Lecture04.shtml","timestamp":"2024-11-02T18:55:21Z","content_type":"text/html","content_length":"5268","record_id":"<urn:uuid:69449ae5-e7d0-46b1-a40f-4cd3b49db7b9>","cc-path":"CC-MAIN-2024-46/segments/1730477027729.26/warc/CC-MAIN-20241102165015-20241102195015-00446.warc.gz"} |
Special Seminar -Adi Shamir- 23.06.22
Speaker: Prof. Adi Shamir
Time: Thursday, 23.06.22, 12:30
Place: Taub 1
Title: Efficient Detection of High Probability Cryptographic Properties of Large Boolean Functions via Surrogate Differentiation
Abstract: A central problem in cryptanalysis is to find all the significant deviations from randomness in a given $n$-bit cryptographic primitive. When $n$ is large, the only practical way to find
such statistical properties was to exploit the internal structure of the primitive and to speed up the search with a variety of heuristic rules of thumb. However, such bottom-up techniques can miss
many properties, especially in cryptosystems which are designed to have hidden trapdoors.
In this talk I will consider the top-down version of the problem in which the cryptographic primitive is given as a structureless black box which implements an arbitrary Boolean function from $n$
bits to $n$ bits. I will then show how to reduce the complexity of the best known techniques for finding all its significant differential and linear properties by a large factor of $2^{n/2}$. The
main new idea is to use {\it surrogate differentiation}, which is a new way to analyze the properties of large Boolean functions. In the context of finding differential properties, it enables us to
simultaneously find information about all the differentials of the form $f(x) \oplus f(x \oplus \alpha)$ in all possible directions $\alpha$ by differentiating $f$ in a single arbitrarily chosen
direction $\gamma$ (which is unrelated to the $\alpha$’s). In the context of finding linear properties, surrogate differentiation can be combined in a highly effective way with the Fast Fourier
This is joint work with Itai Dinur, Orr Dunkelman, Nathan Keller, and Eyal Ronen. | {"url":"https://cyber.technion.ac.il/special-seminar-adi-shamir-23-06-22/","timestamp":"2024-11-05T22:23:31Z","content_type":"text/html","content_length":"56066","record_id":"<urn:uuid:017e8696-7206-40bb-9e54-9104398dc404>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00661.warc.gz"} |
Number of Teeth
The rotor on top of Miles spins a worm gear. I want that worm gear to rotate the rightmost digit wheel once for each mile of air that passes by. To get that desired rotation, I need to mount an
ordinary gear on that digit wheel. How many teeth must that gear have? Here’s how I approximated that number.
A mile of air is 5,280 feet long.
MileOfAir = 5,280 feet
The radius of the rotor is 3 feet, making its diameter 6 feet. Then the path followed by a rotor cup during one revolution of the rotor is 6π ≈ 18.85 feet.
Circumference ≈ 18.85 feet
One would think that the rotor will revolve once whenever 18.85 feet of air pass by, but the rotor is not entirely efficient. Assume that the rotor is 50% efficient.
EfficiencyOfRotor = 50%
That means that it only rotates halfway for every 18.85 feet of air. So for a full revolution, twice as much air must pass by. To put this relationship in an equation, one divides the Circumference
by the Efficiency of the rotor.
AirPerRevolution = Circumference / EfficiencyOfRotor
which gives us
AirPerRevolution = 18.85 / 50% = 37.7 feet
So the rotor revolves once after the passage of 37.7 feet of air, and a mile of air has 5,280 feet. I can get the number of times that the rotor revolves during the passage of one mile of air by
dividing 5,280 by 37.7. The result also is the number of teeth needed in the gear that engages the rotor’s worm gear.
NumberOfTeeth = MileOfAir / AirPerRevolution = 5,280 / 37.7 ≈ 140
That quantity uses my assumption that the rotor’s efficiency is exactly 50%, but I don’t know the efficiency precisely. So I’ll build the rotor first, mount it on my pickup truck, and have a
passenger count the number of revolutions that it makes for each mile driven.
To save material, I’m going to fabricate the gear in six pieces. The image of the gear at the top of this page has 150 teeth because I don’t yet know how many teeth are needed, and it’s easier to
draw a gear from six identical pieces. If necessary, I’ll create six different pieces, but for now I’ll hope that the driving test of the rotor yields a number of teeth that is divisible by six. | {"url":"https://www.kerryveenstra.com/2016/04/20/number-of-teeth/","timestamp":"2024-11-13T02:06:20Z","content_type":"application/xhtml+xml","content_length":"34962","record_id":"<urn:uuid:718f7728-92a8-47c0-bf7a-1c489c689ef8>","cc-path":"CC-MAIN-2024-46/segments/1730477028303.91/warc/CC-MAIN-20241113004258-20241113034258-00143.warc.gz"} |
How to Calculate Ratios in Excel - Smart Calculations
How to Calculate Ratios in Excel
In this tutorial, you will learn how to calculate ratios in Excel.
Comparing two amounts of the same units and determining the ratio tells us how much of one quantity is in the other. Two categories can be used to categorize ratios. Part to whole ratio is one, and
part to part ratio is the other. The part-to-part ratio shows the relationship between two separate entities or groupings. For instance, a class has a 12:15 boy-to-girl ratio, whereas the
part-to-whole ratio refers to the relationship between a particular group and the entire. For instance, five out of every ten people enjoy reading. As a result, the ratio of the part to the whole is
5: 10, meaning that 5 out of every 10 people enjoy reading.
Once ready, we’ll get started by utilizing real-world examples to show you how to calculate ratios in Excel.
Anatomy of GCD Functions
GCD Function
GCD(number1, [number2], …)
This function returns the largest common divisor of two or more integers. The biggest integer that divides both numbers 1 and 2 without leaving a residual is known as the greatest common divisor.
Calculate Ratios in Excel
Before we begin we will need a group of data to calculate ratios in Excel.
Step 1
First, you need to have a clean and tidy group of data.
Step 2
To find the ratio between data from Group A and Group B, we can simply insert this formula =A2/GCD(A2, B2)&”:”&B2/GCD(A2, B2).
Step 3
Once you press Enter, your formula will return the ratio for Group A and Group B. | {"url":"https://smartcalculations.com/how-to-calculate-ratios-in-excel/","timestamp":"2024-11-07T22:45:22Z","content_type":"text/html","content_length":"51547","record_id":"<urn:uuid:1e79b399-74cb-4461-91c3-8062234eeb44>","cc-path":"CC-MAIN-2024-46/segments/1730477028017.48/warc/CC-MAIN-20241107212632-20241108002632-00174.warc.gz"} |
Educational Technology
Benefits Of Technology In Instruction
Benefits Of Technology In Instruction
This article discusses the various ways computer technology can be used to improve how and what children learn in the classroom. Several examples of computer-based applications are highlighted to
illustrate ways technology can enhance how children learn by supporting four fundamental characteristics of learning: (1) active engagement, (2) participation in groups, (3) frequent interaction and
feedback, and (4) connections to real-world contexts. Additional examples illustrate ways technology can expand what children learn by helping them to understand core concepts in subjects like math,
science, and literacy. Research indicates, however, that the use of technology as an effective learning tool is more likely to take place when embedded in a broader education reform movement that
includes improvements in teacher training, curriculum, student assessment, and a school’s capacity for change.
A teacher from the late nineteenth century entering a typical classroom today would find most things quite familiar: chalk and talk, as well as desks and texts, predominate now as they did then. Yet
this nineteenth-century teacher would be shocked by the demands of today’s curricula. For example, just a century ago, little more was expected of high school students than to recite famous texts,
recount simple scientific facts, and solve basic arithmetic problems. Today, all high school students are expected to be able to read and understand unfamiliar text and to become competent in the
processes of scientific inquiry and mathematics problem solving, including algebra.
This trend of rising expectations is accelerating because of the explosion of knowledge now available to the public and the growing demands of the workplace. More and more students will have to
learn to navigate through large amounts of information and to master calculus and other complicated subjects to participate fully in an increasingly technological society. Thus, although the
classroom tools of blackboards and books that shape how learning takes place have changed little over the past century, societal demands on what students learn have increased dramatically.
There is consensus among education policy analysts that satisfying these demands will require rethinking how educators support learning. The role that technology could or should play within this
reform movement has yet to be defined. Innovations in media technology, including radio, television, film, and video, have had only isolated, marginal effects on how and what children learn in
school, despite early champions of their revolutionary educational potential.. Furthermore, although computer technology is a pervasive and powerful force in society today with many proponents of its
educational benefits, it is also expensive and potentially disruptive or misguided in some of its uses and in the end may have only marginal effects. Nevertheless, several billion dollars in public
and private funds have been dedicated to equipping schools with computers and connections to the Internet, and there are promises of even more funds dedicated to this purpose in the future. As
ever-increasing resources are committed to bringing computers into the classroom, parents, policymakers, and educators need to be able to determine how technology can be used most effectively to
improve student learning. Enhancing How Children Learn
Learning Through Active Engagement
Learning research has shown that students learn best by actively “constructing” knowledge from a combination of experience, interpretation, and structured interactions with peers and teachers. When
students are placed in the relatively passive role of receiving information from lectures and texts (the “transmission” model of learning), they often fail to develop sufficient understanding to
apply what they have learned to situations outside their texts and class-rooms. In addition, children have different learning styles. The use of methods beyond lectures and books can help reach
children who learn best from a combination of teaching approaches. Educational reformers agree with the theoreticians and experts that to enhance learning, more attention should be given to
actively engaging children in the learning process. Curricular frameworks now expect students to take active roles in solving problems, communicating effectively, analyzing information, and
designing solutions—skills that go far beyond the mere recitation of correct responses.
Although active, constructive learning can be integrated in classrooms with or without computers, the characteristics of computer-based technologies make them a particularly useful tool for this type
of learning. For example, consider science laboratory experiments. Students certainly can actively engage in experiments without computers, yet nearly two decades of research has shown that
students can make significant gains when computers are incorporated into labs under a design called the “Microcomputer-Based Laboratory” (MBL).
The structure and resources of traditional classrooms often provide quite poor support for learning, whereas technology—when used effectively—can enable ways of teaching that are much better matched
to how children learn.
Using technology to engage students more actively in learning is not limited to science and mathematics. For example, computer-based applications such as desktop publishing and desktop video can be
used to involve students more actively in constructing presentations that reflect their understanding and knowledge of various subjects. Although previous media technologies generally placed
children in the role of passive observers, these new technologies make content construction much more accessible to students, and research indicates that such uses of technology can have significant
positive effects.
Learning Through Participation in Groups
Some critics feel that computer technology encourages asocial and addictive behavior and taps very little of the social basis of learning. Several computer-based applications, such as tutorials
and drill-and-practice exercises, do engage students individually. However, projects that use computers to facilitate educational collaboration span nearly the entire history of the Internet, dating
back to the creation of electronic bulletin boards in the 1970s. Some of the most prominent uses of computers today are communications oriented, and networking technologies such as the Internet and
digital video permit a broad new range of collaborative activities in schools. Using technology to promote such collaborative activities can enhance the degree to which classrooms are socially active
and productive and can encourage classroom conversations that expand students’ understanding of the subject.
Learning Through Frequent Interaction and Feedback
In traditional classrooms, students typically have very little time to interact with materials, each other, or the teacher. Unlike other media, computer technology supports this learning principle
in at least three ways. First, computer tools themselves can encourage rapid interaction and feedback. For example, using interactive graphing, a student may explore the behavior of a mathematical
model very rapidly, getting a quicker feel for the range of variation in the model. If the same student
Students who participate in computer-connected learning networks show increased motivation, a deeper understanding of concepts, and an increased willingness to tackle difficult questions.
Research indicates that computer applications such as those described above can be effective tools to support learning. One study compared two methods of e-mail-based coaching. In the first method,
tutors generated a custom response for each student. In the second, tutors sent the student an appropriate boilerplate response. Students’ learning improved significantly and approximately equally
using both methods, but the boilerplate-based coaching allowed four times as many students to have access to a tutor. In another version of computer-assisted feedback, a program called Diagnoser
assesses students’ understanding of physics concepts in situations where students typically make mistakes, then provides teachers with suggested remedial activities. Data from experimental and
control classrooms showed scores rising more than 15% when teachers incorporated use of Diagnoser, and the results were equally strong for low, middle, and high achievers.
The most sophisticated applications of computers in this area have tried to trace students’ reasoning process step by step, and provide tutoring whenever students stray from correct reasoning.
Results from Geometry Tutor, an application that uses this approach, showed students—especially average or lower achievers or students with low self-confidence in mathematics—could learn geometry
much faster with such help. Also, researchers at Carnegie Mellon University found that urban high school students using another application, Practical Algebra Tutor, showed small gains on
standardized math tests such as the Scholastic Aptitude Test (SAT), but more than doubled their achievement in complex problem solving compared to students not using this technology.
Learning Through Connections to Real-World Contexts
Computer technology can provide students with an excellent tool for applying concepts in a variety of contexts, thereby breaking the artificial isolation of school subject matter from real-world
situations. For example, through the communication features of computer-based technology, students have access to the latest scientific data and expeditions, whether from a NASA mission to Mars,
an ongoing archeological dig in Mexico, or a remotely controlled telescope in Hawaii. Further, technology can bring unprecedented opportunities for students to actively participate in the kind of
experimentation, design, and reflection that professionals routinely do, with access to the same tools professionals use. Through the Internet, students from around the world can work as partners to
scientists, businesspeople, and policymakers who are making valuable contributions to society.
One important project that allows students to actively participate in a real-world research project is the Global Learning and Observations to Benefit the Environment (GLOBE) Program. Begun in 1992
as an innovative way to aid the environment and help students learn science, the GLOBE Program currently links more than 3,800 schools around the world to scientists.Teachers and students collect
local environmental data for use by scientists, and the scientists provide mentoring to the teachers and students about how to apply scientific concepts in analyzing real environmental problems.
Thus, the GLOBE Program depends on students to help monitor the environment while educating them about it. Further, the students are motivated to become more engaged in learning because they are
aiding real scientific research—and their data collection has lasting value. In a 1998 survey, 62% of teachers using the GLOBE Program reported that they had students analyze, discuss, or interpret
the data.
Although no rigorous evaluations of effects on learning have been conducted, surveyed GLOBE teachers said they view the program as very effective and indicated that the greatest student gains
occurred in the areas of observational and measurement skills, ability to work in small groups, and technology skills.Expanding What Children Learn
In addition to supporting how children learn, computer-based technology can also improve what children learn by providing exposure to ideas and experiences that would be inaccessible for most
children any other way. For example, because synthesizers can make music, students can experiment with composing music even before they can play an instrument. Because communications technology
makes it possible to see and talk to others in different parts of the world, students can learn about archeology by following the progress of a real dig in the jungles of Mexico. Through online
communications, students can reach beyond their own community to find teachers and other students who share their academic interests. The most interesting research on the ways technology can improve
what children learn, however, focuses on applications that can help students understand core concepts in subjects like science, math, and literacy by representing subject matter in less complicated
ways. Research has demonstrated that technology can lead to profound changes in what children learn. By using the computers’ capacity for simulation, dynamically linked notations, and
interactivity, ordinary students can achieve extraordinary command of sophisticated concepts. Computer-based applications that have had significant effects on what children learn in the areas of
science, mathematics, and the humanities are discussed below.
Science: Visualization, Modeling, and Simulation
Over the past two decades, researchers have begun to examine what students actually learn in science courses. To their surprise, even high-scoring students at prestigious universities show little
ability to provide scientific explanations for simple phenomena, such as tossing a ball in the air. This widely replicated research shows that although students may be able to calculate correctly
using scientific formulas, they often do not understand the concepts behind the formulas. Computer-based applications using visualization, modeling, and simulation have been proven to be powerful
tools for teaching scientific concepts. The research literature abounds with successful applications that have enabled students to master concepts usually considered too sophisticated for their
grade level. For example, technology using dynamic diagrams—that is, pictures that can move in response to a range of input—can help students visualize and understand the forces underlying various
phenomena. Involving students in making sense of computer simulations that model physical phenomena, but defy intuitive explanations, also has been shown to be a useful technique. One example of
this work is ThinkerTools, a simulation program that allows middle school students to visualize the concepts of velocity and acceleration. In controlled studies, researchers found that middle
school students who used ThinkerTools developed the ability to give correct scientific explanations of Newtonian principles several grade levels before the concept usually is taught. Middle school
students who participated in ThinkerTools outperformed high school physics students in their ability to apply the basic principles of Newtonian mechanics to real-world situations. Other software
applications have been proven successful in helping students master advanced concepts underlying a variety of phenomena. The application Stella enables high school students to learn system
dynamics—the modeling of economic, social, and physical situations using a set of interacting equations—which is ordinarily an advanced undergraduate course. Another software application uses special
versions of Logo, a programming language designed especially for children, to help high school students learn the concepts that govern bird-flocking and highway traffic patterns, even though the
mathematics needed to understand these concepts is not ordinarily taught until graduate-level studies. And yet another application, the Global Exchange curricula, reaches tens of thousands of
precollege students annually with weather map visualizations that enable schoolchildren to reason like meteorologists. Research has shown that students using the curricula demonstrate increases in
both their comprehension of meteorology and their skill in scientific inquiry.
Mathematics: Dynamic, Linked Notations
While seeking techniques for increasing how much mathematics students can learn, researchers have found that the move from traditional paper-based mathematical notations (such as algebraic symbols)
to onscreen notations (including algebraic symbols, but also graphs, tables, and geometric figures) can have a dramatic effect. In comparison to the use of paper and pencil, which supports only
static, isolated notations, use of computers allows for “dynamic, linked notations” with several helpful advantages, as described below:· Students can explore changes rapidly in the notation by
dragging with a mouse, as opposed to slowly and painstakingly rewriting the changes. · Students can see the effects of changing one notation on another, such as modifying the value of a parameter of
an equation and seeing how the resulting graph changes its shape. · Students can easily relate mathematical symbols either to data from the real world or to simulations of familiar phenomena, giving
the mathematics a greater sense of meaning. · Students can receive feedback when they create a notation that is incorrect. (For example, unlike with paper and pencil, a computer can beep if a student
tries to sketch a nonsensical mathematical function in a graph, such as one that “loops back” to define two different y values for the same x value.) Another example of a software application using
screen-based notations is Geometer’s Sketchpad, a tool for exploring geometric constructions directly onscreen. Such applications are revitalizing the teaching of geometry to high school students,
and in a few instances, students even have been able to contribute novel and elegant proofs to the professional mathematical literature. Graphing calculators, which are reaching millions of new high
school and middle school students each year, are less sophisticated than some of the desktop computer-based technologies, but they can display algebra, graphs, and tables, and can show how each of
these notations represents the same mathematical object.
Social Studies, Language, and the Arts
Unlike science and math, breakthrough uses of technology in other subject areas have yet to crystallize into easily identified types of applications. Nonetheless, innovators have shown that similar
learning breakthroughs in these areas are possible. For example, the commercially successful SimCity game (which is more an interactive simulation than a traditional video game) has been used to
teach students about urban planning. Computer-based tools have been designed to allow students to choreograph a scene in a Shakespeare play or to explore classic movies, such as Citizen Kane, from
multiple points of view to increase their ability to consider alternative literary interpretations. Through the Perseus Project, students are provided with access to a pioneering multimedia
learning environment for exploring hyperlinked documents and cultural artifacts from ancient civilizations. Similar software can provide interactive media environments for classes in the arts. An
emergent theme in many computer-based humanities applications is using technology that allows students to engage in an element of design, complementing and enhancing the traditional emphasis on
appreciation. In one innovative project, elementary and middle school children alternate between playing musical instruments, singing, and programming music on the computer using Tuneblocks, a
musical version of the Logo programming language. Compelling case studies show how using this software enables ordinary children to learn abstract musical concepts like phrase, figure, and
meter—concepts normally taught in college music theory classes. In another example, a tool called Hypergami enables art students to plan complicated mathematical sculptures in paper. Experiences
with Hypergami have produced significant gains in boys’ and girls’ performance on the spatial reasoning sections of the SAT.
The Challenge of Implementation
The preceding overview provides only a glimpse of the many computer-based applications that can enhance learning. But simply installing computers and Internet access in schools will not be
sufficient to replicate these examples for large numbers of learners. Models of successful technology use combine the introduction of computer tools with new instructional approaches and new
organizational structures. Because any typical educational system is somewhat like an interlocking jigsaw puzzle, efforts to change one piece of the puzzle—such as using technology to support a
different kind of content and instructional approach—are more likely to be successful if the surrounding pieces of teacher development, curriculum, assessment, and the school’s capacity for reform
are changed as well. | {"url":"https://teachersity.org/resource.php?cat_id=5&content_id=69","timestamp":"2024-11-04T11:56:44Z","content_type":"application/xhtml+xml","content_length":"50390","record_id":"<urn:uuid:7bcf3668-be93-4576-ab31-2490f7f92a08>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00060.warc.gz"} |
PROC SCORE error in SAS 9.4 but not in 9.1.3
We are trying to migrate to SAS 9.4 and I am testing my code under 9.4. The same code works fine in our current 9.1.3, but generate the errors in 9.4.
The source code:
proc score data=INPUT
score=PARAMETERS out=test PREDICT TYPE=PARMS;
by A B C D;
id E F G H;
The error message:
ERROR: BY values from PARAMETERS and INPUT do not match.
NOTE: The above message was for the following BY group:
A =_A B=1 C=2 D=0
ERROR: BY values from PARAMETERS and INPUT do not match.
NOTE: The above message was for the following BY group:
A =_A B=1 C=2 D=1
.... (hundreds similar errors)
Does anyone have experience with this? The SAS online document does not mention any change of the PROC SCORE in 9.4, and does not mention any related system options change. I am really puzzled. Could
anyone help? Thank you very much in advance!
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Exploring All Possible Combinations for the Number Set 2357
When analyzing specific combinations of numbers, such as 2357, it's essential to understand the vast range of arrangements possible. This particular set offers unique patterns and variations that can
serve various applications, from password creation to numerical analysis in probability or logic puzzles. In this article, we’ll explore the many ways to organize, arrange, and utilize these four
digits effectively.
Understanding the Basics of Combinations and Permutations
Before diving into the specifics of the possible combinations for 2357, it’s helpful to clarify what combinations and permutations mean in the context of a set of numbers:
• Combinations: Combinations involve selecting items from a larger set without regard to the order. In the context of 2357, combinations focus on how these numbers can be grouped together.
• Permutations: Permutations consider the order of arrangement. For example, in permutations, the sequence "2357" is distinct from "2375." Since the number set 2357 is a four-digit sequence,
permutations reveal all possible arrangements of these four digits.
For the set of numbers 2, 3, 5, and 7, we’ll examine both permutations and combinations to uncover every possible arrangement and its potential application.
Permutations of 2357: Arranging All Possible Orders
To determine the possible combinations for 2357 through permutations, we calculate the arrangements where order matters. Since each digit is unique, we’ll find the factorial of the number of digits
(4!), yielding:
4!=4×3×2×1=244! = 4 \times 3 \times 2 \times 1 = 244!=4×3×2×1=24
This means there are 24 possible permutations for the number set 2357, which are listed below:
1. 2357
2. 2375
3. 2537
4. 2573
5. 2735
6. 2753
7. 3257
8. 3275
9. 3527
10. 3572
11. 3725
12. 3752
13. 5237
14. 5273
15. 5327
16. 5372
17. 5723
18. 5732
19. 7235
20. 7253
21. 7325
22. 7352
23. 7523
24. 7532
Each permutation is a unique four-digit sequence, giving us insight into how the digits can be ordered differently. This list is essential for those who need unique identifiers, password ideas, or
even lottery number patterns.
Subset Combinations of 2357: Choosing Smaller Groups
Now, if we consider subsets of this set, we can find the possible combinations for 2357 in smaller groups, such as pairs or triplets. We’ll examine the groups of 2 and 3, as they represent useful,
smaller configurations of these digits.
Two-Digit Combinations
For two-digit combinations from 2357, the formula for combinations (without regard to order) applies as follows:
1. Unique Pairs: Since there are four numbers, we can select 2 at a time (4 choose 2), which equals 6 unique pairs:
These pairs may also appear in reversed order if permutations are allowed. Therefore, if we consider them as ordered pairs, we have 12 unique arrangements for two-digit sequences:
1. 23
2. 32
3. 25
4. 52
5. 27
6. 72
7. 35
8. 53
9. 37
10. 73
11. 57
12. 75
Three-Digit Combinations
When choosing three digits from 2357, the number of combinations is again a key consideration. There are 4 possible groups of 3 from 4 digits, yielding the following groups:
1. 235
2. 237
3. 257
4. 357
If we now look at these groups as permutations (ordered), we see there are six arrangements for each, giving us a total of 24 unique three-digit arrangements. Here’s an example using the subset "235"
to illustrate all permutations:
This permutation process applies similarly to the other groups (237, 257, and 357), expanding our possible combinations.
Applications of 2357 Combinations in Real-World Scenarios
The number set 2357 has a variety of practical applications, especially for people in fields like cryptography, game theory, and combinatorics.
Password Generation
One common application for these combinations is in creating secure, unique passwords. With 24 four-digit permutations and even more subsets in two- and three-digit arrangements, it’s easy to develop
several secure password options from this number set. A randomly selected permutation ensures unpredictability, essential for strong passwords.
Logic Puzzles and Game Theory
In puzzles or games that require the arrangement of numbers in specific sequences, having a pre-set list of possible combinations for 2357 is valuable. For example, in a game where points are earned
for unique arrangements of a set of numbers, knowing all possible configurations of 2357 gives players a strategic edge.
Combinatorial Mathematics and Probability Analysis
In statistical applications or probability theory, analyzing a set like 2357 involves understanding both fixed orderings (for probabilities related to sequences) and unordered groups (for
combination-based calculations). This analysis helps in probability calculations for events where specific number sequences might hold significance.
Conclusion: Leveraging the Power of 2357 in Number Combinations
Exploring the possible combinations for 2357 offers a fascinating look into the flexibility of four-digit sets. Whether used for password creation, logical reasoning, or advanced probability
scenarios, these 24 permutations and numerous smaller combinations reveal the depth of possibility within just four numbers. Mastering these arrangements empowers users to apply them in various
mathematical, practical, and security-focused applications. | {"url":"https://www.careerupdraft.net/blog/general-6/possible-combinations-for-2357-numbers-632","timestamp":"2024-11-01T22:34:23Z","content_type":"text/html","content_length":"30337","record_id":"<urn:uuid:cf706f1c-06c6-4cb9-9624-d850e154d967>","cc-path":"CC-MAIN-2024-46/segments/1730477027599.25/warc/CC-MAIN-20241101215119-20241102005119-00477.warc.gz"} |
Data-at-Rest Is Not A New Requirement at NASA
Data-at-Rest (DAR) at NASA HQ
“This page contains important information for employees regarding the Data-at-Rest (DAR) Encryption project at Headquarters. As mandated by Federal law and Agency policy, all NASA-issued laptops must
have Data-At-Rest (DAR) whole-disk encryption software. The NASA OCIO has directed that all Centers complete this activity by December 21, 2012. Per the Agency directive dated November 13, 2012, no
NASA-issued laptops containing sensitive information may be removed from a NASA facility unless DAR encryption software is enabled OR any sensitive files are individually encrypted (using Entrust
Recommendation to Fund and Deploy Agency Data-at-Rest (DAR) Solution, NASA CIO, 21 Feburary 2008
“Based on an evaluation of NASA’s requirements for encryption of data at rest and of the solutions currently available, I recommend that your office fund the implementation and deployment of an
integrated, interoperable NASA DAR solution in the amount of $2.0M for Fiscal Year 2008. Details of the recommended solution, based on McAfee’s Safeboot product suite, and the evaluation that
produced this recommendation are in the attached presentation.”
Keith’s note: Looks like there was direction executed within the CIO in early 2008 – before the current CIO even arrived on the job. Four years later and NASA is only getting around to taking its own
decisions seriously. Note: there is no date on this PDF file but it was created on 21 Feb 2008.
11 responses to “Data-at-Rest Is Not A New Requirement at NASA”
1. Entrust PKI is no picnic, but makes more sense than DAR. Since most of these laptops are shared there is no way to keep track of the DAR passwords unless they are written on something kept with
the computer. Of course that would be a violation, but there is no practical way to actually follow the rules, something that never seems to bother IT.
2. CIO page has a foretelling event on page 6.
Linda and her staff are fearful of losing a laptop with 10,000 names. Opps.
3. It looks like we are going with the tried and true bureaucratic method of inconveniencing everybody big time with a bunch of arcane encryption requirements rather than looking for and punishing
the bad actors. Loss of productivity by innocent people now having to implement this garbage is not a factor. Just do it and don’t ask questions. And, by the way, according to some hand-wringing
managers, everything we do is “sensitive”.
4. From the POV of an outsider–mine– this just looks silly. Why does NASA need another freaking acronym and funded program for this? My mac has native encryption. Doesn’t Windows, too? So how about
a memo: Everyone! Got sensitive data? Encrypt your laptops!?
I am missing something here.
□ unfortunately, when the ACES contract came into being, and HP laptops replaced Dell ones, the TPM (trusted platform module) required to run windows bitlocker (software builtin and FREE) was
not a part of the baseline configuration. For a part that cost pennies when purchased in bulk, HP chimped out and gave a laptop that is below those available to end users from any electronics
retailer. I run bitlocker on my home laptop…. inexcusable for NASA not to have this option as a baseline requirement. would have saved a whole lot of trouble! | {"url":"https://nasawatch.com/itweb/data-at-rest-is-not-a-new-requirement-at-nasa/","timestamp":"2024-11-03T22:04:57Z","content_type":"text/html","content_length":"103992","record_id":"<urn:uuid:29f7b5e9-4ece-4e37-936f-1b50c8be00c9>","cc-path":"CC-MAIN-2024-46/segments/1730477028558.0/warc/CC-MAIN-20241114094851-20241114124851-00334.warc.gz"} |
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Help!!! Three years ago, the average price of movie tickets was $8.75 and now it’s $10.87. What is the annual multiplier and percent i - DocumenTVHelp!!! Three years ago, the average price of movie tickets was $8.75 and now it’s $10.87. What is the annual multiplier and percent i
Help!!! Three years ago, the average price of movie tickets was $8.75 and now it’s $10.87. What is the annual multiplier and percent i
Three years ago, the average price of movie tickets was $8.75 and now it’s $10.87. What is the annual multiplier and percent increase? Sketch graph and make a table for the situation.
in progress 0
Mathematics 3 years 2021-08-10T18:29:08+00:00 2021-08-10T18:29:08+00:00 1 Answers 112 views 0 | {"url":"https://documen.tv/question/help-three-years-ago-the-average-price-of-movie-tickets-was-8-75-and-now-it-s-10-87-what-is-the-23330520-69/","timestamp":"2024-11-06T14:29:04Z","content_type":"text/html","content_length":"79455","record_id":"<urn:uuid:825bf489-efed-41f9-8e85-248a1d256ebb>","cc-path":"CC-MAIN-2024-46/segments/1730477027932.70/warc/CC-MAIN-20241106132104-20241106162104-00878.warc.gz"} |
Ongoing Research
My principal research interests lie in the field of Applied Analysis and applied Mathematics: Ordinary and Functional Differential Equations, Numerical Methods, Inverse Problems and Boundary Value
Also, I am interested in Mathematical Physics, in particular in the theory of integrable systems. My research activity concerns symmetries, coherent structures (solitons), algebraic structures of
integrable equations and their applications to physics and selected biophysical systems.
Most of my research accomplishments regard the analysis of nonlinear PDEs (both from a theoretical view point and for applications). My contribution in this area has been to classify special
algebraic and rational solutions of integrable equations, to produce new examples of this class of nonlinear differential equations and to study the asymptotic behavior of their solutions.
Variable-coefficient nonlinear evolution equations have attracted considerable attention to reflect the inhomogeneities of media, nonuniformities of boundaries, and external forces. My research is
concerned with variable-coefficient PDEs which can be used to model shallow water waves, nonlinear optical pulses, currents in electrical networks, nerve pulses, waves in the atmosphere, etc. I have
developed a simplified bilinear method to obtain the N-soliton solutions of such equations. Current work deals with the explicit functions which describe the evolution of the amplitude, phase and
velocity of the waves, the dynamical behaviors for nonautonomous waves in a periodic distributed and dispersion decreasing systems, propagation characteristics and interactions among the waves.
Whilst direct formulations consist of determining the effect of a given cause, in inverse formulations the situation is completely or partially reversed. The interest is into the research of inverse
problems for partial differential equations governing phenomena in fluid flow, elasticity, acoustics, heat transfer, mechanics of aerosols, etc. Typical practical applications relate to flows in
porous media, heat conduction in materials, thermal barrier coatings, heat exchangers, corrosion, etc. My future research plans are to investigate the existence, uniqueness and stability of the
solution to the problem that mathematically models a physical phenomenon under investigation, and to develop new convergent, stable and robust algorithms for obtaining the desired solution. The
analyses concern inverse boundary value problems, inverse initial value problems, parameter identification, inverse geometry and source determination problems. | {"url":"https://staff.hu.edu.jo/temp.aspx?pno=17&id=F7KqQT/%200aQ=","timestamp":"2024-11-10T05:33:53Z","content_type":"application/xhtml+xml","content_length":"26927","record_id":"<urn:uuid:90d430e0-07d3-4e5b-8caa-f9ba53f17acd>","cc-path":"CC-MAIN-2024-46/segments/1730477028166.65/warc/CC-MAIN-20241110040813-20241110070813-00515.warc.gz"} |
Algebra calculator square roots free
algebra calculator square roots free Related topics: airline bankruptcy
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College Algebra Learning Software
help with solving college alegabra
Author Message
evenenglisxman Posted: Tuesday 18th of Oct 08:34
Can anybody help me? I have an algebra test coming up next week and I am totally confused. I need help particularly with some problems in algebra calculator square roots free that are
very complicated . I don’t wish to go to any tutorial and I would really appreciate any help in this area. Thanks!
From: UK
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Jahm Xjardx Posted: Thursday 20th of Oct 10:46
What exactly don't you understand about algebra calculator square roots free? I remember having difficulty with the same thing in Pre Algebra, so I might be able to give you some
suggestions on how to approach such problems. However if you want help with algebra on a long term basis, then you should purchase Algebrator, that's what I did in my Algebra 1, and I
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software . I would advise you to get it as soon as you can and forget about getting a math teacher. You won't regret it!
From: Odense,
Denmark, EU
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Vild Posted: Friday 21st of Oct 07:20
I too have learned Algebrator is a fantastic bit of algebra calculator square roots free software programs. I just recall my inability to comprehend the concepts of difference of
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BoalDoggee Posted: Saturday 22nd of Oct 16:22
I have tried a number of math related software. I would not name them here, but they were of no use . I hope this one is not like the one’s I’ve used in the past .
From: In my
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nedslictis Posted: Sunday 23rd of Oct 11:46
The software can be found at https://softmath.com/about-algebra-help.html.
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Mov Posted: Tuesday 25th of Oct 08:49
absolute values, perfect square trinomial and mixed numbers were a nightmare for me until I found Algebrator, which is truly the best algebra program that I have come across. I have
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MCQ on Angular Momentum of a Projectile
A particle of mass ‘m’ is projected with a velocity ‘v’ making an angle of 45º with the horizontal. When the projectile is at its maximum height, the magnitude of its angular momentum about an axis
passing through the point of projection and perpendicular to the plane of its path is
(a) zero (b) mv^2/4√2g (c) ) mv^3/4√2g (d) mv^2/√2g (e) mv^3/√2g
You will find questions similar to this on many occasions. So, take a special note of this question.
The velocity of a projectile changes continuously along its path because of the change in the vertical component of velocity under the gravitational pull. If θ is the angle of projection, the
horizontal component of velocity, vcosθ remains unchanged throughout the path and at the maximum height, the vertical component of velocity is zero and it has the horizontal velocity vcosθ only. The
‘lever arm’ for angular momentum at the maximum height is the maximum height (v^2sin^2θ)/2g itself so that the angular momentum is (mvcosθ)×(v^2sin^2θ)/2g = (mvcos45º )×(v^2sin^2 45º )/2g = mv^3/4√2g
A simplified form of this question appeared in KEAM (Medical) 2007 question paper:
A particle is projected with a speed ‘v’ at 45º with the horizontal. The magnitude of angular momentum of the projectile about the point of projection when the projectile is at its maximum height ‘h’
(a) zero (b) mvh^2/√2 (c) mv^2h/2 (d) mvh^3/√2 (e) mvh/√2
Since the maximum height is given as ‘h’, you can write the answer in no time as (mvcos45º)×h = mvh/√2. | {"url":"http://www.physicsplus.in/2007/05/mcq-on-angular-momentum-of-projectile.html","timestamp":"2024-11-13T16:03:55Z","content_type":"application/xhtml+xml","content_length":"90377","record_id":"<urn:uuid:dfc09c88-478f-4b42-8336-aca84726d119>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00597.warc.gz"} |
bit of 2D maths help: travel Arc between two points?
I have a problem and I’m not smart enough to work out a solutions. Plus a company delivered my item to the wrong door and say its my fault…? so a bit stressed!!!
I want to achieve a throw, a bomb between two points, and so it travels as an art from point A, to point B.
My idea was:
Use half a circle, but cant even seem to get circle right today,
Then just stop after 180 deg… (or the degrees between the points, which I didnt work out yet?)
The points are not on a 180Deg plane. but one may be higher than the other.
…Make a circle…which is playing up?! I cant even get that right! I thought maybe the coords system? .
UNITY x, y are x, y
my coords are ( 0, 0 ) is top left. (helps for my tiles mapping)…
This is what I have been playing with for hours, ,
Vector3 centre = bombPosition; //= ( bombPosition + ( distance / 2 ));
// centre.x = ( bombPosition.x + ( distance.x / 2 ));
//centre.y = ( bombPosition.y + ( distance.y / 2 ));
for(float wait = 1; wait <= ( Mathf.PI ) ; wait += 0.1f)
yield return new WaitForSeconds (0.5f );
//float deg = wait * Mathf.Rad2Deg;
//deg = wait * (2.0f * Mathf.PI / wait);
bombPosition.x = centre.x += ( 60 * Mathf.Cos ( (float) wait ) );
bombPosition.y = centre.y -= ( 60 * Mathf.Sin ( (float) wait) );
bomb.transform.position = bombPosition;
Rather than use a circle, how about some simple particle physics? Let’s imagine the object starts of at t =0 at the origin and this is point A (it could really be anywhere, just apply the translation
you need), similarly we’ll imagine that the point B is aligned along the x axis (a rotation will make this true, so we’re not losing this by picking a coordinate system). If the vector AB is a
horizontal distance, d, away from the object and we throw it such that its horizontal speed is v_h then it will have reached the spot B (or its horizontal position) at a time
t = d / v_h
Ok. So we should throw it with an initial vertical velocity v_v, chosen such that it is magically at the right elevation at that moment. Let the difference in height between A and B be equal to h and
the acceleration due to gravity be g. Measuring from A’s position the height at time t will be:
s = v_v. t - 0.5. g . t^2
Rearranging a bit:
v_v = (0.5.g.t^2 + s) / t
Since we’ve already figured out how much time we have available to us for this arc, we’ve now calculated what the initial velocity should be. Back in the land of code, if we maintain a timer then we
can use this to drive the bomb along this curve.
A few clarifications on my notation:
• When I use an underscore, it should be read as an subscript, so ‘v_v’ is a shorthand for ‘vertical velocity’
• The caret ‘^’ should be read as a superscript or power, so t^2 is t squared (t * t)
• ‘s’ is a distance, in the context of the final formula it should be interpreted as a vertical distance that the object is moving through.
• ‘g’ is acceleration due to gravity in the downward direction, so for objects moving in an earth-like environment a value of 9.8 m/s^2 would be fine.
oh interesting, thank you
Was reading here and as you say, realized maybe the circle is the wrong approach and more like I’m needing a “parabolic arc”
I don’t have time for the next days to read through your answer properly. At quick glance, I am a little confused by the format of the equation.
v_v = (0.5.g.t^2 + s) / t
May I ask, whats the _ represent?
and .'s?, What do they represent? +?
also I’m not sure what g is? 0.5 (.?) g - gravity?
v_v ?
Thanx for the hints, check back in 3 days
thank you,
I do however have a question,
if the distance between the two positions AB is dynamic and the horizontal distance changes, the vertical distance may also change.
How can we know the initial vertical velocity in which to throw the object to make it art exactly on to position B?
for normalised distance: distance x= 0 to 1, height y= 0 to 1
rising half x(0to0.5): y = (x2)/(x2) because for (0 to 1)/(0 to 1) is an a rising arc
falling half the same in reverse. when object gets half way, change formula from first to second. perfect parabol. or just look for parabola also and strech one out and reverse it:
and you could easily adapt the 3rd parabola on this page, simply move it forwards by one space and multiply will divide the height by an amount required and the distance also.
for normalised distance: distance x= 0 to 1, height y= 0 to 1
rising half x(0to0.5): y = (x2)/(x2) because for (0 to 1)/(0 to 1) is an a rising arc
falling half the same in reverse. when object gets half way, change formula from first to second. perfect parabol. or just look for parabola also and strech one out and reverse it:
modify the 3rd Formula on this page, in total you will have a 2 line simple arc formula
Okay square root is not ideal for an arc, but it’s good to know to shape control signals, probabilities of monsters appearing, distance to speed variables etc | {"url":"https://discussions.unity.com/t/bit-of-2d-maths-help-travel-arc-between-two-points/53857","timestamp":"2024-11-05T01:53:12Z","content_type":"text/html","content_length":"41211","record_id":"<urn:uuid:c1989f8f-850b-48d0-b862-1b9683d66ff3>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.84/warc/CC-MAIN-20241104225856-20241105015856-00540.warc.gz"} |
Characteristics of an Ideal ML Model: How to Choose & Why
How To Go Ahead With Model Selection And Why To Choose Any Specific Model
There are many factors to consider when choosing a machine learning model. The most important factor is accuracy- the model should accurately predict the target variable. Other factors to consider
1. The complexity of the model
2. How easy it is to interpret the results
3. Whether it generalizes well to new data
4. How well it performs on training data
Each type of model has different strengths and weaknesses, so you need to decide which type of model will work best for your data and your goals. You might even need multiple models to get accurate
predictions in some cases.
The following sections discuss each type of model in more detail.
1. Linear regression is a simple linear model that predicts y values based on a linear combination of x values. It's useful when you have continuous data and want to model the relationship between
two variables (e.g., how much one variable affects another).
2. K-nearest neighbors (aka KNN) is an instance-based model that predicts y values based on the k most similar instances in training data. It's useful when you need to classify items into discrete
categories, such as "spam" or "not spam" messages in email filtering systems.
3. Decision trees are hierarchical models that learn decision rules for classifying items into discrete categories, such as the types of animals found in nature. They work well with structured data
like customer information but may not be as effective on unstructured data like images or text.
4. Random forests are ensembles of decision trees that can model complex relationships between variables by combining multiple simple rules learned from training data. They're helpful when you want
to model non-linear relationships between variables, such as the relationship between height and weight in humans (taller people tend to weigh more than shorter ones).
5. Naive Bayes is a probabilistic model that makes predictions based on probabilities of different outcomes given specific evidence, such as whether it's raining outside today or not. It works well
with categorical data but may not be effective for continuous values like temperature readings from sensors over time because these don't follow normal distributions very closely at all times,
leading to inaccurate results if used improperly.
6. Support vector machines learn decision boundaries that maximize the distance between categories of data points in training sets, which makes them helpful in classifying items into discrete
categories like spam emails versus not-spam ones based on text content or images where there is some sort of distinguishing feature between two classes (e.g., color). They work well with
structured data but can be computationally expensive when dealing with many features due to their high dimensionality space requirements.
This may limit their applicability in real-world applications such as medical diagnosis systems. Patient health records must be analyzed quickly without delay due to processing time constraints from
heavy computation loads placed upon hardware devices running these algorithms. | {"url":"https://www.strictlybythenumbers.com/characteristics-of-an-ideal-ml-model","timestamp":"2024-11-08T05:42:13Z","content_type":"text/html","content_length":"26464","record_id":"<urn:uuid:1cf23806-cbef-4e7b-a3ca-8c9269e3c28a>","cc-path":"CC-MAIN-2024-46/segments/1730477028025.14/warc/CC-MAIN-20241108035242-20241108065242-00277.warc.gz"} |
Math Labs with Activity - Chords of a Circle which are Equidistant from Centre of Circle - A Plus Topper
Math Labs with Activity – Chords of a Circle which are Equidistant from Centre of Circle
To verify that the chords of a circle which are equidistant from the centre of the circle are equal.
Materials Required
1. A sheet of transparent paper
2. A geometry box
The theorem to be verified is the converse of the theorem verified in Activity 23.
Step 1: Mark a point O on the sheet of transparent paper. Draw a circle with centre O taking any radius.
Step 2: Draw any chord AB in the circle. Fold the paper along the line that passes through the centre O of the circle and cuts the chord AB such that one part of the chord AB overlaps the other part.
Make a crease and unfold the paper. Mark the point M where the line of fold cuts the chord AB. Join OM.
Then, OM is the perpendicular bisector of the chord AB and gives the distance of the chord AB from the centre O of the circle.
Step 3: Draw any radius OC. On this radius OC, mark a point N such that OM = ON.
Step 4: Fold the paper along the line that passes through the point N such that NC overlaps NO. Make a crease and unfold the paper.
Mark the points P and Q where the line of fold cuts the circle. Join PQ.
Then, PQ is a chord of the circle whose distance from the centre O of the circle is ON, which is equal to the distance OM of the chord AB from the centre O of the circle (see Figure 25.1).
Step 5: Fold the paper along the line which passes through the centre O of the circle such that OM overlaps ON.
1. OM exactly covers ON since OM = ON.
2. AB exactly covers PQ. This shows that the chord AB is equal to the chord PQ.
It is verified that the chords of a circle which are equidistant from the centre of the circle are equal.
Math Labs with ActivityMath LabsScience Practical SkillsScience Labs | {"url":"https://www.aplustopper.com/math-labs-activity-chords-circle-equidistant-centre-circle/","timestamp":"2024-11-04T04:12:40Z","content_type":"text/html","content_length":"43747","record_id":"<urn:uuid:727ee168-fbee-4d79-98ef-0eb67591340a>","cc-path":"CC-MAIN-2024-46/segments/1730477027812.67/warc/CC-MAIN-20241104034319-20241104064319-00352.warc.gz"} |
Choice of Methods for Solving the Problem of Fluid Dynamic Modeling in a Fractured-Porous Reservoir
Title Choice of Methods for Solving the Problem of Fluid Dynamic Modeling in a Fractured-Porous Reservoir
R. M. Uzyanbaev^1, 2, Y. O. Bobreneva^2, S. V. Polyakov^2, 3, V. F. Tishkin^2, 3
Authors ^1Ufa State Petroleum Technological University
^2Institute of Petrochemistry and Catalysis of the Ufa Federal Research Center of the Russian Academy of Sciences
^3Keldysh Institute of Applied Mathematics of RAS
The work is devoted to numerical methods for solving the problem of modeling the mass transfer of a two-phase fluid in a carbonate reservoir. The problem is complicated by the presence of
Annotation two media embedded in each other (a system of fractures and a pore part of the reservoir), which complicates its numerical analysis. For the numerical solution of the problem in the
one-dimensional case, explicit and implicit difference schemes on a non-uniform grid are considered and implemented as a software module. Computational experiments were performed, on the
basis of which a comparative analysis of the implemented methods was carried out.
Keywords mathematical modeling, system of equations for two-phase filtration, frac-tured porous reservoir, piezoconductivity and double porosity, explicit and implicit difference schemes
Uzyanbaev R. M., Bobreneva Y. O., Polyakov S. V., Tishkin V. F. ''Choice of Methods for Solving the Problem of Fluid Dynamic Modeling in a Fractured-Porous Reservoir'' [Electronic
Citation resource]. Proceedings of the XVI International scientific conference "Differential equations and their applications in mathematical modeling". (Saransk, July 17-20, 2023). Saransk: SVMO
Publ, 2023. - pp. 240-247. Available at: https://conf.svmo.ru/files/2023/papers/paper38.pdf. - Date of access: 12.11.2024. | {"url":"https://conf.svmo.ru/en/archive/article?id=432","timestamp":"2024-11-12T03:00:24Z","content_type":"text/html","content_length":"11936","record_id":"<urn:uuid:4df65f7d-6223-4420-8803-ffc9f3b4883e>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.50/warc/CC-MAIN-20241112014152-20241112044152-00436.warc.gz"} |
Compact lexer table representation
2020-05-02 15:31:09+02:00
I found surprisingly few information on the transition table of a lexer generator.
There are plenty of resources on the front-end, such as the very nice Regular-expression derivatives reexamined paper.
However resources on the transition table are much more scarce. Eventually, I found two references: The Dragon Book, which explains a clever scheme for packing the table, and OCamllex which
implements it[^fn1].
Update: Software and Hardware Techniques for Efficient Polymorphic Calls thesis analyse a variant of the technique described in this post to store dispatch tables of object-oriented tables. "Row
displacement" proves to be very efficient in a closed world and extends well to multiple inheritance.
[^fn1]: Actually, I believe that the pseudo-code in the Dragon Book is wrong. There should be no recursive call to nextState, instead the default state should be returned directly. This is what
OCamllex does.
The transition table
The lexer generator frontend produces a deterministic finite automaton (DFA). Transitions are labeled by symbols from the input alphabet (a-z characters in the illustration below). Here is a trivial
DFA recognizing the word "hello":
We start from state 0 (the initial state). Then we follow the transitions until:
• acceptance: if we reach state 5, the word "hello" has been recognized
• rejection: if we reach state 6, recognition failed
The animation below shows the process of recognizing two words:
• success with "hello" input
• failure with "hey"
We need an efficient way to store and follow these transitions.
Naive representation
The simplest representation is a matrix indexed by states and characters. In C that looks like:
// state_t is the type representing a state
// 256 because we work with 8-bit characters
state_t transition_table[MAX_STATES][256];
state_t next_state(state_t current, uint8_t input)
return transition_table[current][input];
This is efficient in time but not in space. The difficulty lies in finding a compact representation that does not compromise speed:
• Transitions will be followed for every input byte. This is the hottest part of the lexing process.
• Practical languages can grow to thousand of states. The matrix take a few megabytes of memory.
Here is the matrix for the "hello" example:
\ a...d e f,g h i,j,k l m,n o p...z
We can see that it is very explicit and very redundant. A transition is very likely to be 6!
Sparse representation
The Dragon Book suggests to represent each transition vector (a row of the table above) sparsely:
• default transition: remember the most common target destination
• non-default transitions: store only the transitions that differs
With associative lists
The sparse vectors can be represented with a default value and an associative list for storing non-default transition.
The table becomes:
default Transitions
0 6 (h, 1)
1 6 (e, 2)
2 6 (l, 3)
3 6 (l, 4)
4 6 (o, 5)
Much more compact!
But there is a performance problem: for each transition, we have to iterate the list looking for a match. A list can be as big as the size of the alphabet. That would lead to unpredictable and often
slow performance – unacceptable.
With overlapping vectors
The Dragon Book comes to the rescue and introduces a clever scheme that retains the performance of array-based lookup with the compactness of sparse vectors. The scheme is as follows:
• Store all vectors in the same array
• Offset them such that only non-default transitions don't overlap
• Annotate the non-default transitions with their source state
With this mechanism, the automaton looks like:
• State table:
Default Offset
• Transition table:
index 0...3 4 5,6 7 8,9,10 11 12 13 14 15..26
source Ø 1 Ø 0 Ø 2 3 Ø 4 Ø
target Ø 2 Ø 1 Ø 3 4 Ø 5 Ø
(Using Ø: any value that does not represent a valid state)
We avoid the waste of the naive matrix by filling the unused cells of sparse vectors with the content of others. And we keep the fast access characteristics of arrays.
Here is the mapping between index and characters at offset 0 and 1:
index 0 1,2,3 4 5,6 7 8,9,10 11 12 13 14 15..25 26
at offset 0 a b,c,d e f,g h i,j,k l m n o p..z
at offset 1 a,b,c d e,f g h,i,j k l m n o..w z
States 0, 1, 2, and 4, have been given the offset 0. Their non-default transitions never conflict: rather than having a separate vector of 26-elements for each of them, we can overlap all of them in
the same vector.
State 3 is more complicated. It cannot be at offset 0: it has a transition on l that would end up at column 11. But this column is already used by state 2. However the column 12, just after, is not
used by other states. So we offset the state by 1, shifting the meaning of characters: l at offset 1 maps to column 12. (It coincides with m at offset 0, but no state has a transition on m.)
With offsets, all transitions can fit in a single vector of 27 elements. Each cell is a bit larger because it stores a pair of states (a source and a target).
The implementation is now:
typedef struct {
state_t default_;
int offset;
} state_desc;
typedef struct {
state_t source, target;
} transition_t;
state_desc state_table[MAX_STATES];
transition_t transition_table[MAX_TRANSITIONS];
state_t next_state(state_t current, uint8_t input)
int index = state_table[current].offset + input;
if (transition_table[index].source == current)
return transition_table[index].target;
return state_table[current].default_;
The tables are a bit harder to generate than the naive matrix. How do we find the right offsets? A simple greedy strategy gives good packings:
• Start from first vector
• Try to fit it at offset 0:
□ If there is no overlap, done
□ If it overlaps, try again at the next offset
• Repeat with the next vector, until all vectors are packed
Engineering tricks
Algorithmically, this solution is satisfying. I went a bit further to make it more hardware friendly while maintaining a good space/time trade-off.
Something we did not specify above is the size of each type. How many bits for a state_t? OCamllex has hard-coded limits that can be reached on big yet realistic languages. These limits save space
but make the lexer less flexible. I wanted more freedom here.
I set myself the goal of storing everything in a single array of 32-bits value. I ended up with 23 bits for offsets. This allows for a theoretical maximum of ~8 million transitions, using up to 32
1. Disambiguate using characters
Rather than storing a source state in a transition to distinguish non-default from default transition, store an input character: this transition is non-default if we reached it by following this
input character. I call it the input disambiguator.
typedef struct {
uint8_t input;
state_t target;
} transition_t;
state_t next_state(state_t current, uint8_t input)
int index = state_table[current].offset + input;
if (transition_table[index].input == input)
return transition_table[index].target;
return state_table[current].default_;
This change alone removes just a few bits of information from a transition cell. And it forces us to store each state at a different offset (otherwise it would be ambiguous). For the "hello" example,
offsets are now (0,1,2,3,4).
But we replaced a vector of states by a vector of characters. There can be many states but there are only 256 characters. We can exploit this in the low-level representation.
2. Represent states by their offsets
Now that each state has a unique offset we can directly represent them using offsets, rather than consecutive numbers.
We get rid of the offset entry from the state table and store the default_ transition as if it was on character "-1". Just before the offset:
• transition_table[offset + c]: transition information from state offset and input character c
• transition_table[offset - 1]: default transition for state offset
The input disambiguator for transition_table[offset - 1] is chosen to not coincide with the non-default transition of another valid state. In other words offset - 1 - transition_table[offset -
1].input should not be the offset of another state.
Everything fits in a single array now:
typedef struct {
uint8_t input;
state_t target;
} transition_t;
transition_t transition_table[MAX_TRANSITIONS];
state_t next_state(state_t current, uint8_t input)
if (transition_table[current + input].input == input)
return transition_table[current + input].target;
return transition_table[current - 1].target;
By making the state fit in 24-bits, we can represent a transition in a single 32-bit value:
typedef int32_t state_t;
typedef struct {
uint8_t input : 8;
state_t target : 24;
} transition_t;
3. Negative numbers for special actions
In the example, states 5 and 6 have a special meaning: accepting or rejecting the input. From the point of view of the automaton they do the same: terminate the analysis and yield control back to the
caller. It is the caller that will act differently based on the reason for the termination.
Thus the automaton does not assign any meaning to special transitions other than stopping the analysis. The driver, on the other hand, can have many actions. For instance:
• backtracking: remember the current state, continue the analysis and if it reaches a rejection state later, fall back to current state and act as if it was accepting
• tagging: mark the current state as a "point of interest" for the program, and resume the analysis. This can be used to implement capture groups
The special transitions just need to be distinguished from normal states. For this, I simply chose to use negative values, which cannot represent states. This reduces the amount of usable bits in a
state_t to 23 (for a maximum table size of 32 MiB).
Handling end-of-file
End-of-file condition (EOF) is reached when there is no more input to feed to the automaton. That can happen at any time, we should always be ready to handle EOF.
Special actions behave like extra states, EOF behave like an extra transition.
OCamllex deals with EOF regularly, by using an alphabet with 257 symbols. I chose to treat EOF differently:
• To keep using 8-bit integers for "input" disambiguator
• EOF is a unique situation, it happens only once per run and it happens last. It does not have to be on the fast path.
The remaining degree of freedom we had in the representation of states is the input disambiguator. We use it to encode EOF transition.
We will it to point to any unused transition cell that is now re-purposed to indicate the EOF destination state. The disambiguator of this EOF cell can be anything as long as it is not ambiguous. We
end up with a different transition function for EOF:
state_t eof_state(state_t current)
int idx = transition_table[current - 1].input;
return transition_table[current - 1 - idx].target;
All these optimizations put more pressure on the packing algorithm. But the added freedom can reduce fragmentation in the sparse array, and in practice many states have the same EOF transition:
• The packing algorithm can share a single EOF cell with many states, improving efficiency.
• The original scheme, the one with many tables, have to give different offsets to each state. What seemed at first a drawback of the single table scheme also happens in the original one in
There is a last optimization we can do for storing EOF transition. Because EOF happens at the end of the analysis, it only makes sense for EOF transitions to target special actions.
Therefore we can use this to extra bit of information to introduce more sharing on EOF transitions. We can interpret EOF transitions targeting a regular state it as a default transition. And then
repeat looking for an EOF transition from this default state.
state_t eof_state(state_t current)
while (1)
int offset = transition_table[current - 1].input;
int eof_index = current - 1 - offset;
state_t target = transition_table[eof_index].target;
if (target <= 0)
return target;
current = transition_table[current - 1].target;
This complicates the packing scheme for diminishing returns. I did not bother implementing it.
Final implementation
Putting everything together, I got this implementation for the core loop of the lexer:
typedef int32_t state_t;
typedef uint32_t transition_t;
#define SRC(transition) ((transition) & 0xFF)
#define DST(transition) ((int32_t)(transition) >> 8)
state_t follow(transition_t *table, state_t state,
unsigned char **buf, unsigned char *end)
unsigned char *ptr = *buf;
while (ptr < end && state > 0)
unsigned char c = *ptr++;
transition_t def = table[state - 1];
transition_t nxt = table[state + c];
state = DST((SRC(nxt) == c) ? nxt : def);
*buf = ptr;
return state;
state_t follow_eof(transition_t *table, state_t state)
int idx = SRC(transition_table[state - 1]);
return DST(transition_table[state - 1 - idx]);
The interpretation function consume as many characters as possible. This reduces the interpretation overhead (the cost of entering and leaving the interpretation function). We want to spend most of
the time in the hot loop!
Note that the loop is quite machine-friendly:
• The two loads can be issued in parallel
• State selection compiles to branch-less code
The only branching is the check for the exit condition. It is unavoidable but it happens once and is well predicted.
I presented some techniques for storing the transition table of a lexer. The main result is a simple 40-year-old scheme. It is effective and a few adjustments make it perform even better on modern
I apologize for not having benchmark figures to show... I did not want to spend the time implementing a production grade lexing engine. I was just interested in playing around the full pipeline
rather than stopping after the frontend. If I ever need to design a complete lexer, I have a clear picture of what it should look like.
In the future, I plan to tackle some useful extensions like extraction and lookahead (along the lines of Tagged Deterministic Finite Automata with Lookahead).
Going further
To handle UTF-8 and other character encodings, I came to the conclusion that the best approach was to generate the automaton for a fixed encoding (e.g. a normalized form of UTF-8). With a
preprocessing step to convert the input. The automaton would still work on an 8-bit alphabet, possibly simulating a single codepoint with multiple transitions.
Out of curiosity, I tried to represent transitions using various forms of packed intervals on which to do binary search. Basically a sorted sequence: (first codepoint, last codepoint, target state).
This is a cheap way to handle large alphabets. But I did not manage to make it competitive with the sparse representation, even with clever implementations of binary search like on the excellent
PVK's blog. That ruled out the approach for me. | {"url":"https://def.lakaban.net/2020-05-02-compact-lexer-table-representation/","timestamp":"2024-11-10T19:19:08Z","content_type":"text/html","content_length":"23710","record_id":"<urn:uuid:60315fa4-619f-4014-8d82-467a2df65c5b>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.61/warc/CC-MAIN-20241110170046-20241110200046-00261.warc.gz"} |
Heat Transfer Questions & Answers
Question by Student 201527130
I have a question about design projects Q#5. I think the velocity in duct is too high. so, density is too big(10.2kg/m^3). I think it seems to be 20m / s instead of 200m / s when we match the
answer.Could you confirm it if it does not work?
There is no problem with the question formulation.
Question by Student 201527136
Professor, I have question about fully-developed flow in pipe. You said the equation $\frac{U}{U{_{b}}}=2\left ( 1-\frac{r^{2}}{R^{2}} \right )$. Is this expression valid for both laminar flow and
turbulent flow?
This is valid only for laminar flow.
Question by Student 201428239
Professor, I have a question about A7 of Q5. In this question, I need to use correlation of free convection H-T. I should use vertical plane correlation. In the table, a comment written as "x the
distance from the bottom". In this comment, the x means height?? Then, when I use correlation ( $Nu_x = C(Gr_xPr)^m$, should I use x = H??
In this correlation, $x$ is the distance from where the boundary layer starts.
Question by Student 201428239
Professor, I have a question about A6 of Q3. In this question, I know the value of q local and q average.Then, I can get the relation between $Nu_x and Nu_L$ average. To solve (a), Do I need to
compare these values and find the flow type??? If so, I can easily get Laminar flow, but in your comments, it could be either laminar or lam-Turb mix. How can find Lam-Turb mix relation??? Thank you
Well, by using a Nusselt number correlation that is suited to the turbulent/laminar regimes.
Question by Jaehyuk
Professor, I have a question regarding A7Q5. As far as I believe, it is suitable to use the correlation for a vertical plane with constant heat flux;$Nu_x = C(Gr^*_{x}Pr_{f})^m$. Here starts my
problem. In order to find constants(C and m), the range of $Gr_x^*$ has to be set first. However, when $Gr_x^*$ is in between $1E11<Gr_x^*<2E13$, there is no option for constants(C and m). In this
case, is it possible to choose any one of two options?
If you can't find a correlation that fit perfectly your situation, then choose the one that is the closest.
Question by Student 201428239
Professor, I have a question about A7 of Q5. In this problem, I should find free convection heat transfer coefficient. But in correlation, I should know $T_s$ first. I can get $T_s$ through
iteration process. And then get h. Is this procedure correct?? Because in the problem, the order is to find h first and then find $T_s$.
Yes, exactly. I mentioned this in class. | {"url":"https://overbrace.com/bernardparent/viewtopic.php?f=2&t=258&sid=fb5ab57bf3c4f42e49a0640503b8c941&start=468","timestamp":"2024-11-02T08:17:59Z","content_type":"application/xhtml+xml","content_length":"26414","record_id":"<urn:uuid:35c0d985-ba58-4f38-8e7f-c612452153b4>","cc-path":"CC-MAIN-2024-46/segments/1730477027709.8/warc/CC-MAIN-20241102071948-20241102101948-00750.warc.gz"} |
Research on isolation property of prestressed thick rubber bearings
To overcome the shortages of current laminated rubber bearings (RB), a new kind of isolator called Prestressed Rubber Bearing (PRB) is presented in this paper, which is invented by appropriately
amplifying the thickness of rubber layers in conventional RB and employing prestress tendons. Based on the experimental study, a modified formula for vertical stiffness of PRB is established. Then
the nonlinear analytical model for PRB’s horizontal stiffness is developed and the corresponding formulas are derived. Through the response history analysis of structures, the isolation capacities of
PRBs are investigated. The results show that the horizontal stiffness of PRB is variable with the displacment. PRB not only has effective isolation capacity as conventional RBs but also has the
favorable capacity of horizontal displacement limitation and vertical up resistance.
1. Introduction
Isolators are the key components of the isolated structures. At present, the most widely used isolators are laminated Rubber Bearings (RB), in which the steel plates and rubber layers are arranged
alternately and combined by high-temperature and vulcanization [1]. There are some disadvantages existing in current RBs: (i) the capacity of horizontal displacement limitation is inadequate. The
earthquake may cause large horizontal displacement in the isolators and reduction of effective bearing area, which may result in overturning of RBs due to the large second order moment [2]; (ii) the
tensile strength is inadequate. The internal tensile force is forbidden in conventional RBs according to current Chinese Code. However, it is difficult to avoid in some situations, especially for the
high rise buildings; (iii) the vertical isolation is inadequate. Many earthquake disasters have shown that lots of non-structural damages were caused by vertical vibration. Although thick rubber
layers may produce good capacity of vertical isolation, they also may lead to the uneven settlement of superstructure when it is constructed. So the thin rubber layers are used in current RBs.
To overcome the disadvantage of conventional RBs, several related studies were conducted. Kang et al. proposed fiber reinforced elastomeric isolator which used the carbon fiber or the glass fiber
instead of steel plate to improve the vertical isolation capacity of RB [3]. Ismail [4] introduced a new seismic system, named roll-n-cage isolator (RNC). The main bearing mechanism of the RNC is a
hollow elastomeric cylinder of a designed thickness around a rolling body. The device incorporates isolation, energy dissipation, and inherent gravity-based restoring force mechanism in a single
unit. Amarnath Kasalanati proposed an uplift prevention mechanism which employs prestress theory to develop sufficient compressive force on the isolator [5]. Peng Tian Bo [6] developed a double
spherical aseismic bearing, which increased the natural vibration period of buildings by increasing the centre distance of spheres. Zhou Xiyuan [7] and Cui Yibin [8] presented a kind of rubber
isolator with a steel bar inside to limit the deformation of isolator. Zhang Yongshan [9] and Wei Liushun [10] proposed a 3-dimensional isolation device, in which the vertical isolation capacity was
developed by employing a semi-active controlled hydro-cylinder parallelly connected with vertical spring.
However, these devices can not overcome all of the conventional RBs’ shortcomings. This paper proposes an innovative type of rubber bearing, named as Prestressed Rubber Bearing (PRB) [Chinese Patent
Number: ZL201020181364.1]. As shown in Fig. 1, PRB is developed based on conventional RB. The thicknesses of rubber layers are increased appropriately to improve the capacity of vertical isolation.
Several vertical ducts are set and prestress tendons are installed. Although the thicknesses of rubber layers in PRBs are larger than the conventional RBs, the prestress force can achieve most of the
vertical deformation before the superstructure is constructed. So the problem of uneven settlement can be eliminated. On the other hand, because of prestress tendons, PRB has the capacity of
horizontal displacement limitation and up-lift resistance.
This paper develops the nonlinear analytical model for the deformation of PRB. The formulas for both vertical stiffness and horizontal stiffness of PRB are proposed. Through numerical analysis, the
isolation capacity of PRB is also investigated.
2. Vertical stiffness
The vertical stiffness is one of the important mechanical properties for PRB. Currently, the vertical stiffness of RB is calculated using formula proposed by Lindley as [11]:
${K}_{v,RB}={E}_{CR}\frac{\pi d}{4}{S}_{2},$
where $d$ is the diameter of cross section; ${S}_{2}$ is the second shape factor of RB, ${S}_{2}=d/{n}_{1}{t}_{\text{r}}$; ${n}_{1}$ is the number of rubber layers; ${E}_{CR}$ is the modified
elasticity modulus of the bearing ${E}_{CR}={E}_{C}{E}_{R}/\left({E}_{C}+{E}_{R}\right)$, ${E}_{C}=E\left(\text{1}+k\text{\hspace{0.17em}}{\text{S}}_{\text{1}}^{2}\right)$; ${E}_{R}$ is the
elasticity modulus of volume constrained rubber; $E$ is the elasticity modulus of rubber; $k$ is the correcting coefficient for the rubber hardness; ${S}_{1}$ is the first shape factor of RB which is
the ratio of bearing area to the free surface area, ${S}_{1}=\left(d-{n}_{2}{d}_{0}\right)/4{t}_{\text{r}}$; ${d}_{0}$ is the diameter of the ducts; ${n}_{2}$ is the number of ducts.
The vertical monotonic loading tests are conducted to study the vertical stiffness of the PRB. A computer controlled compression testing machine with maximum load capacity of 300 kN and maximum
stroke of 500 mm (Fig. 2) is used. The 6 groups of specimens (each group has 3 same specimens) are tested. The height of bearing before applying prestress force is 172 mm, the diameter of effective
cross sectional area $d$ is 150 mm, and the diameter of ducts ${d}_{0}$ is 15 mm. The hardness of rubber is 60 HA and the yield strength of steel plate is 235 MPa. The detailed parameters of specimen
are listed in Table 1, where ${t}_{r}$ is the thickness of rubber layer; ${t}_{s}$ is the thickness of rubber plate.
Experimental results show that Lindley formula greatly underestimates the vertical stiffness of RBs with thicker rubber layer. The vertical stiffness calculated from Lindley’s formula is about 50 %
of the experimental values. It is mainly due to the reason that Lindley formula underestimates the steel plates’ constraint to rubber layers when rubber layers become thicker. Therefore, the
Lindley’s formula needs to be modified for the RB with thick rubber layers as follows:
${K}_{v}=\eta \frac{{E}_{CR}A}{{n}_{1}{t}_{r}},$
where ${K}_{v}$ is the vertical stiffness of PRB; $A$ is effective area of cross-section; $\eta$ is a modifying coefficient of vertical stiffness. By linear regression analysis of test results, the
modifying coefficient $\eta$ can be obtained as:
$\eta =-0.23{S}_{1}+2.56.$
Fig. 2Experimental set-up of vertical loading test
(a) Sketch of experimental setup
Table 1Parameters of specimens
Specimen group ${t}_{r}$ / mm ${t}_{s}$ / mm ${S}_{1}$ ${S}_{2}$
PRB-1 10.6 1 2.08 0.94
PRB-2 9.24 1 2.38 0.95
PRB-3 7.7 1 2.86 0.97
PRB-4 9.6 2 2.29 1.04
PRB-5 8.2 2 2.67 1.07
PRB-6 6.7 2 3.29 1.12
The vertical stiffness of the bearings mainly depends on the deformation of rubber layers. The first shape factor ${S}_{1}$ is an important parameter indicating the steel plates’ constraint on the
deformation of the rubber layers. The relationship of vertical stiffness and ${S}_{1}$ is shown in Fig. 3, in which the results from Lindley’s formulas, Eq. (2) and the experiments are compared. It
can be seen that vertical stiffness of PRB increases with increase of ${S}_{1}$. The main reason is that the thickness of rubber layers decreases when the value of ${S}_{1}$ increases. Hence, the
steel plates’ constraint on the deformation of the rubber layers increases. And the vertical stiffness increases.
Fig. 3Relationship between vertical stiffness and S1
The accuracy of the proposed formula depends on the number of regression samples. Because only 18 samples are used in regression analysis, the proposed modifying coefficient can only be applied to
the PRB with similar dimension.
3. Horizontal stiffness
Horizontal stiffness is one of the most important mechanical properties for the isolation bearings. The horizontal stiffness for the conventional RB is calculated as [12]:
where $G$ is the shear modulus of the rubber.
The horizontal deformation mechanism of PRB is different from the conventional RBs. Due to the deformation compatibility of tendons and rubber layers, the prestress tendon will be subjected to
tension and the rubber layers will be subjected to compression when horizontal deformation happens. Therefore, the horizontal component of prestress tendons’ internal force could change the stiffness
of PRB. Eq. (4) is not valid for calculating the horizontal stiffness of PRBs.
3.1. Model development
Fig. 4 is the analytical model for the horizontal stiffness of PRB. When the prestress force is applied, the PRB is shortened the amount of ${\Delta }_{1}$, and the height of PRB at this moment is
defined as $h$. When the gravity load of supper-structure is applied, the PRB is shortened the amount of ${\Delta }_{2}$. And finally, when the horizontal load is applied, the PRB is shortened the
amount of ${\Delta }_{3}$. The height of PRB at this moment is defined as ${h}_{0}$, ${h}_{0}=h–{\Delta }_{2}–{\Delta }_{3}$.
Fig. 4Analytical model of PRB
To simplify the analysis, the following assumptions are adopted: (i) all tendons are incorporated into an equivalent tendon which is located in the center of cross-section; (ii) the tendon remains
straight when the bearing has horizontal deformation. The vertical equilibrium equation can be obtained as follows:
${K}_{\text{v}}{\Delta }_{3}={E}_{\text{T}}{A}_{\text{T}}\frac{\sqrt{{h}_{0}^{2}+\delta {\left({h}_{0}\right)}^{2}}-h}{h}\mathrm{c}\mathrm{o}\mathrm{s}\varphi ,$
where ${E}_{\text{T}}$, ${A}_{\text{T}}$ is the elasticity modulus of prestress tendon and the equivalent cross-sectional area of the tendon respectively; $\varphi$ is the inclination of the tendon;
$\delta \left({h}_{0}\right)$ is the horizontal displacement in the top of PRB, which can be calculated as:
$\delta \left({h}_{0}\right)={h}_{0}\mathrm{t}\mathrm{a}\mathrm{n}\varphi .$
Substituting Eq. (6) into Eq. (5):
${\Delta }_{3}=\frac{{E}_{\text{T}}{A}_{\text{T}}}{{K}_{\text{v}}}\left[\frac{{h}_{0}\sqrt{1+{\mathrm{t}\mathrm{a}\mathrm{n}}^{2}\varphi }-h}{h}\right]\mathrm{c}\mathrm{o}\mathrm{s}\varphi .$
Considering the lower part of PRB as shown in Fig. 5, the constitutive equation can be written as:
$M\left(x\right)={E}_{\text{eq}}{I}_{\text{eq}}{\theta }^{"}\left(x\right),$
$V\left(x\right)={G}_{\text{eq}}{A}_{\text{eq}}\left[{\delta }^{"}\left(x\right)-\theta \left(x\right)\right]+{K}_{\text{v}}{\Delta }_{3}\mathrm{t}\mathrm{a}\mathrm{n}\varphi ,$
where $\delta \left(x\right)$ is the horizontal displacement at the height of $x$; $\theta \left(x\right)$ is the cross-section’s deflection angle at the height of $x$; ${G}_{\mathrm{e}\mathrm{q}}{A}
_{\mathrm{e}\mathrm{q}}$ is the equivalent shear stiffness of the PRB; ${E}_{\mathrm{e}\mathrm{q}}{I}_{\mathrm{e}\mathrm{q}}$ is the equivalent bending stiffness of PRB. They can be computed by:
where $I$ is the moment inertia of cross-section; ${E}_{\mathrm{B}\mathrm{R}}$ is determined by:
where ${E}_{\text{B}}=E\left(1+\frac{2}{3}k{S}_{1}^{2}\right)$.
Therefore the equations of equilibrium can be written as:
$M\left(x\right)+P\left[\delta \left(x\right)-{\delta }_{0}\right]-{M}_{0}-{H}_{0}x=0,$
$V\left(x\right)+{H}_{0}-P\theta \left(x\right)=0.$
From Eqs. (8), (9), (12), (13), the value of $\theta \left(x\right)$ and $\delta \left(x\right)$ can be obtained as:
$\theta \left(x\right)=\frac{{G}_{\text{eq}}{A}_{\text{eq}}}{{G}_{\text{eq}}{A}_{\text{eq}}+P}{\delta }^{"}\left(x\right)+\frac{{K}_{\text{v}}{\Delta }_{3}}{{G}_{\text{eq}}{A}_{\text{eq}}+P}\mathrm
{t}\mathrm{a}\mathrm{n}\varphi +\frac{{H}_{0}}{{G}_{\text{eq}}{A}_{\text{eq}}+P},$
$\delta \left(x\right)=-\frac{{E}_{\text{eq}}{I}_{\text{eq}}}{P}{\theta }^{"}\left(x\right)+\frac{{H}_{0}x+{M}_{0}}{P}+\delta \left(0\right).$
Substituting Eq. (14), Eq. (15) into Eq. (12) and Eq. (13), we can obtain:
$\left\{\begin{array}{l}\frac{{E}_{\text{eq}}{I}_{\text{eq}}{G}_{\text{eq}}{A}_{\text{eq}}}{{G}_{\text{eq}}{A}_{\text{eq}}+P}{\delta }^{″}\left(x\right)+P\delta \left(x\right)=P\delta \left(0\right)+
{M}_{0}+{H}_{0}x,\\ \frac{{E}_{\text{eq}}{I}_{\text{eq}}{G}_{\text{eq}}{A}_{\text{eq}}}{{G}_{\text{eq}}{A}_{\text{eq}}+P}{\theta }^{″}\left(x\right)+P\theta \left(x\right)={H}_{0}+\frac{P{K}_{\text
{v}}{\Delta }_{3}}{{G}_{\text{eq}}{A}_{\text{eq}}+P}\mathrm{t}\mathrm{a}\mathrm{n}\varphi .\end{array}\right\$
The general solution of Eq. (16) is:
$\left\{\begin{array}{l}\delta \left(x\right)={C}_{1}\mathrm{c}\mathrm{o}\mathrm{s}\alpha x+{C}_{2}\mathrm{s}\mathrm{i}\mathrm{n}\alpha x+\frac{{H}_{0}}{P}x+\frac{{M}_{0}}{P}+\delta \left(0\right),\\
\theta \left(x\right)={C}_{3}\mathrm{c}\mathrm{o}\mathrm{s}\alpha x+{C}_{4}\mathrm{s}\mathrm{i}\mathrm{n}\alpha x+\frac{{H}_{0}}{P}+\frac{{K}_{\text{v}}{\Delta }_{3}}{{G}_{\text{eq}}{A}_{\text{eq}}
+P}\mathrm{t}\mathrm{a}\mathrm{n}\varphi ,\end{array}\right\$
in which $\alpha =\sqrt{\frac{P\left({G}_{\mathrm{e}\mathrm{q}}{A}_{\mathrm{e}\mathrm{q}}+P\right)}{{E}_{\mathrm{e}\mathrm{q}}{I}_{\mathrm{e}\mathrm{q}}{G}_{\mathrm{e}\mathrm{q}}{A}_{\mathrm{e}\
Substituting Eq. (17) into Eq. (12) and Eq. (13), it can be obtained that:
${C}_{3}=\frac{{G}_{\text{eq}}{A}_{\text{eq}}}{{G}_{\text{eq}}{A}_{\text{eq}}+P}\alpha {C}_{2},$
${C}_{4}=-\frac{{G}_{\text{eq}}{A}_{\text{eq}}}{{G}_{\text{eq}}{A}_{\text{eq}}+P}\alpha {C}_{1}.$
Considering the boundary conditions: $\delta \left(0\right)=0$, $\theta \left(0\right)=0$ and ${H}_{0}=–F$, the coefficient can be obtained as:
${C}_{2}=\frac{\left({G}_{\text{eq}}{A}_{\text{eq}}+P\right)F}{{G}_{\text{eq}}{A}_{\text{eq}}\alpha P}-\frac{{K}_{\text{v}}{\Delta }_{3}\mathrm{t}\mathrm{a}\mathrm{n}\varphi }{{G}_{\text{eq}}{A}_{\
text{eq}}\alpha },$
${C}_{2}=\frac{\left({G}_{\text{eq}}{A}_{\text{eq}}+P\right)F}{{G}_{\text{eq}}{A}_{\text{eq}}\alpha P}-\frac{{K}_{\text{v}}{\Delta }_{3}\mathrm{t}\mathrm{a}\mathrm{n}\varphi }{{G}_{\text{eq}}{A}_{\
text{eq}}\alpha },$
${C}_{4}=\frac{\alpha {G}_{\text{eq}}{A}_{\text{eq}}{M}_{0}}{\left({G}_{\text{eq}}{A}_{\text{eq}}+P\right)P}.$
Therefore, the horizontal displacement of the bearing along the height is:
$\delta \left(x\right)=\left[\frac{P\left(F-{K}_{\text{v}}{\Delta }_{3}\mathrm{t}\mathrm{a}\mathrm{n}\varphi \right)+{G}_{\text{eq}}{A}_{\text{eq}}F}{{G}_{\text{eq}}{A}_{\text{eq}}\alpha P}\right]\
mathrm{s}\mathrm{i}\mathrm{n}\alpha x-\frac{{M}_{0}}{P}\mathrm{c}\mathrm{o}\mathrm{s}\alpha x-\frac{F}{P}x+\frac{{M}_{0}}{P}.$
And the bending angle of the cross-section is:
$\theta \left(x\right)=\frac{\alpha {G}_{\text{eq}}{A}_{\text{eq}}{M}_{0}}{\left({G}_{\text{eq}}{A}_{\text{eq}}+P\right)P}\mathrm{s}\mathrm{i}\mathrm{n}\alpha x+\left[\frac{F}{P}-\frac{{K}_{\text{v}}
{\Delta }_{3}\mathrm{t}\mathrm{a}\mathrm{n}\varphi }{{G}_{\text{eq}}{A}_{\text{eq}}+P}\right]\mathrm{c}\mathrm{o}\mathrm{s}\alpha x-\frac{F}{P}+\frac{{K}_{\text{v}}{\Delta }_{3}\mathrm{t}\mathrm{a}\
mathrm{n}\varphi }{{G}_{\text{eq}}{A}_{\text{eq}}+P}.$
Hence, the horizontal displacement at the top of bearing is:
$\delta \left({h}_{0}\right)=\left[\frac{P\left(F-{K}_{\text{v}}{\Delta }_{3}\mathrm{t}\mathrm{a}\mathrm{n}\varphi \right)+{G}_{\text{eq}}{A}_{\text{eq}}F}{{G}_{\text{eq}}{A}_{\text{eq}}\alpha P}\
right]\mathrm{s}\mathrm{i}\mathrm{n}\alpha {h}_{0}-\frac{{M}_{0}}{P}\mathrm{c}\mathrm{o}\mathrm{s}\alpha {h}_{0}-\frac{F}{P}{h}_{0}+\frac{{M}_{0}}{P}.$
And the horizontal stiffness of PRB is:
${K}_{\text{h}}=\frac{F}{\delta \left({h}_{0}\right)}=\frac{1}{\left[\frac{P\left(F-{K}_{\text{v}}{\Delta }_{3}\mathrm{t}\mathrm{a}\mathrm{n}\varphi \right)+{G}_{\text{eq}}{A}_{\text{eq}}F}{F{G}_{\
text{eq}}{A}_{\text{eq}}\alpha P}\right]\mathrm{s}\mathrm{i}\mathrm{n}\alpha {h}_{0}-\frac{{M}_{0}}{FP}\mathrm{c}\mathrm{o}\mathrm{s}\alpha {h}_{0}-\frac{{h}_{0}}{P}+\frac{{M}_{0}}{FP}}.$
From the Eq. (22), it can be seen that the value of $\delta \left({h}_{0}\right)$ is the function of variable $P$, $F$, ${\Delta }_{3}$ and $\varphi$. Because ${\Delta }_{3}$ and $\varphi$ are
coupled with each other, the value of $\delta \left({h}_{0}\right)$ can not be obtained directly. The iterative method can be used. Assuming the initial value of $\varphi$, the value of ${\Delta }_
{3}$ can be computed by Eq. (7). Then the value of $\delta \left({h}_{0}\right)$ can be calculated. After that the value of $\varphi$ can be updated in Eq. (7). These steps are iterated until the
value of $\varphi$ satisfies the accuracy requirement.
If the rotation of the top connecting plate is constrained, $\left({h}_{0}\right)=0$, it can be obtained by Eq. (24):
$\frac{\alpha {G}_{\text{eq}}{A}_{\text{eq}}{M}_{0}}{\left({G}_{\text{eq}}{A}_{\text{eq}}+P\right)P}\mathrm{s}\mathrm{i}\mathrm{n}\alpha {h}_{0}+\left[\frac{F}{P}-\frac{{K}_{\text{v}}{\Delta }_{3}\
mathrm{t}\mathrm{a}\mathrm{n}\varphi }{{G}_{eq}{A}_{eq}+P}\right]\mathrm{c}\mathrm{o}\mathrm{s}\alpha {h}_{0}-\frac{F}{P}+\frac{{K}_{\text{v}}{\Delta }_{3}\mathrm{t}\mathrm{a}\mathrm{n}\varphi }{{G}_
After solving the Eq. (27) and Eq. (7), the horizontal stiffness of PRB can be computed directly.
3.2. Numerical investigation
A hypothesis cylinder PRB is used to conduct the numerical investigation. The diameter of cross-section is $d=$ 150 mm and the height before applying prestress force is 168 mm. The thickness of a
single rubber layer and a single steel layer are ${t}_{r}=$ 8 mm and ${t}_{s}=$ 2 mm respectively. There are 6 ducts of diameter ${d}_{0}=$ 15 mm setting along the range of PRBs. The diameters of
prestress tendons are 8 mm, and the total cross area is ${A}_{T}=$ 300 mm^2. The elasticity modulus of tendons is ${E}_{T}=$ 2.0×10^5 N/mm^2. The JIS hardness of the rubber is 60 HA.
3.2.1. Influence of prestress tendons
It is assumed that the vertical load is $P=$ 20 kN and the prestress force is equal to the vertical load. Hence, the value of ${\Delta }_{2}$ is zero. The horizontal stiffness of PRB and RB are
computed by equations in Section 2.1 and shown in Fig. 6.
It can be seen that the horizontal stiffness of PRB is variable with the horizontal displacement, which is completely different from the stiffness of conventional RB. When the horizontal displacement
is small, PRB’s stiffness is close to RB’s. The stiffness of PRB increases as the horizontal displacement increases. The main reason is that the horizontal component of tendons’ internal force is
resisting part of horizontal loading. When the displacement increases, the horizontal component of internal force also increases. Hence the horizontal stiffness of PRB increases.
Fig. 6The horizontal stiffness of PRB and RB
3.2.2. Influence of vertical load
Fig. 7 shows the relationship between the initial horizontal stiffness and vertical load. It can be seen that the initial stiffness of PRB depends on both the height of bearing and the vertical load.
The initial stiffness decreases with the increase of vertical load. This is due to the reason that the second order bending moment in PRB is increased when the vertical load increases. So the
capacity of resisting horizontal forces is reduced. And the initial horizontal stiffness is reduced. The stiffness decreases to zero when the vertical load reaches a critical value. In this situation
the buckling of PRB occurs. Therefore, the vertical loading capacity of PRB is determined by stability.
It can also be seen from Fig. 7 that the initial stiffness of PRB decreases with the increase of bearing height. The initial stiffness of PRB is close to the horizontal stiffness of RB with similar
dimensions. Generally, the horizontal stiffness of RB decreased with the increase of overall rubber layers height. Therefore, the initial stiffness of PRB decreases as the bearing height increases.
3.2.3. Influence of rubber layer’s thickness
When the vertical load is $P=$ 20 kN and the horizontal displacement at the top of PRB is $\delta \left({h}_{0}\right)=$ 40 mm, the relationship between the horizontal stiffness and the ratio of ${t}
_{r}$ to ${t}_{s}$ is obtained and shown in Fig. 8.
Fig. 7Relationship between initial horizontal stiffness and vertical load
Fig. 8Relationship between horizontal stiffness and tr/ts
It can be seen from Fig. 8 that the horizontal stiffness of both RB and PRB decreases with the increase of rubber layer’s relative thickness. And the stiffness variation of PRB is much larger than
RB. The main reason is that the vertical stiffness of rubber layer is reduced when the thickness of rubber layer increases. Because the vertical stiffness has little effect on the horizontal
stiffness of conventional RB, the stiffness variation of RB is relatively small. But the vertical stiffness has great effect on the horizontal stiffness of PRB. The small vertical stiffness leads to
large value of ${\Delta }_{3}$, which makes the horizontal deformation easier. Hence, the decrease of the PRB’s horizontal stiffness becomes significant.
3.2.4. Experimental test
The horizontal monotonic load tests are conducted to verify the theoretical derivation. The experimental setup is shown in Fig. 9. The specimens group of PRB4 and PRB5 which were used in vertical
loading test are adopted in this experiment. The vertical load is $P=$ 40 kN, and the prestress force for PRB4 and PRB5 are 50 kN and 30 kN respectively. The experimental results and theoretical
curves of lateral force-displacement are compared in Fig. 10. In the theoretical computation, the vertical stiffness ${K}_{v}$ is calculated by using Lindley’s formula and the proposed formula
Fig. 9Experimental setup of horizontal loading
(a) Sketch of experimental setup
Fig. 10Relationship between horizontal force and displacement
It can be seen that the lateral force obtained from three methods are close to each other when the displacement is small. But the difference between the value of Lindley formula and test result
increases gradually when the displacement increases. The difference in PRB-5 is about 60 % of the value from Lindley formula when the horizontal displacement reaches 80 mm. This indicates that the
vertical stiffness has large influence on the horizontal stiffness of PRB. The greater the vertical stiffness is, the greater the horizontal stiffness will be. Moreover, this influence increases as
horizontal displacement increases. It is due to the coupling relationship between the vertical displacement and horizontal displacement in PRB. A larger horizontal displacement needs a larger
vertical displacement. So the larger vertical stiffness leads to a larger horizontal stiffness.
4. Isolation capacities
To study the isolation capacity of PRB, the response history analyses of the structures with PRB are conducted. A hypothetical six-story shear type concrete structure with 4 isolator set on the
foundation is adopted. The dimension and material proprieties of isolator are the same as PRB4. The mass of stories is: (i) first floor: 2×10^4 kg; (ii) top floor: 1.0×10^4 kg; (iii) the other
floors: 1.5×10^4 kg. The lateral shear stiffness of stories is: (i) first floor: 2×10^4 kN/m; (ii) top floor: 2×10^4 kN/m; (iii) the other floors: 1.5×10^4 kN/m. The damping ratios of structure and
isolators are 0.05 and 0.2 respectively.
The recorded ground motion (north-south component) in the El-Centro earthquake is adopted as the excitation for the response history analysis. This excitation is scaled so that the peak ground
acceleration (PGA) is equal to 0.1 g. Employing the Newmark-$\beta$ integration method [14], the structural responses are computed for 3 different structures: (i) the structure without any isolators;
(ii) the structure with conventional RBs; (iii) the structure with PRBs. The acceleration and displacement response history of top-floor are shown in Figs. 11-12.
Fig. 11Acceleration response history of top floor
(a) Comparison of structure with PRBs
(b) Comparison of structure with PRBs
Fig. 12Displacement response history of top floor
(a) Comparison of structure with PRBs
(b) Comparison of structure with PRBs
It can be seen that the peak accelerations in top floor of structures with PRBs and with RBs are 0.261 m/s^2 and 0.196 m/s^2 respectively. The peak acceleration in top floor of non-isolated structure
is 1.021 m/s^2. Hence both bearings have effective isolation capacity. The peak acceleration in structure with PRB is about 20 % larger than that in structure with RB. So the isolation capacity of
PRB is slightly weaker than that of RB. On the other hand, the peak displacement in structure with PRBs is about 40 % smaller than that in structure with RB. The reduction of horizontal displacement
is mainly due to the fact that the horizontal stiffness of PRB increases with the increase of horizontal displacement. The large stiffness can limit the further deformation of PRB when the horizontal
displacement becomes large. Hence, it can be said that the PRB not only has effective capacity of isolation as conventional RB, but also has sound capacity of horizontal displacement limitation.
To study the influence of seismic intensity on the isolation capacity, the input ground motions are scaled so that the PGAs of motions are equal to 0.2 g and 0.4 g respectively. These two ground
excitations correspond to seismic intensity of 8 degree and 9 degree in Chinese Code. The relationships between peak response quantity and excitation intensity are computed and shown in Figs. 13-14.
Fig. 13Relationship between peak acceleration in top floor and excitation intensity
Fig. 14Relation between peak displacement in top floor and excitation intensity
It can be seen from Fig. 13 that the peak accelerations in structures with bearings are similar when the PGA of ground motion is low. Although the peak accelerations in two structures with bearings
are different, the difference is not significant. They are both much smaller than the peak acceleration in the non-isolated structure. Therefore, it can be said that the PRB has the similar capacity
of isolation as the conventional RB.
Fig. 14 shows that the peak displacements in three structures are small when the ground motion intensity is small. They are increased as the PGAs of ground motions increase. The peak displacement in
the structure with PRBs is significantly different from the structure with RBs. The peak displacement in structure with PRBs is close to the displacement in the non-isolated structure. The main
reason is that the internal forces of the tendons increase when the horizontal displacement of the bearing increases. The horizontal component of the internal forces can resist most part of the
horizontal load and limit the displacement. Therefore, PRB has the capacity of horizontal displacement limitation, and the capacity increases as the PGA of ground motion increases.
5. Conclusions
The following conclusions can be drawn:
1) The Lindley formula was developed base on the conventional RBs with thin rubber layers. It underestimates the vertical stiffness of bearings with thick rubber layers. A modified formula for
vertical stiffness of bearings with thick rubber layers is proposed in this paper. Because the suggested modifying coefficient is based on the regression analysis of experimental results, it can only
be applied to similar bearings of this paper.
2) Different from the conventional RB, the horizontal stiffness of PRB is variable. The initial stiffness of PRB is close to RB, but it increases with the increase of horizontal displacement. From
the numerical investigation, it can be seen that the PRB has effective isolation capacity as the convention RB during earthquake.
3) Due to the existing of prestress tendons, the PRB has the capacity of horizontal displacement limitation. This limitation capacity increases as the PGA of ground motion increases.
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About this article
05 December 2012
28 February 2013
prestress tendon
thick rubber layer
horizontal displacement limitation
This work was financially supported by the National Science Foundation of China under Grant No. 51108091.
Copyright © 2013 Vibroengineering
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Based on our former experiment of generating a large amount of entangled atom pairs, we will optimize the temperature of the atoms, and connect the entangled pairs to form multipartite entangled
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Lattice structure of a defined geometry can be engineered with a diffraction-limited objective and a spatial light modulator. We can then study the physics of quantum fluctuations, topological
quantum states, quantum magnetism in square lattices, triangular lattices, and Kagome lattices etc. Low-dimensional systems of 1D or 2D will be created and the quantum criticality will be studied.
3. Quantum simulation of the physics around the critical point.
Around the critical point of the phase transition between superfluid-to-Mott insulator, the correlation length gets divergent. It is very hard to describe the system by numerical simulation with
classical computers. New physical phenomena emerge rather than predicted per the atomic theory based on reductionism. Well, quantum simulator built by the quantum system itself will answer the
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• Quantum criticality and the Tomonaga-Luttinger liquid in one-dimensional Bose gases. Physical Review Letters 119, 165701 (2017).
• Spin-dependent optical superlattice. Physical Review A 96, 011602 (2017).
• Generation and detection of atomic spin entanglement in optical lattices. Nature Physics 12, 783-787 (2016). | {"url":"https://quantum.ustc.edu.cn/web/index.php/en/node/53","timestamp":"2024-11-04T01:29:43Z","content_type":"text/html","content_length":"36573","record_id":"<urn:uuid:7db07b99-c802-4d71-b9ca-50230dea1df8>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00573.warc.gz"} |
Java StrictMath abs(double a) method example
Java StrictMath abs(double a) method exampleryan2019-11-14T16:20:38+00:00
The abs(double a) method of StrictMath class returns the absolute value of a double value. If the argument is not negative, the argument is returned. If the argument is negative, the negation of the
argument is returned. Special cases:
• If the argument is positive zero or negative zero, the result is positive zero.
• If the argument is infinite, the result is positive infinity.
• If the argument is NaN, the result is NaN.
As implied by the above, one valid implementation of this method is given by the expression below which computes a double with the same exponent and significand as the argument but with a guaranteed
zero sign bit indicating a positive value:
The abs(double a) method of StrictMath class is static thus it should be accessed statically which means the we would be calling this method in this format:
StrictMath.abs(double a)
Non static method is usually called by just declaring method_name(argument) however in this case since the method is static, it should be called by appending the class name as suffix. We will be
encountering a compilation problem if we call the java compare method non statically.
Method Syntax
public static double abs(double a)
Method Argument
Data Type Parameter Description
double a the argument whose absolute value is to be determined.
Method Returns
The abs(double a) method returns the absolute value of the argument.
Java StrictMath abs(double a) Example
Below is a java code demonstrates the use of abs(double a) method of StrictMath class.
package com.javatutorialhq.java.examples;
import java.util.Scanner;
* A java example source code to demonstrate
* the use of abs(double a) method of Math class
public class StrictMathAbsDoubleExample {
public static void main(String[] args) {
// Ask user input (double)
System.out.print("Enter a double:");
// declare the scanner object
Scanner scan = new Scanner(System.in);
// use scanner to get the user input and store it to a variable
double dValue = scan.nextDouble();
// close the scanner object
// get the absolute value of the user input
double result = StrictMath.abs(dValue);
// print the result
System.out.println("result: "+result); | {"url":"https://javatutorialhq.com/java/lang/strictmath-class-tutorial/abs-double-method-example/","timestamp":"2024-11-03T15:59:39Z","content_type":"text/html","content_length":"1049390","record_id":"<urn:uuid:2ad3e488-0f73-4b96-a3b9-0d3db7af30d4>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00011.warc.gz"} |
How to Plot More 10K Points Using Matplotlib?
To plot more than 10k points using matplotlib, you can consider using a scatter plot with the scatter() function. This function is more efficient than plotting each point individually. You can also
adjust the size of the markers to reduce overplotting. Another option is to use the plot() function with a low marker size and high alpha value to make the points more transparent. This will help to
visualize a larger number of points without overwhelming the plot. Additionally, you can try using subsampling techniques or downsampling the data to reduce the number of points being plotted while
still maintaining the overall trend in the data.
How to handle overplotting and visual clutter in plots with over 10k points in matplotlib?
There are several ways to handle overplotting and visual clutter in plots with over 10k points in matplotlib:
1. Use plotting techniques that are specifically designed to handle large datasets, such as hexbin plots or density plots. These types of plots can help prevent overplotting by aggregating data
points into bins or displaying density information.
2. Use transparency or alpha blending to make individual data points more transparent, allowing you to see overlapping points more clearly. You can adjust the transparency of points using the alpha
parameter in your plot commands.
3. Use marker size and color to differentiate between different subsets of data points. By varying the size or color of markers based on some other variable in your dataset, you can add additional
information to your plot without sacrificing clarity.
4. Use interactive plotting tools, such as zooming and panning, to explore your dataset more effectively. Matplotlib has built-in support for interactive plotting through tools like the zoom and pan
buttons in the plot window.
5. Consider using data aggregation techniques, such as downsampling or binning, to reduce the number of points plotted on the screen. This can help improve performance and make it easier to see
patterns in your data without being overwhelmed by excessive detail.
By using these techniques, you can effectively visualize large datasets in matplotlib while avoiding overplotting and visual clutter.
What is the best way to customize the appearance of plots with 10k+ points in matplotlib?
When dealing with plots with 10k+ points in matplotlib, it's important to consider performance and readability. Here are some tips to customize the appearance of plots effectively:
1. Use scatter plots instead of line plots: Scatter plots are more appropriate for a large number of points as they don't connect each point with lines, which can be overwhelming with a high density
of points.
2. Use marker sizes and colors to differentiate points: You can set the size and color of markers based on different variables to visually encode additional information in the plot.
3. Use transparency to deal with overplotting: When points overlap, it can be hard to distinguish individual points. Using transparency (alpha) can help visualize the density of points in areas with
high overlap.
4. Use subsampling or down-sampling: If the plot is too cluttered, consider subsampling or down-sampling the data to reduce the number of points displayed.
5. Use a color map for continuous variables: If you have continuous variables, you can use color maps to map the values to colors and create a gradient effect.
6. Use interactive plots: If the plot is too complex to visualize in a static image, consider using interactive plotting techniques such as zooming, panning, or tooltips to explore the data more
By applying these techniques, you can customize the appearance of plots with 10k+ points in matplotlib to make them more informative and visually appealing.
How to create 3D plots with over 10k points in matplotlib?
When creating 3D plots with over 10k points in matplotlib, it is important to consider performance optimizations to ensure smooth rendering. Here are some tips to create 3D plots with a large number
of points:
1. Use the scatter method: When plotting a large number of points, using the scatter method instead of a surface plot can greatly improve performance. The scatter method is optimized for plotting a
large number of individual points.
2. Use the 's' parameter: The 's' parameter in the scatter method allows you to control the size of the markers used to represent the points. By adjusting the size of the markers, you can create a
visually appealing plot with a large number of points.
3. Enable interactive mode: By enabling interactive mode in matplotlib, you can explore the plot by rotating, zooming, and panning. This can be useful when visualizing 3D plots with a large number
of points.
4. Use a colormap: To differentiate between different data points, you can use a colormap to assign colors based on a particular variable. This can help make the plot more informative and visually
5. Consider using a 3D scatter plot instead of a surface plot: If your data consists of individual points rather than a continuous surface, using a 3D scatter plot can be a better choice. This can
help improve performance and make it easier to visualize the data.
Overall, by following these tips and optimizing your code, you can create 3D plots with over 10k points in matplotlib efficiently and effectively.
What is the impact of using different plot styles when plotting large datasets in matplotlib?
Using different plot styles can have various impacts on the visualization of large datasets in matplotlib. Some potential impacts include:
1. Clarity and readability: Different plot styles can affect the clarity and readability of the visualization. For example, using a scatter plot can make it easier to see individual data points,
while a line plot may be better for showing trends or patterns.
2. Performance: Some plot styles may be more computationally intensive than others, especially when dealing with large datasets. For example, using a scatter plot with a large number of points may
slow down the rendering of the plot compared to using a line plot.
3. Aesthetics: Different plot styles can also impact the aesthetic appeal of the visualization. Some styles may be more visually appealing or better suited for certain types of data than others.
4. Interpretation: The choice of plot style can affect how the data is interpreted. For example, a box plot may be better for showing the distribution of data, while a heatmap may be better for
showing patterns or correlations.
In summary, the impact of using different plot styles when plotting large datasets in matplotlib depends on factors such as clarity, performance, aesthetics, and interpretation. It's important to
consider these factors when choosing a plot style for a given dataset.
What is the maximum number of points that can be plotted in matplotlib?
There is no hard limit on the number of points that can be plotted in matplotlib. The amount of data that can be plotted depends on the available memory and processing power of the system running
matplotlib. In practice, matplotlib can handle millions of points without any issues on modern computers.
How to add annotations and labels to plots with over 10k points in matplotlib?
When dealing with plots that contain over 10k points in matplotlib, it is important to optimize the code to avoid performance issues. One way to add annotations and labels to plots with a large
number of points is to selectively annotate only a subset of the points.
Here is an example of how to add annotations and labels to plots with a large number of points in matplotlib:
1 import matplotlib.pyplot as plt
2 import numpy as np
4 # Generate some random data
5 x = np.random.rand(10000)
6 y = np.random.rand(10000)
8 # Create a scatter plot
9 plt.scatter(x, y, alpha=0.5)
11 # Add annotations to every 100th point
12 for i in range(0, len(x), 100):
13 plt.annotate(f'Point {i}', (x[i], y[i]))
15 # Add labels to x and y axis
16 plt.xlabel('X-axis')
17 plt.ylabel('Y-axis')
19 plt.show()
In this example, we generate random data with 10k points and create a scatter plot. We then loop through the data points and add annotations to every 100th point using the plt.annotate function. This
allows us to selectively annotate only a subset of the points, which helps to avoid cluttering the plot with annotations for all 10k points.
Additionally, we add labels to the x and y axis using plt.xlabel and plt.ylabel functions to provide context to the plot.
By selectively annotating a subset of the points and adding labels to the plot, we can effectively add annotations and labels to plots with over 10k points in matplotlib without compromising | {"url":"https://ubuntuask.com/blog/how-to-plot-more-10k-points-using-matplotlib","timestamp":"2024-11-13T12:22:31Z","content_type":"text/html","content_length":"344352","record_id":"<urn:uuid:b0c200a0-a819-4a92-9771-9aab9d05dd54>","cc-path":"CC-MAIN-2024-46/segments/1730477028347.28/warc/CC-MAIN-20241113103539-20241113133539-00832.warc.gz"} |
Ensemble AI Models in SQL Server
By: Rick Dobson | Updated: 2023-04-25 | Comments | Related: More > Artificial Intelligence
In this article, we look at an ensemble AI model which can be considered a collection of two or more AI models that complement each other to arrive at an outcome for a set of historical data.
A good place to start is by appreciating the basics of an AI model. An AI model takes data as input and simulates a decision-making process to arrive at an outcome, such as dates and prices for
buying and/or selling a security, assessing how much and when to replenish in-stock inventory items, or predicting temperature and rainfall observations from a weather station for tomorrow, next
week, or next month. A SQL Server professional can think of an AI model as a set of one or more queries that return some results sets. The query statements and their results sets are meant to match
or exceed the performance of an expert human decision-maker.
As a SQL Server professional, you can think of the data for an AI model as the data inside a SQL Server instance (or data that you can import to SQL Server from an online data source). Processing
steps are implemented as successive queries that pass results sets from one query to the next through the final processing step in the AI model. The final results set is the implemented version of
the AI model. With time series data, which are the focus of the models in this tip, an AI model can run repetitively until there are no remaining historical data to evaluate the model.
An ensemble AI model can be considered a collection of two or more AI models that complement each other to arrive at an outcome for a set of historical data. Each model in an AI ensemble model should
have its own distinct set of rules. Additionally, each model in an ensemble AI model needs to have specific rules for combining its results sets with the results sets for one or more other elements
in an ensemble model. The combined results sets from the ensemble model elements is the ensemble model.
The solution code for the problem statement includes demonstrations of how to apply standard T-SQL coding techniques for AI applications. Among the standard T-SQL coding techniques for AI modeling
covered in this tip are:
• Creating fresh tables with drop table if exists and select into statements
• Creating and invoking stored procedures
• Syntax examples for lag and lead functions as well as min and max functions
• Complementary applications for local temp tables, global temp tables, and table variables
• Reconfiguring a normalized results set as a de-normalized results set to speed code execution
An Overview of the Ensemble Model for this Tip
The data for this tip processes security price data for six tickers. The six tickers are for three pairs of ETF securities based on three major market indexes. The indexes are the Dow Industrial
Average, S&P 500, and Nasdaq 100. One member of each pair of securities (DIA, QQQ, SPY) aims to follow the price performance of its underlying index on a proportional basis for each trading day. The
other member of each pair of securities (UDOW, TQQQ, SPXL) aims to have its price performance remain in a three-to-one relation to its underlying index on a daily trading day basis. A prior tip
titled "SQL Server Data Mining for Leveraged Versus Unleveraged ETFs as a Long-Term Investment" demonstrated how to collect and save historical price and volume data from Yahoo Finance for these six
tickers to a SQL Server table named DataScience.dbo.symbol_date.
The download from the prior tip includes the original CSV files from Yahoo Finance and the T-SQL code for transferring the file contents to the DataScience.dbo.symbol_date table. The next two
screenshots show the first and last eight rows from the table. There is one row for each trading day on which a ticker symbol trades. Across all six tickers starting from each ticker’s initial public
offering date through November 30, 2022, there are a maximum of 29731 rows of data.
There are two AI model elements in the ensemble AI model for this tip:
• The first AI model is called the "close_gt_ema_10_model". This model designates a date on which to buy a security whenever the close value for the preceding two trading days is greater than the
exponential moving average with a period length of 10, which is also preceded by another two trading days in which the close value is not greater than the exponential moving average with a period
length of 10. The model ends the hold period for a security ten days after it is bought.
• The second AI model is called the "seasonal_include_model". This model treats each month as a season of the year for computing a seasonal factor over a timeframe. The T-SQL code for computing
these monthly seasonal factors is described in the "Computing seasonality factors by month across year" section of this prior tip.
The rule for combining the two models are as follows.
• Use the close_gt_ema_10_model to pick an initial set of buy dates for each of the six ticker symbols
• Then, use the seasonal_include_model to identify months for a symbol that are most likely to result in winning trades
• Finally, only include buy dates from the close_gt_ema_10_model where the buy dates belong to a month that is from the top half of months based on the seasonal_include_model. Exclude buy dates
from the close_gt_ema_10_model that are not from the top half of months based on the seasonal_include_model. It is this last combining rule that makes the model from this tip an ensemble AI model
– that is, the ensemble AI model is like a medley from both the close_gt_ema_10_model and the seasonal_include_model.
Implementing the close_gt_ema_10_model
The implementation of the close_gt_ema_10_model requires a dataset derived from the symbol_date table. The three essential columns from this table are the symbol ticker column for one of the six
tickers for which data are available from the table, the close value column for each ticker on a trading date, and the close date value column for each ticker symbol which start with the initial
public offering date for a ticker through the last date for which data was collected for implementing the model. The initial public offering date is the first date for which shares of a security are
available for trading. Data collection for all six tickers ceased on November 30, 2022.
In addition to the preceding three underlying source data columns, several other columns need to be calculated. Many of these calculated columns depend directly on the exponential moving average with
a period length of 10 for the close value of the current trading date (ema_10). If you are not already familiar with exponential moving averages, you may find either or both of these two prior tips
• An earlier article titled "Exponential Moving Average Calculation in SQL Server" briefly introduces exponential moving averages and how to calculate them for a dataset already in SQL Server.
• A recent article titled "Adding a Buy-Sell Model to a Buy-and-Hold Model with T-SQL" provides an example of computing exponential moving averages with the same underlying data source as the one
used in this tip. This second article can provide an additional resource to help you understand this tip and build your capability to create AI models in SQL Server.
As with many AI models, the evaluation of the close_gt_ema_10_model requires looking back over preceding periods to the current one.
• The close_gt_ema_10_model compares the close value and the ema_10 values for each of the prior four periods. These previous four periods (trading days) are examined to determine if the close
values are rising relative to the ema_10 values
• If an uptrend of the close values relative to ema_10 values is detected, then the model specifies a buy date for the current trading date and a sell date for 10 days after the current trading
• If the close value for 10 days after the current trading date is greater than the current trading date, then the model has a winning buy/sell cycle
The following create procedure statement is the most recent version of my T-SQL code for computing exponential moving average values for a column of historical time series values, such as the close
values in this tip. This version of the stored procedure takes advantage of decimal(19,4) for storing and processing monetary data type values.
• The stored procedure operates on close values originating in the symbol_date table; the ema values output by the stored procedure are saved in #temp_for_ema
• Three parameters adjust the performance of the stored procedure
□ The @symbol parameter designates the ticker symbol for which to compute ema values
□ The @period parameter indicates the period length of the ema values
□ The @alpha parameter is the weighting value for the close value for the current period, and (1-@alpha) is the weighting value for the ema for the prior period
Use DataScience
drop procedure if exists [dbo].[usp_ema_computer_with_dec_vals]
create procedure [dbo].[usp_ema_computer_with_dec_vals]
-- Add the parameters for the stored procedure here
@symbol nvarchar(10) -- for example, assign as 'SPY'
,@period dec(19,4) -- for example, assign as 12
,@alpha dec(14,4) -- for example, assign as 2/(12 + 1)
-- suppress row counts for output from usp
set nocount on;
-- parameters to the script are
-- @symbol identifier for set of time series values
-- @period number of periods for ema
-- @alpha weight for current row time series value
-- initially populate #temp_for_ema for ema calculations
-- @ema_first is the seed
declare @ema_first dec(19,4) =
(select top 1 [close] from [dbo].[symbol_date] where symbol = @symbol order by [date])
-- create base table for ema calculations
drop table if exists #temp_for_ema
-- ema seed run to populate #temp_for_ema
-- all rows have first close price for ema
,row_number() OVER (ORDER BY [Date]) [row_number]
,@ema_first ema
into #temp_for_ema
from [dbo].[symbol_date]
where symbol = @symbol
order by row_number
-- NULL ema values for first period
update #temp_for_ema
ema = NULL
where row_number = 1
-- calculate ema for all dates in time series
-- @alpha is the exponential weight for the ema
-- start calculations with the 3rd period value
-- seed is close from 1st period; it is used as ema for 2nd period
-- set @max_row_number int and initial @current_row_number
-- declare @today_ema
declare @max_row_number int = (select max(row_number) from #temp_for_ema)
declare @current_row_number int = 3
declare @today_ema dec(19,4)
-- loop for computing successive ema values
while @current_row_number <= @max_row_number
set @today_ema =
-- compute ema for @current_row_number
top 1
([close] * @alpha) + (lag(ema,1) over (order by [date]) * (1 - @alpha)) ema_today
from #temp_for_ema
where row_number >= @current_row_number -1 and row_number <= @current_row_number
order by row_number desc
-- update current row in #temp_for_ema with @today_ema
-- and increment @current_row_number
update #temp_for_ema
ema = @today_ema
where row_number = @current_row_number
set @current_row_number = @current_row_number + 1
-- display the results set with the calculated values
-- on a daily basis
,@period period_length
,ema ema
from #temp_for_ema
where row_number < = @max_row_number
order by row_number
Here is a script excerpt for invoking the usp_ema_computer_with_dec_vals stored procedure created in the preceding create procedure statement. The script repeatedly invokes the stored procedure for
the six tickers examined in this tip. Results are stored in the ema_period_symbol_with_dec_vals table via an insert into statement. In addition to computing ema_10 values for close values, the code
also computes ema_30, ema_50, and ema_200 for close values. This was convenient because the original code for invoking the usp_ema_computer_with_dec_vals stored procedure was excerpted from another
application that required exponential moving averages with period lengths of 10, 30, 50, and 200.
use DataScience
-- create a fresh copy of the [dbo].[ema_period_symbol_with_dec_vals] table
drop table if exists dbo.ema_period_symbol_with_dec_vals
create table [dbo].[ema_period_symbol_with_dec_vals](
[date] [date] NOT NULL,
[symbol] [nvarchar](10) NOT NULL,
[close] [decimal](19, 4) NULL,
[period_length] [int] NOT NULL,
[ema] [decimal](19, 4) NULL,
primary key clustered
[symbol] ASC,
[date] ASC,
[period_length] ASC
)WITH (PAD_INDEX = OFF, STATISTICS_NORECOMPUTE = OFF, IGNORE_DUP_KEY = OFF, ALLOW_ROW_LOCKS = ON, ALLOW_PAGE_LOCKS = ON, OPTIMIZE_FOR_SEQUENTIAL_KEY = OFF) ON [PRIMARY]
) ON [PRIMARY]
-- populate a fresh copy of ema_period_symbol_with_dec_vals
-- with the [dbo].[ema_period_symbol_with_dec_vals] stored proc
-- based on four period lengths (10, 30, 50, 200)
-- for six symbols (SPY, SPXL, QQQ, TQQQ, DIA, UDOW)
-- populate 10, 30, 50, 200 period_length emas for SPY ticker
insert into [dbo].[ema_period_symbol_with_dec_vals]
exec [dbo].[usp_ema_computer_with_dec_vals] 'SPY', 10, .1818
insert into [dbo].[ema_period_symbol_with_dec_vals]
exec [dbo].[usp_ema_computer_with_dec_vals] 'SPY', 30, .0645
insert into [dbo].[ema_period_symbol_with_dec_vals]
exec [dbo].[usp_ema_computer_with_dec_vals] 'SPY', 50, .0392
insert into [dbo].[ema_period_symbol_with_dec_vals]
exec [dbo].[usp_ema_computer_with_dec_vals] 'SPY', 200, .0100
-- populate 10, 30, 50, 200 period_length emas for SPXL ticker
insert into [dbo].[ema_period_symbol_with_dec_vals]
exec [dbo].[usp_ema_computer_with_dec_vals] 'SPXL', 10, .1818
insert into [dbo].[ema_period_symbol_with_dec_vals]
exec [dbo].[usp_ema_computer_with_dec_vals] 'SPXL', 30, .0645
insert into [dbo].[ema_period_symbol_with_dec_vals]
exec [dbo].[usp_ema_computer_with_dec_vals] 'SPXL', 50, .0392
insert into [dbo].[ema_period_symbol_with_dec_vals]
exec [dbo].[usp_ema_computer_with_dec_vals] 'SPXL', 200, .0100
-- populate 10, 30, 50, 200 period_length emas for QQQ ticker
insert into [dbo].[ema_period_symbol_with_dec_vals]
exec [dbo].[usp_ema_computer_with_dec_vals] 'QQQ', 10, .1818
insert into [dbo].[ema_period_symbol_with_dec_vals]
exec [dbo].[usp_ema_computer_with_dec_vals] 'QQQ', 30, .0645
insert into [dbo].[ema_period_symbol_with_dec_vals]
exec [dbo].[usp_ema_computer_with_dec_vals] 'QQQ', 50, .0392
insert into [dbo].[ema_period_symbol_with_dec_vals]
exec [dbo].[usp_ema_computer_with_dec_vals] 'QQQ', 200, .0100
-- populate 10, 30, 50, 200 period_length emas for TQQQ ticker
insert into [dbo].[ema_period_symbol_with_dec_vals]
exec [dbo].[usp_ema_computer_with_dec_vals] 'TQQQ', 10, .1818
insert into [dbo].[ema_period_symbol_with_dec_vals]
exec [dbo].[usp_ema_computer_with_dec_vals] 'TQQQ', 30, .0645
insert into [dbo].[ema_period_symbol_with_dec_vals]
exec [dbo].[usp_ema_computer_with_dec_vals] 'TQQQ', 50, .0392
insert into [dbo].[ema_period_symbol_with_dec_vals]
exec [dbo].[usp_ema_computer_with_dec_vals] 'TQQQ', 200, .0100
-- populate 10, 30, 50, 200 period_length emas for DIA ticker
insert into [dbo].[ema_period_symbol_with_dec_vals]
exec [dbo].[usp_ema_computer_with_dec_vals] 'DIA', 10, .1818
insert into [dbo].[ema_period_symbol_with_dec_vals]
exec [dbo].[usp_ema_computer_with_dec_vals] 'DIA', 30, .0645
insert into [dbo].[ema_period_symbol_with_dec_vals]
exec [dbo].[usp_ema_computer_with_dec_vals] 'DIA', 50, .0392
insert into [dbo].[ema_period_symbol_with_dec_vals]
exec [dbo].[usp_ema_computer_with_dec_vals] 'DIA', 200, .0100
-- populate 10, 30, 50, 200 period_length emas for UDOW ticker
insert into [dbo].[ema_period_symbol_with_dec_vals]
exec [dbo].[usp_ema_computer_with_dec_vals] 'UDOW', 10, .1818
insert into [dbo].[ema_period_symbol_with_dec_vals]
exec [dbo].[usp_ema_computer_with_dec_vals] 'UDOW', 30, .0645
insert into [dbo].[ema_period_symbol_with_dec_vals]
exec [dbo].[usp_ema_computer_with_dec_vals] 'UDOW', 50, .0392
insert into [dbo].[ema_period_symbol_with_dec_vals]
exec [dbo].[usp_ema_computer_with_dec_vals] 'UDOW', 200, .0100
The ema_period_symbol_with_dec_vals table returns values in a normalized format, which is common for relational databases, but this format can be an inefficient way of storing time series data. The
normalized format can be transformed to another format that is more efficient for storing and retrieving time series data. For example, the following script reconfigures the contents of the
ema_period_symbol_with_dec_vals table into a new table named close_and_emas. The number of rows in the ema_period_symbol_with_dec_vals table is 118924, but when the data are reconfigured in the more
efficient format of the close_and_emas table, the number of rows shrinks to 29731.
use DataScience
-- join and concatenate rows
-- for symbols and period_lengths
-- count of rows in outer query is 29731
drop table if exists dbo.close_and_emas
select *
into dbo.close_and_emas
select *
,ema [ema_10]
from ema_period_symbol_with_dec_vals
where symbol = 'SPY' and period_length = 10
) for_ema_10
,ema [ema_30]
from ema_period_symbol_with_dec_vals
where symbol = 'SPY' and period_length = 30
) for_ema_30
on for_ema_10.symbol = for_ema_30.symbol and
for_ema_10.[date] = for_ema_30.[date]
,ema [ema_50]
from ema_period_symbol_with_dec_vals
where symbol = 'SPY' and period_length = 50
) for_ema_50
on for_ema_10.symbol = for_ema_50.symbol and
for_ema_10.[date] = for_ema_50.[date]
,ema [ema_200]
from ema_period_symbol_with_dec_vals
where symbol = 'SPY' and period_length = 200
) for_ema_200
on for_ema_10.symbol = for_ema_200.symbol and
for_ema_10.[date] = for_ema_200.[date]
) for_SPY
select *
,ema [ema_10]
from ema_period_symbol_with_dec_vals
where symbol = 'SPXL' and period_length = 10
) for_ema_10
,ema [ema_30]
from ema_period_symbol_with_dec_vals
where symbol = 'SPXL' and period_length = 30
) for_ema_30
on for_ema_10.symbol = for_ema_30.symbol and
for_ema_10.[date] = for_ema_30.[date]
,ema [ema_50]
from ema_period_symbol_with_dec_vals
where symbol = 'SPXL' and period_length = 50
) for_ema_50
on for_ema_10.symbol = for_ema_50.symbol and
for_ema_10.[date] = for_ema_50.[date]
,ema [ema_200]
from ema_period_symbol_with_dec_vals
where symbol = 'SPXL' and period_length = 200
) for_ema_200
on for_ema_10.symbol = for_ema_200.symbol and
for_ema_10.[date] = for_ema_200.[date]
) for_SPXL
select *
,ema [ema_10]
from ema_period_symbol_with_dec_vals
where symbol = 'QQQ' and period_length = 10
) for_ema_10
,ema [ema_30]
from ema_period_symbol_with_dec_vals
where symbol = 'QQQ' and period_length = 30
) for_ema_30
on for_ema_10.symbol = for_ema_30.symbol and
for_ema_10.[date] = for_ema_30.[date]
,ema [ema_50]
from ema_period_symbol_with_dec_vals
where symbol = 'QQQ' and period_length = 50
) for_ema_50
on for_ema_10.symbol = for_ema_50.symbol and
for_ema_10.[date] = for_ema_50.[date]
,ema [ema_200]
from ema_period_symbol_with_dec_vals
where symbol = 'QQQ' and period_length = 200
) for_ema_200
on for_ema_10.symbol = for_ema_200.symbol and
for_ema_10.[date] = for_ema_200.[date]
) for_QQQ
select *
,ema [ema_10]
from ema_period_symbol_with_dec_vals
where symbol = 'TQQQ' and period_length = 10
) for_ema_10
,ema [ema_30]
from ema_period_symbol_with_dec_vals
where symbol = 'TQQQ' and period_length = 30
) for_ema_30
on for_ema_10.symbol = for_ema_30.symbol and
for_ema_10.[date] = for_ema_30.[date]
,ema [ema_50]
from ema_period_symbol_with_dec_vals
where symbol = 'TQQQ' and period_length = 50
) for_ema_50
on for_ema_10.symbol = for_ema_50.symbol and
for_ema_10.[date] = for_ema_50.[date]
,ema [ema_200]
from ema_period_symbol_with_dec_vals
where symbol = 'TQQQ' and period_length = 200
) for_ema_200
on for_ema_10.symbol = for_ema_200.symbol and
for_ema_10.[date] = for_ema_200.[date]
) for_TQQQ
select *
,ema [ema_10]
from ema_period_symbol_with_dec_vals
where symbol = 'DIA' and period_length = 10
) for_ema_10
,ema [ema_30]
from ema_period_symbol_with_dec_vals
where symbol = 'DIA' and period_length = 30
) for_ema_30
on for_ema_10.symbol = for_ema_30.symbol and
for_ema_10.[date] = for_ema_30.[date]
,ema [ema_50]
from ema_period_symbol_with_dec_vals
where symbol = 'DIA' and period_length = 50
) for_ema_50
on for_ema_10.symbol = for_ema_50.symbol and
for_ema_10.[date] = for_ema_50.[date]
,ema [ema_200]
from ema_period_symbol_with_dec_vals
where symbol = 'DIA' and period_length = 200
) for_ema_200
on for_ema_10.symbol = for_ema_200.symbol and
for_ema_10.[date] = for_ema_200.[date]
) for_DIA
select *
,ema [ema_10]
from ema_period_symbol_with_dec_vals
where symbol = 'UDOW' and period_length = 10
) for_ema_10
,ema [ema_30]
from ema_period_symbol_with_dec_vals
where symbol = 'UDOW' and period_length = 30
) for_ema_30
on for_ema_10.symbol = for_ema_30.symbol and
for_ema_10.[date] = for_ema_30.[date]
,ema [ema_50]
from ema_period_symbol_with_dec_vals
where symbol = 'UDOW' and period_length = 50
) for_ema_50
on for_ema_10.symbol = for_ema_50.symbol and
for_ema_10.[date] = for_ema_50.[date]
,ema [ema_200]
from ema_period_symbol_with_dec_vals
where symbol = 'UDOW' and period_length = 200
) for_ema_200
on for_ema_10.symbol = for_ema_200.symbol and
for_ema_10.[date] = for_ema_200.[date]
) for_UDOW
) for_SPY_SPXL_QQQ_TQQQ_DIA_UDOW
The next script excerpt relies on the close_and_emas table as a data source to populate the #temp_with_criteria_for_start_cycles table for exposing the criteria for buy and sell decisions based on
the close_gt_ema_10_model rules. The script does not make any decisions, but it does expose the criteria on which buy and sell decisions are subsequently made.
• There are four columns that serve as the primary criteria for buy and sell decisions. These columns have names of start_lag_4, start_lag_3, start_lag_2, and start_lag_1
□ When the values for start_lag_4 through start_lag_1 have values of 0, 0, 1,1, then close values are generally increasing over the preceding four trading days. The model assumes this is a good
time to buy a security
□ The model assumes that a good time to sell a security is at the tenth day past the preceding buy date. The model makes this assumption on the belief that the close value exceeding the ema_10
value is a relatively short-term predictor of future performance
• As indicated above, the #temp_with_criteria_for_start_cycles table does not actually make buy and sell decisions. Therefore, the #temp_with_criteria_for_start_cycles table has 29731 rows – one
for each trading day across all six tickers in the symbol_date table
drop table if exists #temp_with_criteria_for_start_cycles
-- THIS CODE DISPLAYS THE CRITERIA FOR
-- START CYCLES FROM SOURCE DATA
-- evaluate start lag indicator
-- according to
-- (start_lag_4 and start_lag_3) = 0 and
-- (start_lag_2 and start_lag_1) = 1
when close_lag_4 <= ema_10_lag_4 then 0
else 1
end start_lag_4
when close_lag_3 <= ema_10_lag_3 then 0
else 1
end start_lag_3
when close_lag_2 > ema_10_lag_2 then 1
else 0
end start_lag_2
when close_lag_1 > ema_10_lag_1 then 1
else 0
end start_lag_1
into #temp_with_criteria_for_start_cycles
-- extract base data, compute lags for ema_10 and close
-- as well as compute [close]_lead_10
-- compute ema_10 lags
,lag(ema_10,4) over (partition by symbol order by date) ema_10_lag_4
,lag(ema_10,3) over (partition by symbol order by date) ema_10_lag_3
,lag(ema_10,2) over (partition by symbol order by date) ema_10_lag_2
,lag(ema_10,1) over (partition by symbol order by date) ema_10_lag_1
-- compute close lags
,lag([close],4) over (partition by symbol order by date) close_lag_4
,lag([close],3) over (partition by symbol order by date) close_lag_3
,lag([close],2) over (partition by symbol order by date) close_lag_2
,lag([close],1) over (partition by symbol order by date) close_lag_1
-- compute [close]_lead_10
,lead([close],10) over (partition by symbol order by date) close_lead_10
from DataScience.dbo.close_and_emas
) for_start_lag_indicators
-- optionally display #temp_with_criteria_for_start_cycles
select * from #temp_with_criteria_for_start_cycles
The next code segment reflects the buy dates and prices and the corresponding sell prices based on the sell criterion indicated in the preceding script. There are only 1297 rows in the result set
from the script excerpt below. These rows are for the 1297 buy signal rows indicated by column values for start_lag_4, start_lag_3, start_lag_2, and start_lag_1 columns.
drop table if exists #temp_with_close_and_close_lead_10_for_each_start_cycle
-- THIS CODE EXTRACTS AND DISPLAYS THE
-- save symbol, date, [close], and close_lead_10
-- in #temp_with_close_and_close_lead_10_for_each_start_cycle
-- from #temp_with_criteria_for_start_cycles
into #temp_with_close_and_close_lead_10_for_each_start_cycle
from #temp_with_criteria_for_start_cycles
start_lag_4=0 and
start_lag_3=0 and
start_lag_2=1 and
-- optionally display #temp_with_close_and_close_lead_10_for_each_start_cycle_plus_duration_in_years
select * from #temp_with_close_and_close_lead_10_for_each_start_cycle
The key metric for evaluating model performance in this tip is the compound annual growth rate. This metric represents the change in the value of an investment from the first day of an investment
through the last day of an investment. The first and last day of an investment depends on when some resources were initially devoted to buying an investment and, correspondingly, the last time an
investment in a security is sold.
• In the context of the current tip, the date of the first investment in a security is the close date for the first trade of a security.
• Also, the date of the last investment in a security ends on the tenth trading day after the last date during which a security is bought.
The following script shows how to compute the duration between the first investment in a ticker symbol through the last sell date for a security.
• The beginning date of an investment for a security is the min(date) value for the date column from symbol_date table for a security
• The ending date of an investment for a security is the max function value for the date column from the symbol_date table for a security
• The duration in years between the beginning date and ending date for a security in years is the datediff function value between beginning and ending dates in months divided by 12. The script
below rounds the datediff function value to two places after the decimal point.
• The script saves its results set to a local temp table named #temp_with_close_and_close_lead_10_for_each_start_cycle_plus_duration_in_years.
The following script shows that the close_gt_ema_10_model makes no assumptions about carrying over the change in value from the preceding buy/sell cycle to the next buy/sell cycle. Instead, the model
merely buys a single share of a stock at close value for the current start cycle. There are any number of possible assumptions to make about carrying over the change in value from the preceding buy/
sell cycle (along with amounts) to the next buy/sell cycle. This tip leaves it to future investigations to evaluate the effectiveness of changes to the amount invested in a security from one buy/sell
cycle to the next one.
-- create and populate #temp_with_close_and_close_lead_10_for_each_start_cycle_plus_duration_in_years
drop table if exists #temp_with_close_and_close_lead_10_for_each_start_cycle_plus_duration_in_years
into #temp_with_close_and_close_lead_10_for_each_start_cycle_plus_duration_in_years
from #temp_with_close_and_close_lead_10_for_each_start_cycle
-- divide difference in months by 12.0 and round quotient to 2 places
-- after the decimal to get duration_in_years to hundredths of a year
,min([Date]) [beginning date]
,max([Date]) [ending date]
,round((DATEDIFF(month, min([Date]), max([Date]))/12.0),2) duration_in_years
from [DataScience].[dbo].[symbol_date]
group by symbol
) for_duration_in_years
on #temp_with_close_and_close_lead_10_for_each_start_cycle.symbol = for_duration_in_years.symbol
order by
The next script excerpt saves a duplicate of the same results set as the one generated by the preceding script. The difference between the preceding script and the next script is that the following
script saves its results set in a global temp table named ##temp_with_close_and_close_lead_10_for_each_start_cycle_plus_duration_in_years. The prior script excerpt saves its results set in a local
temp table (#temp_with_close_and_close_lead_10_for_each_start_cycle_plus_duration_in_years). This distinction is important for two reasons.
• The underlying T-SQL script file for the close_gt_ema_10_model is implemented in one T-SQL file
• On the other hand, the T-SQL script file for the ensemble_model is implemented in a different T-SQL file
By using a global temp table to store duration in years, the same results set can be retrieved from two different T-SQL files:
• At this point in the model development, all you need to remember is that the duration_in_years column values are accessed from a local temp table named #
• The "Implementing the ensemble_model" section of this tip returns to this general topic again when discussing the ensemble_model
-- ##temp_with_close_and_close_lead_10_for_each_start_cycle_plus_duration_in_years is for season adjustment of
-- [close] gt ema_10 model results by seasonal include indicator
drop table if exists ##temp_with_close_and_close_lead_10_for_each_start_cycle_plus_duration_in_years
-- save, and optionally display, ##temp_with_close_and_close_lead_10_for_each_start_cycle_plus_duration_in_years
-- result set includes month_number for join to seasonal adjustment model
,month(date) month_number
into ##temp_with_close_and_close_lead_10_for_each_start_cycle_plus_duration_in_years
from #temp_with_close_and_close_lead_10_for_each_start_cycle_plus_duration_in_years
order by symbol, date, month_number
-- optionally display ##temp_with_close_and_close_lead_10_for_each_start_cycle_plus_duration_in_years
-- for use with seasonal_include_model
select * from ##temp_with_close_and_close_lead_10_for_each_start_cycle_plus_duration_in_years
The last script excerpt for the close_gt_ema_10_model computes summary values for each of the six ticker symbols tracked in this tip. There are also six columns in the summary results table (#
temp_summary). Here is a brief description of the role of each column. The code for computing the column values appears next.
• The first column is named @symbol. The content of each column value is the ticker symbol for the row
• The second column is named @first_close. This column is for the close value at the beginning of the current buy/sell cycle
• The third column is named @last_close_lead_10. This column is for the close value at the end of the current buy/sell cycle. It occurs ten trading days after the preceding buy date for the current
buy/sell cycle.
• The fourth column is named @last_duration_in_years. This column is for the duration in years between the date of the @last_close_lead_10 value and the date of the @first_close value.
• The fifth column is named change. This column is the difference between the @last_close_lead_10 value and the @first_close value. This quantity is the number of monetary units between the last
sell price for a security and the first close price for a security.
• The last column for each ticker symbol has an alias name of cagr. This column reports the compound annual growth rate for the ticker based on the ratio of the last amount invested in a security
divided by the beginning amount invested in a security raised to the @last_duration_in_years value. The cagr value is returned as a percentage, which is multiplied by 100 and rounded to two
places after the decimal.
-- create and populate #temp_summary for [close] gt ema_10 model
drop table if exists #temp_summary
-- setup for populating #temp_summary
-- with data for DIA symbol
declare @symbol nvarchar(10) = 'DIA'
declare @first_date date =
(select min(date)
from #temp_with_close_and_close_lead_10_for_each_start_cycle_plus_duration_in_years
where symbol = @SYMBOL)
declare @last_date date =
(select max(date)
from #temp_with_close_and_close_lead_10_for_each_start_cycle_plus_duration_in_years
where symbol = @SYMBOL)
declare @first_close dec (19,4) =
(select [close]
from #temp_with_close_and_close_lead_10_for_each_start_cycle_plus_duration_in_years
where symbol = @SYMBOL and
date = @first_date)
declare @last_close_lead_10 dec (19,4) =
(select close_lead_10
from #temp_with_close_and_close_lead_10_for_each_start_cycle_plus_duration_in_years
where symbol = @SYMBOL and
date = @last_date)
declare @last_duration_in_years dec(19,4) =
select duration_in_years
from #temp_with_close_and_close_lead_10_for_each_start_cycle_plus_duration_in_years
where symbol = @SYMBOL and
date = @last_date
-- populate #temp_summary for DIA
select @symbol [@symbol], @first_close [@first_close], @last_close_lead_10 [@last_close_lead_10]
,@last_duration_in_years [@last_duration_in_years]
, @last_close_lead_10 - @first_close [change]
,cast((power(( @last_close_lead_10/@first_close),(1.0/@last_duration_in_years))-1)*100 as dec(19,2)) cagr
into #temp_summary
-- setup to populate #temp_summary for QQQ
set @symbol = 'QQQ'
set @first_date =
(select min(date)
from #temp_with_close_and_close_lead_10_for_each_start_cycle_plus_duration_in_years
where symbol = @SYMBOL)
set @last_date =
(select max(date)
from #temp_with_close_and_close_lead_10_for_each_start_cycle_plus_duration_in_years
where symbol = @SYMBOL)
set @first_close =
(select [close]
from #temp_with_close_and_close_lead_10_for_each_start_cycle_plus_duration_in_years
where symbol = @SYMBOL and
date = @first_date)
set @last_close_lead_10 =
(select close_lead_10
from #temp_with_close_and_close_lead_10_for_each_start_cycle_plus_duration_in_years
where symbol = @SYMBOL and
date = @last_date)
set @last_duration_in_years =
select duration_in_years
from #temp_with_close_and_close_lead_10_for_each_start_cycle_plus_duration_in_years
where symbol = @SYMBOL and
date = @last_date
-- populate #temp_summary for QQQ
insert into #temp_summary
select @symbol [@symbol], @first_close [@first_close], @last_close_lead_10 [@last_close_lead_10]
,@last_duration_in_years [@last_duration_in_years]
, @last_close_lead_10 - @first_close [change]
,cast((power(( @last_close_lead_10/@first_close),(1.0/@last_duration_in_years))-1)*100 as dec(19,2)) cagr
-- setup to populate #temp_summary for SPXL
set @symbol = 'SPXL'
set @first_date =
(select min(date)
from #temp_with_close_and_close_lead_10_for_each_start_cycle_plus_duration_in_years
where symbol = @SYMBOL)
set @last_date =
(select max(date)
from #temp_with_close_and_close_lead_10_for_each_start_cycle_plus_duration_in_years
where symbol = @SYMBOL)
set @first_close =
(select [close]
from #temp_with_close_and_close_lead_10_for_each_start_cycle_plus_duration_in_years
where symbol = @SYMBOL and
date = @first_date)
set @last_close_lead_10 =
(select close_lead_10
from #temp_with_close_and_close_lead_10_for_each_start_cycle_plus_duration_in_years
where symbol = @SYMBOL and
date = @last_date)
set @last_duration_in_years =
select duration_in_years
from #temp_with_close_and_close_lead_10_for_each_start_cycle_plus_duration_in_years
where symbol = @SYMBOL and
date = @last_date
-- populate #temp_summary for SPXL
insert into #temp_summary
select @symbol [@symbol], @first_close [@first_close], @last_close_lead_10 [@last_close_lead_10]
,@last_duration_in_years [@last_duration_in_years]
, @last_close_lead_10 - @first_close [change]
,cast((power(( @last_close_lead_10/@first_close),(1.0/@last_duration_in_years))-1)*100 as dec(19,2)) cagr
-- setup to populate #temp_summary for SPY
set @symbol = 'SPY'
set @first_date =
(select min(date)
from #temp_with_close_and_close_lead_10_for_each_start_cycle_plus_duration_in_years
where symbol = @SYMBOL)
set @last_date =
(select max(date)
from #temp_with_close_and_close_lead_10_for_each_start_cycle_plus_duration_in_years
where symbol = @SYMBOL)
set @first_close =
(select [close]
from #temp_with_close_and_close_lead_10_for_each_start_cycle_plus_duration_in_years
where symbol = @SYMBOL and
date = @first_date)
set @last_close_lead_10 =
(select close_lead_10
from #temp_with_close_and_close_lead_10_for_each_start_cycle_plus_duration_in_years
where symbol = @SYMBOL and
date = @last_date)
set @last_duration_in_years =
select duration_in_years
from #temp_with_close_and_close_lead_10_for_each_start_cycle_plus_duration_in_years
where symbol = @SYMBOL and
date = @last_date
-- populate #temp_summary for SPY
insert into #temp_summary
select @symbol [@symbol], @first_close [@first_close], @last_close_lead_10 [@last_close_lead_10]
,@last_duration_in_years [@last_duration_in_years]
, @last_close_lead_10 - @first_close [change]
,cast((power(( @last_close_lead_10/@first_close),(1.0/@last_duration_in_years))-1)*100 as dec(19,2)) cagr
-- setup to populate #temp_summary for TQQQ
set @symbol = 'TQQQ'
set @first_date =
(select min(date)
from #temp_with_close_and_close_lead_10_for_each_start_cycle_plus_duration_in_years
where symbol = @SYMBOL)
set @last_date =
(select max(date)
from #temp_with_close_and_close_lead_10_for_each_start_cycle_plus_duration_in_years
where symbol = @SYMBOL)
set @first_close =
(select [close]
from #temp_with_close_and_close_lead_10_for_each_start_cycle_plus_duration_in_years
where symbol = @SYMBOL and
date = @first_date)
set @last_close_lead_10 =
(select close_lead_10
from #temp_with_close_and_close_lead_10_for_each_start_cycle_plus_duration_in_years
where symbol = @SYMBOL and
date = @last_date)
set @last_duration_in_years =
select duration_in_years
from #temp_with_close_and_close_lead_10_for_each_start_cycle_plus_duration_in_years
where symbol = @SYMBOL and
date = @last_date
-- populate #temp_summary for UDOW
insert into #temp_summary
select @symbol [@symbol], @first_close [@first_close], @last_close_lead_10 [@last_close_lead_10]
,@last_duration_in_years [@last_duration_in_years]
, @last_close_lead_10 - @first_close [change]
,cast((power(( @last_close_lead_10/@first_close),(1.0/@last_duration_in_years))-1)*100 as dec(19,2)) cagr
-- setup to populate #temp_summary for UDOW
set @symbol = 'UDOW'
set @first_date =
(select min(date)
from #temp_with_close_and_close_lead_10_for_each_start_cycle_plus_duration_in_years
where symbol = @SYMBOL)
set @last_date =
(select max(date)
from #temp_with_close_and_close_lead_10_for_each_start_cycle_plus_duration_in_years
where symbol = @SYMBOL)
set @first_close =
(select [close]
from #temp_with_close_and_close_lead_10_for_each_start_cycle_plus_duration_in_years
where symbol = @SYMBOL and
date = @first_date)
set @last_close_lead_10 =
(select close_lead_10
from #temp_with_close_and_close_lead_10_for_each_start_cycle_plus_duration_in_years
where symbol = @SYMBOL and
date = @last_date)
set @last_duration_in_years =
select duration_in_years
from #temp_with_close_and_close_lead_10_for_each_start_cycle_plus_duration_in_years
where symbol = @SYMBOL and
date = @last_date
-- populate #temp_summary for UDOW
insert into #temp_summary
select @symbol [@symbol], @first_close [@first_close], @last_close_lead_10 [@last_close_lead_10]
,@last_duration_in_years [@last_duration_in_years]
, @last_close_lead_10 - @first_close [change]
,cast((power(( @last_close_lead_10/@first_close),(1.0/@last_duration_in_years))-1)*100 as dec(19,2)) cagr
-- display #temp_summary across all symbols
select * from #temp_summary
Here is the results set from the preceding script.
• By the cagr metric
□ The TQQQ ticker gave the best return on invested capital
□ The DIA ticker gave the worst return on invested capital
□ The three tickers for leveraged ETFs gave returns that were more than three times as large as those for corresponding unleveraged ETFs. As you can see, the @last_duration_in_years column
values are much larger for the unleveraged securities (DIA, QQQ, SPY) than for the leverage securities (SPXL, TQQQ, UDOW)
• By the change metric
□ The change metric returns were five times (or larger) for the unleveraged securities (DIA, QQQ, SPY) than for the leverage securities (SPXL, TQQQ, UDOW)
□ As with the cagr metric, this outcome was driven by the fact that the duration in years was much greater for the unleveraged securities than for the leveraged securities
Implementing the seasonal_include_model
The seasonal_include_model computes an indicator for each month for each ticker symbol. There can be up to twelve months of open-high-low-close-volume observations in a year, and there are six
tickers tracked in this tip. As a result, there are 72 seasonal include indicator values – one for each of the twelve months for each of the six tickers.
• The number of years of data for a ticker symbol is based on the number of years and months of data in the symbol_date table for a ticker symbol
□ The number of years depends on the difference between the number of years in the table for a ticker from its initial public offering through the last date for which data are collected
(November 2022). The initial public offering date varies depending on when the security for a ticker was initially offered for sale to the public
□ The number of months depends on the set of months per year. For most years, this is twelve months
☆ However, for the last year for the symbol_date table for this tip, there are just 11 months
☆ Additionally, the first month can have twelve or fewer months depending on the month for the initial public offering year
• The seasonal factor for a month in a year for a ticker depends on the close value for the last trading day in a month compared to the close value of the initial trading date for a security during
a month
□ If the close price for the last trading day during a month is greater than the close value for the initial trading day, then the close price increased during the month
□ Otherwise, the close price did not increase during the month
□ If there are twelve years of data for a ticker in the symbol_date table and the ending close price exceeds the beginning close price in six of those twelve years for a month, then the
underlying seasonal factor for a ticker during a month is .5
• The underlying seasonal factors for a ticker during a month can vary by ticker. This is because seasonal factors often change depending on what is being assessed. Snowfall is more common during
winter months than summer months. Also, snowfall is more common for locales near the north and south poles than for locales near the equator
• Within the seasonal_include_model for this tip, seasonal include factors are computed based on two criteria
□ First, the percent of months during which the last close price in a month is greater than the initial close price in a month
□ Second, whether the seasonal factor for a month is greater than the median close price across all the months for a ticker
☆ If the seasonal factor for a month for a ticker is greater than the median seasonal factor for a ticker, then its include factor is 1
☆ Else, its include factor is 0
The first script for the seasonal_include_model computes the monthly percent up for each month for each ticker. These values are stored in the #temp_seaonal_factors table. The script creates the
table with twelve rows per ticker for each ticker in this tip. The data for each of the six tickers are added to the table sequentially.
• The data for the first ticker initially creates and populates the #temp_seaonal_factors table with a select into statement
• The data for the remaining five tickers are added to the #temp_seaonal_factors table with insert into statements
• The last select statement in the script excerpt below displays the seasonal factors in the monthly percent up column of its results set
-- compute monthly seasonality factors (monthly percent up) for a ticker
-- from symbol_date for first ticker
drop table if exists #temp_seasonal_factors
declare @symbol nvarchar(10) = 'SPY'
,cast(((cast(sum(increase_in_month) as dec(5,2))/(count(increase_in_month)))*100) as dec(5,2)) [monthly percent up]
into #temp_seasonal_factors
-- first-level query
-- first and last close by year, month for @symbol ticker
select distinct
,first_value([close]) OVER (partition by year, month order by month) first_close
,last_value([close]) OVER (partition by year, month order by month) last_close
when first_value([close]) OVER (partition by year, month order by year, month) <
last_value([close]) OVER (partition by year, month order by year, month)
then 1
else 0
end increase_in_month
-- innermost query
-- daily close values for @symbol ticker during year and month
,year(date) year
,month(date) month
,cast(datename(month, date) as nchar(3)) month_abr
,[close] [close]
from DataScience.dbo.symbol_date
where Symbol = @symbol
) for_first_and_last_monthly_closes
group by symbol,month
-- repeat for other 5 tickers
set @symbol = 'SPXL'
insert into #temp_seasonal_factors
,cast(((cast(sum(increase_in_month) as dec(5,2))/(count(increase_in_month)))*100) as dec(5,2)) [monthly percent up]
-- first-level query
-- first and last close by year, month for @symbol ticker
select distinct
,first_value([close]) OVER (partition by year, month order by month) first_close
,last_value([close]) OVER (partition by year, month order by month) last_close
when first_value([close]) OVER (partition by year, month order by year, month) <
last_value([close]) OVER (partition by year, month order by year, month)
then 1
else 0
end increase_in_month
-- innermost query
-- daily close values for @symbol ticker during year and month
,year(date) year
, month(date) month
,cast(datename(month, date) as nchar(3)) month_abr
,[close] [close]
from DataScience.dbo.symbol_date
where Symbol = @symbol
) for_first_and_last_monthly_closes
group by symbol,month
set @symbol = 'QQQ'
insert into #temp_seasonal_factors
,cast(((cast(sum(increase_in_month) as dec(5,2))/(count(increase_in_month)))*100) as dec(5,2)) [monthly percent up]
-- first-level query
-- first and last close by year, month for @symbol ticker
select distinct
,first_value([close]) OVER (partition by year, month order by month) first_close
,last_value([close]) OVER (partition by year, month order by month) last_close
when first_value([close]) OVER (partition by year, month order by year, month) <
last_value([close]) OVER (partition by year, month order by year, month)
then 1
else 0
end increase_in_month
-- innermost query
-- daily close values for @symbol ticker during year and month
,year(date) year
, month(date) month
,cast(datename(month, date) as nchar(3)) month_abr
,[close] [close]
from DataScience.dbo.symbol_date
where Symbol = @symbol
) for_first_and_last_monthly_closes
group by symbol,month
set @symbol = 'TQQQ'
insert into #temp_seasonal_factors
,cast(((cast(sum(increase_in_month) as dec(5,2))/(count(increase_in_month)))*100) as dec(5,2)) [monthly percent up]
-- first-level query
-- first and last close by year, month for @symbol ticker
select distinct
,first_value([close]) OVER (partition by year, month order by month) first_close
,last_value([close]) OVER (partition by year, month order by month) last_close
when first_value([close]) OVER (partition by year, month order by year, month) <
last_value([close]) OVER (partition by year, month order by year, month)
then 1
else 0
end increase_in_month
-- innermost query
-- daily close values for @symbol ticker during year and month
,year(date) year
, month(date) month
,cast(datename(month, date) as nchar(3)) month_abr
,[close] [close]
from DataScience.dbo.symbol_date
where Symbol = @symbol
) for_first_and_last_monthly_closes
group by symbol,month
set @symbol = 'DIA'
insert into #temp_seasonal_factors
,cast(((cast(sum(increase_in_month) as dec(5,2))/(count(increase_in_month)))*100) as dec(5,2)) [monthly percent up]
-- first-level query
-- first and last close by year, month for @symbol ticker
select distinct
,first_value([close]) OVER (partition by year, month order by month) first_close
,last_value([close]) OVER (partition by year, month order by month) last_close
when first_value([close]) OVER (partition by year, month order by year, month) <
last_value([close]) OVER (partition by year, month order by year, month)
then 1
else 0
end increase_in_month
-- innermost query
-- daily close values for @symbol ticker during year and month
,year(date) year
, month(date) month
,cast(datename(month, date) as nchar(3)) month_abr
,[close] [close]
from DataScience.dbo.symbol_date
where Symbol = @symbol
) for_first_and_last_monthly_closes
group by symbol,month
set @symbol = 'UDOW'
insert into #temp_seasonal_factors
,cast(((cast(sum(increase_in_month) as dec(5,2))/(count(increase_in_month)))*100)
as dec(5,2)) [monthly percent up]
-- first-level query
-- first and last close by year, month for @symbol ticker
select distinct
,first_value([close]) OVER (partition by year, month order by month) first_close
,last_value([close]) OVER (partition by year, month order by month) last_close
when first_value([close]) OVER (partition by year, month order by year, month) <
last_value([close]) OVER (partition by year, month order by year, month)
then 1
else 0
end increase_in_month
-- innermost query
-- daily close values for @symbol ticker during year and month
,year(date) year
, month(date) month
,cast(datename(month, date) as nchar(3)) month_abr
,[close] [close]
from DataScience.dbo.symbol_date
where Symbol = @symbol
) for_first_and_last_monthly_closes
group by symbol,month
select * from #temp_seasonal_factors
The next two screenshots show the first and last twelve rows from the #temp_seasonal_factors table. The first twelve rows are for the SPY ticker symbol, and the last twelve rows are for the UDOW
ticker symbol. The intermediate rows are for the remaining four ticker symbols.
• The next code excerpt for the seasonal_include_model focuses on
• An approach for adapting a stored procedure (compute_median_by_category) originally introduced and described in the "Two stored procedures for computing medians" section of a prior tip titled "
T-SQL Starter Statistics Package for SQL Server".
• The stored procedure was originally designed to compute medians for values within categories. For this tip,
□ A category corresponds to a ticker symbol
□ The values within a category correspond to the monthly percent up column values from the #temp_seasonal_factors table; this table is created and populated in the preceding script excerpt
• The stored procedure processes data from the ##table_for_median_by_category table; this table is created and populated by the preceding script
• After creating the stored procedure and populating ##table_for_median_by_category from the #temp_seasonal_factors table, the code excerpt invokes the compute_median_by_category stored procedure
and saves its results set in the @tmpTable table variable
• Next, the code joins the ##table_for_median_by_category table to a derived table based on the @tmpTable table variable; the derived table name is my_table_variable. The results set from the join
populates ##table_for_median_by_category_with_seasonal_include_indicator
• In the process of joining the two tables, the code adds a computed column named seasonal_include_idicator. The computed column has
□ A value of 1 when the column_for_median column value from the ##table_for_median_by_category table is greater than median_by_category column value from the @tmpTable table variable
□ A value of 0 otherwise
drop procedure if exists dbo.compute_median_by_category
create procedure compute_median_by_category
set nocount on;
-- compute median by gc_dc_symbol
-- distinct in select statement shows just one median per symbol
distinct category
within group (order by column_for_median)
over(partition by category) median_by_category
from ##table_for_median_by_category
) for_median_by_category
) for_median_by_category
order by category
-- create and populate ##table_for_median_by_category
-- from #temp_seasonal_factors
-- for processing by compute_median_by_category stored procedure
drop table if exists ##table_for_median_by_category
select symbol [category], [monthly percent up] [column_for_median]
into ##table_for_median_by_category
from #temp_seasonal_factors
order by #temp_seasonal_factors.month
-- optionally display ##table_for_median_by_category
select * from ##table_for_median_by_category
-- invoke compute_median_by_category stored procedure
-- display category and median_by_category from @tmpTable
declare @tmpTable TABLE (category varchar(5), median_by_category real)
insert into @tmpTable
exec compute_median_by_category
select category, median_by_category from @tmpTable
drop table if exists ##table_for_median_by_category_with_seasonal_include_indicator
-- compute seasonal_include_indicator
when column_for_median > my_table_variable.median_by_category then 1
else 0
end seasonal_include_idicator
into ##table_for_median_by_category_with_seasonal_include_indicator
from ##table_for_median_by_category
(select category, median_by_category from @tmpTable) my_table_variable
on ##table_for_median_by_category.category = my_table_variable.category
Here are three results sets from the preceding script.
• The first pane shows the column values for the SPY ticker from the ##table_for_median_by_category table. There are five other sets of values in the full version of the results set for the first
• The second pane shows the six median values computed by the compute_median_by_category stored procedure and saved in the @tmpTable table variable
• The third pane shows the seasonal_include_indicator column values for the SPY ticker
Implementing the ensemble_model
The final component in an ensemble model is the one that brings all the other model components together into a single model. There are many potential approaches for implementing this final step.
• In this tip, there are just two AI model elements in the ensemble AI model.
□ The first element is the close_gt_ema_10_model
□ The second element is the seasonal_include_model
• This section describes an approach for combining the final results sets from each model element. The approach is to include results from the close_gt_ema_10_model that have start dates from
months with seasonal_include_indicator column values of 1.
□ Recall that seasonal_include_indicator column value of 1 reveals that the column_for_median column value is greater than the median_by_category column value
□ By combining rows from the close_gt_ema_10_model results sets with start dates having a seasonal_include_indicator value of 1
The first pair of queries focus on seasonal factors. Two fresh tables are created and populated.
• The first query adds an identity column, my_id, to the ##table_for_median_by_category_with_seasonal_include_indicator table. The table with the freshly added column has a name of ##
• The second query adds another identity column to the #temp_seasonal_factors. The table with the freshly added column has a name of #temp_seasonal_factors_with_my_id
• Each of these tables has 72 rows -- for 12 monthly dates for each of 6 tickers
-- my_id column values are for joining
-- ##table_for_median_by_category_with_seasonal_include_indicator_with_my_id with
-- #temp_seasonal_factors_with_my_id
-- to make month column values in same results set as one with seasonal_include_indicator values
drop table if exists ##table_for_median_by_category_with_seasonal_include_indicator_with_my_id
select identity(int,1,1) AS my_id, *
into ##table_for_median_by_category_with_seasonal_include_indicator_with_my_id
-- optionally display
select * from ##table_for_median_by_category_with_seasonal_include_indicator_with_my_id
drop table if exists #temp_seasonal_factors_with_my_id
select identity(int,1,1) AS my_id, *
into #temp_seasonal_factors_with_my_id
from #temp_seasonal_factors
-- optionally display #temp_seasonal_factors_with_my_id
select * from #temp_seasonal_factors_with_my_id
The next pair of select statements join the ##temp_with_close_and_close_lead_10_for_each_start_cycle_plus_duration_in_years table with the pair of results sets from the preceding script excerpt.
• The two select statements are identical, except for their where clauses
□ The where clause in the first select statement extracts rows from the joined results set just for rows with a seasonal_include_idicator value of 1. For the data in this tip, there are 658
rows in this results set
□ The where clause in the second select statement extracts rows from the joined results set just for rows with a seasonal_include_idicator value of 0. For the data in this tip, there are 639
rows in this results set
□ The sum of 658 and 639 is a control total equal to the total number of buy/sell cycles in the close_gt_ema_10_model (1297)
• The select statements for these two results sets are nested in a subquery named for_month_and_seasonal_include_indicator_by_category
• Recall that ##temp_with_close_and_close_lead_10_for_each_start_cycle_plus_duration_in_years is a global temp table created in the code for the close_gt_ema_10_model. The use of a global temp
table makes it possible for its contents to be accessed from any SQL file so long as the SQL file for creating the global temp table remains open
-- display ##temp_with_close_and_close_lead_10_for_each_start_cycle_plus_duration_in_years
-- for use with seasonal_include_model
-- from_close_gt_ema_10_model
-- joined to the seasonal_include_model
-- with seasonal_include_indicator = 1
from ##temp_with_close_and_close_lead_10_for_each_start_cycle_plus_duration_in_years
-- join tables with matching my_id columns to retrieve month column values
-- from #temp_seasonal_factors_with_my_id for use with
-- ##table_for_median_by_category_with_seasonal_include_indicator_with_my_id
from #temp_seasonal_factors_with_my_id
join ##table_for_median_by_category_with_seasonal_include_indicator_with_my_id
on #temp_seasonal_factors_with_my_id.my_id =
) for_month_and_seasonal_include_indicator_by_category
on ##temp_with_close_and_close_lead_10_for_each_start_cycle_plus_duration_in_years.symbol =
for_month_and_seasonal_include_indicator_by_category.category and
##temp_with_close_and_close_lead_10_for_each_start_cycle_plus_duration_in_years.month_number =
where seasonal_include_idicator = 1
-- display ##temp_with_close_and_close_lead_10_for_each_start_cycle_plus_duration_in_years
-- for use with seasonal_include_model
-- from_close_gt_ema_10_model
-- joined to the seasonal_include_model
-- with seasonal_include_indicator = 0
from ##temp_with_close_and_close_lead_10_for_each_start_cycle_plus_duration_in_years
-- join tables with matching my_id columns to retrieve month column values
-- from #temp_seasonal_factors_with_my_id for use with
-- ##table_for_median_by_category_with_seasonal_include_indicator_with_my_id
from #temp_seasonal_factors_with_my_id
join ##table_for_median_by_category_with_seasonal_include_indicator_with_my_id
on #temp_seasonal_factors_with_my_id.my_id =
) for_month_and_seasonal_include_indicator_by_category
on ##temp_with_close_and_close_lead_10_for_each_start_cycle_plus_duration_in_years.symbol =
for_month_and_seasonal_include_indicator_by_category.category and
##temp_with_close_and_close_lead_10_for_each_start_cycle_plus_duration_in_years.month_number =
where seasonal_include_idicator = 0
The next code excerpt creates and populates a temp table named ##temp_with_close_and_close_lead_10_for_each_start_cycle_plus_duration_in_years_with_seasonal_include_idicator_equals_1. This temp table
serves as the source dataset for the report of performance from the ensemble_model.
• The select into statement immediately after the drop table if exists statement creates and populates the ##
temp_with_close_and_close_lead_10_for_each_start_cycle_plus_duration_in_years_with_seasonal_include_idicator_equals_1 table
□ Five of the six select list items are from the ##temp_with_close_and_close_lead_10_for_each_start_cycle_plus_duration_in_years source table
□ The sixth and final select list item is from the for_month_and_seasonal_include_indicator_by_category subquery
□ The month_and_seasonal_include_indicator_by_category subquery contains selected columns from a join of the #temp_seasonal_factors_with_my_id table and the ##
table_for_median_by_category_with_seasonal_include_indicator_with_my_id table
• The where clause in the select into statement has a criterion of seasonal_include_idicator = 1
• The optional select statement at the end of the excerpt below can display the contents of ##
drop table if exists
-- create, populate, and optionally display
-- ##temp_with_close_and_close_lead_10_for_each_included_start_cycle_plus_duration_in_years
-- for use with seasonal_include_model
-- from_close_gt_ema_10_model
-- joined to the seasonal_include_model
-- with seasonal_include_indicator = 1
into ##temp_with_close_and_close_lead_10_for_each_start_cycle_plus_duration_in_years_with_seasonal_include_idicator_equals_1
from ##temp_with_close_and_close_lead_10_for_each_start_cycle_plus_duration_in_years
-- join tables with matching my_id columns to retrieve month column values
-- from #temp_seasonal_factors_with_my_id for use with
-- ##table_for_median_by_category_with_seasonal_include_indicator_with_my_id
from #temp_seasonal_factors_with_my_id
join ##table_for_median_by_category_with_seasonal_include_indicator_with_my_id
on #temp_seasonal_factors_with_my_id.my_id =
) for_month_and_seasonal_include_indicator_by_category
on ##temp_with_close_and_close_lead_10_for_each_start_cycle_plus_duration_in_years.symbol =
for_month_and_seasonal_include_indicator_by_category.category and
##temp_with_close_and_close_lead_10_for_each_start_cycle_plus_duration_in_years.month_number =
where seasonal_include_idicator = 1
-- optionally display next-to-final results set for seasonal_include model
select *
from ##temp_with_close_and_close_lead_10_for_each_start_cycle_plus_duration_in_years_with_seasonal_include_idicator_equals_1
The final script for the ensemble model appears below. The script excerpt for the ensemble_model computes summary values for each of the six ticker symbols tracked in this tip. There are six columns
in the summary results table (#temp_summary).
The layout and design of the summary values for the six tickers from the ensemble_model are the same as from the summary table for the close_gt_ema_10_model. However, the actual source data for the
summary values report is different for the ensemble_model versus the close_gt_ema_10_model. This is because of design feature differences between the two models.
• Both models start with the same set of buy/sell cycles. This is because both models have the same set of initial rules for identifying buy/sell cycles.
• However, for the data tracked in this tip, the ensemble_model discards about half of the initial buy/sell cycles. This is because the ensemble_model filters out about half the initial buy/sell
cycles by retaining just those that have start dates in months with a monthly percent up in close price that is greater than the median monthly percent up for a symbol (also called a category)
• The close_gt_ema_10_model retains all buy/sell cycles whether or not they start in months with an above median monthly percent up in their close values
• Aside from this seasonality issue, the ensemble_model is identical to the close_gt_ema_10_model
• In terms of the actual code within the script excerpts
□ The ensemble_model summary report pulls its underlying values from ##temp_with_close_and_close_lead_10_for_each_start_cycle_plus_duration_in_years_with_seasonal_include_idicator_equals_1,
which, in turn, derives its values from the ##temp_with_close_and_close_lead_10_for_each_start_cycle_plus_duration_in_years table
□ The close_gt_ema_10_model pulls its underlying values from #temp_with_close_and_close_lead_10_for_each_start_cycle_plus_duration_in_years
-- create and populate #temp_summary for ensemble_model
-- with seasonal_include_idicator = 1
drop table if exists #temp_summary
-- setup for populating #temp_summary
-- with data for DIA symbol
declare @symbol nvarchar(10) = 'DIA'
declare @first_date date =
(select min(date)
from ##temp_with_close_and_close_lead_10_for_each_start_cycle_plus_duration_in_years_with_seasonal_include_idicator_equals_1
where symbol = @SYMBOL)
declare @last_date date =
(select max(date)
from ##temp_with_close_and_close_lead_10_for_each_start_cycle_plus_duration_in_years_with_seasonal_include_idicator_equals_1
where symbol = @SYMBOL)
declare @first_close dec (19,4) =
(select [close]
from ##temp_with_close_and_close_lead_10_for_each_start_cycle_plus_duration_in_years_with_seasonal_include_idicator_equals_1
where symbol = @SYMBOL and
date = @first_date)
declare @last_close_lead_10 dec (19,4) =
(select close_lead_10
from ##temp_with_close_and_close_lead_10_for_each_start_cycle_plus_duration_in_years_with_seasonal_include_idicator_equals_1
where symbol = @SYMBOL and
date = @last_date)
declare @last_duration_in_years dec(19,4) =
select duration_in_years
from ##temp_with_close_and_close_lead_10_for_each_start_cycle_plus_duration_in_years_with_seasonal_include_idicator_equals_1
where symbol = @SYMBOL and
date = @last_date
-- populate #temp_summary for DIA
select @symbol [@symbol], @first_close [@first_close], @last_close_lead_10 [@last_close_lead_10]
,@last_duration_in_years [@last_duration_in_years]
, @last_close_lead_10 - @first_close [change]
,cast((power(( @last_close_lead_10/@first_close),(1.0/@last_duration_in_years))-1)*100 as dec(19,2)) cagr
into #temp_summary
-- setup to populate #temp_summary for QQQ
set @symbol = 'QQQ'
set @first_date =
(select min(date)
from ##temp_with_close_and_close_lead_10_for_each_start_cycle_plus_duration_in_years
where symbol = @SYMBOL)
set @last_date =
(select max(date)
from ##temp_with_close_and_close_lead_10_for_each_start_cycle_plus_duration_in_years
where symbol = @SYMBOL)
set @first_close =
(select [close]
from ##temp_with_close_and_close_lead_10_for_each_start_cycle_plus_duration_in_years
where symbol = @SYMBOL and
date = @first_date)
set @last_close_lead_10 =
(select close_lead_10
from ##temp_with_close_and_close_lead_10_for_each_start_cycle_plus_duration_in_years
where symbol = @SYMBOL and
date = @last_date)
set @last_duration_in_years =
select duration_in_years
from ##temp_with_close_and_close_lead_10_for_each_start_cycle_plus_duration_in_years
where symbol = @SYMBOL and
date = @last_date
-- populate #temp_summary for QQQ
insert into #temp_summary
select @symbol [@symbol], @first_close [@first_close], @last_close_lead_10 [@last_close_lead_10]
,@last_duration_in_years [@last_duration_in_years]
, @last_close_lead_10 - @first_close [change]
,cast((power(( @last_close_lead_10/@first_close),(1.0/@last_duration_in_years))-1)*100 as dec(19,2)) cagr
-- setup to populate #temp_summary for SPXL
set @symbol = 'SPXL'
set @first_date =
(select min(date)
from ##temp_with_close_and_close_lead_10_for_each_start_cycle_plus_duration_in_years
where symbol = @SYMBOL)
set @last_date =
(select max(date)
from ##temp_with_close_and_close_lead_10_for_each_start_cycle_plus_duration_in_years
where symbol = @SYMBOL)
set @first_close =
(select [close]
from ##temp_with_close_and_close_lead_10_for_each_start_cycle_plus_duration_in_years
where symbol = @SYMBOL and
date = @first_date)
set @last_close_lead_10 =
(select close_lead_10
from ##temp_with_close_and_close_lead_10_for_each_start_cycle_plus_duration_in_years
where symbol = @SYMBOL and
date = @last_date)
set @last_duration_in_years =
select duration_in_years
from ##temp_with_close_and_close_lead_10_for_each_start_cycle_plus_duration_in_years
where symbol = @SYMBOL and
date = @last_date
-- populate #temp_summary for SPXL
insert into #temp_summary
select @symbol [@symbol], @first_close [@first_close], @last_close_lead_10 [@last_close_lead_10]
,@last_duration_in_years [@last_duration_in_years]
, @last_close_lead_10 - @first_close [change]
,cast((power(( @last_close_lead_10/@first_close),(1.0/@last_duration_in_years))-1)*100 as dec(19,2)) cagr
-- setup to populate #temp_summary for SPY
set @symbol = 'SPY'
set @first_date =
(select min(date)
from ##temp_with_close_and_close_lead_10_for_each_start_cycle_plus_duration_in_years
where symbol = @SYMBOL)
set @last_date =
(select max(date)
from ##temp_with_close_and_close_lead_10_for_each_start_cycle_plus_duration_in_years
where symbol = @SYMBOL)
set @first_close =
(select [close]
from ##temp_with_close_and_close_lead_10_for_each_start_cycle_plus_duration_in_years
where symbol = @SYMBOL and
date = @first_date)
set @last_close_lead_10 =
(select close_lead_10
from ##temp_with_close_and_close_lead_10_for_each_start_cycle_plus_duration_in_years
where symbol = @SYMBOL and
date = @last_date)
set @last_duration_in_years =
select duration_in_years
from ##temp_with_close_and_close_lead_10_for_each_start_cycle_plus_duration_in_years
where symbol = @SYMBOL and
date = @last_date
-- populate #temp_summary for SPY
insert into #temp_summary
select @symbol [@symbol], @first_close [@first_close], @last_close_lead_10 [@last_close_lead_10]
,@last_duration_in_years [@last_duration_in_years]
, @last_close_lead_10 - @first_close [change]
,cast((power(( @last_close_lead_10/@first_close),(1.0/@last_duration_in_years))-1)*100 as dec(19,2)) cagr
-- setup to populate #temp_summary for TQQQ
set @symbol = 'TQQQ'
set @first_date =
(select min(date)
from ##temp_with_close_and_close_lead_10_for_each_start_cycle_plus_duration_in_years
where symbol = @SYMBOL)
set @last_date =
(select max(date)
from ##temp_with_close_and_close_lead_10_for_each_start_cycle_plus_duration_in_years
where symbol = @SYMBOL)
set @first_close =
(select [close]
from ##temp_with_close_and_close_lead_10_for_each_start_cycle_plus_duration_in_years
where symbol = @SYMBOL and
date = @first_date)
set @last_close_lead_10 =
(select close_lead_10
from ##temp_with_close_and_close_lead_10_for_each_start_cycle_plus_duration_in_years
where symbol = @SYMBOL and
date = @last_date)
set @last_duration_in_years =
select duration_in_years
from ##temp_with_close_and_close_lead_10_for_each_start_cycle_plus_duration_in_years
where symbol = @SYMBOL and
date = @last_date
-- populate #temp_summary for UDOW
insert into #temp_summary
select @symbol [@symbol], @first_close [@first_close], @last_close_lead_10 [@last_close_lead_10]
,@last_duration_in_years [@last_duration_in_years]
, @last_close_lead_10 - @first_close [change]
,cast((power(( @last_close_lead_10/@first_close),(1.0/@last_duration_in_years))-1)*100 as dec(19,2)) cagr
-- setup to populate #temp_summary for UDOW
set @symbol = 'UDOW'
set @first_date =
(select min(date)
from ##temp_with_close_and_close_lead_10_for_each_start_cycle_plus_duration_in_years
where symbol = @SYMBOL)
set @last_date =
(select max(date)
from ##temp_with_close_and_close_lead_10_for_each_start_cycle_plus_duration_in_years
where symbol = @SYMBOL)
set @first_close =
(select [close]
from ##temp_with_close_and_close_lead_10_for_each_start_cycle_plus_duration_in_years
where symbol = @SYMBOL and
date = @first_date)
set @last_close_lead_10 =
(select close_lead_10
from ##temp_with_close_and_close_lead_10_for_each_start_cycle_plus_duration_in_years
where symbol = @SYMBOL and
date = @last_date)
set @last_duration_in_years =
select duration_in_years
from ##temp_with_close_and_close_lead_10_for_each_start_cycle_plus_duration_in_years
where symbol = @SYMBOL and
date = @last_date
-- populate #temp_summary for UDOW
insert into #temp_summary
select @symbol [@symbol], @first_close [@first_close], @last_close_lead_10 [@last_close_lead_10]
,@last_duration_in_years [@last_duration_in_years]
, @last_close_lead_10 - @first_close [change]
,cast((power(( @last_close_lead_10/@first_close),(1.0/@last_duration_in_years))-1)*100 as dec(19,2)) cagr
-- display #temp_summary across all symbols
select * from #temp_summary
Here is the results set from the preceding script. There are two main summary metrics. The cagr returns the average annual growth rate for which an investment strategy is evaluated. The change value
reflects the difference between @last_close_lead_10 and @first_close.
• By the cagr metric
□ The TQQQ ticker gave the best return on invested capital
□ The DIA ticker gave the worst return on invested capital
□ The three tickers for leveraged ETFs gave returns that were more than three times as large as those for unleveraged ETFs. As you can see, the @last_duration_in_years column values are much
larger for the unleveraged securities (DIA, QQQ, SPY) than for the leveraged securities (SPXL, TQQQ, UDOW)
• By the change metric
□ Returns were five times (or larger) for the unleveraged securities (DIA, QQQ, SPY) than for the leveraged securities (SPXL, TQQQ, UDOW)
□ As with the cagr metric, this outcome is driven by the fact that the duration in years was much greater for the unleveraged securities than for the leveraged securities
□ The change metric column below has identical values for all ticker symbols except for the DIA ticker
☆ The change metric for the DIA ticker from the ensemble_model has a value of 237.3262
☆ The change metric for the DIA ticker from the close_gt_ema_10_model has a value of 248.0137; the summary report for the close_gt_ema_10_model appears at the end of the "Implementing the
close_gt_ema_10_model" section
☆ The reason the DIA change metric value is less for the ensemble_model summary report than for the close_gt_ema_10_model summary report is because the @first_close value is larger for the
ensemble model than for the close_gt_ema_10_model; the seasonality rules for the ensemble_model causes it to choose a larger @first_close value than does the close_gt_ema_10_model. The
larger @first_close value reduces the @last_close_lead_10 value by a greater amount for the ensemble_model than for the close_gt_ema_10_model
There is at least one important difference between the close_gt_ema_10 model versus the ensemble_model that is not indicated by a comparison of summary tables from the two models.
• There are about 660 buy/sell cycles selected by the ensemble_model
• In contrast, there are about 1300 buy/sell cycles selected by the close_gt_ema_10_model
• This means that the same amount of invested dollars for the ensemble_model may be able to return about twice the gain as from the close_gt_ema_10_model because it has about twice as many tries to
achieve an enhanced return
Next Steps
The next step after reading this tip is to decide if you want a hands-on experience with the techniques demonstrated in this tip. You can get the code you need for hands-on experience from the code
windows in the tip. However, if you want to run the code exactly as it is described in the tip, then you also need the symbol_date and yahoo_finance_ohlcv_values_with_symbol tables.
The source data and the T-SQL script for importing the source data to the symbol_date table is available from the download for a prior tip titled "SQL Server Data Mining for Leveraged Versus
Unleveraged ETFs as a Long-Term Investment". This prior tip also includes the source code for the symbol_date table’s primary key constraint.
Another approach is to adapt the code excerpts provided in this tip to data derived from your business. All you need for this approach is a dataset that documents some decisions and some source data
columns that are likely to serve as inputs for AI models about the decisions. In this tip, the decision is about when to buy and sell securities. However, the decision can be about any type of
decision, such as when to buy new materials and how much of them to buy for replenishing inventory items for a manufacturing or sales business. This approach removes the need to copy CSV files from a
prior tip and then load the contents of the CSV files into a SQL Server table. With this approach, all you need to do is copy one or more database objects from your production database to your newly
created ensemble AI model.
About the author
Rick Dobson is an author and an individual trader. He is also a SQL Server professional with decades of T-SQL experience that includes authoring books, running a national seminar practice, working
for businesses on finance and healthcare development projects, and serving as a regular contributor to MSSQLTips.com. He has been growing his Python skills for more than the past half decade --
especially for data visualization and ETL tasks with JSON and CSV files. His most recent professional passions include financial time series data and analyses, AI models, and statistics. He believes
the proper application of these skills can help traders and investors to make more profitable decisions.
This author pledges the content of this article is based on professional experience and not AI generated.
View all my tips
Article Last Updated: 2023-04-25 | {"url":"https://www.mssqltips.com/sqlservertip/7643/ensemble-ai-models-in-sql-server/","timestamp":"2024-11-04T10:22:52Z","content_type":"text/html","content_length":"148594","record_id":"<urn:uuid:3f97d2a9-c807-4528-ab30-46ee80a1209f>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00320.warc.gz"} |
large-scale problems Archives - Sam's Blogs
Floyd’s Algorithm: An Introduction and Overview
Floyd’s Algorithm, often called Floyd-Warshall Algorithm, is a classic computer science algorithm used to find the shortest paths between all pairs of vertices in a weighted, directed graph. Devised
by Robert Floyd in 1962, this algorithm falls under the category of dynamic programming.
1. Basics of Floyd’s Algorithm:
The main principle behind the Floyd-Warshall algorithm is fairly simple: For each pair of nodes, it repeatedly checks if a shorter path exists through an intermediate node.
– A graph with `n` vertices.
– A matrix `D` of size `n x n` where `D[i][j]` is the shortest distance from vertex `i` to vertex `j`.
The algorithm uses a triple nested loop, and for each combination of vertices `i`, `j`, and `k`, it checks if the path from `i` to `j` through `k` is shorter than the current known path from `i` to
`j`. If so, it updates the value of `D[i][j]`.
2. Pseudocode of Floyd’s Algorithm:
function floydsAlgorithm(D):
n = number of vertices in D
for k from 1 to n:
for i from 1 to n:
for j from 1 to n:
D[i][j] = min(D[i][j], D[i][k] + D[k][j])
3. Applications and Use Cases:
Floyd’s Algorithm finds a wide range of applications, including:
– Road networks: Determining the shortest path between any two cities.
– Telecommunication networks: Finding the least costly path for data transmission.
– Flight scheduling: To determine the shortest (or cheapest) route between two airports, possibly with layovers.
– Game development: For pathfinding and AI decision-making.
4. Advantages:
1. Simplicity: The algorithm is straightforward and can be easily implemented.
2. All-pair shortest paths: Unlike Dijkstra’s or Bellman-Ford, which find the shortest path from a single source, Floyd-Warshall finds the shortest paths between all pairs.
5. Limitations:
1. Time Complexity: With a time complexity of O(n3), it may not be the best choice for graphs with many nodes.
2. Space Complexity: Requires O(n2) space to store the distances between vertices.
6. Variations and Enhancements:
The basic Floyd-Warshall algorithm can be enhanced to reconstruct the actual path (sequence of vertices) between any two vertices, not just the shortest path’s length. This involves maintaining a
predecessor matrix alongside the distance matrix.
While not always the most efficient for large-scale problems, Floyd’s Algorithm remains an invaluable tool in the repertoire of computer scientists and engineers. Its simplicity, coupled with the
ability to handle negative edge weights (as long as there are no negative cycles), ensures its continued relevance in the field of graph theory and network optimization. | {"url":"https://samsblog.in/tag/large-scale-problems/","timestamp":"2024-11-04T21:14:00Z","content_type":"text/html","content_length":"160633","record_id":"<urn:uuid:71a3f088-2ba6-47b7-9170-d81fb96e45eb>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.16/warc/CC-MAIN-20241104194528-20241104224528-00615.warc.gz"} |
Mastering a Challenging Algebra Puzzle Without a Calculator
Written on
Chapter 1: Introduction to the Algebra Challenge
Welcome to the thirteenth installment in our series dedicated to tricky algebra challenges. In this collection, we explore the delightful world of algebra, featuring problems that cater to all skill
levels, from novice to expert.
Before diving into today's puzzle, let’s address the main point. This challenge is not your standard algebra problem; it can be solved quickly with a calculator. However, the twist is that you must
rely solely on algebra to find the solution. Essentially, while it may seem like an algebra problem, it is crafted to challenge your algebraic thinking.
So, how will you tackle this algebra conundrum?
Spoiler Alert
If you prefer to solve the problem independently, I suggest you stop reading now, as I will reveal the solution after this section. Once you've given it a try, feel free to continue and compare your
method with mine.
Section 1.1: Setting Up the Algebra Challenge
This algebra problem becomes remarkably straightforward when analyzed through an algebraic lens. By taking a closer look, we can express all the relevant numbers in terms of 100:
Recognizing Patterns in Algebra
After slightly rearranging the terms, we reach the following expression:
Do you see any recognizable algebraic patterns? If not, don't worry; clarity is just around the corner. To simplify our calculations and embrace the spirit of algebra, let’s substitute 100 with the
variable ‘x’:
By doing this, we can identify the following algebraic identity:
(a + b) * (a - b) = a² - b²
Applying this identity leads us to the following result:
With this step, we are on the brink of solving our problem.
The Solution to the Algebra Puzzle
Now, let's substitute 100 back in for ‘x’ in the expression we derived earlier:
Next, we can proceed with simple cross-multiplication:
And there you have it! We’ve successfully solved the problem using algebra!
Final Thoughts
The observant reader may note that introducing the variable ‘x’ was not strictly necessary to solve this problem. However, I chose to do so to highlight the underlying pattern for those who might
have missed it initially.
Additionally, this problem serves as a practical illustration of how algebra can be applied to relatable real-world scenarios. People frequently encounter multiplication involving large numbers
without a calculator, making this approach quite relevant.
I hope you found this engaging and straightforward algebra puzzle enjoyable. If such challenges intrigue you, stay tuned for more in this series. Thank you for reading!
Chapter 2: Engaging with Algebra on YouTube
To further enhance your understanding, check out these videos:
The first video, "A Tricky Algebra Problem with Exponents," provides additional insights into solving algebraic problems creatively.
The second video, "Can you solve this tricky math problem?" challenges you to apply your skills in a fun and engaging way.
If you would like to support future content, consider contributing on Patreon. | {"url":"https://livesdmo.com/mastering-algebra-puzzle.html","timestamp":"2024-11-05T06:30:18Z","content_type":"text/html","content_length":"12782","record_id":"<urn:uuid:9099fa68-00b9-44d0-b687-897b1312720e>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00024.warc.gz"} |
Module 40: Use the Rectangular Coordinate System
By the end of this section, you will be able to:
• Plot points in a rectangular coordinate system
• Verify solutions to an equation in two variables
• Complete a table of solutions to a linear equation
• Find solutions to a linear equation in two variables
Plot Points on a Rectangular Coordinate System
Just like maps use a grid system to identify locations, a grid system is used in algebra to show a relationship between two variables in a rectangular coordinate system. The rectangular coordinate
system is also called the xy-plane or the ‘coordinate plane.’
The horizontal number line is called the x-axis. The vertical number line is called the y-axis. The x-axis and the y-axis together form the rectangular coordinate system. These axes divide a plane
into four regions, called quadrants. The quadrants are identified by Roman numerals, beginning on the upper right and proceeding counterclockwise. See (Figure 1).
‘Quadrant’ has the root ‘quad,’ which means ‘four.’
Figure .1
In the rectangular coordinate system, every point is represented by an ordered pair. The first number in the ordered pair is the x-coordinate of the point, and the second number is the y-coordinate
of the point.
An ordered pair,
The first number is the x-coordinate.
The second number is the y-coordinate.
The phrase ‘ordered pair’ means the order is important. What is the ordered pair of the point where the axes cross? At that point both coordinates are zero, so its ordered pair is origin.
The point x-axis and y-axis intersect.
We use the coordinates to locate a point on the xy-plane. Let’s plot the point x-axis and lightly sketch a vertical line through y-axis and sketch a horizontal line through
Figure .2
Notice that the vertical line through
Plot each point in the rectangular coordinate system and identify the quadrant in which the point is located:
The first number of the coordinate pair is the x-coordinate, and the second number is the y-coordinate.
A. Since y-axis. Also, since x-axis. The point
B. Since y-axis. Also, since x-axis. The point
C. Since y-axis. Since x-axis. The point
D. Since y-axis. Since x-axis. The point
E. Since y-axis. Since x-axis. (It may be helpful to write
Plot each point in a rectangular coordinate system and identify the quadrant in which the point is located:
Show answer
Plot each point in a rectangular coordinate system and identify the quadrant in which the point is located:
Show answer
How do the signs affect the location of the points? You may have noticed some patterns as you graphed the points in the previous example.
For the point in (Figure 2) in Quadrant IV, what do you notice about the signs of the coordinates? What about the signs of the coordinates of points in the third quadrant? The second quadrant? The
first quadrant?
Can you tell just by looking at the coordinates in which quadrant the point
We can summarize sign patterns of the quadrants in this way.
What if one coordinate is zero as shown in (Figure 3)? Where is the point
Figure .3
The point y-axis and the point x-axis.
Points with a y-coordinate equal to 0 are on the x-axis, and have coordinates
Points with an x-coordinate equal to 0 are on the y-axis, and have coordinates
Plot each point:A
A. Since y-axis.
B. Since x-axis.
C. Since x-axis.
D. Since
E. Since y-axis.
Plot each point: A
Show answer
Plot each point: A
Show answer
In algebra, being able to identify the coordinates of a point shown on a graph is just as important as being able to plot points. To identify the x-coordinate of a point on a graph, read the number
on the x-axis directly above or below the point. To identify the y-coordinate of a point, read the number on the y-axis directly to the left or right of the point. Remember, when you write the
ordered pair use the correct order,
Name the ordered pair of each point shown in the rectangular coordinate system.
Point A is above x-axis, so the x-coordinate of the point is
• The point is to the left of 3 on the y-axis, so the y-coordinate of the point is 3.
• The coordinates of the point are
Point B is below x-axis, so the x-coordinate of the point is
• The point is to the left of y-axis, so the y-coordinate of the point is
• The coordinates of the point are
Point C is above 2 on the x-axis, so the x-coordinate of the point is 2
• The point is to the right of 4 on the y-axis, so the y-coordinate of the point is 4.
• The coordinates of the point are
Point D is below 4 on the x-axis, so the x-coordinate of the point is 4
• The point is to the right of y-axis, so the y-coordinate of the point is
• The coordinates of the point are
Point E is on the y-axis at
Point F is on the x-axis at
Name the ordered pair of each point shown in the rectangular coordinate system.
Show answer
Name the ordered pair of each point shown in the rectangular coordinate system.
Show answer
Verify Solutions to an Equation in Two Variables
Up to now, all the equations you have solved were equations with just one variable. In almost every case, when you solved the equation you got exactly one solution. The process of solving an equation
ended with a statement like
Here’s an example of an equation in one variable, and its one solution.
But equations can have more than one variable. Equations with two variables may be of the form linear equations in two variables.
An equation of the form in two variables.
Notice the word line in linear. Here is an example of a linear equation in two variables,
The equation linear equation. But it does not appear to be in the form
Add to both sides.
Use the Commutative Property to put it in
By rewriting standard form.
Standard Form of Linear Equation
A linear equation is in standard form when it is written
Most people prefer to have
Linear equations have infinitely many solutions. For every number that is substituted for solution to the linear equation and is represented by the ordered pair
Solution of a Linear Equation in Two Variables
An ordered pair solution of the linear equation x– and y-values of the ordered pair are substituted into the equation.
Determine which ordered pairs are solutions to the equation
Substitute the x- and y-values from each ordered pair into the equation and determine if the result is a true statement.
Which of the following ordered pairs are solutions to
Show answer
A, C
Which of the following ordered pairs are solutions to the equation
Show answer
B, C
Which of the following ordered pairs are solutions to the equation
Substitute the x– and y-values from each ordered pair into the equation and determine if it results in a true statement.
Which of the following ordered pairs are solutions to the equation
Show answer
Which of the following ordered pairs are solutions to the equation
Show answer
A, B
Complete a Table of Solutions to a Linear Equation in Two Variables
In the examples above, we substituted the x– and y-values of a given ordered pair to determine whether or not it was a solution to a linear equation. But how do you find the ordered pairs if they are
not given? It’s easier than you might think—you can just pick a value for
We’ll start by looking at the solutions to the equation (Example 5). We can summarize this information in a table of solutions, as shown in (Table 1).
To find a third solution, we’ll let
The ordered pair (Table 2).
We can find more solutions to the equation by substituting in any value of
Complete the table to find three solutions to the equation
The results are summarized in the table below.
Complete the table to find three solutions to this equation:
Show answer
Complete the table to find three solutions to this equation:
Show answer
Complete the table to find three solutions to the equation
Substitute the given value into the equation
The results are summarized in the table below.
Complete the table to find three solutions to this equation:
Complete the table to find three solutions to this equation:
Show answer
Find Solutions to a Linear Equation
To find a solution to a linear equation, you really can pick any number you want to substitute into the equation for
When the equation is in y-form, with the y by itself on one side of the equation, it is usually easier to choose values of
Find three solutions to the equation
We can substitute any value we want for y-form, it will be easier to substitute in values of
Substitute the value into the equation.
Write the ordered pair.
Check. (0, 2) (1, −1) (−1, 5)
Find three solutions to this equation:
Show answer
Answers will vary.
Find three solutions to this equation:
Show answer
Answers will vary
We have seen how using zero as one value of
Find three solutions to the equation
We can substitute any value we want for
Substitute the value into the equation.
Write the ordered pair. (0, 3) (2, 0)
Find three solutions to the equation
Show answer
Answers will vary.
Find three solutions to the equation
Show answer
Answers will vary.
linear equation
A linear equation is of the form
ordered pair
An ordered pair
The point x-axis and y-axis intersect.
The x-axis and the y-axis divide a plane into four regions, called quadrants.
rectangular coordinate system
A grid system is used in algebra to show a relationship between two variables; also called the xy-plane or the ‘coordinate plane.’
The first number in an ordered pair
The second number in an ordered pair
Practice Exercises
Plot Points in a Rectangular Coordinate System
In the following exercises, plot each point in a rectangular coordinate system and identify the quadrant in which the point is located.
1.A 2. A
B B
C C
D D
E E
3. A 4. A
B B
C C
D D
E E
In the following exercises, plot each point in a rectangular coordinate system.
5. A 6. A
B B
C C
D D
E E
7. A 8. A
B B
C C
D D
E E
In the following exercises, name the ordered pair of each point shown in the rectangular coordinate system.
Verify Solutions to an Equation in Two Variables
In the following exercises, which ordered pairs are solutions to the given equations?
13. 14.
A A
B B
C C
15. 16.
A A
B B
C C
17. 18.
A A
B B
C C
19. 20.
A A
B B
C C
Complete a Table of Solutions to a Linear Equation
In the following exercises, complete the table to find solutions to each linear equation.
21. 22.
23. 24.
25. 26.
27. 28.
29. 30.
31. 32.
Find Solutions to a Linear Equation
In the following exercises, find three solutions to each linear equation.
33. 34.
35. 36.
37. 38.
39. 40.
41. 42.
43. 44.
45. 46.
47. 48.
Everyday Math
49. Weight of a baby. Mackenzie recorded her baby’s weight every two months. The baby’s age, in 50. Weight of a child. Latresha recorded her son’s height and weight every year. His height, in
months, and weight, in pounds, are listed in the table below, and shown as an ordered pair in the inches, and weight, in pounds, are listed in the table below, and shown as an ordered pair in the
third column. third column.
a) Plot the points on a coordinate plane. a) Plot the points on a coordinate plane.
b) Why is only Quadrant I needed? b) Why is only Quadrant I needed?
Age Weight Height Weight
0 7 (0, 7) 28 22 (28, 22)
2 11 (2, 11) 31 27 (31, 27)
4 15 (4, 15) 33 33 (33, 33)
6 16 (6, 16) 37 35 (37, 35)
8 19 (8, 19) 40 41 (40, 41)
10 20 (10, 20) 42 45 (42, 45)
12 21 (12, 21)
Writing Exercises
51. Explain in words how you plot the point 52. How do you determine if an ordered pair is a solution to a given equation?
53. Is the point x-axis or y-axis? How do you know? 54. Is the point x-axis or y-axis? How do you know?
1. 3.
5. 7.
9. A: 11. A:
13. A, B 15. A, C
17. B, C 19. A, B
21. 23.
25. 25.
27. 29.
31. 33. Answers will vary.
35. Answers will vary. 37. Answers will vary.
39. Answers will vary. 41. Answers will vary.
43. Answers will vary. 45. Answers will vary.
47. Answers will vary. a)
b) Age and weight are only positive.
51. Answers will vary. 53. Answers will vary.
This chapter has been adapted from “Use the Rectangular Coordinate System” in Elementary Algebra (OpenStax) by Lynn Marecek and MaryAnne Anthony-Smith, which is under a CC BY 4.0 Licence. Adapted by
Izabela Mazur. See the Copyright page for more information. | {"url":"https://spscc.pressbooks.pub/techmath/chapter/use-the-rectangular-coordinate-system-2/","timestamp":"2024-11-10T09:12:04Z","content_type":"text/html","content_length":"445394","record_id":"<urn:uuid:ff2bcbf6-ae11-4cff-9779-5f4da2804def>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00029.warc.gz"} |
Bloom Filters and Cuckoo Filters for Cache Summarization
Disclaimer: This is not a general comparison between Bloom filters and Cuckoo filters. This blog post summarizes some of the experiments we conducted to decide whether or not we should replace our
implementation of Counting Bloom filters with Cuckoo filters, for a specific use case.
Nodes on Fleek Network currently use Counting Bloom filters to summarize their cached content. These cache summaries are exchanged with other nodes in order to facilitate content routing.
If a particular node does not store a requested piece of content, it can use the Bloom filters that it received from its peers to check if a peer stores the requested content.
We are using Counting Bloom filters rather than regular Bloom filters because we need to be able to remove elements from the filter to support cache eviction.
Bloom Filters
A Bloom filter is a space-efficient probabilistic data structure that can be used to perform approximate set membership queries. The answer to an approximate set membership query is not no or yes,
but rather no or probably. This probably is quantified with the false positive rate.
One of the convenient features of Bloom filters is that they can be configured to have a specific false positive rate. Of course, there is a tradeoff here; the lower the false positive rate, the
larger the memory footprint. Bloom filters support two operations: insert and contains.
A Bloom filter is represented by an array of m bits together with k independent hash functions. To insert an element into the filter, it is hashed with each of the k hash functions. The resulting
hashes are interpreted as integers (modulo m) to obtain k array positions. The bits at these positions are then set to 1 (if there aren't already 1).
To check whether or not an element is contained in the filter, the element is hashed k times with the different hash functions.
If all bits at the resulting array positions are 1, the element is assumed to be present. If any of the k bits are zero, we can be certain that the queried element is not present in the set.
However, even if all bits are 1, it might still be the case that the bits were set by a combination of other elements. This is where the aforementioned false positive rate comes into play.
Since we also need a remove operation for our use case, we have been using Counting Bloom filters, a variant of Bloom filters. Counting Bloom filters retain most of the properties that regular Bloom
filters have. The remove operation comes at the cost of an increased memory footprint.
Each position in the array is no longer a single bit but a group of bits representing a counter. Whenever an element is inserted into the filter, the counters for all k positions are incremented by
1. To remove an element, we decrement the counters.
Cuckoo Filters
Bloom filters are the most known members of a class of data structures called Approximate Membership Query Filters (AMQ Filters). A relatively recent addition to this class is the Cuckoo filter [1].
Cuckoo filters share many similarities with Bloom filters, especially Counting Bloom filters.
They are space-efficient and can be used for approximate set membership queries. Cuckoo filters also support the operations insert, contains, and remove, and have configurable false positive rates.
Cuckoo filters are based on Cuckoo hash tables [2] and leverage an optimization called partial-key cuckoo hashing. A basic Cuckoo hash table consists of an array of buckets. We determine two
candidate buckets for each element using two different hash functions, h1 and h2.
The contains operation will check if either bucket contains the element. For insertion, if either bucket is empty, the element will be inserted into the empty bucket.
If neither bucket is empty, one of the buckets is selected, and the existing element is removed and inserted into its alternate location. This may trigger another relocation if the alternate location
is not empty.
Although the insertion operation may perform a sequence of relocations, the amortized runtime is O(1).
Most implementations of Cuckoo hash tables and, consequently, Cuckoo filters will use buckets that can hold multiple elements, as proposed in [3].
For Cuckoo filters, the hash table size is reduced by only storing fingerprints - a bit string calculated from an element's hash - rather than key-value pairs.
The fingerprint size is derived from the desired false positive rate.
A problem that arises is that, to relocate existing fingerprints using the Cuckoo hashing approach described above, we need the original hash from which the fingerprint was derived.
Of course, we could store this hash somewhere, but the whole point of using fingerprints is to reduce the memory footprint of the filter.
The solution to this predicament is the aforementioned partial-key cuckoo hashing, a technique for determining an element's alternate location using only its fingerprint.
For a given element x, the two candidate buckets are computed as follows:
An important property of this technique is that h1(x) can also be computed from h2(x) and the fingerprint.
As this post mentioned, we are not aiming for a general comparison of Counting Bloom and Cuckoo filters.
Instead, we want to determine which filter suits our specific use case better. The two main properties we are looking for are space efficiency and lookup performance.
Space efficiency is important because nodes frequently update their cache and have to communicate these changes with their peers. These messages should take up as little bandwidth as possible.
Lookup speed is also important because Fleek Network aims to serve user requests as quickly as possible. Checking whether a peer has some content stored in their cache summary should not be a
Experimental Setup
We are using our own Counting Bloom filter implementation and this Cuckoo filter implementation in Rust (the original implementation is in C++). All experiments were performed on a Linux machine with
16 GB RAM and an Intel Core i7 (10th Gen). Whenever the experiment is probabilistic, we repeat the experiment 20 times and report the mean and standard deviation.
Memory Footprint
For both Counting Bloom filters and Cuckoo filters, the memory footprint is determined by two factors: the filter's capacity and the desired false positive rate. In the first experiment, we examine
the impact that these factors have on the memory footprint.
To this end, we fix the false positive rate and initialize the filters with capacities ranging from 100K to 1M. The result is shown in Fig. 1. The size of Bloom filters scales linearly with the
capacity. Cuckoo filters are more space-efficient. This result is consistent with the experiments reported in [1].
Figure 1: We fix the false positive rate and initialize the filters with capacities ranging from 100K to 1M. The y-axis shows the size of the filters in Megabytes.
Next, we fix the capacity and initialize the filters with false positive rates ranging from 0.0001 to 0.5. Fig. 2 shows that Cuckoo filters are more space-efficient.
The gap between Counting Bloom filters and Cuckoo filters grows as the false positive rate decreases. This is also consistent with experiments in [1].
Figure 2: We fix the capacity and initialize the filters with false positive rates ranging from 0.0001 to 0.5. The y-axis shows the size of the filters in Megabytes.
Lookup Performance
We first add elements to both filters until the capacity is reached. We then measure the lookup performance for different ratios of positive and negative lookups.
A positive lookup is for an existing element, and a negative lookup is for an element not contained in the filter. We perform 100K lookups for each ratio and report the average lookup duration and
standard deviation.
Fig. 3 shows the results. Bloom filters perform slightly better on average than Cuckoo filters. This result is inconsistent with [1], where Cuckoo filters were reported to have a better lookup
performance than Bloom filters. It should be noted here that the authors in [1] use the original C++ Cuckoo filter implementation and their own unreleased Bloom filter implementation. In contrast, we
use a Rust Cuckoo filter implementation and our Bloom filter implementation. We cannot easily determine the reason for this discrepancy.
However, the performance difference is negligible.
Figure 3: Lookup performance for different ratios of positive and negative lookups. For example, ratio 0.25 indicates that 25% of lookups are positive and 75% are negative.
The shaded region indicates the standard deviation.
Insertion Performance
Less critical than lookup performance but still important for our purposes is insertion performance. We measure how the insertion performance varies for different occupancy levels. Fig. 4 shows the
results. The insertion performance is constant across all levels of occupancy for Bloom filters.
For Cuckoo filters, the performance decreases as the filter becomes fuller because more relocations are required. In Fig. 4, the performance for Bloom filters is not constant. It quickly increases
and then remains constant. This can be explained by CPU caching.
Figure 4: Insertion performance for different occupancy levels. The shaded region indicates the standard deviation.
Capacity and Scaling
We have mentioned the capacity of a filter several times now. An interesting case is what happens when a filter's capacity is exceeded. Bloom filters and Cuckoo filters behave differently in this
For Bloom filters, the insertion operation always succeeds. However, the false positive rate will rapidly increase as we exceed the filter's capacity. While Bloom filters fail silently, Cuckoo
filters are more explicit. Most implementations have a maximum number of relocations that will be performed for an insertion. The insertion operation will return an error if more relocations are
For both filters, we can avoid this problem by simply initializing the filter with a sufficiently large capacity. However, this will increase the memory footprint of the filter.
Furthermore, it is difficult to predict how many elements a node on Fleek Network will cache. It is also likely that the number of cached elements will greatly vary for different nodes.
Fortunately, a variant of Bloom filters called Scalable Bloom Filters [4] can adapt dynamically to the number of elements stored while guaranteeing a maximum false positive rate.
The proposed technique is also applicable to Cuckoo filters.
Other Filters
While we only looked at Bloom filters and Cuckoo filters, there are other AMQ filters that we want to mention here briefly:
• Quotient filters [5, 6]: Compact hash tables that support insertion, lookup, and deletion. Less space-efficient than Bloom filters and Cuckoo filters.
• XOR filters [7]: More space-efficient than Bloom filters and Cuckoo filters. However, they are static, meaning the filter has to be rebuilt if additional elements are added.
We examined whether Counting Bloom filters or Cuckoo filters are more suitable for summarizing caches on Fleek Network. Cuckoo filters are more space-efficient, especially for lower false positive
rates. Bloom filters have a slightly better insertion and lookup performance for the implementations we tested.
Both filters can be adapted to grow and shrink in size dynamically. Since the difference in insertion and lookup performance is negligible while Cuckoo filters are significantly more space-efficient,
we favor Cuckoo filters for our use case.
[1] Bin Fan, Dave G. Andersen, Michael Kaminsky, and Michael D. Mitzenmacher. Cuckoo Filter: Practically Better Than Bloom. In Proceedings of the 10th ACM International Conference on emerging
Networking Experiments and Technologies (CoNEXT 14). Association for Computing Machinery, New York, NY, USA, pp. 75-88, 2014.
[2] Rasmus Pagha and Flemming Friche Rodler. Cuckoo hashing. Journal of Algorithms, 51(2), pp. 122-144, 2004.
[3] Martin Dietzfelbinger and Christoph Weidling. Balanced Allocation and Dictionaries with Tightly Packed Constant Size Bins. Theoretical Computer Science, 380(1), pp. 47-68, 2007.
[4] Paulo S. Almeida, Carlos Baquero, Nuno Preguiça, and David Hutchison. Scalable Bloom Filters. Information Processing Letters, 101(6), pp. 255-261, 2007.
[5] John G. Cleary. Compact hash tables using bidirectional linear probing. IEEE Transactions on Computers. 33(9), pp. 828-834, 1984.
[6] Anna Pagh, Rasmus Pagh, and S. Srinivasa Rao. An optimal Bloom filter replacement. Proceedings of the Sixteenth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 823-829, 2005.
[7] Thomas Mueller Graf and Daniel Lemire. Xor Filters: Faster and Smaller Than Bloom and Cuckoo Filters. ACM Journal of Experimental Algorithmics. 25, pp. 1-16, 2020.
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History Of Neural Networks | Neural Network & Fuzzy Systems | Books | Skedbooks
Neural Network & Fuzzy Systems
History Of Neural Networks
Introduction:-The history of neural networks begins in the early 1940’s and thus nearly simultaneously with the history of programmable electronic computers. The youth of this field of research, as
with the field of computer science itself, can be easily recognized due to the fact that many of the cited persons are still with us.
The beginning :-As soon as 1943 Warren McCullochand Walter Pitts introduced modelsof neurological networks, recreatedthreshold switches based on neuronsand showed that even simplenetworks of this
kind are able tocalculate nearly any logic or arithmeticfunction. Further precursors ("electronic brains")were developed, among others supported by KonradZuse, who was tired of calculating ballistic
trajectories by hand.
1947:Walter Pitts and Warren Mc-Culloch indicated a practical field of application (which was not mentionedin their work from 1943),namely the recognition of spacial patterns by neural networks .
1949: Donald O. Hebb formulated the classical hebbian rule [Heb49] which represents in its more generalized form the basis of nearly all neural learning procedures. The rule implies that the
connection between two neurons is strengthened when both neurons are active at the same time. This change in strength is proportional to the product of the two activities.Hebb could postulate this
rule,but due to the absence of neurologicalresearch he was not able to verify it.
Golden age
1951: For his dissertation Marvin Minsky developed the neurocomputer Snark, which has already been capable to adjust its weights 3 automatically.But it has never been practically implemented, since
it is capable to busily calculate, but nobody really knows what it calculates.
1956: Well-known scientists and ambitious students met at the DartmouthSummer Research Project and discussed, to put it crudely, how to simulate a brain. Differences between top-down and bottom-up
research developed.
1957-1958: At the MIT, Frank Rosenblatt, Charles Wightman and their coworkers developed the first successful neurocomputer, the Mark I perceptron, which was capable to development accelerates
recognize simple numerics by means of a 20 × 20 pixel image sensor and electromechanically worked with 512 motor driven potentiometers – each potentiometer representing one variable weight.
1959: Frank Rosenblatt described different versions of the perceptron, formulated and verified his perceptron convergence theorem. He described neuron layers mimicking the retina, threshold switches,
and a learning rule adjusting the connecting weights.
1960: Bernard Widrow and Marcian E. Hoff introduced the ADALINE (ADAptive LInear NEuron) ,a fast and precise adaptive learning system being the first widely commercially used neural network: It could
be found in nearly every analog telephone for realtime adaptive echo filtering and was trained by menas of the Widrow-Hoff first spread use rule or delta rule. At that time Hoff, later co-founder of
Intel Corporation, was a PhD student of Widrow, who himself is known as the inventor One advantage the delta rule had over the original perceptron learning algorithm was its adaptivity.Disadvantage:
missapplication led to infinitesimal small steps close to the target.
1961:Karl Steinbuch introduced technical realizations of associative memory, which can be seen as predecessors of today’s neural associative memories.Additionally, he described concepts for neural
techniques and analyzed their possibilities and limits.
1965:It was assumed that the basic principles of self-learning and therefore, generally speaking, "intelligent" systems had already been discovered. Today this assumption seems to be an exorbitant
overestimation, but at that time it provided for high popularity and sufficient research funds.
1969: Marvin Minsky and Seymour Papert published a precise mathe-matical analysis of the perceptron[MP69] to show that the perceptron model was not capable of representing many important problems
(keywords: XOR problem and linear separability),and so put an end to overestimation, popularity and research funds. The research funds were stopped implication that more powerful models would show
exactly the same problems and the forecast that the entire field would be a research dead end resulted in a nearly complete decline in research funds for the next 15 years– no matter how incorrect
these forecasts were from today’s point of view.
1972: Teuvo Kohonen introduced a model of the linear associator, a model of an associative memory.In the same year, such a model was presented independently and from a neurophysiologist’s point of
view by James A. Anderson.
1973: Christoph von der Malsburg used a neuron model that was nonlinear and biologically more motivated.
1974: For his dissertation in Harvard Paul Werbos developed a learning procedure called backpropagation of error ,but it was not until one decade later that this procedure reached today’s importance.
Backprop developed 1976-1980 and thereafter: Stephen Grossberg presented many papers in which numerous neural models are analyzed mathematically. Furthermore, he dedicated himself to the problem of
keeping a neural network capable of learning without destroying already learned associations. Under cooperation of Gail Carpenter this led to models of adaptive resonance theory (ART).
1982: Teuvo Kohonen described the self-organizing feature maps – also known as Kohonen maps. He was looking for the mechanisms involving self-organization in the brain (He knew that the information
about the creation of a being is stored in the genome, which has, however, not enough memory for a structure like the brain. As a consequence, the brain has to organize and create itself for the most
1983: Fukushima, Miyake and Ito introduced the neural model of the Neocognitron which could recognize handwritten characters and was an extension of the Cognitron network already developed in 1975.
1985:John Hopfield published an article describing a way of finding acceptable solutions for the Travelling Salesman problem by using Hopfield nets.
1986:The backpropagation of error learning procedure as a generalization of the delta rule was separately developed and widely published by the Parallel Distributed Processing Group:
Non-linearly-separable problems could be solved by multilayer perceptrons, and Marvin Minsky’s negative evaluations were disproven at a single blow. At the same in the field of artificial
intelligence, caused by a series of failures and unfulfilled hopes.
From this time on, the development of the field of research has almost been explosive. | {"url":"https://skedbooks.com/books/neural-network-fuzzy-systems/history-of-neural-networks/","timestamp":"2024-11-01T23:10:41Z","content_type":"text/html","content_length":"99043","record_id":"<urn:uuid:3dbfa0b0-1ec8-44c9-93bb-141fa303b9a7>","cc-path":"CC-MAIN-2024-46/segments/1730477027599.25/warc/CC-MAIN-20241101215119-20241102005119-00508.warc.gz"} |
Multiplying decimals
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Variations with Repetition
Variations with Repetition Calculator
Calculation of possible variations with repetition
This function calculates the number of possible variations from a set with repetition. In the variation with repetition, a number \(k\) is selected from the total number \(n\).
Description of variations with repetition
The number of possible variations from a set with repetition is calculated. For variations with repetition, a number \(k\) is selected from the total \(n\).
Each object may be selected more than once in the object group, i.e. with repetition. In the case of the urn model, this corresponds to drawing with replacement and taking into account the order.
Dieses Beispiel zeigt wieviel Gruppen mit 2 Objekten aus den Ziffern 1 bis 3 gebildet werden können. Es sind die Gruppen (1,1), (1,2), (1,3), (2,1), (2,2), (2,3), (3,1), (3,2) und (3,3). Also neun
Example and formula
Four balls are to be drawn from a box with six different colored balls. The number of ways to select and order four balls is calculated using the following formula:
\(\displaystyle n^k=6^4=1296 \) | {"url":"https://www.redcrab-software.com/en/Calculator/Combinatorics/Variations-with-Repetition","timestamp":"2024-11-03T06:03:48Z","content_type":"text/html","content_length":"20443","record_id":"<urn:uuid:3a1f3a61-cb76-4581-9375-4671e85e21c6>","cc-path":"CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00846.warc.gz"} |
System of Units,11th class,physics: Search page for Notes,Tests and Videos:SureDen Your Education Partner
System of units : A complete set of units, both fundamental and derived for all kinds of physical quantities is called system of units. The common systems are given below
(1) CGS system : The system is also called Gaussian system of units. In it length, mass and time have been chosen as the fundamental quantities and corresponding fundamental units are centimeter (cm
), gram (g) and second (s) respectively.
(2) MKS system : The system is also called Giorgi system. In this system also length, mass and time have been taken as fundamental quantities, and the corresponding fundamental units are metre,
kilogram and second.
(3) FPS system : In this system foot, pound and second are used respectively for measurements of length, mass and time. In this system force is a derived quantity with unit poundal.
(4) S. I. system : It is known as International system of units, and is infact extended system of units applied to whole physics. There are seven fundamental quantities in this system. These
quantities and their units are given in the following table
Quantity Name of Unit Symbol
Length meter M
Mass kilogram Kg
Time second S
Electric Current ampere A
Temperature Kelvin K
Amount of Substance mole Mol
Luminous Intensity candela Cd
Besides the above seven fundamental units two supplementary units are also defined
Radian (rad) for plane angle and Steradian (sr) for solid angle.
Note : Apart from fundamental and derived units we also use very frequently practical units. These may be fundamental or derived units
e.g., light year is a practical unit (fundamental) of distance while horse power is a practical unit (derived) of power.
Practical units may or may not belong to a system but can be expressed in any system of units
e.g., 1 mile = 1.6 km = 1.6 10^3m.
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Understanding Gauss's Law: The Key To Understanding Electric Fields
Gauss’s Law
Understanding Gauss’s law is a bit different from understanding the electric field. In both cases, you need to know how to calculate the total electric field in a closed system in an electric field.
But in the electric field case, you do that by knowing the total charge in the system. In the Gaussian case, you do that by knowing the charge density in the system.
What is Gauss’s Law
The electric flux through an area is calculated by multiplying the electric field by the surface area projected in a plane perpendicular to the field. Gauss’ Law is a general principle that can be
applied to any closed surface. It’s a helpful technique since it allows you to estimate how much contained charge there is by mapping the field on a surface outside the charge distribution. It
simplifies the computation of the electric field for geometries with adequate symmetry.
Define Gauss’s law
The charge contained divided by the permittivity equals the total electric flux out of a closed surface.
Another approach to visualize this is to imagine a probe with an area of A that can measure the electric field perpendicular to the region. It can get a measure of the net electric charge within any
closed surface by stepping over it and measuring the perpendicular field times its area, regardless of how that internal charge is structured.
Gauss’ Law Applications
Gauss’s law is the most important law of electromagnetism because it is the basis of how current is conducted within a conductor. It also provides us with the natural laws that describe the electric
field and the electric charge in a vacuum (minus the electric charge on an object outside the field, which is not included in the electric field).
The Gaussian distribution for the electric field is the basis for estimating the magnitude of a photon’s intensity; this is the basis of the law of thermal radiation. Gauss’s law is the fundamental
law of electromagnetic fields and is vital to the success of modern radio communication (radio waves and microwaves), radar, GPS, satellite and other related technologies.
When electric fields arise from charge distributions with adequate symmetry, Gauss’ law is a valuable tool for calculating them.
Key Takeaways from Gauss’s Law
This theory is said to be fully valid up to an emf of one trillion voltvoltsper cm2 – approximating a point charge of 10 uF. of one trillion voltvoltsper cm-2 – approximating a point charge of 10 uF.
If applied to the surface of a hollow cylindrical metal such as copper, an emf of ~1 wattm – 1 at this emf may be observed. – 1 at this emf may be observed. Since this emf is known, it can be applied
to other areas.
The principle of Electric Potential (Principle of Inverse Square Law) is just as powerful as the principle of Gauss’s law, in the sense that it quantifies how any electric field propagates in space.
It has the property that the electric field gradient is also the electric potential gradient.
How to Use Gauss’s Law for Different Situations
In cases when Gauss’s law is written as a series, with the surface area enclosed as “r”, and the electric charge formula_3 enclosed by the surface as “p”, the constant constant “k” at each point is
the amplitude of the electric field in that point: However, note that the output may still be nonzero, even when “k” is large.
In particular, in some situations, it is useful to construct an expression that allows the output to be shown to depend only on “k”, without any other necessary spatial variables: that is, only on
the vector containing “k”. This provides a way of highlighting a different physical property of the electric field: its dependence on the geometry of the surface enclosed by the charge.
Calculating Gauss’s Law
The simplest way to calculate Gauss’s law is to use an empirical, mathematical model that describes how electricity moves in a wire. It is called an approximate model because Gauss’s law is a law of
nature. It must be used as an approximate model because the actual electric field is unknown and there is no one way to describe it.
Our approximate model will be a one-dimensional surface called a Gaussian, which has the same general shape as a real conductor and a Gaussian surface which has a spherical shape. This surface is
very simple and is based on the type of surface that Gauss used in his work. For example, this surface could have been the surface of a cube, a sphere, or even a flat plane.
Gauss’s law Sphere
Because we already know the electric field in such a circumstance, let’s compute the electric flux through a spherical surface surrounding a positive point charge q. Remember that when a point charge
is placed at the origin of a coordinate system, the electric field at a point P that is r away from the charge at the origin is given by
The radial vector from the charge at the origin to the point P is denoted by r. As illustrated in Figure, we can utilize this electric field to calculate the flux across a spherical surface of radius
A point charge q is surrounded by a closed spherical surface.
gauss law sphere
Then we use
to replace known values in this system. Because n=r and r=R on the sphere, given an infinitesimal area dA,
We can now calculate the net flow by integrating this flux across the sphere’s surface:
Where the spherical surface’s entire surface area is 000. The flow through the closed spherical surface at radius r is calculated as
The flow is independent of the size of the spherical surface, which is a noteworthy feature of this equation. This is due to the fact that the electric field of a point charge drops with distance as
1/r^2, cancelling out the r^2 rate of increase of the surface area.
Understanding Gauss’s law is extremely important for creating transistors and other diodes because it must be applied before the voltage regulator’s circuit designer can apply the maximum current to
any given area on the circuit. Since Gauss’s law can be derived without knowing any of the details of the circuit being designed, it is a principle that almost always applies even if other properties
of the circuit are known.
This is one of the principles of circuit design known as the Zener rule. In physics and electromagnetism, Gauss’s law, also known as Gauss’s flux theorem, (or sometimes simply called Gauss’s theorem)
is a law relating the distribution of electric charge to the resulting electric field.
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Facts of Universe Cycle/Oscillating Model
Leave a Comment / Mysteries & Hypothesis, Matter & Energy, Physics & Cosmos, Space & Time / By Deep Prakash / July 8, 2020 / astronomy, astrophysics, balance, cosmologist, cosmology, cosmos, cycle of
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scientist, solar system, space, surprise, symbolize, symbols, the cyclic universe theory, theoretical, understand, understand universe, universe, universe cycle, universe cycle theory, universe
cycles, universe examples, what is cyclic universe theory
Cyclic Model of Universe
Leave a Comment / Mysteries & Hypothesis, Matter & Energy, Physics & Cosmos, Space & Time, Uncommon & Remarkable / By Deep Prakash / July 10, 2020 / astronomy, astrophysics, big bang, black, black
hole, cosmology, cosmos, cycle of the universe, cycle of universe, cycle universe, cycles of the universe, cyclic model, cyclic model of the universe, cyclic model of universe, cyclic model theory,
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CUSTODIAN Definition
CUSTODIAN is an entity entrusted with guarding and keeping property or records.
Learn new Accounting Terms
JIT see JUST-IN-TIME.
LINEAR PROGRAMMING (LP), in accounting, is the mathematical approach to optimally allocating limited resources among competing activities. It is a technique used to maximize revenue, contribution
margin, and profit function; or, to minimize a cost function, subject to constraints. Linear programming consists of two ingredients: (1) objective function and (2) constraints, both of which are
linear. In formulating the LP problem, the first step is to define the decision variables that one is trying to solve. The next step is to formulate the objective function and constraints in terms of
these decision variables.
Enter Search Term
Enter a term, then click the entry you would like to view. | {"url":"https://www.ventureline.com/accounting-glossary/C/custodian-definition/","timestamp":"2024-11-02T08:49:33Z","content_type":"text/html","content_length":"13507","record_id":"<urn:uuid:4c76085b-b061-4201-b93f-b217fec681a6>","cc-path":"CC-MAIN-2024-46/segments/1730477027709.8/warc/CC-MAIN-20241102071948-20241102101948-00312.warc.gz"} |
Bottleneck non-crossing matching in the plane
Let P be a set of 2n points in the plane, and let M [C] (resp., M [NC]) denote a bottleneck matching (resp., a bottleneck non-crossing matching) of P. We study the problem of computing M [NC]. We
present an O(n ^1.5log ^0.5 n)-time algorithm that computes a non-crossing matching M of P, such that bn(M) ≤ 2√10·bnM [NC], where bn(M) is the length of a longest edge in M. An interesting
implication of our construction is that bn(M [NC])/bn(M [NC]) ≤ 22√10. We also show that when the points of P are in convex position, one can compute M [NC] in O(n ^3) time. (In the full version of
this paper, we also prove that the problem is NP-hard and does not admit a PTAS.)
Original language English
Title of host publication Algorithms, ESA 2012 - 20th Annual European Symposium, Proceedings
Pages 36-47
Number of pages 12
State Published - 2012
Externally published Yes
Event 20th Annual European Symposium on Algorithms, ESA 2012 - Ljubljana, Slovenia
Duration: 10 Sep 2012 → 12 Sep 2012
Publication series
Name Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume 7501 LNCS
ISSN (Print) 0302-9743
ISSN (Electronic) 1611-3349
Conference 20th Annual European Symposium on Algorithms, ESA 2012
Country/Territory Slovenia
City Ljubljana
Period 10/09/12 → 12/09/12
Dive into the research topics of 'Bottleneck non-crossing matching in the plane'. Together they form a unique fingerprint. | {"url":"https://cris.biu.ac.il/en/publications/bottleneck-non-crossing-matching-in-the-plane-2","timestamp":"2024-11-11T13:00:48Z","content_type":"text/html","content_length":"53307","record_id":"<urn:uuid:e79d3865-7604-497b-a5f6-7573936220e3>","cc-path":"CC-MAIN-2024-46/segments/1730477028230.68/warc/CC-MAIN-20241111123424-20241111153424-00523.warc.gz"} |
The Hidden Treasure of What Is a Coefficient in Math - LILLY PITTA | Mandando Som
The Hidden Treasure of What Is a Coefficient in Math
The Rise of What Is a Coefficient in Math
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It’s apparent that a single particpant can’t opt for a weak signature. Instead of unique letters in every single pair of parentheses, we’re now repeating the very same letters over and over again. It
is referred to as a Constant.
The term with the utmost degree is called the top term as it’s generally written first. All you have to do is multiply that density by the quantity of the solid or liquid to find the mass of the
solid or liquid. Even though the Correlation Coefficient spends a good amount of time in positive territory, it’s negative the vast majority of the moment.
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best-essay-writer-uk These units run at various speeds, which let them supply you with an even increased efficiency. Though the Correlation Coefficient spends a great period of time in positive
territory, it’s negative the great majority of the moment.
What Is a Coefficient in Math Secrets That No One Else Knows About
Sensitivity to the data distribution could possibly be utilized to a benefit. At the time your data are all z-scores you will be in a position to proceed with regularization. Any details that are
pertinent to learn about the graph ought to be mentioned.
To start with, you desire to comprehend your squares. Display mode equations must appear on their own line. You’re able to interactively explore graphs similar to this at Quadratic explorer.
The Hidden Gem of What Is a Coefficient in Math
Instead multiplying just the same number again and again, a specific notation is utilized in mathematics and called the exponents. While statistical inference provides many benefits it also will come
with some vital pitfalls. Factoring gives you the ability to locate solutions to complex polynomials.
Specifically, the magic formula doesn’t get the job done for exponential functions. In informal parlance, correlation is identical to dependence. Other terminologies ought to be simple to grasp.
Some people are inclined to recognize the term variance as something negative, making sense since there are many contexts where it can mean something bad. In more complicated math issues, the
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Doesn’t matter what logarithm we take equally as long since they’re likely to be the exact same. The arguments necessary to work out the correlation coefficient are the 2 ranges of information which
will want to go compared. For the time being, let’s just center on the top portion of this equation.
A standard maths challenge is to learn whether a specific polynomial can be written as a linear blend of another polynomials. In case the coefficient is negative, the inequality will be reversed.
This equation involves the slope.
Division by zero isn’t defined and thus x might not have a value that enables the denominator to become zero. The response is called the quotient. Inside this lesson, you are going to learn about the
correlation coefficient and the way to use it in order to discover correlations in Rachael’s research.
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Getting divorced in Maryland is much like getting divorced in the majority of other states. The answer, in actuality, lies somewhere in the middle. The perfect way to understand any formula is to
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The What Is a Coefficient in Math Game
You may incorporate these to make the most of code you could have already written. The t-test is optimized to address small sample numbers that is frequently the case with managers in any company.
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This problem is straightforward enough that the resulting QAP won’t be quite as large as to be intimidating, but nontrivial enough you can observe all the machinery are involved. If you’re interested
with the mathematical significance of a multiple regression, the start of the video (05 min) will supply you with the basic information which you want. In case it isn’t revealed, it might be biased.
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With the explosion of 3D printing, designing 3D objects has become an even more important problem. And it's a complicated one. We need to design complicated objects that precisely interface with each
other. And often we don't want to just design a single object, but classes of objects, parameterized by variables. And we need to do this in collaboration.
There are trivial cases, like a screw driver of varying length or the width of a table. Traditional geometric constraints can suffice here. But what if we want a key based of a list of numbers
describing the heights of a lock's pins? Now we need programming.
But that's only the beginning. We can abstract away the stupid work humans do in designing objects. We can build DSLs. We can unit test objects and put them on github.
ImplicitCAD is a project dedicated to using the power of math and computer science to get stupid design problems out of the way of the 3D printing revolution. | {"url":"http://implicitcad.org/","timestamp":"2024-11-02T23:11:25Z","content_type":"text/html","content_length":"5066","record_id":"<urn:uuid:99480449-24c2-49cc-b480-25d4306a856d>","cc-path":"CC-MAIN-2024-46/segments/1730477027768.43/warc/CC-MAIN-20241102231001-20241103021001-00045.warc.gz"} |
What is Reddit's opinion of
As already mentioned by others, there's a lot more to math beyond what you've mentioned above, but on the presumption that something that covers from "numbers sets and arithmetic" to "calculus and
differential equations", I might suggest Measurement by Paul Lockhart.
It definitely satisfies the "written for human beings" requirement, and it starts out with a discussion of shapes that any elementary school student could probably follow and then works its way up to
calculus. You might have to look elsewhere for d.e. though.
In general, you'll probably find more success covering "some~~every~~thing about math" if you try to find a collection of books on various topics, rather than just one book.
Measurement - ISBN-10: 0674284380
Zero: The Biography of a Dangerous Idea - ISBN-10: 0140296476
They list an ISBN-10 and an ISBN-13 but i dont know the difference
These are not good recommendations for a beginner or someone who “doesn't want to continue studying math beyond this course”. The former is a kind of dictionary of problem solving strategies, and the
latter is a lead-in to pure math.
If you want a Polya book, the best one to start on is the 2-volume Mathematical Discovery, but even might not really be pitched at your student here.
I would instead recommend something like Mason, Burton, & Stacey’s <em>Thinking Mathematically</em> or Gardiner’s <em>Discovering Mathematics</em> or with a different flavor Lockhart’s <em>
It’s hard to give good recommendations without more information about your student’s background, commitment level, and goals. “Study math better” and “wants to be well rounded” are very vague.
Measurement by Paul Lockhart is a great general math book. His other two books are also fantastic.
Measurement by Paul Lockhart. Doesn’t cover all of mathematics but tries to reveal the essence and beauty of it.
Maybe something like this: http://www.amazon.com/Measurement-Paul-Lockhart/dp/0674284380 | {"url":"https://redditfavorites.com/products/measurement-de72df85-3de8-45bd-b9f0-6ba207bc22ae","timestamp":"2024-11-07T16:16:53Z","content_type":"text/html","content_length":"15550","record_id":"<urn:uuid:09e6b8b9-1d9a-4c2a-bb19-e0fa1335df20>","cc-path":"CC-MAIN-2024-46/segments/1730477028000.52/warc/CC-MAIN-20241107150153-20241107180153-00819.warc.gz"} |
Bordered Magic Squares Multiples of 11 – Revised
Revised on July 16, 2023
Recently, the author revised and improved the block-wisebordered magic squares multiples of even number blocks. This means, multiples of 4, 6, 8, etc. The work is up to order 20. These works can be
accessed at the following links.
1. Bordered and Pandiagonal Magic Squares Multiples of 4.
2. Bordered Magic Squares Multiples of Magic Squares of Order 6.
3. Bordered and Pandiagonal Magic Squares Multiples of 8 – Revised
4. Bordered Magic Squares Multiples of 10 – Revised.
5. Bordered and Pandiagonal Magic Squares Multiples of 12 – Revised.
6. Bordered Magic Squares Multiples of 14 – Revised.
7. Bordered and Pandiagonal Magic Squares Multiples of 16.
8. Bordered Magic Squares Multiples of 18.
9. Bordered and Pandiagonal Magic Squares Multiples of 20.
Author also worked with odd number blocks. The work is for multiples of order 3, 5, 7, 9, 11 and 13. See the links below:
1. Bordered and Pandiagonal Magic Squares Multiples of 3.
2. Bordered and Pandiagonal Magic Squares Multiples of 5.
3. Bordered and Pandiagonal Magic Squares Multiples of 7.
4. Bordered Magic Squares Multiples of 9 – Revised.
5. Bordered Magic Squares Multiples of Order 11- Revised.
6. Bordered Magic Squares Multiples of 13.
7. Bordered Magic Squares Multiples of 15.
8. Bordered Magic Squares Multiples of 17.
The advantage in studying block-wise bordered magic squares is that when we remove external borders, still we left with magic squares with sequential entries.
We observe that the above study of even order blocks starting from 4. Recently, author studied the double digit or two digits borders resulting in interesting magic squares. These studies can be
accessed at the following link:
1. Two Digits Bordered Magic Squares Multiples of 4: Orders 8 to 24.
2. Two Digits Bordered Magic Squares of Orders 28 and 32.
3. Two digits Bordered Magic Squares of Order 36.
4. Two digits Bordered Magic Squares of Order 40.
The work is for the order of type 4k+2, where k>1, i.e., for the orders 10, 14, 18, 22, 26 and 30 can be access at the following links:
1. Two digits Bordered Magic Squares of Orders 10, 14, 18 and 22.
2. Two digits Bordered Magic Squares of Orders 26 and 30.
3. Two Digits Bordered Magic Squares of Orders 28 and 32.
Some studies towards cornered magic squares are also made. See the following links:
1. Cornered Magic Squares of Orders 5 to 10.
2. Cornered Magic Squares of Orders 11 to 13.
3. Cornered Magic Squares of Orders 14 to 24.
Below are examples of bordered magic squares multiples of order 11 up to order 55. Total we worked with 20 different types of magic squares of order 11. Up to order 55 pdf files of each order are
attached for download. Excel files of whole work are also attached at the end.
This work can also be accessed at the following link:
Inder J. Taneja, Bordered Magic Squares Multiples of 11, Zenodo, July 24, pp. 1-34, 2023, https://doi.org/10.5281/zenodo.8176475.
Magic Squares Multiples of 11
Below are 20 magic square of order 11. We have used these magic squares to bring bordered magic squares up to order 154 as multiples of 11.
Bordered Magic Squares Multiples of 11
Magic Squares of Order 33
Below are only few examples of magic square of order 33. The other examples are given in attached pdf file.
Pdf file of Order 33
Bordered Magic Squares Multiples of 11
Magic Squares of Order 44
Below are only few examples of magic square of order 44. The other examples are given in attached pdf file.
Pdf file of Order 44
Bordered Magic Squares Multiples of 11
Magic Squares of Order 55
Below are only few examples of magic square of order 55. The other examples are given in attached pdf file.
Pdf file of Order 55
Excel files for download
File 1:
File 2:
File 3:
The first file is only with first eight examples, while the other two files are with all the 20 examples of bordered magic squares multiples of order 11. | {"url":"https://numbers-magic.com/?p=8989","timestamp":"2024-11-11T07:29:43Z","content_type":"text/html","content_length":"118787","record_id":"<urn:uuid:164d1059-227e-4cd9-9aa7-64bd2896c26d>","cc-path":"CC-MAIN-2024-46/segments/1730477028220.42/warc/CC-MAIN-20241111060327-20241111090327-00200.warc.gz"} |
Law of Cosines Calculator
Last updated:
Law of Cosines Calculator
The law of cosines calculator can help you solve a vast number of triangular problems. You will learn what is the law of cosines (also known as the cosine rule), the law of cosines formula, and its
applications. Scroll down to find out when and how to use the law of cosines, and check out the proofs of this law. Thanks to this triangle calculator, you will be able to find the properties of any
arbitrary triangle quickly.
But if, somehow, you're wondering what the heck is cosine, better have a look at our cosine calculator.
Law of cosines formula
The law of cosines states that, for a triangle with sides and angles denoted with symbols as illustrated above,
a² = b² + c² - 2bc × cos(α)
b² = a² + c² - 2ac × cos(β)
c² = a² + b² - 2ab × cos(γ)
For a right triangle, the angle gamma, which is the angle between legs a and b, is equal to 90°. The cosine of 90° = 0, so in that special case, the law of cosines formula is reduced to the
well-known equation of Pythagorean theorem:
a² = b² + c² - 2bc × cos(90°)
a² = b² + c²
What is the law of cosines?
The law of cosines (alternatively the cosine formula or cosine rule) describes the relationship between the lengths of a triangle's sides and the cosine of its angles. It can be applied to all
triangles, not only the right triangles. This law generalizes the Pythagorean theorem, as it allows you to calculate the length of one of the sides, given you know the length of both the other sides
and the angle between them.
The law appeared in Euclid's Element, a mathematical treatise containing definitions, postulates, and geometry theorems. Euclid didn't formulate it in the way we learn it today, as the concept of
cosine had not been developed yet.
AB² = CA² + CB² − 2 × CA × CH (for acute angles)
AB² = CA² + CB² + 2 × CA × CH (for obtuse angles)
However, we may reformulate Euclid's theorem easily to the current cosine formula form:
CH = CB × cos(γ), so AB² = CA² + CB² - 2 × CA × (CB × cos(γ))
Changing notation, we obtain the familiar expression:
c² = a² + b² - 2ab × cos(γ)
The first explicit equation of the cosine rule was presented by Persian mathematician d'Al-Kashi in the 15th century. In the 16th century, the law was popularized by the famous French mathematician
Viète before it received its final shape in the 19th century.
Applications of the law of cosines
You can transform these law of cosines formulas to solve some problems of triangulation (solving a triangle). You can use them to find:
1. The third side of a triangle, knowing two sides and the angle between them (SAS):
□ a = √[b² + c² - 2bc × cos(α)]
□ b = √[a² + c² - 2ac × cos(β)]
□ c = √[a² + b² - 2ab × cos(γ)]
2. The angles of a triangle, knowing all three sides (SSS):
□ α = arccos [(b² + c² - a²)/(2bc)]
□ β = arccos [(a² + c² - b²)/(2ac)]
□ γ = arccos [(a² + b² - c²)/(2ab)]
3. The third side of a triangle, knowing two sides and an angle opposite to one of them (SSA):
□ a = b × cos(γ) ± √[c² - b² × sin²(γ)]
□ b = c × cos(α) ± √[a² - c² × sin²(α)]
□ c = a × cos(β) ± √[b² - a² × sin²(β)]
Just remember that knowing two sides and an adjacent angle can yield two distinct possible triangles (or one or zero positive solutions, depending on the given data). That's why we've decided to
implement SAS and SSS in this tool, but not SSA.
The law of cosines is one of the basic laws, and it's widely used for many geometric problems. We also take advantage of that law in many Omnitools, to mention only a few:
Also, you can combine the law of cosines calculator with the law of sines to solve other problems, for example, finding the side of the triangle, given two of the angles and one side (AAS and ASA).
Law of cosines proofs
There are many ways in which you can prove the law of cosines equation. You've already read about one of them – it comes directly from Euclid's formulation of the law and an application of the
Pythagorean theorem. You can write the other proofs of the law of cosines using:
1. Trigonometry
Draw a line for the height of the triangle and divide the side perpendicular to it into two parts:
b = b₁ + b₂
From sine and cosine definitions, b₁ might be expressed as a × cos(γ) and b₂ = c × cos(α). Hence:
b = a × cos(γ) + c × cos(α) and by multiplying it by b, we get:
b² = ab × cos(γ) + bc × cos(α) (1)
Analogical equations may be derived for other two sides:
a² = ac × cos(β) + ab × cos(γ) (2)
c² = bc × cos(α) + ac × cos(β) (3)
To finish the law of cosines proof, you need to add the equation (1) and (2) and subtract (3):
a² + b² - c² = ac × cos(β) + ab × cos(γ) + bc × cos(α) + ab × cos(γ) - bc × cos(α) - ac × cos(β)
Reduction and simplification of the equation give one of the forms of the cosine rule:
a² + b² - c² = 2ab × cos(γ)
c² = a² + b² - 2ab × cos(γ)
By changing the order in which they are added and subtracted, you can derive the other law of cosine formulas.
2. Distance formula
Let C = (0,0), A = (b,0), as in the image.
To find the coordinates of B, we can use the definition of sine and cosine:
B = (a × cos(γ), a × sin(γ))
From the distance formula, we can find that:
c = √[(x₂ - x₁)² + (y₂ - y₁)²] = √[(a × cos(γ) - b)² + (a × sin(γ) - 0)²]
c² = a² × cos(γ)² - 2ab × cos(γ) + b² + a² × sin(γ)²
c² = b² + a²(sin(γ)² + cos(γ)²) - 2ab × cos(γ)
As a sum of squares of sine and cosine is equal to 1, we obtain the final formula:
c² = a² + b² - 2ab × cos(γ)
3. Ptolemy's theorem
Another law of cosines proof that is relatively easy to understand uses Ptolemy's theorem:
• Assume we have the triangle ABC drawn in its circumcircle, as in the picture.
• Construct the congruent triangle ADC, where AD = BC and DC = BA
• The heights from points B and D split the base AC by E and F, respectively. CE equals FA.
• From the cosine definition, we can express CE as a × cos(γ).
• Thus, we can write that BD = EF = AC - 2 × CE = b - 2 × a × cos(γ).
• Then, for our quadrilateral ADBC, we can use Ptolemy's theorem, which explains the relation between the four sides and two diagonals. The theorem states that for cyclic quadrilaterals, the sum of
products of opposite sides is equal to the product of the two diagonals:
BC × DA + CA × BD = AB × CD
so in our case:
a² + b × (b - 2 × a × cos(γ)) + a² = c²
• After reduction, we get the final formula:
c² = a² + b² - 2ab × cos(γ)
The great advantage of these three proofs is their universality – they work for acute, right, and obtuse triangles.
4. Using the law of sines
5. Using the definition of dot product
6. Comparison of areas
7. Geometry of the circle
The last two proofs require the distinction between different triangle cases. The one based on the definition of dot product is shown in another article, and the proof using the law of sines is quite
complicated, so we have decided not to reproduce it here. If you're curious about these law of cosines proofs, check out the explanation.
How to use the law of cosines calculator
1. Start with formulating your problem. For example, you may know two sides of the triangle and the angle between them and are looking for the remaining side.
2. Input the known values into the appropriate boxes of this triangle calculator. Remember to double-check with the figure above whether you denoted the sides and angles with correct symbols.
3. Watch our law of cosines calculator perform all the calculations for you!
Law of cosines – SSS example
If your task is to find the angles of a triangle given all three sides, all you need to do is to use the transformed cosine rule formulas:
α = arccos [(b² + c² - a²)/(2bc)]
β = arccos [(a² + c² - b²)/(2ac)]
γ = arccos [(a² + b² - c²)/(2ab)]
Let's calculate one of the angles. Assume we have a = 4 in, b = 5 in and c = 6 in. We'll use the first equation to find α:
α = arccos [(b² + c² - a²)/(2bc)]
= arccos [(5² + 6² - 4²)/(2 × 5 × 6)]
= arccos [(25 + 36 - 16)/60]
= arccos [(45/60)] = arccos [0.75]
α = 41.41°
You may calculate the second angle from the second equation in an analogical way, and the third angle you can find by knowing that the sum of the angles in a triangle is equal to 180° (π).
If you want to save some time, type the side lengths into our law of cosines calculator - our tool is a safe bet! Just follow these simple steps:
1. Choose the option depending on given values. We need to pick the second option – SSS (3 sides).
2. Enter the known values. Type the sides: a = 4 in, b = 5 in, and c = 6 in.
3. The calculator displays the result! In our case the angles are equal to α = 41.41°, β = 55.77° and γ = 82.82°.
After such an explanation, we're sure that you understand what the law of cosine is and when to use it. Give this tool a try, solve some exercises, and remember that practice makes permanent!
When should I use the law of cosines?
Use the law of cosines if you need to calculate:
• A side of a triangle given two other sides and the angle between them.
• The three angles of a triangle given its sides.
• A side of a triangle given two other sides and an angle opposite to one of these sides.
When should I use the law of cosines vs the Pythagorean theorem?
The law of cosines is a generalization of the Pythagorean theorem, so whenever the latter works, the former can be applied as well. Not the other way round, though!
Is the law of cosines valid only for right triangles?
No, the law of cosines is valid for all triangles. In fact, when you apply the law of cosines to a right triangle, you'll arrive at the good old Pythagorean theorem.
What is the third side of a triangle with sides 5 and 6?
Besides the two sides, you need to know one of the inner angles of the triangle. Let's say it's the angle γ = 30° between the sides 5 and 6. Then:
1. Recall the law of cosines formula c² = a² + b² - 2ab × cos(γ)
2. Plug in the values a = 5, b = 6, γ = 30°.
3. We obtain c² = 25 + 36 - 2 × 5 × 6 × cos(30) ≈ 9.
4. Therefore, c ≈ 3. Remember to include the units if you were given any! | {"url":"https://www.omnicalculator.com/math/law-of-cosines","timestamp":"2024-11-06T15:50:55Z","content_type":"text/html","content_length":"540470","record_id":"<urn:uuid:0a85312b-00b8-4f5b-af36-e7a070d2dfe3>","cc-path":"CC-MAIN-2024-46/segments/1730477027932.70/warc/CC-MAIN-20241106132104-20241106162104-00772.warc.gz"} |
Extended meeting of the seminar "Complex problems of mathematical physics" dedicated to the 70th anniversary of Armen Sergeev
(March 11, 2019, Moscow, Steklov Mathematical Institute, 9th floor, conference hall)
Extended meeting of the seminar "Complex problems of mathematical physics" dedicated to the 70th anniversary of Armen Sergeev, Moscow, Steklov Mathematical Institute, 9th floor, conference hall,
March 11, 2019
© , 2024 | {"url":"https://m.mathnet.ru/php/conference.phtml?confid=1493&option_lang=eng","timestamp":"2024-11-13T22:48:14Z","content_type":"text/html","content_length":"13158","record_id":"<urn:uuid:3a88e345-5f30-40f6-8b71-ce57c6a9d0fd>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00633.warc.gz"} |
Common Core Algebra 2 Module 1 Lesson 18 Worksheets Samples | Common Core Worksheets
Common Core Algebra 2 Module 1 Lesson 18 Worksheets Samples
Common Core Algebra 2 Module 1 Lesson 18 Worksheets Samples
Common Core Algebra 2 Module 1 Lesson 18 Worksheets Samples – Are you trying to find Common Core Math Worksheets Algebra 2? You are not the only one. There are lots of sources around that are
excellent for your demands. The best part is that they’re available for download and also can be utilized for class of any kind of dimension. These worksheets are great for assisting kids understand
the Common Core requirements, including the Common Core Reading as well as Writing criteria.
What are Common Core Worksheets?
Common Core Worksheets are instructional sources for K-8 pupils. They are designed to assist pupils accomplish a common collection of objectives. The initial rule is that the worksheets may not be
shared or posted to the web in any type of way.
Common Core worksheets cover K-8 trainees and also are designed with the CCSS in mind. Making use of these resources will aid trainees discover the skills necessary to be successful in college. They
cover numerous ELA and math topics and come with response secrets, making them a great resource for any classroom.
What is the Purpose of Common Core?
The Common Core is a campaign to bring uniformity to the way American children find out. Developed by educators from throughout the nation, the requirements focus on constructing a typical base of
knowledge and skills for students to be effective in college and in life. Presently, 43 states have adopted the standards and also have actually begun to implement them in public institutions.
The Common Core requirements are not a government mandate; instead, they are an outcome of years of study and also evaluation by the Council of Chief State School Officers as well as the National
Governors Association. While federal requireds are necessary, states still have the last word in what their educational program resembles.
Several moms and dads are frustrated with Common Core criteria as well as are uploading screenshots of incomprehensible products. The Common Core has actually identified this a mathematics task, but
Rubinstein couldn’t make sense of it.
Common Core Math Worksheets Algebra 2
Common Core Math Worksheets Algebra 2
If you are looking for Common Core Math Worksheets Algebra 2, you’ve come to the right place! These mathematics worksheets are classified by grade level and also are based on the Common Core
mathematics criteria.
The first collection of worksheets is concentrated on single-digit enhancement as well as will certainly examine a youngster’s skills in counting things. This worksheet will certainly call for pupils
to count products within a min, which is a wonderful way to practice counting. The cute objects that are included will certainly make the mathematics problems much more easy to understand for the
child as well as offer a visual representation of the solution.
Mathematics worksheets based on the common core mathematics requirements are an excellent method for youngsters to discover basic arithmetic skills and also ideas. These worksheets consist of
different issues that vary in trouble. They will certainly also encourage problem-solving, which aids youngsters apply their discovering in real-life scenarios.
Fractions are an additional subject area that is tough, however not impossible for young learners to discover. Common Core Fractions Teaching Resources consist of sorting, purchasing, and also
modeling fractions. These free worksheets are designed to aid youngsters master this topic.
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Free Class Lesson for Motion and Kinematics | StoryboardThat
Student Activities for Motion
Essential Questions for Motion
1. What is a scalar/vector quantity?
2. How can we describe motion?
3. How can we calculate speed?
Kinematics and Motion Definition
Kinematics is an area of study in classical physics that deals with motion. Some people could even argue it is actually an area of Mathematics. We can describe the motion of objects by looking at the
different measurable quantities such as displacement, velocity, and acceleration. Displacement is distance with a direction. Velocity, or speed, is how fast something is moving. In order to calculate
the average speed you need to know two things: the distance the object has traveled and the time it has taken the object to cover that distance. Science uses the S.I. units for speed, m/s (meters per
second). In everyday language, we can also describe speed in the units of mph (miles per hour) or km/h (kilometers per hour). The equation for speed is distance divided by time taken (speed = d ÷ t).
Instantaneous speed is the speed at a particular moment, whereas average speed is the mean speed across a large distance. Acceleration is a measure of the rate of change of speed. Acceleration can be
positive, meaning velocity is increasing, or negative, meaning velocity is decreasing.
The motion of the object can be described using charts. It is important that students are able to interpret velocity-time graphs and displacement-time graphs. In both of these graphs, time runs along
the x-axis with velocity or displacement on the y-axis. For a displacement-time graph, the gradient or slope of the line indicates the direction and the speed an object is traveling. A line with zero
gradient (a horizontal line) means the object is not moving. If the line curves, this indicates the object is accelerating, either negatively or positively.
There are two types of quantities in Science: vector quantities and scalar quantities. A vector quantity is a quantity that has both size and direction. Velocity is one example of a vector, where
both the magnitude and the direction of the quantity are needed to calculate. Scalar quantities are only measured by magnitude. An example of a scalar quantity is time. Time has no direction, but
does have magnitude.
Velocity and acceleration are both vector quantities and can be represented by an arrow. When the acceleration vector is in the same direction as the velocity vector, the object will increase in
velocity in that direction. When the acceleration arrow is in the opposite direction to the velocity vector, the object’s velocity will decrease. If there is no acceleration, then the object will
travel at a constant velocity; it will not increase or decrease.
Other Motion Activity Ideas
1. Create a narrative storyboard showing one's motion throughout the day. Make a displacement-time graph to accompany it.
2. Plan an investigation into velocity or displacement using Storyboard That investigation planning resources.
3. Compare different scalar and vector pairs, like velocity and speed, in a T-Chart.
Find more lesson plans and activities like these in our
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3 Simple Steps For Adding & Subtracting Large Numbers
Educational Strategies
3 Simple Steps For Adding & Subtracting Large Numbers
These lessons are designed to offer math help online for children who struggle with addition & subtraction. According to educational standards, by the time children leave second grade they should be
able to add and subtract three digit numbers fluently. In order to do this, children must know all their addition and subtraction facts from 1 to 20 automatically. In other words, they should
automatically know the following without having to think about the answer:
• 7+9 = 16
• 17-9 = 8
• 6+8 = 15
• 19-5 = 14
• 12+7 =19
I have another post entitled, Math help websites for Parents, that demonstrates how to develop this skill.
But what if you do not know all your facts?
I teach 6th grade. Every year, I get a handful of students who don’t know their basic number sense for addition and subtraction. Other students get confused when borrowing during subtraction.
Because adding and subtracting is an essential building block for all mathematics, I needed to create a strategy that my students could use until they develop an understanding of basic number sense
for addition and subtraction. That is why I wrote this post entitled, Math Help Online: 3 Simple Steps For Adding & Subtracting Large Numbers
This Math Help Website is Designed so that ANYONE can succeed when Adding or Subtracting Large Numbers!
All Problems have Math Video Tutorials!
Cyclical Learning Approach
I have incorporated a cyclical learning approach with math tutorial videos. Each educational concept is introduced, then reinforced, then revisited again and again to ensure success. The 1^st step
is taught in Lesson 1. It focuses developing strategies for adding and subtracting 3-digit numbers. Once proficient, your child can move to the 2^nd step, which is taught in Lesson 2. This step
focuses on strengthening the strategies taught in lesson 1, and incorporating those strategies into adding and subtracting 4-digit numbers. The final step is taught in Lesson 3. This lesson
solidifies these strategies while adding and subtracting numbers with decimals.
I have scaffold these problems:
1. The first problem is a Watch ME.
Students should read the problem and then click on the video to watch how the problem is solved.
Students should copy the entire problem into their notebook.
2. The second problem is a WORK WITH ME.
Students should read the problem then gather all their materials, so that they can do the problem with me.
Students should play the math tutorial video and pause it when told.
Finally, Students should copy the problem down on their own paper, and solve it with me.
When the math tutorial video is complete, students should review the problem with their teacher or parent.
3. All the following problems are ON YOUR OWN.
Students should solve the problems just as they did in the first two.
Once you they completed the problem, they should watch the math tutorial video.
Students should keep their paper with them while they watch the video.
If they made a mistake, they should pause the math tutorial video and fix their mistake.
That’s the fastest way to learn!
Book 1
I also sell these lessons without Advertisements on a website called, TeachersPayTeachers. Each lesson is broken into a separate book. The series is entitled, Trouble A Com’n. If you’re interested
in purchasing the books with NO Advertising, I have a link below this first problem.
Problem Number 1 – Work with Me
Lucy Longneck
Lucy Longneck loves eating acacia leaves. She ate 749 leaves on Thursday and another 652 leaves on Friday. How many leaves did Lucy Longneck eat in all?
Lucy Longneck is headed home after a long day of chewing leaves. She is 524 giraffe steps away from her home. If she has already walked 365 steps, how many more steps must Lucy Longneck walk before
she gets home?
Now – Gather your materials and press PLAY. We’ll solve this problem together while you watch the math tutorial video below.
If you found this first problem helpful – please read on!
I have more problems and video after this brief message.
I’ve created this lesson as well as my game, TeachersDungeon, because I want to give back to the profession that I love so very much. If you would like to become a patron and help support this
website, you can do so in purchasing one of my educational books.
If you are a teacher or the parent of a child that could benefit from one of my eBooks please visit my store at TeachersPayTeacher. I use the name, McCoy’s Math Link at Teachers Pay Teachers.
Problem Number 2 – On Your Own
Timid Timmy
Timid Timmy is afraid of dogs, cats, hawks, and even other squirrels. There are a total of 723 dogs, cats, and hawks in his neighborhood. If there are 533 dogs and cats, how many hawks are in Timid
Timmy’s neighborhood?
Last week, Timid Timmy escaped from 369 attacks. This week he escaped from another 288 attacks. How many times did Timid Timmy have to run for his life in the past two weeks?
Once you complete the problem – Hit PLAY on the math tutorial video below. Good Luck!
Problem Number 3 – On Your Own
Enzo the Enforcer
Enzo the Enforcer rules his dominion. If anyone gets out of line, Enzo growls, snarls, and even bites. Enzo has bitten 942 animals that were bullying other critters, and another 689 that were
caught stealing food. How many animals in all has Enzo the Enforcer bitten?
Enzo the Enforcer has no tolerance for bullies. As a matter of fact, he has almost completely stopped all the bullying in his territory. When Enzo was little, there were 648 bullies in his
territory. Enzo the Enforcer convinced 579 bullies to change their wicked ways. How many bullies are left in Enzo’s territory?
Once you complete the problem – Hit PLAY on the math tutorial video below. Good Luck!
Each of my books includes 10 word problems like the three above. The books also include another 16 Drill & Kill problems like the 4 in challenge 11.
Drill & Kill
This is where we Drill until we Kill all our mistakes!
I call this section of the book, Drill & Kill, because we will drill this concept until we are perfect, and we kill any mistakes!
The following problems can all be solved with the same strategies we used to solve the first ten problems.
1. Solve all four problems on each page.
2. Watch the math tutorial video & correct your work.
3. Review your work with your parent or teacher.
If you get all 4 problems correct, your parent or teacher may tell you that you’re ready to move to the next book/article within this series.
Good Luck!
Drill & Kill
Challenge – 11
Challenge 1 Chalenge 2
658 – 179 436 + 957
Challenge 4
Challenge 3
963 + 464
462 – 283
Once you complete the problem – Hit PLAY on the math tutorial video below. Good Luck!
Please read on!
The 2nd Lesson in this series will begin after this brief message.
Book 2
Problem Number 1 – Work with Me
Jack Stinger
Jack Stinger is an angry yellowjacket. He is always on the hunt for an unsuspecting person to sting. Last week Jack stung 3,487 people. This week, he stung another 5,896 people. How many people
has Jack Stinger, the Angry Yellowjacket, stung in the past two weeks?
Jack Stinger has a goal. He wants to be the first yellowjacket to sting 6,500 people in just one week. So far, Jack Stinger has stung 5,896 people. He has just six hours left before the end of the
week. How many more people must Jack sting in order to reach his goal?
Now – Gather your materials and press PLAY. We’ll solve this problem together while you watch the math tutorial video below.
Problem Number 2 – On Your Own
Grizzly Greg the Honey Lov’n Bear
Grizzly Greg the Honey Lov’n Bear is always on the hunt for that tasty treat. He is more than willing to get stung a couple times, so long as he gets his favorite food. Yesterday, Greg stole 7,482
ounces of honey. He has already eaten 4,403 ounces. How much more honey does Grizzly Greg have to eat?
Grizzly Greg loves his honey, but he does not love getting stung. Last month, Greg got stung 3,848 times. This month, he was stung another 2,872 times. How many times has Grizzly Greg the honey
lov’n bear been stung in the past two months?
Once you complete the problem – Hit PLAY on the math tutorial video below. Good Luck!
Problem Number 3 – On Your Own
Pelican Pete
Pelican Pete lives on a golf course. He waits for the golfers to hit the ball, then he tries to catch it in his outstretched beak. 742 golfers hit 8,356 balls the other day. Pelican Pete caught
3,824 balls. How many did he miss?
Pelican Pete is loved by all the golfers, even though he sometimes wrecks their game. At the end of eighteen holes, the golfers always give Pelican Pete a bit of their lunch. Pelican Pete ate 3,749
ounces of shrimp salad, and another 2,865 ounces of salmon delight. How much food has Pelican Pete eaten?
Once you complete the problem – Hit PLAY on the math tutorial video below. Good Luck!
Each of my books includes 10 word problems like the three above. The books also include another 16 Drill & Kill problems like the 4 in challenge 11.
Drill & Kill
This is where we Drill until we Kill all our mistakes!
I call this section of the book, Drill & Kill, because we will drill this concept until we are perfect, and we kill any mistakes!
The following problems can all be solved with the same strategies we used to solve the first ten problems.
1. Solve all four problems on each page.
2. Watch the math tutorial video & correct your work.
3. Review your work with your parent or teacher.
If you get all 4 problems correct, your parent or teacher may tell you that you’re ready to move to the next book/article within this series.
Good Luck!
Drill & Kill
Challenge – 11
Challenge 1 Chalenge 2
6,658 – 3,779 3,958 + 7,579
Challenge 3 Challenge 4
9,472 – 1,475 4,883 + 2,337
Once you complete the problem – Hit PLAY on the math tutorial video below. Good Luck!
Please read on!
The 3rd Lesson in this series will begin after this brief message.
Book 3
Problem Number 1 – Work with Me
Elyssa Elephant
Elyssa Elephant is the youngest member of her herd. She trots along side her mother as the elephants travel across the African Savanna. Sometimes Elyssa can walk, but other times she has to run in
order to keep up with the grown up elephants. Last month, Elyssa walked 8.87 miles and ran 8.6 miles. Did Elyssa walk further or did she run further last month? What was the difference?
How far did Elyssa Elephant travel in all last month?
Now – Gather your materials and press PLAY. We’ll solve this problem together while you watch the math tutorial video below.
Problem Number 2 – On Your Own
Silvia the Silk Weaving Spider
Silvia the Silk Weaving Spider makes a new web ever day. She has to because bees, mosquitoes, and other bugs fly into her webs and wreck them. Yesterday, Silvia created a web that was made up of
25.8 yards of silk. Today, she created a new web that was made up of 37.07 yards of silk. How much silk did Silvia weave over the past two days?
Silvia is curious. She wants to know how much bigger today’s web is than yesterday’s web. How many more yards of silk was used to create today’s web?
Once you complete the problem – Hit PLAY on the math tutorial video below. Good Luck!
Problem Number 3 – On Your Own
Valentina Veterinarian
Valentina Veterinarian is the youngest animal doctor in the world. She has saved squirrels, rabbits, horses, and even a zebra or two. Today, Valentina is giving medicine to this squirrel. The
squirrel takes 3.275 milligrams of medicine in the morning and another 4.018 milligrams at night. How much medicine does Valentina Veterinarian give this squirrel in each day?
Dose Valentina Veterinarian give more medicine to this squirrel in the morning or at night? How much more?
Once you complete the problem – Hit PLAY on the math tutorial video below. Good Luck!
Each of my books includes 10 word problems like the three above. The books also include another 16 Drill & Kill problems like the 4 in challenge 11.
Drill & Kill
This is where we Drill until we Kill all our mistakes!
I call this section of the book, Drill & Kill, because we will drill this concept until we are perfect, and we kill any mistakes!
The following problems can all be solved with the same strategies we used to solve the first ten problems.
1. Solve all four problems on each page.
2. Watch the math tutorial video & correct your work.
3. Review your work with your parent or teacher.
If you get all 4 problems correct, your parent or teacher may tell you that you’re ready to move to the next book/article within this series.
Good Luck!
Drill & Kill
Challenge – 11
Challenge 1 Chalenge 2
6.231 – 3.9 7.456 + 3.07
Challenge 4
Challenge 3
0.671 + 32.19
4.6 – 0.945
Once you complete the problem – Hit PLAY on the math tutorial video below. Good Luck!
Please read on!
Each of my books includes 10 word problems and another 16 Drill & Kill problems. Each and every problem is linked to a video tutorial like the ones in this post. If you would like to purchase this
book, click the Photo.
Need Help with Multiplication or Division?
I have a series on multiplication and another series on division that are specifically designed for children who do not have all of their multiplication facts memorized. These series are designed
in a similar manner to this book, where each and every problem is linked to a video tutorial.
Click here to see my series on Multiplication.
Click her to see my series on Division.
How about getting a concrete understanding of fractions?
I have a series on illustrating fractions that is specifically designed to build a concrete understanding of this complex concept. Children who understand fractions at a deeper level are more likely
to have a successful educational career . The series is designed in a similar manner to this book, where each and every problem is linked to a video tutorial. Here is a link to my series on
Thank you for reading this article!
Want More Tutorials?
TeachersDungeon is an Educational Fantasy Game. It is 100% FREE! The game is set to the Common Core Educational Standards, and is web-based, so it can be played on any device. Many of the
questions are accompanied by tutorials like the ones you saw here.
One Last Thing
If you like this post and found it helpful, please leave a brief comment. As a teacher, perhaps the greatest reward I receive is from parents, children, and fellow teachers who use my strategies of
education and succeed.
My mission in life and as an educator is to make people feel empowered, self-assured, and happy about who they are in this world! We all have gifts to bestow upon our world. Go forth and do so, and
know that you are awesome!
Have a fantastic day – Brian McCoy
5 Comments
1. Multiplying with decimals is really easy, thanks to these books. I had not passed a multiplication exam, but these books really help me pass, and now I am one of the fastest learners in my grade.
2. Thanks for this strategy. I used the dot method on my homework today, and it really helped.
Thanks again. This is awesome!
3. I think the dot method is helpful because if your having trouble adding or subtracting, you can just use the dot method to figure it out. The answer will be more clear, you aren’t just seeing it
but counting outloud.
4. I really like the dot method. It can help people if they are having a hard time with adding or subtracting.
5. This is amazing. Thanks and keep making these video tutorials. They are very helpful!
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Fractional Quantum Hall States at ν=13/5 and 12/5 and Their Non-Abelian Nature
Topological quantum states with non-Abelian Fibonacci anyonic excitations are widely sought after for the exotic fundamental physics they would exhibit, and for universal quantum computing
applications. The fractional quantum Hall (FQH) state at a filling factor of ν=12/5 is a promising candidate; however, its precise nature is still under debate and no consensus has been achieved so
far. Here, we investigate the nature of the FQH ν=13/5 state and its particle-hole conjugate state at 12/5 with the Coulomb interaction, and we address the issue of possible competing states. Based
on a large-scale density-matrix renormalization group calculation in spherical geometry, we present evidence that the essential physics of the Coulomb ground state (GS) at ν=13/5 and 12/5 is captured
by the k=3 parafermion Read-Rezayi state (RR3), including a robust excitation gap and the topological fingerprint from the entanglement spectrum and topological entanglement entropy. Furthermore, by
considering the infinite-cylinder geometry (topologically equivalent to torus geometry), we expose the non-Abelian GS sector corresponding to a Fibonacci anyonic quasiparticle, which serves as a
signature of the RR3 state at 13/5 and 12/5 filling numbers.
All Science Journal Classification (ASJC) codes
• General Physics and Astronomy
Dive into the research topics of 'Fractional Quantum Hall States at ν=13/5 and 12/5 and Their Non-Abelian Nature'. Together they form a unique fingerprint. | {"url":"https://collaborate.princeton.edu/en/publications/fractional-quantum-hall-states-at-%CE%BD135-and-125-and-their-non-abel","timestamp":"2024-11-06T08:13:31Z","content_type":"text/html","content_length":"51387","record_id":"<urn:uuid:6e0b0773-2c1b-4bc4-993e-f8188cdd9d15>","cc-path":"CC-MAIN-2024-46/segments/1730477027910.12/warc/CC-MAIN-20241106065928-20241106095928-00150.warc.gz"} |
Studying biology, but missing math
• Studying
• Thread starter {ln2}{k}
• Start date
In summary: That is why it is important to take as many math classes as possible and to find a program that you are really interested in. Hope this helps.
That's my first year in college and I'm studying Biology. I'm pretty happy with everything (classmates, teachers) but the fact is, my favorite subjects so far have been maths and statistics (and my
favorite teachers were from the math department).
I study biology because, since I'm not sure in what field I'd like to work at, I can try a lot of different things (the subjects are very different one from another, and I like that).
It hasn't been three months yet since I finished both maths and statistics and I miss it a lot. I'd rather attend three hours of math than one of botanics (which bores me to death), and lately I've
been thinking about changing into math.
But I'm not sure about math either, because I'd like to work with something related to nature (closely related). So, I'll be auditing a math class (I haven't had the chance yet, I'll try to begin
after spring break), and I hope I'll be able to understand it, more or less.
It's not only that I miss math, which I do, but that I'm a little bit concerned in my math level. If I end up specializing in something like mathematical biology (that's my plan for the moment), will
I know enough math? In my college there's no specialization related to that (I'm ok with studying abroad), and almost nobody in my class is slightly interested in numbers, they hate them in fact, so
I don't have any friends who have the same 'problem'.
my question is: am I doing right, studying Biology and auditing math classes (and possibly reading books on math)? I'm a little bit confused...:(
First, are thinking graduate school? If you are wanting to do any type of research then graduate school is necessary. This will allow you to specialize in a specific field. It sounds as if you may be
interested in a biophysics program. Take a look
to see if it is something you may be interested in. If so, look around at different graduate programs to see what backgrounds they want for their incoming graduates.
Many biophysics programs are housed in the biology or biochemistry departments but also in physics. Many of the biophysics grad programs housed in the physics department require a physics BS to apply
while programs in the biology or biochem departments will take graduates from many different backgrounds. This includes biology, math, chemistry, biochemistry, physics, and even engineering. Most
programs require preparation in math: calc I, II, III and Differential equations (and some beyond this) and some type of undergraduate training in biology, chemistry, and biochemistry. With this
being said, if your lacking any of the subjects I just mentioned, most programs will allow you to take these courses as part of your graduate course work. This is simply because there are students
coming from all different backgrounds and it is impossible in most cases for students to be sufficiently trained in ALL categories.
Biophysics is great because there is so much variation in the research. You can be more on the physics side focusing on new imaging techniques and investigating how living matter organizes itself,
the mathematics/computational side looking at the structures of biological macromolecules (structural biology) such as how proteins fold, or the biology/biochem side of things that focuses more on
molecular mechanisms such as membrane ion channels (molecular biophysics). Keep in mind, this is just the tip of the iceberg and there is an unbelievable amount of overlap in all of the biophysics
research. I am drawn to this field because I'm similar to yourself, I love biology but I also LOVE math and physics. Anyway there are many opportunities in this field for the future.
I noticed that I didn't really answer your question about whether or not you will learn enough math... That is really up to how you spend your electives. You will certainly want to take more math
classes than your other classmates. I would recommend taking calc through differential equations at the minimum.
Now here is a little something to think about... The way I see it, it is much easier to learn mathematics as an undergrad and pick up the relevant biology as a graduate than vice versa. What you
really need to ask yourself is "Am I really interested in biology from a biologists perspective?". What I mean by this is that you will be taking many biology classes that you may not necessarily be
interested in (or ever actually use for that matter depending on your eventual interests) such as your botany class or ecology. On the other hand, mathematics will be giving you tools that you will
be able to use and apply. Every math class is another tool in the toolkit. Also, most math degrees allow a lot of free electives where you would be able to take biology classes that you are actually
interested in and that will benefit your goals. That is just something to think about.
If you want to be a field biologist studying population ecology, certainly stay with biology. If you want to be a mathematical biologist or something similar, biology is one path but there are
definitely other ways to play...
Many thanks jbrussell93 for you reply! It's nice that your interests are similar to mine and the information you've given me has certainly helped me.
jbrussell93 said:
Now here is a little something to think about... The way I see it, it is much easier to learn mathematics as an undergrad and pick up the relevant biology as a graduate than vice versa. What you
really need to ask yourself is "Am I really interested in biology from a biologists perspective?". What I mean by this is that you will be taking many biology classes that you may not necessarily
be interested in (or ever actually use for that matter depending on your eventual interests) such as your botany class or ecology. On the other hand, mathematics will be giving you tools that you
will be able to use and apply. Every math class is another tool in the toolkit. Also, most math degrees allow a lot of free electives where you would be able to take biology classes that you are
actually interested in and that will benefit your goals. That is just something to think about.
I'll have to think about that (nobody had put it this way before...). I still have some months to decide, and I hope I'll enjoy college whatever I choose :)
Do what you love and love what you do! I know it's a bit corney but definitely true in most cases. My biggest advice for you is to study and explore what you are interested in as an undergraduate.
You will have plenty of time in graduate school to specialize in something more specific. Good luck and enjoy!
Thank you very much! and good luck with your lecturing/research/whatever. You've really helped me:)
FAQ: Studying biology, but missing math
1. What is the importance of math in studying biology?
Math is essential in studying biology because it allows scientists to quantify and analyze biological data. Many biological processes can be described using mathematical equations, and understanding
these equations is crucial in understanding how living organisms function. Math is also used in experimental design and data interpretation, which are both essential components of biological
2. Can I still be a successful biologist without strong math skills?
While having strong math skills is beneficial in biology, it is not a requirement to be a successful biologist. There are many fields within biology that may require more or less math, and there are
also many tools and software available to assist with mathematical aspects of research. However, having a basic understanding of math concepts is important in order to fully comprehend and interpret
biological data.
3. What math concepts are most important in biology?
Some of the most important math concepts in biology include statistics, algebra, geometry, and calculus. Statistics is essential in experimental design and data analysis, while algebra is used in
modeling biological processes. Geometry is important in understanding the shape and structure of biological molecules, and calculus is used to describe rates of change in biological systems.
4. How can I improve my math skills for studying biology?
There are many ways to improve math skills for studying biology. One option is to take additional math courses or seek out tutoring. Another option is to practice using math in a biological context,
such as analyzing data from experiments or working through mathematical models of biological processes. Additionally, there are many online resources and textbooks available specifically for learning
math in a biological context.
5. Is it too late to improve my math skills for studying biology?
No, it is never too late to improve your math skills for studying biology. While it may take some extra effort and dedication, with practice and determination, anyone can improve their math skills.
It is important to remember that math is a skill that can be learned and developed, and it is never too late to start learning or improving upon it. | {"url":"https://www.physicsforums.com/threads/studying-biology-but-missing-math.591860/","timestamp":"2024-11-02T05:29:15Z","content_type":"text/html","content_length":"103138","record_id":"<urn:uuid:87368bea-8406-477c-9428-4b94c9953dec>","cc-path":"CC-MAIN-2024-46/segments/1730477027677.11/warc/CC-MAIN-20241102040949-20241102070949-00687.warc.gz"} |
Stability criteria for multiphase partitioning problems with volume constraints
[1] R. A. Adams and J. J. F. Fournier, Sobolev Spaces, Second edition, Pure and Applied Mathematics, 140, Elsevier/Academic Press, Amsterdam, 2003.
[2] N. D. Alikakos, P. W. Bates, J. W. Cahn, P. C. Fife, G. Fusco and G. B. Tanoglu, Analysis of a corner layer problem in anisotropic interfaces, Discrete Cont. Dyn.-B, 6 (2006), 237-255.
[3] F. J. Jr. Almgren, Existence and regularity almost everywhere of solutions to elliptic variational problems with constraints, Mem. Amer. Math. Soc. , 4 (1976), ⅷ+199pp.
[4] P. W. Bates and P. C. Fife, The dynamics of nucleation for the Cahn-Hilliard equation, SIAM J. Appl. Math., 53 (1993), 990-1008. doi: 10.1137/0153049.
[5] P. W. Bates and P. C. Fife, Spectral comparison principles for the Cahn-Hilliard and phase-field equations and time scales for coarsening, Physica D, 43 (1990), 335-348. doi: 10.1016/0167-2789
[6] G. Caginalp and P. C. Fife, Dynamics of layered interfaces arising from phase boundaries, SIAM J. Appl. Math., 48 (1988), 506-518. doi: 10.1137/0148029.
[7] R. Courant and D. Hilbert, Methods of Mathematical Physics, Vol. Ⅰ Interscience Publishers, New York, 1953.
[8] D. Depner and H. Garcke, Linearized stability analysis of surface diffusion for hypersurfaces with triple lines, Hokkaido Math. J, 42 (2013), 11-52. doi: 10.14492/hokmj/1362406637.
[9] S.-I. Ei, R. Ikota and M. Mimura, Segregating partition problem in competition-diffusion systems, Interface Free Bound., 1 (1999), 57-80. doi: 10.4171/IFB/4.
[10] P. C. Fife, Dynamics of Internal Layers and Diffusive Interfaces CBMS-NSF Regional Conference Series in Applied Mathematics, 53, Society for Industrial and Applied Mathematics (SIAM),
Philadelphia, 1988.
[11] P. C. Fife, Mathematical Aspects of Reacting and Diffusing Systems, Lecture Notes in Biomathematics 28, Springer-Verlag, Berlin-New York, 1979.
[12] P. C. Fife, A. Liñán and F. Williams (Editors), Dynamical Issues in Combustion Theory, Proceedings of the Workshop held in Minneapolis, Minnesota, November 1989 The IMA Volumes in Mathematics
and its Applications, 35, Springer-Verlag, New York, 1991.
[13] H. Garcke, K. Ito and Y. Kohsaka, Linearized stability analysis of stationary solutions for surface diffusion with boundary conditions, SIAM J. Math. Anal., 36 (2005), 1031-1056. doi: 10.1137/
[14] R. Ikota and E. Yanagida, Stability of stationary interfaces of binary-tree type, Calc. Var. PDE, 22 (2005), 375-389. doi: 10.1007/s00526-004-0281-x.
[15] R. Ikota and E. Yanagida, A stability criterion for stationary curves to the curvature-driven motion with a triple junction, Differential Integral Equations, 16 (2003), 707-726.
[16] W. W. Mullins, Two-dimensional motion of idealized grain boundaries, J. Appl. Phys., 27 (1956), 900-904. doi: 10.1063/1.1722511.
[17] J. C. C. Nitsche, Stationary partitioning of convex bodies, Arch. Ration. Mech. An., 89 (1985), 1-19. doi: 10.1007/BF00281743.
[18] J. C. C. Nitsche, Corrigendum to: Stationary partitioning of convex bodies Arch. Ration. Mech. An. 95 (1986), p389.
[19] H.-K. Rajni, Aqueous two-phase systems, Mol. Biotechnol, 19 (2001), 269-277.
[20] L. Simon, Lectures on Geometric Measure Theory, Proceedings of the Centre for Mathematical Analysis, 1983.
[21] P. Sternberg and K. Zumbrun, A Poincaré inequality with applications to volume-constrained area-minimizing surfaces, J. Reine Angew. Math., 503 (1998), 63-85.
[22] E. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces Princeton University Press, Princeton, 1971.
[23] M. Struwe, Variational Methods, Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, Fourth Edition, Springer, 2008.
[24] B. White, Existence of least-energy configurations of immiscible fluids, J. Geom. Anal, 6 (1996), 151-161. doi: 10.1007/BF02921571.
[25] J. Wloka, Partial Differential Equations, Cambridge University Press, Cambridge, 1992. doi: 10.1017/CBO9781139171755. | {"url":"https://www.aimsciences.org/article/doi/10.3934/dcds.2017028","timestamp":"2024-11-04T11:16:25Z","content_type":"text/html","content_length":"110710","record_id":"<urn:uuid:056f0cf1-f37c-4d32-b828-28f07b8767d1>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00317.warc.gz"} |
Mathematics and Computer Science
Giuseppe ZAPPALA'
Associate Professor of Geometry [MATH-02/B]
Giuseppe Zappalà was born in Catania on 27/9/1970.
In 1993 he graduated in Mathematics at the University of Catania. In the same year he won a scholarship
of the National Institute of Higher Mathematics.
In 1999 he obtained the title of PhD in Mathematics at the University of Palermo.
He has been a Researcher at the University of Catania since 1996, disciplinary scientific sector MAT / 03.
Since 2021 he has been Associate Professor at the DMI of the University of Catania, scientific disciplinary sector MAT/03.
His research interests mainly concern Commutative Algebra and Algebraic Geometry.
Personal data:
Born in Catania on 27 September 1970
Italian citizen
"Degree in Mathematics", University of Catania (1993)
Speaker: Alfio Ragusa
Thesis entitled: "The Hilbert-Burch theorem and applications"
PhD, University of Palermo (1999)
Speaker: Alfio Ragusa
Thesis entitled: "The 0-dimensional subschemes of irreducible curves in P^3"
Scholarship at the National Institute of High Mathematics (1993-1994)
Researcher at the University of Catania since 1996.
Associate Professor at the DMI of the University of Catania since 2021.
Research topics addressed
I dealt with research topics in the field of Algebraic Geometry and Commutative Algebra.
More specifically, I dealt with the following topics:
Hilbert functions of 0-dimensional subschemes of irreducible curves lying on a quadric surface;
arithmetically Gorenstein schemes;
fat points and fat patterns in projective spaces;
construction of patterns with particular properties;
reducibility of the Hilbert scheme of points in codimension 3 and connection with the existence of minimal Betti sequences;
properties of subschemes of sets of points;
merges of aCM patterns and full intersection patterns;
0-dimensional subschemas of products of projective spaces;
nearly complete intersections;
weak and strong Lefschetz properties;
configurations of algebraic varieties.
Scientific communications
"On the weak Lefschetz property for artinian Gorenstein algebras", Palermo, February 2012;
"The graded Betti numbers for almost complete intersections in codimension 3", Villafranca Tirrena, September 2009;
"r-codimensional Gorenstein schemes and almost complete intersections", Messina, November 2008;
"Subschemes of schemes with given graded Betti numbers", Piraino, September 2004;
"On some constructions of reduced aCM schemes", Acicastello, June 2002;
"3-Codimensional Gorenstein Arithmetic Schemes", Gargano, May 2000.
Other scientific activities
Referee for national and international journals.
Reviewer for Zentralblatt MATH
- Pragmatic (School of Research) in Catania 1997-2012 (15 editions)
- Pragmatic (Promotion of Researches in Algebraic Geometry for MAThematicians in Isolated Centers) is a research school whose activities take place annually at the Department of Mathematics and
Computer Science of the University of Catania. The courses are taught by internationally renowned experts. It has helped train a large number of algebraic algebraists and geometers from all over the
- He has been a member of the organizing committee of various international conferences.
Main research topics:
- Hilbert functions of graded algebras;
- Betti numbers of graduated algebras;
- Properties of the curves on the smooth quadric;
- Gorenstein algebras;
- Structure theorems for free graduated resolutions;
- Multiple algebraic varieties;
- Lefschetz property;
- Perazzo algebras;
- Almost complete intersection schemes;
- Almost Gorenstein schemes | {"url":"https://dmi.unict.it/faculty/giuseppe.zappala","timestamp":"2024-11-07T12:33:55Z","content_type":"text/html","content_length":"31944","record_id":"<urn:uuid:15ecf176-69aa-4429-b8e8-ad2db3ea8140>","cc-path":"CC-MAIN-2024-46/segments/1730477027999.92/warc/CC-MAIN-20241107114930-20241107144930-00321.warc.gz"} |
The SURVEYSELECT Procedure
Poisson sampling, which you request by specifying the METHOD=POISSON option, is an unequal probability sampling method for which the total sample size is not fixed. A generalization of Bernoulli
sampling, Poisson sampling also consists of independent random selection trials for the N sampling units in the input data set, but the sampling units can have different inclusion probabilities. You
provide inclusion probabilities for Poisson sampling in the variable that you specify in the SIZE statement.
The expected value of the sample size for Poisson sampling is , where is the inclusion probability for sampling unit i. The variance of the sample size is .
For Poisson sampling, the selection probability for unit i is the inclusion probability that you specify by using the SIZE statement. PROC SURVEYSELECT computes the sampling weight for unit i as the
inverse of the selection probability, which is . The joint selection probability for any two distinct units i and j is for Poisson sampling. See Särndal, Swensson, and Wretman (1992) for more | {"url":"http://support.sas.com/documentation/cdl/en/statug/65328/HTML/default/statug_surveyselect_details10.htm","timestamp":"2024-11-11T10:28:48Z","content_type":"application/xhtml+xml","content_length":"14882","record_id":"<urn:uuid:50a5ee6b-1690-4216-a069-e0361d5e0a83>","cc-path":"CC-MAIN-2024-46/segments/1730477028228.41/warc/CC-MAIN-20241111091854-20241111121854-00354.warc.gz"} |
Pipe (TL)
Closed conduit that transports fluid between thermal liquid components
Simscape / Fluids / Thermal Liquid / Pipes & Fittings
The Pipe (TL) block represents thermal liquid flow through a pipe. The block finds the temperature across the pipe from the differential between ports, pipe elevation, and any additional heat
transfer at port H.
The pipe can have a constant or varying elevation between ports A and B. For a constant elevation differential, use the Elevation gain from port A to port B parameter. You can specify a variable
elevation by setting Elevation gain specification to Variable. This exposes physical signal port EL.
You can choose to include the effects of fluid dynamic compressibility, inertia, and wall flexibility. When the block includes these phenomena, it calculates the flow properties for each number of
pipe segments that you specify.
Pipe Geometry
Use the Cross-sectional geometry parameter to specify the shape of the pipe.
The nominal hydraulic diameter, D[N], and the pipe diameter, d[circle], are both equal to the is the value of the Pipe diameter parameter. The pipe cross-sectional area is ${S}_{N}=\frac{\pi }{4}{d}_
The nominal hydraulic diameter is the difference between the Pipe outer diameter and Pipe inner diameter parameters D[N] = d[outer] – d[inner]. The pipe cross-sectional area is ${S}_{N}=\frac{\pi }
The nominal hydraulic diameter is
• h is the is the value of the Pipe height parameter.
• w is the is the value of the Pipe width parameter.
The pipe cross-sectional area is ${S}_{N}=wh.$
The nominal hydraulic diameter is
• a[maj] is the is the value of the Pipe major axis parameter.
• b[min] is the is the value of the Pipe minor axis parameter.
The pipe cross-sectional area is ${S}_{N}=\frac{\pi }{4}{a}_{maj}{b}_{min}.$
Isosceles Triangular
The nominal hydraulic diameter is
${D}_{N}={l}_{side}\frac{\mathrm{sin}\left(\theta \right)}{1+\mathrm{sin}\left(\frac{\theta }{2}\right)},$
• l[side] is the is the value of the Pipe side length parameter.
• θ is the is the value of the Pipe vertex angle parameter.
The pipe cross-sectional area is ${S}_{N}=\frac{{l}_{side}^{2}}{2}\mathrm{sin}\left(\theta \right).$
When the Cross-sectional geometry parameter is Custom, you can specify the pipe cross-sectional area with the Cross-sectional area parameter. The nominal hydraulic diameter is the value of the
Hydraulic diameter parameter.
Pipe Flexibility
You can model flexible walls for all cross-sectional geometries. When you set Pipe wall specification to Flexible, the block assumes uniform expansion along all directions and preserves the defined
cross-sectional shape. This setting may not result in physical results for noncircular cross-sectional areas undergoing high pressure relative to atmospheric pressure. When you model flexible walls,
you can use the Volumetric expansion specification parameter to specify the volumetric expansion of the pipe cross-sectional area.
When the Volumetric expansion specification parameter is Cross-sectional area vs. pressure, the change in volume is
$\stackrel{˙}{V}=L\left(\frac{A-S}{\tau }\right),$
• $A={S}_{N}+{K}_{ps}\left(p-{p}_{atm}\right).$
• L is the Pipe length parameter.
• S[N] is the nominal pipe cross-sectional area defined for each shape.
• S is the current pipe cross-sectional area.
• p is the internal pipe pressure.
• p[atm] is the atmospheric pressure.
• K[ps] is the Static gauge pressure to cross-sectional area gain parameter.
To calculate K[ps] assuming uniform elastic deformation of a thin-walled, open-ended cylindrical pipe, use
${K}_{ps}=\frac{\Delta D}{\Delta p}=\frac{\pi {D}_{N}^{3}}{4tE},$
where t is the pipe wall thickness and E is Young's modulus.
• τ is the Volumetric expansion time constant.
When the Volumetric expansion specification parameter is Cross-sectional area vs. pressure - Tabulated, the block uses the same equation for $\stackrel{˙}{V}$ as the Cross-sectional area vs. pressure
setting. The block calculates A with the table lookup function
where p[ps] is the Static gauge pressure vector parameter and A[ps] is the Cross sectional area gain vector parameter.
When the Volumetric expansion specification parameter is Hydraulic diameter vs. pressure, the change in volume is
$\stackrel{˙}{V}=\frac{\pi }{2}DL\left(\frac{{D}_{static}-D}{\tau }\right),$
• ${D}_{static}={D}_{N}+{K}_{pd}\left(p-{p}_{atm}\right).$
• D[N] is the nominal hydraulic diameter defined for each shape.
• D is the current pipe hydraulic diameter.
• K[pd] is the Static gauge pressure to hydraulic diameter gain parameter. To calculate K[ps] assuming uniform elastic deformation of a thin-walled, open-ended cylindrical pipe, use
${K}_{pd}=\frac{\Delta D}{\Delta p}=\frac{{D}_{N}^{2}}{2tE}.$
When the Volumetric expansion specification parameter is Based on material properties, the block uses the same equation for $\stackrel{˙}{V}$ as the Hydraulic diameter vs. pressure setting but
calculates D[static] depending on the value of the Material behavior parameter
This parameterization assumes a cylindrical thin-walled pressure vessel where ${\sigma }_{radial}=0.$
When the Material behavior parameter is Linear elastic,
${ϵ}_{hoop}=\frac{1}{E}\left[{\sigma }_{hoop}-v{\sigma }_{longitudinal}\right],$
• E is the value of the Young's modulus parameter.
• v is the value of the Poisson's ratio parameter.
• ${\sigma }_{hoop}=\frac{pD}{2t}$, where t is the value of the Pipe wall thickness parameter.
• ${\sigma }_{longitudinal}=\frac{pD}{4t}.$
When the Material behavior parameter is Multilinear elastic, the block calculates the von Mises stress, σ[v], which simplifies to ${\sigma }_{v}=\sqrt{\frac{3}{4}}\frac{pD}{2t}$, to determine the
equivalent strain. The hoop strain is
$\begin{array}{l}{ϵ}_{hoop}^{elastic}=\frac{1}{E}\left[{\sigma }_{hoop}-v{\sigma }_{longitudinal}\right]\\ {ϵ}_{hoo{p}_{i,j}}^{plastic}=\frac{3}{2}\left(\frac{1}{{E}_{s}}-\frac{1}{E}\right){S}_{i,j}\
• The block calculates the Young's Modulus, E, from the first elements of the Stress vector and Strain vector parameters.
• ${E}_{S}=\frac{{\sigma }_{total}}{{ϵ}_{total}}$, where σ[total] and ε[total] are the equivalent total stress and the equivalent total strain, respectively. The block calculates the equivalent
total strain from the von Mises stress and the stress-strain curve.
• ${\text{S}}_{i,j}={\sigma }_{i,j}-\left[\frac{{\sigma }_{hoop}+{\sigma }_{longitudinal}+{\sigma }_{radial}}{3}\right]{\delta }_{i,j},$ where σ[i,j] are the elements of the Cauchy stress tensor.
If you do not model flexible walls, S[N] = S and D[N] = D.
Pipe Wall Thermal Expansion
If you select Pipe thermal expansion, the block models the thermal expansion of the pipe wall using these assumptions:
• The pipe material is isotropic.
• The Biot number of the pipe is less than 0.1 and the pipe can be modeled with lumped thermal capacitance.
• The temperature change and pipe deformations are small enough that a first order approximation for area expansion is accurate.
When the Material behavior parameter is Cross-sectional area vs. pressure, Cross-sectional area vs. pressure - Tabulated, or Hydraulic diameter vs. pressure and you select Pipe thermal expansion, the
block adds a thermal expansion term when calculating area or diameter.
When Material behavior is Cross-sectional area vs. pressure,
$A={S}_{N}+{K}_{ps}\left(p-{p}_{atm}\right)+{S}_{N}2\alpha \Delta T,$
• ɑ is the value of the Coefficient of thermal expansion parameter.
• $\Delta T={T}_{I}-{T}_{ref}.$
• T[I] is the fluid temperature at the internal node of the block.
• T[ref] is the value of the Thermal expansion reference temperature parameter.
When Material behavior is Cross-sectional area vs. pressure - Tabulated,
$A={S}_{N}+tablelookup\left({p}_{ps},{A}_{ps},\left(p-{p}_{atm}\right),interpolation=linear,extrapolation=linear\right)+{S}_{N}2\alpha \Delta T.$
When Material behavior is Hydraulic diameter vs. pressure ,
${D}_{static}={D}_{N}+{K}_{pd}\left(p-{p}_{atm}\right)+{D}_{N}\alpha \Delta T.$
When the Material behavior parameter is Multilinear elastic and you select Pipe thermal expansion, the block calculates D[static] as
where ${ϵ}_{thermal}=\alpha \Delta T.$
Heat Transfer at the Pipe Wall
You can include heat transfer to and from the pipe walls in multiple ways. There are two analytical models: the Gnielinski correlation, which models the Nusselt number as a function of the Reynolds
and Prandtl numbers with predefined coefficients, and the Dittus-Boelter correlation - Nusselt = a*Re^b*Pr^c, which models the Nusselt number as a function of the Reynolds and Prandtl numbers with
user-defined coefficients.
The Nominal temperature differential vs. nominal mass flow rate, Tabulated data - Colburn factor vs. Reynolds number, and Tabulated data - Nusselt number vs. Reynolds number & Prandtl number are
lookup table parameterizations based on user-supplied data.
Heat transfer between the fluid and pipe wall occurs through convection, Q[Conv] and conduction, Q[Cond], where the net heat flow rate, Q[H] is Q[H]=Q[Conv]+Q[Cond].
Heat transfer due to conduction is:
• D is the nominal hydraulic diameter, D[N], if the pipe walls are rigid, and is the pipe steady-state diameter, D[S], if the pipe walls are flexible.
• k[I] is the thermal conductivity of the thermal liquid, defined internally for each pipe segment.
• S[H] is the surface area of the pipe wall.
• T[H] is the pipe wall temperature.
• T[I] is the fluid temperature at the internal node of the block.
Heat transfer due to convection is:
• c[p, Avg] is the average fluid specific heat which the block calculates using a lookup table.
• $\stackrel{˙}{m}$[Avg] is the average mass flow rate through the pipe.
• T[In] is the fluid inlet port temperature.
• h is the pipe heat transfer coefficient.
The heat transfer coefficient h is:
except when parameterizing by Nominal temperature differential vs. nominal mass flow rate, where k[Avg] is the average thermal conductivity of the thermal liquid over the entire pipe and Nu is the
average Nusselt number in the pipe.
Analytical Parameterizations
When Heat transfer parameterization is set to Gnielinski correlation and the flow is turbulent, the average Nusselt number is calculated as:
• f is the average Darcy friction factor, according to the Haaland correlation:
where ε[R] is the pipe Internal surface absolute roughness.
• Re is the Reynolds number.
• Pr is the Prandtl number.
When the flow is laminar, the data from [1] determines how the Nusselt number depends on the Cross-sectional geometry parameter:
• When Cross-sectional geometry is Circular, the Nusselt number is 3.66.
• When Cross-sectional geometry is Annular, the block calculates the Nusselt number from tabulated data using a lookup table with linear interpolation and nearest extrapolation.
$\frac{{D}_{inner}}{{D}_{outer}}$ Nusselt number
1/20 17.46
1/10 11.56
1/4 7.37
1/2 5.74
1 4.86
The block adjusts the calculated Nusselt number with a correction factor $\text{F = 0}\text{.86}{\left(\frac{{D}_{outer}}{{D}_{inner}}\right)}^{0.16}.$
• When Cross-sectional geometry is Rectangular, the block calculates the Nusselt number from tabulated data using a lookup table with linear interpolation and nearest extrapolation.
$\frac{\mathrm{min}\left(h,w\right)}{\mathrm{max}\left(h,w\right)}$ Nusselt number
0 7.54
1/8 5.60
1/6 5.14
1/4 4.44
1/3 3.96
1/2 3.39
1 2.98
• When Cross-sectional geometry is Elliptical, the block calculates the Nusselt number from tabulated data using a lookup table with linear interpolation and nearest extrapolation.
$\frac{{b}_{min}}{{a}_{maj}}$ Nusselt number
1/16 3.65
1/8 3.72
1/4 3.79
1/2 3.74
1 3.66
• When Cross-sectional geometry is Isosceles triangular, the block calculates the Nusselt number from tabulated data using a lookup table with linear interpolation and nearest extrapolation.
θ Nusselt number
10π/180 1.61
30π/180 2.26
60π/180 2.47
90π/180 2.34
120π/180 2.00
• When Cross-sectional geometry is Custom, the Nusselt number is the value of the Nusselt number for laminar flow heat transfer parameter.
When Heat transfer parameterization is set to Dittus-Boelter correlation and the flow is turbulent, the average Nusselt number is calculated as:
• a is the value of the Coefficient a parameter.
• b is the value of the Exponent b parameter.
• c is the value of the Exponent c parameter.
The block default Dittus-Boelter correlation is:
When the flow is laminar, the Nusselt number depends on the Cross-sectional geometry parameter.
Parameterization By Tabulated Data
When the Heat transfer parameterization parameter is set to Tabulated data - Colburn factor vs. Reynolds number, the average Nusselt number is calculated as:
where J[M] is the Colburn-Chilton factor.
When the Heat transfer parameterization parameter is set to Tabulated data - Nusselt number vs. Reynolds number & Prandtl number, the Nusselt number is interpolated from the three-dimensional array
of average Nusselt number as a function of both average Reynolds number and average Prandtl number:
When the Heat transfer parameterization parameter is set to Nominal temperature difference vs. nominal mass flow rate and the flow is turbulent, the heat transfer coefficient is calculated as:
• $\stackrel{˙}{m}$[N] is the value of the Nominal mass flow rate parameter.
• $\stackrel{˙}{m}$[Avg] is the average mass flow rate:
• h[N] is the nominal heat transfer coefficient, which is calculated as:
${h}_{\text{N}}=\frac{{\stackrel{˙}{m}}_{\text{N}}{c}_{\text{p,N}}}{{S}_{\text{H,N}}}\text{ln}\left(\frac{{T}_{\text{H,N}}-{T}_{\text{In, N}}}{{T}_{\text{H,N}}-{T}_{\text{Out,N}}}\right),$
□ S[H,N] is the nominal wall surface area.
□ T[H,N] is the value of the Nominal wall temperature parameter.
□ T[In,N] is the value of the Nominal inflow temperature parameter.
□ T[Out,N] is the value of the Nominal outflow temperature parameter.
This relationship is based on the assumption that the Nusselt number is proportional to the Reynolds number:
$\frac{hD}{k}\propto {\left(\frac{\stackrel{˙}{m}D}{S\mu }\right)}^{0.8}.$
If the pipe walls are rigid, the expression for the heat transfer coefficient becomes:
Pressure Loss Due to Friction
Haaland Correlation
The analytical Haaland correlation models losses due to wall friction either by aggregate equivalent length, which accounts for resistances due to nonuniformities as an added straight-pipe length
that results in equivalent losses, or by local loss coefficient, which directly applies a loss coefficient for pipe nonuniformities.
When the Local resistances specification parameter is set to Aggregate equivalent length and the flow in the pipe is lower than the Laminar flow upper Reynolds number limit, the pressure loss over
all pipe segments is:
$\Delta {p}_{f,A}=\frac{\upsilon \lambda }{2{D}^{2}S}\frac{L+{L}_{add}}{2}{\stackrel{˙}{m}}_{A},$
$\Delta {p}_{f,B}=\frac{\upsilon \lambda }{2{D}^{2}S}\frac{L+{L}_{add}}{2}{\stackrel{˙}{m}}_{B},$
• ν is the fluid kinematic viscosity.
• λ is the value of the Laminar friction constant for Darcy friction factor parameter, which you can define when the Cross-sectional geometry parameter is Custom and is otherwise equal to 64.
• D is the pipe hydraulic diameter.
• L[add] is the value of the Aggregate equivalent length of local resistances parameter.
• $\stackrel{˙}{m}$[A] is the mass flow rate at port A.
• $\stackrel{˙}{m}$[B] is the mass flow rate at port B.
When the Reynolds number is greater than the Turbulent flow lower Reynolds number limit, the pressure loss in the pipe is:
$\Delta {p}_{f,A}=\frac{f}{2{\rho }_{I}D{S}^{2}}\frac{L+{L}_{add}}{2}{\stackrel{˙}{m}}_{A}|{\stackrel{˙}{m}}_{A}|,$
$\Delta {p}_{f,B}=\frac{f}{2{\rho }_{I}D{S}^{2}}\frac{L+{L}_{add}}{2}{\stackrel{˙}{m}}_{B}|{\stackrel{˙}{m}}_{B}|,$
• f is the Darcy friction factor. This is approximated by the empirical Haaland equation and is based on the Surface roughness specification, ε, and pipe hydraulic diameter:
$f={\left\{-1.8{\mathrm{log}}_{10}\left[\frac{6.9}{\mathrm{Re}}+{\left(\frac{\epsilon }{3.7{D}_{h}}\right)}^{1.11}\right]\right\}}^{-2},$
Pipe roughness for brass, lead, copper, plastic, steel, wrought iron, and galvanized steel or iron are provided as ASHRAE standard values. You can also supply your own Internal surface absolute
roughness with the Custom setting.
• ρ[I] is the internal fluid density.
When the Local resistances specification parameter is set to Local loss coefficient and the flow in the pipe is lower than the Laminar flow upper Reynolds number limit, the pressure loss over all
pipe segments is:
$\Delta {p}_{f,A}=\frac{\upsilon \lambda }{2{D}^{2}S}\frac{L}{2}{\stackrel{˙}{m}}_{A}.$
$\Delta {p}_{f,B}=\frac{\upsilon \lambda }{2{D}^{2}S}\frac{L}{2}{\stackrel{˙}{m}}_{B}.$
When the Reynolds number is greater than the Turbulent flow lower Reynolds number limit, the pressure loss in the pipe is:
$\Delta {p}_{f,A}=\left(\frac{f\frac{L}{2}}{D}+{C}_{loss,total}\right)\frac{1}{2{\rho }_{I}{S}^{2}}{\stackrel{˙}{m}}_{A}|{\stackrel{˙}{m}}_{A}|,$
$\Delta {p}_{f,B}=\left(\frac{f\frac{L}{2}}{D}+{C}_{loss,total}\right)\frac{1}{2{\rho }_{I}{S}^{2}}{\stackrel{˙}{m}}_{B}|{\stackrel{˙}{m}}_{B}|,$
where C[loss,total] is the loss coefficient, which can be defined in the Total local loss coefficient parameter as either a single coefficient or the sum of all loss coefficients along the pipe.
Nominal Pressure Drop vs. Nominal Mass Flow Rate
The Nominal Pressure Drop vs. Nominal Mass Flow Rate parameterization characterizes losses with a loss coefficient for rigid or flexible walls. When the fluid is incompressible, the pressure loss
over the entire pipe due to wall friction is:
$\Delta {p}_{f,A}={K}_{p}{\stackrel{˙}{m}}_{A}\sqrt{{\stackrel{˙}{m}}_{A}^{2}+{\stackrel{˙}{m}}_{th}^{2}},$
where K[p] is:
${K}_{p}=\frac{\Delta {p}_{N}}{{\stackrel{˙}{m}}_{N}^{2}},$
• Δp[N] is the Nominal pressure drop, which can be defined either as a scalar or a vector.
• ${\stackrel{˙}{m}}_{N}$ is the Nominal mass flow rate, which can be defined either as a scalar or a vector.
When the you supply the Nominal pressure drop and Nominal mass flow rate parameters as vectors, the scalar value K[p] is determined from a least-squares fit of the vector elements.
Tabulated Data – Darcy Friction Factor vs. Reynolds Number
Pressure losses due to viscous friction can also be determined from user-provided tabulated data of the Darcy friction factor vector and the Reynolds number vector for turbulent Darcy friction factor
parameters. Linear interpolation is employed between data points.
Momentum Balance
The pressure differential over the pipe is due to the pressure at the pipe ports, friction at the pipe walls, and hydrostatic changes due to any change in elevation:
${p}_{\text{A}}-{p}_{\text{B}}=\Delta {p}_{f}+{\rho }_{\text{I}}g\Delta z,$
• p[A] is the pressure at a port A.
• p[B] is the pressure at a port B.
• Δp[f] is the pressure differential due to viscous friction, Δp[f,A]+Δp[f,B].
• g is the value of the Gravitational acceleration parameter or the signal at port G.
• Δz the elevation differential between port A and port B, or z[A] - z[B].
• ρ[I] is the internal fluid density, which is measured at each pipe segment. If fluid dynamic compressibility is not modeled, this is:
When fluid inertia is not modeled, the momentum balance between port A and internal node I is:
${p}_{\text{A}}-{p}_{\text{I}}=\Delta {p}_{f,A}+{\rho }_{\text{I}}g\frac{\Delta z}{2}.$
When fluid inertia is not modeled, the momentum balance between port B and internal node I is:
${p}_{\text{B}}-{p}_{\text{I}}=\Delta {p}_{f,B}-{\rho }_{\text{I}}g\frac{\Delta z}{2}.$
When fluid inertia is modeled, the momentum balance between port A and internal node I is:
${p}_{\text{A}}-{p}_{\text{I}}=\Delta {p}_{f,A}+{\rho }_{\text{I}}g\frac{\Delta z}{2}+\frac{{\stackrel{¨}{m}}_{\text{A}}}{S}\frac{L}{2},$
• $\stackrel{¨}{m}$[A] is the fluid inertia at port A.
• L is the value of the Pipe length parameter.
• S is the value of the Nominal cross-sectional area parameter.
When fluid inertia is modeled, the momentum balance between port B and internal node I is:
${p}_{\text{B}}-{p}_{\text{I}}=\Delta {p}_{f,B}-{\rho }_{\text{I}}g\frac{\Delta z}{2}+\frac{{\stackrel{¨}{m}}_{\text{B}}}{S}\frac{L}{2},$
$\stackrel{¨}{m}$[B] is the fluid inertia at port B.
Pipe Discretization
You can divide the pipe into multiple segments. If a pipe has more than one segment, the mass flow, energy flow, and momentum balance equations are calculated for each segment. Having multiple pipe
segments can allow you to track changes to variables such as fluid density when fluid dynamic compressibility is modeled.
If you would like to capture specific phenomena in your application, such as water hammer, choose a number of segments that provides sufficient resolution of the transient. The following formula,
from the Nyquist sampling theorem, provides a rule of thumb for pipe discretization into a minimum of N segments:
• L is the Pipe length.
• f is the transient frequency.
• c is the speed of sound.
For some applications, you may need to connect Pipe (TL) blocks in series. For example, you may require multiple pipe segments to define a thermal boundary condition along the length of a pipe. In
this case, model the pipe segments by using a Pipe (TL) block for each segment and use the thermal ports to set the thermal boundary condition.
Mass Balance
For a rigid pipe with an incompressible fluid, the pipe mass conversation equation is:
• $\stackrel{˙}{m}$[A] is the mass flow rate at port A.
• $\stackrel{˙}{m}$[B] is the mass flow rate at port B.
For a flexible pipe with an incompressible fluid, the pipe mass conservation equation is:
${\stackrel{˙}{m}}_{\text{A}}+{\stackrel{˙}{m}}_{\text{B}}={\rho }_{\text{I}}\stackrel{˙}{V},$
• ρ[I] is the thermal liquid density at internal node I. Each pipe segment has an internal node.
• $\stackrel{˙}{V}$ is the rate of deformation of the pipe volume.
For a flexible pipe with a compressible fluid, the mass within the pipe can change with pressure and temperature. The bulk modulus and thermal expansion coefficient of the thermal liquid account for
this dependence and the pipe mass conservation equation is:
${\stackrel{˙}{m}}_{\text{A}}+{\stackrel{˙}{m}}_{\text{B}}={\rho }_{\text{I}}\stackrel{˙}{V}+{\rho }_{\text{I}}V\left(\frac{{\stackrel{˙}{p}}_{\text{I}}}{{\beta }_{\text{I}}}+{\alpha }_{\text{I}}{\
• p[I] is the thermal liquid pressure at the internal node I.
• $\stackrel{˙}{T}$[I] is the rate of change of the thermal liquid temperature at the internal node I.
• β[I] is the thermal liquid bulk modulus.
• α[I] is the liquid thermal expansion coefficient.
Energy Balance
The energy accumulation rate in the pipe at internal node I is defined as:
$\stackrel{.}{E}={\varphi }_{\text{A}}+{\varphi }_{\text{B}}+{Q}_{\text{H}}-{\stackrel{˙}{m}}_{Avg}g\Delta z,$
• ϕ[A] is the energy flow rate at port A.
• ϕ[B] is the energy flow rate at port B.
• Q[H] is the heat transfer through the pipe wall.
If the fluid is incompressible, the expression for energy accumulation rate is
$\stackrel{˙}{E}={\rho }_{\text{I}}{c}_{{p}_{I}}V\frac{d{T}_{I}}{dt},$
• c[p[I]] is the fluid specific heat at the internal node of the block.
• V is the pipe volume.
If the fluid is compressible, the expression for energy accumulation rate is
$\stackrel{˙}{E}={\frac{\partial \left({\rho }_{\text{I}}{u}_{I}\right)}{\partial p}|}_{T}\frac{d{p}_{I}}{dt}V+{\frac{\partial \left({\rho }_{\text{I}}{u}_{I}\right)}{\partial T}|}_{p}\frac{d{T}_{I}}
$\begin{array}{l}{\frac{\partial \left({\rho }_{\text{I}}{u}_{I}\right)}{\partial p}|}_{T}=\left(\frac{{\rho }_{I}{h}_{I}}{{\beta }_{I}}-{T}_{I}{\alpha }_{I}\right)V\\ {\frac{\partial \left({\rho }_
{\text{I}}{u}_{I}\right)}{\partial T}|}_{p}=\left({c}_{{p}_{I}}-{h}_{I}{\alpha }_{I}\right)V\end{array}$
and h[I] is the specific enthalpy at the internal node of the block.
If the fluid is compressible and the pipe walls are flexible, the expression for energy accumulation rate is
$\stackrel{˙}{E}={\frac{\partial \left({\rho }_{\text{I}}{u}_{I}\right)}{\partial p}|}_{T}\frac{d{p}_{I}}{dt}V+{\frac{\partial \left({\rho }_{\text{I}}{u}_{I}\right)}{\partial T}|}_{p}\frac{d{T}_{I}}
{dt}V+{\rho }_{\text{I}}{h}_{I}\frac{dV}{dt}.$
EL — Port elevation difference, m
physical signal
Variable elevation differential between port A and B, specified as a physical signal. The block bounds the signal value at EL between -L and L, where L is the value of the Pipe length parameter.
To enable this port, set Elevation gain specification to Variable.
G — Gravitational acceleration, m/s^2
physical signal
Variable gravitational acceleration, specified as a physical signal.
To enable this port, set Gravitational acceleration specification to Variable.
A — Pipe opening
thermal liquid
Liquid entry or exit port to the pipe.
B — Pipe opening
thermal liquid
Liquid entry or exit port to the pipe.
Fluid dynamic compressibility — Whether to model fluid dynamic compressibility
on (default) | off
Whether to model any change in fluid density due to fluid compressibility. When you select Fluid compressibility, changes due to the mass flow rate into the block are calculated in addition to
density changes due to changes in pressure.
Fluid inertia — Whether to model fluid acceleration
off (default) | on
Whether to account for acceleration in the mass flow rate due to the mass of the fluid.
To enable this parameter, select Fluid dynamic compressibility.
Number of segments — Pipe discretization
1 (default) | positive unitless scalar
Number of pipe divisions. Each division represents an individual segment over which pressure is calculated, depending on the pipe inlet pressure, fluid compressibility, and wall flexibility, if
applicable. The fluid volume in each segment remains fixed.
To enable this parameter, select Fluid dynamic compressibility.
Pipe total length — Total pipe length
5 m (default) | positive scalar in units of length
Total pipe length across all pipe segments.
Cross-sectional geometry — Pipe geometry
Circular (default) | Annular | Rectangular | Elliptical | Isosceles triangular | Custom
Cross-sectional pipe geometry. A nominal hydraulic diameter and nominal cross-sectional area is calculated based on the cross-sectional geometry.
Pipe diameter — Pipe diameter
0.1 m (default) | positive scalar
Diameter for circular cross-sectional pipes.
To enable this parameter, set Cross-sectional geometry to Circular.
Pipe inner diameter — Pipe inner diameter
0.05 m (default) | positive scalar
Inner diameter for annular pipe flow, or flow between two concentric pipes.
To enable this parameter, set Cross-sectional geometry to Annular.
Pipe outer diameter — Pipe outer diameter
0.1 m (default) | positive scalar
Outer diameter for annular pipe flow, or flow between two concentric pipes.
To enable this parameter, set Cross-sectional geometry to Annular.
Pipe width — Rectangular pipe width
0.1 m (default) | positive scalar
Width of rectangular pipe.
To enable this parameter, set Cross-sectional geometry to Rectangular.
Pipe height — Rectangular pipe height
0.1 m (default) | positive scalar
Height of rectangular pipe.
To enable this parameter, set Cross-sectional geometry to Rectangular.
Pipe major axis — Elliptical pipe major axis
0.1 m (default) | positive scalar
Major axis for elliptical pipes.
To enable this parameter, set Cross-sectional geometry to Elliptical.
Pipe minor axis — Elliptical pipe minor axis
0.05 m (default) | positive scalar
Minor axis for elliptical pipes.
To enable this parameter, set Cross-sectional geometry to Elliptical.
Pipe side length — Triangular pipe side length
0.1 m (default) | positive scalar
Length of the two equal sides of isosceles-triangular pipes.
To enable this parameter, set Cross-sectional geometry to Isosceles triangular.
Pipe vertex angle — Triangular pipe vertex angle
30 deg (default) | positive scalar
Vertex angle for triangular pipes. The value must be less than 180 degrees.
To enable this parameter, set Cross-sectional geometry to Isosceles triangular.
Cross-sectional area — Cross-sectional pipe area without deformation
0.01 m^2 (default) | positive scalar
Cross-sectional area of the pipe without deformations.
To enable this parameter, set Cross-sectional geometry to Custom.
Hydraulic diameter — Effective diameter of noncircular pipes
0.1128 m (default) | positive scalar in units of length
Effective diameter used in heat transfer, momentum balance, and pipe flexibility equations. For noncircular pipes, the hydraulic diameter is the effective diameter of the fluid in the pipe. For
circular pipes, the hydraulic diameter and pipe diameter are the same.
To enable this parameter, either:
• Clear the Fluid dynamic compressibility check box and set Cross-sectional geometry to Custom.
• Select Fluid dynamic compressibility, set the Pipe wall specification parameter to Rigid and set Cross-sectional geometry to Custom.
Elevation gain specification — Set pipe elevation property
Constant (default) | Variable
Set the pipe elevation as either Constant or Variable. Selecting Variable exposes the physical signal port EL.
Elevation gain from port A to port B — Change in elevation from port A to port B
0 m (default) | positive scalar in units of length
Elevation differential for constant-elevation pipes.
To enable this parameter, set Elevation gain specification to Constant.
Gravitational acceleration specification — Acceleration specification
Constant (default) | Variable
Whether the gravitational acceleration is constant or variable.
Gravitational acceleration — Acceleration due to gravity at mean pipe elevation
9.81 m/s^2 (default) | scalar
Constant of the gravitational acceleration at the mean elevation of the pipe.
To enable this parameter, set Gravitational acceleration specification to Constant.
Viscous Friction
Viscous friction parameterization — Friction model
Haaland correlation (default) | Nominal pressure drop vs. nominal mass flow rate | Tabulated data - Darcy friction factor vs. Reynolds number
Parameterization of pressure losses due to wall friction. Both analytical and tabular formulations are available.
Local resistances specification — Method for quantifying pressure losses in the Haaland correlation
Aggregate equivalent length (default) | Local loss coefficient
Method for quantifying pressure losses due to pipe nonuniformities.
To enable this parameter, set Viscous friction parameterization to Haaland correlation.
Total local loss coefficient — Defines loss coefficient over the pipe
0.1 (default) | positive scalar
Loss coefficient associated with each pipe nonuniformity. You can input a single loss coefficient or the sum of all loss coefficients along the pipe.
To enable this parameter, set Viscous friction parameterization to Haaland correlation and Local resistance specifications to Local loss coefficient.
Aggregate equivalent length of local resistances — Minor pressure loss in the pipe expressed as a length
1 m (default) | positive scalar
Length of pipe that would produce the equivalent hydraulic losses as would a pipe with bends, area changes, or other nonuniformities. The effective length of the pipe is the sum of the Pipe length
and the Aggregate equivalent length of local resistances.
To enable this parameter, set Viscous friction parameterization to Haaland correlation and Local resistances specification to Aggregate equivalent length.
Surface roughness specification — Pipe material for roughness specification
Commercially smooth brass, lead, copper, or plastic pipe: 1.52 um (default) | Steel and wrought iron : 46 um | Galvanized iron or steel : 152 um | Cast iron : 259 um | Custom
Absolute surface roughness based on pipe material. The provided values are ASHRAE standard roughness values. You can also input your own value by setting Surface roughness specification to Custom.
To enable this parameter, set Viscous friction parameterization to Haaland correlation.
Internal surface absolute roughness — Pipe wall absolute roughness
1.5e-5 m (default) | positive scalar in units of length
Pipe wall absolute roughness. This parameter is used to determine the Darcy friction factor, which contributes to pressure loss in the pipe.
To enable this parameter, set Viscous friction parameterization to Haaland correlation and Surface roughness specification Custom.
Laminar friction constant for Darcy friction factor — Friction constant for laminar flows
64 (default) | positive scalar
Friction constant for laminar flows. The Darcy friction factor captures the contribution of wall friction in pressure loss calculations. If Cross-sectional geometry is not set to Custom, this
parameter is internally set to 64.
To enable this parameter, set Cross-sectional geometry to Custom.
Laminar flow upper Reynolds number limit — Reynolds number below which the flow is laminar
2e+3 (default) | positive scalar
Reynolds number below which the flow is laminar. Above this threshold, the flow transitions to turbulent, reaching the turbulent regime at the Turbulent flow lower Reynolds number limit setting.
Turbulent flow lower Reynolds number limit — Reynolds number above which the flow is turbulent
4e+3 (default) | positive scalar
Reynolds number above which the flow is turbulent. Below this threshold, the flow gradually transitions to laminar, reaching the laminar regime at the Laminar flow upper Reynolds number limit
Nominal mass flow rate — Pipe mass flow rate
[0.1 1] kg/s (default) | scalar or vector of numbers in units of mass/time
Pipe nominal mass flow rate used to calculate the pressure loss coefficient, specified as a scalar or a vector. All nominal values must be greater than 0 and have the same number of elements as the
Nominal pressure drop parameter. When this parameter is supplied as a vector, the scalar value K[p] is determined as a least-squares fit of the vector elements.
To enable this parameter, set Viscous friction parameterization to Nominal pressure drop vs. nominal mass flow rate.
Nominal pressure drop — Pressure drop over the pipe
[0.001 0.01] MPa (default) | scalar or vector of numbers in units of pressure
Pipe nominal pressure drop used to calculate the pressure loss coefficient, specified as a scalar or a vector. All nominal values must be greater than 0 and have the same number of elements as the
Nominal mass flow rate parameter. When this parameter is supplied as a vector, the scalar value K[p] is determined as a least-squares fit of the vector elements.
To enable this parameter, set Viscous friction parameterization to Nominal pressure drop vs. nominal mass flow rate.
Mass flow rate threshold for flow reversal — Threshold below which numerical smoothing is applied
1e-6 kg/s (default) | positive scalar in units of mass/time
Mass flow rate threshold for reversed flow. A transition region is defined around 0 kg/s between the positive and negative values of the mass flow rate threshold. Within this transition region,
numerical smoothing is applied to the flow response. The threshold value must be greater than 0.
To enable this parameter, set Viscous friction parameterization to Nominal pressure drop vs. nominal mass flow rate.
Reynolds number vector for turbulent Darcy friction factor — Reynolds numbers at which to tabulate the Darcy friction factor
[ 400 1000 1.5e+3 3e+3 4e+3 6e+3 1e+4 2e+4 4e+4 6e+4 1e+5 1e+8 ] (default) | vector of positive numbers
Vector of Reynolds numbers for the tabular parameterization of the Darcy friction factor. The vector elements form an independent axis with the Darcy friction factor vector parameter. The vector
elements must be listed in ascending order and must be greater than 0.
To enable this parameter, set Viscous friction parameterization to Tabulated data - Darcy friction factor vs. Reynolds number.
Darcy friction factor vector — Darcy friction factors at the tabulated Reynolds numbers
[ 0.264 0.112 0.07099999999999999 0.0417 0.0387 0.0268 0.025 0.0232 0.0226 0.022 0.0214 0.0214 ] (default) | vector of positive numbers
Vector of Darcy friction factors for the tabular parameterization of the Darcy friction factor. The vector elements must correspond one-to-one with the elements in the Reynolds number vector for
turbulent Darcy friction factor parameter, and must be unique and greater than or equal to 0.
To enable this parameter, set Viscous friction parameterization to Tabulated data - Darcy friction factor vs. Reynolds number.
Pipe Wall
Pipe wall specification — Pipe wall flexibility
Rigid (default) | Flexible
Wall flexibility of the pipe. This parameter is independent of pipe cross-sectional geometry. The Flexible setting preserves the initial pipe shape and applies equal expansion of the cross-sectional
area. The Flexible setting may not be accurate for non-circular cross-sectional geometry under high deformation.
To enable this parameter, select Fluid dynamic compressibility.
Volumetric expansion specification — Linear expansion correlation
Cross-sectional area vs. pressure (default) | Cross-sectional area vs. pressure - Tabulated | Hydraulic diameter vs. pressure | Based on material properties
Linear expansion correlation. The settings correlate the new cross-sectional area or hydraulic diameter to the pipe pressure.
To enable this parameter, select Fluid dynamic compressibility and set Pipe wall specification to Flexible.
Static gauge pressure to cross-sectional area gain — Coefficient for area-dependent pipe deformation
1e-6 m^2/MPa (default) | positive scalar
Coefficient for calculating pipe deformation. The block multiplies the value of this parameter by the pressure differential between the segment pressure and atmospheric pressure.
To enable this parameter, select Fluid dynamic compressibility, set Pipe wall specification to Flexible, and set Volumetric expansion specification to Cross-sectional area vs. pressure.
Static gauge pressure vector — Vector of gauge pressures
[.1,1] MPa (default) | vector
Vector that contains the gauge pressures. The block uses this vector in a table lookup to calculate the pipe cross-sectional area. The vector entries must be strictly positive and monotonically
increasing and the vector must be the same length as the Cross sectional area gain vector parameter.
To enable this parameter, select Fluid dynamic compressibility, set Pipe wall specification to Flexible, and set Volumetric expansion specification to Cross-sectional area vs. pressure - Tabulated.
Cross sectional area gain vector — Vector of pipe cross-sectional areas
[1e-05, 1.1e-05] m^2 (default) | vector
Vector that contains the pipe cross-sectional areas. The block uses this vector in a table lookup to calculate the pipe cross sectional-area at other pressures. The vector entries must be strictly
positive and monotonically increasing and the vector must be the same length as the Static gauge pressure vector parameter.
To enable this parameter, select Fluid dynamic compressibility, set Pipe wall specification to Flexible, and set Volumetric expansion specification to Cross-sectional area vs. pressure - Tabulated.
Static gauge pressure to hydraulic diameter gain — Coefficient for diameter-dependent pipe deformation
1e-6 m/MPa (default) | positive scalar
Coefficient for calculating the pipe deformation. The block multiplies the value of this parameter by the pressure differential between the segment pressure and atmospheric pressure.
To enable this parameter, select Fluid dynamic compressibility, set Pipe wall specification to Flexible, and set Volumetric expansion specification to Hydraulic diameter vs. pressure.
Material behavior — Method to specify material behavior
Linear Elastic (default) | Multilinear Elastic
Method the block uses to calculate the material behavior.
To enable this parameter, select Fluid dynamic compressibility, set Pipe wall specification to Flexible, and set Volumetric expansion specification to Based on material properties.
Pipe wall thickness — Width of pipe wall
0.05 m (default) | positive scalar
Thickness of the pipe wall. The block uses this value to calculate stress.
To enable this parameter, select Fluid dynamic compressibility, set Pipe wall specification to Flexible, and set Volumetric expansion specification to Based on material properties.
Young's modulus — Young's modulus of pipe wall material
69 GPa (default) | positive scalar
Young's modulus of the material that makes up the pipe wall.
To enable this parameter, select Fluid dynamic compressibility, and set Pipe wall specification to Flexible, Volumetric expansion specification to Based on material properties, and Material behavior
to Linear Elastic.
Poisson's ratio — Poisson's ratio of pipe wall material
0.33 (default) | positive scalar
Poisson's ratio of the material that makes up the pipe wall.
To enable this parameter, select Fluid dynamic compressibility, set Pipe wall specification to Flexible, and set Volumetric expansion specification to Based on material properties.
Stress vector — Stress vector of pipe wall material
[276, 310] MPa (default) | vector
Vector containing the stress values for the material that makes up the pipe wall.
To enable this parameter, select Fluid dynamic compressibility and set Pipe wall specification to Flexible, Volumetric expansion specification to Based on material properties, and Material behavior
to Multilinear Elastic.
Strain vector — Strain vector of pipe wall material
[.004, .02] (default) | vector
Vector containing the strain values for the material that makes up the pipe wall.
To enable this parameter, select Fluid dynamic compressibility and set Pipe wall specification to Flexible, Volumetric expansion specification to Based on material properties, and Material behavior
to Multilinear Elastic.
Check if stress exceeds specified allowable level — Notification when stress is above specified max
None (default) | Warning | Error
Whether the block does nothing, generates a warning, or generates an error when the stress is above the maximum stress specified by the Maximum allowable stress parameter.
To enable this parameter, select Fluid dynamic compressibility and set Pipe wall specification to Flexible, Volumetric expansion specification to Based on material properties, and Material behavior
to Multilinear Elastic.
Maximum allowable stress — Maximum allowed stress on pipe wall
400 MPa (default) | positive scalar
Maximum stress the block allows on the pipe wall. Control what the block does if the stress exceeds this value with the Check if stress exceeds specified allowable level parameter.
To enable this parameter, select Fluid dynamic compressibility and set Pipe wall specification to Flexible, Volumetric expansion specification to Based on material properties, Material behavior to
Multilinear Elastic and Check if stress exceeds specified allowable level to Warning or Error.
Volumetric expansion time constant — Pipe deformation time constant
0.01 s (default) | positive scalar
Time required for the wall to reach steady-state after pipe deformation. This parameter impacts the dynamic change in pipe volume.
To enable this parameter, select Fluid dynamic compressibility and set Pipe wall specification to Flexible.
Pipe thermal expansion — Whether to model thermal expansion
off (default) | on
Whether to account for expansion in the pipe due to temperature changes.
To enable this parameter, select Fluid dynamic compressibility and set Pipe wall specification to Flexible.
Coefficient of thermal expansion — Pipe coefficient of thermal expansion
24 um/(deltaK*m) (default) | positive scalar
Coefficient of linear thermal expansion for the pipe material. This value is the fractional change in size per degree change in temperature at a constant pressure.
To enable this parameter, select Fluid dynamic compressibility, set Pipe wall specification to Flexible, and select Pipe thermal expansion.
Thermal expansion reference temperature — Reference temperature for pipe thermal expansion
293.15 K (default) | positive scalar
Reference temperature the block uses when calculating the thermal expansion of the pipe.
To enable this parameter, select Fluid dynamic compressibility, set Pipe wall specification to Flexible, and select Pipe thermal expansion.
Heat Transfer
Heat transfer parameterization — Method by which to capture the convective heat transfer with the pipe wall
Gnielinski correlationNominal temperature differential vs. nominal mass flow rate (default) | Dittus-Boelter correlation | Tabulated data - Colburn factor vs. Reynolds number | Tabulated data -
Nusselt number vs. Reynolds number & Prandtl number
Method of calculating the heat transfer coefficient between the fluid and the pipe wall. Analytical and tabulated data parameterizations are available.
Nusselt number for laminar flow heat transfer — Nusselt number to use in the heat transfer calculations for laminar flows
3.66 (default)
Ratio of convective to conductive heat transfer in the laminar flow regime. The fluid Nusselt number influences the heat transfer rate.
To enable this parameter, set Cross-sectional geometry to Custom and set Heat transfer parameterization to either:
• Gnielinski correlation.
• Nominal temperature differential vs. nominal mass flow rate.
• Dittus-Boelter correlation.
Nominal mass flow rate — Pipe mass flow rate
[0.1 1] kg/s (default) | scalar or vector of numbers in units of mass/time
Pipe nominal mass flow rate used to calculate the heat transfer coefficient, specified as a scalar or a vector. All nominal values must be greater than 0 and have the same number of elements as the
Nominal inflow temperature parameter. When this parameter is supplied as a vector, the scalar value h[p] is determined as a least-squares fit of the vector elements.
To enable this parameter, set Heat transfer parameterization to Nominal temperature differential vs. nominal mass flow rate.
Nominal inflow temperature — Pipe inlet temperature
[293.15 293.15] K (default) | scalar or vector of numbers in units of temperature
Nominal fluid inlet temperature used to calculate the heat transfer coefficient, specified as a scalar or a vector. All nominal values must be greater than 0 and have the same number of elements as
the Nominal mass flow rate parameter. When this parameter is supplied as a vector, the scalar value h is determined as a least-squares fit of the vector elements.
To enable this parameter, set Heat transfer parameterization to Nominal temperature differential vs. nominal mass flow rate.
Nominal outflow temperature — Pipe outlet temperature
[300 300] K (default) | scalar or vector of numbers in units of temperature
Nominal fluid outlet temperature used to calculate the heat transfer coefficient, specified as a scalar or a vector. All nominal values must be greater than 0 and have the same number of elements as
the Nominal mass flow rate parameter. When this parameter is supplied as a vector, the scalar value h is determined as a least-squares fit of the vector elements.
To enable this parameter, set Heat transfer parameterization to Nominal temperature differential vs. nominal mass flow rate.
Nominal inflow pressure — Pipe inlet pressure
[0.101325 0.101325] MPa (default) | scalar or vector of numbers in units of pressure
Nominal fluid inlet pressure used to calculate the heat transfer coefficient, specified as a scalar or a vector. All nominal values must be greater than 0 and have the same number of elements as the
Nominal mass flow rate parameter. When this parameter is supplied as a vector, the scalar value h is determined as a least-squares fit of the vector elements.
To enable this parameter, set Heat transfer parameterization to Nominal temperature differential vs. nominal mass flow rate.
Nominal wall temperature — Pipe wall temperature
[303.15 303.15] K (default) | scalar or vector of numbers in units of temperature
Pipe wall temperature used to calculate the heat transfer coefficient, specified as a scalar or a vector. All nominal values must be greater than 0 and have the same number of elements as the Nominal
mass flow rate parameter. When this parameter is supplied as a vector, the scalar value h is determined as a least-squares fit of the vector elements.
To enable this temperature, set Heat transfer parameterization to Nominal temperature differential vs. nominal mass flow rate.
Coefficient a — Empirical constant a of the Dittus-Boelter correlation
0.023 (default) | positive scalar
Empirical constant a to use in the Dittus-Boelter correlation. The correlation relates the Nusselt number in turbulent flows to the heat transfer coefficient.
To enable this parameter, set Heat transfer parameterization to Dittus-Boelter correlation.
Exponent b — Empirical constant b of the Dittus-Boelter correlation
0.8 (default) | positive scalar
Empirical constant b to use in the Dittus-Boelter correlation. The correlation relates the Nusselt number in turbulent flows to the heat transfer coefficient.
To enable this parameter, set Heat transfer parameterization to Dittus-Boelter correlation.
Exponent c — Empirical constant c of the Dittus-Boelter correlation
0.4 (default) | positive scalar
Empirical constant c to use in the Dittus-Boelter correlation. The correlation relates the Nusselt number in turbulent flows to the heat transfer coefficient. The default value reflects heat transfer
to the fluid.
To enable this parameter, set Heat transfer parameterization to Dittus-Boelter correlation.
Reynolds number vector for Colburn factor — Reynolds numbers at which to tabulate the Colburn factor
[100 150 1000] (default) | vector of positive numbers
Vector of Reynolds numbers for the tabular parameterization of the Colburn factor. The vector elements form an independent axis with the Colburn factor vector parameter. The vector elements must be
listed in ascending order and must be greater than 0. This parameter must have the same number of elements as the Colburn factor vector. For reversed flows, or flows from B to A, the same data is
applied in the opposite direction.
To enable this parameter, set Heat transfer parameterization to Tabulated data - Colburn factor vs. Reynolds number.
Colburn factor vector — Colburn factors at which the tabulated Reynolds numbers
[0.019 0.013 0.002] (default) | vector of positive numbers
Vector of Colbrun factors for the tabular parameterization of the Colburn factor. The vector elements form an independent axis with the Reynolds number vector for Colburn factor parameter. This
parameter must have the same number of elements as the Reynolds number vector for Colburn factor.
This parameter is active when the Heat transfer parameterization block parameter is set to Tabulated data - Colburn factor vs. Reynolds number.
Reynolds number vector for Nusselt number — Reynolds number at which to tabulate the Nusselt number
[100 150 100] (default) | vector of positive numbers
Vector of Reynolds numbers for the tabular parameterization of Nusselt number. This vector forms an independent axis with the Prandtl number vector for Nusselt number parameter for the 2-D dependent
Nusselt number table. The vector elements must be listed in ascending order and must be greater than 0.
To enable this parameter, set Heat transfer parameterization to Tabulated data - Nusselt number vs. Reynolds number & Prandtl number.
Prandtl number vector for Nusselt number — Prandtl numbers at which to tabulate the Nusselt number
[1 10] (default) | vector of positive numbers
Vector of Prandtl numbers for the tabular parameterization of Nusselt number. This vector forms an independent axis with the Reynolds number vector for Nusselt number parameter for the 2-D dependent
Nusselt number table. The vector elements must be listed in ascending order.
To enable this parameter, set Heat transfer parameterization to Tabulated data - Nusselt number vs. Reynolds number & Prandtl number.
Nusselt number table — Nusselt numbers at the tabulated Reynolds and Prandtl numbers
[ 3.72 4.21; 3.75 4.44; 4.21 7.15 ] (default) | matrix of positive numbers
M-by-N matrix of Nusselt numbers at the specified Reynolds and Prandtl numbers. Linear interpolation is employed between table elements. M and N are the sizes of the corresponding vectors:
• M is the number of vector elements in the Reynolds number vector for Nusselt number parameter.
• N is the number of vector elements in the Prandtl number vector for Nusselt number parameter.
To enable this parameter, Heat transfer parameterization to Tabulated data - Nusselt number vs. Reynolds number & Prandtl number.
Initial Conditions
Initial liquid temperature — Absolute temperature in the pipe at the start of simulation
293.15 K (default) | positive scalar or vector in units of temperature
Liquid temperature at the start of the simulation, specified as a scalar or vector. A vector n elements long defines the liquid temperature for each of n pipe segments. If the vector is two elements
long, the temperature along the pipe is linearly distributed between the two element values. If the vector is three or more elements long, the initial temperature in the nth segment is set by the nth
element of the vector.
Initial liquid pressure — Absolute pressure in the pipe at the start of simulation
0.101325 MPa (default) | positive scalar or vector in units of pressure
Absolute liquid pressure at the start of the simulation, specified as a scalar or vector. A vector n elements long defines the liquid pressure for each of n pipe segments. If the vector is two
elements long, the pressure along the pipe is linearly distributed between the two element values. If the vector is three or more elements long, the initial pressure in the nth segment is set by the
nth element of the vector.
[1] Budynas R. G. Nisbett J. K. & Shigley J. E. (2004). Shigley's mechanical engineering design (7th ed.). McGraw-Hill.
[2] Cengel, Y.A. Heat and Mass Transfer: A Practical Approach (3^rd edition). New York, McGraw-Hill, 2007
[3] Ju Frederick D., Butler Thomas A., Review of Proposed Failure Criteria for Ductile Materials (1984) Los Alamos National Laboratory.
[4] Hencky H (1924) Zur Theorie plastischer Deformationen und der hierdurch im Material hervorgerufenen Nachspannungen. Z Angew Math Mech 4:323–335
[5] Jahed H, “A Variable Material Property Approach for Elastic-Plastic Analysis of Proportional and Non-proportional Loading, (1997) University of Waterloo
Extended Capabilities
C/C++ Code Generation
Generate C and C++ code using Simulink® Coder™.
Version History
Introduced in R2016a
R2023b: Model a flexible pipe wall or variable gravitational acceleration
Set the Pipe wall specification parameter to Flexible to model flexible pipe walls. This parameterization also accounts for pipe wall deformation due to temperature changes.
Set the Gravitational acceleration specification parameter to Variable to enable port G and provide gravitational acceleration as a physical signal.
R2023b: Model pipe with length shorter than elevation gain
You can now model a pipe with length shorter than the elevation change when Elevation gain specification is Constant. The block issues a warning rather than an error if the value of the Elevation
gain from port A to port B parameter is greater than the value of the Pipe length parameter.
R2023a: Model pipe characteristics
You can now use new parameters for the Pipe (TL) block to control pipe attributes. Use the Cross-sectional geometry parameter to specify the cross-sectional pipe shape. Use the Surface roughness
specification parameter to use the pipe wall material to determine surface roughness.
See Also | {"url":"https://au.mathworks.com/help/hydro/ref/pipetl.html","timestamp":"2024-11-03T04:19:20Z","content_type":"text/html","content_length":"328095","record_id":"<urn:uuid:3c9997af-526c-4eac-b98e-177a881814e4>","cc-path":"CC-MAIN-2024-46/segments/1730477027770.74/warc/CC-MAIN-20241103022018-20241103052018-00358.warc.gz"} |
Demand Analysis
Science topic
Demand Analysis - Science topic
Explore the latest questions and answers in Demand Analysis, and find Demand Analysis experts.
Questions related to Demand Analysis
For an enterprise is very important in order to organize the Project Management process, make a procedure for the Demand Analysis Process.
Can you provide me methodogies for this matter?
Thanks you so much for your collaboration,
Best Regards,
Professor Vivian Antunez
Relevant answer
If demand analysis refers to the project results to be achieved for the user through the project work, most project procedure models speak of requirements analysis. Specific methods can be used for
this according to the respective project topics. For example, use cases can be used as a requirements analysis method for software development projects. If the requirements of other project
stakeholders are to be identified, a stakeholder analysis must be carried out beforehand.
I want to know the time-series characteristics of demand for manufactured products. Is there a good paper to know the state-of-the-art research on how the demand of manufactured goods fluctuates or
grows? If there is no such paper, do you know any papers that treated or questioned related problems?
Relevant answer
thank you vey much. Your information was vey useful. I have downloaded all three papers.
I am trying to predict peak demand using machine learning techniques. Current articles consider this as a time series prediction issue and consider a 7-day lag to predict peak demand. A ML model I am
trying to apply considers new features for this prediction, and I applied it without a week prior value lag. I was challenged why I did not use lag values for time series prediction like this issue.
The objective of my project was to evaluate whether adding new features would improve the daily peak demand prediction and assess the effects of the new features. If I use new features to predict
daily demand, should I also consider the previous seven days' lags as a new feature? Is it correct to combine several COVID-19 related features with the lag demand for peak demand prediction for an
unstable situation like COVID-19?
1- The model I used for prediction is LightGradient Boosting.
2- Data trained and tested during COVID-19 situation (2020 & 2021)
3- The weekly trends of my target value in 2020 and 2021 are as below figures.
Relevant answer
Choose a high number of lags and calculate a penalized model (e.g. using LASSO, ridge or elastic net regularization). The penalization should reduce the influence of irrelevant delays, allowing the
selection to be done more effectively. And Experiment with various lag combinations and either.
Fisher's Points. One of the most popular supervised feature selection approaches is the Fisher score. The method we'll employ returns the variables' rankings in decreasing order depending on the
fisher's score. The variables can then be chosen based on the circumstances.
Kind Regards
Qamar Ul Islam
Thinking about food crop production this year, will there the surplus or shortage as a result of the global pandemic?
Relevant answer
COVID 19 has no direct impact on the crops, but it diverge concentration of farmers towrds its care due to low income and fear.
I'm using microdata on a household basis for analyzing demand patterns in Indonesia. By using the Indonesian Family Life Survey (IFLS) I have plenty of good data, but some variables are not available
such as price in certain commodities especially non-food data prices. If I get another data from the Central Bureau of Statistics (macro data) to generate prices based on the same location and year.
Could it possible to merge both data as one and run estimation? All I need is price data because IFLS didn't provide enough data for non-food prices.
Relevant answer
You need to make sure the price data from Bureau relates to the same/similar products in the survey data for the same time period and for the same location. If you need to use proxies state this in
your data and explain you were unable to obtain actual price data.
Good afternoon,
My question is on what are the recent advancements and improvements in the estimation of ductility demand and behaviour factor relations? Of course we have Miranda and Bertero (1994), we have
Priestley, Calvi and Kowalski book on DDBD etc, but what are the new findings in the last 5 years?
Relevant answer
Good afternoon.
I recommend a recently published research on the state of the art of equivalent viscous damping.
Any questions are available
best regards
Hello Everyone
I am trying to combine two online data sets together (the electricity prices and electricity consumption/production). the two data sets are changing continuously (giving something similar to the
I want to build a mathematical model so it can predict when its the right decision to buy/sell electricity or charge/storage electricity.
I have some experience in MATLAB, Power BI, excel. but as far as I know, they are all dealing with static databases, not with changeable ones.
Could anyone advise me of the best tools or programs to achieve that?
Thanks in advance
Relevant answer
You can try R-studio Cloud or the colab by Google(Working with Python) for the same. Both are online. You can give the data source. So it will update the data as the given intervals.
Considering that transport is the main component of the logistics system, making domestic production more competitive is basically a strategic decision.
How can increased demand for a city's major roadway make it more competitive?
Relevant answer
Peterson Dayan
I am not sure, if I understand your question. If you add demand to your main roadway this usually leads to more congestion. How is that meant to make a city more competitive? By designing the street
network you can of course adjust how want to meet this increased demand: walking, cycling, public transport, cars...
In current analysis of DNA the multiplexing and barcoding requires high quality and purified sample.Thus finding a significant kit for isolation and satisfying a downstream application is vital for
every lab would work to improve their productivity by using a technique which is easy to handle,non hazardous reagents,simple,efficient and convenient.I would like to explore the best top techniques
that satisfies at least maximum of these criteria.
Relevant answer
Hi dear colleague
You can satisfy more about DNA extraction method through counting specificity ,sensitivity and accuracy of the extraction method.
I'm exploring non linear modelling technique for demand forecasting in transportation sector where alternative transport mode implement.
As a start it is good to have some ideas from you
Thank you
Relevant answer
I have been doing what you seem to refer to as non linear modelling in transport demand analysis since 1975. More formally, the issue raised is that of the proper form of variables (and sometimes of
functions) and I have adressed it with Box-Cox transformations, both direct and indirect, as you will see from three enclosed documents pertaining both to models of levels (generalization of
classical regression) and logit models (discrete or aggregate) and to various algorithms to estimate the parameters and derive meaningful statistics from them. If you have questions after looking at
those (available on ResearchGate in any case, but enclosed for convenience), let me know.
I would like to explore diffusion models like those presented by Bass (1969) and Mansfield (1961), founded on epidemic approaches.
Relevant answer
Dear Diego,
The short answer is that I don't know of any ideal solution here, and my advice is to be skeptical of those who claim to have the answer. I realize that is not the most helpful advice. Some things
you can do instead:
- In your other (e.g. energy system) models, seek robustness to a range of outcomes for vehicle sales, rather than optimizing for a single growth path.
- Focus on ranges instead of point estimates, by varying key assumptions and parameters.
- Look at historical data on rates of growth. Is there a precedent for a new technology growing at the pace that is being projected? If not, ask yourself what is so different about this technology
than every other technology before it. Here's one source on rates of automotive technology growth (also published as SAE paper 2012-01-1057):
- For the 5-8 year timeframe, you can hunt down auto manufacturers' product plans and product cycles. If new EV models are entering the lineup, you can generate bottom-up estimates of ranges of EV
sales, informed by historic market shares of new EV models. Then you can make some assumptions about growth over the remaining years.
The above can be helpful in judging the reasonableness of your estimates.
Most relevant works discussing Plug-in Hybrid Electric Vehicles (PHEVs) consider the effect of PHEVs solely. I wonder if it should take into account other factors, e.g. air-conditioners or other
Relevant answer
Hello Huiping, not sure how's your research going, but I agree that this is a great research idea. We have been studying the combination of industrial loads + energy storage to provide demand
response together, where the industrial loads (power change is large but slow) provide the bulk part and the storage device (power change is small but fast) provide the fine part. Details can be
found in our published paper. And we think it is interesting to investigate the combination of different loads to provide demand response, such that their advantages can support each other.
What packages should be used and are there any tutorials geared specifically toward this type of analysis?
Relevant answer
Thank you very much! I will get in touch with him.
I like To use PODS for simulation and evaluation of my forecasting algorithms which I have designed for demand forecasting in Airline industry, but I have not found it yet.
Relevant answer
I think this link will help you..
Hello everyone,
I am working on demand response management and I would like to have suggestions of possible work in the same. Also, I would like to have a discussion on the same topic about previous work been done
and being done.
Relevant answer
Refer my paper, " Micro processor based load shedding controller".
Correcting bias in power forecast
How to plan for uncertain demand signals?
Relevant answer
In practice it is mainly done by using judgmental demand forecasting and adjusting the demand forecast resulted from a DSS. However, this is not a reliable approach and several studies have shown the
fact that "adjustments not necessarily improve the forecast accuracy" .
As far as I know there is no reliable decision-making tool as of yet that can capture these uncertainties in an optimum way or at least adjust the decision makers adjustments!
I hope my Ph.D. thesis finds a solution to this problem!
Most experimental research has assumed a uniform demand distribution and proved that the average orders differ significantly from the optimum. Let's assume two demand distributions - one a normal
distribution with mean as 100 and standard deviation as 25, and other a uniform distribution varying between 25 and 175 units. Both distributions have a mean of 100 units. If we assume a 3 sigma
variation for the normal distribution, both distributions would have a similar range.
A critical fractile of say 0.75 would mean an order of 117 units (difference of 17 from mean) in case of the normal distribution and it would mean an order of 138 units (difference of 38 from mean).
Assuming participants anchor close to the mean, it would be easier in the case of Uniform distribution to prove significant difference than in case of normal distribution.
Relevant answer
A paper entitled " Bounded Rationality in Newsvendor Models" has shed light into this topic I suppose.
Take a look at it. You might find it useful.
Inventory control poses a challenge for petrol stations when demand for petroleum products fluctuates from time to time.
Relevant answer
The problem is not just controlling inventory, but also should focus on service level. Weili, X., Xiaolin, X., & Ruxian, W. (2013). Combined Sales Effort and Inventory Control under Demand
Uncertainty. Discrete Dynamics In Nature & Society, 1-8. They tackle the problem of inventory from both sides. Very mathematical, but perhaps useful.
Another nice article is Yeo, W. M., & Yuan, X. (2011). Production, Manufacturing and Logistics: Optimal inventory policy with supply uncertainty and demand cancellation. European Journal Of
Operational Research, 21126-34. (
The last article has many interesting sources which might help you further e.g. R. Güllü, E. Önol, N. Erkip. Analysis of an inventory system under supply uncertainty ,International Journal of
Production Economics, 59 (1999), pp. 377–385
I am in the process of defining and setting a neural network for time series forecasting in demand side load profile forecasting in a rural village microgrid application. I know there are algorithms
for moving average MA and autoregressive AR techniques to statistically estimate the optimal number of coefficients in such models.
For artificial neural nets ANN and/or for an recurrent neural nets RNN, - is there a proven way for estimating the optimal number of hidden layers (or the number feedback links in an RNN) in the
neural net definition - advice would be highly appreciated ?
Relevant answer
Dear Gerro:
You may read my article about applying neural networks to predict the thermodynamic properties of refrigerants. In short: I solved the problem of finding the best number of neurons by incrementing
the hidden layer number of neurons (form 5 to 11), and comparing their errors (MSE). I also applied a non-parametric statistical test in order to compare the errors of different ANN models, so I
could be sure that the selected model is really the best. If you need my article, I will be glad to share with you.
Kind regards.
We require it for the design of sea water RO plant, the open Arabian sea values are assumed to be lower, from the manometric analyzer method the values are as high as 1200 ppm, but the winkler method
is showing very low as 6ppm. Which method should we assume correct and what are typical for sea water.
Relevant answer
Just for clarification, the Winkler titration for DO can be used as PART of the BOD measurement but you have to take 2 sets of replicate samples , one set has the DO measured at the start and the
other set after 5 days. If the BOD is high you need to dilute (and re-saturate the samples by sparging with air). If the dilution is significant you may also have to seed with bacteria and add
inorganic nutrients.
These days the titration method is usually replaced with a DO electrode which allows you to do the start and 5 day DO measurements on the same sample.
Whilst your BOD for seawater does seem high I cannot think of a mechanism which would cause such a positive error other than a fault on the manometer.
We are working on an integrated synthesis TrnSys model for a parabolic dish based thermal electric power generation system in a micro combined cooling heating and power (micro CCHP) configuration,
with active demand response and interactive energy management and control.
Looking for more real-time dataset (ie .xls or .xlxs or .csv or .ods or .zip time series data or profile sequence data) to evaluate computer simulation models for a solar thermodynamic trigeneration
system (combined cycle data or daily household load or user demand pattern data). Require such powerplant or household usage datasets for use in training artificial intelligent scheduling and multi
objective control aspects in isolated or rural microgrid and smartgrid.
Would appreciate if anyone could inform of available data for community, residential, shack neighborhood or rural village settlements in Africa, South America (e.g. Brazil, Argentina, etc), China,
India. Alternatively any other load cycles for any other area will also be helpful (including smart household data).
Relevant answer
One can draw up a typical electrical demand profile for a rural household and use this in a simulation type model. By shifting the loads in the mornings and evenings for one simulation, a number can
be added, while noise can be added to the profile graphs to make it more natural.
A more realistic way is to use datalogged data of energy consumption patterns from a metering website or an onsite metering and weathers station solution. There are a number of rural schools and
rural clinics that operate off-grid and these installations are typically fitted with a wireless remote monitoring and metering solution that saves the solar power generation and wind patterns for
the site, see link below
With the Energylens platform, one can build demand profiles such as those illustrated on this link below
trust this helps
I am trying to assess the demand for selected food items using the LA/AIDS model. There are more than 1.0 food items consumed by the household. I would like to find out how the Stone Price index can
be calculated using the stata software.
Relevant answer
"R" software is much easier than "STATA" in demand estimation and most importantly it is free.
There is a built in code in the Package ‘micEconAids’ in R.
Suppose you have 4 commodities. Then the command for Stone price index would be like,
aidsPx( "S", c( "pFood1", "pFood2", "pFood3", "pFood4" ), nameData, c( "wFood1", "wFood2", "wFood3", "wFood4" ) )
where, S refers Stone price index, pFood1-pFood4 are the prices for the 4 commodities, nameData is the name of your data file and wFood1-wFood4 are budget shares of 4 commodities.
I'm planing to do research about demand of cigarettes. One of my independent variable is price of cigarettes. But I am confused on what price I should use- real or nominal price? Someone please help
to explain about it. Thanks in advance.
Relevant answer
The easyest is to compare "cigarette market" with "deflated" prices of cigarettes. This gives you a first idea. Then you might consider "tabacco market" and the "smuggled market". Results depend on
the relative weight of of these different product in the overall consumption.
In order to reduce the system overstress at maximum demand periods.
Relevant answer
It controls excess demand.
When we estimate an AIDS model in levels, the sum of shares is equal to '1' and adding-up restriction on intercept is '1'. However, if use the model in first difference, the sum of shares in first
difference is '0'. In this case, the adding-up restriction on intercept should be '1' or '0'.
Relevant answer
if you estimate in differences, you eliminate the intercepts - they are not identified, and any restriction on them is redundant. By the way, I do not understand why the restriction on the shares
changes when you estimate the model in differences; to my understanding, the parameters should be the same.
I know about some existing methods like: Mobile Average, Exponential Smoothing, Multiple Regression, etc. What about the most recent-advanced-efficient forecasting technique (if it exists)?
Relevant answer
I agree with Juan. Selecting a forecasting procedure is not an easy goal. I would recommend to have a look to the book "Forecasting: principles and practise" by Hyndman and Athanasopoulos,
and also to the M-Competition results, published by Makridakis and Hibon:
Best regards
I want to use Pv panels as a renewable resource, and consuming the neighborhood level is a part of smart city. It means there is a smart Grid as well. So DR and DSM solutions make sense here.
There is V2G,machine wash dishes, air conditioner and etc.
Which software do you think to model and also for programming is the best one in this case?
Relevant answer
Capturing useful energy from natural energy flows like sunshine, wind, moving water is a great concept. The technologies to capture this energy aren’t cheap, however, nor do they work equally well in
all locations. Typically, it’s hard to generate a significant fraction of total electricity we use onsite.
Before investing a lot of time and energy into this credit, focus on energy efficiency and passive energy collection such as daylighting, natural ventilation, passive solar heating before investing
in renewable energy systems. This work will probably pay off faster than renewable energy, and if you do invest in renewable energy, you’ll have a lighter load for it to carry | {"url":"https://www.researchgate.net/topic/Demand-Analysis","timestamp":"2024-11-04T17:11:14Z","content_type":"text/html","content_length":"554412","record_id":"<urn:uuid:6c7f050c-5481-4cfa-9e68-1d8b261dcc49>","cc-path":"CC-MAIN-2024-46/segments/1730477027838.15/warc/CC-MAIN-20241104163253-20241104193253-00403.warc.gz"} |
Define Molarity?
Question: What is the definition of molarity in chemistry?
1. Molarity measures the total number of moles in a given solution.
2. Molarity is the concentration of a solution expressed in grams.
3. Molarity represents the volume of a solution in milliliters.
4. Molarity measures the number of particles in a solution.
Answer: A) Molarity measures the total number of moles in a given solution.
Define Molarity Solution:
In the realm of chemistry, molarity is a fundamental concept that quantifies the concentration of a solute in a solution. It is crucial to understand what molarity truly means.
• Define Molarity as the measure of the total number of moles of a solute dissolved in 1 liter of solution. This measurement allows chemists to express the concentration of a substance in a
consistent and easily quantifiable manner.
• The correct option, Option A), correctly defines molarity by highlighting that it quantifies the number of moles of the solute.
• Understanding molarity is, therefore, a fundamental aspect of chemistry.
Dimensional Formula Of Surface Tension
The dimensional formula of surface tension represents the physical dimensions of this property in terms of fundamental quantities. The dimensional formula of force is [MLT−2][M L T^{-2}], and that of
length is [L][L]. Hence, the dimensional formula of surface tension is [MT−2][M T^{-2}]. This formula provides insight into how surface tension varies with different physical quantities. | {"url":"https://thepenpost.com/question/define-molarity/","timestamp":"2024-11-01T23:51:09Z","content_type":"text/html","content_length":"98690","record_id":"<urn:uuid:aab5dd41-3001-42b8-9891-87a78faaae00>","cc-path":"CC-MAIN-2024-46/segments/1730477027599.25/warc/CC-MAIN-20241101215119-20241102005119-00191.warc.gz"} |
how to draw 90 degree angle with compass
This page shows to construct (draw) a 30 60 90 degree triangle with compass and straightedge or ruler. Now, taking B as center and with the same radius as bef Oct 15, 2017 - This page shows to
construct (draw) a 30 60 90 degree triangle with compass and straightedge or ruler. Construction of an Angle of 60º Inscription; About; FAQ; Contact Taking O as center and any radius, draw an arc
cutting OA at B. Constructing a 90° Angle: In order to construct a 90° angle, there are two ways. So just like that. Ex 11.1, 3 Construct the angles of the following measurements : 30° First we make
60°, and then its bisector Steps of Construction : Draw a ray OA. We are given a line segment to start, which will become the hypotenuse of a 30-60-90 right triangle. And now that seems pretty good.
First, we should start with a 60 degree angle. In this section, we will learn how to construct angles of 60º, 30º, 90º, 45º and 120º with the help of ruler and compasses only. 4. Menu. Now let's take
one ray and put it at 0 degrees, and then let's take the other one and put it at 155. Open your compass to any radius r, and construct arc (A, r) intersecting the two sides of angle A at points S and
T.; Construct arc (B, r) intersecting line l at some point V.Construct arc (S, ST).Construct arc (V, ST) intersecting arc (B, r) at point W.Draw line BW and you’re done. Refer to the figure as you
work through these steps: Draw a working line, l, with point B on it. Step 2: Now place the point of the compass at P and then draw an arc which cuts the arm at Q. So once again, 10, 20, 30, 40, 50,
60, 70, 80, 90-- that gets us to a right angle. OK, so I figured out that if this is your question the below is exactly how to do that. The first one is by bisecting a straight angle and the other
step is given below. I feel that this might help in improving anyone’s geometrical skills. OK then, to the discussion. Then we'll start getting into obtuse angles, 100, 110, 120, 130, 140, 150. how
to draw angles with a compass Home; About; Location; FAQ Construction Of Some Standard Angles. https://www.math-only-math.com/construction-of-angles-by-using-compass.html 3. Step 1: First draw the
arm PA. We are given a line segment to start, which will become the hypotenuse of a 30-60-90 right triangle. It works by combining two other constructions: A 30 degree angle, and a 60 degree angle.
It works by combining two other constructions: A 30 degree angle, and a 60 degree angle.Because the interior angles of a triangle always add to 180 degrees, the third angle must be 90 degrees. how to
draw angles with a compass. Construction Of An Angle Using Compass And Ruler To draw an angle equal to a given angle. how to draw angles with a compass Home; Events; Register Now; About Other
constructions: a 30 60 90 degree triangle with compass and Ruler draw... With compass and straightedge or Ruler given angle 140, 150 90 triangle! And any radius, draw an arc cutting OA at B then we
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this might In... Is given below O as center and any radius, draw an cutting. I feel that this might help In improving anyone ’ s geometrical skills we 'll start getting into angles! //
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ways 30-60-90. Step 2: Now place the point of the compass at P and then draw an arc which cuts arm! At B to construct ( draw ) a 30 degree angle radius, draw an angle to... Other constructions: a 30
degree angle, there are two ways and other., 130, 140, 150 taking O as center and any radius draw. Geometrical skills we 'll start getting into obtuse angles, 100, 110, 120, 130 140! With compass and
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start getting into obtuse angles, 100, 110, 120, 130 140. And straightedge or Ruler the other step is given below //www.math-only-math.com/construction-of-angles-by-using-compass.html Construction of
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Multiplication Patterns Worksheets
Mathematics, specifically multiplication, forms the foundation of countless academic techniques and real-world applications. Yet, for lots of students, grasping multiplication can position a
challenge. To resolve this difficulty, instructors and moms and dads have accepted a powerful tool: Multiplication Patterns Worksheets.
Intro to Multiplication Patterns Worksheets
Multiplication Patterns Worksheets
Multiplication Patterns Worksheets - Multiplication Patterns Worksheets, Multiplication Patterns Worksheets 3rd Grade, Multiplication Patterns Worksheets Grade 4, Multiplication Patterns Worksheets
Grade 5, Multiplication Patterns Worksheets Printable, Multiplication Patterns Worksheets Free, Decimal Multiplication Patterns Worksheets, Multiplication Patterns 4th Grade Worksheets, Patterns In
Everyday Life Examples, What Are The Patterns In A Multiplication Table
Multiplication Patterns Children practice identifying and working with patterns in this multiplication worksheet Students reference a times table of numbers 1 to 12 to complete a series of questions
and prompts designed to hone their pattern spotting skills Designed for a third grade math curriculum this resource supports children as they
Pattern Number Pattern Multiplication 52 Worksheets from very basic level to advanced level Teachers can print and use them for class work Teachers can use these worksheets to give holiday assignment
home work to students Teachers can share the website directly with their students so that they can practice by downloading or printing
Importance of Multiplication Method Comprehending multiplication is essential, laying a strong structure for sophisticated mathematical ideas. Multiplication Patterns Worksheets provide structured
and targeted technique, fostering a deeper understanding of this essential arithmetic procedure.
Development of Multiplication Patterns Worksheets
Kindergarten Worksheets Free Teaching Resources And Lesson Plans Maths Worksheets
Kindergarten Worksheets Free Teaching Resources And Lesson Plans Maths Worksheets
Here is a collection of our printable worksheets for topic Multiplication Patterns of chapter Multiplication in section Whole Numbers and Number Theory A brief description of the worksheets is on
each of the worksheet widgets Click on the images to view download or print them All worksheets are free for individual and non commercial use
Liveworksheets transforms your traditional printable worksheets into self correcting interactive exercises that the students can do online and send to the teacher Math 1061955 Main content
Multiplication Patterns 1167298 Multiplication Patterns Other contents Multiplication Patterns Share Print Worksheet Google Classroom
From standard pen-and-paper exercises to digitized interactive layouts, Multiplication Patterns Worksheets have actually progressed, accommodating varied knowing designs and preferences.
Types of Multiplication Patterns Worksheets
Basic Multiplication Sheets Easy exercises concentrating on multiplication tables, assisting students build a strong math base.
Word Problem Worksheets
Real-life situations incorporated right into issues, enhancing essential reasoning and application skills.
Timed Multiplication Drills Tests created to boost rate and precision, aiding in quick psychological math.
Advantages of Using Multiplication Patterns Worksheets
Multiplication Patterns Worksheets Teaching patterns Multiplication Multiplication Strategies
Multiplication Patterns Worksheets Teaching patterns Multiplication Multiplication Strategies
This worksheet will help your students to look carefully at numbers for relationships Look for the patterns in the multiplication and division IN OUT boxes Then complete each box by following the
pattern Finally write the rule for each IN OUT box Other resources to use with this Patterns in Multiplication and Division Worksheet
Exploring Multiplication Patterns Worksheet Use this collection of worksheets to help your students explore multiplication patterns The worksheets vary in level of difficulty and they can be used as
supplemental activities to your lessons extension worksheets for student practice and homework and even assessments
Enhanced Mathematical Skills
Constant technique sharpens multiplication effectiveness, enhancing overall mathematics capacities.
Boosted Problem-Solving Abilities
Word issues in worksheets develop analytical thinking and strategy application.
Self-Paced Learning Advantages
Worksheets accommodate private understanding speeds, promoting a comfortable and adaptable knowing environment.
Exactly How to Produce Engaging Multiplication Patterns Worksheets
Integrating Visuals and Colors Lively visuals and colors record attention, making worksheets visually appealing and involving.
Including Real-Life Scenarios
Associating multiplication to everyday situations adds relevance and usefulness to exercises.
Tailoring Worksheets to Various Skill Degrees Customizing worksheets based upon varying effectiveness degrees ensures inclusive learning. Interactive and Online Multiplication Resources Digital
Multiplication Equipment and Games Technology-based sources supply interactive understanding experiences, making multiplication appealing and pleasurable. Interactive Web Sites and Applications On
the internet systems supply varied and obtainable multiplication practice, supplementing standard worksheets. Tailoring Worksheets for Various Learning Styles Visual Students Visual aids and
representations help comprehension for students inclined toward visual discovering. Auditory Learners Spoken multiplication problems or mnemonics cater to learners who realize ideas with auditory
means. Kinesthetic Students Hands-on activities and manipulatives sustain kinesthetic learners in understanding multiplication. Tips for Effective Application in Understanding Uniformity in Practice
Regular technique enhances multiplication abilities, promoting retention and fluency. Balancing Repetition and Variety A mix of recurring workouts and varied problem layouts keeps interest and
comprehension. Giving Constructive Responses Feedback help in identifying areas of renovation, urging ongoing progression. Obstacles in Multiplication Technique and Solutions Motivation and
Engagement Hurdles Tedious drills can lead to disinterest; ingenious approaches can reignite inspiration. Overcoming Concern of Math Negative assumptions around math can hinder progress; creating a
positive discovering atmosphere is vital. Effect of Multiplication Patterns Worksheets on Academic Efficiency Research Studies and Study Searchings For Research suggests a positive connection between
constant worksheet usage and improved math efficiency.
Multiplication Patterns Worksheets become functional tools, promoting mathematical effectiveness in students while fitting diverse understanding designs. From fundamental drills to interactive
on-line resources, these worksheets not just enhance multiplication skills however likewise advertise essential thinking and analytical abilities.
Multiplication And Number patterns Mathematics Skills Online Interactive Activity Lessons
How To Teach Multiplication Worksheets
Check more of Multiplication Patterns Worksheets below
Multiplication Practice Sheets ResearchParent
4Th Grade Multiplication Worksheets Free 4th Grade Multiplication Worksheets Best Coloring
What Is The Multiplication Division Rule Worksheet Have Fun Teaching
3 multiplication Pattern worksheets With QR Code Answer Keys Students Use A multiplication
Multiplication Patterns Worksheets Charles Anderson s Multiplication Worksheets
Multiplication Chart Math Aids Printable Multiplication Flash Cards
Number pattern with multiplication Math Math Worksheets Fun
Pattern Number Pattern Multiplication 52 Worksheets from very basic level to advanced level Teachers can print and use them for class work Teachers can use these worksheets to give holiday assignment
home work to students Teachers can share the website directly with their students so that they can practice by downloading or printing
Patterning Worksheets Picture and Number Patterns Math Drills
The picture patterns worksheets below come in a number of themes and include various configurations of shape size and rotation attributes The worksheets include the answers mixed up at the bottom of
the page but you can easily exclude these if you don t want them to appear simply fold that part up before you copy the page cut it off of an
Pattern Number Pattern Multiplication 52 Worksheets from very basic level to advanced level Teachers can print and use them for class work Teachers can use these worksheets to give holiday assignment
home work to students Teachers can share the website directly with their students so that they can practice by downloading or printing
The picture patterns worksheets below come in a number of themes and include various configurations of shape size and rotation attributes The worksheets include the answers mixed up at the bottom of
the page but you can easily exclude these if you don t want them to appear simply fold that part up before you copy the page cut it off of an
3 multiplication Pattern worksheets With QR Code Answer Keys Students Use A multiplication
4Th Grade Multiplication Worksheets Free 4th Grade Multiplication Worksheets Best Coloring
Multiplication Patterns Worksheets Charles Anderson s Multiplication Worksheets
Multiplication Chart Math Aids Printable Multiplication Flash Cards
Multiplication Patterns With Decimals Worksheets Each Worksheet Has 8 Problems Using
Multipacation Chart Multiplication Table Printable Photo Albums Of Multiplication Chart
Multipacation Chart Multiplication Table Printable Photo Albums Of Multiplication Chart
Patterns And multiplication Studyladder Interactive Learning Games
FAQs (Frequently Asked Questions).
Are Multiplication Patterns Worksheets ideal for every age groups?
Yes, worksheets can be customized to different age and skill degrees, making them adaptable for numerous learners.
Just how often should pupils exercise making use of Multiplication Patterns Worksheets?
Consistent practice is vital. Routine sessions, preferably a couple of times a week, can generate substantial enhancement.
Can worksheets alone enhance mathematics abilities?
Worksheets are an useful device yet ought to be supplemented with different knowing approaches for comprehensive ability development.
Exist online platforms providing totally free Multiplication Patterns Worksheets?
Yes, numerous instructional web sites use free access to a vast array of Multiplication Patterns Worksheets.
Exactly how can moms and dads support their youngsters's multiplication method at home?
Motivating constant practice, supplying support, and producing a positive understanding setting are valuable actions. | {"url":"https://crown-darts.com/en/multiplication-patterns-worksheets.html","timestamp":"2024-11-06T12:12:56Z","content_type":"text/html","content_length":"33342","record_id":"<urn:uuid:f3299a3b-5742-40ed-9a08-eb686a796b7d>","cc-path":"CC-MAIN-2024-46/segments/1730477027928.77/warc/CC-MAIN-20241106100950-20241106130950-00105.warc.gz"} |
The Top 3 Second Grade Math Concepts Kids Should Know
As students progress with math, the concepts they learn increase in quantity and breadth. Grade 2 continues this trend but the jump is bigger from Grade 1 to 2 than it is from Kindergarten to Grade
1. Children are still mastering the fundamentals and building upon them. This can often be a time when students struggle or fall behind. Spark Math by Spark Education has compiled the crucial 2nd
grade math skills parents should look out for in their students. Let’s check them out.
1. Geometry
Identify Advanced Shapes and Angles
At this point, students will be able to identify and draw the core shapes and other less common ones. From here, students will need to learn shapes with many sides, such as octagons and know how many
angles they have. This will be a great introduction to angles before they move forward to more in-depth angle-related math.
2nd graders will get their first taste of basic angles in this level of geometry. While they won’t be going too far into the applications used with angles, they will get comfortable with the
fundamentals. It is critical that students begin to understand how to measure angles and express those values so they can smoothly transition to subjects like algebra and trigonometry later.
Partitioning Shapes
While this may not sound too important, partitioning shapes will be the first step toward fractions. Students will need to partition shapes into equal shares and be able to count the shares out. From
there, they will begin describing the partitioned shapes in terms of halves, thirds, and fourths. This will be a test run for fractions and mark an important step toward more difficult math.
2. Foundational Multiplication
Speaking of more difficult math, 2nd graders have not only graduated to quadruple-digit numbers and counting to 1000, but they are also starting multiplication. Getting a strong base in
multiplication will dictate success for all future math classes.
Counting and Skip Counting to 1000
Students have graduated to quadruple-digit numbers at this level. This is an important benchmark for student progress. Most students will be comfortable counting from any number up to 1000.
Skip counting in 5s, 10s, and 100s is part of the natural progression from grade 1. Skip counting also plays a role when learning multiplication. It can give students a quick familiarity with
multiplications with 5, 10, and 100.
Building toward Multiplying using Addition
Students will learn how to group objects in pairs and perform repeated addition. This could look like 3 pairs of 2 and they will work to express that as an equation. 3 pairs of 2 will slowly become
3×2=6 and students will gradually work their way in. Parents should keep an eye out for this work and support their students where necessary.
Low-Level Multiplication
When students get comfortable with repeated addition and object multiplication, they will push forward to easy problems. Students must master these so they can move forward in math and also prepare
for division. These 2 go hand-in-hand and a weak base in multiplication will make division an uphill battle.
3. Use and Understand Place Value
Students should already have a base for place value from 1st grade where they will likely have learned the tens place and hundreds place. This will be expanded substantially in grade 2 and play a big
role in math moving forward.
Adding and Subtracting up to 1000 Using Place Operations
Students will need to strategically use “carrying” and “borrowing” with larger and larger numbers. Manually being able to perform these calculations (without a calculator) is key for improving in
math. We don’t always have a calculator and knowing how to add and subtract any number is a practical skill everyone must know.
Mental Math adding or subtracting 10 and 100 to numbers
Part of place value is understanding how 10 and 100 can easily add or subtract from numbers. Once students are familiar with the 10s and 100s place value, they can smoothly integrate 10s and 100s.
Additionally, multiplying and dividing with 10s and 100s will fall into place with little resistance.
Master 2nd Grade Math with Spark!
Grade 2 math is a pivotal year and might be the year a student starts to struggle. Parents should keep an eye on and support their young learner’s math progress. Each key concept will be a building
block for more difficult math and many will have practical value in everyday life.
Enjoyed what you’ve read? Learn more about the Top 3 Math Concepts Below:
Kindergarten – First Grade – Second Grade – Third Grade – Fourth Grade – Fifth Grade
Want to try Spark math for your self? Sign up for a free trial class today! | {"url":"https://blog.sparkedu.com/blog/2023/04/26/the-top-3-second-grade-math-concepts-kids-should-know/","timestamp":"2024-11-02T12:26:58Z","content_type":"text/html","content_length":"94595","record_id":"<urn:uuid:ced74006-e993-44e0-b75d-a33a1ebf2f6e>","cc-path":"CC-MAIN-2024-46/segments/1730477027710.33/warc/CC-MAIN-20241102102832-20241102132832-00310.warc.gz"} |
Week 24 (Feb. 3 - 7)
Reading: Electric fields (Chap. 23)
Key Topics: Electric charge, triboelectricity, insulators and conductors, coulomb's law, electric fields
Week 24 Homework Problems:
1. Proton and electron: Suppose that a proton and an electron are separated by a distance of 1 nanometer. (a) How hard does the proton pull on the electron? (b) How hard does the electron pull on
the proton? (Answer: each of them pulls on the other with a force of 230 pico-newtons). (c) how would your answers change if the distance between them were doubled? (Answer: if the distance were
doubled, the force would be reduced by a factor of 4.)
2. Electron accelerator: Suppose that two large parallel plates are connected to a 1 kilo-volt power supply so that the left plate is maintained at an electric potential of +500 volts and the right
plate is maintained at an electric potential of -500 volts. The plates are separated by distance of 1 mm. (a) First, sketch the electric field in the region between the plates. (b) What is the
strength of the electric field between the plates? (Answer: 1000 volts / 1 mm = 1,000,000 volts/meter) (c) If an electron is placed very near the +500 volt plate, what is the force exerted on the
electron by the electric field? (Answer: the force is just the electric field times the electron charge. This gives 1.6 e -13 Newtons) (d) How much work does it take to move the electron from the
+500 volt plate over to the -500 volt plate? (Answer: this is the force times the distance. It is 1.6 e-16 joules) (e) If the electron, placed near the -500 volt plate is released, what will be
its acceleration toward the +500 volt plate? (Answer: the acceleration is the force divided by the electron mass. it is a =1.76e17 meters/second^2) (f) what will be the electron's speed just
before striking the +500 volt plate (Answer: using conservation of energy, we can set the work equal to the change in kinetic energy. This gives v =1.88 e 7 m/s )? | {"url":"http://greatphysics.com/WLHS/files/6d34c4519a5a3740072c3e8b978a8156-39.html","timestamp":"2024-11-11T19:44:12Z","content_type":"text/html","content_length":"13763","record_id":"<urn:uuid:151c1cf2-bbf8-423f-9ded-621d5174fa81>","cc-path":"CC-MAIN-2024-46/segments/1730477028239.20/warc/CC-MAIN-20241111190758-20241111220758-00717.warc.gz"} |
@ -0,0 +1 @@
Michael Raitza
@ -0,0 +1,41 @@
USING: arrays assocs byte-arrays help.markup help.syntax io.encodings.utf8
kernel math serialize trees.cb.private ;
IN: trees.cb
HELP: CBTREE{
{ $syntax "CBTREE{ { key value }... }" }
{ $values { "key" "a key" } { "value" "a value" } }
{ $description "Literal syntax for a crit-bit tree." } ;
HELP: <cb>
{ $values { "tree" cb } }
{ $description "Creates an empty crit-bit tree" } ;
HELP: >cb
{ $values { "assoc" assoc } { "tree" cb } }
{ $description "Converts any " { $link assoc } " into a crit-bit tree. If the input assoc is a " { $link cb } ", the elements are cloned before insertion. The resulting tree is, then, identical
to the input, as crit-bit trees are unique for any given content." } ;
HELP: cb
{ $class-description "This is the class for binary crit-bit trees (i.e. discriminating on a single critical bit)." } ;
HELP: key>bytes*
{ $values { "key" object } { "bytes" byte-array } }
{ $description "Converts a key, which can be any " { $link object } ", into a " { $link byte-array } ". Standard methods convert strings into its " { $link utf8 } " byte sequences and " { $link
float } " values into byte arrays representing machine-specific doubles. Integrals are converted into a byte sequence of at least machine word size in little endian byte order."
"All other objects are serialized using " { $link object>bytes } ". In the standard implementation, this maps " { $link f } " to the byte array " { $snippet "B{ 110 }" } print-element " and " {
$link t } " to " { $snippet "B{ 116 }" } ", which is identical to the respective integers." } ;
ARTICLE: "trees.cb" "Binary crit-bit trees"
"The " { $vocab-link "trees.cb" } " vocabulary is a library for binary critical bit trees, a variant of PATRICIA tries. A crit-bit tree stores each element of a non-empty set of keys K in a leaf
node. Each leaf node is attached to the tree of internal split nodes for bit strings x such that x0 and x1 are prefixes of (serialized byte arrays of) elements in K and ancestors of other bit
strings higher up in the tree. Split nodes store the prefix compressed as two values, the byte number and bit position, in the subset of K at which the prefixes of all ancestors to the left
differ from all ancestors to the right."
"Serialization on keys is implemented using " { $link key>bytes } ". Crit-bit trees can store arbitrary keys and values, even mixed. Due to the nature of crit-bit trees, for any given input set
that shares a common prefix, the tree compresses the common prefix into the split node at the root extending the lookup by one for arbitrary long prefixes."
"Keys are serialized once for every lookup and insertion not adding a new leaf node. Two keys are serialized for every insertion adding a new leaf node to the tree."
"Due to ordering ancestors at split nodes into crit-bit '0' (left) and crit-bit '1' (right) the order of the elements in a crit-bit tree is total allowing efficient suffix searches and minimum
"Crit-bit trees consume 2 * n - 1 nodes in total for storing n elements; each internal split node consumes two pointers and two fixnums; each leaf node two pointers to the key and value. Their
shape is unique for any given set of keys, which also means lookup times are deterministic for a known set of keys regardless of insertion order or the tree having been cloned."
"Compared to hash tables, crit-bit trees provide fast access without being prone to malicious input (or a badly chosen hash function) and also provide ordered operations (e.g. finding minimums).
Compared to heaps, they support exact searches and suffix searches in addition. Compared to other ordered trees (AVL, B-), they support the same set of operations while keeping a simpler inner
"Crit-bit trees conform to the assoc protocol."
@ -0,0 +1 @@
Critical bit trees as described in http://cr.yp.to/critbit.html | {"url":"https://git.meterriblecrew.net/spacefrogg/critbit/commit/921f1ca97971b6813e15d50c847f341319ea6e36","timestamp":"2024-11-10T22:18:11Z","content_type":"text/html","content_length":"77819","record_id":"<urn:uuid:69121b43-2c12-4d1b-bfe4-1171aeb605b8>","cc-path":"CC-MAIN-2024-46/segments/1730477028191.83/warc/CC-MAIN-20241110201420-20241110231420-00537.warc.gz"} |
Symmetric union diagrams and refined spin models
Published Paper
Inserted: 18 sep 2018
Last Updated: 21 may 2019
Journal: Canadian Mathematical Bulletin
Year: 2018
Doi: doi:10.4153/S0008439518000115
An open question akin to the slice-ribbon conjecture asks whether every ribbon knot can be represented as a symmetric union. Next to this basic existence question sits the question of uniqueness of
such representations. Eisermann and Lamm investigated the latter question by introducing a notion of symmetric equivalence among symmetric union diagrams and showing that inequivalent diagrams can be
detected using a refined version of the Jones polynomial. We prove that every topological spin model gives rise to many effective invariants of symmetric equivalence, which can be used to distinguish
infinitely many Reidemeister equivalent but symmetrically inequivalent symmetric union diagrams. We also show that such invariants are not equivalent to the refined Jones polynomial and we use them
to provide a partial answer to a question left open by Eisermann and Lamm. | {"url":"https://calcio.math.unifi.it/paper/308/","timestamp":"2024-11-02T01:31:46Z","content_type":"text/html","content_length":"4621","record_id":"<urn:uuid:e9455073-a56a-434b-9a96-336b51f05ff0>","cc-path":"CC-MAIN-2024-46/segments/1730477027632.4/warc/CC-MAIN-20241102010035-20241102040035-00717.warc.gz"} |
Exploring entanglement in finite-size quantum systems with degenerate ground state
We develop an approach for characterizing non-local quantum correlations in spin systems with exactly or nearly degenerate ground states. Starting with linearly independent degenerate eigenfunctions
calculated with exact diagonalization we generate a finite set of their random linear combinations with Haar measure, which guarantees that these combinations are uniformly distributed... Show more | {"url":"https://synthical.com/article/Exploring-entanglement-in-finite-size-quantum-systems-with-degenerate-ground-state-7fa5727b-9b46-4645-a31a-cecafea7555f?","timestamp":"2024-11-04T11:09:28Z","content_type":"text/html","content_length":"65208","record_id":"<urn:uuid:50eb731a-a70a-4be7-80db-05e20ef4c759>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00083.warc.gz"} |
Apportioned Load
A parser can use more than one database node to load a single input source in parallel. This approach is referred to as apportioned load. Among the parsers built into Vertica, the default (delimited)
parser supports apportioned load.
Apportioned load, like cooperative parse, requires an input that can be divided at record boundaries. The difference is that cooperative parse does a sequential scan to find record boundaries, while
apportioned load first jumps (seeks) to a given position and then scans. Some formats, like generic XML, do not support seeking.
To use apportioned load, you must ensure that the source is reachable by all participating database nodes. You typically use apportioned load with distributed file systems.
It is possible for a parser to not support apportioned load directly but to have a chunker that supports apportioning.
You can use apportioned load and cooperative parse independently or together. See Combining Cooperative Parse and Apportioned Load.
How Vertica Apportions a Load
If both the parser and its source support apportioning, then you can specify that a single input is to be distributed to multiple database nodes for loading. The SourceFactory breaks the input into
portions and assigns them to execution nodes. Each Portion consists of an offset into the input and a size. Vertica distributes the portions and their parameters to the execution nodes. A source
factory running on each node produces a UDSource for the given portion.
The UDParser first determines where to start parsing:
• If the portion is the first one in the input, the parser advances to the offset and begins parsing.
• If the portion is not the first, the parser advances to the offset and then scans until it finds the end of a record. Because records can break across portions, parsing begins after the first
record-end encountered.
The parser must complete a record, which might require it to read past the end of the portion. The parser is responsible for parsing all records that begin in the assigned portion, regardless of
where they end. Most of this work occurs within the process() method of the parser.
Sometimes, a portion contains nothing to be parsed by its assigned node. For example, suppose you have a record that begins in portion 1, runs through all of portion 2, and ends in portion 3. The
parser assigned to portion 1 parses the record, and the parser assigned to portion 3 starts after that record. The parser assigned to portion 2, however, has no record starting within its portion.
If the load also uses Cooperative Parse, then after apportioning the load and before parsing, Vertica divides portions into chunks for parallel loading.
Implementing Apportioned Load
To implement apportioned load, perform the following actions in the source, the parser, and their factories.
In your SourceFactory subclass:
• Implement isSourceApportionable() and return true.
• Implement plan() to determine portion size, designate portions, and assign portions to execution nodes. To assign portions to particular executors, pass the information using the parameter writer
on the plan context (PlanContext::getWriter()).
• Implement prepareUDSources(). Vertica calls this method on each execution node with the plan context created by the factory. This method returns the UDSource instances to be used for this node's
assigned portions.
• If sources can take advantage of parallelism, you can implement getDesiredThreads() to request a number of threads for each source. See SourceFactory Class for more information about this method.
In your UDSource subclass, implement process() as you would for any other source, using the assigned portion. You can retrieve this portion with getPortion().
In your ParserFactory subclass:
• Implement isParserApportionable() and return true.
• If your parser uses a UDChunker that supports apportioned load, implement isChunkerApportionable().
In your UDParser subclass:
• Write your UDParser subclass to operate on portions rather than whole sources. You can do so by handling the stream states PORTION_START and PORTION_END, or by using the ContinuousUDParser API.
Your parser must scan for the beginning of the portion, find the first record boundary after that position, and parse to the end of the last record beginning in that portion. Be aware that this
behavior might require that the parser read beyond the end of the portion.
• Handle the special case of a portion containing no record start by returning without writing any output.
In your UDChunker subclass, implement alignPortion(). See Aligning Portions.
The SDK provides a C++ example of apportioned load in the ApportionLoadFunctions directory:
• FilePortionSource is a subclass of UDSource.
• DelimFilePortionParser is a subclass of ContinuousUDParser.
Use these classes together. You could also use FilePortionSource with the built-in delimited parser.
The following example shows how you can load the libraries and create the functions in the database:
=> CREATE LIBRARY FilePortionSourceLib as '/home/dbadmin/FP.so';
=> CREATE LIBRARY DelimFilePortionParserLib as '/home/dbadmin/Delim.so';
=> CREATE SOURCE FilePortionSource AS
LANGUAGE 'C++' NAME 'FilePortionSourceFactory' LIBRARY FilePortionSourceLib;
=> CREATE PARSER DelimFilePortionParser AS
LANGUAGE 'C++' NAME 'DelimFilePortionParserFactory' LIBRARY DelimFilePortionParserLib;
The following example shows how you can use the source and parser to load data:
=> COPY t WITH SOURCE FilePortionSource(file='g1/*.dat') PARSER DelimFilePortionParser(delimiter = '|',
record_terminator = '~'); | {"url":"https://www.vertica.com/docs/11.0.x/HTML/Content/Authoring/ExtendingVertica/UDx/UDL/ApportionedLoad.htm","timestamp":"2024-11-12T06:14:56Z","content_type":"text/html","content_length":"46885","record_id":"<urn:uuid:0ee609ca-d094-4145-bb79-fae56a4824e4>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.58/warc/CC-MAIN-20241112045844-20241112075844-00421.warc.gz"} |
Medicine is packed in boxes, each weighing 4 kg 500g. How many such boxes can be loaded in a van which cannot carry beyond 800 kg?Medicine is packed in boxes, each weighing 4 kg 500g. How many such boxes can be loaded in a van which cannot carry beyond 800 kg?
Medicine is packed in boxes, each weighing 4 kg 500g. How many such boxes can be loaded in a van which cannot carry beyond 800 kg?
To find out how many boxes of medicine can be loaded in a van that cannot carry beyond 800 kg, you need to divide the maximum weight the van can carry by the weight of each box.
1 box of medicine weighs 4 kg 500 g. First, convert 500 g to kg (1 kg = 1000 g):
500 g = 500 g / 1000 g/kg = 0.5 kg
So, 1 box of medicine weighs 4 kg + 0.5 kg = 4.5 kg.
Now, calculate how many boxes can be loaded:
Number of boxes = Maximum weight the van can carry / Weight of each box Number of boxes = 800 kg / 4.5 kg/box
Number of boxes ≈ 177.78 boxes
Since you cannot have a fraction of a box, you can load a maximum of 177 boxes of medicine in the van.
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A Klein Page
An important part of the study of any geometry is the identification of the
congruence classes. We shall consider triangles, quadrilaterals and conics.
triangles in affine geometry
In euclidean geometry, we define the triangle ABC as the subset of R^2
consisting of the vertices A, B and C and the sides consisting of the
points on the segments AB, BC and CA. To avoid degenerate cases,
we insist that A, B and C are non-collinear.
In affine geometry, we also have the concept of a line segment. Hence,
we can use the same formal definition of triangle, and conclude that
affine transformations map triangles to triangles
From the affine group page, we have
The Fundamental Theorem of Affine Geometry
If L =(A,B,C) and L' = (A',B',C') are lists of non-collinear points of R^2,
then there is a unique element of A(2) mapping L to L'.
This has two interesting corollaries in the present context:
Corollary 1
All triangles are affine congruent.
Suppose that ABC and PQR are triangles, so (A,B,C) and (P,Q,R) are lists
of non-collinear points. By the Fundamental Theorem there is an affine
transformation t mapping A, B, C to P, Q, R, respectively. Then t maps
triangle ABC to triangle PQR, so the triangles are A(2)-congruent.
Note. The triangle ABC consists of the points of the segments AB, BC, CA.
The set of points may equally be described as the triangle ACB, BAC, BCA,
CAB, or CBA! In euclidean geometry, even if the triangles ABC and PQR are
E(2)-congruent, we can usually map A,B,C to P,Q,R only in one particular
order. Here it does not matter!
In the klein view section, we meet the idea of symmetries of a figure
in any geometry. For the case of triangles in affine geometry, we have,
as a consequence of the uniqueness clause in the fundamental theorem,
Corollary 2
For any triangle T, the A(2)-symmetry group of T is isomorphic to S[3].
proof of corollary 2
Note that the situation here is quite different to that in euclidean and
similarity geometry. In those geometries, only an equilateral triangle
has six symmetries. This is so since we must be able to map any side
to any other, so the ratio of lengths must be 1 in all cases. We can also
observe that, in euclidean and similarity geometry, an isosceles triangle
has two symmetries - the identity and reflection in the bisector of the
apex angle. All other triangles have only the identity symmetry.
Once we have looked at the points fixed by a finite subgroup of A(2),
we will have a nice affine proof of the Medians Theorem.
This is rather more group theory
than geometry - it could easily
be omitted at first reading. | {"url":"https://www.maths.gla.ac.uk/wws/cabripages/klein/affine5.html","timestamp":"2024-11-10T15:51:03Z","content_type":"text/html","content_length":"4344","record_id":"<urn:uuid:57c72132-3a03-4f74-98ef-dd2e633e8eae>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.60/warc/CC-MAIN-20241110134821-20241110164821-00702.warc.gz"} |
Inertial base construction for vibration measurements in an oven
SDTools regularly performs Experimental Modal Analyses (EMA) in an oven to control temperature dependency. One of the objectives of such test is to identify the constitutive properties of various
• 3D woven composites
• Tuned-mass dampers
• Polymer materials
• …
From the EMA, an inverse problem can be resolved to minimize the error between a model synthetizing the constitutive law parameters and the experimental results, as part of the model updating
techniques available at SDTools.
The key to constitutive law parameter identification is to design an experiment that generates a sufficient deformation in the associated directions of work. Exploiting structural modes allows an
optimal ratio between energy input and deformation amplitude.
For this application, the sample must thus be placed in the oven with the following constraints:
• Mode shapes of interest must lie in a target frequency band
👉 Tune boundary conditions: connection of the sample to an inertial base
• Isolate the system from the oven structure to avoid measurement pollution
👉 Suspend the base over springs
• Fit in the oven
👉 It is kind of obvious, but this limits the base size (dimension and weight)
• Be of a reasonable price
Regularly, a large base weight is required for the mode shapes of interest to be low enough in frequency, but it needs to be tuned as function of the sample. We have thus decided to build our own
evolutive inertial base for such use cases.
Below is the story of the conception of this base divided in two main steps:
1️⃣ Inertial base design with FEM to check technical specifications
2️⃣ Inertial base building
I – Inertial base design with FEM to check technical specifications
The inertial base technical specifications are set up using simulation-based estimations. A very simple geometry is defined, considering the constraints.
From our past measurement campaigns and our oven characteristics, the following specs are derived
• 300mm width and 300mm depth
👉 Oven fitting constraint: with a 600×1000 width and depth frame, we want to ensure comfortable handling in the oven.
• Base mass < 20kg
👉 The base must be handled by a single person.
• Up to 100kg
👉 Possibility to add weights to the base side for tunability, once in position.
• No deformation mode below 500Hz (base fitted with 4 weights of 10kg)
👉 Aluminum beam truss to balance rigidity and mass limit.
👉 Compact structure: it will be a 300mm cube shape.
Considering cost and optimal rigidity, the aluminum beam profiles are chosen to be the standard EU norm of 40x40mm².
From this initial concept a model can be built to refine the design. Using the beam properties provided by the manufacturer, a cube shaped of 300mm edge truss is defined. The first mode is estimated
at 637Hz.
Fastening features must then be added. We should be able to add weights around the cube sides (as bodybuilding training weights) so that a fastener at the middle of the faces is required. We also
need fasteners to connect the system on the base top, and to suspend the whole with supporting springs.
Beam crosses are thus added to each cube face, leading to an increase of the first mode frequency at 1204Hz.
We then add 4 punctual masses at the middle of each side to coarsely represent the addition of 4*10 kg. The first mode shape is now down to 415Hz, below the 500Hz specifications.
To increase the first mode frequency, beams are placed inside the cube to constraint the observed breathing pattern, resulting in a frequency of 627Hz.
This refined concept meets the specifications it is retained for a physical build.
II – Inertial base building
From the model, the base device is built from
• Saw-cut aluminum profile sections (40 x 40 mm²) for the base truss
• 144 x 90° angle brackets (beam connectors)
• 432 x M8 nuts + bolts
• 4 x training weight holders for walls
• 4 x 10 kg training weight discs
Below are the main building steps :
1️⃣ Cut everything (beams, weight wall supports)
2️⃣ Build the first face
Place 4 x M8 nuts in each beam:
• On the inside of the cube: to be connected to other beams through brackets
• On the outside of the cube: to open connection for evolutions, as port to connect body training weight discs, further base modifications, base suspensions, system clamping to the base…
3️⃣ Raise first, second and third floors! 😊
4️⃣ Put the suspension frame, the cube, the training weights, and the system into the oven.
III Conclusions
We are now happy to physically see our “hypercube” base!
It is customizable
• Weight from 20kg to 100kg
• Fasteners to fit any system on top
• Fasteners to fit a suspension device at the bottom
• Fasteners to further evolution on each side of the cube.
The next step is to validate the realized system regarding the initial specifications. The critical question is whether the prototype actually dynamically behaves as the (maybe too 😉) coarse model.
The unknown mostly resides in beam connections. Are nodal equivalence constraints at beam ends representative of the bolted 90° bracket connection stiffness? The assumption only has to be valid for
the first mode at low amplitudes!
Only measurements will tell! But this is another story… 😏 | {"url":"https://www.sdtools.com/inertial-base-construction-for-vibration-measurements-in-an-oven/","timestamp":"2024-11-08T04:37:12Z","content_type":"text/html","content_length":"295250","record_id":"<urn:uuid:7f3ce7fc-fb58-4a7c-8713-8eeda551e95a>","cc-path":"CC-MAIN-2024-46/segments/1730477028025.14/warc/CC-MAIN-20241108035242-20241108065242-00578.warc.gz"} |
vortex crossfire ii 2 7x32 scope
• Is R≠ a transitive relation? See also. Learn about the world's oldest calculator, Abacus. See more. I'm trying to figure out the transitive relation, and the composite relation. In simple terms,
The transitive property, sometimes, misapplies the transitive property to non-numerical things to reach illogical conclusions or false equivalencies. A homogeneous relation R on the set X is a
transitive relation if, [1]. Suppose that a metal sample X is heavier than a metal sample Y, and that Y is heavier than a sample Z. b Answering a major conception of students of "Is trigonometry
hard?". Is the relation transitive? c , and indeed in this case Solution: The relation R is transitive as for every (a, b) (b, c) belong to R, we have (a, c) ∈ R i.e, (1, 2) (2, 1) ∈ R ⇒ (1, 1) ∈ R.
Note1: The Relation ≤, ⊆ and / are Learn Vedic Math Tricks for rapid calculations. A = {a, b, c} Let R be a transitive relation defined on the set A. but (1,1) is not an element of R. • Now
Relation Rfun on A = {1,2,3,4} defined as: For example, the relation defined by xRy if xy is an even number is intransitive,[11] but not antitransitive. ∈ ∈ Effective way of Digital Learning you
should know? I think the following would be a good example: Let X = {x,y,z} and the binary relation on X, R = { (x,y)} (that is, xRy), This is transitive, since only two elements are related. A
relation is a transitive relation if, whenever it relates some A to some B, which B to some C, it also relates that A thereto C. Some authors call a relation intransitive if it's not transitive.
Check transitive To check whether transitive or not, If (a , b ) ∈ R & (b , c ) ∈ R , then (a , c ) ∈ R Here, (1, 2) ∈ R and (2, 3) ∈ R and (1, 3) ∈ R ∴ R is transitive Hence, R … A homogeneous
relation R on the set X is a transitive relation if,[1]. R "Is greater than", "is at least as great as", and "is equal to" (equality) are transitive relations on various sets, for instance, the set
of real numbers or the set of natural numbers: The empty relation on any set The relation defined by xRy if x is the successor number of y is both intransitive[14] and antitransitive. Or any partial
equivalence relation; Reflexive and symmetric: The relation R on Z, defined as aRb ↔ "a − b is divisible by at least one of 2 or 3." • Answer: Yes. What is more, it is antitransitive: Alice can never
be the birth parent of Claire. is vacuously transitive. R The mother carried the baby. Now let us move onto some transitive properties and what they imply. Of Course not. and Transitive Phrasal Verbs
fall into three categories, depending on where the object can occur in relation to the verb and the particle. The transitive property of equality is for any elements a, b and c if a=b and b=c then a
=c. A relation from a set A to itself can be though of as a directed graph. It’s quite trivially symmetric, transitive, and even anti-reflexive. This blog deals with domain and range of a parabola.
It is not a transitive relation since (1,2) R and (2,1) R Examples of Intransitive Verb. The converse of a transitive relation is always transitive: e.g. Example of a relation that is reflexive,
symmetric, antisymmetric but not transitive. a For instance, knowing that "is a subset of" is transitive and "is a superset of" is its inverse, we can say that the latter is transitive as well.
Understand how the values of Sin 30, Cos 30, Tan 30, Sec 30, Cosec 30, Cot 30 & sine of -30 deg... Understanding what is the Trigonometric Table, its values, tricks to learn it, steps to make it
by... Line of best fit refers to a line that best expresses the relationship between a scatter plot of... How to Find the Areas of Various Shapes in Geometry? There are several examples of relations
which are symmetric but not transitive & refelexive . {\displaystyle aRc} Learn about Circles, Tangents, Chords, Secants, Concentric Circles, Circle Properties. x Also some other car c of the same
model will also be equal to car a and b. For example, in the set A of natural numbers if the relation R be defined by ‘x less than y’ then a < b and b < c imply a < c, that is, aRb and bRc ⇒ aRc. b
{\displaystyle (x,x)} As a nonmathematical example, the relation "is an ancestor of" is transitive. • Rfun = {(1,2),(2,2),(3,3)}. See examples in this entry! An intransitive relation is one which
will or may not hold between a and c if it also holds between a and b and between b and c, counting on the objects substituted for a, b, and c. In other words, there's a minimum of one substitution
on which the relation between a and c does hold and a minimum of one substitution on which it doesn't. The example just given exhibits a trend quite typical of a substantial part of Recursion Theory:
given a reflexive and transitive relation ⩽r on the set of reals, one steps to the equivalence relation ≡ r generated by it, and partitions the reals into r -degrees (usually indicated by boldface
letters such as a, b, c, …); then one studies the structure Dr of the r-degrees under the partial ordering ⩽ induced by ⩽ r, with the goal … The converse of a transitive relation is always
transitive: e.g. For instance, "was born before or has the same first name as" is not a transitive relation, since e.g. We know that if then and are said to be equivalent with respect to .. knowing
that "is a subset of" is transitive … The union of two transitive relations need not be transitive. Examples. What are naturally occuring examples of relations that satisfy two of the following
properties, but not the third: symmetric, reflexive, and transitive. This seems quite obvious, but it's also very important. To achieve the normalization standard of Third Normal Form (3NF), you must
eliminate any transitive dependency. , Solution: Let us consider x ∈ A. • R≠={(1,2),(1,3),(1,4),(2,1),(2,3),(2,4),(3,1),(3,2),(3,4),(4,1),(4,2),(4,3)} a [8] However, there is a formula for finding
the number of relations that are simultaneously reflexive, symmetric, and transitive – in other words, equivalence relations – (sequence A000110 in the OEIS), those that are symmetric and transitive,
those that are symmetric, transitive, and antisymmetric, and those that are total, transitive, and antisymmetric. In math, if A=B and B=C then A=C. Why operations and algebraic thinking is important.
TRANSITIVE RELATION. Solution: Since all cars of the same design are same in shape and size, we can say that for every, .Therefore it represents a reflexive relation. On the other hand, "is the birth
parent of" is not a transitive relation, because if Alice is the birth parent of Brenda, and Brenda is the birth parent of Claire, then Alice is not the birth parent of Claire. The relations
``…loves…” and “… isn't adequate to …” are examples. A transitive relation need not be reflexive. Now, consider the relation "is an enemy of" and suppose that the relation is symmetric and satisfies
the condition that for any country, any enemy of an enemy of the country is not itself an enemy of the country. Learn Vedic Math Tricks for rapid calculations. Transitive relations and examples. This
page was last edited on 19 December 2020, at 03:08. , while if the ordered pair is not of the form Transitive law, in mathematics and logic, any statement of the form “If aRb and bRc, then aRc,”
where “R” may be a particular relation (e.g., “…is equal to…”), a, b, c are variables (terms that which will get replaced with objects), and the result of replacing a, b, … [12] The relation defined
by xRy if x is even and y is odd is both transitive and antitransitive. Helping Students with Learning Disabilities. Examples of transitive in a sentence, how to use it. More examples of transitive
relations: "is a subset of" (set inclusion) "divides" (divisibility) "implies" (implication) Properties Closure properties. As a nonmathematical example, the relation "is an ancestor of" is
transitive. A transitive relation is which objects of a similar nature are the same. Since y = (x + a)(x + b), and y also equals x2 + (a + b)x + ab, then those two quantities must be equal to each
other! 2. Solved example of transitive relation on set: 1. If whenever object A is related to B and object B is related to C, then the relation at that end transitive provided object A is also
related to C. Being a child is a transitive relation, being a parent is not. for some This blog helps students identify why they are making math mistakes. [6] For example, suppose X is a set of
towns, some of which are connected by roads. One such example is the relation of perpendicularity in the set of all straight lines in a plane. Such a relation is reflexive if and only if it is
serial, that is, if ∀a∃b a ~ b. b Hence, R is symmetric. For example, if Amy is an ancestor of Becky, and Becky is an ancestor of Carrie, then Amy, too, is an ancestor of Carrie. Our tech-enabled
learning material is delivered at your doorstep. Let R be a transitive relation defined on set A. The reason is of course that the same object may appear in different ways whose identity may not be
either obvious or a priori known. This blog deals with the common ratio of an geometric sequence. This post covers in detail understanding of allthese Here's an example of how we could use this
transitive property. Transitive Relation. {\displaystyle a,b,c\in X} The complement of a transitive relation need not be transitive. a The Life of an Ancient Astronomer : Claudius Ptolemy. So let \(A
\) be a nonempty set and let \(R\) be a relation on \(A\). This may include any relation that's not a transitive relation, or the stronger property of antitransitivity, which describes a relation
that's never a transitive relation. , C } let R be a property of binary relations that are n't true... I 'm trying to figure out the contrast between these operations… transitive relations and
examples of Distance Learning of—math! Renames the object mentioned next to it first of his Common Notions 's what mathematics all! A community that is reflexive symmetric and transitive which
objects of a transitive relation is transitive! Similar to the meaning with equivalence relation is so natural that Euclid it... To determine whether the defined relation is said to be transitive
only its! Be equivalent with respect to to divide two numbers using Abacus examples for verbs of cases! Onto some transitive Properties and Applications about Operations and Algebraic Thinking for
Grade 2 life always... Three categories, depending on where the object that appears before it not! Onto some transitive Properties and what they imply two transitive relations and.. Mentioned next to
it given by caris congruent to car a and b was last edited on 19 December,.? `` examples for transitive relation intersection of two transitive relations need not be transitive always transitive: e.g
choice... Or renames the object can occur in relation to the relation defined by xRy if x heavier! Metal sample x is one for which objects of a relation to the and... Trigonometry Hard? `` to itself
can be easily understood within the sentence, Secants, Concentric,. Is so natural that Euclid stated it as the first place about and! Is heavier than z 1st to 10th Grade kids is even and y is odd is
intransitive. R 2 are equivalence relation proof and its examples list, the relation among life forms is,. Instance, then would you expect a to itself can be used both as transitive and
antitransitive,. Transitive relations on a finite set ( sequence A006905 in the collection objects. How to properly determine if reflexive, symmetric, and b=a as the first place a... < y and y < z
then x < z if ∀a∃b ~! Of intransitivity arise in situations such as political questions or group preferences Cash Prizes worth Rs.50 lakhs * up grabs...: let R be a transitive relation let a be any
set an even number is intransitive in! To subtract two numbers using Abacus only if it is, if x is a transitive relation reflexive... Same model will also be 5 by the transitive closure of the equals
sign must be equal to one.... What they seem in the OEIS ) is known sin 30, Cos,... 2X + 3x = 5x, which are equal to car a and b = C, y. A\ ) then y is related by R to y, and that y is heavier than a
sample!: Yes, it can-not be regarded as generally transitive or generally intransitive nature may stand to each other things. Likes Cath it does n't necessarily follow that Ann likes Cath it n't...
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property, sometimes, misapplies the transitive relation birth parent of Claire a. ) may be alternatively defined as a directed graph ] for example, an equivalence.... A major conception of students
of `` is a subset of '' is transitive … of., Cos pi/3, Sec pi/3, Cosec 30, Cot pi/3 the baby ” is mother! < y and y is heavier than z 2 are equivalence relation. [ ]..., C } let R be a transitive
relation let a = b b. |A|=1\ ) could use this transitive property but not transitive of a parabola the separation of the equals must... Of y is related by R to x math, if A=5 instance... Calculator,
Abacus relation defined on the set a as given below Prizes worth Rs.50 *... That if then and are said to be transitive are n't always true, so by the transitive of! Guide to Preparing for Exams,
Environment, Mind-set, Location, Material and Diet students & 300+ Pan... Form ( 3NF ), you must eliminate any transitive dependency relation let a be any set given an of! Similar to the relation
'greater than ' for numbers interpret the csc Sec Cot... Tangent:... What is more, it is the result of applying the particle Movement Rule transitive relations and examples 'm to. Such a relation on
set: 1 determine if reflexive, symmetric transitive. Follow that Ann likes Cath it as the first two statements are true does make... Oeis ) is known Prizes worth Rs.50 lakhs * up for grabs that
appears before is... Is transitive and intransitive according to the meaning are also equal to car determine or... A similar nature may stand to each other 's similar to the relation 'greater than '
for numbers ” the! This blog helps students identify why they are making math mistakes this seems quite obvious, not! Voters need to rank them so as to preference is usually transitive 5,00,000+... }
let R be a transitive relation is reflexive symmetric and transitive a sample z, at 03:08 always! Let \ ( A\ ) nor transitive on 19 December 2020, at 03:08 is related by R to.. They seem in the set a
) of a transitive relation is another generalization ; it a. Whether or not sets of tuples have a mathematical result you could be wrong some special part-whole cases, are! A sample z you think you
have a certain type of relation. [ 5.! And yRz always implies that xRz does not have any cycles related to Algebraic Thinking exactly! 12,028 6,344 Lexington, MA ( USA ) Oct 22, 2008 # 2 Hello
terr13! Relation examples that does not hold transitive property to non-numerical things to reach illogical conclusions or equivalencies... But ( 1,3 ) ∉ R 2 but ( 1,3 ) ∉ R 2 but ( 1,3 ) R! And some
other, which are connected by roads not exactly the same model also! Is for any elements a, b and b = C, then b and if. Transitive Properties and Applications to reach illogical conclusions or
examples for transitive relation equivalencies of such are! Grade 5 grass, so when you think you have a mathematical result you could be wrong if R and... Defined as a nonmathematical example, the
relation 'greater than ' for numbers that Euclid stated it as cars. 1,2,3 }: let R be a transitive relation if, [ ]... The action verb in this sense appears before it is true in—a property! Not
negatively transitive because ¬ zRy and ¬ xRz but xRy xRy if xy is an example an! As obvious as what they imply the relations `` …loves… ” and “ … n't! In mathematics, transitive, and some other car
C of the objects in collection! Is even and y is both intransitive [ 14 ] and antitransitive assume in some context a always C! Them so as to preference the manager discussed the company strategies
with his employees: reflexive symmetric. Sign must examples for transitive relation true that x is heavier than z of which are to! Does not make the final “ conclusion ” true a to itself can be
denoted:! = C, then would you expect a to itself can be easily understood within the sentence:.... ) be a relation in knockout tournaments subtract two numbers using Abacus highlighted words are the
verbs in OEIS... Is asymmetric if and only if it is also trivial that it is required to transitive! Prove that R is called equivalence relation. [ 7 ], a relation is! Are connected by roads two
numbers using Abacus what they seem in the collection of objects in the OEIS is. Set ( sequence A006905 in the first place list, the transitive property to each.. Understand and interpret the csc Sec
Cot... Tangent Function: Domain, Range, Properties and what they.... The collection of objects in the first two statements are true does not have any cycles examples. Always prove a result before you
can be easily understood within the sentence if, 11. Are intransitive for numbers page was last edited on 19 December 2020, at 03:08 defined xRy! 1,2,3 }: let R be a transitive relation. [ 5 ] is
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fitcharge (c49b1)
The Charge and Drude Polarizability Fitting
By V.Anisimov and G.Lamoureux, December 2004
Editions By E. Harder 2007
The commands of this section solve the task of charge fitting to
QM electrostatic potential (ESP) maps. In the case of classical Drude
polarizable systems both ESP fitted charges and atomic polarizabilities
will be determined in the single fitting step. The polarizability
determination is based on Drude charge fitting to the series of perturbed
ESP maps obtained in presence of perturbation charges. See DRUDE.DOC for
a description of the classical Drude polarizable model. The citations given
in the references section give further details about the charge fitting
procedure. See FITCHARGE test for the practical sample of charge fitting
and Drude polarizability determination.
The fitcharge routine can be used for charge fitting for the additive
model. A single unperturbed QM ESP is used in this case.
The program supports lone-pairs in either additive or Drude
polarizable model. The QM ESP maps and fitcharge instruction set are
independent of the presence of lone pairs.
| Syntax of charge fitting commands
| Introduction to charge fitting
| Purpose of the commands
| Input example
| Known limitations
Syntax of charge fitting commands
[SYNTAX FITCharge - charge fitting]
FITCharge { [EQUIvalent atom-selection]
[RESTraint [PARAbolic|HYPErbolic]
[BHYP real] [DHYP real] [FLAT real] [DFLAt real]
atom-selection-1 atom-selection-2
[NITEr int] [TOLErance real] [COUNter int]
NCONf int UPOT int UOUT int NPERt int [int]
[TEST] UPPOt int UCOOrd int [ALTInput]
[UPRT int] [ASCAle real] [RDIPole real] [VTHOLe] }
atom-selection ::= see
» select
Introduction to charge fitting
Unrestrained charge fitting:
The electrostatic properties of a molecular mechanics model with
Drude polarizabilities are represented by atomic partial charges {q_i}
and Drude charges {delta_i}. The Drude charges are related to atomic
polarizabilities {alpha_i = q_i^2/k_D}, where k_D is a uniform
harmonic coupling constant between each atom and its Drude particle.
These charges are ajusted to give the best agreement with the ab
initio molecular electrostatic potential phi^AI, computed on a set of
gridpoints {r_g} around the molecule.
Although partial charges of a nonpolarizable model can be extracted
from a single potential map, adjusting the polarizabilities requires a
series of /perturbed/ potential maps {phi^AI_p}, each one representing
the molecule in the presence of a small point charge at a given
position r_p. The molecular mechanics model for the molecule under
the influence of perturbation p is a collection of point charges {q_i
- delta_i} at atomic positions {r_i} and Drude charges {delta_i} at
positions {r_i + d_pi}. The model electrostatic potential for the
p-th perturbation, at the g-th gridpoint, is
phi_pg({q}) = sum_i [ (q_i-\delta_i)/(|r_i-r_g|)
+ (delta_i)/(|r_i+d_pi-r_g|} ]
The optimal displacements {d_pi} depend on the position r_p of the
perturbating charge, as well as on the atomic and Drude charges. All
charges are adjusted to minimize the discrepancy between the ab initio
and model potential maps, i.e., we find the charges that minimize the
following chi^2 function:
chi^2 = chi^2_\phi({q}) = sum_pg [ phi^AI_pg - phi_pg({q}) ]^2
Because of the implicit charge-dependence of the displacements
{d_pi}, the system of equations
(@ chi^2)/(@ q) = 0
where q designates either {q_i} or {delta_i}, has to be solved
iteratively. We use the Levenberg-Marquardt algorithm, specially
designed to minimize chi^2-type functions.
Restrained charge fitting:
Solving equations for chi^2 = chi^2_phi, one usually ends up
with partial charges and polarizabilities having poor chemical significance
(e.g. charges on carbon > 1). For nonpolarizable models, it was shown
that fitted charges of neighboring atoms were highly correlated, and,
more generally, that the atomic point charges model of the potential
was largely overparametrized. It is therefore desirable to either
remove charge contributions that have a negligible effect on the
potential, or to penalize any deviation from some ``intuitive'' (or
``conservative'') reference charge, given that the restraint doesn't
significantly deteriorate the quality of the fit.
The original RESP scheme of Bayly et al. minimizes chi^2 =
chi^2_phi + chi^2_r, with either
chi^2_r = A sum_i (q_i - qbar_i)^2
chi^2_r = A sum_i [ sqrt(q_i^2 + b^2) - b ]
The first restraint is forcing the charges q_i to their ``reference''
values qbar_i, and the second restraint is favoring smaller
charges. The force constant A is chosen so that undesirable charge
deviations are penalized while chi^2_phi stays close to its
unrestrained value. It assumes a uniform restraint force A, independent
of the atom type. A more flexible scheme would allow various A's, but
this has not been implemented.
Although the RESP scheme was formulated for nonpolarizable,
partial charges models, it is generalizable to models with Drude
polarizabilities. Equation may be written
chi^2_r = sum_i [ A ( sqrt(q_i^2 + b^2) - b )
+ A'( sqrt(delta_i^2 + b'^2) - b' ) ]
where distinct force constants A and A' are used for the atomic and
Drude charges, along with distinct hyperbolic stiffnesses b and b'. We
thus separately penalize the net atomic charges {q_i} and the Drude
charges {delta_i = sqrt(alpha_i / k_D)}.
The restraint function has the form
chi^2_r = N_p N_g sum_i [
w_i S(q_i - qbar_i) + w'_i S(delta_i - deltabar_i) ],
where N_p is the number of perturbations and N_g is the number of
The restraints are not applied directly to the charges of the
particles, but to the net charges {q_i} and dipoles
{-delta_i,delta_i}. The weights {w_i} and {w'_i} are read from the
WMAIN array and the initial atomic charges are taken as reference
charges {qbar_i} and {deltabar_i}.
The function S(q) describes the shape of the penalty as the deviation
increases. Two basic shapes are available:
PARA Parabolic shape, S(q) = q^2
HYPE Hyperbolic shape, S(q) = sqrt(q^2+b^2)-b
Parameter b (keyword BHYP) is 0.1 electron by default. The additional
keyword DHYP, with default value B, is used for Drude charges. To
produce S(q) = |q|, set b=0.
The FLAT keyword modifies the shape:
S(q+FLAT) if q < -FLAT,
S'(q) = 0 if -FLAT < q < FLAT,
S(q-FLAT) if FLAT < q.
The default value is FLAT=0. The additional keyword DFLAt, with
default value FLAT, is used for Drude charges.
Purpose of the commands
EQUIvalent atom-selection
This block allows explicit equivalences between atoms to be stated.
Default value is no equivalences, i.e. each atom is unique in the fitting
procedure. Multiple EQUIvalence keywords are allowed. For each EQUI keyword,
the selected atoms are made equivalent.
[ RESTraint [PARAbolic|HYPErbolic]
[BHYP real] [DHYP real] [FLAT real] [DFLAt real] ]
RESTraint keyword invokes RESP restrained fitting. Not specifying
the RESTraint keyword causes unrestrained fitting to be performed. The
charges and polarizabilities are restrainted to their initial values
(for parabolic penalty function, invoked by keyword PARA, which is also
default) or to zero (in the case of the hyperbolic restraint, HYPE keyword).
The restraint forces (penalty weight) are taken from WMAIN array. They can
be assigned to individual atoms but in practice a uniform stiffness
parameter works well for the whole system (see example below).
A choice between PARAbolic or HYPERbolic function can be made for the
penalty function in the case of the restrained fitting. The PARAbolic shape
introduces the penalty function in the form S(q) = q^2 where q is charge
deviation from the restrained value. The HYPErbolic penalty function is
S(q) = sqrt(q_^2 + B^2) - B, where B is the parabola stiffness parameter.
BHYP keyword sets the stiffness for atomic charges.
DHYP keyword penalizes the atomic polarizability (i.e. the Drude charges).
FLAT keyword introduces a flat well potential,i zeroing the penalty for
the charge deviation in the range from -FLAT to +FLAT.
DFLAT keywords has simular effect for atomic polarizabilites (i.e. Drude
atom-selection-1 atom-selection-2
SELEct ... END The first atom selection specifies the atoms to fit.
This is an obligatory keyword.
SELEct ... END The second atom selection specifies the atoms
contributing to the electrostatic potential. This is an obligatory keyword.
In most common cases both selections should be pointing to all atoms of the
system excluding the perturbation ion. All other (non-selected) atoms are
contributing to the potential energy, and are considered as a perturbation
(this is how the CALcium perturbation atom is handled).
[NITEr int] [TOLErance real] [COUNter int]
NITEr - maximum number of Levenberg-Marquardt (least square) iterations.
Default value NITE=50. If the program does not converge in 50 iterations most
likely something is wrong with the input data.
TOLErance - relative tolerance on the convergence of the minimized
function (chi^2 corresponding to ESP deviation and penalty contribution) for
Levenberg-Marquardt algorithm. Default value TOLE=1.0E-4. Setting bigger
value is not advised. Smaller values may cause convergence problems.
COUNter is number of iterations under the tolerance it needs to
converge. Default value COUN=2. In most cases setting COUN=1 will result in
the fitting requiring less number of LM steps but the results may be highly
questionable. COUN=2 is proven to be safe. Greater values can be used to test
convergence to assue that the real minimum is identified though this is not
necessary. Inspection of "lambda" variable (an equivalent to level shifting
in QM) from the program output having values 0.05 and below is usually a good
indication of convergence. Smaller final value for "lambda" indicates better
result of the fitting.
NCONf int UPOT int UOUT int NPERt int [int]
NCONf specifies the number of conformations to be used in the
electrostatic fit. Typically 1 conformation is used.
UPOT is the file unit number from which to read the unperturbed ESP
map. The format of this file is: Number of lines: ngrid(iconf) Format:
Xgrid Ygrid Zgrid Potential (4f15.6) For NCONf > 1, units UPOT+1,
UPOT+2, ..., UPOT+NCONf-1 will also be read. These files should have
been open before FITCharge execution.
UOUT is the scratch file unit. The file is used for temporary storage of
NPERt is the number of perturbations for each conformation, e.g. NPERT 40
indicates that 40 perturbation ESP maps are calculated in QM jobs and
provided for charge fitting. NPERT 40 42 indicates that 40 perturbed ESP maps
are available for the first conformation and 42 maps are available for th
second one.
This is a test case to compare CHARMM Drude and QM electrostatic potentials
generated in the position of perturbation ions. This requires perturbation
ions and grid points being placed at the same locations, giving equal number
of perturbation ions and grid points. No fitting will be performed in this
case. CHARMM and QM potential along with differences in static and perturbed
potential will be printed out on the unit specified by UOUT keyword. The order
of columns is the following: perturbation ion numer, QM static ESP in the
position of the specified perturbation ion, CHARMM static ESP, QM perturbed
ESP, CHARMM perturbed ESP, QM polarization component, CHARMM polarization
UPPOt int UCOOrd int [ALTInput]
UPPOt is input unit for the perturbed ESP maps. The file format is
Number of lines: npert(iconf)*NGRID(iconf) Format: Potential (1f15.6)
For NCONF > 1 (multiple conformation fitting), units UPPOT+1, UPPOT+2, ...,
UPPOT+NCONf-1 will also be read.
UCOOrd is the unit number of the first file with model compound coordinates
and a perturbation ion. Coordinates are in CHARMM format. NPERt number of such
files has to be provided. All files have to be opened before invoking
ALTInput switches on the alternative input for coordinates. In this mode,
the coordinates of the atoms of the second selection are read from UCOOrd, for
each conformation and perturbation.
[UPRT int] [ASCAle real] [RDIPole real]
UPRT is file unit for final printout of the FITCharge results. The data are
printed in the form of a CHARMM stream file.
ASCAle is the polarizability (alpha) scaling factor. Useful to scale
gas-phase polarizabilities. The scaling keeps atomic charges intact.
RDIPole is reference dipole for charge scaling. The charges will be
scaled to reproduce the reference dipole.
VTHOLe allows the fitting of chemical type dependent thole parameters
in addition to the charges. If this flag is not included a constant value of
a_i = 1.3 for all chemical types will be used to fit the charges.
This corresponds to a parameter a = a_i + a_j = 2.6 which is the THOLE parameter
in the old Drude command syntax.
set residue cyt
! potential for unperturbed system will be read from this file
open read card unit 11 name @residue.pot0
! potential for perturbed systems
open read card unit 21 name @residue.pot
! ESP calculated by CHARMM; a scratch file
open write card unit 30 name @residue.scratch
! fitcharge results will be stored here
open write card unit 90 name @residue.charge.optimized
! all the positions of the 0.5 charge; for alternative input
open read card unit 31 name @residue.all
! set weighting factor for restraints
scalar wmain set 1.0d-5 select segid @residue end
equivalent select type H4* end - ! make atoms H41 and H42 equivalent
select segid @residue end - ! atoms to fit
select segid @residue end - ! ESP contributing atoms
restraint para - ! invoke restrained fitting
flat 0.0 dflat 0.1 - ! use flat well potential for polarizability
upot 11 uout 30 -
NITE 50 - ! look for input errors if job does not converge in 50 steps
NCONF 1 - ! 1 conformation will be used in fitting
NPERT 57 - ! 57 perturbed QM ESP maps were given on input
uppot 21 -
ucoord 31 altinput - ! use alternative input
ascale 0.742 - ! Scale polarizability in analogy with SWM4P water model
rdipole 6.72 - ! Cytosize B3LYP/aug-cc-pVDZ gas-phase dipole moment
uprt 90 ! results will be saved in the form of a CHARMM script
Send questions or comments about this document to CHARMM forum or to
Victor Anisimov at victor@outerbanks.umaryland.edu
1) Bayly et al, JPC 97 (40), 10269, 1993
2) V.M.Anisimov, G.Lamoureux, I.V.Vorobyov, N.Huang, B.Roux,
A.D.MacKerell,Jr. JCTC, 2004, Vol.1, No.1
Task required for charges/polarizabilility fitting not yet included in CHARMM:
- ion placement and grid generation around the model compound
- QM ESP calculation
- extraction of ESP data from the QM output files
Scripts to perform these functions may be requested on the CHARMM forum.
1. Unperturbed QM ESP map (static) is not included into charge and
polarizability fitting when Drude model is employed.
2. In the Lone-Pair case "altinput" keyword is mandatory. | {"url":"https://academiccharmm.org/documentation/version/c49b1/fitcharge","timestamp":"2024-11-08T18:35:33Z","content_type":"text/html","content_length":"31898","record_id":"<urn:uuid:3e579e8c-a2b1-4a69-9897-7c7a6e6a5943>","cc-path":"CC-MAIN-2024-46/segments/1730477028070.17/warc/CC-MAIN-20241108164844-20241108194844-00629.warc.gz"} |
Nonparametric model checking for k-out-of-n systems
It is an important problem in reliability analysis to decide whether for a given k-out-of-n system the static or the sequential k-out-of-n model is appropriate. Often components are redundantly added
to a system to protect against failure of the system. If the failure of any component of the system induces a higher rate of failure of the remaining components due to increased load, the sequential
k-out-of-n model is appropriate. The increase of the failure rate of the remaining components after a failure of some component implies that the effects of the component redundancy are diminished. On
the other hand, if all the components have the same failure distribution and whenever a failure occurs, the remaining components are not affected, the static k-out-of-n model is adequate. In this
paper. we consider nonparametric hypothesis tests to make a decision between these two models. We analyze test statistics based on the profile score process as well as test statistics based on a
multivariate intensity ratio and derive their asymptotic distribution. Finally, we compare the different test statistics.
• Hypothesis testing
• Multivariate intensity ratio
• k-out-of-n model
• Profile score process
• Sequential k-out-of-n model
• Static k-out-of-n model
• INFERENCE
• RATES
Dive into the research topics of 'Nonparametric model checking for k-out-of-n systems'. Together they form a unique fingerprint. | {"url":"https://cris.maastrichtuniversity.nl/en/publications/nonparametric-model-checking-for-k-out-of-n-systems","timestamp":"2024-11-07T02:19:58Z","content_type":"text/html","content_length":"54469","record_id":"<urn:uuid:c5fb2684-5902-4006-966f-bff55537afc0>","cc-path":"CC-MAIN-2024-46/segments/1730477027951.86/warc/CC-MAIN-20241107021136-20241107051136-00661.warc.gz"} |
Chapter 6: Exploring Data: Relationships Lesson Plan - ppt video online download
1 Chapter 6: Exploring Data: Relationships Lesson PlanFor All Practical Purposes Displaying Relationships: Scatterplots Regression Lines Correlation Least-Squares Regression Interpreting Correlation
and Regression Mathematical Literacy in Today’s World, 7th ed. 1 © 2006, W.H. Freeman and Company
2 Chapter 6: Exploring Data: Distributions Displaying RelationshipsRelationship Between Two Variables Examine data for two variables to see if there is a relationship between the variables. Does one
influence the other? Study both variables on the same individual. If a relationship exists between variables, typically one variable influences or causes a change in another variable. Explanatory
variable explains, or causes, the change in another variable. Response variable measures the outcome, or response to the change. Response variable – A variable that measures an outcome or result of a
study (observed outcome). Explanatory variable – A variable that explains or causes change in the response variable. 2
3 Chapter 6: Exploring Data: Distributions Displaying Relationships: ScatterplotsData to Be Used for a Scatterplot A scatterplot is a graph that shows the relationship between two numerical
variables, measured on the same individual. Explanatory variable, x, is plotted on the horizontal axis, (x). Response variable, y, is plotted on the vertical axis (y). Each pair of related variables
(x, y) is plotted on the graph. Example: A study done to see how the number of beers that a young adult drinks predicts his/her blood alcohol content (BAC). Results of 16 people: Young Adult 1 2 3 4
5 6 7 8 Beers 9 BAC 0.10 0.03 0.19 0.12 0.04 0.095 0.07 0.06 10 11 12 13 14 15 16 0.02 0.05 0.85 0.09 0.01 Explanatory variable, x = beers drunk Response variable, y = BAC level 3
4 Chapter 6: Exploring Data: Distributions Displaying Relationships: ScatterplotsExample continued: The scatterplot of the blood alcohol content, BAC, (y, response variable) against the number of
beers a young adult drinks (x, explanatory variable). The data from the previous table are plotted as points on the graph (x, y). BAC vs. number of beers consumed Examining This Scatterplot… 1. What
is the overall pattern (form, direction, and strength)? Form – Roughly a straight-line pattern. Direction – Positive association (both increase). Strength – Moderately strong (mostly on line). 2. Any
striking deviations (outliers)? Not here. Outliers – A deviation in a distribution of a data point falling outside the overall pattern. 4
5 Chapter 6: Exploring Data: Distributions Regression LinesA straight line that describes how a response variable y changes as an explanatory variable x changes. Regression lines are often used to
predict the value of y for a given value of x. BAC vs. number of beers consumed A regression line has been added to be able to predict blood alcohol content from the number of beers a young adult
drinks. Graphically, you can predict that if x = 6 beers, then y = 0.95 BAC. (Legal limit for driving in most states is BAC = 0.08.) 5
6 Chapter 6: Exploring Data: Distributions Regression LinesUsing the Equation of the Line for Predictions It is easier to use the equation of the line for predicting the value of y, given the value
of x. Using the equation of the line for the previous example: predicted BAC = − ( )(beers) y = − (x) For a young adult drinking 6 beers (x = 6): predicted BAC = − (6) = 0.095 Straight Lines A
straight line for predicting y from x has an equation of the form: predicted y = a + b x In this equation, b is the slope, the amount by which y changes when x increases by 1 unit. The number a is
the intercept, the value of y when x =0. 6
7 Chapter 6: Exploring Data: Distributions CorrelationMeasures the direction and strength of the straight-line relationship between two numerical variables. Correlation is usually written as r. A
correlation r is always a number between −1 and 1. It has the same sign as the slope of a regression line. r > 0 for positive association. r < 0 for negative association. Perfect correlation r = 1 or
r = −1 occurs only when all points lie exactly on a straight line. The correlation moves away from 1 or −1 (toward zero) as the straight-line relationship gets weaker. Correlation r = 0 indicates no
straight-line relationship. 7
8 Chapter 6: Exploring Data: Distributions CorrelationCorrelation is strongly affected by a few outlying observations. (Also, the mean and standard deviation are affected by outliers.) Equation of
the Correlation To calculate the correlation, suppose you have data on variable x and y for n individuals. From the data, you have the values calculated for the means and standard deviations x and y.
The means and standard deviations for the two variables are ¯ and sx, for the x-values, and ¯ and sy for the y-values. The correlation r between x and y is: (x1 − ¯) (y1 − ¯) + (x2 − ¯) (y2 − ¯) + …
+ (xn − ¯) (yn − ¯) n – sx sy sx sy sx sy x y ( ) x y x y x y r = 8
9 Chapter 6: Exploring Data: Distributions CorrelationThe scatterplots below show examples of how the correlation r measures the direction and strength of a straight-line association. 9
10 Chapter 6: Exploring Data: Distributions Least Squares RegressionLeast-Squares Regression Line A line that makes the sum of the squares of the vertical distances of the data points from the line
as small as possible. Equation of the Least-Squares Regression Line From the data for an explanatory variable x and a response variable y for n individuals, we have calculated the means ¯ , ¯ , and
standard deviations sx , sy , as well as their correlation r. x y The least-squares regression line is the line: Predicted y = a + bx With slope … b = r sy/sx And intercept … a = ¯ − b ¯ This
equation was used to calculate the line for predicting BAC for number of beers drunk. Predicted y = − x y x 10
11 Chapter 6: Exploring Data: Distributions Interpreting Correlation and RegressionA Few Cautions When Using Correlation and Regression Both the correlation r and least-squares regression line can be
strongly influenced by a few outlying points. Always make a scatterplot before doing any calculations. Often the relationship between two variables is strongly influenced by other variables. Before
conclusions are drawn based on correlation and regression, other possible effects of other variables should be considered. A strong association between two variables is not enough to draw conclusions
about cause and effect. Sometimes an observed association really does reflect cause and effect (such as drinking beer causes increased BAC). Sometimes a strong association is explained by other
variables that influence both x and y. Remember, association does not imply causation. 11 | {"url":"https://slideplayer.com/slide/4515264/","timestamp":"2024-11-02T11:22:43Z","content_type":"text/html","content_length":"189170","record_id":"<urn:uuid:7fb8feb1-0cac-43aa-81ea-8c551425cf9f>","cc-path":"CC-MAIN-2024-46/segments/1730477027710.33/warc/CC-MAIN-20241102102832-20241102132832-00244.warc.gz"} |
Stochastic Oscillator - Trading IndicatorStochastic Oscillator - Trading Indicator
Stochastic Oscillator
The Stochastic Oscillator is one of the most popular trading indicators. Generally when prices begin rising Stochastic rises and when price falls the Stochastic indicator falls. However psychology is
important in trading, fear and greed rule the markets and fear and greed generate momentum in prices. Stochastic is created by a formula that judges momentum.
Markets always rise and fall, in uptrends they rise more than they fall and in downtrends they fall more than they rise, however a trend always has pullbacks. In a ranging market there are swings in
prices within a certain range. This movement is like a heartbeat to the market.
Markets make lows and they make highs. In uptrends there are higher highs and higher lows, in downtrends there are lower lows and lower highs. In ranges prices duck and dive making lows and highs
within the range. Stochastic will measure momentum within these moves.
Momentum trading indicator
Think of throwing a ball to a friend. You launch the ball into the air. It seems to hang in the air for a period of time (still rising) before plunging downwards into the hands of your friend. One of
the trickiest things to judge in trading is, when is price running out of momentum? When is the ball going to start falling?
The time to exit a buy trade is when momentum is reaching its peak. Not after it has already lost its momentum. When momentum is lost the ball will fall rapidly. The trouble is that if you just look
at prices then you will find it really difficult to exit a trade at this stage. All you will see is that prices are still rising. Greed could prevent you from exiting. Also the fear of missing out on
further possible profit could prevent you from exiting. However if you could see price slowing, momentum gone, just before price begins to fall then this would be useful, wouldn’t it? This is one
thing that the Stochastic indicator can help you with.
What is the Stochastic indicator?
Like all indicators Stochastic is based on price, however it gives a slightly different interpretation of price action than just looking at prices alone.
Stochastic takes into consideration Closing Prices, Low Prices and High Prices within a period.
If price within a period keeps making similar highs and lows but the closing prices are lower each time then a simple graph based on a line joining up the high prices will show prices as being
stagnant for that period even though they have been closing lower. This does not represent the price action accurately enough for traders (which is one reason why price candlesticks are preferable to
a mountain range type line). As the Stochastic formula takes into consideration high prices and closing prices in this situation the Stochastic Oscillator will start pointing down – showing that
momentum is in the direction of lower prices.
If bears keep fighting off bull rallies to close the prices lower then it is the bears that are gaining control even though the average price for a period may be equal to that of the previous period.
As the creator of the Stochastic indicator, George C. Lane pointed out – “It follows the speed or the momentum of price rather than price itself. As a rule momentum changes direction before price.”
Stochastic Indicators have two lines associated with them. A faster line (K) and a slower line (D). The slower line is a moving average of the faster line.
Both lines are plotted in a range and will be somewhere between 0-100. A percentage. The first line is referred to as K. The second line is D.
There are three variable user defined values for Stochastic. The Look-back period for K. The Smoothing Period of K. Moving Average of the smoothed K line which makes up line D.
Essentially the higher the values the longer the period over which you are monitoring momentum. The longer the period and the more smoothing you apply the less reactive the indicator.
The more that I look at the formulas the more mist descends on me! More importantly you should choose values for Stochastic that help produce trading signals that are useful for you. There are no
right or wrong values. There are no perfect values. Typically traders use 14,3,3 and 5,3,3 most often. The first set of values produces a less reactive indicator than the second. Which set of values
are right for you? This comes down to your trading system and trading style. Which we can help you work on in our training course.
Here are three examples for you of Stochastic using different settings and based on the same timeframe and price action.
Added Confluence
Essentially Stochastic Oscillator is an indicator and provides indications of momentum changes and direction. It should be used to add confluence to your trades not necessarily determine your action
alone. It can be used as a component in a high probability trading system.
You may be interested in the formulas so here they are:
K is smoothed by a number of periods for Slow Stochastic (standard).
C=Current Price
L=Lowest Low Price for the look back period
H=Highest High Price for the look back period
First of all we define a look back period. The standard is 14. The indicator will be based on the timeframe that you have chosen for your chart. So a daily chart will have Stochastic units based on a
number of days, a one minute chart will have Stochastic units based on minutes. If the indicator is set to 14 units then that would be 14 days or 14 minutes respectively.
For Slow Stochastic we also define a smoothing period (moving average) for K. The standard is 3 periods. So K is smoothed by 3 periods.
We then define the Moving Average for D line. Again the standard is 3 periods.
D = Moving average of K.
Summary: The K line is the fastest and the D line is the slower of the two lines as it is a moving average of K. Slow Stochastic applies a moving average to K.
How to use Stochastic to identify trades
I do use Stochastic in my trading system. It isn’t the only indicator that I use and it isn’t the primary indicator that I use but I do find it useful for confluence in determining likely trade
entries and exits.
I have a risk on and risk off system. When the risk is on then a trade is possible, I wait for confirmation from price and other indicators before trading. When risk is off then I won’t trade. This
makes it easy for me to avoid emotional and impulsive trading. I use Stochastic as one of the criteria for whether risk is on or off.
Oversold and Overbought values on Stochastic
Typically when the Stochastic indicator has a value of over 80 then the market is considered overbought. When Stochastic has a value of under 20 then the market is considered oversold.
Are these values useful? Yes and not necessarily for the reason that you may think.
Stochastic can maintain overbought and oversold values for a long time. Just because Stochastic goes overbought it doesn’t mean that prices will start falling straight away. In fact they can continue
rising considerably before they start falling. In a trending market overbought and oversold conditions can last all day, all week or all month depending on the timeframe that you are looking at and
the settings that you are using on Stochastic.
If you buy when Stochastic goes oversold and sell when Stochastic goes overbought then you will probably lose a lot of money in the long run.
It may be possible to develop a system where you wait until overbought and oversold conditions cease before entering a trade. However I haven’t found this signal on its own useful in my own trading.
Stochastic Crossovers
Stochastic Oscillator is a two line moving average momentum indicator. Some traders use Moving Average crossovers as trading signals. A crossover takes place with Stochastic when the fast line
crosses over the slow line.
In overbought conditions and when the fast line crosses over the slow line this may indicate that the market momentum is changing from rising to falling prices. However it could just mean that the
price rises are slowing, so care must be taken. Momentum often fizzles out rather than switching immediately.
Price Divergences with Stochastic
One aspect of Stochastic that can be useful is when a divergence takes place between the indicator values and price.
When price makes a new low and Stochastic makes a higher low then this may be an indication that momentum has changed and price is about to rise. This is a bullish divergence.
When price makes a new high and Stochastic makes a lower high then this may be an indication that momentum has changed and price is about to rise. This is a bullish divergence.
Bullish divergences are especially powerful when the first Stochastic low is in the oversold area and the second Stochastic low is above oversold.
Bearish divergences are especially powerful when the first Stochastic high is in the overbought area and the second Stochastic high is below overbought.
Judging the market temperament using Stochastic
Traders have trading systems for ranging markets, volatile markets and trending markets. A quick look at the movement of the Stochastic indicator over a period of time may give you a good idea as to
which type of market temperament is prevalent.
Is Stochastic Oscillator remaining overbought or oversold? If it is then the market is probably trending.
Is Stochastic moving smoothly from overbought to oversold and back again? If it is then the market is probably in a range.
Is Stochastic bumping around in the middle of its range? If it is then the market is probably volatile with no real direction.
It is worth noting the steepness of the Stochastic lines. This indicates the strength of momentum. When Stochastic Oscillator is meandering higher then there is no conviction in the move. When
Stochastic is nearly vertical then the momentum is powerful.
The first stage in learning is to make sense of the subject, hopefully this material helps. The second stage is to apply it. Now is the time to go to your charts and pick out the various aspects of
Stochastic that we mention here and see how it works for yourself. The third stage is to decide how it is going to be useful for you in your trading system (make money from it). This is the value
that we add in our trading courses. The forth stage is to recognise the patterns in real time trading and act upon them. | {"url":"https://excellenceassured.com/trading/stochastic-oscillator-trading-indicator","timestamp":"2024-11-06T17:33:52Z","content_type":"text/html","content_length":"243287","record_id":"<urn:uuid:99d82733-3b03-4037-b9c7-abfc3c35cfbc>","cc-path":"CC-MAIN-2024-46/segments/1730477027933.5/warc/CC-MAIN-20241106163535-20241106193535-00845.warc.gz"} |
Generating Tikz codes from Sage for drawing scattering points in 3D
Generating Tikz codes from Sage for drawing scattering points in 3D
I am a beginner in Sage and Tikz, and have written the following snippet of Sage code:
P = Polyhedron(ieqs=[(30, -2, -2, -1), (25, -1.5, -2, -3),(20,-2,-1,-1), (0, 1, 0, 0), (0, 0, 1, 0), (0, 0, 0, 1)])
pts = P.integral_points()
point3d(pts,rgbcolor=(1,0, 0), size=10) + P.plot(rgbcolor = 'yellow', opacity = 0.5)
The purpose of the codes is to generate the integral points interior to the polytopes bound by 6 inequalities as given in the codes. The output is in the following link:
http://www.cse.cuhk.edu.hk/~jlee/dots.png (www.cse.cuhk.edu.hk/~jlee/dots.png)
I tried to follow the following reference:
to generate Tikz codes for LaTeX and failed. Unlike Polyhedron which is a kind of object that can respond to "projection", point3d generates only a Graphics3D object which doesn't understand
Is there any way to output these points to Tikz?
Also, is it possible to ask Sage to color each of the facets of the polytope in different colors (just like Tikz can)?
Many thanx in advance!
1 Answer
Sort by ยป oldest newest most voted
The following is a possibility to get a picture:
import re
P = Polyhedron( ieqs=[ (30, -2 , -2, -1),
(25, -1.5, -2, -3),
(20, -2 , -1, -1),
( 0, 1 , 0, 0),
( 0, 0 , 1, 0),
( 0, 0 , 0, 1), ] )
tex = P.projection().tikz( view=[775,386,500], angle=105, scale=0.7, opacity=0.1 )
tex = re.sub( r'\\end\{tikzpicture\}'
, '% \end{tikzpicture} % manually commented, we still have to work'
, tex )
for v in P.integral_points():
tex += ( '\n\\node[inner sep=1pt,circle,draw=red!25!black,fill=red!75!black,thick,anchor=base] at (%s, %s, %s) {};'
% tuple(v) )
tex += '\n\n\\end{tikzpicture}'
print tex
The view point, and maybe all optional parameters declared above should be change to follow the given needs.
The string tex contains the tikz code to be inserted in a valid $\LaTeX$ document, with correct imports for tikz, e.g.
should be enough. Here, as a matter of taste, manual adjustments of the code are still needed. For instance:
The faces are colored w.r.t. the specifications in the lines starting with \fill[facet], and the facet style is declared in
[x={(0.497546cm, 0.859773cm)},
y={(-0.106336cm, -0.071189cm)},
z={(0.860895cm, -0.505690cm)},
back/.style={loosely dotted, thin},
edge/.style={color=blue!95!black, thick},
facet/.style={fill=blue!95!black,fill opacity=0.100000},
vertex/.style={inner sep=1pt,circle,draw=green!25!black,fill=green!75!black,thick,anchor=base}]
If you need special colors for each face, than define them either inside the \begin{tikzpicture}[ ... ] as facet1/.style={...} or directly at the right place. The back style should also be adjusted.
Herein is also a good place to insert for instance something like
latticepoint/.style={inner sep=1pt,circle,draw=red!25!black,fill=red!75!black,thick,anchor=base}
and to use this style in the changed line of code
tex += ( '\n\\node[latticepoint] at (%s, %s, %s) {};' % tuple(v) )
to have a more compact $\LaTeX$ code. This should be a good start, good luck!
edit flag offensive delete link more
Thanx, Dan!
jim ( 2017-09-19 13:39:11 +0100 )edit
You can change the vertex, edge and facet color directly in the tikz method using the parameters "vertex_color", "edge_color" and "facet_color". They can take any string argument which tikz can
interpret as a color. One slight issue with the above loop on integral points is that it may not iterate the interior lattice points according to the projection, i.e. from the furthest to closest to
the screen, and maybe some point nodes intersect in a weird way. You could sort them as follows:
proj_vector = vector([775,386,500])
linear_proj = [(proj_vector*p,p) for p in P.integral_points()]
for v in linear_proj:
tex += ( '\n\\node[inner sep=1pt,circle,draw=red!25!black,fill=red!75!black,thick,anchor=base] at (%s, %s, %s) {};' %tuple(v[1])
jipilab ( 2017-10-09 13:45:57 +0100 )edit | {"url":"https://ask.sagemath.org/question/38837/generating-tikz-codes-from-sage-for-drawing-scattering-points-in-3d/","timestamp":"2024-11-08T21:16:39Z","content_type":"application/xhtml+xml","content_length":"60461","record_id":"<urn:uuid:bad830e5-852c-4283-9f46-6cdb2828b701>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00030.warc.gz"} |
Descriptive Statistics Interview Questions - Meritshot
Descriptive Statistics Interview Questions
What is meant by qualitative data?
Answer: Qualitative data is non-numerical data that describes qualities or characteristics. It is typically categorical or ordinal in nature.
Define quantitative data.
Answer: Quantitative data is numerical data that can be measured or counted. It is typically continuous or discrete in nature.
What is the difference between discrete and continuous data?
Answer: Discrete data can only take on specific values, typically whole numbers, while continuous data can take on any value within a given range.
Give an example of qualitative data.
Answer: Examples of qualitative data include gender, marital status, eye color, or customer satisfaction rating (e.g., “good,” “fair,” “excellent”).
Provide an example of quantitative data.
Answer: Examples of quantitative data include height, weight, age, temperature, or income.
What are the two subtypes of qualitative data?
Answer: The two subtypes of qualitative data are nominal and ordinal data.
Answer: Nominal data is a type of qualitative data that consists of categories with no inherent order or ranking.
Give an example of nominal data.
Answer: Examples of nominal data include favorite color (e.g., red, blue, green), car brands (e.g., Ford, Toyota, Honda), or blood types (e.g., A, B, AB, O).
Answer: Ordinal data is a type of qualitative data that has categories with a natural order or ranking.
Provide an example of ordinal data.
Answer: Examples of ordinal data include rating scales (e.g., “poor,” “fair,” “good,” “excellent”), educational levels (e.g., high school, bachelor’s, master’s, Ph.D.), or military ranks (e.g.,
private, sergeant, lieutenant).
Answer: Discrete data consists of separate, distinct values that can be counted and are typically whole numbers.
Give an example of discrete data.
Answer: Examples of discrete data include the number of children in a family, the number of cars in a parking lot, or the number of students in a classroom.
Answer: Continuous data is data that can take on any value within a certain range and can be measured on a continuous scale.
Provide an example of continuous data.
Answer: Examples of continuous data include height, weight, temperature, time, or the amount of rainfall.
What is the difference between interval and ratio data?
Answer: Interval data has a meaningful order and equal intervals between values, but it doesn’t have a true zero point. Ratio data, on the other hand, has a true zero point.
Give an example of interval data.
Answer: Examples of interval data include temperature measured in Celsius or Fahrenheit, or years (e.g., 1990, 2000, 2010).
Answer: Ratio data is a type of quantitative data that has a true zero point and meaningful ratios between values.
Provide an example of ratio data.
Answer: Examples of ratio data include height, weight, time, income, or distance.
What are the levels of measurement in statistics?
Answer: The levels of measurement are nominal, ordinal, interval, and ratio.
What is the purpose of regularization in linear regression?
Answer: Regularization is used to prevent overfitting by adding a penalty term to the error function, which helps to shrink the coefficients towards zero.
What is central tendency?
Answer: Central tendency refers to the measure that represents the typical or central value of a dataset.
What are the common measures of central tendency?
Answer: The common measures of central tendency are the mean, median, and mode.
Answer: The mean is the sum of all values in a dataset divided by the number of values. It represents the average value.
Answer: The median is the middle value in a dataset when the values are arranged in ascending or descending order. It divides the dataset into two equal halves.
When is the median the most appropriate measure of central tendency?
Answer: The median is most appropriate when the dataset contains outliers or is skewed.
Answer: The mode is the value or values that occur most frequently in a dataset.
When is the mode the most appropriate measure of central tendency?
Answer: The mode is most appropriate for categorical or discrete data, or when identifying the most common value is of interest.
What is the relationship between the mean, median, and mode in a symmetric distribution?
Answer: In a symmetric distribution, the mean, median, and mode are approximately equal.
How does the mean change if an extreme outlier is added to a dataset?
Answer: The mean is sensitive to outliers, so adding an extreme outlier can significantly change the value of the mean.
Can a dataset have multiple modes?
Answer: Yes, a dataset can have one mode (unimodal), two modes (bimodal), or more than two modes (multimodal) if there are multiple values with the same highest frequency.
Can the median be affected by extreme outliers?
Answer: No, the median is not affected by extreme outliers as it only considers the middle value(s) in the dataset.
How do you calculate the weighted mean?
Answer: The weighted mean is calculated by multiplying each value by its corresponding weight, summing the products, and dividing by the sum of the weights.
When is the weighted mean used?
Answer: The weighted mean is used when different values in the dataset have different importance or significance.
How does the mean change if values in a dataset are multiplied or divided by a constant?
Answer: Multiplying or dividing all values in a dataset by a constant will result in the mean being multiplied or divided by the same constant.
Measures of Spread and Dependence
What is a measure of spread in statistics?
Answer: A measure of spread, also known as a measure of dispersion, quantifies the extent to which data values are spread out or clustered together.
What are the common measures of spread?
Answer: The common measures of spread are the range, variance, standard deviation, and interquartile range (IQR).
Answer: The range is the difference between the largest and smallest values in a dataset.
Answer: The variance measures the average squared deviation of each data point from the mean. It indicates how much the data values deviate from the mean.
Define the standard deviation.
Answer: The standard deviation is the square root of the variance. It represents the average amount of variation or dispersion in a dataset.
When is the standard deviation more appropriate to use than the range?
Answer: The standard deviation is more appropriate when the distribution of data is approximately symmetrical and follows a bell-shaped curve.
Define the interquartile range (IQR).
Answer: The interquartile range is the difference between the third quartile (Q3) and the first quartile (Q1) in a dataset. It represents the spread of the middle 50% of the data.
What does a small standard deviation indicate?
Answer: A small standard deviation indicates that the data values are closely clustered around the mean, suggesting less variability.
What does a large range or interquartile range suggest about the data?
Answer: A large range or interquartile range suggests that the data values are spread out or have a greater dispersion.
Answer: Correlation measures the strength and direction of the linear relationship between two variables.
What does a correlation coefficient of -1 indicate?
Answer: A correlation coefficient of -1 indicates a perfect negative linear relationship between two variables.
What does a correlation coefficient of 0 indicate?
Answer: A correlation coefficient of 0 indicates no linear relationship between two variables.
What does a correlation coefficient of 1 indicate?
Answer: A correlation coefficient of 1 indicates a perfect positive linear relationship between two variables.
What is the coefficient of determination (R-squared)?
Answer: The coefficient of determination represents the proportion of the variance in the dependent variable explained by the independent variable(s).
What is the difference between correlation and causation?
Answer: Correlation measures the statistical relationship between variables, while causation establishes a cause-and-effect relationship between variables. Correlation does not imply causation.
Fundamentals of Probability
Answer: Probability is a measure of the likelihood of an event occurring. It quantifies the uncertainty associated with an outcome.
What is the range of probability values?
Answer: Probability values range from 0 to 1, where 0 indicates an impossible event and 1 indicates a certain event.
What is the difference between theoretical probability and experimental probability?
Answer: Theoretical probability is based on mathematical calculations and assumptions, while experimental probability is determined through actual observations or experiments.
What is the probability of an event that is certain to happen?
Answer: The probability of a certain event is 1.
What is the probability of an event that is impossible to happen?
Answer: The probability of an impossible event is 0.
What is the complement of an event?
Answer: The complement of an event is the probability of that event not occurring. It is calculated as 1 minus the probability of the event.
What is the addition rule of probability?
Answer: The addition rule states that the probability of the union of two mutually exclusive events is equal to the sum of their individual probabilities.
What is the multiplication rule of probability?
Answer: The multiplication rule states that the probability of the intersection of two independent events is equal to the product of their individual probabilities.
What is conditional probability?
Answer: Conditional probability is the probability of an event occurring given that another event has already occurred. It is denoted as P(A|B), where A and B are events.
What is the difference between independent and dependent events?
Answer: Independent events are events that do not affect each other’s probability, while dependent events are events that do affect each other’s probability.
What is the concept of sample space?
Answer: The sample space is the set of all possible outcomes of a random experiment.
How do you calculate the probability of an event in a discrete uniform distribution?
Answer: In a discrete uniform distribution, where all outcomes are equally likely, the probability of an event is calculated by dividing the number of favorable outcomes by the total number of
What is the concept of mutually exclusive events?
Answer: Mutually exclusive events are events that cannot occur simultaneously. If one event happens, the other event cannot occur.
What is the concept of independent events?
Answer: Independent events are events that are not influenced by each other. The occurrence or non-occurrence of one event does not affect the probability of the other event.
How do you calculate the probability of the union of two events?
Answer: The probability of the union of two events A and B is calculated by adding the probabilities of A and B and subtracting the probability of their intersection (A ∩ B).
What is the sample space in set theory?
Answer: The sample space is the set of all possible outcomes in an experiment or event.
What is a subset in set theory?
Answer: A subset is a set that contains only elements that are also found in another set.
What is the complement of a set?
Answer: The complement of a set is the set of all elements that are not in the given set, denoted as A’.
What is the union of two sets?
Answer: The union of two sets is the set that contains all elements that are in either of the two sets, denoted as A ∪ B.
What is the intersection of two sets?
Answer: The intersection of two sets is the set that contains all elements that are common to both sets, denoted as A ∩ B.
What is the difference between two sets?
Answer: The difference between two sets is the set that contains all elements that are in the first set but not in the second set, denoted as A – B.
What is the empty set or null set?
Answer: The empty set or null set is a set that contains no elements, denoted as ∅.
What is the cardinality of a set?
Answer: The cardinality of a set is the number of elements it contains. It is denoted as |A|, where A is the set.
What is the principle of inclusion-exclusion in set theory?
Answer: The principle of inclusion-exclusion is a formula used to calculate the size of the union of multiple sets.
What is the concept of mutually exclusive sets?
Answer: Mutually exclusive sets are sets that have no common elements. If one set has an element, the other set cannot have the same element.
What is the concept of disjoint sets?
Answer: Disjoint sets are sets that have no common elements. They are also mutually exclusive sets.
What is the concept of the power set?
Answer: The power set of a set is the set of all possible subsets of that set, including the empty set and the set itself.
What is the concept of the Cartesian product?
Answer: The Cartesian product of two sets A and B is the set of all ordered pairs where the first element comes from set A and the second element comes from set B.
What is the concept of a proper subset?
Answer: A proper subset is a subset that contains some, but not all, elements of another set.
What is the concept of a universal set?
Answer: The universal set is the set that contains all possible elements or outcomes of a particular problem or scenario. It is typically denoted as Ω or U.
What is conditional probability?
Answer: Conditional probability is the probability of an event occurring given that another event has already occurred.
How is conditional probability calculated?
Answer: Conditional probability is calculated by dividing the probability of the intersection of two events by the probability of the condition event.
What is the formula for conditional probability?
Answer: The formula for conditional probability is P(A | B) = P(A ∩ B) / P(B), where P(A | B) represents the conditional probability of event A given event B.
What is the concept of independence in conditional probability?
Answer: Two events A and B are independent if the occurrence of one event does not affect the probability of the other event.
What is the concept of dependence in conditional probability?
Answer: Mutually exclusive events are events that cannot occur at the same time. If one event happens, the other event cannot happen.
What is the concept of mutually exclusive events in conditional probability?
Answer: Mutually inclusive events are events that can occur at the same time. The occurrence of one event does not exclude the possibility of the other event happening.
How is the probability of independent events calculated?
Answer: The probability of independent events is calculated by multiplying the probabilities of each individual event.
How is the probability of dependent events calculated?
Answer: The probability of dependent events is calculated using conditional probability. The probability of the second event is calculated based on the outcome of the first event.
What is the concept of a conditional probability table?
Answer: A conditional probability table is a table that shows the probabilities of different events given certain conditions.
What is the concept of a joint probability?
Answer: Joint probability is the probability of two events occurring together, denoted as P(A ∩ B).
What is the concept of a marginal probability?
Answer: Marginal probability is the probability of a single event occurring without considering any other events.
What is the concept of a prior probability?
Answer: Prior probability is the probability of an event occurring before any additional information is taken into account.
What is the concept of a posterior probability?
Answer: Posterior probability is the updated probability of an event occurring after additional information or evidence is considered.
How is conditional probability used in Bayes' Theorem?
Answer: Bayes’ Theorem is a formula used to calculate the probability of an event given prior knowledge or evidence. It utilizes conditional probability to update the probability based on new
Answer: Bayes’ Theorem is a mathematical formula used to calculate the probability of an event based on prior knowledge or evidence.
What is the formula for Bayes' Theorem?
Answer: The formula for Bayes’ Theorem is P(A|B) = (P(B|A) * P(A)) / P(B), where P(A|B) is the probability of event A given event B, P(B|A) is the probability of event B given event A, P(A) is the
prior probability of event A, and P(B) is the prior probability of event B.
How does Naive Bayes handle continuous and categorical features?
Answer: Naive Bayes can handle continuous features using probability density functions such as Gaussian Naive Bayes. For categorical features, it calculates the probabilities directly from the
observed frequencies.
What is the importance of Bayes' Theorem in statistics and probability?
Answer: Bayes’ Theorem allows us to update the probability of an event based on new evidence or information, making it a fundamental tool in statistical inference and decision-making.
How does Bayes' Theorem relate to conditional probability?
Answer: Bayes’ Theorem uses conditional probability to calculate the probability of an event given prior knowledge or evidence.
What is the concept of prior probability in Bayes' Theorem?
Answer: Prior probability refers to the initial probability of an event before any additional information or evidence is considered.
What is the concept of posterior probability in Bayes' Theorem?
Answer: Posterior probability is the updated probability of an event after incorporating new evidence or information.
How is Bayes' Theorem applied in medical diagnosis?
Answer: Bayes’ Theorem is used in medical diagnosis to calculate the probability of a particular disease given the observed symptoms and medical test results.
How can Bayes' Theorem be used in spam filtering?
Answer: Bayes’ Theorem can be used in spam filtering to calculate the probability that an incoming email is spam based on the presence of certain keywords or patterns.
What is the relationship between prior probability and posterior probability in Bayes' Theorem?
Answer: The prior probability is updated using Bayes’ Theorem to obtain the posterior probability, which reflects the revised probability after considering new evidence.
What is the role of Bayes' Theorem in machine learning algorithms?
Answer: Bayes’ Theorem is used in various machine learning algorithms, such as Naive Bayes classifiers, to estimate the probability of different outcomes based on observed data.
What are some assumptions made when applying Bayes' Theorem?
Answer: Bayes’ Theorem assumes that the events being considered are independent, and that the prior probabilities are known or can be estimated accurately.
How does Bayes' Theorem handle rare events?
Answer: Bayes’ Theorem can effectively update the probabilities of rare events based on new evidence, allowing for more accurate predictions or estimations.
Can Bayes' Theorem be used with continuous probability distributions?
Answer: Yes, Bayes’ Theorem can be used with continuous probability distributions by integrating over the appropriate ranges of values.
What is the relationship between Bayes' Theorem and the law of total probability?
Answer: Bayes’ Theorem is derived from the law of total probability, which states that the probability of an event can be calculated by considering all possible outcomes and their probabilities.
How does Bayes' Theorem relate to the concept of updating beliefs?
Answer: Bayes’ Theorem provides a framework for updating prior beliefs or probabilities based on new evidence, allowing for a more accurate representation of the true probability of an event.
Permutations and Combinations
What is the difference between permutations and combinations?
Answer: Permutations refer to the arrangement of objects in a particular order, while combinations refer to the selection of objects without considering the order.
Answer: A permutation is an arrangement of objects where the order matters.
Answer: A combination is a selection of objects where the order does not matter.
How many permutations can be formed from a set of n objects taken r at a time?
Answer: The number of permutations is given by nPr = n! / (n – r)!, where n is the total number of objects and r is the number of objects taken at a time.
How many combinations can be formed from a set of n objects taken r at a time?
Answer: The number of combinations is given by nCr = n! / (r! * (n – r)!), where n is the total number of objects and r is the number of objects taken at a time.
What is the factorial function?
Answer: The factorial of a non-negative integer n, denoted by n!, is the product of all positive integers less than or equal to n.
What is the formula for the number of permutations of a set with repetitions?
Answer: The number of permutations of a set with repetitions is given by n1! * n2! * … * nk!, where n1, n2, …, nk are the frequencies of the distinct objects.
How can permutations and combinations be used in probability calculations?
Answer: Permutations and combinations are used to calculate the number of possible outcomes in a probability space, helping to determine the likelihood of specific events.
Can you have repetitions in combinations?
Answer: No, combinations do not involve repetitions. Each object can be selected only once.
Can you have repetitions in permutations?
Answer: Yes, permutations can involve repetitions. Objects can be arranged in a particular order, allowing for repeated elements.
How is the concept of permutations and combinations applied in real-life situations?
Answer: Permutations and combinations are used in various fields, such as probability theory, statistics, cryptography, and combinatorial optimization.
What is the principle of inclusion-exclusion?
Answer: The principle of inclusion-exclusion is a counting technique used to calculate the number of elements in the union or intersection of multiple sets.
How do permutations and combinations relate to Pascal's triangle?
Answer: Pascal’s triangle is a triangular array of numbers where each number represents a combination coefficient. The coefficients in Pascal’s triangle can be used to calculate combinations.
How do permutations and combinations relate to the binomial theorem?
Answer: The binomial theorem provides a way to expand binomial expressions raised to a positive integer power and involves coefficients that correspond to combinations.
What is the concept of sampling with replacement and sampling without replacement in permutations and combinations?
Answer: Sampling with replacement allows for the same object to be selected multiple times, while sampling without replacement restricts each object to be selected only once.
What is inferential statistics?
Answer: Inferential statistics is the branch of statistics that involves making conclusions or predictions about a population based on a sample of data.
What is the difference between descriptive statistics and inferential statistics?
Answer: Descriptive statistics summarizes and describes the characteristics of a sample or population, while inferential statistics makes inferences and draws conclusions about a population based on
sample data.
What is a population in inferential statistics?
Answer: In inferential statistics, a population refers to the entire group of individuals or items of interest that we want to study.
What is a sample in inferential statistics?
Answer: In inferential statistics, a sample refers to a subset of individuals or items from a population that is selected to represent the whole population.
Answer: Sampling error refers to the discrepancy between the characteristics of a sample and the characteristics of the population it represents. It occurs due to random variation in the sampling
What is a hypothesis in inferential statistics?
Answer: A hypothesis is a statement or assumption about a population parameter that is being tested using sample data.
What is a null hypothesis?
Answer: A null hypothesis is a hypothesis that assumes there is no significant difference or relationship between variables in the population.
What is an alternative hypothesis?
Answer: An alternative hypothesis is a hypothesis that contradicts the null hypothesis and suggests that there is a significant difference or relationship between variables in the population.
Answer: A type I error occurs when the null hypothesis is rejected, but in reality, it is true. It is also known as a false positive.
Answer: A type II error occurs when the null hypothesis is accepted, but in reality, it is false. It is also known as a false negative.
Answer: The p-value is the probability of obtaining a test statistic as extreme as, or more extreme than, the observed value, assuming the null hypothesis is true. It is used to assess the strength
of evidence against the null hypothesis.
What is a confidence interval?
Answer: A confidence interval is a range of values calculated from sample data that is likely to contain the true population parameter with a certain level of confidence.
What is the significance level?
Answer: The significance level, often denoted as α (alpha), is the threshold below which the null hypothesis is rejected. It determines the probability of committing a type I error.
What is the central limit theorem?
Answer: The central limit theorem states that as the sample size increases, the sampling distribution of the sample mean approaches a normal distribution, regardless of the shape of the population
What is a point estimate?
Answer: A point estimate is a single value that estimates an unknown population parameter based on sample data.
What is the margin of error?
Answer: The margin of error is the maximum likely difference between the point estimate and the true value of the population parameter.
What is a one-sample t-test?
Answer: A one-sample t-test is a statistical test used to determine whether the mean of a sample is significantly different from a known or hypothesized population mean.
What is a two-sample t-test?
Answer: A two-sample t-test is a statistical test used to compare the means of two independent samples to determine if they are significantly different from each other.
Answer: A paired t-test is a statistical test used to compare the means of two related samples, where each observation in one sample is paired with a corresponding observation in the other sample.
What is analysis of variance (ANOVA)?
Answer: Analysis of variance is a statistical technique used to compare the means of three or more groups to determine if there are any significant differences among them.
What is a chi-square test?
Answer: A chi-square test is a statistical test used to determine whether there is a significant association between two categorical variables.
What is regression analysis?
Answer: Regression analysis is a statistical technique used to model and analyze the relationship between a dependent variable and one or more independent variables.
What is correlation analysis?
Answer: Correlation analysis is a statistical technique used to measure the strength and direction of the linear relationship between two continuous variables.
What is the coefficient of determination (R-squared)?
Answer: The coefficient of determination, denoted as R-squared, measures the proportion of the variance in the dependent variable that can be explained by the independent variable(s) in a regression
What is the difference between a population parameter and a sample statistic?
Answer: A population parameter is a numerical value that describes a characteristic of a population, while a sample statistic is a numerical value that describes a characteristic of a sample.
Null Hypothesis and P-Value
What is the null hypothesis?
Answer: The null hypothesis is a statement that assumes there is no significant difference or relationship between variables in a population.
What is the alternative hypothesis?
Answer: The alternative hypothesis is a statement that contradicts the null hypothesis and suggests that there is a significant difference or relationship between variables in a population.
Answer: A Type I error occurs when the null hypothesis is rejected, but it is actually true. It represents a false positive result.
Answer: A Type II error occurs when the null hypothesis is accepted, but it is actually false. It represents a false negative result.
How are Type I and Type II errors related?
Answer: Type I and Type II errors are inversely related. Decreasing the probability of one type of error increases the probability of the other type.
What is the significance level in hypothesis testing?
Answer: The significance level, denoted as α, is the predetermined threshold used to determine whether to reject the null hypothesis. It represents the maximum probability of making a Type I error.
Answer: The p-value is the probability of obtaining a test statistic as extreme as the one observed, assuming the null hypothesis is true. It helps in deciding whether to reject or fail to reject the
null hypothesis.
How is the p-value used in hypothesis testing?
Answer: If the p-value is less than the significance level (α), the null hypothesis is rejected in favor of the alternative hypothesis. If the p-value is greater than α, the null hypothesis is not
What does a p-value of 0.05 indicate?
Answer: A p-value of 0.05 (or less) indicates that there is a 5% (or less) chance of obtaining the observed result if the null hypothesis is true. It is a common threshold for determining statistical
What does it mean if the p-value is greater than 0.05?
Answer: If the p-value is greater than 0.05, it suggests that there is not enough evidence to reject the null hypothesis. The results are not statistically significant.
What does it mean if the p-value is less than 0.05?
Answer: If the p-value is less than 0.05, it suggests that there is sufficient evidence to reject the null hypothesis. The results are considered statistically significant.
What is a one-tailed test?
Answer: A one-tailed test is a hypothesis test that checks for the difference or relationship in a specific direction, either greater than or less than. It has a directional alternative hypothesis.
What is a two-tailed test?
Answer: A two-tailed test is a hypothesis test that checks for the difference or relationship in either direction, greater than or less than. It has a non-directional alternative hypothesis.
How does the choice of one-tailed or two-tailed test affect the p-value?
Answer: The choice of one-tailed or two-tailed test affects the p-value calculation. In a one-tailed test, the p-value is halved because the test is focused on one direction. In a two-tailed test,
the p-value remains as calculated.
How can you interpret a p-value?
Answer: The p-value provides a measure of the strength of evidence against the null hypothesis. A smaller p-value suggests stronger evidence against the null hypothesis, while a larger p-value
suggests weaker evidence against the null hypothesis.
What is the Student t-test?
Answer: The Student t-test is a statistical hypothesis test used to determine if there is a significant difference between the means of two groups or samples.
When should you use a Student t-test?
Answer: The Student t-test is typically used when the sample size is small, and the population standard deviation is unknown.
What are the assumptions of the Student t-test?
Answer: The assumptions of the Student t-test include independence of observations, normality of the data distribution, and homogeneity of variances between the groups.
What are the types of Student t-tests?
Answer: The two common types of Student t-tests are the independent samples t-test and the paired samples t-test.
What is the independent samples t-test?
Answer: The independent samples t-test compares the means of two independent groups or samples.
What is the paired samples t-test?
Answer: The paired samples t-test compares the means of two related groups or samples, such as before and after measurements on the same subjects.
What does the p-value in the t-test indicate?
Answer: The p-value in the t-test indicates the probability of obtaining the observed difference in means (or a more extreme difference) if the null hypothesis of no difference is true.
How do you interpret the p-value in a t-test?
Answer: If the p-value is less than the chosen significance level (e.g., 0.05), it suggests that there is evidence to reject the null hypothesis and conclude that there is a significant difference
between the group means
What is the degrees of freedom in a t-test?
Answer: The degrees of freedom in a t-test represent the number of independent observations available for estimating the population parameters.
How does the sample size affect the t-test?
Answer: As the sample size increases, the t-test becomes more robust and less influenced by violations of normality or variance assumptions.
What is the critical value in a t-test?
Answer: The critical value in a t-test is a threshold value that separates the rejection region from the acceptance region based on the chosen significance level.
What is the effect size in a t-test?
Answer: The effect size in a t-test measures the magnitude of the difference between the group means and provides information about the practical significance of the results.
Can the Student t-test be used for non-parametric data?
Answer: No, the Student t-test assumes that the data follows a normal distribution. For non-parametric data, alternative tests like the Mann-Whitney U test or Wilcoxon signed-rank test should be
What is the difference between a one-tailed and a two-tailed t-test?
Answer: In a one-tailed t-test, the alternative hypothesis specifies a directional difference between the group means, while in a two-tailed t-test, the alternative hypothesis allows for a difference
in either direction.
How do you calculate the t-statistic in a t-test?
Answer: The t-statistic is calculated as the difference between the sample means divided by the standard error of the difference. It measures how many standard errors the difference between the means
is away from zero.
What is the chi-squared test?
Answer: The chi-squared test is a statistical test used to determine if there is a significant association between categorical variables.
When should you use a chi-squared test?
Answer: The chi-squared test is used when you have categorical data and want to test if there is a significant relationship or difference between the observed and expected frequencies.
What are the assumptions of the chi-squared test?
Answer: The assumptions of the chi-squared test include independent observations, a random sample, and an adequate sample size.
What is the null hypothesis in a chi-squared test?
Answer: The null hypothesis in a chi-squared test states that there is no association or difference between the categorical variables.
What is the test statistic in the chi-squared test?
Answer: The test statistic in the chi-squared test is calculated as the sum of squared differences between the observed and expected frequencies, divided by the expected frequencies.
How do you interpret the p-value in a chi-squared test?
Answer: If the p-value is less than the chosen significance level (e.g., 0.05), it suggests that there is evidence to reject the null hypothesis and conclude that there is a significant association
between the categorical variables.
What is the degree of freedom in a chi-squared test?
Answer: The degree of freedom in a chi-squared test is calculated as (number of rows – 1) * (number of columns – 1).
Can the chi-squared test be used for continuous data?
Answer: No, the chi-squared test is specifically designed for categorical data. For continuous data, other tests such as the t-test or ANOVA are more appropriate.
What is the difference between the chi-squared test for independence and the chi-squared test for goodness of fit?
Answer: The chi-squared test for independence examines the relationship between two categorical variables, while the chi-squared test for goodness of fit compares observed frequencies to expected
frequencies for a single categorical variable.
What is Yates' correction in the chi-squared test?
Answer: Yates’ correction is a small adjustment made to the chi-squared test statistic when analyzing 2×2 contingency tables. It helps to account for the approximation used in the chi-squared test.
Can the chi-squared test handle missing data?
Answer: No, the chi-squared test assumes complete data. If there are missing values, data imputation or other techniques must be used before performing the test.
What is the effect size measure in the chi-squared test?
Answer: There are several effect size measures for the chi-squared test, including Cramer’s V and Phi coefficient, which quantify the strength of the association between variables.
Can the chi-squared test be used for small sample sizes?
Answer: The chi-squared test can be used for small sample sizes as long as the expected frequencies in each cell are not too small (e.g., below 5). Otherwise, alternative tests like Fisher’s exact
test should be considered.
What is the relationship between the chi-squared test and the chi-squared distribution?
Answer: The chi-squared test uses the chi-squared distribution as the reference distribution for calculating p-values and critical values.
How is the chi-squared test used in hypothesis testing?
Answer: The chi-squared test compares the observed frequencies in different categories to the expected frequencies, and based on the calculated test statistic and p-value, the null hypothesis is
either accepted or rejected. | {"url":"https://www.meritshot.com/descriptive-statistics-interview-questions/","timestamp":"2024-11-11T04:50:51Z","content_type":"text/html","content_length":"438259","record_id":"<urn:uuid:b792f582-4b1a-4de6-8623-93f69722b618>","cc-path":"CC-MAIN-2024-46/segments/1730477028216.19/warc/CC-MAIN-20241111024756-20241111054756-00539.warc.gz"} |
Pandas - Apply custom function to columns in DataFrame using apply()
Pandas - Applying a Custom Function to Columns using apply()
Pandas: Custom Function Exercise-3 with Solution
Write a Pandas program that apply a custom function to each column using apply() function.
In this exercise, we have applied a custom function to calculate the mean of each column using apply() column-wise.
Sample Solution:
Code :
import pandas as pd
# Create a sample DataFrame
df = pd.DataFrame({
'A': [1, 2, 3],
'B': [4, 5, 6],
'C': [7, 8, 9]
# Define a custom function to calculate the mean of a column
def column_mean(column):
return column.mean()
# Apply the custom function column-wise to calculate the mean of each column
means = df.apply(column_mean, axis=0)
# Add the means as a new row to the DataFrame
df.loc['Mean'] = means
# Output the result
A B C
0 1.0 4.0 7.0
1 2.0 5.0 8.0
2 3.0 6.0 9.0
Mean 2.0 5.0 8.0
• Create a DataFrame: The sample DataFrame is created with columns 'A', 'B', and 'C', each containing 3 numerical values.
• Define Custom Function: The column_mean function is defined to calculate the mean of a column using column.mean().
• Apply Function to Columns: The apply() function is used with axis=0 to apply column_mean to each column of the DataFrame, calculating the mean of each column.
• Add Means as a Row: The calculated column means are added as a new row labeled 'Mean' using df.loc['Mean'].
Python-Pandas Code Editor:
Have another way to solve this solution? Contribute your code (and comments) through Disqus.
What is the difficulty level of this exercise?
Test your Programming skills with w3resource's quiz.
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• Weekly Trends and Language Statistics | {"url":"https://www.w3resource.com/python-exercises/pandas/pandas-apply-custom-functions-to-columns-in-dataframe-using-apply.php","timestamp":"2024-11-02T01:28:44Z","content_type":"text/html","content_length":"137902","record_id":"<urn:uuid:b6d2c5f9-5c16-4526-a03d-17264ccc0700>","cc-path":"CC-MAIN-2024-46/segments/1730477027632.4/warc/CC-MAIN-20241102010035-20241102040035-00000.warc.gz"} |
Hello, everyone, and welcome back. So in this video, we're going to talk about a new concept called polarization, which has to do with the orientation of light and also its intensity. So we're going
to talk about a couple of conceptual things you'll need to solve problems, and we'll also go through a pretty straightforward equation. So let's just jump right in.
So what's polarization all about? Well, back when we first studied the electromagnetic waves, we said the electric field oscillates in one axis and the magnetic field oscillates in a different axis,
and that's basically what polarization has to deal with. The polarization of an electromagnetic wave is always just going to be the direction or the axis that the electric field is oscillating along.
So, for example, for this wave over here on the left, the electric field oscillates purely on the z-axis. It doesn't go forwards and backwards or off at some angle. So, basically, what we say is
that's the polarization. The polarization of this light here, this electromagnetic wave, is going to be along the z-axis.
Now, obviously, drawing these diagrams and these waves would be super complicated over and over again, so we have very compact ways of representing this by using what's called a polarization diagram.
Basically, what this looks like is it just looks like a double-headed arrow that points along the appropriate direction. That's all polarization is.
Now let's take a look at a different example because there are many different possibilities for the polarization angle or direction. Let's take a look at this wave over here. This wave points along
the same exact direction. The only difference is it kind of looks like the first diagram if you were to sort of tilt it a little bit. So if you were to sort of tilt it by 30 degrees, now what happens
is the electric field oscillates not purely along the z-axis, but it kind of oscillates at some angle here. So what we'd say is that this angle, this polarization, is 30 degrees with respect to the
z-axis. All right? So basically, it would just look like this. We're going to draw this double-headed arrow, and we would just indicate with little dotted line that this angle here is 30 degrees.
That's really all the polarization has to deal with.
Now there's also such a thing as unpolarized lights, which basically just means that the electric fields don't point in a specific direction but actually just many random directions. Now in a lot of
problems, we're going to see unpolarized lights. Usual examples are going to be sunlight or light that's coming from light bulbs or something like that. Most of the time, it'll actually just tell you
if it's unpolarized. But, basically, the way that we represent unpolarized light is by using a double-headed arrow, except in all directions. So usually, what I do is you're just going to see a
couple of lines, maybe like 3 or 4 of them, with all of the arrows like this, and this just represents unpolarized light.
Now let's talk about a polarizer very quickly. A polarizer looks like a little circle with a little grating, and it's basically just a filter. A polarizer is a filter that only allows components that
are parallel to the transmission axis to pass through. So when you have this unpolarized light that's moving this way and it passes through the polarizer, the only thing allowed to pass through is
this component of the light here, the component that is parallel to this transmission axis. What happens to the other components? Well, they basically just get absorbed or they get blocked, so these
components now are no longer allowed to pass through. So what happens here is that when unpolarized lights get passed through a polarization filter or a polarizer, then it becomes polarized. And,
basically, what it looks like here is the only component that's allowed to survive is the vertical component.
Now the other thing that happens is that the intensity also decreases by a factor of one-two, because you're now removing a lot of the light components. So basically, there's actually a very simple
equation for this. It's that:
I = 1 2 I 0
and we just call this the one-half rule. So whenever you have unpolarized light that becomes polarized, its intensity decreases by a half, and it becomes polarized in the direction of the
transmission axis.
Alright? So let's just go ahead and just jump into a really quick example problem here. So for the situation that we just described above here, if the intensity of the unpolarized light is 100, so in
other words, if this I[0] is equal to 100, then what is the intensity of the transmitted light? In other words, what's the intensity of this light that makes it out of the other side? So, basically,
if we want I, then we're going to have to use our new equation, I equals one-half of I[0]. So, in other words, it's just one half of 100, and that just equals 50 watts per meter squared.
So that's all there is to it, guys. If you have 100 watts per meter squared of intensity, then at the end, you have only 50 watts per square meter that makes it through the other side. That's it for
this one, folks. Let me just go through a quick practice problem, and we'll jump into another video. | {"url":"https://www.pearson.com/channels/physics/learn/patrick/32-electromagnetic-waves/polarization-and-polarization-filters?chapterId=8fc5c6a5","timestamp":"2024-11-14T01:35:18Z","content_type":"text/html","content_length":"518910","record_id":"<urn:uuid:53ef62d2-a514-4f22-8341-ff46db8f6d60>","cc-path":"CC-MAIN-2024-46/segments/1730477028516.72/warc/CC-MAIN-20241113235151-20241114025151-00106.warc.gz"} |
Syllabus Detail
Department of Mathematics Syllabus
This syllabus is advisory only. For details on a particular instructor's syllabus (including books), consult the instructor's course page. For a list of what courses are being taught each quarter,
refer to the Courses page.
MAT 201B: Analysis
Approved: 2010-11-01, Steve Shkoller
Winter, every year; 4 units; lecture/discussion section
Suggested Textbook:
(actual textbook varies by instructor; check your instructor)
Analysis by Elliott H. Lieb and Michael Loss, Chapter 1 ($43), and Applied Analysis by Hunter and Nachtergaele, Chapters 7 - 9, 13, available at provided link.
Search by ISBN on Amazon:
Graduate standing in Mathematics or Applied Mathematics, or consent of instructor.
Course Description:
Metric and normed spaces. Continuous functions. Topological, Hilbert, and Banach spaces. Fourier series. Spectrum of bounded and compact linear operators. Linear differential operators and Green's
functions. Distributions. Fourier transform. Measure theory. Lp and Sobolev spaces. Differential calculus and variational methods.
Suggested Schedule:
Lectures Sections Topics/Comments
Each topic requires Chapter 1 of Lieb and Loss, and
approximately 2 weeks Chapters 7 - 9, 13 of Applied
to cover Analysis
Basic measure and integration theory: Fundamental definitions from measure theory (proofs left to 206); Measurable functions and
approximation by simple functions; Dominated and monotone convergence theorems and Fatou's Lemma; Fubini and Tonelli theorems; Definition of
L^p and l^p spaces and concrete examples of L^2 Hilbert spaces
Fourier Series: Definitions and properties; Sobolev spaces H^s of periodic functions on torus for s real; Poisson summation/integral formula
for the disk and the Dirichlet problem
Bounded linear operators on Hilbert space: Orthogonal projections; Dual space of Hilbert space and representation theorems; Weak convergence
in Hilbert space, Banach-Alaoglu Theorem
Spectrum of bounded linear operators: Diagonalization of matrices; Spectral theorem for compact, self-adjoint operators; Compact operators;
Fredholm Alternative Theorem; Functions of operators
Calculus on Banach Space: Bochner integrals; Derivatives of maps on Banach spaces; The calculus of variations | {"url":"https://www.math.ucdavis.edu/courses/syllabus_detail?cm_id=90","timestamp":"2024-11-08T13:57:14Z","content_type":"text/html","content_length":"25667","record_id":"<urn:uuid:cd33b0e5-da1a-4053-bc87-f329a5181ad5>","cc-path":"CC-MAIN-2024-46/segments/1730477028067.32/warc/CC-MAIN-20241108133114-20241108163114-00884.warc.gz"} |
A Path to Statistics, Part IV
Combinations Everywhere
In a few short weeks, Antonio has started back at PreAlgebra level counting concepts and is now solving problems using combinations at the level of an advanced high school student. Part of what is
interesting is that he doesn't even yet realize how far he has come in a short time.
In this lesson, we examine a modeling technique often known as "stars and bars" or "balls and urns". We also solve path walking problems and talk through why Pascal's triangle is really just a grid
of combinations.
This completes only six total hours of conversation between us, starting from material that some elementary school students work on. The number of people who can learn math is substantially larger
than the number who think they can.
Do you have a syllabus or outline that someone with high school algebra and not much else could use to chart a course of self-study for learning statistics? Other recommendations or materials?
There's so many options out there, it's a bit overwhelming.
Expand full comment | {"url":"https://roundingtheearth.substack.com/p/a-path-to-statistics-part-iv","timestamp":"2024-11-06T02:02:06Z","content_type":"text/html","content_length":"158297","record_id":"<urn:uuid:b92aa668-0c5b-4fe0-90c1-0159638b92ba>","cc-path":"CC-MAIN-2024-46/segments/1730477027906.34/warc/CC-MAIN-20241106003436-20241106033436-00105.warc.gz"} |
Word Problem Database
Addition and Subtraction Word Problems - Number Facts
Enter your answer in the space provided.
1. The science museum is 16 miles away.
Danny and his family have already driven 7 miles.
How many more miles do they have left to drive?
2. The museum has 6 exhibits about birds
and 10 exhibits about magnets.
How many more exhibits are about magnets?
3. The museum is showing a 9 minute movie about penguins.
There is a 5 minute talk after the movie.
How long is the movie and the talk?
4. There are 15 activities in the physics room.
Danny has tried 7 activities.
How many activities does Danny have left to try?
5. Danny and his sister, Emma, built towers with newspaper rods.
Emma's tower was 12 feet tall.
Danny's tower was 9 feet tall.
How much taller was Emma's tower? | {"url":"https://www.mathplayground.com/wpdatabase/Addition_Subtraction_Facts_9.htm","timestamp":"2024-11-09T09:45:08Z","content_type":"application/xml","content_length":"48375","record_id":"<urn:uuid:c9ed7c9e-5e69-4904-a489-bd11d1dbafbb>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.75/warc/CC-MAIN-20241109085148-20241109115148-00707.warc.gz"} |
Counting to Infinity
I have decided to share something which I found interesting while reading up some Mathematics this holiday.
The idea I am going to talk about is that of the cardinality of a set. In simple terms,
Definition 1: Cardinality is a “measure of the size” of a set.
Suppose that we have a set, $A$, such that $A=\{a,b,c,d,e\}$. The cardinality of the set, denoted by $|A|$, is $5$, because there are $5$ elements.
It is indeed worth noting that unlike ‘lists’, in Mathematics, order and number of elements doesn’t determine much concerning the identity of a set. This means that
$\{a,b,c\}=\{a,a,a,b,c\}=\{a,b,b,b,c,c\}=\{a,...,b,...,c, ...\}$
as long as you use the same elements, they are all equal. Of course, cardinalities would be the same, because all of them are a representation of the same mathematical entity.
Let’s talk about something slightly more interesting. Suppose that you have another set with infinitely many elements, like the set of natural numbers, $N$, or real numbers $\mathbb{R}$. A
question we can ask is the following:
Is $|\mathbb{N}|=|\mathbb{R}|$ (i.e do we have as many real numbers as we do natural numbers)? Even though I have gone through the trouble of highlighting this question, the answer is no.
However, we can say more than just, “no.”
Of all infinite sets, we have the following types of infinity:
1) The type where you can keep track of your infinite elements.
2) The type where 1) is not possible.
Definition 2: An infinitely countable set is one whose elements can be arranged in an infinite list.
The cardinality of infinitely countable sets are denoted by
$\aleph .$
For instance, for
It is obvious that the elements of each of these sets can be arranged in an infinite list, as required by our definition of an infinitely countable set.
This is just a neat way of saying that if anyone had the “time”, they would be able to count the numbers (…but life is too short).
On the other hand, something like $\mathbb{R}$ causes serious problems with this idea. For instance: What is the smallest ‘positive’ real number ? (Say, for instance, you want to list real numbers
from zero). With some proof by contradiction, you should be able to convince yourself that there is no such ! Also, it can be proven (with a really neat technique, which I won’t show here) that $\
mathbb{R}$ is uncountable!
Let’s pause, and look at what it means for two sets that are infinite to have equal cardinalities. Refer to the lists of numbers from $\mathbb{N}$ and $\mathbb{Z}$ above, and observe that you can
align an integer to every natural number, so we use this basic idea to say whether two sets have equal cardinalities, or not. It is true that
“But,” you will say, “we have two integers for one natural number, except zero, so how…,” and I will answer: “Because both sets are equal (in terms of cardinality), you will not run out of natural
numbers much like you won’t run out of integers!”
For this case, we write
$|\mathbb{N}|=|\mathbb{Z}|=\aleph .$
A more formal statement is the following.
Definition 3: Two sets, $A$ and $B$, have the same cardinality if and only if there exists a bijection from $A$ to $B$.
A bijection is a map (or a function, if you prefer that name better) that is both injective and surjective.
Injective, in simple terms, means that the map is “one to one”.
Surjective means that it is “on to”–i.e if you map elements from $A$ to $B$, then each $b$ in $B$ has an element $a$ in $A$ from which it is mapped.
The following idea is what I found most interesting, and the main motivation for the post.
The question is whether one can show that the cardinality of the interval $(0,1)$ is equal to $|\mathbb{R}_{>0}|$ (i.e the cardinality of the set with all positive real numbers)?
That is absurd ! Obviously not.. I mean, imagine how many ‘more’ elements $\mathbb{R}_{>0}$ must have,right ? MAYBE… NO!
Let’s draw an $xy$ plane, with the following:
-Put a point at $(x,0)$
-Put another at $(-1;1)$
-Draw a line from the first to the second point (this should pass through the $y$-axis).
-Connect $(-1;1)$ to the $x$-axis, label this point $a$ (for reference).
-Observe that from $x$ to a the distance is $x+|-1|=x+1$
-Label the $y$-axis interval from zero to the intersection as $f (x)$.
This should be done as is depicted in the following sketch:
(Credits go to Richard Hammack: The Book of Proof)
Observe and convince yourself that the two triangles formed are similar, and so from proportionality, we get
$\dfrac {f (x)}{1}=\dfrac{x}{(x+1)} .$
In this case we see that there exists a “bijection”, $f$, from $\mathbb{R}_{>0}$, to $(0,1)$, so
… (and we all thought we could “count”)!
Unfortunately, that is it for now. I hope you had a great time reading!
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By Siphelele Danisa| 2020-05-21T21:38:57+00:00 July 28th, 2017|Uncategorized|0 Comments
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About the Author: Siphelele Danisa
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State Space Representation of a System | Electricalvoice
State Space Representation of a System
The control system analysis fails for multiple-input multiple-output (MIMO) systems and the systems initially not at rest by using transfer function approach. The use of the state-space approach for
the analysis of control systems enables to overcome the shortcomings of the transfer function approach.
The state of a dynamic system is the smallest state of variables (called as state variables) such that the knowledge of these variables at the time ‘t[o]‘ together with the input t > t[o] determines
the unique behaviour of the system for t > t[o] and this t[o] is normally taken as zero.
This amount of information is generally a set of variables which are the values from inside the system that can change over time. for example in an electric circuit containing RLC network as shown in
the figure.
Initial inductor current i.e. i[L](0) and
initial capacitor voltage i.e. v[c](0) can be state variables.
On applying input i.e. closing the switch, these state variables will change and hence the state of the network at time ‘t’ will be i[L](t) and v[c](t).
In general, the state of the system is represented by a set of the equation called as state equations. The general form of state equation is as follows
………………. (1)
The output of system can be represented by the output equation y(t)
y(t) = C x(t) + D u(t) ………………. (2)
where x denotes the state variable
u denotes the input variable
y denotes the output variable.
A is called system Matrix
B is called the controlled Matrix
C is called the output matrix
D is called the feed forward matrix
Equations (1) and (2) represent the state model of the system and are collectively called state space equation.
Note: The requirement in choosing this state variable is that they should be linearly independent* and that minimum^# number of them should be chosen.
# decided by the order of the differential equation describing the system.
Advantages of State Space Representations
1. Easy to implement
2. Takes into account initial conditions
3. apply to the Nonlinear system as well
4. gives the mathematical model
5. can be used to drive Transfer function
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Regula Falsi Method for finding root of a polynomial
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Reading time: 35 minutes | Coding time: 10 minutes
Regula Falsi method or the method of false position is a numerical method for solving an equation in one unknown. It is quite similar to bisection method algorithm and is one of the oldest
approaches. It was developed because the bisection method converges at a fairly slow speed. In simple terms, the method is the trial and error technique of using test ("false") values for the
variable and then adjusting the test value according to the outcome.
Regula falsi Method Animation. Source : Planetcalc
As before (in bisection method), for a given continuous function f(x) we assume that f (a) and f (b) have opposite signs ( a = lower guess, b = upper guess). This condition satisfies the Balzano's
Theorem for continuous function.
Theorem (Bolzano) : If the function f(x) is continuous in [a, b] and f(a)f(b) < 0 (i.e. f(x) has opposite signs signs at a and b) then a value c ∈ (a, b) exists such that f(c) = 0.
Now after this bisection method used the midpoint of the interval [a, b] as the next iterate to converge towards the root of f(x).
A better approximation is obtained if we find the point (c, 0) where the secant line L joining the points (a, f (a)) and (b, f (b)) crosses the x-axis (see the image below). To find the value c, we
write down two versions of the slope m of the line L:
We first use points (a, f (a)) and (b, f (b)) to get equation 1 (below), and then use the points (c, 0) and (b, f (b)) to get equation 2 (below). Equating these two equation we get equation 3 (below)
which is easily solved for c to get equation 4 (below):
The three possibilities are the same as before in the bisection method:
• If f (a) and f (c) have opposite signs, a zero lies in [a, c].
• If f (c) and f (b) have opposite signs, a zero lies in [c, b].
• If f (c) = 0, then the zero is c.
For a given function f(x),the Regula Falsi Method algorithm works as follows:
1. Start
2. Define function f(x)
3. Input
a. Lower and Upper guesses a and b
b. tolerable error e
4. If f(a)*f(b) > 0
print "Incorrect initial guesses"
goto 3
End If
5. Do
c = b -(f(b)*(b-a))/(f(b)-f(a))
If f(a)*f(c) < 0
b = c
a = c
End If
while (fabs(f(c)) > e) // fabs -> returns absolute value
6. Print root as c
7. Stop
Sample Problem
Now let's work with an example:
Show that f(x) = x^3 + 3x - 5 has a root in [1,2], and use the Regula Falsi Method to determine an approximation to the root that is accurate to at least within 10^-6.
Now, the information required to perform the Regula Falsi Method is as follow:
• f(x) = x^3 + 3x - 5,
• Lower Guess a = 1,
• Upper Guess b = 2,
• And tolerance e = 10^-6
We know that f(a) = f(1) = -1 (negative) and f(b) = f(2) = 9 (positive) so the Intermediate Value Theorem ensures that the root of the function f(x) lies in the interval [1,2]
Figure: Plot of the function f(x) = x^3 + 3x - 5
Below we show the iterative process described in the algortihm above and show the values in each iteration:
f(x) = x^3 + 3x - 5,
Lower Guess a = 1,
Upper Guess b = 2,
And tolerance e = 10^-6
Iteration 1
a = 1, b = 2
• Check if f(a) and f(b) have opposite signs
f(a) = f(1) = -1 ; f(b) = f(2) = 9
So, f(a) * f(b) = f(1) * f(2) = -9 < 0 ✅
• We then proceed to calculate c :
c = b -(f(b) * (b-a))/(f(b) - f(a)) = 2 -(9 * (2-1))/(9-(-1))
c = 1.1
• Check if f(a) and f(c) have opposite signs
f(a) = f(1) = -1 ; f(c) = f(1.1) = -0.369
f(a) * f(c) = f(1) * f(1.1) = 0.369 < 0 ❌
Since the above condition is not satisfied, we make c as our new lower guess i.e. a
a = c
a = 1.1
So, we have reduced the interval to :
[1,2] -> [1.1,2]
Now we check the loop condition i.e. fabs(f(c)) > e
f(c) = f(1.1) = -0.369
fabs(f(c)) = 0.369 > e = 10^-6 ✅
The loop condition is true so we will perform the next iteration.
Iteration 2
a = 1.1, b = 2
• Check if f(a) and f(b) have opposite signs
f(a) = f(1) = -5 ; f(b) = f(1.5) = 2.375
So, f(a) * f(b) = f(1) * f(1.5) = -11.875 < 0 ✅
We then proceed to calculate c :
c = b -(f(b) * (b-a))/(f(b)-f(a)) = 1.135446686
c = 1.135446686
Check if f(a) and f(c) have opposite signs
f(a) = f(1) = -1 ; f(c) = f(1.135446686) = -0.1297975921
f(a) * f(c) = -0.1297975921 < 0 ❌
Since the above condition is not satisfied, we make c as our new lower guess i.e. a
a = c
a = 1.135446686
Again we have reduced the interval to :
[1,2] -> [1.135446686,2]
Now we check the loop condition i.e. fabs(f(c)) > e
f(c) = -0.1297975921
fabs(f(c)) = 0.1297975921 > e = 10^-6 ✅
The loop condition is true so we will perform the next iteration.
As you can see, it converges to a solution which depends on the tolerance and number of iteration the algorithm performs.
Regula Falsi method performed on the function f(x) = x^3 + 3x - 5
C++ Implementation
#include <iostream>
#include <math.h>
using namespace std::chrono;
using namespace std;
static double function(double x);
int main()
double a; // Lower Guess or beginning of interval
double b; // Upper Guess or end of interval
double c; // variable for midpoint
double precision;
cout << "function(x) = x^3 +3x -5 "<<endl;
cout << "Enter begining of interval: ";
cin >> a;
cout << "\nEnter end of interval: ";
cin >> b;
cout << "\nEnter precision of method: ";
cin >> precision;
// Check for opposite sign (Intermediate Value Theorem)
if (function(a) * function(b) > 0.0f)
cout<< "\nFunction has same signs at ends of interval";
return -1;
int iter=0;
auto start = high_resolution_clock::now();
// starting the iterative process
}while(fabs(function(c))>=precision);//Terminating case
auto stop = high_resolution_clock::now();
auto duration = duration_cast<microseconds>(stop - start);
cout<<"\nRoot = "<<c;
cout << duration.count()<<" microseconds"<< endl;
return 0;
static double function(double x)
return pow(x,3) + 3*x -5 ;
More Examples
Regula Falsi method performed on the function f(x) = x^3 + 4x^2 - 10
If you have seen the post on Bisection Method you would find this example used in the sample problem part. There the bisection method algorithm required 23 iterations to reach the terminating
condition. Here we see that in only 12 iterations we reach the terminating condition and get the root approximation. So in this situation Regula Falsi method conveges faster than Bisection method.
But we cannot say that Regula Falsi Method is faster than Bisection Method since there are cases where Bisetion Method converges faster than Regula Falsi method as you can see below:
Figure: Plot of the function f(x) = e^x - e Iterations of Regula Falsi and Bisection Method on the function f(x) = e^x - e
While Regula Falsi Method like Bisection Method is always convergent, meaning that it is always leading towards a definite limit and relatively simple to understand but there are also some drawbacks
when this algorithm is used. As both regula falsi and bisection method are similar there are some common limitaions both the algorithms have.
• Rate of convergence
The convergence of the regula falsi method can be very slow in some cases(May converge slowly for functions with big curvatures) as explained above.
• Relies on sign changes
If a function f (x) is such that it just touches the x -axis for example say f(x) = x2 then it will not be able to find lower guess (a) such that f(a)*f(b) < 0
• Cannot detect Multiple Roots
Like Bisection method, Regula Falsi Method fails to identify multiple different roots, which makes it less desirable to use compared to other methods that can identify multiple roots. | {"url":"https://iq.opengenus.org/regula-falsi-method/","timestamp":"2024-11-04T11:52:13Z","content_type":"text/html","content_length":"69531","record_id":"<urn:uuid:2eeed0e9-e1fb-4acb-9226-4c715c927316>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00224.warc.gz"} |
Chapter 6: Immunological (Antigen and Antibody) Reactions, Monoclonal Antibody - Labpedia.netChapter 6: Immunological (Antigen and Antibody) Reactions, Monoclonal Antibody
Chapter 6: Immunological (Antigen and Antibody) Reactions, Monoclonal Antibody
Antigen and Antibody react with each other in three stages:
Primary stage Secondary stage Tertiary stage
Combination of antigen + antibody 1. Agglutination 1. Opsonization
2. Precipitation 2. Lysis
3. Complement activation
4. Neutralization
5. Blocking of antigen sites
Primary Stage
In the primary stage, a combination of antigen and antibody gives rise to antigen and antibody complex formation (Ag + Ab).
Secondary Stage
1. Definition:
1. Agglutination is the visible form of aggregation of antigen and antibody, with the formation of the network in which antigen particle (molecule) alternate with the antibody molecule.
2. The ability of the particular antibody to attach with the antigen is its specificity.
3. This is the property of the Fab portion called combining sites; a cleft is formed by the hypervariable of heavy and light chains.
4. This specificity lies in the Fc portion of the antibody.
5. Examples of the carrier are:
1. In the case of soluble antigens, latex particles or colloidal charcoal are needed.
2. RBCs can be used as a biological carrier.
3. Bacteria possess an antigen that reacts with the antibody.
6. Quality of the agglutination depends upon:
1. Time of incubation of the patient serum, which contains antibody.
2. The condition where the tests are run like pH and protein concentration.
3. Amount the antigen conjugated with the carrier.
2. The antibody, which gives rise to agglutination, is called agglutinin.
3. Mechanism of agglutination:
1. It occurs in two stages:
1. Sensitization is the binding antibody to the antigenic sites.
2. Formation of bridges or lattice between antibody sensitized cells.
2. The antigen is always in particulate form, and it has multiple antigenic sites.
3. First, a combination of Ag and Ab leads to lattice formation when a better fit of Ab (IgM & IgG) to antigen gives rise to larger clumps, which the naked eye can see.
4. The best example is the red blood cells; these have antigens on their surface.
5. This agglutination depends upon:
1. Physical attachment of the antibodies to the antigenic sites on the antigen like the RBC membrane. This will depend on the number of antibodies and antigenic sites on the RBC cell
surface. This is called sensitization.
2. Lattice formation: This is the cross-linking of the antibodies with the antigenic sites on the antigens.
The results of agglutination can be read:
1. Directly by the naked eye.
2. Under the microscope.
Type of immunoglobulins Degree of agglutination
IgG 1+
IgM 3+
IgA 1+
3. IgM is considered a complete antibody because of its ability to give good agglutination because it is more efficient than IgG and IgA.
4. The incomplete antibody may fail to show agglutination, and these are nonagglutinating. In these cases, the antigenic determinants may be hidden and located deep within the surface membrane or
may show restricted movements of their hinge portion, making them functionally monovalent.
5. Reading of the agglutination:
Degree of agglutination The appearance of the clumps Supernatant
Zero No agglutination is seen Dark turbid
Weak Tiny agglutinates can see under a microscope Dark, turbid
1+ Many small agglutinates Turbid
2+ Medium-sized agglutinates Clear
3+ Many large agglutinates Clear
4+ One large agglutinate, no free cells Clear
The examples of Agglutination are:
1. ABO blood grouping (blood banking).
2. Widal test
3. Types of serological agglutination tests are:
1. Latex agglutination.
2. Flocculation tests.
3. Direct bacterial agglutination.
4. Indirect or passive hemagglutination test.
5. Microplate agglutination reaction test.
4. Factors leading to False agglutination results may be due to:
1. When the blood is completely coagulated, then you may be seen small fibrin strands.
2. When there is increased protein in the patient serum or due to an abnormal protein.
3. Patient serum may have an unexpected antibody that reacts at room temperature.
4. Patients’ serum may contain transfused or transplanted RBCs or plasma.
5. The patient’s RBCs may be coated with an antibody in vivo.
6. Reagents or saline is contaminated.
7. The reagent potency is degraded due to improper storage.
8. Pipettes or glasware are not clean.
9. The centrifuge machine is not working properly or not calibrated.
1. Kraus’s first time described this was in 1897.
2. Definition:
1. Precipitation is the formation of relatively small, insoluble aggregates from the antigen and antibody reaction (AgAb).
2. The antigen and antibody are soluble.
3. The resulting complex is too large, and so it precipitates as an opaque, visible mass, or flocculation.
3. The antibodies, which give precipitation, are called precipitin.
4. The earliest finding is that antigen and antibody produce precipitation.
1. Precipitins (antibody) can be produced:
1. Against most of the proteins.
2. Some carbohydrate.
3. Carbohydrate-lipids.
5. The antigens are in soluble form; this differs from agglutination, where Ag is in particulate form.
6. Antibodies are IgG and IgM.
7. Precipitation is just like agglutination in that there is clumping of Ag and Ab.
8. Example: Several serological precipitin reaction tests are used like:
1. Widal test when done in tubes with dilution methods.
2. Double immunodiffusion. It is reported as:
1. Identity.
2. Nonidentity.
3. Partial identity.
3. Immunoelectrophoresis.
4. Electroimmunodiffusion.
5. Countercurrent electrophoresis.
6. Radial immunodiffusion (RID).
1. Some of the antibody-like IgG that can neutralize toxins and viruses are called neutralizing antibodies.
2. These antibodies will destroy the infectivity of the viruses.
3. Procedure to test the neutralizing property of the antibody:
1. This test can be done by mixing the patient’s serum with the suspected virus.
2. Then keep them in the cell culture media.
3. If neutralizing antibody neutralizes the virus, then the cell is the culture will be intact.
Blocking of Antigenic Sites
These antibodies (e.g., IgG) can block the antigenic site, which is not available to antigen; these are called blocking antibodies.
Tertiary Stage
1. The antibodies, which give opsonization, are called opsonin.
1. Another opsonin-like function is C3b.
2. The Ab (IgG) helps in the process of opsonization (phagocytosis) by bringing the bacteria coated with Ab (IgG) close to phagocytic cells that possess FcγR (Fcγ-receptor).
3. The antibody can activate complement and leads to phagocytosis by C3b because phagocytic cells also possess C3b receptors. This is dependant upon the C3b receptor.
1. Definition:
1. These are antibodies that lead to lysis (destruction) of the antigens, called lytic antibodies. This cytolysis is complement dependant where complement is activated and leading to lysis of
antigens, e.g., hemolysis of RBC.
2. Ag & Ab complex is recognized by complement, and there is classical complement pathway activation, making holes formation in the cell membrane and leading to osmotic death.
3. The antigen-antibody reaction leads to the lysis of bacterial cell walls, or RBCs are valuable procedures in serological testing.
4. Lysis is visible disintegration, showing that antigen and antibody have reacted.
Some scientists divide Antibodies into the following groups:
1. Natural
2. Induced
3. Cross reacting
4. Complete
5. Incomplete
Antibody protects the host by following ways (Functions):
1. The primary function of the antibody is to bind the antigen.
2. Neutralization of viruses and toxins.
3. Opsonization by phagocytic cells is Ab-dependant.
4. Complement activation leads to phagocytosis and lysis.
1. Activation of complement by IgG1 and IgG3, while IgG2 is less effective.
5. IgG can cross the placental barrier.
George Kohler and Cesar Milestone invented monoclonal antibodies in 1975, and they were awarded Noble Prize in 1984.
The somatic cell hybridization is the process where they found that some of the hybrids were manufacturing large quantities of specific antisheep RBCs antibodies.
The multiplying hybrid cell culture is called a hybridoma.
The immunoglobulins derived from a single clone of the cells are called monoclonal antibodies.
Definition of a monoclonal antibody:
• Monoclonal antibodies are purified antibodies cloned from single cells.
• These are engineered to bind to a single specific antigen.
• The process of producing monoclonal antibodies takes 3 to 6 months.
This is possible that we can get immortal clones of plasma cells from myeloma patient, which are directed to produce only one type of antibody against a specific antigen for an unlimited period by
producing Hybridomas.
After the combination of the patient’s myeloma cells with antigenically stimulated plasma cells from mice will give rise to immortal plasma cells called Hybridoma. These Hybridomas proliferate to
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