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dalam Peningkatan Kemampuan Penalaran Matematis Siswa SMA
Ditasona, Candra (2017) Penerapan Pendekatan Differentiated Instruction dalam Peningkatan Kemampuan Penalaran Matematis Siswa SMA. Jurnal EduMatSains, 2 (1). pp. 43-54. ISSN 2527-7235
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This research is based on the problem of low mathematical reasoning ability of high school students. This study aims to examine the improvement of mathematical reasoning ability among students who
get Differentiated Instruction approach and conventional learning. The research design was quasi experiment with non-equivalent control group research design using Purposive Sampling technique.
Instruments used include mathematical prior knowledge test, mathematical reasoning tests, observation sheets, and interview guides. Quantitative analysis was performed by using t-test and two- way
anova test. The result of the research shows that (1) the improvement of student’s mathematical reasoning ability through DI learning is better than conventional learning, (2) The improvement of
student’s mathematical reasoning ability through DI learning is better than the students who follow the conventional learning in terms of Mathematical Prior Knowledge. Keywords: Differentiated
Instruction, Mathematical Prior Knowledge, Mathematical Reasoning.
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Corporate Finance
FIN 300 Corporate Finance
Slides are meant as a template for lectures and should not be used in lieu of coming to class. Slides are constructed based on topics from Ross, Westerfield, and Jordan Fundamentals of Corporate
Exam 4 Topics Exam 4 Practice Formula Sheet
Chapter Eight
This chapter covers stock valuation. We will use valuation tools such as Dividend Growth Model and Multiples. Common terms associated with the stocks are also covered.
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Chapter Twelve
An introduction to risk and return. Average vs Geometric return and a brief introduction to efficient market hypothesis.
View Slides
Chapter Thirteen
More in-depth discussion of risk and returns. Includes calculation of portfolio returns and variances. Introduction to diversification and capital asset pricing model (CAPM).
View Slides
Chapter Fourteen
A brief introduction to cost of capital and the calculation of weighted average cost of capital (WACC).
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International Journal for Simulation and Multidisciplinary Design Optimization (IJSMDO)
Issue Int. J. Simul. Multidisci. Des. Optim.
Volume 15, 2024
Article Number 18
Number of page(s) 8
DOI https://doi.org/10.1051/smdo/2024017
Published online 25 October 2024
Int. J. Simul. Multidisci. Des. Optim.
, 18 (2024)
Research Article
Numerical modelling of hygro-mechanical behavior of Rhecktophyllum Camerunense vegatable fibers
^1 Laboratory of Mechanic, University of Douala Cameroon, Douala, Cameroon
^2 National Higher Polytechnic Institute, Department of Mechanical and Industrial Engineering, Universality of Bamenda Cameroon, Bamenda, Cameroon
^* e-mail: borisnoutegomo@yahoo.fr; noutegomo@gmail.com
Received: 31 August 2023
Accepted: 20 August 2024
Indeed the influence of humidity on the mechanical behavior of Rhecktophyllum Camerunense (RC) vegatable fiber was studied using numerical modelling and simulation. This method was used to
investigate the hygromechanical caracteristics of the vegetable fiber that should be difficult to obtain experimentally. Another goal is to compare the results obtained experimentally and
analytically in others works. The Finite Element Analysis using ANSYS sofware enables us to discretize a continuous problem and obtain an approximated solution. The Modeling Geometry adopted was an
assembly of three concentric cylinders representing the sublayers S1, S2 and S3 of the fiber. The numerical model developed is a decoupled hygromechanical model. Thus the first stage of this model
concerns the consideration of hygroscopy by simulating the diffusion of water within the material. Secondly, the mechanical calculation is carried out by taking as loading the results of the purely
hygroscopic calculation, expressed in the form of hygroscopic fields. Identification of input parameters for the numerical simulation was found in literature. The results coroborated with the ones of
literature and shown that humidity decrease the mechanical properties of RC vegetable fiber.
Key words: RC fibers / hygroscopic / swelling strain / FEM / mechanical properties / ANSYS
© B. Noutegomo et al., Published by EDP Sciences, 2024
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution,
and reproduction in any medium, provided the original work is properly cited.
1 Introduction
The demand for bio-based materials in semi-structural and structural application is constantly growing to conform to new environmental policies enacted in Europe and worldwide which try to replace
conventional oil-based polymers and composites. Natural fibers reinforced composites meet an important success because of their interesting specific mechanical properties, availability and reasonable
price compared to conventional glass fiber reinforced composites [1]. The interest in natural fibre reinforced polymer composites is growing rapidly due to its high mechanical performances, low cost
and ease of processing. Natural fibers are relatively cheap, pose no health hazards and offer solution to environmental pollution by finding new uses for waste materials. However, the hydrophilic
nature of these vegatable fibers results in a poor durability when exposed to humidity and temperature [2]. Their applications are still limited due to several factors like moisture absorption, poor
wettability and large scattering in mechanical properties [3,4]. Natural fibers and its composites are hydrophilic materials. Moisture content in fiber composites significantly affects their physical
and mechanical properties. The absorbed moisture results in to the deterioration of mechanical properties since the water not only affects the unfilled polymer matrices physically and/or chemically
but also attacks the hydrophilic natural fiber as well as the fiber-matrix interface. To understand this phenomenon, many authors used experimental investigations to study the behavior of mechanical
properties of some vegetables fibers exposed to relative humidities of 30%, 40%, 50%, 60%, 70% as flax and nettle [5], flax and sisal [6], flax and Hemp [7], hemp, sisal, flax, jute, agava to the
relative humidities of 10%, 25%, 50%, 80% [8] and flax on the relative humidities of 33% and 60% [9]. Many of those works leads to the conclusion that moisture decreases the mechanical properties of
natural fibers. Few of them mentionned that the moisture doesn't affect the mechanical properties of natural fibers. The experimental investigations are very important because they are closed to the
reality. Experiments also have some limits because of the high cost. They require specialized ultramodern laboratories with well-trained staff. They also consume a lot of time and energy. The other
limitation lies in the fact that majority of the hygro-mechanical characteristics are only known at the macroscopic scale of the fiber and its microstructure at the scale of chemical constituents and
sub-layers are difficult to obtain. To complete these shortcomings resulting from the experimental method, the use multiscale modeling method whether analytical or numerical os very important.
Despite the fact that this method is based on idealized fibres, a better agreement has always been observed in comparison with the experimental results. Modelling method is highly dependent on
experiments because the datas needed for simulation comes from experimental results. Besides, the evolution of their mechanical properties in service conditions can be predicted with numerical
simulations that rely on experimental results [10,11]. Having a numerical model offer the opportunity to assess the influence of many parameters during aging. Plant fibres exhibit a hierarchical
structure, leading to complex mechanical. Deciphering the origins of this behaviour in terms of physical phenomena requires exploring different scales from millimetres to the molecular scale.
Consequently, numerical approaches such as the finite element analysis (FEA) are gaining increasing interest as complements to tedious experimental characterization. The great variability of plant
fibres and the need to take into account a complex hierarchical structure led to the use of numerical methods such as the finite element analysis. The FEA enables us to discretize a continuous
problem and obtain an approximated solution. Analytical and numerical work on plant fibres originated from wood in the 1950s and was extended to plant fibres such as flax and hemp in more recent
years. Industrial interests have most likely been conducting the research in the area of wood, and emerging numerical works on plant fibres is still constrained by the difficulty of obtaining
experimental datas to corroborate the models. Researchs on hygromechanical behavior of composite based on vegetables are abondants while the modelling of plant fibres remain an open challenge. The
littérature on the domain is very poor. The first step is to define the physical system to be analysed and convert in a mathematical model that can be either mesh-based (FEA) or meshless (molecular
dynamics). The discretisation of the mathematical model leads to a finite element model. Within the context of fibre mechanical modelling, the procedure is the following: defining the fibre geometry,
meshing the geometry with structural elements, defining the boundary and loading conditions (displacement-based and force-based), specifying the material model (isotropic, anisotropic, elastic,
elasto-plastic, etc.), and performing the analysis (static, dynamic, transient). Under the assumption of linear elasticity, the calculations are based on the generalized Hooke's law predicting
deformations caused by an arbitrary combination of stresses in a material. The young modulus is the initial slope of the curve stress in fonction of strain as presented in equation (1).
Δσ: variation of stress in MPa; Δε: variation of strain in %.
The number of independent variables is reduced depending on the symmetry of the system. More complex behaviour laws, taking into account viscoelasticity or plasticity, for instance, can be
implemented depending on the material. After solving the equation system using either direct or indirect solvers, the convergence and correctness of the model are checked, and these steps might
require refinement. The final output is an approximate solution of the initial problem. Experimental data are required at different steps to strengthen the model: at the nanoscopic or microscopic
scale to help define a realistic model and at the macroscopic scale to check the correctness of the model. Moreover, fibre geometries have to be implemented in the model. Neagu et al. [12] developed
a multilayer finite element model to investigate the link between the MFA and hygroelastic behaviour of wood fibres. They studied different boundary conditions and found that constrained fibres
exhibit a stiffer response, resembling the behaviour of plant fibres constrained by their neighbours. Changes in the MFA were correlated with changes in the compliance values. The dominating
deformation mechanism under moisture content changes was the twisting of the fibres. The model was further developed by Joffre et al. [13] using a 3D reconstruction of the S2 layer obtained by X-ray
microtomography. The hygroexpansion coefficients were estimated by comparing the predicted and experimental geometries in the wet state. They minimized the geometrical approximation, but only the
elastic behaviour was studied. Finally, a multi-scale finite element analysis was developed by Saavedra Flores et al. [14], covering the tensile behaviour from microfibrils to bulk Palmetto wood. The
evaluation of moisture content is based on the equation (2). Specimen are dried in an oven and expose to humidity. The mass is mesured at progressive intervals of time untill saturation. Therefore
the kinetic absorbtion of the specimen can be realized by a curve of moisture content in function of time.
m[o] is the mass of the dry specimen; m[i] is the mass of the wet specimen.
Research are still going on new fibers in order to understand their caracteristics such as the hygromechanical behaviour. Amoungt them we have Rhecktophyllum Camerunence fibers on which the researchs
started in 2008. The physical, chemical and microstructural characterization were carry out [15]. The modelling of the moisture sorption isotherm of the Rhecktophyllum Camerunence (RC) fiber at 23°C
by using the BET, GAB and DLP models where also studied [16] and his diffusive behavior was investigated to determine the diffusion coefficient [17]. The experimental hygromechanical study of this
fiber were also carry out [18]. The mathematical modeling of hygromechanical were also studied at the level of chemical constituant of the fiber to undetstand the behavior of the sub-layers [19].
This modeling was based on the works of Marklund [20,21]. The aim of this study is to undesrtand the hygro-mechanical behavior of RC fibers using the FEA sofware ANSYS in order to validate the
results obtained experimentally.
2 Methodology
2.1 Modeling geometry adopted
This paragraph presents in detail the numerical model designed under the finite element simulation software ANSYS. The numerical model developed is a decoupled hygromechanical model. Thus the first
stage of this model concerns the consideration of hygroscopy by simulating the diffusion of water within the material. For this, we used the diffusion properties and the boundary conditions imposed
inside and outside the fiber and which correspond to the water content at saturation of the material. Secondly, the mechanical calculation is carried out by taking as loading the results of the
purely hygroscopic calculation, expressed in the form of hygroscopic fields. They are obtained at saturation of an elementary fiber.
For the fiber we consider an ideal geometry (Fig. 1) corresponding to an assembly of concentric cylinders representing the sub-layers S1, S2 and S3 taking into account the fact that the RC fiber has
a large circularity. The average diameter of the fiber is 20μm, it comes from the literature [15]. The lumen represents 20% of the radius, the sub-layers S1, S2 and S3 represent respectively 10%,
85% and 5% of the total fiber thickness. The circles, thus modeled are extruded over a length of 0.1mm.
Between each sub-layer we have three interfaces represented by the radius r[2], and r[4]. For interior and exterior are represented by the radius r[1], and r[5] which are respectivelly the radius of
lumen and fiber. For boundary conditions we have continuities of displacements and stresses in the interfaces S3−S2 and S2−S1 to avoid slips on x, y, z directions.
Fig. 1
Modeling of an RC fiber, S1 is in brown, S2 in green and S3 in red (cylindrical geometry).
2.2 Mesh and hygroscopic loading
As soon as plant fibers are brought into contact with the ambient air, an absorption or desorption phenomenon which depends on the water content within the fibers and on the ambient humidity occurs
at the level of the fibres. But this phenomenon is not instantaneous. This is why the hygroscopic loading is carried out by a transient model. The element type chosen is the Solid70 element, which is
an 8-node 3D element and has temperature as its degree of freedom. We assume that there is a slight variation in temperature between the inner and outer walls. In our case, the temperature
corresponds to the humidity for the fiber. The finished geometry is then meshed by choosing all the areas (Fig. 2). The elements have a length of 0.1mm in the longitudinal direction. The
corresponding sub-layer is then associated with each volume. When the mesh has been carried out as presented on Figure 2, the hygroscopic loading is applied to the nodes of the mesh belonging to the
interior and exterior edges of the geometry studied.
Fig. 2
Mesh structure of the elementary fiber.
2.3 Obtaining macroscopic flexibility constants
The purpose of the numerical model is to determine the macroscopic flexibility tensor of the RC fiber from the hygro-mechanical properties of the sub-layers. For this, it is necessary to apply
different boundary conditions to the fiber according to the constants to obtain. Then the homogenization is done according to a mechanical calculation whose loading is the hygroscopic field
determined in the preceding paragraph. At the beginning of the mechanical calculation, it is necessary to pass from the hygroscopic part of Ansys to the structure part thus changing the Solid70
element to Solid185, which is a 3D structural element with 8 nodes or a layered element.
2.4 Identification of input parameters for the numerical simulation
Mechanical properties of the sub-layers S1, S2 and S3 that will be used in our model to have the properties at the macroscopic level of the fiber should ccme from the literature rewiew. Some of them
have been determined experimentally and others by modelling and simulation. One of the parameter is the ticknesses of sub-layers as mention in Table 1. The thicknesses mentioned in this table are
those obtained from TEM observations of the RC fiber. The density is also needed and the value is 0.9g/cm^3 determined experimentally [15].
The mechanical properties of fibers were determined in the dry state. They are presented in Table 2.
For simulation it is necessary to know the diffusion coefficient of water in the RC fibers and the datas are presented in Table 3. They have been obtained experimentally using Fick's law [17].
Another parameter is the microfibril angle (MFA) of each layer regrouped in Table 4.
The only data in the literature concerning the microfibrils angle of the RC fiber is the value of the S2 sub-layer [15]. The value of angle in the S1 and S3 sublayers are taken from the wood fiber
data [20]. We also need to know the water content to apply to the inner and outer edges of the geometry. This water content was determined experimentally following hygroscopic tests carried out on
the RC fiber [16]. The second part of the numerical model is based on the mechanics and therefore on the properties of each sub-layer in its orthotropy reference frame. These properties were
determined in literature [15].
Table 2
Mechanical properties of the sub-layers plant fiber constituents in the dry state [21].
Table 3
Diffusion coefficients of RC fiber as function of relative humidity [18].
Table 4
Microfibril angle of sub layers according to longitudinal axis [16,21].
3 Results and discussion
3.1 Hygroscopic properties at the macroscopic scale for different relative humidities
The applied humidity corresponds to the water content at saturation for a given relative humidity. Figure 3 gives the adsorption kinetics for an elementary fiber modeled at a relative humidity of
75%, which corresponds to a water content at saturation of 10.84%. This water content is close to the 10% that was determined experimentally [16]. The saturation time is instantaneous. It is 0.15s
instead of 7h determined experimentally. Experimentally the fibers are dried in an oven and exposed to humidity. The moisture taken are mesure with a balance at intervals of time untill the mass does
not vary again. We observed that no matter the relative humidities, they increase from zero which is the dry state untill the saturated point and from there the curve is flat because the mass does'nt
vary again. The same evolution is observed in the modelling curve and it just take few secondes. only the curve of 75̈% relative humidity is presented.
The hygroscopic distribution represented by Figure 4 for the relative humidities of 23, 54, 75% is not uniform throughout the fiber, but all the values are close to the water contents determined
experimentally and therefore the fiber will be considered as saturated in water. Table 5 presents the water content values obtained by our modelling.
Fig. 3
Adsorption kinetics for an elementary fiber (a) Modeled at a relative humidity of 75% and (b) Experimentally in litterature for 23%, 54% and 75% humidity [17].
Fig. 4
Hygroscopic distribution within the elementary fiber modeled for the relative humidities of (a) 23, (b) 54 and (c) 75%.
Table 5
Water content of RC fiber as a function of RH by the numerical model.
Table 6
Comparison between the tensile behavior of the RC fiber obtained by numerical simulation and experimentally from reference [19].
3.2 Hygro-mechanical properties at the macroscopic scale for different relative humidities
In order to validate the numerical model, different sub-layers that constitute the plant wall have been studied for different behaviors transverse mechanical and hygroscopic. Figure 5 below presents
the microfibril angles of the S1, S2 and S3 sublayers as they are respectivelly −80°, +40°, +60° modelled with ansys.
Figure 6 shows the stress and strain distributions on the fiber at 23% relative humidity obtained by Ansys software while Figure 7 shows the fiber tensile behavior curves at 23%, 54% and 75% relative
humidities obtained by simulation in Ansys. It is noticed that whatever the relative humidity the tensile behavior curve presents the same shape. We have a very strong elasticity then, a great
ductility before reaching the rupture. This is in accordance with the experimental results presented in the literature which present the RC fiber as having an elasto-ductile behavior [15,18,19,22].
That curve is based on the Hooke's law as presented in equation (1). The same shape is observed on Figure 8 presenting the work that was previously done experimentally on the RC fibers.
From this table we observe a good correlation between the numerical and experimental results for all the mechanical characteristics properties at the different humidities with regard to the
variations. As regards the stress at rupture and the elastic modulus, very high values are noted for the experimental tests compared to the results of the numerical simulation. This could be
explained by the fact that the fibers show quite high dispersion due to harvest location, climate, geometry, extraction, experimental conditions and many others factors. This can also be justified by
the fact that many input parameters of RC fiber are not yet known so far and they have been replaced by those of wood which already have a lot of works in that direction. Globally those results are
with accordance to the one found in literature concernind others fibers with relative humidities of 30, 40, 50, 60, 70% for flax and nettle [5], flax and sisal [6], flax and Hemp [7], hemp, sisal,
flax, jute, agava to the relative humidities of 10%, 25%, 50%, 80% [8] and flax on the relative humidities of 33% and 60% [9] deternined experimentally and flax on the relative humidities of 33% and
60% [9] determined by numerical simulation.
Fig. 5
Representation of the microfibril angles of the S1, S2 and S3 sublayers on ansys.
Fig. 6
Distribution on the fiber at a Relative Humidity of 23%: a- stress; b- strain
Fig. 7
Model of Tensile behavior curve of the fiber at a relative humidity of (a) 23%; (b) 54% and (c) 75%.
Fig. 8
Tensile behavior of the RC fiber obtained experimentally from litterature [19].
4 Conclusion
Modelling and numerical simulation of hygro-mechanical behavior of Rhecktophyllum Camerunense vegatable fibers using ANSYS sofware were used.. We generally noted that the mechanical properties of RC
fibers drop with the moisture as many of the vegetables fibers encounter in the literature and confirm the result got experimentally. The RC fiber remain and elasto-ductile fiber no matter the
relative humidity. That bring us to the conclusion that RC fibers should be treated before using them in composite materials to conserve its mechanical properties.
This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.
Conflicts of interest
The authors declare no conflict of interest.
Data availability statement
The authors state the availability of any data at submission.
Author contribution statement
Boris Noutegomo: Conceptualization, Data curation, Formal analysis, Investigation, Methodology, Resources, Software, Visualization, Writing – original draft, Writing – review and Editing. Fabien
Betene Ebanda: Investigation, Methodology, Project administration, Resources, Supervision, Validation, Visualization, Writing – review and editing. Ateba Atangana: Conceptualization, Formal analysis,
Methodology, Project administration, Resources, Supervision, Validation, Visualization, Writing – review and Editing.
Cite this article as: Boris Noutegomo, Fabien Betene Ebanda, Ateba Atangana, Numerical modelling of hygro-mechanical behavior of Rhecktophyllum Camerunense vegatable fibers, Int. J. Simul.
Multidisci. Des. Optim. 15, 18 (2024)
All Tables
Table 2
Mechanical properties of the sub-layers plant fiber constituents in the dry state [21].
Table 3
Diffusion coefficients of RC fiber as function of relative humidity [18].
Table 4
Microfibril angle of sub layers according to longitudinal axis [16,21].
Table 5
Water content of RC fiber as a function of RH by the numerical model.
Table 6
Comparison between the tensile behavior of the RC fiber obtained by numerical simulation and experimentally from reference [19].
All Figures
Fig. 1
Modeling of an RC fiber, S1 is in brown, S2 in green and S3 in red (cylindrical geometry).
In the text
Fig. 2
Mesh structure of the elementary fiber.
In the text
Fig. 3
Adsorption kinetics for an elementary fiber (a) Modeled at a relative humidity of 75% and (b) Experimentally in litterature for 23%, 54% and 75% humidity [17].
In the text
Fig. 4
Hygroscopic distribution within the elementary fiber modeled for the relative humidities of (a) 23, (b) 54 and (c) 75%.
In the text
Fig. 5
Representation of the microfibril angles of the S1, S2 and S3 sublayers on ansys.
In the text
Fig. 6
Distribution on the fiber at a Relative Humidity of 23%: a- stress; b- strain
In the text
Fig. 7
Model of Tensile behavior curve of the fiber at a relative humidity of (a) 23%; (b) 54% and (c) 75%.
In the text
Fig. 8
Tensile behavior of the RC fiber obtained experimentally from litterature [19].
In the text
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136 MATHEMATICS
In fact, the problems of the above types are solved by applying the following
principle known as the fundamental principle of counting, or, simply, the multiplication
principle, which states that
“If an event can occur in m different ways, following which another event
can occur in n different ways, then the total number of occurrence of the events
in the given order is m×n.”
The above principle can be generalised for any finite number of events. For
example, for 3 events, the principle is as follows:
‘If an event can occur in m different ways, following which another event can
occur in n different ways, following which a third event can occur in p different ways,
then the total number of occurrence to ‘the events in the given order is m × n × p.”
In the first problem, the required number of ways of wearing a pant and a shirt
was the number of different ways of the occurence of the following events in succession:
(i) the event of choosing a pant
(ii) the event of choosing a shirt.
In the second problem, the required number of ways was the number of different
ways of the occurence of the following events in succession:
(i) the event of choosing a school bag
(ii) the event of choosing a tiffin box
(iii) the event of choosing a water bottle.
Here, in both the cases, the events in each problem could occur in various possible
orders. But, we have to choose any one of the possible orders and count the number of
different ways of the occurence of the events in this chosen order.
Example 1
Find the number of 4 letter words, with or without meaning, which can be
formed out of the letters of the word ROSE, where the repetition of the letters is not
There are as many words as there are ways of filling in 4 vacant places
by the 4 letters, keeping in mind that the repetition is not allowed. The
first place can be filled in 4 different ways by anyone of the 4 letters R,O,S,E. Following
which, the second place can be filled in by anyone of the remaining 3 letters in 3
different ways, following which the third place can be filled in 2 different ways; following
which, the fourth place can be filled in 1 way. Thus, the number of ways in which the
4 places can be filled, by the multiplication principle, is 4 × 3 × 2 × 1 = 24. Hence, the
required number of words is 24. | {"url":"https://daily-class-notes.b-cdn.net/NCERT/Maths%20Class%2011/07%20PERMUTATIONS%20AND%20COMBINATIONS.html","timestamp":"2024-11-06T11:21:49Z","content_type":"application/xhtml+xml","content_length":"1049257","record_id":"<urn:uuid:61074381-bf67-4954-944d-b05ac8924477>","cc-path":"CC-MAIN-2024-46/segments/1730477027928.77/warc/CC-MAIN-20241106100950-20241106130950-00632.warc.gz"} |
HclustParam-class: Hierarchical clustering in bluster: Clustering Algorithms for Bioconductor
Run the base hclust function on a distance matrix within clusterRows.
1 HclustParam(
2 metric = "euclidean",
3 method = "complete",
4 cut.fun = NULL,
5 cut.dynamic = FALSE,
6 cut.height = NULL,
7 cut.number = NULL,
8 ...
9 )
11 ## S4 method for signature 'ANY,HclustParam'
12 clusterRows(x, BLUSPARAM, full = FALSE)
HclustParam( metric = "euclidean", method = "complete", cut.fun = NULL, cut.dynamic = FALSE, cut.height = NULL, cut.number = NULL, ... ) ## S4 method for signature 'ANY,HclustParam' clusterRows(x,
BLUSPARAM, full = FALSE)
metric String specifying the distance metric to use in dist.
method String specifying the agglomeration method to use in hclust.
cut.fun Function specifying the method to use to cut the dendrogram. The first argument of this function should be the output of hclust, and the return value should be an atomic vector specifying
the cluster assignment for each observation. Defaults to cutree if cut.dynamic=FALSE and cutreeDynamic otherwise.
cut.dynamic Logical scalar indicating whether a dynamic tree cut should be performed using the dynamicTreeCut package.
cut.height Numeric scalar specifying the cut height to use for the tree cut when cut.fun=NULL. If NULL, defaults to half the tree height. Ignored if cut.number is set.
cut.number Integer scalar specifying the number of clusters to generate from the tree cut when cut.fun=NULL.
... Further arguments to pass to cut.fun, when cut.dynamic=TRUE or cut.fun is non-NULL.
x A numeric matrix-like object where rows represent observations and columns represent variables.
BLUSPARAM A HclustParam object.
full Logical scalar indicating whether the hierarchical clustering statistics should be returned.
Function specifying the method to use to cut the dendrogram. The first argument of this function should be the output of hclust, and the return value should be an atomic vector specifying the cluster
assignment for each observation. Defaults to cutree if cut.dynamic=FALSE and cutreeDynamic otherwise.
Logical scalar indicating whether a dynamic tree cut should be performed using the dynamicTreeCut package.
Numeric scalar specifying the cut height to use for the tree cut when cut.fun=NULL. If NULL, defaults to half the tree height. Ignored if cut.number is set.
Integer scalar specifying the number of clusters to generate from the tree cut when cut.fun=NULL.
Further arguments to pass to cut.fun, when cut.dynamic=TRUE or cut.fun is non-NULL.
A numeric matrix-like object where rows represent observations and columns represent variables.
Logical scalar indicating whether the hierarchical clustering statistics should be returned.
To modify an existing HclustParam object x, users can simply call x[[i]] or x[[i]] <- value where i is any argument used in the constructor.
The HclustParam constructor will return a HclustParam object with the specified parameters.
The clusterRows method will return a factor of length equal to nrow(x) containing the cluster assignments. If full=TRUE, a list is returned with clusters (the factor, as above) and objects (the
hclust output).
dist, hclust and cutree, which actually do all the heavy lifting.
cutreeDynamic, for an alternative tree cutting method to use in cut.fun.
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Be Knowledgeable in Statistics Just by Following 4 Easy Steps of Statistics Homework Help for Students – University Homework Help
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How to Perform T-Tests in Python (One- and Two-Sample) • datagy
In this post, you’ll learn how to perform t-tests in Python using the popular SciPy library. T-tests are used to test for statistical significance and can be hugely advantageous when working with
smaller sample sizes.
By the end of this tutorial, you’ll have learned the following:
• What the different t-tests are and when they should be applied
• How to perform a one-sample t-test and a two-sample t-test in Python
• How to interpret the results from your statistical tests
Understanding the T-Test
The t-test, or often referred to as the student’s t-test, dates back to the early 20th century. An Irish statistician working for Guinness Brewery, William Sealy Gosset, introduced the concept.
Because the brewery was working with small sample sizes and was under strict orders of confidentiality, Gosset published his findings under the pseudonym “Student”. His seminal work, “The Probable
Error of a Mean,” laid the groundwork for what we now know as Student’s t-test.
This leads us to one of the primary benefits of the t-test: the t-test is able to make reliable inferences about a population using a small sample size. Let’s explore how this works by discussing the
theory behind the t-test in the following section.
Understanding the Student’s T-Test
Statistical tests are used to make assumptions about some population parameters. For example, it lets us test whether or not the average test score for any given group of students is 70%. The T-Test
works in two different ways:
1. The one-sample t-test allows us to test whether or not the population mean is equal to some value
2. The two-sample t-test allows us to test whether or not two population means are equal
Let’s explore these in a little more depth.
Understanding the One-Sample T-Test
The one-sample t-test is used to test the null hypothesis that the population mean inferred from a sample is equal to some given value. It can be described as below:
H0: μ = μ0 (population mean is equal to some hypothesized value μ0)
There are actually three different alternative hypotheses:
1. Two-tailed: The population mean is not equal to some given value
2. Left-tailed: The population mean is less than some given value
3. Right-tailed: The population mean is greater than some given value
We can use the following formula to calculate our test statistic:
t = (x – μ) / (s/√n)
• x: the sample mean
• μ[0]: a hypothesized population mean
• s: the sample standard deviation
• n: the sample size
We then need to calculate the p-value using degrees of freedom equal to n – 1. If the p-value is less than your chosen significance level, we can reject the null hypothesis and say that the means
Understanding the Two-Sample T-Test
The two-sample t-test is used to test whether two population means are equal (or if they differ in a significant way). In this case, the null hypothesis assumes that the two population means are
When we sample two different groups, we are almost guaranteed that their sample means will differ. But the t-test allows us to test whether or not this difference is different in a statistically
significant way.
Similar to the one-sample t-test, there are three different alternative hypotheses:
1. Two-tailed: The two means are not equal
2. Left-tailed: Population mean #1 is less than population mean #2
3. Right-tailed: Population mean #1 is greater than population mean #2
The formula for the two-sample t-test can be written as:
• X1 and X2 are the sample means of the two groups.
• s1 and s2 are the sample variances of the two groups.
• n1 and n2 are the sample sizes of the two groups.
We then need to calculate the p-value using degrees of freedom equal to (n[1]+n[2]-1). If the p-value is less than your chosen significance level, we can reject the null hypothesis and say that the
means differ.
Requirements for the Student T-Test
Both types of t-tests follow a key set of assumptions, including:
1. Observations should be independent of one another
2. The data should be relatively normally distributed
3. The samples should have approximately equal variances (this only applies to the two-sample t-test)
4. The samples were collected using random sampling
It’s easy to test for these assumptions using Python (and I have included links to tutorials covering how to do this). Let’s take a look at example walkthroughs of how to conduct both of these tests
in Python.
Perform a One-Sample T-Test in Python
In this section, you’ll learn how to conduct a one-sample t-test in Python. Suppose you are a teacher and have just given a test. You know that the population mean for this test is 85% and you want
to see whether the score of the class is significantly different from this population mean.
Let’s start by importing our required function, ttest_1samp() from SciPy and defining our data:
from scipy.stats import ttest_1samp
# Sample data (exam scores of a class)
sample_scores = [75, 82, 88, 78, 95, 89, 92, 85, 88, 79]
# Population mean (hypothetical mean of all students' scores)
population_mean = 85
In the code block above, we first imported our required library. We then defined our sample as a list of values and defined our population mean as its own variable.
We can now pass these values into the function, as shown below:
# Perform one-sample t-test
t_statistic, p_value = ttest_1samp(sample_scores, population_mean)
# Output the results
print(f"t-statistic: {t_statistic}")
print(f"P-value: {p_value}")
# Returns:
# t-statistic: 0.04886615700133708
# P-value: 0.9620932123799038
The function returns a test statistic and the corresponding p-value. We can print these values out using f-strings to simplify the labeling, as shown above.
Finally, we can write a simple if-else statement to evaluate whether or not our sample mean is significantly different from the population mean:
# Check if the result is statistically significant (using a common significance level of 0.05)
if p_value < 0.05:
print("The average exam score is significantly different from the population mean.")
print("There is no significant difference in the average exam score.")
# Returns:
# There is no significant difference in the average exam score.
We can see that by running this if-else statement, that our test indicates that there is no significant difference in the exam scores.
In order to calculate the different one-sample t-test alternative hypotheses, we can use the alternative= parameter:
• alternative='two-sided' is the default value, checking for a two-sided alternative hypothesis
• alternative='less' checks whether the provided mean is less than the population mean
• alternative='greater' checks whether the provided mean is greater than the population mean
Now that you have a strong understanding of how to perform a one-sample t-test, let’s dive into the exciting world of two-sample t-tests!
Perform a Two-Sample T-Test in Python
A two-sample t-test is used to test whether the means of two samples are equal. The test requires that both samples be normally distributed, have similar variances, and be independent of one another.
Imagine that we want to compare the test scores of two different classes. This is the perfect example of when to use a t-test. Let’s begin by running a two-tailed test, which only evaluates whether
or not the two means are equal. It begins with the null hypothesis, which states that the two means are equal.
Let’s take a look at how we can run a two-tailed t-test in Python:
# Running a Two-Tailed Two-Sample T-Test in Python
from scipy.stats import ttest_ind
# Generate two independent samples (example: exam scores of two classes)
class1_scores = [64, 58, 66, 75, 57, 57, 75, 67, 55, 65]
class2_scores = [80, 80, 87, 65, 67, 79, 74, 88, 75, 70]
# Perform two-sample t-test
t_statistic, p_value = ttest_ind(class1_scores, class2_scores)
# Output the results
print(f"t-statistic: {t_statistic}")
print(f"P-value: {p_value}")
# Returns:
# t-statistic: -3.747537032729207
# P-value: 0.001474009849334239
We can see that the ttest_ind() function returns both a test statistic and a p-value. We can run a simple if-else statement to check whether or not we can reject or fail to reject the null
# Check if the result is statistically significant (using a common significance level of 0.05)
if p_value < 0.05:
print("There is a significant difference between the exam scores of the two classes.")
print("There is no significant difference between the exam scores of the two classes.")
# Returns:
# There is a significant difference between the exam scores of the two classes.
We can see that there is a significant difference between the two sets of scores. However, the two-tailed test doesn’t tell us in which direction.
In order to do this, we need to use a right- or left-tailed two-sample t-test. To do this in SciPy, we use the alternative= parameter. By default, this is set to 'two-sided'. However, we can modify
this to either 'less' or 'greater', if we want to evaluate whether or not the mean for one sample is less than or greater than another.
Let’s see how we can check if the mean of class 2 is significantly higher than that of class 1:
# Running a Right-Tailed Two-Sample T-Test in Python
from scipy.stats import ttest_ind
# Generate two independent samples (example: exam scores of two classes)
class1_scores = [64, 58, 66, 75, 57, 57, 75, 67, 55, 65]
class2_scores = [80, 80, 87, 65, 67, 79, 74, 88, 75, 70]
# Perform two-sample t-test
t_statistic, p_value = ttest_ind(class2_scores, class1_scores, alternative='greater')
# Output the results
print(f"t-statistic: {t_statistic}")
print(f"P-value: {p_value}")
# Returns:
# t-statistic: 3.747537032729207
# P-value: 0.0007370049246671195
Because our p-value is less than our defined value of 0.05, we can say that the mean of class 2 is higher with statistical significance.
In conclusion, this comprehensive guide has equipped you with the knowledge and practical skills to perform t-tests in Python using the SciPy library. T-tests are invaluable tools for assessing
statistical significance, particularly when working with smaller sample sizes.
Throughout this tutorial, you’ve gained insights into:
1. The different types of t-tests and their applications.
2. How to conduct one-sample and two-sample t-tests in Python.
3. Interpretation of results obtained from statistical tests.
Remember that t-tests come with certain assumptions, and it’s crucial to validate them before applying these tests to your data. Python provides tools to check these assumptions, ensuring the
robustness and reliability of your statistical analyses.
To learn more about these functions, check out the official documentation for the one-sample t-test and for the two-sample t-test in SciPy. | {"url":"https://datagy.io/t-test-python/","timestamp":"2024-11-14T16:43:12Z","content_type":"text/html","content_length":"154508","record_id":"<urn:uuid:ca119894-7b3e-4eeb-9514-904f15c6755b>","cc-path":"CC-MAIN-2024-46/segments/1730477393980.94/warc/CC-MAIN-20241114162350-20241114192350-00612.warc.gz"} |
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Doing Math in Different Language
11-19-2015, 04:18 AM
Post: #1
Bill (Smithville NJ) Posts: 471
Senior Member Joined: Dec 2013
Doing Math in Different Language
I will be showing my ignorance with this post.
I am reading the book "The Martians of Science: Five Physicists Who Changed the Twentieth Century" by Istvan Hargittai. I am currently reading about the early upbringing of Theodore von Karman. As I
was reading the following caught my attention:
"As an adult, even though he could add and subtract in several languages, he could multiply only in Hungarian."
I have never thought of numbers, addition, subtraction and multiplication as being "done in a particular language."
I know there is a wide cross section of nationalities and languages here and that many of you are mathematicians.
So my question is - do you do mathematics in a particular language or is mathematics just universal - independent of any particular language.
Like I said, I am exposing my total ignorance on this.
Smithville, NJ
11-19-2015, 04:46 AM
Post: #2
Katie Wasserman Posts: 640
Super Moderator Joined: Dec 2013
RE: Doing Math in Different Language
I always thought that mathematics *is* a language. I suppose that there are different dialects of mathematics -- certainly there are over history -- but there's no need for it to be translated into a
natural language to solve a problem (unless it's a so-called "word problem").
11-19-2015, 04:50 AM
Post: #3
Didier Lachieze Posts: 1,658
Senior Member Joined: Dec 2013
RE: Doing Math in Different Language
Multiplication in English :
eight times seven equals fifty-six
In French :
huit fois sept égal cinquante six
When I was a child I learned the multiplication tables in French, later on I learned English. Now as an adult it's faster for me to think to numbers and to calculate in French than in English.
11-19-2015, 05:15 AM
Post: #4
Thomas Klemm Posts: 2,271
Senior Member Joined: Dec 2013
RE: Doing Math in Different Language
Multiplication in English :
8 x 7 = 56
In French :
8 x 7 = 56
In German :
8 x 7 = 56
In Arabic :
٧ = ٥٦ x ٨
11-19-2015, 05:57 AM
(This post was last modified: 11-19-2015 05:58 AM by Didier Lachieze.)
Post: #5
Didier Lachieze Posts: 1,658
Senior Member Joined: Dec 2013
RE: Doing Math in Different Language
(11-19-2015 05:15 AM)Thomas Klemm Wrote: Multiplication in English :
8 x 7 = 56
In French :
8 x 7 = 56
In German :
8 x 7 = 56
In Arabic :
٧ = ٥٦ x ٨
Well, you're not using the listed languages, just a graphical representation of the numbers. But when you talk about them you have to use names in one specific language.
Over the phone I cannot say 8, I have to say eight or huit or acht depending on who I am talking to.
And the way I learned the multiplication tables is deeply tied to the French names of the numbers. Other people may have their brain setup differently.
11-19-2015, 10:31 AM
Post: #6
walter b Posts: 1,957
On Vacation Joined: Dec 2013
RE: Doing Math in Different Language
Based on reading directions, it looks like two-digit numbers are "spoken" in Arabic like in German, sequence-wise. Compare fifty-six and sechsundfünfzig, for instance. Is anybody on this forum who
can confirm or deny that?
11-19-2015, 10:38 AM
Post: #7
ggauny@live.fr Posts: 582
Senior Member Joined: Nov 2014
RE: Doing Math in Different Language
For calcuations I thing all of us think in their mother language. I speak very well polish, but if I have to calcul, I calcul in french.
11-19-2015, 10:49 AM
(This post was last modified: 11-19-2015 10:51 AM by Thomas Klemm.)
Post: #8
Thomas Klemm Posts: 2,271
Senior Member Joined: Dec 2013
RE: Doing Math in Different Language
(11-19-2015 10:31 AM)walter b Wrote: Is anybody on this forum who can confirm or deny that?
Can confirm:
ستة وخمسو
sitah wa khamsun: sechs und fünfzig
But it's similar to German for bigger numbers:
tausend und eine Nacht
ألف ليلة وليلة
alf lailat wa laila: tausend Nächte und eine Nacht
11-19-2015, 10:50 AM
(This post was last modified: 11-19-2015 10:57 AM by Vtile.)
Post: #9
Vtile Posts: 406
Senior Member Joined: Oct 2015
RE: Doing Math in Different Language
It is different:
1 + 1 = 2
1 1 +
There is also the cryptography called higher mathematics where everything is discuised with odd hieroglypth based on where the writer have found his education. Universal and exact .. nope.
11-19-2015, 11:33 AM
Post: #10
walter b Posts: 1,957
On Vacation Joined: Dec 2013
RE: Doing Math in Different Language
(11-19-2015 10:49 AM)Thomas Klemm Wrote: But it's similar to German for bigger numbers:
tausend und eine Nacht
ألف ليلة وليلة
alf lailat wa laila: tausend Nächte und eine Nacht
(emphasis added)
Why "but"? It applies for smaller numbers as well AFAICS.
P.S.: Quoting without proper filing is also an interesting experience. Chaos rules ...
11-19-2015, 12:09 PM
Post: #11
Thomas Klemm Posts: 2,271
Senior Member Joined: Dec 2013
RE: Doing Math in Different Language
(11-19-2015 11:33 AM)walter b Wrote: Why "but"?
I referred to "sequence-wise": you cold have the idea that the digits are just read from right to left. The example 1001 shows that this is not the case.
11-19-2015, 04:49 PM
Post: #12
Alberto Candel Posts: 169
Member Joined: Dec 2013
RE: Doing Math in Different Language
It is interesting that not only I must add in Spanish, but even arithmetical operations have a different "feeling" in different languages. Multiplication in Spanish "feels' like a 2-dimensional
object, as if measuring area: 7 x 9 is a rectangle of size 7 by 9 in Spanish (7 por 9), but it is a linear object resulting from placing a length of 9 seven times (7 times 9, which would be "7 veces
9" in Spanish). So the commutative law for multiplication is implicit in Spanish, but requires a proof in English. I wonder if this is just an idiosyncrasy.
11-20-2015, 06:39 PM
(This post was last modified: 11-20-2015 06:40 PM by Julián Miranda (Spain).)
Post: #13
Julián Miranda (Spain) Posts: 4
Junior Member Joined: Aug 2014
RE: Doing Math in Different Language
It would be interesting if someone with a mother tonge which is written from right to left like arabic or hebrew could explain how they write a technical paper with formulas. You are writting right
to left, switch to left to right for formulas and continue right to left?
For the original post of this thread I agree with Didier or ggauny, we calculate in our mother tongue. I remember when I started studying English, reading simple sentences with numbers in it, like an
address, was a problem because I read the numbers in spanish. It took an aditional effort to "translate" numbers, as Didier said, 8 was "ocho" (in Spanish) for me. The additional effort was to
convert 8 in eight.
Very interesting also the "feeling" Alberto explains in the previous post.
11-20-2015, 07:26 PM
Post: #14
Thomas Klemm Posts: 2,271
Senior Member Joined: Dec 2013
RE: Doing Math in Different Language
(11-20-2015 06:39 PM)Julián Miranda (Spain) Wrote: You are writting right to left, switch to left to right for formulas and continue right to left?
Modern Arabic mathematical notation
Quote:The most remarkable of those features is the fact that it is written from right to left following the normal direction of the Arabic script.
11-20-2015, 07:53 PM
Post: #15
rprosperi Posts: 6,632
Super Moderator Joined: Dec 2013
RE: Doing Math in Different Language
(11-19-2015 04:18 AM)Bill (Smithville NJ) Wrote: So my question is - do you do mathematics in a particular language or is mathematics just universal - independent of any particular language.
Yet another seemingly obvious concept leads to another fascinating thread - thanks for asking this Bill; like you, it never occurred to me...
--Bob Prosperi
11-20-2015, 11:36 PM
Post: #16
jebem Posts: 1,355
Senior Member Joined: Feb 2014
RE: Doing Math in Different Language
While we mentally resolve maths using our mother language, mathematics is an example of a universal language, if you accept Latin characters as being "universal" (well, it is universal at least in
the Occident anyway).
But, unfortunately for us occidentals, Latin alphabet is not the universal world standard, as there are other strong competitors like Cyrillic, Greek, Arabic, Hebrew and Chinese among others, and
this creates a barrier when trying to read a book written in those alphabets and languages.
There are other universal languages in different fields, be it pure science or technological ones.
As I see it, the essence behind all things is pure logic. Logic thinking is common among all human cultures, independently of the mother language or alphabet, and this lead us to tell that
mathematics is a universal language.
For instance, in electrical engineering a schematics diagram is instantaneously recognized and more or less understood by anyone in the field, independently of the used language or alphabet.
Again, the used alphabet and language will create obstacles to fully understand the message.
I remember too well my first contact with a Russian professional navigation system´s service guide more than 35 years ago.
At the time I knew nothing about Russian language or Cyrillic, but I could understood most of the schematics contents to be able to do my job, only needing assistance to translate some text parts.
Later on, after learning the Cyrillic alphabet, I was much more confident when consulting the Russian technical guides, as the electrical and physic units are basically the same in Latin and in
Cyrillic, but written on a different alphabet.
Just my 2 Cents anyway.
Jose Mesquita
RadioMuseum.org member
11-21-2015, 12:57 AM
(This post was last modified: 11-21-2015 01:46 AM by Vtile.)
Post: #17
Vtile Posts: 406
Senior Member Joined: Oct 2015
RE: Doing Math in Different Language
Is there btw. a study made (and written) of the history of mathematical notation used along the years?
I must add to above example that there is still a lot of variation, while normally people in field knows what some diagram or drawing means, it is a lot based to the knowledge that there is
variation. We can take example of logic gates which are different in different "standard" regions, same goes with ie. resistor markings. Then there is whole lot of style draw relays and how the
wiring logics goes. If the reader knows enough the subject he can fill the voids.
Same in the maths different people use different notation of symbols for same stuff, ie. how something like voltage or current is written ie. is it peak or rms and how are the super or subscripts
used etc. etc. Same goes with pure mathematics, there is different styles to write ie. derivative f'(x) or d/dx which makes reading even relative simple mathematics bizarre until you get enough
knowledge in notation and get knowledge to regognice patterns in mathematical sentences, oh this is Eulers formula and the next line looks like it does because he did this and that trick to get
there. So basicly the maths is as much pattern recognition as ie. reading a wiring diagram and there is also variation of writing what you are saying, like how complex numbers are written.
Then there is the so called basic math vs. so called higher maths language differences, which are total sidetrack. Ie. the classic square root with negative numbers is there a solution or not. For
person of knowledge of only basic math there is no solution of square root of negative number in his/her world, while for person who have knowledge of more advanced math there is solution(s) on
complex domain for that mathematical operation.
Logics and rules are pretty universal I would assume.
11-21-2015, 02:15 AM
Post: #18
Bill (Smithville NJ) Posts: 471
Senior Member Joined: Dec 2013
RE: Doing Math in Different Language
(11-20-2015 07:53 PM)rprosperi Wrote:
(11-19-2015 04:18 AM)Bill (Smithville NJ) Wrote: So my question is - do you do mathematics in a particular language or is mathematics just universal - independent of any particular language.
Yet another seemingly obvious concept leads to another fascinating thread - thanks for asking this Bill; like you, it never occurred to me...
Hi Bob,
I almost didn't start this thread - I was afraid I had missed the obvious.
But from what I have read here, how people think about math can be affected by the particular languages they may know.
Fascinating. Thanks to everyone for posting.
Smithville, NJ
11-21-2015, 02:25 AM
Post: #19
Bill (Smithville NJ) Posts: 471
Senior Member Joined: Dec 2013
RE: Doing Math in Different Language
(11-21-2015 12:57 AM)Vtile Wrote: Is there btw. a study made (and written) of the history of mathematical notation used along the years?
Following is list of a few books on Math Notation.
NOTE: I have not read or reviewed any of these, but some do look interesting:
A History of Mathematical Notations: Vol I
by Florian Cajori
A History of Mathematical Notations, Volume II
by Florian Cajori
Enlightening Symbols: A Short History of Mathematical Notation and Its Hidden Powers
by Joseph Mazur
Writing the History of Mathematical Notations: 1483-1700
by Sr. Mary Leontius Schulte and Albrecht Heeffer
Numerical Notation: A Comparative History
by Stephen Chrisomalis
Maybe some of the members here have seen some of these and may want to comment.
Smithville, NJ
11-21-2015, 06:19 AM
Post: #20
Gerald H Posts: 1,627
Senior Member Joined: May 2014
RE: Doing Math in Different Language
(11-20-2015 06:39 PM)Julián Miranda (Spain) Wrote: It would be interesting if someone with a mother tonge which is written from right to left like arabic or hebrew could explain how they write a
technical paper with formulas. You are writting right to left, switch to left to right for formulas and continue right to left?
For the original post of this thread I agree with Didier or ggauny, we calculate in our mother tongue. I remember when I started studying English, reading simple sentences with numbers in it,
like an address, was a problem because I read the numbers in spanish. It took an aditional effort to "translate" numbers, as Didier said, 8 was "ocho" (in Spanish) for me. The additional effort
was to convert 8 in eight.
Very interesting also the "feeling" Alberto explains in the previous post.
The feeling may be the same as a Spaniard adding two multidigit numbers starting from the right & proceeding leftwards who then has to divide two multidigit numbers & starts from the left proceeding
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Fundamental Counting Principle and Probability
A class is taking a multiple choice exam. There...
Solved on Sep 18, 2023
Exam with 7 multiple-choice questions, 3 answers each. Find the number of possible answer combinations and the probabilities of a student randomly guessing all correct, all incorrect, or 2 correct
and the rest incorrect.
1) How many ways can the exam be answered? $3^7$ 2) Probability of all answers correct: $\left(\frac{1}{3}\right)^7$ or $0.0000\%$ 3) Probability of all answers incorrect: $\left(\frac{2}{3}\right)^
7$ or $0.4629\%$ 4) Probability of 2 correct (questions 5 and 6) and the rest incorrect: $\left(\frac{1}{3}\right)^2 \left(\frac{2}{3}\right)^5$ or $0.0123\%$
STEP 1
Assumptions1. There are7 questions in the exam. . Each question has3 possible answers, with exactly one correct answer.
3. The student is guessing randomly, so each answer has an equal chance of being chosen.
STEP 2
First, we need to find the total number of different ways to answer all the questions on the exam. We can do this by using the Fundamental Counting Principle, which states that if there are n ways to
do one thing and m ways to do another, then there are n*m ways to do both.
$Total\, ways = Number\, of\, choices^{Number\, of\, questions}$
STEP 3
Now, plug in the given values for the number of choices and the number of questions to calculate the total ways.
$Total\, ways =3^{7}$
STEP 4
Calculate the total number of ways to answer all the questions.
$Total\, ways =3^{7} =2187$
STEP 5
Now, we can calculate the probability of the student getting all the answers correct. Since there is only one correct answer for each question, the number of ways to get all the answers correct is1^
$Probability\, of\, all\, correct = \frac{Number\, of\, ways\, to\, get\, all\, correct}{Total\, ways}$
STEP 6
Plug in the values for the number of ways to get all correct and the total ways to calculate the probability.
$Probability\, of\, all\, correct = \frac{1^{}}{218}$
STEP 7
Calculate the probability of getting all answers correct.
$Probability\, of\, all\, correct = \frac{1^{7}}{2187} = \frac{1}{2187}$
STEP 8
Convert the probability to a percentage and round to four decimal places.
$Probability\, of\, all\, correct = \frac{1}{2187} \times100\% =0.0457\%$
STEP 9
Now, we can calculate the probability of the student getting all the answers wrong. Since there are2 wrong answers for each question, the number of ways to get all the answers wrong is2^7.
$Probability\, of\, all\, wrong = \frac{Number\, of\, ways\, to\, get\, all\, wrong}{Total\, ways}$
STEP 10
Plug in the values for the number of ways to get all wrong and the total ways to calculate the probability.
$Probability\, of\, all\, wrong = \frac{2^{7}}{2187}$
STEP 11
Calculate the probability of getting all answers wrong.
$Probability\, of\, all\, wrong = \frac{^{7}}{2187} = \frac{128}{2187}$
STEP 12
Convert the probability to a percentage and round to four decimal places.
$Probability\, of\, all\, wrong = \frac{128}{2187} \times100\% =5.8577\%$
STEP 13
Finally, we can calculate the probability of the student getting questions5 and6 correct and the rest of the answers wrong. There is way to get each of these questions correct and2 ways to get each
of the other questions wrong.
$Probability\, of\,5\, and\,6\, correct\, and\, rest\, wrong = \frac{Number\, of\, ways\, to\, get\,5\, and\,6\, correct\, and\, rest\, wrong}{Total\, ways}$
STEP 14
Plug in the values for the number of ways to get and6 correct and the rest wrong and the total ways to calculate the probability.
$Probability\, of\,\, and\,6\, correct\, and\, rest\, wrong = \frac{2^{}}{2187}$
STEP 15
Calculate the probability of getting questions5 and correct and the rest of the answers wrong.
$Probability\, of\,5\, and\,\, correct\, and\, rest\, wrong = \frac{2^{5}}{2187} = \frac{32}{2187}$
STEP 16
Convert the probability to a percentage and round to four decimal places.
$Probability\, of\,5\, and\,6\, correct\, and\, rest\, wrong = \frac{32}{218} \times100\% =.4634\%$a) The probability the student got all of the answers correct is0.045%. b) The probability the
student got all of the answers wrong is5.857%. c) The probability the student got questions5 and6 correct and the rest of the answers wrong is.4634%. | {"url":"https://studdy.ai/learning-bank/problem/the-problem-asks-about-the-funda-JhxleclITR_0CtS6","timestamp":"2024-11-04T20:50:54Z","content_type":"text/html","content_length":"206079","record_id":"<urn:uuid:5df30d8b-78c9-45a9-b77e-92f25b468d6d>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.16/warc/CC-MAIN-20241104194528-20241104224528-00852.warc.gz"} |
Trap 1 - Type Cast
◀ Can You Spot Bugs?▶ Trap 2 - cin.get()Amazon T
ype cast refers to the way a variable of a type changes to another type implicitly or explicitly. As an exercise to see how much you know about type casting predict the output of the following
using namespace std;
int main() {
double result;
int numOfPies, numOfPeople;
numOfPies = 14;
numOfPeople = 3;
result = numOfPies/numOfPeople;
cout<<numOfPies<<" pies split up evenly between "<<numOfPeople<<" people.\n";
cout<<"Therefore, each person gets "<<result<<" pies.\n";
return 0;
: The output looks like this:
14 pies split up evenly between 3 people.
Therefore, each person gets 4 pies
The intention of this program is to divide 14 pies evenly between 3 people, so each one should get 4 2/3 pies, but why does the output say each person gets only 4 pies?
First we check the type of result, which is double, so there should not be any problem. However, numOfPies and numOfPeople are int. The result of numOfPies/numOfPeople, therefore, is 4. Now because
the types of both operands should match, 4 is transformed into double.
By examining this example, it is not hard to see that many bugs come from lack of understanding of the programming language. Therefore, the first step to improve your programming skills is to
completely understand the programming language you are using.
Let’s look at the next exercise!
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previous index next
74 Particle Drift in Nonuniform Static Magnetic Field: Guiding Center Motion
Michael Fowler UVa
Jackson 12.4
Guiding Center
We’ve found that in a uniform magnetic field a particle moves in a path that is a combination of circling around a field line (in a circle with radius proportional to the speed of the particle) and a
steady speed of the center of the circle$—$the guiding center$—$in the direction of the field.
We’ll now consider a slowly spatially varying (static) magnetic field: in particular, one that varies little over the radius of the particle’s circular motion. (So we’re also restricting the energy
range of the particle.) The first approximation to the motion is still that of spiraling around a field line, but that neglects other important effects.
Here we’ll examine two kinds of field variation that give rise to guiding center drift perpendicular to the local field direction, that is, sideways:
1. A gradient in field magnitude, and
2. Curvature of field lines.
Particle Moving in (x,y) Plane, Varying Strength z-Direction Field: Gradient Drift
We’ll begin with a very crude example to illustrate the essential physics, and for simplicity take the nonrelativistic limit.
Suppose the magnetic field points in the $z$-direction, and is almost constant, but is linearly increasing in magnitude in the $x$-direction. To simplify even further, we suppose the field has
magnitude ${B}_{1}$ for $x<0$ and ${B}_{2}={B}_{1}+\Delta B$ for $x>0:$ just replacing the linear increase with a simple small step.
Now consider a charged particle circling in the field in the $x,y$ plane. Suppose we start at the bottom point of the curve shown in the diagram, on the $y$-axis, the particle moving in the negative
$x$-direction at speed ${v}_{\perp }$ (which stays constant throughout). Moving in the weaker field ${\stackrel{\to }{B}}_{1},$ the particle traces a half-circle of radius ${r}_{1},$ where
${v}_{\perp }={\omega }_{{B}_{1}}{r}_{1},\text{ }{\omega }_{{B}_{1}}=\frac{e{B}_{1}}{mc}.$
The particle crosses into the incrementally stronger field at a point $2{r}_{1}$ above the original entry point. After a downward half-circle in the stronger field, the particle is close to the
original entry point, but vertically displaced by an amount
$\Delta y=2\left({r}_{1}-{r}_{2}\right)=2{v}_{\perp }\left(\frac{1}{{\omega }_{{B}_{1}}}-\frac{1}{{\omega }_{{B}_{2}}}\right)\approx 2{v}_{\perp }\frac{\Delta {\omega }_{B}}{{\omega }_{B}^{2}}\approx
\frac{2{v}_{\perp }mc}{e}\frac{\Delta B}{{B}^{2}}.$
The particle’s velocity at this point is identical in magnitude and direction to its initial value, so the path will repeat, just displaced vertically. That is, after $\Delta t=2\pi /{\omega }_{B}$
the model predicts that the orbit will have moved in the $y$-direction by $\Delta y$ given above, and writing ${v}_{\perp }=\omega r$ and $mc/eB=1/\omega ,$ we find to first order the obvious result
$\Delta y\approx 2r\Delta B/B$
and therefore a net drift velocity perpendicular to the field
${v}_{\text{G model}}=\frac{\Delta y}{\Delta t}=\frac{2r\Delta B/B}{2\pi /\omega }=r\omega \frac{\Delta B}{B}.$
The crude part of our model is just taking two possible magnetic field values, to represent a smoothly varying field. To compare our result with Jackson’s more precise analysis we write
$\Delta B\approx rabla B,$
to give
${v}_{\text{G model}}\approx {r}^{2}{\omega }_{B}abla B/B\approx \left({v}_{\perp }^{2}/{\omega }_{B}\right)abla B/B.$
Comparing this naïve result with Jackson’s 12.55, ${v}_{G}=\left(\omega {r}^{2}/2{B}^{2}\right)\left(\stackrel{\to }{B}×{abla }_{\perp }B\right),$ we see that the simple model automatically gives the
direction of the drift velocity (see diagram) as perpendicular to the magnetic field and to the direction of its gradient. However, the magnitude is off by a factor of 2: the proper way to find the
net vertical displacement is to integrate around the orbit using the linearly varying magnetic field. It’s fairly straightforward, see Jackson for details, the corrected result is:
${v}_{\text{G}}=\left({v}_{\perp }^{2}/2{\omega }_{B}\right)abla B/B.$
Notice that the direction of the drift is also perpendicular to the gradient (hence subscript G) of the magnetic field strength. This means the particle will drift along an “equipotential” of
magnetic field strength. Suppose now the magnetic field strength is zero beyond a certain distance $R$ from the origin. Then the “equipotentials” must be closed curves, and an electron moving within
this region will never get out! Of course, it has to be moving slowly enough for our approximations to be valid. This problem has been fully discussed by a famous physicist: Edward Witten, Annals
of Physics 120, 72 (1979).
Drift Velocity from Field Lines Curvature
Obviously, the field lines in general curve, taking them straight in the example above was an approximation. To see what difference arises from curved field lines, Jackson (page 591) takes as an
example field lines from a line of current along the $z$-axis, that is, circles in the $\left(x,y\right)$ plane, the same irrotational vector field as the fluid velocity field around a whirlpool.
Now assume with Jackson that the particle is tightly spiraling around a field line which is an arc of a circle of radius $R,$ with net velocity ${v}_{\parallel }$ along the field line. This net
motion has acceleration ${v}_{\parallel }^{2}/R$ towards the $z$-axis, equivalent to motion in an effective radial electric field ${E}_{\text{eff}}=\left(m/eR\right){v}_{\parallel }^{2}.$ Recalling
that crossed electric and magnetic fields give a drift velocity $c\stackrel{\to }{E}×\stackrel{\to }{B}/{B}^{2},$ we find the drift velocity from this field curvature ${v}_{C}$ to be (using ${\omega
}_{B}=eB/mc$ ) in the $z$-direction, with magnitude
${v}_{C}={v}_{\parallel }^{2}/{\omega }_{B}R.$
For the effectively two-dimensional magnetic field we are considering, $\stackrel{\to }{abla }×\stackrel{\to }{B}=0$ means ${abla }_{\perp }B/B=-\stackrel{\to }{R}/{R}^{2},$ and recalling from the
previous section that the drift velocity from a field gradient is ${v}_{\text{G}}=\left({v}_{\perp }^{2}/2{\omega }_{B}\right)abla B/B,$ evidently in the $z$-direction here (since $\stackrel{\to }{B}
$ is azimuthal and $\stackrel{\to }{abla }B$ is radial), the two drift velocities can be added, yielding total
${v}_{\text{drift}}=\left(1/{\omega }_{B}R\right)\left({v}_{\parallel }^{2}+\frac{1}{2}{v}_{\perp }^{2}\right).$
Note that the drift depends on the sign of the charge through ${\omega }_{B}$ so in a plasma charges would separate, in contrast to movement induced by crossed electric and magnetic fields.
Containing Hot Plasma
These drifts are problematic in machines designed to contain a hot plasma, such as attempts at nuclear fusion, where ions are typically inside some kind of toroid. Billions of dollars have been spent
designing magnetic containment chambers, but so far none have been successful in holding significant nuclear fusion. For an interesting attempt, click stellarator.
Earth’s Ring Current
As we’ll discuss later, this sideways drift acts on the particles in the van Allen radiation belts to generate a ring current around the Earth, which is enough, during magnetic storms, to partially
cancel the Earth’s magnetic field, and thus allow more cosmic radiation to reach the Earth’s surface.
previous index next | {"url":"https://galileoandeinstein.phys.virginia.edu/Elec_Mag/2022_Lectures/EM_74_Guiding_Center_Drift.html","timestamp":"2024-11-03T19:37:27Z","content_type":"text/html","content_length":"24177","record_id":"<urn:uuid:07d676a1-73c4-43c0-b502-ff3f039867e4>","cc-path":"CC-MAIN-2024-46/segments/1730477027782.40/warc/CC-MAIN-20241103181023-20241103211023-00291.warc.gz"} |
=DOLLARDE formula | Convert dollar prices entered with a special notation to decimal numbers.
Formulas / DOLLARDE
Convert dollar prices entered with a special notation to decimal numbers. =DOLLARDE(fractional_dollar,fraction)
• fractional_dollar - required argument that is the dollar component in special fractional notation
• fraction - required argument that is the numerator in the fractional unit
• =DOLLARDE(1.02,16)
The DOLLARDE function can be used to convert fractions to decimals. For example, the formula can be used to convert 1.02 to a decimal. Since 1.02 is the same as 1 and 2/16, the function will
return 1.125.
• =DOLLARDE(1.125,16)
The DOLLARDE function can also be used to convert decimals to fractions. For example, the formula can be used to convert 1.125 to a fraction. The function will return 1.02, which is the same as 1
and 2/16.
• =DOLLARDE(1.08,16)
The DOLLARDE function can also be used to convert currency fractions to decimals. For example, the formula if you have a currency fraction of $1 and 8/16, you can use to convert it to 1.5. This
is because 1.08 is equal to $1.50 when expressed as 1 and 8/16.
• =DOLLARDE(1.5,16)
The DOLLARDE function can also be used to convert decimals to currency fractions. For example, the formula if you have a decimal of 1.5, you can use to convert it to a currency fraction. The
function will return 1.08, which is the same as 1 and 8/16 or $1.50.
The DOLLARDE function is a useful tool for converting values pricing as a fraction or dollar to decimal format. It allows for easy and efficient calculations.
• The DOLLARDE function converts a dollar price into a decimal number and the DOLLARFR function converts a decimal number into a dollar price.
• Fractional_dollar and fraction arguments are used to specify the dollar component and the numerator in the fractional unit, respectively.
Frequently Asked Questions
What is the DOLLARDE function?
The DOLLARDE function is a mathematical formula that converts a dollar price expressed as an integer part and a fraction part into a dollar price expressed as a decimal number.
What is the dollar price?
The dollar price is a decimal number.
What is the fractional part of the dollar price?
The fractional part of the dollar price is divided by an integer that is specified.
Can the DOLLARDE function be used to convert a dollar price expressed as a fractional part and decimal number?
No, the DOLLARDE function is only used to convert a dollar price expressed as an integer part and fractional part.
What are some examples of how to use the DOLLARDE function?
The DOLLARDE function can be used to convert a dollar price expressed as an integer part and fractional part into a dollar price expressed as a decimal number. For example, if the integer part is 2
and the fractional part is 3/4, the DOLLARDE function can be used to convert this into 2.75. | {"url":"https://sourcetable.com/formula/dollarde","timestamp":"2024-11-11T05:07:17Z","content_type":"text/html","content_length":"58520","record_id":"<urn:uuid:ac64efb8-2883-4a54-8822-e6f89746f01c>","cc-path":"CC-MAIN-2024-46/segments/1730477028216.19/warc/CC-MAIN-20241111024756-20241111054756-00435.warc.gz"} |
How I wish someone would explain SHAP values to me -
Have you ever struggled to interpret the decisions of an AI? SHAP was created to help you overcome these issues. The acronym stands for SHapley Additive exPlanations, a relatively recent method (less
than 10 years old) that seeks to explain the decisions of artificial intelligence models in a more direct and intuitive way, avoiding “black box” solutions.
Its concept is based on game theory with robust mathematics. However, a complete understanding of the mathematical aspects is not necessary to use this methodology in our daily lives. For those who
wish to delve deeper into the theory, I recommend reading this publication in English.
In this text, I will demonstrate practical interpretations of SHAP, as well as understanding its results. Without further ado, let’s get started! To do this, we’ll need a model to interpret, right?
I will use as a basis the model built in my notebook (indicated by the previous link). It is a tree-based model for binary prediction of Diabetes. In other words, the model predicts people who have
this pathology. For the construction of this analysis, the shap library was used, initially maintained by the author of the article that originated the method, and now by a vast community.
First, let’s calculate the SHAP values following the package tutorials:
# Library
import shap
# SHAP Calculation - Defining explainer with desired characteristics
explainer = shap.TreeExplainer(model=model)
# SHAP Calculation
shap_values_train = explainer.shap_values(x_train, y_train)
Note that I defined a TreeExplainer. This is because my model is based on a tree, so the library has a specific explainer for this family of models. In addition, up to this point, what we did was:
• Define an explainer with the desired parameters (there are a variety of parameters for TreeExplainer, I recommend checking the options in the library).
• Calculate the SHAP values for the training set.
With the set of SHAP values already defined for our training set, we can evaluate how each value of each variable influenced the result achieved by the predictive model. In our case, we will be
evaluating the results of the models in terms of probability, i.e., the X percentage that the model presented to say whether the correct class is 0 (no diabetes) or 1 (has diabetes).
It is worth noting that this may vary from model to model: If you use an XGBoost model, probably your default result will not be in terms of probability as it is for the random forest of the sklearn
To make the value in terms of probability, you can define it through the TreeExplainer, using parameters.
But the burning question is: How can I interpret SHAP values? To do this, let’s calculate the prediction probability result for the training set for any sample that predicted a positive value:
# Prediction probability of the training set
y_pred_train_proba = model.predict_proba(x_train)
# Let's now select a result that predicted as positive
print('Probability of the model predicting negative -', 100*y_pred_train_proba[3][0].round(2), '%.')
print('Probability of the model predicting positive -', 100*y_pred_train_proba[3][1].round(2), '%.')
The above code generated the probability given by the model for the two classes. Let’s now visualize the SHAP values for that sample according to the possible classes:
# SHAP values for this sample in the positive class
array([-0.01811709, 0.0807582 , 0.01562981, 0.10591462, 0.11167778, 0.09126282, 0.05179034, -0.10822825])
# SHAP values for this sample in the negative class
array([ 0.01811709, -0.0807582 , -0.01562981, -0.10591462, -0.11167778, -0.09126282, -0.05179034, 0.10822825])
Simplified formula for SHAP, where i refers to the category that those values represent (in our case, category 0 or 1).
Let’s check this in code:
# Sum of SHAP values for the negative class
print('Sum of SHAP values for the negative class in this sample:', 100*y_pred_train_proba[3][0].round(2) - 100*expected_value[0].round(2))
# Sum of SHAP values for the positive class
print('Sum of SHAP values for the positive class in this sample:', 100*y_pred_train_proba[3][1].round(2) - 100*expected_value[1].round(2))
"Sum of SHAP values for the negative class in this sample: -33.0
Sum of SHAP values for the positive class in this sample: 33.0"
And as a lesson from home office, here’s the following question: The sum of SHAP values for a class x added to the base value of that class will exactly give the probability value of the model found
at the beginning of this section!
Note that the SHAP values match the result presented earlier. But what do the individual SHAP values represent? For this, let’s use more code, using the positive class as a reference:
for col, vShap in zip(x_train.columns, shap_values_train[1][3]):
print('###################', col)
print('SHAP Value associated:', 100*vShap.round(2))
################### Pregnancies
SHAP Value associated: -2.0
################### Glucose
SHAP Value associated: 8.0
################### BloodPressure
SHAP Value associated: 2.0
################### SkinThickness
SHAP Value associated: 11.0
################### Insulin
SHAP Value associated: 11.0
################### BMI
SHAP Value associated: 9.0
################### DiabetesPedigreeFunction
SHAP Value associated: 5.0
################### Age
SHAP Value associated: -11.0
Here we evaluate the SHAP values for the positive class for sample 3. Positive SHAP values like Glucose, BloodPressure, SkinThickness, BMI, and DiabetesPedigreeFunction influenced the model in
predicting the positive class correctly. In other words, positive values imply a tendency towards the reference category.
On the other hand, negative values like Age and Pregnancies aim to indicate that the true class is negative (the opposite). In this example, if both were also positive, our model would result in a
100% prediction for the positive class. However, since that did not happen, they represent the 17% that goes against the choice of the positive class.
In summary, you can think of SHAP as contributions to the model’s decision between classes:
• In this case, the sum of SHAP values cannot exceed 50%.
• Positive values considering a reference class indicate favorability towards that class in prediction.
• Negative values indicate that the correct class is not the reference one but another class.
Additionally, we can quantify the contribution of each variable to the final response of that model in percentage terms by dividing by the maximum possible contribution, in this case, 50%:
for col, vShap in zip(x_train.columns, shap_values_train[1][3]):
print('###################', col)
print('SHAP Value associated:', 100*(100*vShap.round(2)/50).round(2),'%')
################### Pregnancies
SHAP Value associated: -4.0 %
################### Glucose
SHAP Value associated: 16.0 %
################### BloodPressure
SHAP Value associated: 4.0 %
################### SkinThickness
SHAP Value associated: 22.0 %
################### Insulin
SHAP Value associated: 22.0 %
################### BMI
SHAP Value associated: 18.0 %
################### DiabetesPedigreeFunction
SHAP Value associated: 10.0 %
################### Age
SHAP Value associated: -22.0 %
Here, we can see that Insulin, SkinThickness, and BMI together had an influence of 62%. We can also notice that the variable Age can nullify the impact of SkinThickness or Insulin in this sample.
Now that we’ve seen many numbers, let’s move on to the visualizations. In my perception, one of the reasons why SHAP has been so widely adopted is the quality of its visualizations, which, in my
opinion, surpass those of LIME.
Let’s make an overall assessment of the training set regarding our model’s prediction to understand what’s happening among all these trees:
# Graph 1 - Variable Contributions
shap.summary_plot(shap_values_train[1], x_train, plot_type="dot", plot_size=(20,15));
Graph 1: Summary Plot for SHAP Values.
Before assessing what this graph is telling us about our problem, we need to understand each feature present in it:
• The Y-axis represents the variables of our model in order of importance (SHAP orders this by default, but you can choose another order through parameters).
• The X-axis represents the SHAP values. As our reference is the positive category, positive values indicate support for the reference category (contributes to the model predicting the positive
category in the end), and negative values indicate support for the opposite category (in this case of binary classification, it would be the negative class).
• Each point on the graph represents a sample. Each variable has 800 points distributed horizontally (since we have 800 samples, each sample has a value for that variable). Note that these point
clouds expand vertically at some point. This occurs due to the density of values of that variable in relation to the SHAP values.
• Finally, the colors represent the increase/decrease of the variable’s value. Deeper red tones are higher values, and bluish tones are lower values.
In general, we will look for variables that:
• Have a clear color division, i.e., red and blue in opposite places. This information shows that they are good predictors because only by changing their value can the model more easily assess
their contribution to a class.
• Associated with this, the larger the range of SHAP values, the better that variable will be for the model. Let’s consider Glucose, which in some situations presents SHAP values around 0.3,
meaning a 30% contribution to the model’s result (because the maximum any variable can reach is 50%).
The variables Glucose and Insulin exhibit these two mentioned characteristics. Now, note the variable BloodPressure: Overall, it is a confusing variable as its SHAP values are around 0 (weak
contributions) and with a clear mix of colors. Moreover, you cannot see a trend of increase/decrease of this variable in the final response. It is also worth noting the variable Pregnancies, which
does not have as large a range as Glucose but shows a clear color division.
Through this graph, you can get an overview of how your model arrives at its conclusions from the training set and variables. The following graph shows an average contribution from the previous plot:
Graph 2 - Importance Contribution of Variables
shap.summary_plot(shap_values_train[1], x_train, plot_type="bar", plot_size=(20,15));
Graph 2: Variable Importance Plot based on SHAP Values.
Essentially, as the title of the X-axis suggests, each bar represents the mean absolute SHAP values. Thus, we evaluate the average contribution of the variables to the model’s responses. Considering
Glucose, we see that its average contribution revolves around 12% for the positive category.
This graph can be created in relation to any of the categories (I chose the positive one) or even all of them. It serves as an excellent graph to replace the first one in explanations to managers or
individuals more connected to the business area due to its simplicity.
Interpretation of Prediction for the Sample
In addition to general visualizations, SHAP provides more individual analyses per sample. Graphs like these are interesting to present specific results. For example, suppose you are working on a
customer churn problem, and you want to show how your model understands the departure of the company’s largest customer.
Through the graphs presented here, you can effectively demonstrate in a presentation what happened through Machine Learning and discuss that specific case. The first graph is the Waterfall Plot built
in relation to the positive category for the sample 3 we studied earlier.
# Graph 3 - Impact of variables on a specific prediction of the model in Waterfall Plot version
shap.plots._waterfall.waterfall_legacy(expected_value=expected_value[1], shap_values=shap_values_train[1][3].reshape(-1), feature_names=x_train.columns, show=True)
Graph 3: Contribution of Variables to the Prediction of a Sample.
In this graph, you can see that your prediction starts at the bottom and rises to the probability result.
Each variable contributes positively (model predicting the positive category) and negatively (model predicting another class). In this example, we see, for instance, that the contribution of
SkinThickness is offset by the contribution of Age.
Also, in this graph, the X-axis represents the SHAP values, and the arrow values indicate the contributions of these variables.
In the next graph, we have a new version of this visualization:
# Graph 4 - Impact of variables on a specific prediction of the model in Line Plot version
shap.decision_plot(base_value=expected_value[1], shap_values=shap_values_train[1][3], features=x_train.iloc[3,:], highlight=0)
Graph 4: Contribution of Variables to the Prediction of a Sample through “Path”.
This graph is equivalent to the previous one. As our reference category is positive, the model’s result follows towards more reddish tones (on the right), indicating a prediction for the positive
class, and towards the left, a prediction for the negative class. In this graph, values close to the arrow indicate the values of the variables (for the sample) and not the SHAP values.
SHAP emerges as a tool capable of explaining, in a graphical and intuitive way, how artificial intelligence models arrive at their results. Through the interpretation of the graphs, it is possible to
understand the decision-making in Machine Learning in a simplified manner, allowing for explanations to be presented and knowledge to be conveyed to people who do not necessarily work in this area.
Throughout this text, we were able to assess the key concepts about SHAP values, as well as their visualizations. From SHAP values, we understand how the values of each variable influenced the
model’s outcome. In this case, we evaluated the results in terms of probability. Analyzing the visualizations, it was possible to perceive that SHAP allows us to interpret specific and individual
results, as well as understand what the scheme expresses about the problem.
Despite the robust mathematics, understanding this methodology is simpler than it seems. The SHAP technology does not stop here! There are many things that can be done with this technique, and that’s
why I strongly recommend:
Do you want to discuss other applications of SHAP? Do you want to implement data science and make decision-making more accurate in your business? Get in touch with us! Let’s schedule a chat to
discuss how technology can help your company!
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A computer is worth $2000 when it is new. After each year it is worth half what it was the previous year. What will its worth be after 5 years? | Socratic
A computer is worth $2000 when it is new. After each year it is worth half what it was the previous year. What will its worth be after 5 years?
1 Answer
It's a simple geometric progression of ratio $r = \frac{1}{2}$
After the first year, the value will be given by:
#V_1 =$2000*(1/2)=$1000#
In the second:
In the ${n}^{t h}$ year:
So, after 5 years:
Impact of this question
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数学代写|MATH1111 Calculus 👍 👍 - 👍代考代写：100%准时可靠 您的作业代写专家数学代写|MATH1111 Calculus
MY-ASSIGNMENTEXPERT™可以为您提供sydney MATH1111 Calculus微积分的代写代考和辅导服务！
数学代写|MATH1111 Calculus
This unit is an introduction to the calculus of one variable. Topics covered include elementary functions, differentiation, basic integration techniques and coordinate geometry in three dimensions.
Applications in science and engineering are emphasised.
At the completion of this unit, you should be able to:
• LO1. Apply mathematical logic and rigour to solving problems, and express mathematical ideas coherently in written and oral form;
• LO2. Demonstrate fluency in manipulating real numbers, their symbolic representations, operations, and solve associated algebraic equations and inequalities;
• LO3. Develop fluency with lines, coordinate geometry in two dimensions, the notion of a function, its natural domain, range and graph;
• LO4. Become conversant with elementary functions, including trigonometric, exponential, logarithmic and hyperbolic functions and be able to apply them to real phenomena and to yield solutions of
associated equations;
• LO5. Perform operations on functions and be able to invert functions where appropriate;
• LO6. Understand the definitions of a derivative, definite and indefinite integral and be able to apply the definitions to elementary functions;
• LO7. Develop fluency in rules of differentiation, such as the product, quotient and chain rules, and use them to differentiate complicated functions;
• LO8. Understand and apply the Fundamental Theorem of Calculus; and develop fluency in techniques of integration, such as integration by substitution, the method of partial fractions and
integration by parts;
• LO9. Develop some fluency with coordinate geometry in three dimensions, planes, surfaces, ellipsoids, paraboloids, level curves and qualitative features such as peaks, troughs and saddle points.
MATH1111 Calculus HELP（EXAM HELP， ONLINE TUTOR）
问题 1.
Show that $f(x)=x^3-4 x^2+1$ has exactly two roots in $(-1,1)$ and use Newton’s method with $x_1= \pm 1$ to approximate these roots within two decimal places.
To prove existence using Bolzano’s theorem, we note that $f$ is continuous with
f(-1)=-1-4+1<0, \quad f(0)=1>0, \quad f(1)=1-4+1<0 .
In view of Bolzano’s theorem, $f$ must then have a root in $(-1,0)$ and another root in $(0,1)$, so it has two roots in $(-1,1)$. Suppose that it has three roots in $(-1,1)$. Then $f^{\prime}$ must
have two roots in this interval by Rolle’s theorem. On the other hand,
f^{\prime}(x)=3 x^2-8 x=x(3 x-8)
has only one root in $(-1,1)$. This implies that $f$ can only have two roots in $(-1,1)$.
To use Newton’s method to approximate the roots, we repeatedly apply the formula
x_{n+1}=x_n-\frac{f\left(x_n\right)}{f^{\prime}\left(x_n\right)}=x_n-\frac{x_n^3-4 x_n^2+1}{3 x_n^2-8 x_n} .
Starting with the initial guess $x_1=-1$, one obtains the approximations
x_2=-0.6364, \quad x_3=-0.4972, \quad x_4=-0.4735, \quad x_5=-0.4728 .
Starting with the initial guess $x_1=1$, one obtains the approximations
x_2=0.6, \quad x_3=0.5398, \quad x_4=0.5374, \quad x_5=0.5374 .
This suggests that the two roots are roughly -0.47 and 0.53 within two decimal places.
问题 2.
A rectangle is inscribed in an equilateral triangle of side length $a>0$ with one of its sides along the base of the triangle. How large can the area of the rectangle be?
Let $x, y$ be the two sides of the rectangle and assume that $x$ lies along the base of the triangle. Then one can relate the two sides $x, y$ by noting that
\tan 60^{\circ}=\frac{y}{(a-x) / 2} \quad \Longrightarrow \quad \sqrt{3}=\frac{2 y}{a-x} \quad \Longrightarrow \quad y=\frac{\sqrt{3}}{2}(a-x) .
We need to maximise the area $A$ of the rectangle and this is given by
A(x)=x y=\frac{\sqrt{3}}{2} x(a-x)=\frac{\sqrt{3}}{2}\left(a x-x^2\right), \quad 0 \leq x \leq a .
Since $A^{\prime}(x)=\frac{\sqrt{3}}{2}(a-2 x)$, the only points at which the maximum value may occur are the points $x=0, x=a$ and $x=\frac{a}{2}$. Since $A(0)=A(a)=0$, the maximum is $A\left(\frac
{a}{2}\right)=\frac{a^2 \sqrt{3}}{8}$.
问题 3.
A ladder $5 \mathrm{~m}$ long is resting against a vertical wall. The bottom of the ladder slides away from the wall at the rate of $0.2 \mathrm{~m} / \mathrm{s}$. How fast is the angle $\theta$
between the ladder and the wall changing when the bottom of the ladder lies $3 \mathrm{~m}$ away from the wall?
Let $x$ be the horizontal distance between the base of the ladder and the wall, and let $y$ be the vertical distance between the top of the ladder and the floor. We must then have
x(t)^2+y(t)^2=5^2 \quad \Longrightarrow \quad 2 x(t) x^{\prime}(t)+2 y(t) y^{\prime}(t)=0 .
At the given moment, $x^{\prime}(t)=0.2=1 / 5$ and also $x(t)=3$, so it easily follows that
y^{\prime}(t)=-\frac{x(t) x^{\prime}(t)}{y(t)}=-\frac{x(t) x^{\prime}(t)}{\sqrt{5^2-x(t)^2}}=-\frac{3 / 5}{\sqrt{5^2-3^2}}=-\frac{3}{20} .
We now need to determine $\theta^{\prime}$. Using the chain rule along with the quotient rule, we get
\tan \theta=\frac{x}{y} \quad \Longrightarrow \quad \sec ^2 \theta \cdot \theta^{\prime}=\frac{x^{\prime} y-y^{\prime} x}{y^2} \quad \Longrightarrow \quad \theta^{\prime}=\frac{x^{\prime} y-y^{\
prime} x}{y^2} \cdot \cos ^2 \theta .
Since $\cos \theta=y / 5$ and the other variables are already known, we may conclude that
\theta^{\prime}=\frac{x^{\prime} y-y^{\prime} x}{y^2} \cdot \cos ^2 \theta=\frac{4(1 / 5)-3(-3 / 20)}{4^2} \cdot\left(\frac{4}{5}\right)^2=\frac{1}{20} .
数学代写|MATH1111 Calculus
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Chad Fulton
My research focuses on rational inattention and applied time series econometrics.
Published papers
Choosing what to pay attention to (2022)
Theoretical Economics, 2022
This paper studies static rational inattention problems with multiple actions and multiple shocks. We solve for the optimal signals chosen by agents and provide tools to interpret information
processing. By relaxing restrictive assumptions previously used to gain tractability, we allow agents more latitude to choose what to pay attention to. Our applications examine the pricing problem of
a monopolist who sells in multiple markets and the portfolio problem of an investor who can invest in multiple assets. The more general models that our methods allow us to solve yield new results. We
show conditions under which the multimarket monopolist would optimally choose a uniform pricing strategy, and we show how optimal information processing by rationally inattentive investors can be
interpreted as learning about the Sharpe ratio of a diversified portfolio.
Forecasting US inflation in real time (2022)
Econometrics, 2022, with Kirstin Hubrich
We analyze real-time forecasts of US inflation over 1999Q3–2019Q4 and subsamples, investigating whether and how forecast accuracy and robustness can be improved with additional information such as
expert judgment, additional macroeconomic variables, and forecast combination. The forecasts include those from the Federal Reserve Board’s Tealbook, the Survey of Professional Forecasters, dynamic
models, and combinations thereof. While simple models remain hard to beat, additional information does improve forecasts, especially after 2009. Notably, forecast combination improves forecast
accuracy over simpler models and robustifies against bad forecasts; aggregating forecasts of inflation’s components can improve performance compared to forecasting the aggregate directly; and
judgmental forecasts, which may incorporate larger and more timely datasets in conjunction with model-based forecasts, improve forecasts at short horizons.
Bayesian Estimation and Forecasting of Time Series in statsmodels (2022)
Proceedings of the 21st Python in Science Conference, 2022
Statsmodels, a Python library for statistical and econometric analysis, has traditionally focused on frequentist inference, including in its models for time series data. This paper introduces the
powerful features for Bayesian inference of time series models that exist in statsmodels, with applications to model fitting, forecasting, time series decomposition, data simulation, and impulse
response functions.
SciPy 1.0: fundamental algorithms for scientific computing in Python (2020)
Nature methods, 2020, with Pauli Virtanen, Ralf Gommers, and 108 others
SciPy is an open-source scientific computing library for the Python programming language. Since its initial release in 2001, SciPy has become a de facto standard for leveraging scientific algorithms
in Python, with over 600 unique code contributors, thousands of dependent packages, over 100,000 dependent repositories and millions of downloads per year. In this work, we provide an overview of the
capabilities and development practices of SciPy 1.0 and highlight some recent technical developments.
Working papers
Mechanics of static quadratic Gaussian rational inattention tracking problems (2018)
This paper presents a general framework for constructing and solving the multivariate static linear quadratic Gaussian (LQG) rational inattention tracking problem. We interpret the nature of the
solution and the implied action of the agent, and we construct representations that formalize how the agent processes data. We apply our approach to a price-setting problem and a portfolio choice
problem - two popular rational inattention models found in the literature for which simplifying assumptions have thus far been required to produce a tractable model. In contrast to prior results,
which have been limited to cases that restrict the number of underlying shocks or their correlation structure, we present general solutions. In each case, we show that imposing such restrictions
impacts the form and interpretation of solutions and implies suboptimal decision-making by agents.
Mechanics of linear quadratic Gaussian rational inattention tracking problems (2017)
Note: This is an previous version of the working paper Mechanics of static quadratic Gaussian rational inattention tracking problems, although it contains some sections not included there. In
particular, it expands on the dynamic case and provides more detail on the equilibrium solution to the rational inattetion price-setting problem.
This paper presents a general framework for constructing and solving the multivariate static linear quadratic Gaussian (LQG) rational inattention tracking problem. We interpret the nature of the
solution and the implied action of the agent, and we construct representations that formalize how the agent processes data. We apply this infrastructure to the rational inattention price-setting
problem, confirming the result that a conditional response to economics shocks is possible, but casting doubt on a common assumption made in the literature. We show that multiple equilibria and a
social cost of increased attention can arise in these models. We consider the extension to the dynamic problem and provide an approximate solution method that achieves low approximation error for
many applications found in the LQG rational inattention literature.
Estimating time series models by state space methods in Python: Statsmodels (2015)
This paper describes an object oriented approach to the estimation of time series models using state space methods and presents an implementation in the Python programming language. This approach at
once allows for fast computation, a variety of out-of-the-box features, and easy extensibility. We show how to construct a custom state space model, retrieve filtered and smoothed estimates of the
unobserved state, and perform parameter estimation using classical and Bayesian methods. The mapping from theory to implementation is presented explicitly and is illustrated at each step by the
development of three example models: an ARMA(1,1) model, the local level model, and a simple real business cycle macroeconomic model. Finally, four fully implemented time series models are presented:
SARIMAX, VARMAX, unobserved components, and dynamic factor models. These models can immediately be applied by users.
Other research
Index of Common Inflation Expectations (2020)
with Hie Joo Ahn
This note develops a new index of common inflation expectations that summarizes the comovement of various inflation expectation indicators based on a dynamic factor model. This index suggests that
inflation expectations were relatively stable between 1999 and 2012, and then experienced a downward shift that persisted, despite some fluctuations, at least through the beginning of the COVID-19
pandemic in early 2020. Since then it has successfully captured pandemic-driven concerns, first falling on fears of a prolonged recession, and then rising as the US economy has recovered and anxiety
about inflation has grown. | {"url":"http://www.chadfulton.com/research.html","timestamp":"2024-11-13T16:01:51Z","content_type":"text/html","content_length":"20161","record_id":"<urn:uuid:88f23fcb-e8db-4ffc-9ea3-14708d3cc1f5>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00720.warc.gz"} |
Why lower win rates are better
In the world of the ad exchange, your "win rate" is the ratio of bids to wins. This seems like an intuitive metric: if you want to win more, bid higher. Even with all of the crazy multi-tier
pseudo-auctions we see these days, this is still generally the case.
I recently heard a client say "DSP XXX has higher win rates than DSP YYY... so therefore, DSP XXX is better". Counterintuitively, it's probably the opposite.
Let's assume a very simple scenario. You want to spend $100 on a simple audience-targeted campaign across some defined set of inventory, and you're flexible on the CPM that you pay. For this example,
let's assume that the audience is fairly large, so there's plenty of inventory available. You log into DSP XXX and DSP YYY and set up the campaign, run it for a day, and look at the results.
DSPSpendCPMWin Rate XXX$100$2.503.1% YYY$100$1.100.6%
DSP XXX had a higher win rate at a higher CPM. But wait - I thought a higher bid would mean winning more impressions?
Pacing and Win Rate
Inside every DSP is a pacing algorithm that decides how to spend a budget over the course of some time interval. The basic idea is that if you want to spend $100 a day, you want to spend $4.16 per
hour or about $0.07 per minute. For a typical audience-targeted campaign, a bidder sees around 10,000 eligible impressions a minute. If you buy every impression at a $2.50 CPM, you'll spend $25 per
minute. So bidders bid on only a fraction of the impressions - a fraction determined by the win rate. If your win rate is 3.1%, you need to bid 32 times for every impression you win; to win 28
impressions you need to bid 900 times a minute.
So if you need to bid 900 times a minute to spend your budget at a $2.50 CPM, what does the DSP do for the other 9100 bid requests? It ignores them, or, "goes to sleep".
Let's look at DSP YYY and see what it's doing. With the exact same inventory, this DSP bids $1.10 and wins only 0.6% of the impressions. Let's do the math. At a lower CPM, you need to win 63
impressions a minute to hit our budget. At a win rate of 0.6%, you need to bid 150 times for every impression you win. This means you need to bid on around 9500 impressions to hit our budget -
effectively, all of them. There's no need to go to sleep.
Gross Win Rate
Let's put these results in a new table to see them in a different light.
DSPCPMImpressions PurchasedImpressions TargetableGross Win Rate XXX$2.5040,0009,000,0000.4% YYY$1.1090,9009,000,0001.1%
If you calculate the win rate based on the total targetable volume instead of the impressions where DSP decided to submit a bid, the story is completely different. DSP YYY actually won more
impressions on the same traffic.
1. Require DSPs to provide Gross Win Rate instead of Net Win Rate in their reporting and analytics (AppNexus just added this in the "Bid Analyzer" tool)
2. Don't adjust bidding strategy based on Net Win Rate; you may be lowering your actual win rate by raising your bids! (AppNexus v8 introduces automatic gross win rate optimization to find the
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Starlight U.S. Multi-Family (No. 5) Core Fund completes successful acquisition of the Starlight U.S. Multi-Family Core Funds and Campar Capital Corporation by way of a plan of arrangement - REIT REPORT
TORONTO, Oct. 17, 2016 /CNW/ – Starlight U.S. Multi-Family Core Fund (TSX.V: UMF.A, UMF.U) (“Fund1“), Starlight U.S. Multi-Family (No. 2) Core Fund (TSX.V: SUD.A, SUD. U) (“Fund2“), Starlight U.S.
Multi-Family (No. 3) Core Fund (TSX.V: SUS.A, SUS.U) (“Fund3“) and Starlight U.S. Multi-Family (No. 4) Core Fund (TSX.V: SUF.A, SUF.U) (“Fund4” and collectively, the “Existing Starlight Funds“),
Campar Capital Corporation (TSX.V: CHK.P) (“Campar“) and Starlight U.S. Multi-Family (No. 5) Core Fund (TSX.V: SUA.A, SUA.U) (“Fund5“) announced today that they have completed their previously
announced plan of arrangement (the “Arrangement“) pursuant to which, among other things, Fund5 acquired all of the outstanding units of the Existing Starlight Funds and all of the outstanding common
shares of Campar. For details on the effective exchange ratios for a particular class of units for each Existing Starlight Fund, see the Exchange Values and Total Distribution Increase table at the
end of this news release.
In connection with the closing of the Arrangement, former unitholders of each of the Existing Starlight Funds will receive a stub period distribution for the period from October 1, 2016 to October
17, 2016. The distributions will be paid on or about October 28, 2016 to unitholders of record of each Existing Starlight Fund on October 14, 2016, the effective date of the Arrangement.
The distribution amounts for Fund1 will be as follows:
i. C$0.03199 per class A unit, representing approximately C$0.70 per unit on an annualized basis;
ii. C$0.03516 per class C unit, representing approximately C$0.77 per unit on an annualized basis;
iii. C$0.03437 per class F unit, representing approximately C$0.75 per unit on an annualized basis;
iv. C$0.03331 per class I unit, representing approximately C$0.73 per unit on an annualized basis; and
v. US$0.03199 per class U unit, representing approximately US$0.70 per unit on an annualized basis.
The distribution amounts for Fund2 will be as follows:
i. C$0.03199 per class A unit, representing approximately C$0.70 per unit on an annualized basis;
ii. C$0.03199 per class C unit, representing approximately C$0.70 per unit on an annualized basis;
iii. C$0.03199 per class D unit, representing approximately C$0.70 per unit on an annualized basis;
iv. C$0.03199 per class F unit, representing approximately C$0.70 per unit on an annualized basis; and
v. US$0.03199 per class U unit, representing approximately US$0.70 per unit on an annualized basis.
The distribution amounts for Fund3 will be as follows:
i. C$0.03199 per class A unit, representing approximately C$0.70 per unit on an annualized basis;
ii. C$0.03199 per class C unit, representing approximately C$0.70 per unit on an annualized basis;
iii. C$0.03199 per class D unit, representing approximately C$0.70 per unit on an annualized basis;
iv. C$0.03199 per class F unit, representing approximately C$0.70 per unit on an annualized basis; and
v. US$0.03199 per class U unit, representing approximately US$0.70 per unit on an annualized basis.
The distribution amounts for Fund4 will be as follows:
i. C$0.03199 per class A unit, representing approximately C$0.70 per unit on an annualized basis;
ii. C$0.03199 per class C unit, representing approximately C$0.70 per unit on an annualized basis;
iii. C$0.03199 per class D unit, representing approximately C$0.70 per unit on an annualized basis;
iv. US$0.03199 per class E unit, representing approximately US$0.70 per unit on an annualized basis;
v. C$0.03199 per class F unit, representing approximately C$0.70 per unit on an annualized basis;
vi. C$0.02285 per class H unit, representing approximately C$0.70 per unit on an annualized basis less a portion of the cost of the derivative instrument purchased by Fund4 to provide the holders of
class H units with some protection against any weakening of the U.S. dollar as compared to the Canadian dollar on termination and liquidation of Fund4; and
vii. US$0.03199 per class U unit, representing approximately US$0.70 per unit on an annualized basis.
The class A units and class U units of Fund5 were listed on the TSX Venture Exchange on October 14, 2016 and it is expected that these units will begin trading on the TSX Venture Exchange on October
18, 2016. Each of the Existing Starlight Funds and Campar is expected to be delisted from the TSX Venture Exchange effective October 17, 2016 and each of the Existing Starlight Funds and Campar
intend to apply to cease to be reporting issuers under applicable Canadian securities laws.
Forward-looking Statements
Certain statements made in this news release are forward-looking statements, including, but not limited to, the expected stub period distribution date, the expected trading date of the listed Fund5
units, the expected delisting of the units of each Existing Starlight Fund and the common shares of Campar and other statements that are not historical facts. Forward-looking statements, by their
very nature, are subject to inherent risks and uncertainties and are based on several assumptions, both general and specific, which give rise to the possibility that actual results or events could
differ materially from our expectations expressed in or implied by such forward-looking statements. As a result, readers are cautioned against placing undue reliance on any of these forward-looking
These forward looking statements are made as of the date of this news release and, except as expressly required by law, the Existing Starlight Funds and Campar undertake no obligation to update or
revise publicly any forward-looking statements, whether as a result of new information, future events or otherwise, after the date on which the statements are made or to reflect the occurrence of
unanticipated events.
About Fund5
Fund5 is a limited partnership formed under the Limited Partnerships Act (Ontario) for the primary purpose of indirectly acquiring, owning and operating a portfolio of diversified income producing
rental properties in the U.S. multi-family real estate market.
Neither the TSX Venture Exchange nor its Regulation Service Provider (as that term is defined in policies of the TSX Venture Exchange) accepts responsibility for the adequacy or accuracy of this
Exchange Values and Total Distribution Increase
Initial Value of Initial Implied Fund5 Increase in Yield on
Fund Investment Unit at Exchange Annual Pro Forma Annual Initial
per Unit Exchange Ratio^^1 Distribution Annual Distribution Investment
Starlight U.S. Multi-Family Core Fund
Class A – CDN$ $10.00 $24.19 2.4187x $0.70 $1.57 124.3% 15.7%
Class C – CDN$ $10.00 $25.52 2.5515x $0.77 $1.66 115.6% 16.6%
Class F – CDN$ $10.00 $24.94 2.4941x $0.75 $1.62 116.0% 16.2%
Class I – CDN$ $10.00 $24.18 2.4175x $0.73 $1.57 115.1% 15.7%
Class U – US$ $10.00 $18.32 1.8324x $0.70 $1.19 70.0% 11.9%
Starlight U.S. Multi-Family (No. 2) Core Fund
Class A – CDN$ $10.00 $24.62 2.4615x $0.70 $1.60 128.6% 16.0%
Class C – CDN$ $10.00 $26.19 2.6191x $0.70 $1.70 142.9% 17.0%
Class F – CDN$ $10.00 $25.56 2.5558x $0.70 $1.66 137.1% 16.6%
Class D – CDN$ $10.00 $24.70 2.4697x $0.70 $1.61 130.0% 16.1%
Class U – US$ $10.00 $19.08 1.9081x $0.70 $1.24 77.1% 12.4%
Starlight U.S. Multi-Family (No. 3) Core Fund
Class A – CDN$ $10.00 $17.80 1.7804x $0.70 $1.16 65.7% 11.6%
Class C – CDN$ $10.00 $19.01 1.9010x $0.70 $1.24 77.1% 12.4%
Class F – CDN$ $10.00 $18.55 1.8545x $0.70 $1.21 72.9% 12.1%
Class D – CDN$ $10.00 $17.92 1.7924x $0.70 $1.17 67.1% 11.7%
Class U – US$ $10.00 $14.07 1.4074x $0.70 $0.92 31.4% 9.2%
Starlight U.S. Multi-Family (No. 4) Core Fund
Class A – CDN$ $10.00 $13.53 1.3532x $0.70 $0.88 25.7% 8.8%
Class C – CDN$ $10.00 $14.40 1.4404x $0.70 $0.94 34.3% 9.4%
Class D – CDN$ $10.00 $13.59 1.3591x $0.70 $0.88 25.7% 8.8%
Class E – US$ $10.00 $12.87 1.2873x $0.70 $0.84 20.0% 8.4%
Class F – CDN$ $10.00 $13.79 1.3788x $0.70 $0.90 28.6% 9.0%
Class H – CDN$ $10.00 $13.33 1.3334x $0.50 $0.47 -6.0% 4.7%
Class U – US$ $10.00 $12.80 1.2801x $0.70 $0.83 18.6% 8.3%
^1 The exchange ratios for a particular class of Existing Units for a particular Existing Starlight Fund is determined to be the quotient equal to: (i) the net equity value (which is based on the
aggregate appraised value (as determined by an independent appraiser) of the properties owned by the applicable Existing Starlight Fund less the applicable “carried interest” of each Existing
Starlight Fund) of such Existing Starlight Fund allocable to such class, calculated on the basis of the corresponding “proportionate class interest” definition set out in the applicable Existing
Starlight Fund limited partnership agreement (provided that in the case of units other than class E units of Fund4 and class U units of any Existing Starlight Fund, the value is converted into
Canadian dollars using the effective exchange rate) divided by the total outstanding units of such class, divided by (ii) the issue price of the corresponding class of units of Fund5 (being US$10.00
in the case of Fund5 class E units and Fund5 class U units and CDN$10.00 in the case of all other classes). The exchange ratio for Campar is equal to (i) Campar’s equity value (which is based on 80%
of the appraised value of the San Antonio, Texas property to be contributed by Campar) divided by the number of outstanding shares of Campar, divided by (ii) CDN$10.00.
SOURCE Starlight U.S. Multi-Family Core Fund | {"url":"https://reitreport.ca/starlight-u-s-multi-family-no-5-core-fund-completes-successful-acquisition-of-the-starlight-u-s-multi-family-core-funds-and-campar-capital-corporation-by-way-of-a-plan-of-arrangement/","timestamp":"2024-11-08T12:33:27Z","content_type":"text/html","content_length":"79972","record_id":"<urn:uuid:d2cca490-1b0e-4a4e-84d7-1b2270194a83>","cc-path":"CC-MAIN-2024-46/segments/1730477028059.90/warc/CC-MAIN-20241108101914-20241108131914-00050.warc.gz"} |
A basketball starts from rest and rolls without slipping down a
hill. The radius of the...
A basketball starts from rest and rolls without slipping down a hill. The radius of the...
A basketball starts from rest and rolls without slipping down a hill. The radius of the basketball is 0.23 m, and its 0.625 kg mass is evenly distributed in its thin shell. The hill is 50 m long and
makes an angle of 25° with the horizontal. How fast is it going at the bottom of the hill?
Group of answer choices
10.7 m/s
12.3 m/s
15.8 m/s
14.4 m/s
17.2 m/s | {"url":"https://justaaa.com/physics/86986-a-basketball-starts-from-rest-and-rolls-without","timestamp":"2024-11-02T12:33:18Z","content_type":"text/html","content_length":"38404","record_id":"<urn:uuid:0da4b724-af8d-447b-815c-cf6bc99d14aa>","cc-path":"CC-MAIN-2024-46/segments/1730477027710.33/warc/CC-MAIN-20241102102832-20241102132832-00234.warc.gz"} |
Burst Edit Distance
In this paper we define two types of burst edit errors that occur in Text Editing scenarios when the communication speed is unstable within a wireless keyboard usage: (1) A Burst of Errors (BE)
involves a sequence of erroneous identical symbols and allows a single edit operation applied to a sequence of identical symbols; (2) A Burst of Operations (BO) involves a sequence of erroneous
symbols that are not necessarily identical and allows a single edit operation applied to a sequence of symbols. In both burst types, every burst operation has a penalty, which is a cost function F
(k), where k is the burst length. The burst edit distance of two strings S and T is: (1) The minimum cost of a sequence of BE operations that transforms S into T in the bursts of errors variant
(EDBE); (2) The minimum cost of a sequence of BO operations that transforms S into T in the bursts of operations variant (EDBO). We describe solutions to both problems for general natural penalty
functions families. A conditional lower bound for the EDBE problem is also given. The K-bounded versions of the problems are considered as well.
Publication series
Name Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume 14899 LNCS
ISSN (Print) 0302-9743
ISSN (Electronic) 1611-3349
Conference 31st International Symposium on String Processing and Information Retrieval, SPIRE 2024
Country/Territory Mexico
City Puerto Vallarta
Period 23/09/24 → 25/09/24
Bibliographical note
Publisher Copyright:
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2025.
• Burst Errors
• Edit Distance
• String Similarity
Dive into the research topics of 'Burst Edit Distance'. Together they form a unique fingerprint. | {"url":"https://cris.biu.ac.il/en/publications/burst-edit-distance","timestamp":"2024-11-13T02:16:06Z","content_type":"text/html","content_length":"57355","record_id":"<urn:uuid:7ea6b06b-42b1-4415-8120-5f9a064ae990>","cc-path":"CC-MAIN-2024-46/segments/1730477028303.91/warc/CC-MAIN-20241113004258-20241113034258-00332.warc.gz"} |
Curves and Abelian Varieties: International Conference March by Valery Alexeev, Arnaud Beauville, C. Herbert Clemens, Elham
By Valery Alexeev, Arnaud Beauville, C. Herbert Clemens, Elham Izadi
This booklet is dedicated to contemporary development within the examine of curves and abelian kinds. It discusses either classical facets of this deep and gorgeous topic in addition to vital new
advancements, tropical geometry and the idea of log schemes. as well as unique learn articles, this ebook includes 3 surveys dedicated to singularities of theta divisors, of compactified Jacobians of
singular curves, and of ""strange duality"" between moduli areas of vector bundles on algebraic types
Read Online or Download Curves and Abelian Varieties: International Conference March 30-april 2, 2007 University of Georgia Athens, Georgia PDF
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Two elements r1 and r2 of R are called associates if there is a unit u ∈ R such that r1 = ur2. 2. The prime elements of an integral domain are irreducible. Proof. Let p be a prime element of an
integral domain R. Then, for any a, b ∈ R such that p = ab we have that p|a or p|b. Assume, without loss of generality, that p|a. Then there is c ∈ R such that a = pc. Hence, is a unit and therefore
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3 Riemann-Roch Space Definition Let F/K(x) be a function field and D a divisor of F. The set is called Riemann-Roch space of D. The following lemma justifies why L (D) is called a space. 1. Let F/K
(x)be a function field. For any divisor D the Riemann-Roch space L(D) is a K-vector space. Proof. Let a, b ∈ L(D) and k ∈ K. For every place of F we have that and Thus, a + b and ka belong to L(D)
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Rated of 5 – based on votes | {"url":"http://en.magomechaya.com/index.php/epub/curves-and-abelian-varieties-international-conference-march-30-april-2-2007","timestamp":"2024-11-12T20:08:42Z","content_type":"text/html","content_length":"29553","record_id":"<urn:uuid:2992ad1c-b557-45b4-bc03-df168f49d2eb>","cc-path":"CC-MAIN-2024-46/segments/1730477028279.73/warc/CC-MAIN-20241112180608-20241112210608-00848.warc.gz"} |
What is the time complexity of generating a matrix using zeros(n). ...
Answers (2)
What is the time complexity of generating a matrix using zeros(n). Is it n^2.
4 views (last 30 days)
What is the time complexity of generating a matrix using zeros(n). Is it n^2.
I don't know the theoretical answer to this.
Practically, though, you can run this code to see that the generation time grows very slowly.
Here's some code that will generate matrices of growing size (up to the default maximum size, when it will error out).
N = 0;
STEP = 50;
KILO = 1000;
MILLI = 0.001;
while true
N = N + STEP;
x = zeros(N);
t(N/STEP) = toc;
clear x
title('Time to generate zeros(N,N)')
ylabel('Time [milliseconds]')
You'll get a different curve every time you try this, but a typical one I got looks like this:
You can see the growth. It's not easy to discern the shape, but regardless of the shape, the time to generate the largest matrix MATLAB can store (46350x46350 on my machine) is still only about 0.3
2 Comments
Walter Roberson on 30 Jan 2016
Note that MATLAB accelerator might notice that the array is not used before it is cleared and so might not bother to do the allocation. You need to time the allocation and then you need to fetch from
the memory so that MATLAB can't null out the operation. You should probably check the time after fetching from it too in case it delays allocation until then. You would compare that time to what is
required to fetch from a matrix that was fully initialized and not being trashed to determine if it is indeed taking extra time for the first allocation.
Benchmarks are tough to get right to be sure you are measuring the right thing in a JIT environment.
the cyclist on 30 Jan 2016
I did not show the code here, but I actually also did some tests where I included a line like
x(N) = 1;
in there as well. I think that forces the allocation, right? The timings were not noticeably different.
Not necessarily. Provided there is readily available memory, the first overhead is constant time. After that the question becomes how the memory gets zeroed. In some memory allocation systems, memory
is zeroed when it is added to the pool of memory, but more common would be that the memory would be zeroed when it was allocated. But zeroing memory is often delegated to special purpose hardware
instructions, some of which will work in parallel for sufficiently large chunks of memory or memory that meets certain address requirements. Zeroing memory at time of allocation is sometimes handled
by the memory management hardware, the part that maps between physical addresses and virtual addresses. Sometimes the contents of an entire block of physical memory is zeroed in constant time by the
simple method of turning off power to the chip; sometimes an entire block of physical memory is zeroed by a special strobe line that tells the block to ground itself so as to zero the contents. A
relevant term from my large computer days was "demand zero memory", which is virtual memory that (with hardware assistance) is always initialized as zero when the memory is received from the
operating system.
Thus, for the special case of initializing with 0,sometimes it is constant time, sometimes it is grows proportional to ceiling(n^2/Blocksize), sometimes it is pure n^2, sometimes it would be a mix...
The one thing that it would not be would be order(n). | {"url":"https://in.mathworks.com/matlabcentral/answers/265856-what-is-the-time-complexity-of-generating-a-matrix-using-zeros-n-is-it-n-2?s_tid=prof_contriblnk","timestamp":"2024-11-14T17:35:30Z","content_type":"text/html","content_length":"128934","record_id":"<urn:uuid:d56eacf9-561e-49f8-9479-317404816511>","cc-path":"CC-MAIN-2024-46/segments/1730477393980.94/warc/CC-MAIN-20241114162350-20241114192350-00258.warc.gz"} |
Data Visualization with Matplotlib III
Hi and welcome
This is the third post on my series of post on Data Visualization with Matplotlib, if this is your first read and you're new to matplotlib , do well to read the Part I and part II of this series of
In this post , we shall be considering different
• How to control the location of the legend on your graph
• Customization of the ticks and their orientations on the axes
• How to utilize the inbuilt custom styles of pyplot package
Have a lovely read
Placing the legend
The legend by default looks for the best possible location and sit, however we can specify a location we want using the loc argument of the legend. The possible values of the loc are can be in terms
of strings or corresponding codes. The values are
Below are illustration of these concepts
1 import matplotlib.pyplot as plt
2 import numpy as np
3 x=np.linspace(-10,10,1000)
4 sin=np.sin(x)
5 cos=np.cos(x)
6 plt.plot(x,sin,label="Sine")
7 plt.plot(x,cos,label="Cosin")
8 plt.legend()
In the plot above , the best location was used , which is the default case of the loc parameter. Coincidentally , the best location happened to be the upper left.
1 import matplotlib.pyplot as plt
2 import numpy as np
3 x=np.linspace(-10,10,1000)
4 sin=np.sin(x)
5 cos=np.cos(x)
6 plt.plot(x,sin,label="Sine")
7 plt.plot(x,cos,label="Cosine")
8 plt.legend(loc="lower right")
In the graph above, a string is used to specify the location.
1 import matplotlib.pyplot as plt
2 import numpy as np
3 x=np.linspace(-10,10,1000)
4 sin=np.sin(x)
5 cos=np.cos(x)
6 plt.plot(x,sin,label="Sine")
7 plt.plot(x,cos,label="Cosine")
8 plt.legend(loc=7)
In the plot above , an integer is used to specify the location.
Customizing the ticks
The ticks are the values that appear on the axes. We can take control of what appears by using the .xticks() and .yticks() methods.
Below is the plot the sine graph without specifying the ticks that appears
1 import matplotlib.pyplot as plt
2 import numpy as np
3 x=np.linspace(-10,10,1000)
4 sin=np.sin(x)
5 cos=np.cos(x)
6 plt.plot(x,sin,label="Sine")
7 plt.grid(True)
In the graph below, only five points appear on the x axis and three on the y axis because we specified so.
1 import matplotlib.pyplot as plt
2 import numpy as np
3 x=np.linspace(-10,10,1000)
4 sin=np.sin(x)
5 cos=np.cos(x)
6 plt.plot(x,sin,label="Sine")
7 plt.xticks([-10,-5,0,5,10])# specifies ticks on the x-axis
8 plt.yticks([-1,0,1])# specifies ticks on the y-axis
9 plt.grid(True)
The labels appearing on can also be replaced using a second list of values called the labels , as shown below
1 import matplotlib.pyplot as plt
2 import numpy as np
3 x=np.linspace(-10,10,1000)
4 sin=np.sin(x)
5 cos=np.cos(x)
6 plt.plot(x,sin,label="Sine")
7 plt.xticks([-10,-5,0,5,10],["a","b","c","d","e"])# specifies ticks on the x-axis
8 plt.yticks([-1,0,1])# specifies ticks on the y-axis
9 plt.grid(True)
The orientation of these values can also be changed , using the rotation parameter, as shown below
1 import matplotlib.pyplot as plt
2 import numpy as np
3 x=np.linspace(-10,10,1000)
4 sin=np.sin(x)
5 cos=np.cos(x)
6 plt.plot(x,sin,label="Sine")
7 plt.xticks([-10,-5,0,5,10],["a","b","c","d","e"],rotation=45)#rotates the ticsks to an angle of 45 degrees
8 plt.yticks([-1,0,1])# specifies ticks on the y-axis
9 plt.grid(True)
Using the inbuilt custom styles of pyplot
Pyplot comes with a use set of custom styling which helps to improve the appearance of our graph/plots . There are different styles, each which a name. To see the lists of available styles, we can
use the .style.available attribute , as shown below
1 print(plt.style.available)
The output is
['bmh', 'classic', 'dark_background', 'fast', 'fivethirtyeight', 'ggplot', 'grayscale',
'seaborn-bright', 'seaborn-colorblind', 'seaborn-dark-palette', 'seaborn-dark',
'seaborn-darkgrid', 'seaborn-deep', 'seaborn-muted', 'seaborn-notebook',
'seaborn-paper', 'seaborn-pastel', 'seaborn-poster', 'seaborn-talk',
'seaborn-ticks', 'seaborn-white', 'seaborn-whitegrid', 'seaborn', 'Solarize_Light2',
'tableau-colorblind10', '_classic_test']
To use any of these styles , we pass it in as the argument of the .style.use() method, as shown below
1 plt.hist([12,34,42,56,23,65,12,34,5,4,34,5,44,34,3,4,6,5,4,3,12,34,23])
Above is a plot without specifying any style
In the graphs below , three of the available styles are used
1 plt.style.use("ggplot")
2 plt.hist([12,34,42,56,23,65,12,34,5,4,34,5,44,34,3,4,6,5,4,3,12,34,23])
The result is
1 plt.style.use('seaborn-dark-palette')
2 plt.hist([12,34,42,56,23,65,12,34,5,4,34,5,44,34,3,4,6,5,4,3,12,34,23])
The result is
1 plt.style.use('classic')
2 plt.hist([12,34,42,56,23,65,12,34,5,4,34,5,44,34,3,4,6,5,4,3,12,34,23])
The result is
Play around with other styles on other plots and see the difference.
Having fun , right? Stick around and see more customization and plots types in the posts to come.
This post is part of a series of blog post on the Matplotlib library , based on the course Practical Machine Learning Course from The Port Harcourt School of AI (pmlcourse). | {"url":"https://blog.phcschoolofai.org/data-visualization-with-matplotlib-iii","timestamp":"2024-11-02T01:36:08Z","content_type":"text/html","content_length":"213056","record_id":"<urn:uuid:eaf3d861-fbf4-4956-b80d-805536c9efee>","cc-path":"CC-MAIN-2024-46/segments/1730477027632.4/warc/CC-MAIN-20241102010035-20241102040035-00070.warc.gz"} |
An ICME Trilogy is a selection of papers and other inputs produced for the 10th International Congress on Mathematical Education, ICME 10, in Denmark in 2004; and for ICME 11 in Mexico as well
for ICME 12 in Souh Korea in 2012.
The MADIFpapers is a selection of papers written for the MADIF conferences arranged by the Swedish Society for Research in Mathematics Education. It is held the day before the Swedish Biennale where
mathematics teachers from kindergarten to college meet to share knowledge through exhibitions and to inform themselves about new trends and ideas, and listen to foreign or local researchers having
met the day before at the MADIF confernce.
Diagnozing Poor PISA Performance is a selection of papers aimed at diagnozing and curing poor PISA performance. Increased research has let to decreasing PISA math results as In Sweden caused by a
goal/means confusion. Grounded as a means to an outside goal, mathematics becomes a natural science about the physical fact Many. This ManyMatics differs from the school’s Mathematics. However,
ManyMatics might be the cure against poor PISA performance.
Foucault and Mathematics
Two discourses about the natural fact Many exist, both called Mathematics. One suppresses the other. Two discourses about learning exist, both called education. One is based upon European Bildung,
the other upon North-American Enlightenment.
The French post-structuralist thinker Michel Foucault gives one understanding of competing discourses.
Postmodern Contingency Research
A paper that answers three questions: What is the postmodern? Is there a postmodern research paradigm? To what field of mathematics education research can postmodern research contribute, and what are
examples of postmodern studies?
ICME13 papers is a selection of papers written for the ICME 13, the 13^th International Congress on Mathematics Education, ask how mathematics education would look like from the viewpoint of the
existentialist principle ‘existence precedes essence’.
Miscellaneous contains papers and PowerPointPresentations written for various conferences or lectures | {"url":"http://mathecademy.net/various/papers/","timestamp":"2024-11-13T14:37:50Z","content_type":"text/html","content_length":"21975","record_id":"<urn:uuid:ce8da262-19bf-4ac9-bac4-941a49711e19>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00515.warc.gz"} |
1.8 Truth Tables: Conditionals and Biconditionals (2024)
Learning Objectives
• Basic Truth Tables for
• Working with the Conditional Statement
□ Converse
□ Inverse
□ Contrapositive
This is sometimes called an implication. A conditional is a logical compound statement in which a statement p, called the antecedent, implies a statement q, called the consequent.
A conditional is written as p → q and is translated as “if p, then q”.
The English statement “If it is raining, then there are clouds is the sky” is a conditional statement. It makes sense because if the antecedent “it is raining” is true, then the consequent “there are
clouds in the sky” must also be true.
Notice that the statement tells us nothing of what to expect if it is not raining; there might be
clouds in the sky, or there might not. If the antecedent is false, then the consequent becomes
Suppose you order a team jersey online on Tuesday and want to receive it by Friday so you can
wear it to Saturday’s game. The website says that if you pay for expedited shipping, you will
receive the jersey by Friday. In what situation is the website telling a lie?
There are four possible outcomes:
1) You pay for expedited shipping and receive the jersey by Friday
2) You pay for expedited shipping and don’t receive the jersey by Friday
3) You don’t pay for expedited shipping and receive the jersey by Friday
4) You don’t pay for expedited shipping and don’t receive the jersey by Friday
Only one of these outcomes proves that the website was lying: the second outcome in which you pay for expedited shipping but don’t receive the jersey by Friday. The first outcome is exactly what was
promised, so there’s no problem with that. The third outcome is not a lie because the website never said what would happen if you didn’t pay for expedited shipping;
maybe the jersey would arrive by Friday whether you paid for expedited shipping or not. The fourth outcome is not a lie because, again, the website didn’t make any promises about when the jersey
would arrive if you didn’t pay for expedited shipping.
It may seem strange that the third outcome in the previous example, in which the first part is false but the second part is true, is not a lie. Remember, though, that if the antecedent is false, we
cannot make any judgment about the consequent. The website never said that paying for expedited shipping was the only way to receive the jersey by Friday.
A friend tells you “If you upload that picture to Facebook, you’ll lose your job.” Under what conditions can you say that your friend was wrong?
There are four possible outcomes:
1) You upload the picture and lose your job
2) You upload the picture and don’t lose your job
3) You don’t upload the picture and lose your job
4) You don’t upload the picture and don’t lose your job
There is only one possible case in which you can say your friend was wrong: the second outcome in which you upload the picture but still keep your job. In the last two cases, your friend didn’t say
anything about what would happen if you didn’t upload the picture, so you can’t say that their statement was wrong. Even if you didn’t upload the picture and lost your job anyway, your
friend never said that you were guaranteed to keep your job if you didn’t upload the picture; you might lose your job for missing a shift or punching your boss instead.
Aconditional statementtells us that if the antecedent is true, the consequent cannot be false. Thus, a conditional statement is only false when a true antecedent implies a false consequent.
Another example is living in an apartment and paying rent.p → q where p is I live in an apartment and q is then I pay rent. What are the outcomes?
1. I do live in an apartment and I pay rent, then the situation is true (no eviction!)
2. I live in an apartment and I don’t pay rent, then the situation is false (eviction, broken promise)
3. I don’t live in an apartment but I do pay rent, then the situation is true (though why would you do it?)
4. I don’t live in an apartment and I don’t pay rent, then the situation is true (no promise broken)
With conditional situations, we also have the following:
Related Statements
The original conditional is “if p, then q” p → q
The converse is “if q, then p” q → p
The inverse is “if not p, then not q” ~p →~ q
The contrapositive is “if not q, then not p” ~q → ~p
Consider the conditional “If it is raining, then there are clouds in the sky.” It seems reasonable to assume that this is true.
The converse would be “If there are clouds in the sky, then it is raining.” This is not always true.
The inverse would be “If it is not raining, then there are not clouds in the sky.” Likewise, this is not always true.
The contrapositive would be “If there are not clouds in the sky, then it is not raining.” This
statement is true, and is equivalent to the original conditional.
Suppose this statement is true: “If I eat this giant cookie, then I will feel sick.” Which of the following statements must also be true?
a. If I feel sick, then I ate that giant cookie.
b. If I don’t eat this giant cookie, then I won’t feel sick.
c. If I don’t feel sick, then I didn’t eat that giant cookie.
a. This is the converse, which is not necessarily true. I could feel sick for some other reason, such as drinking sour milk.
b. This is the inverse, which is not necessarily true. Again, I could feel sick for some other reason; avoiding the cookie doesn’t guarantee that I won’t feel sick.
c. This is the contrapositive, which is true, but we have to think somewhat backwards to explain it. If I ate the cookie, I would feel sick, but since I don’t feel sick, I must not have eaten the
Notice again that the original statement and the contrapositive have the same truth value (both
are true), and the converse and the inverse have the same truth value (both are false).
A biconditional is a logical conditional statement in which the antecedent and consequent are interchangeable.
A biconditional is written as p ↔ q and is translated as “p if and only if q”.
Because a biconditional statement p ↔ q is equivalent to (p → q) ⋀ (q → p), we may think of it as a conditional statement combined with its converse: if p, then q and if q, then p. The double-headed
arrow shows that the conditional statement goes from left to right and from right to left. A biconditional is considered true as long as the antecedent and the consequent have the same truth value;
that is, they are either both true or both false.
Thebiconditionaltells us that, “Either both are the case, or neither is… ” Thus, a biconditional statement is true when both statements are true, or both are false.
Suppose this statement is true: “The garbage truck comes down my street if and only if it is Thursday morning.” Which of the following statements could be true?
a. It is noon on Thursday and the garbage truck did not come down my street this morning.
b. It is Monday and the garbage truck is coming down my street.
c. It is Wednesday at 11:59PM and the garbage truck did not come down my street today.
a. This cannot be true. This is like the second row of the truth table; it is true that I just experienced Thursday morning, but it is false that the garbage truck came.
b. This cannot be true. This is like the third row of the truth table; it is false that it is Thursday, but it is true that the garbage truck came.
c. This could be true. This is like the fourth row of the truth table; it is false that it is Thursday, but it is also false that the garbage truck came, so everything worked out like it should.
Working with the Conditional Statement
Conditional statements play a very big role in logic and one of the ways we can learn more about them is to study the three related statements.
Consider again the valid implication “If it is raining, then there are clouds in the sky.”
Write the related converse, inverse, and contrapositive statements.
Try It
Conditional Converse Inverse Contrapositive
p q [latex]p\rightarrow{q}[/latex] [latex]q{\rightarrow}p[/latex] [latex]\sim{p}\rightarrow\sim{q}[/latex] [latex]\sim{q}\rightarrow\sim{p}[/latex]
T T T T T T
T F F T T F
F T T F F T
F F T T T T
This is the end of the section. Close this tab and proceed to the corresponding assignment. | {"url":"https://archeryhut.net/article/1-8-truth-tables-conditionals-and-biconditionals","timestamp":"2024-11-07T03:12:50Z","content_type":"text/html","content_length":"123318","record_id":"<urn:uuid:c7dd6705-0dbf-46b2-9b78-5f67f5308d13>","cc-path":"CC-MAIN-2024-46/segments/1730477027951.86/warc/CC-MAIN-20241107021136-20241107051136-00565.warc.gz"} |
The Steiner Ratio for the Obstacle-Avoiding Steiner Tree Problem
University of Waterloo
This thesis examines the (geometric) Steiner tree problem: Given a set of points P in the plane, find a shortest tree interconnecting all points in P, with the possibility of adding points outside P,
called the Steiner points, as additional vertices of the tree. The Steiner tree problem has been studied in different metric spaces. In this thesis, we study the problem in Euclidean and rectilinear
metrics. One of the most natural heuristics for the Steiner tree problem is to use a minimum spanning tree, which can be found in O(nlogn) time . The performance ratio of this heuristic is given by
the Steiner ratio, which is defined as the minimum possible ratio between the lengths of a minimum Steiner tree and a minimum spanning tree. We survey the background literature on the Steiner ratio
and study the generalization of the Steiner ratio to the case of obstacles. We introduce the concept of an anchored Steiner tree: an obstacle-avoiding Steiner tree in which the Steiner points are
only allowed at obstacle corners. We define the obstacle-avoiding Steiner ratio as the ratio of the length of an obstacle-avoiding minimum Steiner tree to that of an anchored obstacle-avoiding
minimum Steiner tree. We prove that, for the rectilinear metric, the obstacle-avoiding Steiner ratio is equal to the traditional (obstacle-free) Steiner ratio. We conjecture that this is also the
case for the Euclidean metric and we prove this conjecture for three points and any number of obstacles.
Computer Science, Computational geometry, Steiner tree, Steiner ratio | {"url":"https://uwspace.uwaterloo.ca/items/d955d65a-22e8-477a-b2d7-7473e9ed1b0c","timestamp":"2024-11-12T00:58:12Z","content_type":"text/html","content_length":"425418","record_id":"<urn:uuid:b46dd4d6-a6a3-4d9d-8d16-18ab316f470a>","cc-path":"CC-MAIN-2024-46/segments/1730477028240.82/warc/CC-MAIN-20241111222353-20241112012353-00592.warc.gz"} |
The Stacks project
Lemma 36.39.1. Let $X$ be a scheme. There is a functor
\[ \det : \left\{ \begin{matrix} \text{category of perfect complexes} \\ \text{with tor amplitude in }[-1, 0] \\ \text{morphisms are isomorphisms} \end{matrix} \right\} \longrightarrow \left\{ \begin
{matrix} \text{category of invertible modules} \\ \text{morphisms are isomorphisms} \end{matrix} \right\} \]
In addition, given a rank $0$ perfect object $L$ of $D(\mathcal{O}_ X)$ with tor-amplitude in $[-1, 0]$ there is a canonical element $\delta (L) \in \Gamma (X, \det (L))$ such that for any
isomorphism $a : L \to K$ in $D(\mathcal{O}_ X)$ we have $\det (a)(\delta (L)) = \delta (K)$. Moreover, the construction is affine locally given by the construction of More on Algebra, Section 15.122
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RE: st: stata code for two-part model
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RE: st: stata code for two-part model
From "Shehzad Ali" <[email protected]>
To <[email protected]>
Subject RE: st: stata code for two-part model
Date Fri, 22 Aug 2008 09:43:13 +0100
Thank you, Austin. These are very helpful comments.
Truly appreciate your help.
-----Original Message-----
From: [email protected]
[mailto:[email protected]] On Behalf Of Austin Nichols
Sent: 20 August 2008 14:18
To: [email protected]
Subject: Re: st: stata code for two-part model
Shehzad et al.--
I think a plug for -fmm- (findit fmm) also belongs in this thread,
with a mention of the mixtureof(density) option, e.g.
mixtureof(gamma). Perhaps the package's author will comment on the
preferred mixture model for hospital expenditures, or you can consult
one of the refs listed in -help fmm-:
Deb, P. and P. K. Trivedi (1997), Demand for Medical Care by the
Elderly: A Finite Mixture Approach, Journal of Applied Econometrics,
12, 313-326.
-tobit- for expenditures works if willingness to pay for a good is
normally distributed and there is an observed market price, so those
with WTP<p spend nothing, but this is clearly not the case for
hospital expenditures. In fact, it's not clear whether we should
consider the patient or the doctor the "consumer" --I suspect this is
true even outside the US health care system, unlike much of the
concern about the effects of health insurance and incentives, but
perhaps to a lesser extent... in the British NHS for example, is it
you or your doctor who makes decisions about tests and treatments? I
think a simplified resolution of this problem is (a largely unstated)
part of the motivation for a two-part model: you make the decision
about whether or not to seek care, and conditional on seeking care,
your physician makes decisions about tests and treatments, so you no
longer control expenditures, to a first approximation.
On Tue, Aug 19, 2008 at 4:33 PM, Stas Kolenikov <[email protected]> wrote:
> -heckman- and -zip- are both trying to deal with too many zeroes (and
> so does -tobit-, but it puts just too many assumptions in... although
> originally it was developed for the expenditure models). -zip- says
> that for some reason, there is a probability of hitting zero before
> the rest of Poisson kicks in. -heckman- says that there is selection
> and (unobserved) utility functions at work. The selection models are
> more of the behavioral flavor, while zip models are more of the
> descriptive, if not population-averaging, nature, without trying to
> explain why certain people did or did not participate in <whatever>.
> Arguably, you can put a model similar to Heckman's model to hospital
> expenditure, too: if a person does not have (good enough) insurance,
> they may not be able to afford hospitalization, and choose not to go.
> If the (total discounted) budget is less than the predicted hospital
> bill, then we observe zero hospitalization costs. So there is a
> similar utility / budget interplay, and arguably Mills' ratio does
> belong in the linear regression part.
> Alternatively one can say that there are healthy people and sick
> people -- the former are spending zero on hospitals, and others spend
> some non-zero amounts, with the implicit assumption of perfect markets
> and absence of budget constraints. This does not seem quite right to
> me, but I can imagine there are occasions where that's how things
> might be working.
> In reality, both things should be at play: "too low" expenditure for
> the healthy, and "too high" expenditure for the poor. Ideally both
> should be modelled (and neither "true" expenditure is observed), but I
> am not aware of any models that are aimed specifically at that.
> On Tue, Aug 19, 2008 at 2:55 PM, Austin Nichols <[email protected]>
>> Shehzad Ali <[email protected]>:
>> An approach using -heckman- is discussed in the Mullahy ref mentioned
>> earlier (http://www.nber.org/papers/t0228), I believe, along with
>> -tobit-.
>> If the conditional distribution of y seems to fall in two large
>> groups, one at zero and one at higher values, with zero density in
>> between, there may be more justification for one of the two-part types
>> of models where a case is either zero or nonzero, and then the nonzero
>> values are determined by a possibly different process.
>> If you want to model ln(y) as a function of X, so ln(y) for y=0 is
>> missing, then you might prefer -heckman-; if you want to model y as a
>> function of X in one of those models, so y=0 is the lower limit, then
>> you might prefer -tobit-, but both models incorporate a normality
>> assumption that is usually violated in practice... see the Stata
>> reference manuals and cited works for more discussion of the
>> identifying assumptions.
>> Presumably your two sets of expenditure data are for the same
>> individuals, and exhibit correlated errors, so -nlsur- rather than
>> -glm- may be in order.
>> On Tue, Aug 19, 2008 at 1:33 AM, Shehzad Ali <[email protected]> wrote:
>>> Thank you all for your very useful thoughts on this issue.
>>> I am running regression on two separate sets of expenditure data: one
>>> general health expenditure which includes all costs including those for
>>> self-medication etc., and second for expenditure related to formal
>>> care, including primary and hospital care but excluding self-medication.
>>> I agree that two-part model is not the best option but is -heckman-
model a
>>> resaonable alternative if the selection step is for zero/non-zero
>>> expenditure and outcome for the positive expenditure? Looking at
>>> argument, I understand that -heckman- run into similar problem as
>>> model. Is that right?
* For searches and help try:
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How to Sort Comma Delimited Time Values In Pandas?
To sort comma delimited time values in pandas, you can first read the data into a pandas DataFrame using the pd.read_csv() function with the sep=',' parameter to specify that the values are delimited
by commas. Once you have the data loaded, you can use the pd.to_datetime() function to convert the time values to datetime objects.
After converting the time values to datetime objects, you can use the sort_values() function to sort the values in ascending or descending order based on the time values. You can specify the column
containing the time values as the by parameter in the sort_values() function.
Finally, you can use the to_csv() function to save the sorted data back to a CSV file if needed. Overall, by following these steps, you can easily sort comma delimited time values in pandas.
How to create a pivot table in pandas?
To create a pivot table in pandas, you can use the pivot_table function. Here's an example of how to create a pivot table from a sample DataFrame:
1 import pandas as pd
3 # Create a sample DataFrame
4 data = {
5 'Date': ['2021-01-01', '2021-01-01', '2021-01-02', '2021-01-02', '2021-01-03'],
6 'Category': ['A', 'B', 'A', 'B', 'A'],
7 'Value': [10, 20, 15, 25, 30]
8 }
10 df = pd.DataFrame(data)
12 # Create a pivot table
13 pivot_table = df.pivot_table(index='Date', columns='Category', values='Value', aggfunc='sum')
15 print(pivot_table)
In this example, we are creating a pivot table from the df DataFrame with the Date column as the index, the Category column as the columns, and the Value column as the values to be aggregated. We are
specifying the aggregation function as sum.
You can customize the pivot table by changing the index, columns, values, and aggregation function according to your requirements.
How to calculate the mean of a column in a pandas dataframe?
You can calculate the mean of a specific column in a pandas dataframe by using the mean() method on the column of interest. Here is an example:
1 import pandas as pd
3 # Create a sample dataframe
4 data = {'A': [1, 2, 3, 4, 5],
5 'B': [10, 20, 30, 40, 50]}
6 df = pd.DataFrame(data)
8 # Calculate the mean of column 'A'
9 mean_A = df['A'].mean()
11 print(mean_A)
This will output:
In this example, we calculate the mean of column 'A' in the dataframe df and store the result in the variable mean_A. You can replace 'A' with the name of the column for which you want to calculate
the mean.
What is the use of the sample function in pandas?
The sample function in pandas is used to randomly select a specified number or fraction of items from a dataframe or series. It is often used for creating a subset of the original data for further
analysis or visualization. This function can help in creating randomized samples for testing or training machine learning models, or for conducting statistical analyses on a subset of the data.
How to convert a pandas dataframe to a numpy array?
You can convert a pandas DataFrame to a NumPy array using the values attribute of the DataFrame. Here is an example:
1 import pandas as pd
2 import numpy as np
4 # Create a sample DataFrame
5 data = {'A': [1, 2, 3, 4], 'B': [5, 6, 7, 8]}
6 df = pd.DataFrame(data)
8 # Convert the DataFrame to a NumPy array
9 array = df.values
11 print(array)
This will output:
1 [[1 5]
2 [2 6]
3 [3 7]
4 [4 8]]
Now array is a NumPy array containing the values of the DataFrame.
How to calculate the correlation between columns in a pandas dataframe?
To calculate the correlation between columns in a pandas dataframe, you can use the .corr() method. Here's an example:
1. Load the pandas library:
1. Create a sample dataframe:
1 data = {
2 'A': [1, 2, 3, 4, 5],
3 'B': [5, 4, 3, 2, 1],
4 'C': [3, 3, 3, 3, 3]
5 }
7 df = pd.DataFrame(data)
1. Calculate the correlation between columns:
1 correlation = df.corr()
2 print(correlation)
This will output a correlation matrix where each element represents the correlation between the corresponding columns. Positive values indicate a positive correlation, negative values indicate a
negative correlation, and a value of 1 represents a perfect correlation.
You can also use the method parameter of the corr() method to specify the correlation method to use. The default is the Pearson correlation coefficient, but you can also use 'spearman' or 'kendall'.
For example:
1 correlation = df.corr(method='spearman')
2 print(correlation) | {"url":"https://mybloggg.wayner.ca/blog/how-to-sort-comma-delimited-time-values-in-pandas","timestamp":"2024-11-02T06:19:45Z","content_type":"text/html","content_length":"175041","record_id":"<urn:uuid:68a0942f-dd46-45fd-9943-924f6ab24912>","cc-path":"CC-MAIN-2024-46/segments/1730477027677.11/warc/CC-MAIN-20241102040949-20241102070949-00614.warc.gz"} |
Fluorophore Colocalization
Fluorophore Colocalization - Java Tutorial
Two or more fluorescence emission signals can often overlap in digital images recorded by confocal microscopy due to their close proximity within the specimen. This effect is known as colocalization
and usually occurs when fluorescently labeled molecules bind to targets that lie in very close or identical spatial positions. This interactive tutorial explores the quantitative analysis of
colocalization in a wide spectrum of specimens that were specifically designed either to demonstrate the phenomenon, or to alternatively provide examples of fluorophore targets that lack any
significant degree of colocalization.
The tutorial initializes with a randomly chosen confocal microscopy dual or triple labeled fluorescence image appearing in the Specimen Image window and the accompanying red-green (or red-blue)
scatterplot graphed two-dimensionally in the adjacent Colocalization Scatterplot coordinate system. Plots of the available channel permutations (Red-Green, Red-Blue, and Green-Blue) can be displayed
using the Colocalization Channels set of radio buttons. In addition, each channel in the Specimen Image window can be toggled on or off using the check boxes in the Channels menu. A three-dimensional
rendering of the colocalization scatterplot (number of pixels plotted on the z axis) can be obtained by activating the 3D radio button. This view can be rotated within the window using the mouse
cursor. Colocalization coefficients automatically displayed beneath the scatterplot graph include Pearson's, Overlap, and Global (k1 and k2), as described below.
In order to operate the tutorial, use the mouse cursor to draw a region of interest in the Colocalization Scatterplot graph. The default area selection tool generates rectangular regions, but
elliptical and freehand areas can be chosen with the appropriate Region of Interest radio buttons. Once a region has been selected, an overlay of the colocalized pixels is displayed in the Specimen
Image window and the Global colocalization coefficient display changes into the Local (M1 and M2) value calculated within the region of interest. The image Colocalization Overlay view can be toggled
between Full Color and Binary views using check boxes. At any point, a new specimen can be selected using the Choose A Specimen pull-down menu. Details of the specimen fluorophore staining protocol
and the potential for colocalization are described in the yellow text box at the bottom of the tutorial window.
A quantitative assessment of fluorophore co-localization in confocal optical sections can be obtained using the information obtained from scatterplots and selected regions of interest. Several values
are generated using information from the entire scatterplot, while others are derived from pixel values contained within a selected region of interest. Among the variables used to analyze the entire
scatterplot is Pearson's correlation coefficient (R(r)), which is one of the standard techniques applied in pattern recognition for matching one image to another in order to describe the degree of
overlap between the two patterns. Pearson's correlation coefficient is calculated according to the equation:
where S1 is the signal intensity of pixels in the first channel and S2 is the signal intensity of pixels in the second channel. The values S1(average) and S2(average) are the average values of pixels
in the first and second channel, respectively. In Pearson's correlation, the average pixel intensity values are subtracted from the original intensity values. As a result, the value of this
coefficient ranges from -1 to 1, with a value of -1 representing a total lack of overlap between pixels from the images, and a value of 1 indicating perfect image registration. Pearson's correlation
coefficient accounts only for the similarity of shapes between the two images, and does not depend upon image pixel intensity values. When applying this coefficient to co-localization analysis,
however, the potentially negative values are difficult to interpret, requiring another approach to clarify analysis results.
A simpler technique often employed to calculate an alternative correlation coefficient involves eliminating the subtraction of average pixel intensity values from the original intensities. Defined
formally as the Overlap coefficient (R), this value ranges between 0 and 1 and is not sensitive to intensity variations in the image analysis. The Overlap coefficient is defined as:
The product of channel intensities in the numerator returns a significant value only when both values belong to a pixel involved in co-localization (if both intensities are greater than zero). As a
result, the numerator in equation (2) is proportional to the number of co-localizing pixels. In a similar manner, the denominator of the Overlap equation is proportional to the number of pixels from
both components in the image, regardless of whether co-localization is present (Note: the components are defined as the red and green images or the pixel arrays from channel 1 and channel 2,
respectively). A major advantage of the Overlap coefficient is its relative insensitivity to differences in signal intensities between various components of an image, which are often produced by
fluorochrome concentration fluctuations, photobleaching, quantum efficiency variations, and non-equivalent electronic channel settings.
The most important disadvantage of using the Overlap coefficient is the strong influence of the ratio between the number of image features in each channel. To alleviate this dependency, the Overlap
coefficient is divided into two different sub-coefficients, termed k(1) and k(2) in order to express the degree of co-localization as two separate parameters:
The overlap coefficients, k(1) and k(2), describe the differences in intensities between the channels, with k(1) being sensitive to the differences in the intensity of channel 2 (green signal), while
k(2) depends linearly on the intensity of the pixels from channel 1 (red signal). The equations described thus far are able to generate information about the degree of overlap and can account for
intensity variations between the color channels. In order to estimate the contribution of one color channel in the co-localized areas of the image to the overall amount of co-localized fluorescence,
an additional set of co-localization coefficients, m(1) and m(2), are defined:
The co-localization coefficient m(1) is employed to describe the contribution from channel 1 to the co-localized area, while the coefficient m(2) is used to describe the same contribution from
channel 2. Note that the variable S1(i,coloc) is equal to S1(i) if S2(i) is greater than zero and vice versa for the variable S2(i,coloc). These coefficients are proportional to the amount of
fluorescence of the co-localizing fluorophores in each channel of the composite image, relative to the total fluorescence in that channel. Co-localization coefficients m(1) and m(2) can be determined
even when the signal intensities in the two image channels have significantly different levels.
A second pair of co-localization coefficients can be calculated for pixel intensity ranges defined by an area of interest delineated on the scatterplot. The coefficient M(1) is utilized to describe
the contribution of the channel 1 fluorophore to the co-localized area, while M(2) is used to describe the contribution of the channel 2 fluorophore. These co-localization coefficients are defined
where S1(i,coloc) equals S1(i) if S2(i) lies within the region of interest thresholds (left and right sides of a rectangular ROI) and equals zero if S2(i) represents a pixel outside the threshold
levels. Similarly, S2(i,coloc) equals S2(i) if S1(i) lies within the region of interest thresholds (top and bottom sides of a rectangular ROI) and equals zero if S1(i) is outside the region of
interest. In other words, for each channel, the numerator represents the sum of all pixel intensities in that channel that also have a component from the other channel, whereas the denominator
represents the sum of all intensities from the channel. These coefficients are proportional to the amount of fluorescence of co-localizing objects in each channel of the composite image, relative to
the total fluorescence in that channel.
A majority of the co-localization software analysis programs available commercially are able to calculate the parameters described above, including Pearson's correlation coefficient, the total
overlap coefficient, as well as the individual k(x), m(x), and M(x) co-localization coefficients. In addition, many programs contain algorithms to apply background subtraction corrections, generate
scatterplots of the entire image, and/or perform the calculations using selected regions of interest on single dual channel composite images or optical stacks along the axial plane. The most
important data output from these software packages is the co-localization coefficient, which indicates the relative degree of overlap between signals. For example, a co-localization coefficient value
of 0.75 for the fluorophore in channel 1 indicates that the ratio for all channel 1 intensities that have a channel 2 component, divided by the sum of all channel 1 intensities, is 75 percent. This
is a relatively high degree of co-localization. Likewise, a value of 0.25 for the channel 2 fluorophore indicates a significantly diminished level of co-localization (equal to one-third of the
channel 1 fluorophore). | {"url":"https://www.olympus-lifescience.com/en/microscope-resource/primer/java/colocalization/","timestamp":"2024-11-13T23:15:58Z","content_type":"application/xhtml+xml","content_length":"54183","record_id":"<urn:uuid:0f3f8a4f-78e8-47ff-bf70-1a9df7a8a789>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00826.warc.gz"} |
Algebraic Expressions Worksheet: Practice & Answers (PDF)
translate algebraic expressions worksheet with answers pdf
Translate Algebraic Expressions Worksheet with Answers PDF is a valuable resource for students learning basic algebra․ These worksheets provide practice in converting verbal phrases into algebraic
expressions․ The worksheets are typically designed for middle school students and can be used in classrooms or for independent study․ They are often available in PDF format, making them easy to print
and use․
These worksheets help students develop their understanding of how mathematical concepts can be represented in different ways․ They also help students build essential skills for solving algebraic
equations and problems․ By providing practice in translating verbal phrases into algebraic expressions, these worksheets equip students with the tools they need to succeed in algebra and beyond․
Algebraic expressions worksheets are a fundamental tool in the learning process of algebra․ They serve as a bridge between the abstract world of mathematical symbols and the concrete world of
everyday language․ By translating verbal phrases into algebraic expressions, students gain a deeper understanding of how mathematical concepts are applied in real-world situations․ This process of
translation not only enhances their algebraic skills but also strengthens their critical thinking and problem-solving abilities․
These worksheets are particularly beneficial for students who are new to algebra, as they provide a structured approach to learning the basics․ They introduce students to essential concepts such as
variables, coefficients, and operations, laying a solid foundation for more advanced algebraic concepts․ The worksheets often include examples and explanations, guiding students through the process
of translating verbal phrases into algebraic expressions․ By working through these exercises, students build confidence in their ability to manipulate and solve algebraic equations․
The inclusion of answers in these worksheets is crucial for self-assessment and learning․ Students can check their work and identify areas where they need further practice․ This feedback loop is
essential for reinforcing concepts and ensuring that students are progressing at a steady pace․ The availability of answers also encourages independent learning, allowing students to work at their
own pace and seek clarification as needed․
Types of Algebraic Expressions Worksheets
Algebraic expressions worksheets come in a variety of formats, each designed to address specific learning objectives and cater to different learning styles․ Some common types of worksheets include⁚
• Basic Expressions Worksheets⁚ These worksheets focus on translating simple verbal phrases into algebraic expressions involving one or two variables․ They typically involve basic operations such
as addition, subtraction, multiplication, and division․
• Simplifying Expressions Worksheets⁚ These worksheets challenge students to simplify algebraic expressions by combining like terms, using the distributive property, and applying the order of
operations․ They help students develop fluency in manipulating algebraic expressions․
• Evaluating Expressions Worksheets⁚ These worksheets require students to substitute given values for variables in an algebraic expression and then evaluate the expression․ They help students
understand the relationship between variables and their numerical values․
• Generating Expressions Worksheets⁚ These worksheets encourage students to create their own algebraic expressions based on given scenarios or word problems․ This type of worksheet promotes
creative thinking and problem-solving skills․
• Solving Equations Worksheets⁚ These worksheets introduce students to solving simple algebraic equations by isolating the variable․ They build upon the foundation of translating verbal phrases
into expressions and extend the concept to solving equations․
The specific types of worksheets used will vary depending on the grade level and the curriculum being taught․ However, all of these worksheets aim to enhance students’ understanding of algebraic
concepts and provide them with the tools they need to succeed in algebra․
Simplifying Algebraic Expressions Worksheets
Simplifying algebraic expressions worksheets play a crucial role in helping students master the fundamental operations and techniques used in algebra․ These worksheets focus on reducing complex
expressions to their simplest forms, a skill essential for solving equations, inequalities, and more advanced algebraic concepts․
The worksheets typically involve tasks such as⁚
• Combining Like Terms⁚ Students learn to identify terms with the same variable and exponent and combine their coefficients, simplifying the expression․ For example, 2x + 3x can be simplified to
• Using the Distributive Property⁚ Students apply the distributive property to multiply a constant or variable by a sum or difference within parentheses․ For instance, 2(x + 3) can be simplified to
2x + 6․
• Applying the Order of Operations (PEMDAS)⁚ Students practice following the order of operations to simplify expressions involving parentheses, exponents, multiplication, division, addition, and
subtraction․ This ensures that expressions are simplified consistently and accurately․
Simplifying algebraic expressions worksheets build a solid foundation for further algebraic studies, enabling students to manipulate equations, solve for unknowns, and work with more complex
mathematical concepts with confidence․
Evaluating Algebraic Expressions Worksheets
Evaluating algebraic expressions worksheets provide students with the opportunity to practice substituting numerical values for variables within an expression and then calculating the resulting
value․ This skill is crucial for understanding how algebraic expressions represent relationships between variables and how they can be used to solve real-world problems․
These worksheets typically involve⁚
• Substituting Values⁚ Students are given an algebraic expression and a set of values for the variables within the expression․ They need to substitute these values into the expression correctly․
For example, if the expression is 2x + 3y and x = 4 and y = 2, students would substitute 4 for x and 2 for y, resulting in 2(4) + 3(2)․
• Applying Order of Operations⁚ After substituting values, students must follow the order of operations (PEMDAS) to simplify the expression․ This ensures that they perform operations in the correct
sequence, leading to the accurate evaluation of the expression;
• Interpreting Results⁚ Evaluating algebraic expressions helps students understand how changing the values of variables affects the outcome of the expression․ This reinforces the concept of
variables as placeholders for unknown values and how they can be manipulated to solve problems․
Evaluating algebraic expressions worksheets provide a crucial link between symbolic representation and numerical calculations, laying a solid foundation for more complex algebraic operations and
problem-solving in later math courses․
Generating Algebraic Expressions Worksheets
Generating algebraic expressions worksheets involves creating exercises that challenge students to translate verbal descriptions or real-world scenarios into mathematical expressions․ This process
helps students develop a deeper understanding of how algebraic expressions represent relationships between quantities and how they can be used to model real-world problems․
Here are some common elements found in generating algebraic expressions worksheets⁚
• Verbal Phrases⁚ Worksheets often present verbal phrases that describe mathematical operations, such as “the sum of a number and 5,” “twice a number minus 3,” or “the product of two numbers․”
Students must translate these phrases into algebraic expressions, using variables to represent unknown quantities․
• Real-World Scenarios⁚ Worksheets may include real-world scenarios that require students to identify the relevant variables and relationships, and then express them algebraically․ For example, a
problem might involve calculating the cost of buying a certain number of items at a given price, or determining the area of a rectangle given its length and width․
• Combinations of Operations⁚ As students progress, worksheets may introduce more complex scenarios that involve combinations of arithmetic operations, such as addition, subtraction,
multiplication, and division․ These challenges require students to carefully analyze the relationships between quantities and express them accurately using algebraic expressions․
Generating algebraic expressions worksheets help students build a strong foundation in algebra by bridging the gap between verbal language and mathematical symbols․ They encourage students to think
critically about how mathematical expressions can be used to represent and solve real-world problems․
Solving Algebraic Equations Worksheets
Solving algebraic equations worksheets provide students with practice in finding the values of unknown variables that make an equation true․ These worksheets are crucial for developing algebraic
proficiency and are typically used in middle school and high school math courses․
Here are some key features of solving algebraic equations worksheets⁚
• Linear Equations⁚ Worksheets often focus on linear equations, which involve variables raised to the first power․ Students learn techniques like combining like terms, isolating the variable, and
using inverse operations to solve for the unknown․
• Multi-Step Equations⁚ As students progress, worksheets introduce multi-step equations that require a series of operations to solve․ These exercises reinforce understanding of the order of
operations and the importance of applying inverse operations strategically․
• Word Problems⁚ Some worksheets include word problems that require students to translate real-world scenarios into algebraic equations and then solve them․ This helps students apply their
algebraic skills to practical situations and develop problem-solving strategies․
• Equations with Fractions and Decimals⁚ Advanced worksheets may include equations with fractions or decimals, requiring students to apply additional techniques like multiplying by the least common
multiple or converting decimals to fractions․
Solving algebraic equations worksheets are essential for building a strong foundation in algebra․ They help students develop critical thinking skills, problem-solving strategies, and a deep
understanding of how equations represent relationships between variables․
Algebraic Expressions Worksheets for Specific Grade Levels
Algebraic expressions worksheets are carefully designed to cater to the specific developmental stages and learning objectives of students at different grade levels․ The complexity and scope of these
worksheets increase as students progress through their math education․
Here’s a glimpse into how algebraic expressions worksheets are tailored for specific grade levels⁚
• Elementary Grades (3rd-5th)⁚ Introductory worksheets at this level focus on basic concepts like identifying variables, understanding simple expressions, and substituting values into expressions․
They often use visual aids and real-world examples to make the concepts relatable․
• Middle School (6th-8th)⁚ Worksheets at this level delve deeper into combining like terms, simplifying expressions with parentheses, and solving basic equations․ They may introduce concepts like
distributive property and factoring, laying the foundation for advanced algebraic concepts․
• High School (9th-12th)⁚ Algebraic expressions worksheets in high school are more rigorous, covering topics like polynomial expressions, rational expressions, and systems of equations․ They
prepare students for advanced mathematics, including calculus and linear algebra․
By providing grade-specific worksheets, teachers can ensure that students are presented with concepts and challenges that are appropriate for their developmental level․ This approach helps students
build a strong understanding of algebraic expressions and progress smoothly through their mathematical journey․
Algebraic Expressions Worksheets for Different Concepts
Algebraic expressions worksheets are not limited to a single concept; they encompass a wide range of topics within algebra․ These worksheets cater to various learning objectives and skill levels,
ensuring a comprehensive understanding of algebraic expressions․ Here’s a breakdown of how worksheets address different concepts within algebra⁚
• Simplifying Expressions⁚ Worksheets focusing on this concept guide students in combining like terms, applying the distributive property, and simplifying expressions with parentheses and
exponents․ These exercises build a strong foundation for solving equations and inequalities․
• Evaluating Expressions⁚ These worksheets involve substituting specific values for variables within given expressions and calculating the resulting value․ This practice helps students understand
the relationship between variables and their values, preparing them for working with functions and equations․
• Writing Expressions from Word Problems⁚ Worksheets designed for this purpose challenge students to translate verbal descriptions into algebraic expressions․ This skill is crucial for applying
algebra to real-world scenarios and solving practical problems․
• Factoring Expressions⁚ Worksheets covering factoring introduce students to techniques like factoring out common factors, factoring quadratics, and factoring by grouping․ These skills are
essential for solving equations, finding roots, and working with rational expressions․
• Operations with Expressions⁚ These worksheets involve adding, subtracting, multiplying, and dividing algebraic expressions․ They emphasize the rules of operations and the use of distribution and
simplification techniques․
By providing worksheets that address diverse concepts, educators ensure that students gain a multifaceted understanding of algebraic expressions․ These worksheets contribute to a strong foundation
for further exploration of advanced algebra topics and mathematical applications․
Algebraic Expressions Worksheets with Answers
Algebraic expressions worksheets with answers are invaluable resources for both students and educators․ These worksheets provide practice problems and solutions, allowing learners to check their
understanding and identify areas that require further attention; The inclusion of answers empowers students to take ownership of their learning, fostering independent exploration and self-assessment․
For teachers, these worksheets offer a convenient tool for assigning homework, classwork, or review activities․ The readily available answers streamline the grading process, allowing educators to
focus on providing personalized feedback and addressing individual learning needs․ The presence of answers also encourages students to engage actively with the material, as they can verify their
solutions and gain immediate reinforcement․
The availability of answers promotes a culture of self-directed learning․ Students can work through the problems at their own pace, checking their answers along the way․ This process helps build
confidence and a deeper understanding of the concepts․ Moreover, by providing a clear path to correct solutions, these worksheets encourage perseverance and reduce the frustration often associated
with learning new mathematical skills․
In conclusion, algebraic expressions worksheets with answers serve as a valuable tool for supporting both student learning and teacher instruction․ They foster independent exploration, encourage
active engagement, and facilitate a more effective learning experience․
Online Resources for Algebraic Expressions Worksheets
The internet has become a treasure trove of resources for educators and students alike, and algebraic expressions worksheets are no exception․ Numerous websites offer a vast collection of free
printable worksheets, covering various aspects of algebraic expressions, from basic concepts to more advanced topics․ These online resources provide convenience and flexibility, allowing users to
access and download worksheets anytime and anywhere․
One of the key advantages of online resources is their accessibility․ Users can easily search for specific topics or grade levels, ensuring that they find worksheets that are appropriate for their
needs․ Moreover, the availability of answer keys often accompanies these worksheets, providing immediate feedback and facilitating self-directed learning․ This accessibility empowers students to take
charge of their learning, working at their own pace and addressing their individual needs․
Online resources also offer a wide range of customization options․ Some websites allow users to generate worksheets with specific parameters, such as the number of problems, difficulty level, and
specific concepts to be covered․ This customization feature allows educators to tailor worksheets to meet the unique requirements of their students and curriculum․ The ability to create personalized
worksheets further enhances the value of these online resources․
In conclusion, online resources have revolutionized the way educators and students access and utilize algebraic expressions worksheets․ They offer convenience, accessibility, and customization
options, making them an invaluable tool for supporting effective learning and teaching;
Benefits of Using Algebraic Expressions Worksheets
Algebraic expressions worksheets offer a multitude of benefits for students learning algebra, contributing to a deeper understanding of the subject and enhancing their problem-solving abilities․
These worksheets provide structured practice, allowing students to reinforce essential concepts and develop fluency in manipulating algebraic expressions․
The repetitive nature of worksheets helps students solidify their understanding of key concepts, such as identifying like terms, combining terms, and simplifying expressions․ This repetition promotes
mastery of basic algebraic operations, building a strong foundation for more advanced topics․ Moreover, worksheets provide a visual representation of algebraic concepts, making them easier to grasp
and internalize․
Furthermore, algebraic expressions worksheets foster independent learning and self-assessment․ Students can work through the problems at their own pace, identifying areas where they need further
practice or clarification․ The inclusion of answer keys empowers students to check their work and gain immediate feedback, promoting self-directed learning and encouraging active engagement․
In conclusion, algebraic expressions worksheets provide a valuable tool for reinforcing concepts, developing fluency, and fostering independent learning․ By incorporating these worksheets into their
learning process, students can enhance their understanding of algebra and build a solid foundation for future success in mathematics․ | {"url":"https://joshuawagneronline.com/translate-algebraic-expressions-worksheet-with-answers-pdf/","timestamp":"2024-11-06T01:07:24Z","content_type":"text/html","content_length":"57766","record_id":"<urn:uuid:8d6d07b0-4369-413c-875a-2b28a6659229>","cc-path":"CC-MAIN-2024-46/segments/1730477027906.34/warc/CC-MAIN-20241106003436-20241106033436-00340.warc.gz"} |
Vertical spreads are created by same kind of options (call or put) on the same underlying security and with the same maturity date but different strike prices. For instance, Iron Condor strategy
that’s been demonstrated in the previous post is a strategy that is comprised of 2 vertical spreads.
An investor would typically invest in vertical spread strategies when a limited directional movement is expected. If he or she has a strong expectation of the limited move in the underlying price
vertical spreads can be a perfect strategy to employ while providing protection in case the investor is caught in the wrong direction at a cost of capping the potential profits. We will explain this
phenomenon further with examples and demonstrations.
Let’s take a closer look at all the different type of spreads that are usually utilized in the investment world. There are commonly two bull spreads and two bear spreads. Bull spreads are
bull call spreads and bull put spreads.
Bull Spreads
A bull call spread is created by purchasing a call option with a lower strike price and selling a call option with a higher strike price
Bull put spread: A bull put spread is created by purchasing a put option with a lower strike price and selling a put option with a higher strike price.
You might realize both bull spreads involve purchasing the option with lower strike price and selling the option with higher strike price. In terms of premium cost and gain this has different
implications. Since call options get more expensive as the strike price goes lower (as call options give the right to purchase at the strike price purchasing right at a lower price creates higher
intrinsic value), bull call spread will result in a negative cash flow (Low premium gained – High premium paid). Bull put spread, however, will result in a positive cash flow since put option with
the lower strike price is going to cost cheaper than put option with the higher strike price.
│ Maximum Profit and Loss │
│Vertical Spread:│ Bull Call │ Bull Put │
│Maximum Profit: │ the spread between the │ Premium collected (credit) │
│ │strike prices minus the initial debit│ │
│ │ │ the spread between the │
│Maximum Loss: │ Premium paid (debit) │strike prices minus the initial credit │
│ │ │ │
│ │ │ │
Bull Call Spread - Option Trading
Bull Put Spread - Option Trading
Both bull call spreads and bull put spread will be profitable if the underlying price increases. There will be a cap on the profit though limiting the profit at a certain value no matter how much the
underlying price increases. That’s why it is rather suitable if the investor believes the rise potential is limited. Since the maximum possible loss is also limited bull spreads are not as risky as
naked option investments or plain vanilla option investments. (Plain vanilla is a term used in finance to signify most basic, standard or simplest version of financial instruments such as options,
bonds, swaps or futures.)
Bear Spreads
A bear call spread is created by purchasing a call option with a higher strike price and selling a call option with a lower strike price
Bear call spread can be a valuable investment strategy if the market insight is limited downside regarding the underlying price. In this situation bear call spread will allow capped profits based on
the falling underlying prices while providing limited loss profile in case the insight is wrong and the underlying price starts rising.
A bear put spread is created by purchasing a put option with a higher strike price and selling a put option with a lower strike price.
You might realize both bear spreads involve purchasing the option with higher strike price and selling the option with lower strike price. In terms of premium cost and gain this has different
implications similar to what we saw in bull spreads. Since call options get more expensive as the strike price goes lower (as call options give the right to purchase at the strike price purchasing
right at a lower price creates higher intrinsic value), bear call spread will result in a positive cash flow (High premium gained – Low premium paid). Bear put spread, however, will result in a
negative cash flow since put option with the lower strike price is going to cost cheaper than put option with the higher strike price.
Bear Call Spread - Option Trading
Both bear call spreads and bear put spread will be profitable if the underlying price decreases. There will be a cap on the profit limiting the profit at a certain value no matter how much the
underlying price decreases. That’s why it is rather suitable if the investor believes the fall potential is limited. Since the maximum possible loss is also limited bear spreads are not as risky as
naked option investments or plain vanilla option investments. (Plain vanilla is a term used in finance to signify most basic, standard or simplest version of financial instruments such as options,
bonds, swaps or futures.)
│ Maximum Profit and Loss │
│Vertical Spread:│ Bear Call │ Bear Put │
│Maximum Profit: │ Premium collected (credit) │ the spread between the │
│ │ │strike prices minus the initial debit│
│ Maximum Loss: │ the spread between the │ Premium paid (debit) │
│ │strike prices minus the initial credit │ │
Vertical spreads are one of the most convenient ways to bring in regular income from option investments. Since you are involved in both buying and selling and option pair the whole strategy is about
obtaining a capped profit or recurring limited losses. It is never easy to be right 100% of the time when it comes to the markets. Even the most professional investors and traders stay on the wrong
side every now and then. This is why the protection provided by vertical spreads can be very handy for a portfolio with a stable income.
Another point about the usefulness of vertical spreads is by adjusting the strike prices carefully you can design the exact spread you would like to achieve. If it’s a bull spread, you
can profit from increasing underlying prices and if it’s a bear spread, you can profit from decreasing underlying prices. But if your insight is wrong your loss will have a maximum limit. Let’s look
at some examples that will make you have a much better understanding of 4 different vertical spreads that are commonly utilized by most experienced finance professionals.
Bear Put Spread - Option Trading
Example 1: Bull Call Spread
While Amazon is trading at $1696…
│ Bull Call Spread (@AMZN) │
│ Long call │ Short call │
│Strike $1665, term: 3 months │Strike $1725, term: 3 months │
│ Price: $140.00 │ Price: $97.60 │
Long call; Strike: $1665, maturity: 3 months ahead (option trading at $140)
Short call; Strike: $1725, maturity: 3 months ahead (option trading at $97.60)
Net premium result: $97.60 – $140 = -$42.40 (debit: net investment required)
Scenario 1: Amazon share price increases to $1750. Both call options would be in the money in this scenario, but the long call option would be deeper in the money as following:
$1750 – $1665 = $85 (long call option position)
$1750 – $1725 = $25 (short call option position)
Vertical Spread is now worth $85 – $25 = $60
After adjusting with the initial investment total net result is: $60 – $42.40 = $17.60 (Profit)
Scenario 2: Amazon share price decreases to $1550. Both call options would expire worthless and hence not be exercised in this scenario, meaning the whole amount of initial investment would be lost
Total result: -$42.40 (Loss)
Example 2: Bull PUT Spread
While Amazon is trading at $1696…
│ Bull Put Spread (@AMZN) │
│ Long put │ Short put │
│Strike $1645, term: 3 months │Strike $1785, term: 3 months │
│ Price: $90.00 │ Price: $149.25 │
Long put: Strike: $1645, maturity: 3 months ahead (option trading at $90)
Short put: Strike: $1785, maturity: 3 months ahead (option trading at $149.25)
Net premium result: $149.25 – $90 = $59.25 (credit: net premium collected)
Scenario 1: Amazon share price decreases to $1500. Both put options would be in the money in this scenario, but the short put option would be deeper in the money as following:
$1500 – $1785 = -$285 (short put option position)
$1645 – $1500 = $145 (long put option position)
Vertical Spread is now worth $145 – $285 = -$140
After adjusting with the initial premium gains total net result is: -$140 + $59.25 = -$-80.75 (Loss)
You can also see that this result is equal to strike price difference + premiums collected initially.
Scenario 2: Amazon share price increases to $1850. Both put options would expire worthless and hence not be exercised in this scenario, meaning the whole amount of initial collected premiums would be
realized profit.
Total result: $59.25 (Profit)
Example 3: bear Call Spread
While Amazon is trading at $1696…
│ Bear Call Spread (@AMZN) │
│ Long call │ Short call │
│Strike $1800, term: 3 months │Strike $1600, term: 3 months │
│ Price: $64.50 │ Price: $167.00 │
Long call: Strike: $1800, maturity: 3 months ahead (trading at $64.50)
Short call: Strike: $1600, maturity: 3 months ahead (trading at $167.00)
Net premium result: $167 – $64.50 = $102.50 (credit: net premium collected)
Scenario 1: Amazon share price increases to $1810. Both call options would be in the money in this scenario, but the short call option would be deeper in the money as following:
$1810 – $1800 = $10 (long call option position)
$1810 – $1600 = $210 (short call option position)
Vertical Spread is now worth $10 – $210 = -$200
After adjusting with the initial investment total net result is: $102.50 – $200 = -$97.50 (Loss)
Please note how loss is equal to difference between strike prices plus the initial net premium gain.
Scenario 2: Amazon share price decreases to $1550. Both call options would expire worthless and hence not be exercised in this scenario, meaning the whole amount of initial investment would be lost.
Total result: -$42.40 (Loss)
Example 4: bear put Spread
While Amazon is trading at $1696…
│ Bear Put Spread (@AMZN) │
│ Long put │ Short put │
│Strike $1780, term: 3 months │Strike $1640, term: 3 months │
│ Price: $135.30 │ Price: $77.10 │
Long put: Strike: $1780, maturity: 3 months ahead (trading at $135.30)
Short put: Strike: $1640, maturity: 3 months ahead (trading at $77.10)
Net premium result: $77.10 – $135.30 = -$58.20 (debit: net investment required)
Scenario 1: Amazon share price decreases to $1550. Both put options would be in the money in this scenario, but the long put option would be deeper in the money as following:
$1780 – $1550 = $230 (long put option position)
$1780 – $1640 = $140 (short put option position)
Vertical Spread is now worth $230 – $140 = $90
After adjusting with the initial investment total net result is: $90 – $58.20 = $31.80 (Profit)
Scenario 2: Amazon share price increases to $1880. Both put options would expire worthless and hence not be exercised in this scenario, meaning the whole amount of initial investment would be lost.
Total result: -$58.20 (Loss) | {"url":"https://www.coldalmond.com/straddles-strangles-and-butterflies-2/","timestamp":"2024-11-15T04:13:36Z","content_type":"text/html","content_length":"118294","record_id":"<urn:uuid:314b29e3-35c1-4455-8573-70f3e21fc831>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00812.warc.gz"} |
Union SpatVector or SpatExtent objects — union
Union SpatVector or SpatExtent objects
If you want to append polygon SpatVectors use rbind instead of union. union will also intersect overlapping polygons between, not within, objects. Union for lines and points simply combines the two
data sets; without any geometric intersections. This is equivalent to rbind. Attributes are joined.
If x and y have a different geometry type, a SpatVectorCollection is returned.
If a single SpatVector is supplied, overlapping polygons are intersected. Original attributes are lost. New attributes allow for determining how many, and which, polygons overlapped.
SpatExtent: Objects are combined into their union; this is equivalent to +.
# S4 method for class 'SpatVector,SpatVector'
union(x, y)
# S4 method for class 'SpatVector,missing'
union(x, y)
# S4 method for class 'SpatExtent,SpatExtent'
union(x, y)
See also
merge and mosaic to union SpatRasters.
crop and extend for the union of SpatRaster and SpatExtent.
merge for merging a data.frame with attributes of a SpatVector.
aggregate to dissolve SpatVector objects.
e1 <- ext(-10, 10, -20, 20)
e2 <- ext(0, 20, -40, 5)
union(e1, e2)
#> SpatExtent : -10, 20, -40, 20 (xmin, xmax, ymin, ymax)
v <- vect(system.file("ex/lux.shp", package="terra"))
v <- v[,3:4]
p <- vect(c("POLYGON ((5.8 49.8, 6 49.9, 6.15 49.8, 6 49.65, 5.8 49.8))",
"POLYGON ((6.3 49.9, 6.2 49.7, 6.3 49.6, 6.5 49.8, 6.3 49.9))"), crs=crs(v))
values(p) <- data.frame(pid=1:2, value=expanse(p))
u <- union(v, p)
plot(u, "pid")
b <- buffer(v, 1000)
u <- union(b)
u$sum <- rowSums(as.data.frame(u))
plot(u, "sum") | {"url":"https://rspatial.github.io/terra/reference/union.html","timestamp":"2024-11-06T20:58:43Z","content_type":"text/html","content_length":"14171","record_id":"<urn:uuid:803335cc-c5e8-444f-8bc5-87c803f384fd>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.47/warc/CC-MAIN-20241106194801-20241106224801-00515.warc.gz"} |
sle.score: Score impact of each sample on sparse leading eigen-value in GabrielHoffman/decorate: Differential Epigenetic Coregulation Test
Score impact of each sample on sparse leading eigen-value. Compute correlation using all samples (i.e. C), then compute correlation omitting sample i (i.e. Ci). The score the sample i is based on
sparse leading eigen-value of the diffrence between C and Ci.
sle.score( Y, method = c("pearson", "kendall", "spearman"), rho = 0.05, sumabs = 1 )
Y data matrix with samples on rows and variables on columns
method specify which correlation method: "pearson", "kendall" or "spearman"
rho a positive constant such that cor(Y) + diag(rep(rho,p)) is positive definite.
sumabs regularization paramter. Value of 1 gives no regularization, sumabs*sqrt(p) is the upperbound of the L_1 norm of v,controling the sparsity of solution. Must be between 1/sqrt(p) and 1.
data matrix with samples on rows and variables on columns
a positive constant such that cor(Y) + diag(rep(rho,p)) is positive definite.
regularization paramter. Value of 1 gives no regularization, sumabs*sqrt(p) is the upperbound of the L_1 norm of v,controling the sparsity of solution. Must be between 1/sqrt(p) and 1.
# load iris data data(iris) # Evalaute score on each sample sle.score( iris[,1:4] )
For more information on customizing the embed code, read Embedding Snippets. | {"url":"https://rdrr.io/github/GabrielHoffman/decorate/man/sle.score.html","timestamp":"2024-11-06T07:29:20Z","content_type":"text/html","content_length":"33108","record_id":"<urn:uuid:2f6e70c1-4c8a-4c23-8c31-01af2bfd4723>","cc-path":"CC-MAIN-2024-46/segments/1730477027910.12/warc/CC-MAIN-20241106065928-20241106095928-00081.warc.gz"} |
EWK: iShares MSCI Belgium ETF | Logical Invest
What do these metrics mean?
'Total return, when measuring performance, is the actual rate of return of an investment or a pool of investments over a given evaluation period. Total return includes interest, capital gains,
dividends and distributions realized over a given period of time. Total return accounts for two categories of return: income including interest paid by fixed-income investments, distributions or
dividends and capital appreciation, representing the change in the market price of an asset.'
Applying this definition to our asset in some examples:
• Compared with the benchmark SPY (101.5%) in the period of the last 5 years, the total return, or performance of 17.6% of iShares MSCI Belgium ETF is smaller, thus worse.
• Compared with SPY (29.7%) in the period of the last 3 years, the total return, or performance of -1.2% is lower, thus worse.
'The compound annual growth rate (CAGR) is a useful measure of growth over multiple time periods. It can be thought of as the growth rate that gets you from the initial investment value to the ending
investment value if you assume that the investment has been compounding over the time period.'
Applying this definition to our asset in some examples:
• The annual return (CAGR) over 5 years of iShares MSCI Belgium ETF is 3.3%, which is lower, thus worse compared to the benchmark SPY (15.1%) in the same period.
• During the last 3 years, the annual performance (CAGR) is -0.4%, which is smaller, thus worse than the value of 9.1% from the benchmark.
'Volatility is a rate at which the price of a security increases or decreases for a given set of returns. Volatility is measured by calculating the standard deviation of the annualized returns over a
given period of time. It shows the range to which the price of a security may increase or decrease. Volatility measures the risk of a security. It is used in option pricing formula to gauge the
fluctuations in the returns of the underlying assets. Volatility indicates the pricing behavior of the security and helps estimate the fluctuations that may happen in a short period of time.'
Using this definition on our asset we see for example:
• Looking at the volatility of 22.2% in the last 5 years of iShares MSCI Belgium ETF, we see it is relatively greater, thus worse in comparison to the benchmark SPY (20.9%)
• During the last 3 years, the 30 days standard deviation is 18.6%, which is higher, thus worse than the value of 17.6% from the benchmark.
'Risk measures typically quantify the downside risk, whereas the standard deviation (an example of a deviation risk measure) measures both the upside and downside risk. Specifically, downside risk in
our definition is the semi-deviation, that is the standard deviation of all negative returns.'
Applying this definition to our asset in some examples:
• Compared with the benchmark SPY (14.9%) in the period of the last 5 years, the downside volatility of 16.2% of iShares MSCI Belgium ETF is greater, thus worse.
• Compared with SPY (12.3%) in the period of the last 3 years, the downside volatility of 13.1% is greater, thus worse.
'The Sharpe ratio was developed by Nobel laureate William F. Sharpe, and is used to help investors understand the return of an investment compared to its risk. The ratio is the average return earned
in excess of the risk-free rate per unit of volatility or total risk. Subtracting the risk-free rate from the mean return allows an investor to better isolate the profits associated with risk-taking
activities. One intuition of this calculation is that a portfolio engaging in 'zero risk' investments, such as the purchase of U.S. Treasury bills (for which the expected return is the risk-free
rate), has a Sharpe ratio of exactly zero. Generally, the greater the value of the Sharpe ratio, the more attractive the risk-adjusted return.'
Applying this definition to our asset in some examples:
• The Sharpe Ratio over 5 years of iShares MSCI Belgium ETF is 0.04, which is smaller, thus worse compared to the benchmark SPY (0.6) in the same period.
• Looking at Sharpe Ratio in of -0.16 in the period of the last 3 years, we see it is relatively lower, thus worse in comparison to SPY (0.37).
'The Sortino ratio, a variation of the Sharpe ratio only factors in the downside, or negative volatility, rather than the total volatility used in calculating the Sharpe ratio. The theory behind the
Sortino variation is that upside volatility is a plus for the investment, and it, therefore, should not be included in the risk calculation. Therefore, the Sortino ratio takes upside volatility out
of the equation and uses only the downside standard deviation in its calculation instead of the total standard deviation that is used in calculating the Sharpe ratio.'
Applying this definition to our asset in some examples:
• Compared with the benchmark SPY (0.84) in the period of the last 5 years, the downside risk / excess return profile of 0.05 of iShares MSCI Belgium ETF is smaller, thus worse.
• During the last 3 years, the downside risk / excess return profile is -0.22, which is lower, thus worse than the value of 0.53 from the benchmark.
'The ulcer index is a stock market risk measure or technical analysis indicator devised by Peter Martin in 1987, and published by him and Byron McCann in their 1989 book The Investors Guide to
Fidelity Funds. It's designed as a measure of volatility, but only volatility in the downward direction, i.e. the amount of drawdown or retracement occurring over a period. Other volatility measures
like standard deviation treat up and down movement equally, but a trader doesn't mind upward movement, it's the downside that causes stress and stomach ulcers that the index's name suggests.'
Which means for our asset as example:
• The Ulcer Index over 5 years of iShares MSCI Belgium ETF is 15 , which is larger, thus worse compared to the benchmark SPY (9.32 ) in the same period.
• Looking at Ulcer Ratio in of 14 in the period of the last 3 years, we see it is relatively greater, thus worse in comparison to SPY (10 ).
'Maximum drawdown is defined as the peak-to-trough decline of an investment during a specific period. It is usually quoted as a percentage of the peak value. The maximum drawdown can be calculated
based on absolute returns, in order to identify strategies that suffer less during market downturns, such as low-volatility strategies. However, the maximum drawdown can also be calculated based on
returns relative to a benchmark index, for identifying strategies that show steady outperformance over time.'
Applying this definition to our asset in some examples:
• Compared with the benchmark SPY (-33.7 days) in the period of the last 5 years, the maximum DrawDown of -38.2 days of iShares MSCI Belgium ETF is smaller, thus worse.
• During the last 3 years, the maximum DrawDown is -33.1 days, which is smaller, thus worse than the value of -24.5 days from the benchmark.
'The Drawdown Duration is the length of any peak to peak period, or the time between new equity highs. The Max Drawdown Duration is the worst (the maximum/longest) amount of time an investment has
seen between peaks (equity highs) in days.'
Applying this definition to our asset in some examples:
• Compared with the benchmark SPY (488 days) in the period of the last 5 years, the maximum days below previous high of 852 days of iShares MSCI Belgium ETF is higher, thus worse.
• Compared with SPY (488 days) in the period of the last 3 years, the maximum time in days below previous high water mark of 716 days is greater, thus worse.
'The Drawdown Duration is the length of any peak to peak period, or the time between new equity highs. The Avg Drawdown Duration is the average amount of time an investment has seen between peaks
(equity highs), or in other terms the average of time under water of all drawdowns. So in contrast to the Maximum duration it does not measure only one drawdown event but calculates the average of
Which means for our asset as example:
• The average time in days below previous high water mark over 5 years of iShares MSCI Belgium ETF is 325 days, which is larger, thus worse compared to the benchmark SPY (123 days) in the same
• During the last 3 years, the average time in days below previous high water mark is 344 days, which is higher, thus worse than the value of 177 days from the benchmark. | {"url":"https://logical-invest.com/app/etf/ewk/ishares-msci-belgium-etf","timestamp":"2024-11-04T10:50:17Z","content_type":"text/html","content_length":"59237","record_id":"<urn:uuid:a8f8002e-ebf7-4018-b3b0-17cf2e3c1356>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00203.warc.gz"} |
Quantum Field Theory, String Theory and Predictions (Part 8)
Last year, in a series of posts, I gave you a tour of quantum field theory, telling you some of what we understand and some of what we don’t. I still haven’t told you the role that string theory
plays in quantum field theory today, but I am going to give you a brief tour of string theory before I do.
What IS String Theory? Well, what’s Particle Theory?
What is particle theory? It’s nothing other than a theory that describes how particles behave. And in physics language, a theory is a set of equations, along with a set of rules for how the things
in those equations are related to physical objects. So a particle theory is a set of equations which can be used to make predictions for how particles will behave when they interact with one
Now there’s always space for confusion here, so let’s be precise about terminology.
• “Particle theory” is the general category of the equations that can describe particles, of any type and in any combination.
• “A particle theory” is a specific example of such equations, describing a specific set of particles of specific types and interacting with each other in specific ways.
For example, there is a particle theory for electrons in atoms. But we’d need a different one for atoms with both electrons and muons, or for a bottom quark moving around a bottom anti-quark, even
though the equations would be of a quite similar type.
Most particle theories that one can write down aren’t relevant (or at least don’t appear to be relevant) to the real world; they don’t describe the types of particles (electrons, quarks, etc.) that
we find (so far) in our own universe. Only certain particle theories are needed to describe aspects of our world. The others describe imaginary particles in imaginary universes, which can be fun,
or even informative, to think about.
Modern particle theory was invented in the early part of the 20th century in response to — guess what? — the discovery of particles in experiments. First the electron was discovered, in 1897; then
atomic nuclei, then the proton, then the photon, then the neutrino and the neutron, and so on… Originally, the mathematics used in particle theory was called “quantum mechanics”, a set of equations
that is still widely useful today. But it wasn’t complete enough to describe everything physicists knew about, even at the time. Specifically, it couldn’t describe particles that move at or near the
speed of light… and so it wasn’t consistent with Einstein’s theory (i.e. his equations) of relativity.
What is Quantum Field Theory?
To fix this problem, physicists first tried to make a new version of particle theory that was consistent with relativity, but it didn’t entirely work. However, it served as an essential building
block in their gradual invention of what is called quantum field theory, described in much more detail in previous posts, starting here. (Again: the distinction between “quantum field theory” and “a
quantum field theory” is that of the general versus the specific case; see this post for a more detailed discussion of the terminology.)
In quantum field theory, fields are the basic ingredients, not particles. Each field takes a value everywhere in space and time, in much the same way that the temperature of the air is something you
can specify at all times and at all places in the atmosphere. And in quantum field theory, particles are ripples in these quantum fields.
More precisely, a particle is a ripple of smallest possible intensity (or “amplitude”, if you know what that means.) For example, a photon is the dimmest possible flash of light, and we refer to it
as a “particle” or “quantum” of light.
We call such a “smallest ripple” a “particle” because in some ways it behaves like a particle; it travels as a unit, and can’t be divided into pieces. But really it is wave-like in many ways, and
the word “quantum” is in some ways better, because it emphasizes that photons and electrons aren’t like particles of dust.
To sum up:
• particles were discovered in experiments;
• physicists invented the equations of particle theory to describe their behavior;
• but to make those equations consistent with Einstein’s special relativity (needed to describe objects moving near or at the speed of light) they invented the equations of quantum field theory, in
which particles are ripples in fields.
• in this context the fields are more fundamental than the particles; and indeed it was eventually realized that one could (in principle) have fields without particles, while the reverse is not
true in a world with Einstein’s relativity.
• thus, quantum field theory is a more general and complete theory than particle theory; it has other features not seen in particle theory.
Now what about String Theory?
In some sense, strings also emerged from experiments — experiments on hadrons, back before we knew hadrons were made from quarks and gluons. The details are a story I’ll tell soon and in another
context. For now, suffice it to say that in the process of trying to explain some puzzling experiments, physicists were led to invent some new equations, which, after some study, were recognized to
be equations describing the quantum mechanical behavior of strings, just as the equations of particle theory describe the quantum mechanical behavior of particles. (One advantage of the string
equations, however, is that they were, from the start, consistent with Einstein’s relativity.) Naturally, at that point, this class of equations was named “string theory”.
An aside: theories of non-relativistic strings have appeared in the literature, for instance in Gimon et al. from 2002. There must be earlier versions, but I haven’t found them. [Experts will think
of light-cone gauge.]
A few more years of study of these equations led to a number of realizations about the simplest forms of string theory. Note my use of the vague term “simplest forms”. I’ve used this to alert you
that what I’m about to say about string theory is not always true. It is true of what people knew about string theory back in 1985 or so. But keep in mind this is not the final word on what “string
theory” means; it was, rather, just the first attempt. I’ll make this concept much less vague in my next post.
Anyway, here are some things that people learned about this theory in the 1970s and 1980s, presented in a somewhat ahistorical order.
Just as particles are ripples in fields, strings can be seen as a sort of ripple in string fields. (See Figure 1.) But unlike particle theory, which is actually incomplete unless one takes a field
point of view, the mathematics of the string equations make it easier to understand strings on their own, without the use of string fields. And string field theory is very complicated, and still very
poorly understood even today. This is part of why, for strings, people usually talk about “string theory” and not “string field theory”, while for particles, people usually talk about “field theory”,
not “particle theory”.
Fig. 1: Adding relativity to the original theory of particles (1920s) led to quantum field theory (1940s-1970s), in which particles are ripples in the fields. Note that not all quantum field theories
have particles, however. String theory (1960s-1970s) describes relativistic strings; its generalization to string field theory (1980s) has had limited, though interesting, applications. For this
reason one most often hears of “quantum field theory” and “string theory”.
In its simplest context, a theory of strings is equivalent to a theory of a huge number of fields and their particles. Roughly, even though there’s only one type of string, a string can move or
vibrate in different ways. A string vibrating in one way will appear in an experiment as though it is one type of particle; a string vibrating in a different way will appear to be a different
particle. In short, a single type of string, though so small it seems like a particle in current experiments, has many types of vibrations, and these would appear in current experiments to be many
types of particles. See Figure 2. The masses and other properties of these particles, and the forces by which they interact with each other, are arranged in special patterns, a point I’ll return to
in a moment.
Fig. 2: In a simple string theory, there are few types of string, perhaps only one. But a string can wiggle many ways (above). Viewed from afar, or by a macroscopic observer too large to see the
string has a size, the many vibrations of the strings appear (below) to be many types of particles.
The simplest string theories have only boson particles (particles which, like Higgs particles, photons and gravitons, have spin = 0,1,2…). The next simplest are superstring theories, which also have
fermion particles (which have spin 1/2, 3/2, …; electrons and quarks have spin 1/2.) To describe our world, then, superstrings are necessary.
Moreover, for a theory of strings to be consistent and stable (remember a theory is a set of equations — it has to make mathematical sense, or you can’t use it to make predictions of any sort) the
strings have to be superstrings, at least approximately.
In addition, for a theory of superstrings to be consistent, the strings have to [with a few subtle caveats] move in a space with 9 spatial dimensions. That is, adding on time as one more dimension,
they can exist only in universes with a total of 10 space-time dimensions. That sounds bad… but it isn’t. It might be that only a few of those dimensions are like the ones we know; the others might
be so short that we might not notice them, much as a sheet of paper appears two-dimensional if you don’t notice its thickness (Figure 3). So our world could be described by string theory as long as
there are six unseen “extra dimensions”, which are too short for us to detect with current experiments.
Fig. 3: To an observer who can’t see too well, a sheet of paper appears two-dimensional, but is really three-dimensional when investigated closely. Similarly, our world, which appears to us to have
three dimensions of space, may reveal more when investigated, in the future, with experiments that can probe very small distances.
In a (simple) superstring theory which describes a world similar to the one we live in, with three very large spatial dimensions, the pattern of masses for the particles in that universe would be
something vaguely like that shown in Figure 4. There would be huge numbers of types of heavy particles, with masses so large that we aren’t even close to producing them in our experiments, and in
the near term we have no hope of checking whether they can exist. Only a small number of types of particles, massless or with rather small masses, will be easily observable. These include (Figure 4)
• particles and fields similar in type, though with details that may vary widely, to those that we find in the Standard Model of particle physics (the quantum field theory that describes the known
particles and non-gravitational forces.) The equations that describe them would be those of quantum field theory.
• a graviton: a spin-two particle that is a ripple in a gravitational field. The equations for this field are those of Einstein’s general relativity, in which gravity is an effect of the curvature
of space and time. But these equations are generalized into a quantum mechanical form, which we call “quantum gravity”.
Let me say that again, because it’s a prediction — not a hugely impressive one, because it is vague and after-the-fact, but nevertheless, something deserving of the name.
Fig. 4: A vague prediction of string theory in its simplest forms: the world would have many types of very heavy particles, but an observer who cannot produce them would observe only a few types of
massless or low-mass particles in experiments. These particles and their fields would mostly be described by quantum field theory and gravity. This vague prediction roughly agrees with nature —
giving proponents of the theory hope in the 1980s that string theory provides a fully quantum description of our world, including quantum gravity, along with much else as yet unknown.
String Theory’s First Prediction (Vague, and with Loopholes)
In its simplest forms, string theory, at least naively, predicts that in a universe whose basic objects are superstrings, one will probably observe a quantum version of something similar to (and
possibly identical to) Einstein’s vision of gravity, and probably other very lightweight and massless particles not entirely dissimilar from the ones we observe, along with additional forces like the
ones we observe… all of which can be described using quantum field theory.
Now this is a remarkable prediction of the theory [again, I remind you, of the theory in its simplest form, i.e. as it was understood in 1985], because this prediction agrees with data.
But how excited should we be about this success? The prediction is quite vague; it’s analogous to the fact that (as I explained here) the simplest forms of quantum field theory (before you choose a
particular example) make only a few, very vague, predictions: that there will be particles in the world; these will be fermions or bosons; and any two particles of the same type will be literally
identical. This prediction of string theory is so vague that its success is hardly convincing; one could imagine there are lots of other theories out there that predict the same thing. And just as
complicated quantum field theories don’t always have particles, this simple prediction isn’t necessarily true when string theory gets complicated.
Furthermore, it’s a prediction of something we already knew about the world, which physicists sometimes call a postdiction. It is a lot easier to make a prediction about nature when you already know
what the answer has to be! An example of a postdiction is Einstein’s calculation that his theory of general relativity, which he was developing at the time, predicted the small observed shift in
Mercury’s orbit. The shift was already known from data, so he had a target to aim at… and that’s part of why people were mildly impressed (his theory could have failed at this step by giving the
wrong answer) but hardly convinced. It was (and is) much more impressive that the theory correctly predicted things that had never previously been observed — the deflection of light by the sun, the
gravitational slowing-down of clocks, the gravitational redshift of light, energy loss by radiation of gravitational waves, etc.
But historically and sociologically, this first prediction of string theory was a very important one. People had been trying for many years to find a quantum theory of gravity that would also be
consistent with the quantum field theories used to describe other particles… or perhaps something more general that would contain both Einstein’s theory of gravity and the quantum field theory that
describes the other forces and particles. It was a sort of “holy grail”, one that Einstein himself spent his last 30 years seeking fruitlessly. So you have to understand that it was quite
remarkable that a theory of strings, once the equations were worked out carefully, simply dumped the holy grail on the table without anyone asking or looking for it to do so.
Of course, we didn’t and still don’t know it is the only holy grail; maybe there are others. But at worst, a simple form of superstring theory is a very interesting example of how a quantum theory
containing both gravity and field theory might work!
Is Our World a World of Strings?
In short, back when physicists were still new to string theory, they found that simple forms of string theory could potentially describe universes much like the one we live in. And this fact
generated a lot of optimism that maybe there is a string theory that describes our own universe. But if so, which one?
By 1983, it turned out that most string theories anyone had written down were mathematically inconsistent (or had to be very complicated to make them consistent), but not all of them. A short list of
5 simple superstring theories survived all mathematical requirements, and only one seemed relevant for our own universe. I’ll say a bit more about them in my next post. And since there was only one
of these that seemed directly relevant, a huge wave of over-optimism swept across the string theory experts [I was an undergraduate, watching with some skepticism, at the time] that they were on the
verge of figuring out the complete theory of elementary particles, forces and space-time.
I should note that at this same time the string theory folks took the unfortunate step of annoying all the other physicists, along with chemists, biologists, social scientists of all sorts, artists,
writers, lawyers, and chefs by calling this potentially complete theory of particles, forces and spacetime a “Theory of Everything”. We won’t use that term here.
But This Was Just The Beginning of a Saga
Let me pause at this moment to make some important cautionary remarks.
Up to this point, the methods that people had used to study string theories were of a similar type used (as I described here) in the simplest quantum field theories. This is a method known as
“perturbation theory”, which is a technique of successive approximation: the full calculation is written as a simple estimate, plus a small but more complicated correction to the estimate, plus a
smaller but even more complicated correction to the correction… etc.
The problem is that — just as in quantum field theory — successive approximation only works when all relevant forces are relatively weak (in a technical sense — see this post). But there are some
questions in string theory — most notably, how does the world end up with 3 large spatial dimensions rather than 9? and why don’t the strengths of all forces end up being zero? — for which the method
of successive approximation is not good enough. When this was realized in the mid-1980s, progress in string theory slowed down markedly, and the goal of a complete theory of particles, forces and
space-time receded for the moment.
Nevertheless, some experts continued to explore string theory, and made slow but steady progress. And then, spurred in part by advances in quantum field theory in 1993-1994, a major set of
realizations occurred in string theory studies during the period 1994-1998. By the time that mini-revolution was over, what string theorists knew about this theory had dramatically expanded and
changed. And that will be the subject of my next post in this series.
109 Responses
3. Thanks for the link to your discussion of extra dimensional space. I was unaware of the prior tutorial. Two quick follow-ups if I may.
First, why are the Euclidean spatial dimensions seemingly unbounded (astronomical scale awareness), but the conceptual subatomic scale spatial dimensions are thought to be bounded and very
uniquely shaped? Why would nature be so fickle.
Second, the “time” dimension can be thought of as a surrogate parameter for the cosmological motion that has been ongoing since the Big Bang, and consequently has mathematical analog to movement
within the 3 spatial dimensions. As such, could the known spatial dimensions fluctuate minutely, thereby opening voids that may account for extra dimensional phenomena?
4. Just a quick question relating to the meaning “dimensions” in this context.
It seems to me that there are at least two possible conceptions of the idea that there may more than 3 spacial dimensions.
The first is that we (as human thinkers) lack the intuitive ability to visualize space in more than 3 spacial dimensions, and therefore develop confusing analogies or rationalizations which imply
validity and justification.
The second is that all spacial dimensions in excess of the 3 Euclidean dimensions simply overlap with the original 3, but we treat them differently because the math is easier within the context
of the newly envisioned space. Another way to say this is that atomic space overlaps astronomical space, but you cannot efficiently use astronomical math to solve problems in atomic space, so a
new thought paradigm is necessary. In this sense, the term “dimension” would really mean parameter.
1. No, dimensions really means dimensions of space. There is no sense in which they overlap with the ones we know; they are orthogonal to ones we know, in the strict logical, mathematical and
physical sense of “orthognal”, and in the same way that the thickness of a table adds to and is orthogonal to the surface area of a table. Read the articles in http://profmattstrassler.com/
articles-and-posts/some-speculative-theoretical-ideas-for-the-lhc/extra-dimensions/ . These dimensions are not some abstract concept; these are directions into which sufficiently small or
sufficiently free objects can really move, do dances, pirouettes, and all the things you can do in the diemnsions we know. In particular, an object that moves in the direction of one of those
dimensions has motion-energy, just as objects that move along one of the dimensions we’re used to — with an analogous Einsteinian formula relating energy, momentum and mass.
2. @TomA,
We could agree on the fact that the average layperson does not have the proper tools to visualize a space with any abritrary number N of dimensions, with N being any natural number larger
than 4, but humans are capable of visualizing such a thing, at least any human being that also happens to be a subject matter expert.
Just as Matt has described, dimensions on any metric space are all perpendicular (“orthogonal” is the proper word in the jargon) to each other, so, as they have no projections on each other
(they don’t cast a shadow on each other), there is not a chance that they could overlap.
The molecules, atoms and particles exist in the very same space that we do, within the very same dimensions, there is no “special” space for atoms, particles and quantum fields.
Kind regards, GEN
5. while googling for “string theory dark matter” i came across this site:
wich seems to be devoted to explain stringtheory to the non-experts too.
6. And it’s one of the reasons Einstein took such issues with all flavors of Quantum Mechanics and QED, in addition to his objections re. determinism and causality. So the whole concept of a “spin-2
graviton” gets to be problematical. And from my limited understanding, I believe that the equations of most of these “simple” versions of string theory still assume that the spacetime dimensions
act like a Minkowsky flat spacetime.
This is Smolin’s complaint about string theory, it’s lack of background independence. Is this really still true in M theory?
7. Matt, thanks for another excellent article.
Just one more suggestion re. being more specific. Several times you refer to “relativity”, when you should specify “Special Relativity” or “General Relativity”. For example, the quantum field
theory of the Standard Model is based only on the Minkowsky spacetime of Special Relativity; it is completely incompatible with the core concepts of General Relativity. It’s not just a technical
detail that GR can’t be quantized the way other forces are. In General Relativity, there is NO “force of gravity”, just curvature of spacetime due to the presence of energy and momentum (and
“stress”) within that spacetime. (We don’t feel a “force of gravity”; in open space, we just move along an inertial path through a curved spacetime: along a geodesic path according to the local
metric of that spacetime. We only feel a force that prevents us from moving along that path — e.g., the electromagnetic forces between the atoms in us and those in the surface of the Earth that
impede that motion, rather than letting us continue in free-fall.) All of the equations of QM and (most) QFT assume an underlying linear, flat spacetime — where, for example, “parallel transport”
between points in spacetime is independent of the path taken. (In GR, things like the rate at which time evolves and the angles between vectors between to two points can depend on the path taken
between them, in non-linear ways. Even concepts such as “velocity” can get tricky and subtle.) This, along with the concepts of general covariance and the non-linearity of Einstein’s equations,
are what keep GR from being equivalent to a perturbative theory of gravity. And it’s one of the reasons Einstein took such issues with all flavors of Quantum Mechanics and QED, in addition to his
objections re. determinism and causality. So the whole concept of a “spin-2 graviton” gets to be problematical. And from my limited understanding, I believe that the equations of most of these
“simple” versions of string theory still assume that the spacetime dimensions act like a Minkowsky flat spacetime.
However, one should also note that Einstein’s Equations in General Relativity are notoriously difficult to solve and there are a wide range of possible solutions, most of which result in
universes nothing like our own. We can describe the observable universe on large scales very well with a very simple Friedmann-LeMaitre-Robertson-Walker solution with a few minor adjustments for
things like dark matter, but describing things like the spacetime around black holes requires completely different solutions. However, useful solutions and approximations for GR were found and
exploited very early on to make testable predictions, and led to its adoption. But I would argue that we still probably don’t know what “the” correct General Relativity solution is to fully
describe our real universe. So, just as with QFT, to be precise you often need to specify whether you are referring to general properties of GR (such as the above), or specify _which_ solution of
General Relativity you are talking about.
1. @Wlm,
If you check on many other posts by Matt in this very same blog, he’s very precise in mentioning that when he uses the term “Relativity” in the context of talking about some detail of a
Quantum Field Theory (any given QFT, actually), he always means the “Special Relativity” (SR) kind that is the one that applies to quantum mechanical effects.
Besides, from a purely relativistic perspective, SR is completely contained within GR, or we could say that SR is just a sub-set of GR, and this is simple to prove: if we use the complete set
of equations for GR for the case of inertial reference frames (the only types of reference frames considered by SR), which it would mean to assign zero to certain terms in the complete set of
equations, and them we simplify the equations by eliminating the terms that cancel out, voilá!, we end up having a simpler set of equations that are exactly the SR equations.
Regarding gravity not being a force from a GR perspective while at the same time from a Standard Model perspective it is a quantum mechanical force, with its own vector boson as mediator,
there is no issue with that.
The simplest way to show that there is no real issue with that is by remembering Niels Bohr’s Correspondence Principle.
Kind regards, GEN
1. A graviton (spin-2), is it not a tensorboson?
1. It’s a tensor boson… but specifically a spin-two tensor boson. There are many types of tensor bosons, because the term “tensor” is very general, much more general than “vector” [DEL:
of:DEL] or “scalar”.
1. Thank you.
BTW of is Dutch for or 🙂
2. Gaston,
Yes, you are right that Special Relativity is, by design, a subset of General Relativity. One of Einstein’s criteria when devising GR was that SR is what is obtained in the “limit” of
zero curvature. But it is a very proscribed subset (with a specific, constant Lorentzian metric) that does not reflect the complexities and subtleties of GR, either mathematically or
geometrically. This is especially true when you have equations with derivatives based on a linear time parameter. So basing a QFT just on SR, and not GR, has some fairly radical
consequences — like being able to write consistent QFT equations from a Hamiltonian and solve them perturbatively (or non-perturbatively), which still hasn’t been accomplished for a
realistic curved universe in 3+1 dimensions.
But I wrote my post because, from some comments above, some people seem to be confusing the term “relativity” with “Einsteins’s theory of gravity”, when Standard Relativity (that was
incorporated into QED and later QFTs) has nothing at all to do with gravity. Hence my suggestion.
1. @Wlm,
You have a point with that.
I’m not an expert on these matters, but it is my understanding that the problem that you describe is mostly significant for those events of our universe where you need to combine both
QFT and GR equations to describe it, like say, the collapse of a massive star, including black holes but not limiting to them.
Besides this, the combination of QFT and GR equations might also be significant to further the study of the hierarchy problem (this is a guess of mine).
Differential geometry is a relatively new segment of mathematics, it is a rather complex subject to tackle, in particular when you consider its use to apply it to describe GR, since
you have to find a way to work out how to mix and match constraints from non-euclidean spaces (the types that are useful for GR) and the physical constraints required for GR, such
that it all makes sense.
Kind regards, GEN
8. String theory is good because it gives you a “holy grail”? Well, I guess that’s consistent with my general impression of string theory: a quasi-religious enterprise pushed by a small band of
die-hard enthused zealots. Evidence-wise, it’s a complete bust. String theory is based on SUSY, and SUSY is on life-support, it’s predictions not having been confirmed by LHC.
1. @M Mahin,
Even though the LHC data collected so far has put into question certain variants of SUSY, there are other variants of SUSY that are still into the competition in rather good shape, so, it is
not proper to assert that the LHC has ruled out SUSY en bloc.
9. Why must there be finite dimensions? Yes, I believe every variable (of nature) has a finite range or else the universe would not exist, i.e. either drift (tear itself apart) to infinite space or
collapse (dissipate) to “absolute zero” (as opposed to ZPE state).
We perceive ourselves living in a 3D world because that is how we elected to write our equations, x,y & z. An astronaut in orbit cannot make the distinction between linear and rotational motion
without an external reference frame, he will only see changes of speeds (velocities w/o directional cues). Is it because the dimensions are infinite and we and everything else are moving in
curved space? Every point in space is moving relative to the next, adjacent point, in infinite directions, but to an external observer you see a continuum of space (3D volume).
10. “The biggest problem in this whole business — and that is why I am writing these posts — is that no one who is prominent in the public string debate (Greene, Woit, etc.) is sufficiently precise
about what they are talking ”
so why do you think these brilliant physicists are not being precise.?
covering their bases ?
11. “Each field takes a value everywhere in space and time, in much the same way that the temperature of the air is something you can specify at all times and at all places in the atmosphere.”
Except that temperature can be measured.
Writing F(x,y,z,t) does not imply that F has physical existence. A field thus takes values at points of the spacetime coordinate domain, not in real space. In the same sense that the lattice of
lattice-QCD is not a physical latttice.
The fact that QFT is only an effective theory implies directly that the fields do not really exist. A Lagrangian is not something that flies around in space, either. In QFT particles do not even
have a position. That is why everything is described in terms of impulse p.
And therefore particles cannot be said to be ripples in a field; they are described (approximately) by field ripples, that is something else.
1. Martin,
Well said. I had the same response when reading Matt’s description. “Temperature” came about because macro-sized things were discovered to expand or contract in a way that corresponded to how
“hot” something seemed. It was something that could be used consistently to get useful experimental results. It was only later that it became incorporated into a methematical theory of heat.
And then it turns out that it really corresponds to a statistical, probabilistic average related to how often and how hard (with what “kinetic energy”, in Newtonian physics terms) a large
group of molecule-sized objects bang into each other (or a measuring probe). Does it exist “at every point”? Only theoretically. What’s the “temperature” inside a proton? Does the space
between molecules in a very rarified gas have a real temperature, or only a theoretical expected value?
This seems like an example of the “physics is math; math is physics” logical error. Another one: that Fourier analysis shows that every wave is the sum of an infinite number of harmonics of
waves up to those with an infinitessimally small wavelength. Does a sound wave — a group of a large but finite number molecules shoved closer together then farther apart for a brief period of
time — vibrate with all possible wavelengths at every point? It’s an excellent model, but is it “real”? Of course not. But that’s what you need to assume to make the math work out right, and
to get answers that correspond to things on the scale that we can measure.
The added problem with QFT is the very nature of the “fields”. These are very abstract things that are “vibrating”: an infinite space of an infinite series of complex-valued functions with
some very specific properties under some very specific kinds of mathematical operators. That’s why Matt can’t quite explain “What’s vibrating” — because the answer is hard to even describe,
much less translate into anything someone without detailed knowledge of the math can understand.
Do these “fields” actually exist? The correct answer is that no one knows. It seems clear that, whatever subatomic “particles” really are, they are absolutely nothing like tiny little
billiard balls. We may have absolutely nothing in the macro world that is anything like them, and the descriptive label “particle” is a complete misnomer. But just like our very early
understanding of temperature, that does not mean that our current mathematical way of describing their action will turn out to be of much use in describing what is actually going on and what
they actually are. It may only provide some of the criteria to help us tell whether future efforts to better describe them are getting things right or not.
12. I also have a question:
The Higgs Boson, a ripple in the Higgs field, has spin-0.
The Photon, a ripple in the electromagnetic field, has spin-1.
The Graviton, a ripple in the gravitational field, has spin-2.
Different fields, different ripples, different functional specs , different technical specs?
13. To Matt Strassler: “a graviton: a spin-two particle that is a ripple in a gravitational field.” -> why is a graviton predicted to be “spin-two” particle? instead of any other spin:0, 1, 3, 4..
14. Oh! I am sorry. Rick mentions the same thing in the comment just above his last comment. I noticed it after I posted the above comment and there was no way to cancel my comment. Anyway ,when you
get time, I would also like to see your opinion on this.
15. If the critical dimension in bosonic string theory is
D = 2 – 2/(1+2+3+…), then doesn’t that mean D = 2 since the sum in the denominator diverges? Since D has to be at least 4, doesn’t that disprove string theory?
1. Your premise is wrong, so your conclusion is too. There are some 2 (i.e. 1 + time) dimensional string theories, but the critical dimension in what people usually mean by bosonic string theory
is 26 (i.e. 25 + time). There are other problems with bosonic string theory, which is why superstrings (with critical dimension 9+time = 10) are generally what you hear about.
1. Matt: There is a discussion going on another blog that actually the apparently divergent seies 1+2+3+…. should be interpreted as -1/12 by regularization etc. Then the formula quoted by
Rick would work OK as 26. What is your opinion on this procedure?
1. Kashyap,
See my response to Rick above. I think the formula is ok for total number of dimensions (assuming 1 of time the rest of space) but doesn’t this only hold for certain String theories?
Regardless, it is bad form to write it as D = 2 -2/1+2+3… (unless you are trying to impress/confuse someone), rather D = 2 – 2/ζ(-1) = 26.
1. @S. Dino. You do not have to believe in String Theory to be amazed by such results. I am not sure but the theory may give 1+2+3..before giving zeta function. Perturbation theory
if you take literally as it comes diverges anyway. For one thing, the summation of such divergent series was given by all time great mathematicians, such as Euler, Borel,
Ramanujan and others who did not know anything about ST. Motl’s blog has several references and one excellent video lecture by Carl Bender explaining this stuff. Good
mathematicians accept these results. Analytic continuation is also acceptable in general. By the way, I asked a question about D for fermions and such infinite series give also a
correct answer D=9 (10 with time). Disputes about validity of ST is altogether a different question. For me the most amazing thing is that such amusing mental gymnastics gives
agreement with experiments (sometime) which people perform using things which they can hold in their hands and are as real as one can get!!
16. Can you please shed light on this whole thing about why 1 + 2 + 3 + 4 +… equals -1/12 and its relation to string theory? I’m sorry, but I have no idea why that divergent sum must equal -1/12. Any
light shed on this issue would be greatly appreciated! I’ve seen many people on many different blogs talk about this (eg. Lubos).
1. Rick,
Of course 1+2+3+… = infinity. However 1+2+3+… = 1/1^-1 + 1/2^-1 + 1/3^-1 = ζ(-1). IF the extended definition of the Zeta function is used ζ(-1) = -1/12. You can extend the definition of ζ(s)
to s < 1 using a process called ‘analytic continuation’. Using this in a consistent manner (regularization of divergent series) it is possible to have some unwanted nasty infinities cancel
out. This is deeply linked to the renormalization of divergent integrals in String Theory. Hope this helps…
1. Oops, meant 1+2+3+… = 1/1^-1 + 1/2^-1 + 1/3^-1 +… = ζ(-1). Forgot the +… after the 1/3^-1 above.
1. S. Dino,
Thanks very much! That was helpful. Do you, by any chance, know why one can get the same answer (-1/12) by doing funny tricks with the series 1 – 1 + 1 -… = 1/2. I don’t know if you
have seen the Numberphile youtube video which shows a tricky way to evaluate 1 + 2 + 3 +… without using the zeta function.
17. I am glad that you continue to help to explain the basis of the physics approach as a fundamental description. This is important in maintaining some hold on the purpose of this foundational math
as a attribute. To me, it is very important in light of the mathematics chosen, as a descriptor of the given reality.
18. I’ve found your expositions to be remarkably clear and hugely illuminating, and I’m deeply grateful for you efforts. Thank you.
19. “Vague and with loopholes”
I guess that is what I don’t like about string theory. Its predictions, as you noted, are vague and with loopholes. To me that is not a very satisfying construct. To me a useful scientific
theory, regardless of how it is arrived at, should be testable with todays’ technology or technology we can strive for in the near future. Theories that do not meet this criterion should be put
on the shelf until they do become testable.
It helps physics ADVANCE when theories that make very specific predictions and without loopholes are found to be in error by experiment or discovery thus forcing a scientific revolution and
ultimately the development of a new theory whose equations reduce to the older theory under the conditions it was successful but also predicts new phenomena that accounts for the experiment or
discovery that the older theory could not. And so it goes…advance after advance…scientific revolution after scientific revolution…
If anybody thinks the physics of the 31st Century will look like the physics of the 21st Century … good luck with that.
1. By the equivalence principle, it must look *almost exactly* like it! 😉
2. People apply a double-standard to String Theory, asking it to predict everything, and complaining when it can only make predictions once assumptions are made. Well, field theory also makes
only vague predictions with loopholes, until you make assumptions: that there are electrons and photons and quarks, that they have masses with certain values, that gravity is not important
for particle physics, that the universe has a roughly flat space-time with slow expansion — etc. Newton’s laws do NOT predict the planets’ orbits, until you first measure the current location
and velocity of the sun and the planets. Every theory requires some level of input before it can make precise predictions; if you don’t put in any inputs you get vague predictions with
loopholes. For example, without inputs, Newton’s laws only predict that you *can* have planets orbiting stars; they don’t tell you whether any given star actually has planets, and they don’t
tell you the radii of the orbits or whether there are several planets orbiting in a plane or anything like that.
This is what I’m talking about when I say that people are not precise in their thinking. They apply one standard to one theory and a different standard to a second theory, and complain that
the first theory is worse than the second because it fails their expectations, even though the second theory would fail the same set of expectations if those expectations were applied to it.
1. Just in case you mean people like me when you say “some people”; I am not applying a double-standard to String Theory, asking it to predict everything. I am asking it to predict
something, anything, that is testable beyond what GR and QFT predicts. Input whatever assumptions seem reasonable to you – do you get any testable predictions beyond GR and QFT?
Also you note that the string equations are consistent with Einstein’s Relativity. So let me ask you a simple question. If Relativity turns out to be wrong is String Theory then also
3. I think the ability of a theory to make many precise predictions, and the value of the theory that it gets as close as we can get to describe our world, are two different things. A theory may
have difficulties to make predictions, at least on our current incomplete level of understanding, and with experimental technology confined to our present capabilities, but still may be, in
the long run, the best way to describe (or even “explain”, althoug the meaning of this word is not precisely defined here) our physical world.
Einstein thought that some of the predictions of his theories might never be testable because the effects are really small, but this did not stop him in believing his theory is correct. Only
some decades later technology and ideas for experiments have sufficiently advanced to experimentally proof the existance of gravitational lenses predicted by Einstein et al.
This suggests that theories with predictions that are difficult to test should not be thrown out too early. They might be the best theory available, and a clever idea for a new experiment or
an unexpected technological advance may put new ways of verification more rapidly into reach than expected.
There is no reason to expect that Nature´s laws are, in the set of all possible laws, those that are most easily understood or experimentally detected by human beings.
1. Hi Markus,
Einstein put forth GR in late 1915. Observations of the deflection of starlight by the Sun in accordance with the theory came in 1919. I never intended that EVERY prediction of a new
useful theory must be testable by the technology of the day or near future .
20. Newton used Kepler’s Laws of planetary motion to inductively reason the equations for gravity, that is, Newton started from the facts determined by previous experiments.
As far as I can remember, the only idea that Newton “inserted” out of nowhere into the theory was the inverse-square law: Kepler’s Laws gave him no insights regarding this, and it was the only
way he found to make gravity work in such a way that he could calculate the orbits and periods for the known planets.
21. More and more I’m not buying it. As pretty as it is, it is pure inductive logic. Statistical probabilities based on mathematical models doesn’t cut it for me. There is something inherently wrong.
Rather than using deductive reasoning we have devolved into quasi inductive reasoning science. It could be that I’m still thinking 19th century physics of cause and effect, but I don’t think so.
What does the evidence of experiment show and what does it mean without preconceptions?
1. Physics has never used deductive reasoning, since it became successful. It’s always been inductive; you inductively reason from experiments.
22. It is my understanding that getting the Mercury perihelion correct was more impressive than the light deflection experiments, even though the former was long known. In truth, it is the very fact
that the Mercury anomaly (for Newton) was known and that scientists had long tried to account for it (through means determined to be ad hoc) that made it so impressive when GTR predicted it
directly. I’ve written this quickly in passing, but have discussed it in a book of mine….
1. Part of this balance is that the initial light-deflection measurements weren’t very convincing. Yes, the other suggestions for how to fix Mercury’s orbit tended to be partly ad hoc, but they
showed that there were multiple possible explanations, not all of which required something quite so radical as a rewriting of Newton’s gravity! Dust was another possibility, though the case
became weaker over time.
In any case, many people did not believe Einstein’s theory through the 1920s, despite the postdiction of Mercury’s orbit and the weak confirmation of the prediction of the bending of light.
The evidence wasn’t viewed as strong. Only in the 1960s, when more tests became available, did the case become more convincing.
1. Let me say this more clearly.
Mercury’s orbit was shifting by a certain amount per year — that is, by a certain angle. Many people came up with models for where that shift could come from. Einstein’s theory predicted
it in a single shot. That’s impressive. But still, it was a prediction of one number.
Supersymmetric grand unification, similarly, predicts one number… which you can phrase, if you like, as the mixing angle known as theta_weak that tells you how electromagnetism is a mix
of hypercharge and weak isospin. Do we therefore believe in supersymmetric grand unification? Some people do. Some people don’t. In this case, the prediction wasn’t even entirely a
postdiction, because the data got better and the prediction survived the improvements.
Generally, we don’t start believing theories as a community until they start predicting many things… many numbers, or even entire functions, which we obtain through many different types
of measurements. It’s one thing to be impressed, another thing to be convinced. When Einstein’s theory starting giving correct answers for multiple types of measurements, including some
that humans could set up themselves, that’s when the balance really tipped.
1. My model might be under investigation at Fermilab. Have you tipped them?
1. No.
2. Just to support Matt’s point regarding this.
The existence of the planet Neptune was first predicted by Le Verrier during the first half of the XIX century, based on the irregularities measured in the orbit of Uranus.
Then Neptune was discovered just where Le Verrier had predicted that it should be.
Also, during the XIX century, Bessel had predicted that (Alpha) Sirius (the star) might be actually a binary star system, accompanied by a smaller (and dimmer) but very dense star, by
doing calculations based on the “wobliness” of the trajectory of (Alpha) Sirius.
Beta Sirius was discovered in 1971, and Bessel’s predictions were validated.
Because of these events, some astronomers during the XIX century proposed that the anomaly in Mercury’s orbit could be due to the gravitational influence of an unknown planet.
The truth is that Einstein’s postdiction of the actual deviation was not that impressive. What was really impressive at the time (1916) was the qualitative and quantitative prediction
that heavy bodies like stars can warp space-time in such a way that the path followed by light is not a straight line in the vicinity of said massive objects, or that heavy bodies
like stars or planets can warp space-time in such a way that time runs slower when measured closer to the massive object.
The slowing down of time by massive bodies could not be detected by experiments until the 1970s, with the use of atomic clocks.
23. It seems pretty clear that any expert in HEP may realize, sooner rather than later, from a conversation with somebody else if that other person is or is not an expert in the field, too.
24. Many theories predict mass of particles in much more explicit way, than the string theory does already (www.blacklightpower.com/wp-content/uploads/theory/TheoryPresentationPt3.pdf, arxiv.org/ftp/
arxiv/papers/1308/1308.1849.pdf, http://www.cosmology-particles.pl/files/LP.pdf, quantoken.blogspot.cz/2005/02/guitar-predictions-of-muon-mass.html) We shouldn’t forget Heim’s theory in this
1. String theory doesn’t predict the masses of the known particles at all, or even what the types of particles that we find in our universe should be, so you’re setting a very low bar.
25. I do hope you will say a few words about the cancellation of gravitational anomalies. It was the first instance of a recurring phenomenon appearing all across string theory. Namely the fact that
string theory is overly constrained and could prove inconsistent in dozens of ways, but never does.
1. You’ve said this in a strange way, and I’m not sure I completely understand or agree with the way you said it — maybe you can clarify it? I would have said that most string theories proved
inconsistent. And so instead of your statement that “string theory is overly constrained and could prove inconsistent in dozens of ways, but never does“, I would have said “… and usually (but
not quite always) does.” The amazing thing is that after all the dust settled, and most string theories were completely dead, there were five left over that were, remarkably, consistent… and
that even more amazing (next post) they all turned out to be related to one another.
1. Indeed, my formulation was not precise. By “string theory” I had in mind the 5 five remaining consistent string theories. My point is that after the worldsheet analysis which selects
them, there is a long list of consistency constraints, starting with the cancellation of the local gravitational anomalies in the effective spacetime supergravity, which in all likelihood
should have failed. The fact that they did not is what really launched the subject. For instance the anomalies of type IIB supergravity cancel because an over-constrained system of 3
linear equations in 3 variables has a non-trivial solutions. Type I looks naively inconsistent, but, the theory miraculously cures itself through the Green-Schwarz mechanism.
1. Thanks…
26. Matt: your remark” in particular, whether such theories can really be stable, and thus really describe any long-lived universes.’ Tell me if this is correct. Without SUSY Higgs mass is so high
and/or Higgs is unstable so that universe is not stable. Is this correct?
1. It’s not correct, because there are several different notions here, and because you’re making an error.
First there is the issue of the hierarchy problem, also known as the naturalness problem. A universe with the Standard Model and a low-mass Higgs, with nothing else nearby, is, among similar
theories, extremely unusual. That does not mean that it is extremely unstable.
Supersymmetry at the LHC scale is ONE OF MANY ways to solve or mitigate this problem. You are making a big mistake if you think supersymmetry is the only possible solution.
Then there is, separately, the issue of *metastability*. It turns out that for a Higgs of mass 125 GeV/c^2 and a top quark of about 173 GeV/c^2, the Standard Model may be very slightly
unstable (“metastable”). This is not a big deal since the universe will still probably last far, far longer than it has so far. In short, this is not a problem that requires a solution — it’s
perfectly fine, as far as our universe is concerned, if this is the case.
When I say that these problematic string theories don’t describe stable long-lived universes, I mean that the universes that they could host would last less than a tiny tiny fraction of a
tiny tiny fraction of a tiny tiny fraction of a second. THAT’s instability.
1. Thanks for clarification. Unless I misunderstood there is a possible typo in “In the simplest contexts, superstring theories in more than 9 spatial dimensions have anomalies (and string
theories that aren’t superstrings have anomalies in more than 25 spatial dimensions.) Shouldn’t “more” be “less” ? i.e to avoid anomaly you have to go at least 9 and 25 dim resp?
1. No, there’s no typo. Roughly speaking: you need to have 9 and only 9 to avoid anomalies; 6 is bad, 13 is bad, etc. You can have less than 9 only in a subtle way, by replacing the
missing dimensions with something that isn’t really geometry but leaves a comparable number of particles in the theory (very technical subject.) You certainly cannot have more than 9
— within weakly interacting string theory. You can have 10, in fact, if interactions are strong, but again only in a subtle context to be described soon. You cannot have 11 or more in
a superstring-like context.
1. Thanks again. I see that there must be equations where you have to put 9 or 25!!. This proves for the nth time that it is too much to hope for that one can understand theoretical
physics without math!!
1. In the end, yes, the math really matters.
2. Theories in physics must follow certain rules:
They must be mathematically consistent, as they are deduced out of the use of a certain “section” of math, then they also have to be consistent with existing principles and
concepts of theoretical physics, and if the theories are “lucky” enough, they are also favored by nature, that is, they are validated by experiments.
Once you take into account the constraints to the math applied by the consistency with physical principles and concepts and any additional constraints that could come from the
experiments, you can just use the math to deduce new properties, concepts and equations to the theory.
So, math is the language of science, but in science, the math that is used is bound by certain constraints.
27. Excellent article, as always, very exciting stuff and well described, helps even the layperson, like myself, visualize and dream, thank you Prof.
Even, after reading (and studying, playing mind games with various permutations of all ideas, theories) I still converge to a universe which is merely an ensemble of distorted space-time. Here is
one of my dreams (illusions? :-));
A large, circular flat (2D), uniform, flexible fabric … stretch it tight, up to but not beyond it’s plastic deformation stress-strain point, … fix the circumference solid in a complete 360 degree
clamp. Now go in center of this enormously large surface area fabric and pinch a 1-2 inches in length and fold it once but careful not yield (go beyond plastic deformation) and release, … it will
“spring” back to it’s original shape. Now fold it again us many times necessary to yield the two ends (pinched ends) so it stays folded.
Here is the dream part, :-), assume the fabric is space and the “yielding”, the change of parameters, characteristics, is time. Yielding of the boundaries of this small section of the fabric is
like time dilation caused by
the rotations of this affected space and at some point when it yields, reaches a critical velocity (since a section of space is changing at different times, vectors will be created) it will take
hold and keep those parameters, characteristics.
This critical velocity would be the “constant” speed of light, the minimum rate of change of space to maintain stability.
Finally, if you assume this pinched section to be the quantum of “distorted space” apply it in a 3D world and an infinite array, in all 3 directions, of these quanta. … (gravitons?)
Do I need professional counselling? 🙂
28. An additional point. Even without String theory the existence of a landscape is strongly supported in cosmology. So I don’t mean to imply that the landscape counts against string theory, quite
the opposite in fact. One could argue, I think . that our current best understanding in cosmology which is well supported by observation counts as support for string theory. I think many
cosmologists think this way.
Strongly implied might be better.
29. . An additional point. Even without String theory the existence of a landscape is strongly supported in cosmology. So I don’t mean to imply that the landscape counts against string theory, quite
the opposite in fact. One could argue, I think . that our current best understanding in cosmology which is well supported by observation counts as support for string theory. I think many
cosmologists think this way.
1. That may be; but I would argue that the landscape fails to explain some obvious things about LHC physics, so that cuts the other direction.
30. In response to Matt
I agree Matt that not “all” aspects of the laws of physics can be environmental, there has to exist a meta theory of the landscape itself. Perhaps the Noether symmetries are fundamental in every
vacua state.
I still think the landscape is an unfortunate development though it is believable ., especially based on our best cosmological theory of origins , eternal inflation and quantum tunneling in third
Let me clarify my point about SUSY. I know SUSY isn’t direct evidence for String theory, but String theory needs SUSY, so the lack of SUSY at CERN leaves us without something String theory needs.
I think that’s not good news for String theory. It’s true that SUSY can break outside the reach of the LHC in string theory, but a SUSY breaking at such high energy means it can’t do the things
we expect it to like solve the Hierarchy problem. This will cast a shadow over SUSY for sure and consequently string theory which needs SUSY. So I don’t think I am in contradiction here.
1. I still don’t agree with the logic of your third paragraph.
2. Ruling out SUSY that can solve the hierarchy problem is a bigger problem for SUSY on its own. It would make SUSY not as interesting. That does not necessarily mean a String Theory which spits
out a non-hierarchy-solving SUSY is also not interesting, because SUSY is not the only feature of String Theory.
31. Matt–
Just a terminological/semantic issue that might be a source of confusion for readers whom I’d like to point to this article — it regards the way that you say that “quantum mechanics” was
historically not capable of handling relativity and was replaced by quantum field theory.
As you well know, quantum mechanics is the overarching framework of Hilbert spaces, operators, Born’s rule, etc., that contains as examples nonrelativistic systems, quantum field theories, string
theory, etc. We often refer to (0+1)-dimensional QFTs as “quantum mechanics,” when what we really mean is “nonrelativistic quantum mechanics”, i.e., a quantum-mechanical system with finitely many
degrees of freedom.
Quantum field theory is a different model inside quantum mechanics that improved on the nonrelativistic quantum mechanics of finitely many point particles, but QFT was not a replacement for
quantum mechanics as a whole, which has survived for a century.
I just don’t want lay readers (including young people who don’t this stuff yet but might be interested in going into physics) to get the idea that quantum mechanics was overturned by quantum
field theory, or, at least so far, by anything else.
1. @Matt297,
It was Paul Dirac’s equation of the electron that offered the best and closed approach of Quantum Mechanics (QM) to integrate itself with special relativity.
It was a very good interim solution, but it was far away from being able to explain how matter interacts with the electromagnetic force bosons or photons (that would require QED to really
solve and explain).
QM was able to predict quantum entanglement, but it did not predict the infinite stories that pertain to the paths of particles … That would have to wait for QED to predict that.
It took the genius of Dick Feynman’s intuition to think about the double slit experiment from a completely different perspective to fully grasp this concept of a particle running down all
possible paths at the same time to unleash and let free the concepts behind QED.
Kind regards, GEN
2. Your point is a fair one. Let me think about how to handle it. I usually use “quantum theory” to mean “quantum mechanics the framework” and use “quantum mechanics” to mean “quantum mechanics
the particle theory and as taught in undergraduate courses.” In this post I am indeed in danger of misleading readers about this point… though the fact that the word “quantum” appears in
“quantum field theory” and in my discussion of string theory should help mitigate this.
1. I concur with you on these caveats: QM is a particle theory and not a QFT.
2. QM (the particle theory) and QED (the QFT) make the same numerical predictions of many experiments, but the equations and explanations are very different.
QM, the particle theory is a nice first try, but is neither as complete a theory nor a proper approach to really understand and describe all the experiments.
1. GEN– With all due respect, what are you talking about?
Look, I have other work to do, but on the off-chance that this discussion leads to confusion among readers, classical field theory is a branch of classical mechanics, and quantum
field theory is a branch of quantum mechanics. QED is a kind of QFT which is a kind of quantum-mechanical system, just as Maxwell’s electromagnetism is a kind of classical field
theory which is a kind of classical-mechanical system.
Every quantum-mechanical theory, including any QFT, has a Hilbert space, satisfies a Schrodinger equation over that Hilbert space (even if the wave-function is actually a
wave-functional), and has the Born rule for computing probabilities of measurement outcomes. Every quantum-mechanical theory with continuous degrees of freedom, whether
nonrelativistic point-particle theories or quantum fields, has Feynman path integrals in it. The nonrelativistic harmonic oscillator has path integrals just as does QED.
The discussion with Matt Strassler is just over terminology—whether one should use “quantum theory” for the overarching class that contains nonrelativistic quantum-mechanical systems,
QFTs, and string theory, or whether one should use the term “quantum mechanics” for that overarching class. I have argued that to avoid confusion or ambiguity, one should not use
“quantum mechanics” to refer solely to nonrelativistic systems of quantum particles, but that one should explicitly say “nonrelativistic quantum mechanics.” That’s the entire point of
this discussion.
1. I agree about the terminological discussion as the main issue that you were arguing.
Factical sciences and other professions (like, say, engineering) that derive from factical sciences need the use of unambiguous terms and definitions.
2. Matt297 is correct, GEN, that this is purely a matter of terminology. Some people would say “classical field theory” is a branch of “classical mechanics”; others would say it is a
branch of “classsical physics”; and this is purely an issue of definition of terms. There is a similar issue about how the words “quantum mechanics” are used; as I said, I use
“quantum theory” whereas Matt297 uses “quantum mechanics” to describe the full range of quantum phenomena. No one is right or wrong; there is simply an issue of needing to be
clear to non-experts.
3. Matt–
So I’m basically just suggesting a slight clarification — maybe something as simple as modifying your statement to “But nonrelativistic quantum theory [my edit] wasn’t complete
enough to describe everything physicists knew about, even at the time. Specifically, it couldn’t describe particles that move at or near the speed of light… and so it wasn’t
consistent with Einstein’s theory (i.e. his equations) of relativity.”
Maybe also in parentheses as note about this terminological ambiguity, something like “(A note on terminology: I use the term “quantum mechanics” here to refer just to the
specific nonrelativistic equations that needed to be replaced by quantum field theory, whereas other use “quantum mechanics” to mean the whole framework of using quantum theories,
in which case quantum field theory is just an example.)”
4. I mean, Sakurai’s classic book on quantum field theory is called “Advanced Quantum Mechanics.”
Meanwhile, Wikipedia is rather schizophrenic about all of this. (http://en.wikipedia.org/wiki/Quantum_mechanics) It says that “quantum mechanics” and “quantum theory” and “quantum
physics” are all synonymous, but then also says of quantum mechanics that “It is the non-relativistic limit of quantum field theory (QFT), a theory that was developed later that
combined quantum mechanics with relativity.” The Wikipedia entry for “quantum theory” is just a menu of redirects to other entries.
32. As I understand it, the uncertainty principle is a natural consequence of certain properties of the math (the equations) used in the quantum theories.
The equations used have a general form that contains eigen values and eigen functions.
The quantum numbers that determine the (quantum) state of the system are contained within the eigen values.
The eigen values determine the quantized properties of the system. Certain combinations of properties have such behavior that both the values of their quantum properties cannot be determined
(measured) with certainty at the same time, even though they do allow to be measured with certain one at a time.
These certain pairs of quantum properties that show this “weird” behavior are described as non-commuting operators.
Again, it is a natural consequence of the math used by the equations. Just as Benjamin Peirce said: “Mathematics is the science that draws necessary conclusions.”
33. Great series of posts. I have a question about how the uncertainty principle comes into play in both String theory and QFT.
Does the uncertainty principle have to be included ad hoc into both these theories? Or does is “pop out” of the equations somewhere? For example, in QFT, if I write down a Lagrangian, how do I
know it obeys the uncertainty principle, or does it not have to?
1. I think an uncertainty principle arises in pretty much any theory with waves. It’s really a property of Fourier Transforms, which are pretty fundamental to the study of anything that
A function that looks like a lump of some kind can be written as a sum of sinusoids at different frequencies. The narrower you want to make that lump, the wider the set of frequencies of
different sinusoids you have to add together to get it. So how precisely the lump occupies a particular position is inversely proportional to how precisely it occupies a particular frequency.
That’s the gist of the uncertainty principle.
1. @Xezlec,
Any classical physics phenomenon that can be described by waves and it includes wave packets will present a behaviour identical to the uncertainty principle, as it pertains to nature of
But there is an additional aspect to the uncertainty principle in all quantum mechanical theories that is not present in classical theories, and that is the behaviour of non-commuting
operators. I will leave the mathetical side of this for now, and will describe this aspect from a more “physical” perspective.
The state of any quantum particle is described by a certain set of integer numbers, aptly called quantum numbers. These are the ones responsible for the fact that certain physical
properties of the particle are quantized, that is, those properties can only have a discrete set of possible values as their magnitude.
With experiments, we can measure any one of these quantized properties and get a precise measurement of its value at the moment of the experiment.
Many pairs of these properties can be measured at the same time and we can measure very precises values of these properties, but there are certain pairs of these quantized properties that
cannot be measured at the same time with precision: if we measure one of them with precision, we get an uncertain value for the measurement of the other properties of the pair.
This is a rather unique behaviour present in all quantum mechanical theories and it is not present in classical physics.
We can use the math of wave packets to obtain the widely known expressions for the uncertainty principle, the expression with the “thresholds” to the uncertainty.
But with the classical physics behaviour alone we can’t predict these pairs of properties that cannot be measured with precision at the same time.
This is a very important aspect that sets apart quantum mechanical theories from classical physics theories.
Kind regards, GEN
34. Do I understand that even in absence of supersymmetry, both fermions and bosons can be understood as vibration modes of the strings with no essential difference between the modes?
1. No, you still need something on the string which distinguishes fermions from bosons. This is a bit subtle, and I didn’t describe it carefully in this post. Maybe I’ll think of a way to do a
better job, but what I hinted at — that there might be supersymmetry in nature but it may be profoundly hidden and not look anything like what you might naively have expected — was an attempt
to paper over some subtleties.
35. “In addition, for a theory of superstrings to be consistent, the strings have to [with a few subtle caveats] move in a space with 9 spatial dimensions.”
Interesting- I’d always been under the impression that the extra dimensions were necessary for a theory that could reduce to the Standard Model, but not for the mathematical consistency of the
theory itself. Are you able to give any sort of account of how string theory falls apart in fewer than 9+1 dimensions?
1. There is something called a “gauge anomaly” which afflicts certain field theories. A field theory with a gauge anomaly has the property that everything you try to calculate — say, the
annihilation of two particles to make two other particles — gives you zero divided by zero. Obviously, a theory where every prediction for experiment is 0/0 is completely useless and
describes nothing.
The tricky part is that this gauge anomaly is not visible if you do classical physics. It is a property that can only arise in quantum mechanics. As a result, there are quantum field theories
that seem to exist (because they have a classical analogue) but do not (because they have a gauge anomaly.)
String theories, and other theories of quantum gravity, can have gauge anomalies and gravitational anomalies. Similarly to field theories, a string theory with such an anomaly doesn’t make
any sense; it may seem to have a classical analogue, but as a quantum theory, every prediction it makes is 0/0.
In the simplest contexts, superstring theories in more than 9 spatial dimensions have anomalies (and string theories that aren’t superstrings have anomalies in more than 25 spatial
dimensions.) In fewer than 9, you have to be a little careful about what you mean by dimensions, but essentially the result is the same: if you want to avoid anomalies, you either need 9
dimensions or you need to replace some of those dimensions by something more subtle.
But in any case, this has nothing to do with getting the Standard Model out of the theory. It’s a mathematical consistency condition, without which the theory cannot be used to predict
anything at all, not even about imaginary universes.
36. Very informative post indeed (at least to me), thank you very much! How’s the winter school? What did *you* learn from it (even though you were teaching there)? 🙂
1. Hmm. I learned a lot of things (as I always do) from Raman Sundrum, one of the other lecturers, but not about what he was teaching. I learned some things that I hadn’t thought through
carefully from Gino Isidori about certain models of quark and lepton masses. And I learned how to present the Higgs discovery to students, and about the subtleties that one needs to discuss
to make sure they get a clear picture. Maybe other things will occur to me…
37. If you’re envisioning a distorted lattice, you’re actually envisioning a distortion of space itself — a gravitational wave, whose quantum is a graviton, not a photon.
Are you saying in the above sentence that “space” is the “field” in which a graviton may be found to be a distortion? That is the gravitational wave is propogating through the field of space?
1. Roughly speaking yes, but I was not entirely precise. In Einstein’s gravity, you have to be careful because you can choose different points of view. The “field” in which the graviton is a
distortion may be thought of as the distance-measuring field (called the “metric”). But a distortion in the distances between grid points can also be viewed as a distortion in the grid
What’s independent of your point of view is that energy and momentum can be carried from one place to another, and the amount transferred and what happens when it hits something doesn’t
depend on whether you view the wave as a distortion in the grid or in how you think of the distance between grid points.
38. What clarity! A perspective like no other. Will be looking forward to your further posts.
39. Thanks Matt for this very informative post. String-M theory may very well describe our universe, but as I have argued before, it’s not very predictive. It was hoped early on that String theory
would make a unique prediction that looked just like the universe we live in. These hopes were dashed with the discovery of the landscape. Both cosmology and String theory point to the TOE being
a environmental model. This puts physics in an awkward position with regard to the falsification possibilities of these models. I think the sense that string theory is correct would be greatly
boosted if we find SUSY during the next run at CERN. You can have SUSY without string theory, but not the other way around. SUSY might break outside the range of the LHC and String theory could
still be correct, but the lack of SUSY would raise doubts if physics took the right turn with String -M theory in my opinion.
1. I don’t entirely agree with these statements.
“It was hoped early on that String theory would make a unique prediction that looked just like the universe we live in. These hopes were dashed with the discovery of the landscape.” Many of
us never had much hope and the hopes died more slowly than you think… because the `landscape’ of possibilities was suspected long, long before it was “discovered” within the theory. I’ll
describe this later.
“Both cosmology and String theory point to the TOE [Theory of Everything] being a environmental model.” I don’t agree. In any theory, some things are environmental, others are not. Nothing in
either cosmology or string theory indicates that all aspects of the laws of nature are environmental.
” I think the sense that string theory is correct would be greatly boosted if we find SUSY during the next run at CERN. You can have SUSY without string theory, but not the other way around.”
Your second sentence logically contradicts the first one. Yes, you can have SUSY without string theory, so finding SUSY tells you nothing about string theory; and even if you can’t have
string theory without supersymmetry, that supersymmetry may be greatly hidden from you in many different ways, so not finding it tells you, again, nothing about string theory.
1. Is there really no hope of string theory making any set of testable predictions? Surely there must be something detectable in our world that would be different if strings exist. Black
Hole behavior, very early universe structure or behavior maybe.
1. You’re asking a profoundly ill-defined question. The reason your question is bad is that you’ve not specified it. Are you talking about a prediction that string theory makes that is
true IN ALL ITS FORMS (VACUA)? Are you talking about a prediction that string theory makes that is true IN ITS VACUA THAT ARE DESCRIBED BY SUCCESSIVE APPROXIMATION (perturbation
theory) but not in others? Are you talking about a prediction that is true only in those vacua where strings that have lengths near the apparent 4 dimensional Planck scale? Are you
talking about a SPECIFIC VACUUM? and if so, which one?
If you make your question sufficiently specific, then the answer may be provided, and it may be “yes” or “not likely in the near future” or “unknown”. But if you don’t make your
question precise, then the answer depends completely on how I choose to interpret your question — which is probably not what you wanted.
For example, in certain (not very credible) vacua of string theory, people have already made predictions, which have been tested, and those vacua have been ruled out by data… see
Figure 1 of http://arxiv.org/pdf/1010.0203.pdf .
The biggest problem in this whole business — and that is why I am writing these posts — is that no one who is prominent in the public string debate (Greene, Woit, etc.) is
sufficiently precise about what they are talking about. And that’s confusing all non-experts, and some experts too.
1. I have been following the extremes and the mainstreams on this subject for a very long time. Your exposition here is outstanding. You answer the questions put to you. In short you
are to be applauded .
I am somewhat of a “frayed knot” at my age. So if if I have it tangled excuse me.
I do hope I read correctly that you seem pretty convinced – as reflected in answer to M. Rally – that the “experts” are imprecise in how they explain this to the non-experts.
If so, I would only add that experts- at least some – benefit tangibly and Intangibly by their positions, it is hard for some to separate themselves from their vested interests.
Thanks for staying above it.
1. Your last paragraph is so true 🙂 Points to Matt.
40. Great post!
Matt, I hope that at some point in this series of posts, you could describe how String Theories and Super Symmetric Theories are related.
1. That’s easy; they’re separate issues.
You can have supersymmetric field theories that have nothing to do with string theory.
You can (in principle) have string theories that have no supersymmetry. However, it can be hard to prove things about them — in particular, whether such theories can really be stable, and
thus really describe any long-lived universes.
The only string theories that we are *fully confident* make sense for describing an interesting universe have some sort of supersymmetry built in, at least in some sort of approximate way.
However, even in such theories, there is no guarantee that this supersymmetry has any near-term experimental consequences. In particular, you may not see any sign of it until you begin to
produce the very heavy particles indicated in Figure 4, and even then it may be far, far from obvious.
1. Hi Matt,
It’s interesting to note that if we forget about string theory, extra dimensions, branes, and even gravity, and consider just Yang-Mills theory in regular old 3+1 dimensions souped up
with some (N=4) supersymmetry, then, as you well know, it turns out the Hilbert space is equivalent to that of a 10D string theory. So in some cases the two ideas are connected, although
the usual N=1 supersymmetry we’re looking for in accelerators may really be unrelated to string theory.
41. Matt: I seem to recall you depicting a quantum field as a lattice. Then when you say a photon is a ripple in the field, I can envisage a distorted lattice. No problem there. But nowhere have I
seen anything to the effect that “the lattice is made out of string”. String theory seems to be a whole different animal. Can you comment on that, and can you say what the strings in string
theory are made of?
1. A quantum field is not to be depected as a lattice; you’re confused. When people do computer simulations of quantum fields, they make *space* into a lattice, not the fields. If you’re
envisioning a distorted lattice, you’re actually envisioning a distortion of space itself — a gravitational wave, whose quantum is a graviton, not a photon.
Instead, you should view a quantum field in these computer simulation in the following way. A nice example of a (non-elementary) field is the density of air. We can measure the density of air
at each point in the room and figure out how it will change over time. If we want to simulate this on a computer, however, we might set up a grid of points in the room, one centimeter apart
in each of the vertical and two horizontal directions. Then the field becomes the density of the air in the vicinity of each grid point. The density can still have any value, but we only know
what is on a spatial grid. And a wave in the density will involve the density increasing and then decreasing at each grid point, but displaced in time from one grid point to the next. The
density is changing, not the grid.
String fields are somewhat of a different animal, but for a lot of reasons… one of them being that quantum gravity is included and so space itself is part of the story, rather than being a
spectator, as it is for a field theory that does not include gravity. No one knows how to do computer simulations of string fields using a similar approach to the lattice approach to
simulating ordinary fields.
1. It is my understanding that the lattice in the computer simulation is a consequence of the numerical method chosen for the simulation.
If that is so, in itself it has nothing to do with the physics model that is the subject of the simulation.
Kind regards, GEN
2. Thanks for all that Matt. Yes I was envisioning a distortion of space, but for an electromagnetic wave. A photon is associated with displacement current. Displacement is associated with
density variation as per a sound wave. And field is the derivative of potential so integrate your sine wave to a hump going through a lattice. (There was some CERN video like this, but I
can’t find it any more). A horizontal lattice line is like a guitar string. Your pluck is like Planck’s constant. Action is associated with reaction. Any cell that’s skewed is a spin1
virtual photon. Any cell that’s flattened is a spin2 virtual graviton. Skewed squares are flattened. A string vibration on a closed path is a fermion. And so on. It doesn’t sound like QFT
or string theory, but to me it sounds like their child. To you it doubtless sounds like an ass, so no reply necessary.
1. John, your statement that “field is the derivative of potential” is not generally true. You are probably thinking of the electromagnetic field strength (the “E” and “B” fields) that
can indeed be expressed in terms of the derivatives of the vector potential “A”. Interestingly enough, in quantum electrodynamics it is “A” itself that serves as the field to be
quantized, the quanta being called photons. The field “A” has only indirect physical significance and quantum mechanics is subtle, so it is extremely difficult to build an intuition
for the “nature” of a photon. (I don’t know if the experts succeed in this – I for sure have not.) That’s why it is so important to cling to the equations. Any simple intuitions,
especially mechanical ones, are quite certainly wrong.
1. I think you will find that Mr. Duffield is interested in his own opinions more than in the facts.
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Digital Math Resources
Display Title
Math Example--Polygons--Quadrilateral Classification: Example 25
Math Example--Polygons--Quadrilateral Classification: Example 25
This example showcases a quadrilateral labeled ABCD with angles at B and D marked as 60 degrees. Sides AD and BC are labeled y. The shape is identified as a parallelogram. This classification is
based on the fact that both pairs of opposite angles are congruent, which means opposite sides are parallel, a defining property of parallelograms.
Quadrilateral classification is a fundamental concept in geometry that helps students understand the properties and relationships of four-sided shapes. By analyzing various examples, students learn
to identify key characteristics such as side lengths, angle measures, and the relationships between them. This collection of examples provides a comprehensive overview of different quadrilateral
types, allowing students to develop their analytical skills and geometric reasoning.
Studying multiple worked-out examples is essential for students to fully comprehend the concept of quadrilateral classification. Each example presents a unique set of characteristics, challenging
students to apply their knowledge and critically evaluate the given information. This approach helps reinforce understanding, improve pattern recognition, and develop problem-solving skills in
Teacher's Script: In this example, we're given information about two angles and two sides of the quadrilateral. How does the congruence of opposite angles relate to the classification of this shape
as a parallelogram? Can you explain why having congruent opposite angles means that opposite sides are parallel? What can we infer about the other two angles that aren't marked? How does the labeling
of sides AD and BC as y support the parallelogram classification?
For a complete collection of math examples related to Quadrilateral Classification click on this link: Math Examples: Quadrilateral Classification Collection.
Common Core Standards CCSS.MATH.CONTENT.7.G.B.5, CCSS.MATH.CONTENT.4.G.A.2, CCSS.MATH.CONTENT.5.G.B.4, CCSS.MATH.CONTENT.3.G.A.1
Grade Range 3 - 7
Curriculum Nodes • Quadrilaterals
• Definition of a Quadrilateral
Copyright Year 2013
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Discuss Everything About The Walking Dead: No Man's Land Wiki | Fandom
I noticed that the Special Equipment Page doesn't actually list the names of the weapons. Is there a reason for that?
The GW Planner is not up to date!
Is there an update?
Which is better on a warrior perseverance or strong?
Welcome to The Walking Dead No Man's Land Wiki. Use the navigation pane at the top to find all kinds of useful info about the game we all love.
Hey everyone, I'm here to explore a little bit about this forum discussion tab. This can be a good place to discuss certain topics and show off various topics about the game. And for starters, I come
with a question:
What is the maximum amount of damage a single main attack can achieve on a single target?
First, some observations and rules before we start digging into it:
• The attack we are going to consider should be a main attack. That is, action points were used to directly attack a specific target. With this rule, we cannot consider Overwatch, Prowl's attacks,
Legal Authority attacks or any other type of attack that is not tied to using an action to attack a specific target (This could be considered in another discussion, perhaps). However, we can use
Charge attacks, for example.
• In this discussion, we can only apply a single attack, with a single value and one target. So, things that consider damage separately like Coup de Grâce or Yumiko's LT cannot be considered, as
they deal damage separately from the main one; For multiple attacks at once, we should only consider an attack for one target only.
That said, let's start checking all of that out.
To find out what the best possible value would be, we will first have to assemble our *Perfect Team* to achieve this. And currently, in Update 5.8, we have a lot of variety of information that can
influence the damage done. So let's separate the thread, to think about the bests combinations.
Choosing the perfect Support for this case is pretty obvious in this case. Currently, (Update 5.8), there are only 3 Supports that have anything to do with damage: Shiva, Hwacha, and Carol's Cookies.
However, as we stipulated, the attack must be your main attack, and only Carol's Cookies can increase the damage to your main attack, giving even more damage if it is a charge attacks. So well,
Carol's Cookies is our choice.
Well, now we have to check which would be our ideal Opponent to cause the most damage possible. As there are some heroes that increase damage based on the Enemy's health, so we have to select an
enemy with very high health. The walkers that have the highest health are The Exploder, Metalhead, Goo, Spiked and Tank. It wouldn't be a good idea to use Metalhead Walker, since he has a Damage
Reduction inside him, so maybe, depending on the ideal team we decide, it could be harmful to use against him. So, that leaves Exploder, Goo, Spiked and Tank. We can use any of them for our
experiment. Let's take their level as 60, to maximize health (each one of them has 6.226.246 Health in Level 60).
Now, we have the choice that is key for us to imagine the most damage possible: The team of survivors. Currently, we can only add up to 3 survivors to our team in a mission, so we can only decide 3
survivors for our team. We might consider using regular Survivors, but given that heroes have a natural boost to their damage and unique traits that may or may not be more useful than normal traits,
so let's just consider heroes in this case, and all of them will be 10 Stars and Maximum Level (30), just to maximize the results.
Thinking a little, we can separate our team as follows:
We have our leader, through which your Leader Trait can be distributed to all other members of your team; We have the booster, whose Leader Trait can help other users to deal more damage; and our
Champion, who will be responsible for dealing the Maximum damage we are looking for so much.
For our booster, there’s not so much Heroes that can help other survivors to deal more damage when he/she’s not the leader. I think the only ones that can make it are Beth, Connie and Negan. Negan’s
Leader Trait adds 30% more total damage to the survivor’s attacks towards the target, and Connie, when she’s not the leader, can add until 75% more total damage to our Champion, but Beth add Extra
Damage equals to 25% of the current HP of the walker. If our target is on current Health, then it will be ¼ of their health (which should be 1.556.561 damage), gone just by her leader trait. I doubt
that Negan and Connie can deal so much damage. So, Beth is our pick for booster.
So, for our leader and our Champion, we need to see the 2 most powerful heroes that can deal the most damage to all. For that, we have 4 Heroes that are known for doing massive damage from 1 hit:
Beth, Connie, Mercer, Survivalist Rick, and Rosita. Since we already choose Beth as our booster, we will not include her.
• Mercer’s powerful Leader Traits depend on the current Health of the enemy, so, if we consider our enemy in full health, he will deal 45% (or 60% for melee) of the health, which will be 2.801.810
(or 3.735.747 for melee) damage. It’s hard to compete against this.
• Survivalist Rick increases the total damage to the charge attack by 95%, which is a lot, mainly because it's a charge attack, and there aren't many traits that can boost that.
• Connie and Rosita’s Leader Traits buffs the total damage by an elevate percentage. In the maximum Leader Trait Level, Rosita will cause 150% more total damage to a single target. For Connie, it
depends on how much stacks she can get: if she’s the leader, she can get 4 stacks, and that means she will get 4 × 75% = 300% more total damage, and 2 x 75% = 150% If she’s not the Leader.
Based on all this, as you can see that Mercer, Rick, Rosita, and Connie are our greatest damage dealers, but we can only choose two of them. Mercer deal’s massive damage compared to the others, so he
has a guaranteed spot, leaving only the other 3. So here, we use a game theory to discover what is the best choice for Team formation and which one should be our last pick for the team. We will use
the possibilities with Mercer as Leader, and Mercer as Champion (making Connie/Rick/Rosita as the Leader). We will imagine a best-case scenario to maximize the possible damage.
As you can see, with Connie we will always deal more damage than if Rosita was in the team instead, regardless that Mercer is the Leader or the Champion. Consequently, we can confirm that Rosita will
be eliminated from this pick. Goodbye, Rosita!
We can also see that if we put Mercer as the leader, for either Connie or Rick as the other pick, we have an addition of +15% of the enemy's current health as Damage. This damage addition corresponds
to 933,936 damage. It's hard to cover that damage if Mercer was the Champion instead of the leader. So, we conclude that Mercer must be our Leader.
Now, we have an argument between Connie and Rick. Although Connie, in theory, has a higher percentage in terms of Leader Trait, we must consider that Rick’s Leader Trait applies for Charge Attack’s
damage, which, depending on how the things goes, could have a bigger impact than Connie's damage.
So, the choice of which Survivor is best to be our Champion will depend on what their impact will be on the overall formula.
Knowing that the Charge attack formula is:
((Base Damage × (1 + Normal Stars Modifier + Hero Boost Modifier) + Weapons × (1 + Weapon Damage Variation) + Flat Damage Badges + Flat Critical Damage Badges) × (1 + Damage Traits Buffs) × (1 +
Damage Badges) × (1 + Class Charge Damage Boost + Critical Damage Traits Buffs + Critical Damage Badges) × (1 + Charged Attacks Damage Traits Buffs) + Extra Damage) × (1 + Carol's Cookies)
Knowing that Connie's Leader trait applies to Damage Traits Buffs, and Scout Rick to Charged Attacks Damage Traits Buffs, we see that (1 + Damage Traits Buffs) and (1 + Charged Attacks Damage Traits
Buffs) multiply.
So, to maximize the highest possible damage, we have to make this multiplication have the highest possible value. And for that, we have to know what the values of other Traits are before we make that
So, we're going to have to postpone that decision a bit until we have the full picture. On the good side, both Scout Rick and Connie are Scouts, so they are capable of getting the same possible
survivor traits, weapons, and armors which makes our job a lot easier to calculate the best combination of Traits.
Now, we need to decide what should be the other 4 survivor traits for our Champion. Considering we will use his charge attack, we need to check the traits that help do more damage. But a Scout only
has 3 traits that create more damage: Strong, Power Strike and Ruthless. Good to see that we didn’t need to change the Initial traits!
For our 4° trait, We could use the way we want. Maybe Lucky, for the Mercer’s Leader trait? Or perhaps weakening? Your choice, it will not make much difference in our thread.
Now, we have to see what would be the perfect equipment for our Champion, to maximize his damage. We are not going to limit our thinking to created weapons, but we are going to think about weapons
not yet created, with the purpose of having the best possible result (obviously, respecting the knowledge we have collected during all this playing time). We will consider all of them in the Maximum
For the weapons, we need to create a powerful weapon with all their traits helping to deal the most damage possible. First, we see what is the maximum weapon value we have seen for Scouts.
According to the Equipment page, the base damage from a level 33 weapon is 5100. Considering a legendary weapon with Buffed damage (175% Multiplier), we have:
5107 × (40% (Legendary Weapon) + 175%) = 5107 × 215% = 10.980 damage.
That would be the case if the multiplier of a Scout Weapon in Charge attack would be the same that in normal attacks. But that’s not the case with a Scout Weapon. Instead, he got 180% Multiplier
instead of 175%. Not so much, but we're aiming for the maximum damage. So:
5107 × (40% (Legendary Weapon) + 175%) = 5107 × 215% = 11.235 damage.
Now for the weapon traits, we have 4 weapon traits for scouts that increase damage: Charging, Destructive, First Strike and Lethal. We could create a weapon with these traits, one of them being
Infused. Now, which one of them should be the silver one?
• Charging is one of the few traits that influence the Total Damage for Charge attacks, so it wouldn’t be a good idea to decrease that trait to a silver trait.
• Destructive effects Critical Damage, which more rare than Charge Damage, so is also not a good idea to decrease that trait.
So, is between First Strike and Lethal. Since First Strike has a much more percentage bonus than Lethal, so Lethal will be our Silver trait.
So, it will be a weapon with Silver Trait Lethal, and Golden traits being Destructive, First Strike and Charging.
For the armor traits, the only trait that increases damage somehow is Ruthless. Therefore, we will consider Ruthless as a golden trait, and we will ignore the Silver trait for now, since it will not
interact with the damage.
Now that we get the values of all the traits that will affect Total Damage and Charge Attacks’ Damage, we can now decide which one will be our champion: Connie, with his +150% Total Damage, or
Survivalist Rick, with his +95% of Charged Damage. First, let's clarify which variables are influencing each factor:
• Impacting Total Damage, we have Strong (20%), Power Strike (53%), Silver Trait Lethal (10%), and Golden Trait First Strike (50%). So, is 20% + 53% + 10% + 50% = 133% More Total Damage.
• Now, impacting Charged Damage, we got the Survivor Trait Ruthless (45%), and Golden Traits Charging (15%) and armor trait Ruthless (30%), Totalizing 45% + 15% + 30% = 90% More Charge Damage.
With that, we got on our formula (1 + 133%) × (1 + 90%) = 233% × 190%, and we need to make this multiplication active the highest value possible. So, we test:
• If we Add Connie, we get +150% Total Damage, so (233% + 150%) × 190% = 233% × 190% + 150% × 190%.
• If we Add Survivalist Rick, we get +95% of Charged Damage, so 233% × (190% + 95%) = 233 × 190% + 233% × 95%.
So, we just need to see which is higher: 150% × 190% or 233% × 95%. If we do the math, we see that 150% + 190% is higher, so Connie is more effective for the team than Survivalist Rick. That was so
Therefore, here’s our team: Mercer as Leader, Connie as Champion and Beth as Booster. What a team!
Finally, we reached Badges, which is the simplest one. For our champion, we will consider the maximum values of the Damage and Critical Damage values, all of them of the same set, to get that 20%
bonus. Therefore, we calculate the total value:
For Damage: (18% + 5%) × 3 (Number of Damage Badges) × 1.2 (Set Bonus) = 23% × 3 × 1.2 = 69% × 1.2% = 82.8%.
Observation: The 18% + 5% damage is displayed as 18% + 4%, but the displayed value is wrong. It’s actually 18% + 5%.
For Critical Damage: (22% + 6%) × 3 (Number of Damage Badges) × 1.2 (Set Bonus) =
28% × 3 (Number of Damage Badges) × 1.2 (Set Bonus) = 84% × 1.2 = 100.8%.
We finally reached the last part of this thread. This is a massive journey.
Here we are going to create the perfect scenario for the traits to be activated and have nothing to stop him from doing the highest possible damage. We'll also consider the range of damage at its
Picture this: “It's the last turn before the threat counter resets, and more walkers spawn. You're with your team: Mercer, Beth and Connie, Connie is with Carol's cookies as her support. Her charge
points are full, and Beth just killed a walker on Connie's side. You place your Mercer on Connie's side as well, to activate her exclusive Trait for up to 2 stacks. The counter reaches zero, and more
walkers spawn. A level 60 Tank Walker is born right next to Connie. It's the perfect opportunity. You activate Carol’s cookies and the charge attack, and then you are ready for the kill.”
That said, let's move on to the part I, personally, was excited about the most: The final damage calculation.
The Charge Attack’s Damage Formula is:
((Base Damage × (1 + Normal Stars Modifier + Hero Boost Modifier) + Weapons × (1 + Weapon Damage Variation) + Flat Damage Badges + Flat Critical Damage Badges) × (1 + Damage Traits Buffs) × (1 +
Damage Badges) × (1 + Class Charge Damage Boost + Critical Damage Traits Buffs + Critical Damage Badges) × (1 + Charged Attacks Damage Traits Buffs) + Extra Damage) × (1 + Carol's Cookies).
You can see more details about the formulas here.
((3942 × (1 + 40% + 25% + 25%) + 11.235 × (1 + 20%) + 0 + 0) × (1 + 150% + 20% + 53% + 10% + 50%) × (1 + 82.8%) × (1 + 100% + 60% + 100.8%) × (1 + 45% + 15% + 30%)) + (25% + 60%) × 6.226.246) × (1 +
=((3942 × 190% + 11.235 × 120%) × (1 + 283%) × (1 + 82.8%) × (1 + 260.8%) × (1 + 90%) + 85% × 6.226.246) × (1 + 180%)
= ((7.489.8 + 13.482) × 383% × 182.8% × 360.8% × 190% + 5.292.309.1) × 280%
= (20.971 × 700.124% × 685.52% + 5.292.309.1) × 280%
= (1.006.501.0573 + 5.292.309.1) × 280%
= 6.298.810.1573 × 280%
= 17.636.668
So, 17.636.668 damage. That’s the maximum damage that you could get from a single Charge attack. That’s simply incredible.
So, that’s it guys, what an adventure. Thanks for reading! | {"url":"https://twdnml.fandom.com/f","timestamp":"2024-11-12T22:33:59Z","content_type":"text/html","content_length":"811858","record_id":"<urn:uuid:b9aa81d2-36ff-4ff9-a2b8-d3f9ca22345c>","cc-path":"CC-MAIN-2024-46/segments/1730477028290.49/warc/CC-MAIN-20241112212600-20241113002600-00583.warc.gz"} |
David Hilditch (CENTRA, IST)
When solving general relativity numerically for a given physical problem we must use a formulation of the field equations for which the resulting partial differential equation problem is well-posed.
Building such a good formulation usually requires making a coordinate choice. This leads to the standard statement that `gauge freedom in general relativity is the choice of coordinates'. The latter
two facts have long bothered me, because one of the first lessons in relativity is that coordinates should in some sense not matter. In my talk I will explain the solution to my earlier confusion.
Time permitting I will also describe ongoing work to exploit the solution for practical calculations. | {"url":"https://centra.tecnico.ulisboa.pt/events/?id=713","timestamp":"2024-11-14T11:28:46Z","content_type":"text/html","content_length":"6935","record_id":"<urn:uuid:13534857-b9bd-4e66-b021-483f3968d573>","cc-path":"CC-MAIN-2024-46/segments/1730477028558.0/warc/CC-MAIN-20241114094851-20241114124851-00024.warc.gz"} |
More Less is More
In each of these games, you will need a little bit of luck, and your knowledge of place value to develop a winning strategy.
More Less is More printable sheet - game instructions
More Less is More printable sheet - blank grids
These challenges follow on from Less is More.
This video below introduces these challenges:
You can have a go at the four different versions using this interactivity:
If you are working away from a computer, you could treat this as a game for two people, or play in two teams of two.
You will need a 1-6 or 0-9 dice. Our dice interactivity can be used to simulate throwing different dice.
Each team should draw some cells that look like the pictures below.
In Version 1, you place the numbers after each throw of the dice.
You will need to throw the dice eight times in total. After
throw of the dice, each team decides which of their cells to place that number in.
When all the cells are full, each team will be able to check if their number sentence is correct.
In all cases, you score if the sentence is correct. The score is the result of the calculation on the left of the inequality sign.
See the hint for some examples of scoring.
The winner is the team with the higher score.
In between rounds, teams might try to find the highest possible score they could have achieved, if they had known the eight numbers in advance. Their new scores could be added to their running
In Version 2, have a go at playing the game in a similar way to Version 1, but this time, note down all eight dice rolls before deciding where to place them.
Keep a running total of your scores.
Who is the winner after ten rounds?
Who is the first to reach 500 points?
Final challenge:
Imagine that you have thrown the numbers 1-8.
What is the highest possible score for each of the games above?
Can you provide a convincing argument that you have got the highest possible score?
A clue is given in the hints.
You may like to check whether you have indeed got the maximum score by typing the numbers 1 to 8 (without commas and with no spaces between them) into the 'Values' box in the Settings of the
interactivity above, and then testing your solution.
Getting Started
Score = 42 + 16 = 58
With 1-8, the maximum score is more than 130 points
Score = 55 - 35 = 20
With 1-8, the maximum score is more than 55 points
Score = 42 - 16 = 26
With 1-8, the maximum score is more than 70 points
Score = 12 + 24 = 36
With 1-8, the maximum score is more than 70 points
Student Solutions
Gerard from Frederick Irwin Anglican School in Australia inserted numbers that make the inequality true:
32+36 is less than 65+26 this is because 32+36=68 while 65+26=91
Gerard's score is 68.
Leticia and Amelie from Halstead Prep School and Rishaan, Swarnim and Eshaan from Ganit Kreeda in India worked out how to get the highest score using the numbers 1 to 8. This is Leticia's work:
I got the four highest numbers 5, 6, 7 and 8. I then arranged them do that I had 5 and 8 on the left hand side and 6 and 7 on the right hand side. I then had the other four numbers and I then figured
out that I could do 1+3=4 and 4+2=6 and you can’t have 3+2 and 1+4 because these both add up to five so they are equal. You can’t do 2+1 on the left hand side because that is not the highest
combination. So I then did 51 and 83 on the left and 64 and 72. If you did this the other way round the inequality would be incorrect. Then I added 51 and 83 and then I got 134 which is the highest
Gowri from Ganit Kreeda explained how to maximise your score given any 8 numbers:
Gerard inserted numbers that make the inequality true:
54-56 is less than 44-26 as 54-56= -2
Gerard's numbers give a negative score of -2.
Eshaan, Rishaan, Samaira, Gowri, Vibha, Viha, Vansh, Vraj, Arya, Swarnim, Rudraraj and Renah from Ganit Kreeda worked together to find the best score possible using the numbers 1 to 8:
Swarnim, Vansh and Eshaan had used the strategy of greatest number – smallest number to get the maximum difference.
Rishaan tried the numbers at tens place such as the difference will be maximum and equal for both the sides.
As 8-2=7-1 = 6
He used 8_ - 2_ < 7 _ - 1_
The kids then used the remaining digits at ones place. Again, they used the same strategy as used in the first challenge. They tried to get the difference of 1.
85 – 24 < 76 – 13 or 61 < 63
The Highest Score is 61.
Amelie used very similar reasoning, expressed in a different way:
Gerard inserted numbers that make the inequality true:
31-45 is a negative while 34+14 is positive
Again, Gerard's score is negative (this time -14).
Amelie combined the numbers 1 to 8 to create a higher score:
Note that Amelie probably meant 87-13 rather than 87-14.
The students from Ganit Kreeda managed to get a slightly higher score:
87-12 will give the maximum score we can get on the left side which is 75 and now to get a number bigger than this on the right side we can use the leftover numbers.
87 - 12 < 35 + 46
Gerard inserted numbers that make the inequality almost true:
12+11 is less than 46-23
In fact, now both sides are equal to 23, so this is not actually true!
Amelie used the numbers 1 to 8 to create a high score:
The students from Ganit Kreeda also got a score of 71:
First, we tried to use maximum difference which is 75 on the right side.
But then using remaining digits, the smallest possible addition was 35+46 which is 81, and 81>75.
So, we tried to make right side 74 by changing 12 to 13 as 87-13.
With remaining digits we got 26+45=71 and 87–13=74 OR 46+25<87-13
The Highest Score is 71.
It is actually possible to get a score of 72. Can you see how? | {"url":"http://nrich.maths.org/problems/more-less-more","timestamp":"2024-11-07T12:55:09Z","content_type":"text/html","content_length":"59070","record_id":"<urn:uuid:ce2c4946-382b-4652-84ee-1abd35e0b3bc>","cc-path":"CC-MAIN-2024-46/segments/1730477027999.92/warc/CC-MAIN-20241107114930-20241107144930-00605.warc.gz"} |
Find the ratio of the black cell phone covers sold to the total number of cell phone covers sold last week
Find an answer to your question 👍 “Find the ratio of the black cell phone covers sold to the total number of cell phone covers sold last week ...” in 📗 Mathematics if the answers seem to be not
correct or there’s no answer. Try a smart search to find answers to similar questions.
Search for Other Answers | {"url":"https://cpep.org/mathematics/415955-find-the-ratio-of-the-black-cell-phone-covers-sold-to-the-total-number.html","timestamp":"2024-11-07T14:11:09Z","content_type":"text/html","content_length":"23458","record_id":"<urn:uuid:815ddca5-306d-484d-a076-beadafb2d254>","cc-path":"CC-MAIN-2024-46/segments/1730477027999.92/warc/CC-MAIN-20241107114930-20241107144930-00147.warc.gz"} |
Up-sample pupil data — pupil_upsample
Increase the sampling frequency to 1000Hz. See https://dr-jt.github.io/pupillometry/ for more information.
Inserts additional rows into the data with missing pupil and gaze values. Adds a column, `UpSampled` to identify whether the data has been up-sampled.
Up-sample to 1000Hz
There are some advantages to up-sampling the data to a sampling frequency of 1000Hz, and is even a recommended step in preprocessing by Kiret & Sjak-Shie (2019).
Up-sampling, should occur before smoothing and interpolation. In general, it is safer to apply smoothing before interpolation (particularly if cubic-spline interpolation is to be used). However, if
up-sampling is to be used, interpolation needs to occur first in order to fill in the missing up-sampled values. The question, then, is how can we apply smoothing first while still doing up-sampling?
This is resolved in this package by first up-sampling with `pupil_upsample()` and then smoothing `pupil_smooth()`. `pupil_upsample()` will not interpolate the missing up-sampled values. Instead, a
linear interpolation will be done in`pupil_smooth()`, if `pupil_upsample()` was used prior, followed by smoothing and then after smoothing, originally missing values (including the missing up-sampled
values and missing values due to blinks and other reasons) will replace the linearly interpolated values (essentially undoing the initial interpolation). After `pupil_smooth()`, interpolation can
then be applied to the up-sampled-smoothed data with `pupil_interpolate()`.
This is all to say that, the intuitive workflow can still be used in which, `pupil_upsample()` is used, followed by `pupil_smooth()`, followed by `pupil_interpolate()`.
Alternatively, to interpolate before smoothing, `pupil_upsample()` is used, followed by `pupil_interpolate()`, followed by `pupil_smooth()`. The difference being that, in this case, no interpolation
and then replacing the missing values back in the data is done in `pupil_smooth()` because interpolation was performed first anyways. | {"url":"https://dr-jt.github.io/pupillometry/reference/pupil_upsample.html","timestamp":"2024-11-08T09:21:43Z","content_type":"text/html","content_length":"11488","record_id":"<urn:uuid:86d7221f-6660-4574-b135-e42151e35f28>","cc-path":"CC-MAIN-2024-46/segments/1730477028032.87/warc/CC-MAIN-20241108070606-20241108100606-00813.warc.gz"} |
Events Calendar
Select a calendar:
Filter December Events by Event Type:
Events for December 16, 2016
• Fri, Dec 16, 2016 @ 11:00 AM - 12:00 PM
Information Sciences Institute
Conferences, Lectures, & Seminars
Speaker: Mason Porter, UCLA
Talk Title: Multilayer Networks
Series: AI Seminar
Abstract: Networks arise pervasively in biology, physics, technology, social science, and myriad other areas. Traditionally, a network is modeled as a graph and consists of a time-independent
collection of entities (the nodes) that interact with each other via a single type of edge. However, most networks include multiple types of connections (which could represent, for example,
different modes of transportation), multiple subsystems, and nodes and/or edges that change in time. The study of "multilayer networks", which is perhaps the most popular area of
network science, allows one to investigate networks with such complexities. In this talk, I'll give an introduction to multilayer networks and their applications.
Biography: Mason Porter earned a B.S. in applied mathematics from Caltech in 1998 and a Ph.D. from the Center for Applied Mathematics from Cornell University in 2002. He was a postdoc at
Georgia Tech (math), Mathematical Sciences Research Institute, and Caltech (physics) before joining the faculty of the Mathematical Institute at University of Oxford in 2007. He was named
Professor of Nonlinear and Complex Systems in 2014. A few months ago, he took up a position as Professor of Mathematics at UCLA. Porter is known for the diversity and interdisciplinarity of
his research (and for his sharp wit). In networks and complex systems, Porter has contributed to myriad topics, including community structure in networks, core--periphery structure, social
contagions, political networks, granular force networks, multilayer networks, temporal networks, and navigation in transportation systems. Other subjects he has studied include granular
crystals, Bose--Einstein condensates, nonlinear optics, numerical evaluation of hypergeometric functions, quantum chaos, and synchronization of cows. Porter's awards include the 2014 Erd\H{o}
s--R\'{e}nyi Prize in network science, a Whitehead Prize (London Mathematical Society) in 2015, the Young Scientist Award for Socio- and Econophysics (German Physical Society) in 2016, and
teaching awards from University of Oxford in recognition of his lecturing and student mentorship. Porter was named a Fellow of the American Physical Society in October 2016.
Host: Emilio Ferrara
Webcast: http://webcastermshd.isi.edu/Mediasite/Play/ef4957a6864d4e1db06e15cba71b9b021d
Location: Information Science Institute (ISI) - 1135 - 11th fl Large CR
WebCast Link: http://webcastermshd.isi.edu/Mediasite/Play/ef4957a6864d4e1db06e15cba71b9b021d
Audiences: Everyone Is Invited | {"url":"https://viterbi.usc.edu/calendar/?date=12/16/2016&","timestamp":"2024-11-07T09:21:24Z","content_type":"text/html","content_length":"22917","record_id":"<urn:uuid:5416720a-7f9c-45b5-b9c6-5d58da431927>","cc-path":"CC-MAIN-2024-46/segments/1730477027987.79/warc/CC-MAIN-20241107083707-20241107113707-00709.warc.gz"} |
\]. Then number of ordered triplets \
Hint: Here we are given to find the number of possible ordered triplets that can be formed such that product of those three values is equal to \[105\]. To do this we will use the method of prime
numbers, where we write prime numbers with their powers, needed for the factorization of \[105\]. After finding the factors we find the possible triplets. Then we use those prime numbers to find
other factors also and find the total possible triplets.
Formula used: Total numbers of possible ways to organize three things \[a,b,b\] where two things are common is given by \[\dfrac{{3!}}{{2!}}\].
Total numbers of possible ways to organize three things \[a,b,c\] are \[3!\].
Complete step-by-step solution:
Here we have to factorize \[105\] using prime numbers. We can see that we can write \[105\] as,
\[105 = {3^1}{5^1}{7^1}\]
Hence we get the factors as \[3,5,7\]. Now we have to find the possible triplets that can be formed by these numbers. We see that,
\[3\] can be written at any place be it \[x,y\,or\,z\]. Same way numbers \[5and7\] can also be written.
Hence, total possible triplets \[ = 3!\].
For \[105 = 35 \times 3 \times 1\], we get possible triplets as \[3! = 6\]
For \[105 = 21 \times 5 \times 1\], we get possible triplets as \[3! = 6\]
For \[105 = 7 \times 15 \times 1\], we get possible triplets as \[3! = 6\]
For \[105 = 105 \times 1 \times 1\], as two factors are same, we get possible triplets as \[\dfrac{{3!}}{{2!}} = 3\]
Hence we get possible triplets as, \[6 + 6 + 6 + 6 + 3 = 27\]
So answer is B).
Note: Whenever we have to find the number of factors of any big number, we should always try to find them through the prime number method and then find other factors after multiplying them with one
another as we have done here. | {"url":"https://www.vedantu.com/question-answer/let-x-cdot-y-cdot-z-105-where-xyz-in-n-then-class-8-maths-cbse-60a3b7cdd4529b28fc9a7d73","timestamp":"2024-11-10T11:57:41Z","content_type":"text/html","content_length":"151901","record_id":"<urn:uuid:b9f120ae-5316-4e9a-8498-e7d690d2875b>","cc-path":"CC-MAIN-2024-46/segments/1730477028186.38/warc/CC-MAIN-20241110103354-20241110133354-00042.warc.gz"} |
Zero-Knowledge: Theoretical Foundations II
This post is a continuation of a previous post, and will proceed with the material from Bar Ilan’s 2019 Winter School.
In this post, our objective is to show that there exists a zero-knowledge interactive proof system for any language in NP. In order to do so, we will need to loosen our definition of zero-knowledge
slightly. We will define statistical zero-knowledge and computational zero-knowledge, both of which are relaxations of perfect zero-knowledge (which we defined in the previous post). We will then
take an NP-complete language, \(HAM\), and give an interactive proof system which satisfies computational zero-knowledge. It will then follow that any language in NP has a computational
zero-knowledge proof, since any language in NP can reduced to an instance of \(HAM\).
The problem with perfect zero-knowledge for NP
Why do we need relaxations on our definition of perfect zero-knowledge at all? Why can’t we just construct proof system which satisfies perfect zero-knowledge for an NP-complete language?
Well it’s very very strongly believed that such a proof system cannot exist. In 1987, just a couple years after the original discovery of the concept of zero-knowledge, Fortnow showed that if such a
proof system exists, then the polynomial hierarchy collapses. Don’t worry if you don’t know what this means - just take my word that theoretical computer scientists believe very very strongly that
the polynomial hierarchy does not collapse (quite similar to the strong belief that P \(\neq\) NP).
Thus, if we want to achieve zero-knowledge protocols for all languages in NP, we’re going to need to relax our definition of zero-knowledge.
In order to do tinker with the definition of zero-knowledge, we need to discuss some different notions of indistinguishability.
Notions of indistinguishability
Consider two random variables \(X, Y\) over the domain \(\Omega = \{ 0,1\}^{n}\). We want to formally define some notions of indistinguishability between these two variables.
The first notion is that of perfect indistinguishability: the distributions \(X\) and \(Y\) are exactly the same. Thus any algorithm will behave exactly the same whether its input is \(X\) or \(Y\).
\(X,Y\) are perfectly-indistinguishable if for any algorithm \(A\) (even a computationally unbounded algorithm), \(\lvert \Pr[A(X)=1] - \Pr[A(Y)=1] \rvert = 0\).
Relaxing this definition a little, we get statistical indistinguishability: the distributions \(X\) and \(Y\) are almost the same, and their difference is bounded by a negligible value. Any algorithm
will behave almost the same whether its input is \(X\) or \(Y\).
\(X,Y\) are statistically-indistinguishable if for any algorithm \(A\) (even a computationally unbounded algorithm), \(\lvert \Pr[A(X)=1] - \Pr[A(Y)=1] \rvert \leq \epsilon(n)\), for some negligible
function \(\epsilon(n)\).
In the real world, it is reasonable to assume adversaries have limited computation power. We can apply this idea to get computational indistinguishability: any polynomially-bounded algorithm will
behave almost the same whether its input is \(X\) or \(Y\).
\(X,Y\) are computationally-indistinguishable if for any PPT algorithm \(A\), \(\lvert \Pr[A(X)=1] - \Pr[A(Y)=1] \rvert \leq \epsilon(n)\), for some negligible function \(\epsilon(n)\).
Relaxing perfect zero-knowledge
Now that we have these relaxed notions of indistinguishability, we can use them to define a relaxed notion of zero-knowledge.
First recall the definition of perfect zero-knowledge:
An interactive proof system \(P, V\) for \(L\) is perfect zero-knowledge if for all PPT \(V^*\), there exists a PPT simulator \(S\) such that \(\forall x \in L\), \(S(x) \cong (P,V^*)(x)\).
Note that this definition requires the simulator’s output and the transcript to be perfectly-indistinguishable.
We can relax this to only require them to be statistically-indistinguishable:
An interactive proof system \(P, V\) for \(L\) is statistical zero-knowledge if for all PPT \(V^*\), there exists a PPT simulator \(S\) such that \(\forall x \in L\), \(S(x) \cong_S (P,V^*)(x)\).
However, it turns out that this definition is still not loose enough to use for NP. Fortnow additionally showed that if there exists statistical zero-knowledge proofs for all problems in NP, then the
polynomial hierarchy collapses.
Thus, we need another level of relaxation. This time, we allow the simulator’s output and the transcript to be computationally-indistinguishable:
An interactive proof system \(P, V\) for \(L\) is computational zero-knowledge if for all PPT \(V^*\), there exists a PPT simulator \(S\) such that \(\forall x \in L\), \(S(x) \cong_C (P,V^*)(x)\).
This definition will be loose enough, as we will be able to show that every language in NP has a computational zero-knowledge proof system. We’re almost ready to show such a construction, but we need
to first discuss an important tool: commitment schemes.
Commitment schemes
A commitment scheme is a two-phase protocol between two parties: a committer, and a receiver. The idea is to have the committer commit to some value \(m\) without revealing to receiver what \(m\) is.
To achieve this, the committer computes some value \(c = Com(m,r)\) (where \(r\) is some randomness) and sends \(c\) to the receiver. At a later time, the committer can “decommit,” revealing the the
original message \(m\) and randomness \(r\text{:}\) \(Dec(c) = (m, r)\). The receiver can verify that \(c = Com(Dec(c))\).
There a couple properties that are desirable for commitment schemes to have:
1. Hiding: the receiver should not be able to learn any information about the committed message \(m\) from seeing the commit value \(c\)
2. Binding: the committer should be bound to the originally committed message \(m\)
For each of these properties, we can define them using a computationally unbounded model (“statistical”), or a computationally bounded model (“computational”).
A commitment scheme \(Com\) is statistically-hiding if for all pairs of distinct messages \(m_1, m_2\), \(Com(m_1) \cong_S Com(m_2)\).
A commitment scheme \(Com\) is computationally-hiding if for all pairs of distinct messages \(m_1, m_2\), \(Com(m_1) \cong_C Com(m_2)\).
To formally define binding, we’ll describe the “binding game.” Given two distinct messages \(m_1, m_2\), an algorithm \(C\) “wins the binding game” if \(C\) generates values \(c, r_1, r_2\) such that
\(Com(m_1, r_1) = c = Com(m_2, r_2)\).
A commitment scheme \(Com\) is statistically-binding if for any algorithm \(C\) (even a computationally unbounded one), for all pairs of distinct messages \(m_1, m_2\), \(\Pr[C \text{ winning the
binding game}] \leq \epsilon(n)\), where \(\epsilon(n)\) is some negligible function.
A commitment scheme \(Com\) is computationally-binding if for any PPT \(C\), for all pairs of distinct messages \(m_1, m_2\), \(\Pr[C \text{ winning the binding game}] \leq \epsilon(n)\), where \(\
epsilon(n)\) is some negligible function.
In an ideal world, it’d be great to have a commitment scheme which is both statistically-hiding and statistically-binding. However, such an awesome commitment scheme cannot exist. In fact, it’s a
nice exercise to show that a scheme can never satisfy both properties simultaneously.
As a result, we are limited to working with two flavors of commitment schemes:
• statistically-hiding schemes, which are statistically-hiding and computationally-binding
• statistically-binding schemes, which are statistically-binding and computationally-hiding
Ok, we’re just about done with the dry definitions. Good job if you stuck through it. We’re almost at the finish line.
Hamiltonian cycles
Define \(HAM = \{ G | G \text{ contains a Hamiltonian cycle}\}\). A Hamiltonian cycle is a cycle through the graph which visits each vertex exactly once. This language \(HAM\) is NP complete - any
language in NP can be reduced to \(HAM\) in polynomial time.
For this post, we will consider a graph with \(n\) vertices as being represented by an \(n \times n\) adjacency matrix, where the \((i,j)^{th}\) entry is a 1 if the graph contains the edge \(i \
rightarrow j\), and 0 otherwise.
Adjacency matrix representation of a graph G with a Hamiltonian cycle w (bolded)
[Source: Lecture 2, Slide 25]
ZK proof for HAM
Computationally zero-knowledge proof for HAM
[Source: Lecture 2, Slide 27]
• \(P\) draws a random permutation \(\Pi \in S_n\)
• \(P\) commits to \(c = Com(\Pi(G))\), and sends it to \(V\)
□ \(\Pi(G)\) represents the original graph \(G\) after permuting the vertices according to \(\Pi\)
□ \(P\) can commit to \(\Pi(G)\) by committing to each bit of the adjacency matrix for \(\Pi(G)\)
• \(V\) then draws a random challenge bit \(b \in \{ 0, 1\}\), and sends it to \(P\)
• If \(b=0\text{:}\)
□ \(P\) only reveals the cycle within the permuted graph, i.e. that \(u = \Pi(w) \in Dec(c)\)
☆ \(P\) does this by revealing only the particular bits in \(\Pi(G)\) which correspond to the cycle \(u = \Pi(w)\)
□ \(V\) verifies that \(u \in Dec(c)\) and that \(u\) is a cycle
• If \(b=1\text{:}\)
□ \(P\) sends the permutation \(\Pi\) and reveals the permuted graph \(H = \Pi(G)\)
☆ \(P\) does this by revealing all the bits of \(H=\Pi(G)\)
□ \(V\) verifies that \(H = Dec(c)\), and that \(H = \Pi(G)\)
Completeness: Completeness is straightforward. Suppose that \(G\) has a Hamiltonian cycle \(w\). A prover \(P\) who follows the protocol honestly by drawing a random permutation \(\Pi\) and sending \
(c = Com(\Pi(G))\) will be able to answer both challenges correctly:
• \(u = \Pi(w)\) is a cycle in \(\Pi(G)\) since \(w\) is a cycle in \(G\). \(u = \Pi(w)\) is contained in \(H = \Pi(G)\), and therefore \(u \in Dec(c)\). So the \(b=0\) challenge is successful.
• \(H = Dec(c)\) where \(H=\Pi(G)\) by construction, so the \(b=1\) challenge is successful.
Soundness: The claim is that if \(Com\) is statistically-binding, then soundness holds.
Assume that \(G \notin HAM\). Now suppose for contradiction that \(\Pr_b[(P^*, V) \text{ accepts } G] > 1/2\). Then it must be the case that both challenges succeed, and hence \(u\) is a Hamiltonian
cycle in \(H\) and \(H = \Pi(G)\). But then \(\Pi^{-1}(u)\) would give a valid Hamiltonian cycle in \(G\), which would imply \(G \in HAM\). This is a contradiction, and hence we conclude that \(\Pr_b
[(P^*, V) \text{ accepts } G] \leq 1/2\).
Note that this argument only holds if was assume \(Com\) to be statistically-binding. If not, then \(P^*\) could commit to some value \(c\), and then reveal different values (both compatible with \(c
\)) depending on which challenge bit was sent by \(V\). Remember that in this model, the computation of \(P^*\) is not bounded. That’s why a computationally-binding commitment is not good enough
Computational zero-knowledge: We’ll construct a simulator \(S^{V^*}(G)\) as follows:
1. Set \(G_0 = u\) for some random cycle \(u\) over \(n\) vertices
2. Set \(G_1 = \Pi(G)\) for some random permutation \(\Pi \in S_n\)
3. Sample \(b\) randomly from \(\{ 0, 1 \}\)
□ If \(b=0\), set \(c = Com(G_0)\)
□ If \(b=1\), set \(c = Com(G_1)\)
4. If \(V^*(c) = b\)
□ If \(b=0\), output \((c, b, u)\)
□ If \(b=1\), output \((c, b, (\Pi, G_1))\)
5. Otherwise, repeat
The output of this simulator \(S^{V^*}(G)\) is computationally-indistinguishable from a real transcript \((P, V)(G)\text{:}\)
• The two distributions of the commit message \(c\) are computationally-indistinguishable because the commitment scheme \(Com\) is computationally-hiding.
□ Note that if \(Com\) were not computationally-hiding, then a PPT adversary might be able to distinguish between \(Com(G_0)\) and \(Com(G_1)\), and could therefore distinguish between the
commit \(c\)’s simulated distribution vs its transcript distribution (the simulator commits to the cycle \(G_0\) about half of the time, while the real protocol always commits to the fully
permuted graph \(G_1\)).
• The two distributions of the challenge bit \(b\) are identical, as \(S\) matches the distribution of \(V^{*}\).
• The distributions of \(u\) and \((\Pi, G_1)\), as they are drawn in effectively the same way (uniformly random) in both the simulator and the real protocol.
Next, we argue that if \(Com\) is computationally-hiding, then \(\Pr_{c, b}[V^*(Com(G_b)) = b] \approx 1/2\) (where the approximation sign \(\approx\) indicates that the difference between the two
values is negligible). If this were not the case, then \(V^*\)’s output distribution would be non-negligibly different when running on input \(Com(G_0)\) and \(Com(G_1)\), and therefore \(V^*\) could
distinguish between the two inputs \(Com(G_0)\) and \(Com(G_1)\). But this violates the assumption that \(Com\) is computationally-hiding.
From \(\Pr_{c, b}[V^*(Com(G_b)) = b] \approx 1/2\), it follows that the expected number of repetitions the simulator makes is 2 (this analysis is similar to that done in the previous post for the
quadratic residuosity proof).
Alright! That’s it, we’re done!
We started by establishing a limitation of our definition of perfect zero-knowledge, namely that there cannot exist (well, we really strongly believe that there cannot exist) perfect zero-knowledge
proofs for all languages in NP. This motivated us to relax our definition to computational zero-knowledge: zero-knowledge that holds for polynomially-bounded machines. We then defined commitment
schemes and their useful properties, all in preparation for our zero-knowledge proof of \(HAM\).
\(HAM\) is an NP-complete language, meaning any language in NP can be mapped to \(HAM\) in polynomial time. Thus, if we can create a computational zero-knowledge proof for \(HAM\), then this proof
can be used to prove any language in NP! And that’s indeed what we did! Using statistically-hiding commitment schemes, we constructed an elegant interactive proof for \(HAM\) satisfying computational | {"url":"https://andyrdt.com/posts/zk2","timestamp":"2024-11-02T00:18:31Z","content_type":"text/html","content_length":"25736","record_id":"<urn:uuid:82e37078-afbd-4792-a2f3-aa4c2ab50048>","cc-path":"CC-MAIN-2024-46/segments/1730477027599.25/warc/CC-MAIN-20241101215119-20241102005119-00861.warc.gz"} |
Lesson 25
Summing Up
• Let’s figure out a better way to add numbers.
Problem 1
The formula for the sum \(s\) of the first \(n\) terms in a geometric sequence is given by \(s = a \left( \frac{1-r^{n}}{1-r}\right)\), where \(a\) is the initial value and \(r\) is the common ratio.
A drug is prescribed for a patient to take 120 mg every 12 hours for 8 days. After 12 hours, 6% of this drug is still in the body. How much of the drug is in the body after the last dose?
Problem 2
The formula for the sum \(s\) of the first \(n\) terms in a geometric sequence is given by \(s = a \left( \frac{1-r^{n}}{1-r}\right)\), where \(a\) is the initial value and \(r\) is the common ratio.
If a sequence has \(a=10\) and \(r=0.25\),
1. What are the first 4 terms of the sequence?
2. What is the sum of the first 17 terms of the sequence?
Problem 3
Jada drinks a cup of tea every morning at 8:00 a.m. for 14 days. There is 40 mg of caffeine in each cup of tea she drinks. 24 hours after she drinks the tea, only 6% of the caffeine is still in her
1. How much caffeine is in her body right after drinking the tea on the first, second, and third day?
2. When will the total amount of caffeine in Jada be the highest during the 14 days? Explain your reasoning.
Problem 4
Select all polynomials that have \((x+1)\) as a factor.
(From Unit 2, Lesson 15.)
Problem 5
A car begins its drive in heavy traffic and then continues on the highway without traffic. The average cost (in dollars) of the gas this car uses per mile for driving \(x\) miles is \(c(x)=\frac
{0.65+0.15x}{x}\). As \(x\) gets larger and larger, what does the end behavior of the function tell you about the situation?
(From Unit 2, Lesson 18.)
Problem 6
Write a rational equation that cannot have a solution at \(x=2\).
(From Unit 2, Lesson 22.)
Problem 7
For \(x\)-values of 0 and -1, \((x+1)^3 = x^3+1\). Does this mean the equation is an identity? Explain your reasoning.
(From Unit 2, Lesson 24.) | {"url":"https://im.kendallhunt.com/HS/students/3/2/25/practice.html","timestamp":"2024-11-02T18:53:37Z","content_type":"text/html","content_length":"72109","record_id":"<urn:uuid:3d635bab-66e6-402b-9fe4-5860c3305809>","cc-path":"CC-MAIN-2024-46/segments/1730477027729.26/warc/CC-MAIN-20241102165015-20241102195015-00428.warc.gz"} |
Pattern Puzzles to Build Algebraic Thinking | DIGITAL Puzzles for 3rd-5th
Pattern Puzzles to Build Algebraic Thinking | DIGITAL Puzzles
Do your kids see the connection between arithmetic and algebra? Build fluency with operations and problem solving skills with these engaging pattern puzzles for grades 3-5.
Mathematics is often referred to as the science of patterns. Noticing patterns in numbers, operations and the world around us is what math is all about. So I love finding unique ways to practice
looking for patterns with my kids. These triangle pattern puzzles introduce the idea of a function or ‘rule’ in a fun and non-threatening way, while challenging kids to see patterns and practice
basic operations. Once kids determine the ‘rule’ of the puzzle they can solve for the missing number. It’s such a fun and unique challenge, they won’t even realize they’re doing math!
How the Pattern Puzzles Work:
Each puzzle shows three triangles.
The numbers in the triangle (the top, left and right values) follow a particular rule to give the solution in the middle of the triangle.
The goal is for students to figure out what operations they can use to get the middle number.
For example, in puzzle #1, the numbers in the first triangle are 12, 6, & 3 and the solution in the middle is 9. Using addition & subtraction, we can use the triangle numbers to find the solution: 12
– 6 + 3 = 9.
To test this pattern, we can follow the same rule with the middle triangle to see if the rule holds true. This gives us: 8 – 4 + 2 = 6 (this is true, so we have a pattern).
Following this pattern, we can now find the missing number in the last triangle: 6 – 3 + 1 = 4.
To write a general rule for this pattern, we can use the letters T (for the top number), L (for the bottom left number) and R (for the bottom right number).
This makes the rule for this pattern: T – L + R
An additional example, with directions is included in the Google Slides resource, helping to clarify the directions for students (especially if you are not completing these together in person).
Completing the Pattern Puzzles in Google Slides:
This free download includes 5 different pattern puzzles in a digital format. By grabbing the Google Slides, you can assign one or more puzzles in Google Classroom or you can display it virtually with
your class to discuss as a whole group.
To complete them, there is space on each slide for students to type a general rule to represent the pattern they see and a box to type in the missing number.
Each of these 5 puzzles can be solved using only addition & subtraction.
An answer key is also included.
These would make fun math warm-ups or weekly puzzle challenges. You could also use these as enrichment for advanced students or early finishers.
Note: Although these are meant to be a no-print activity, where students type the rule and missing number onto each slide, you can print the slides out if you prefer.
To do this, make a copy of the resource in your Google Drive.
Then delete the text box on each slide where it says “Type your rule here.”
Then go to File–>Print and select your print settings.
Pattern Puzzles for the Whole Year:
If you and your students enjoy these math challenges, you may be interested in the whole set of puzzles.
The complete set includes 20 different puzzles that increase in difficulty.
These incorporate more math operations (multiplication & division), a “create you own puzzle” challenge and come in both digital + printer-friendly formats.
Learn more about the set of Triangle Pattern Puzzles HERE
To try out this sample set of triangle pattern puzzles, use the link below to grab it from my shop!
And find more missing number puzzles at the links below:
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Trading and understanding the US Volatility Index with Pepperstone | Pepperstone
Trading and understanding the US Volatility Index with Pepperstone
Aug 25, 2022
Volatility is one of the most essential inputs in trading – it affects how much risk we take in a position, arguably the core element of any trading system. Volatility (vol) should be the key
determinant of position size for each trade, but also our emotional state, where higher vol regimes can see traders lose discipline, act recklessly and often overtrade the account.
But how should we view and measure volatility?
Volatility can be split into ‘realised’ and ‘implied’ volatility.
Realised volatility (vol) is a statistical measure of how far prices move (on each bar) away from the average of a set period – the 20-day average is commonly used here. The further, or more
dispersed prices move away from the average, the higher the measured volatility (vol).
Importantly, the greater the vol, the wider the stop loss should be on the trade. As a consequence of taking on more risk, a trader could reduce the position size and/or leverage rate, and vice
The majority of volatility indicators used by traders are based on realised volatility and are determined by past price moves – some of the most popular volatility measures include Bollinger Bands,
ATR (Average True Range), Keltner Bands, standard deviation, and even pivot points.
We can loosely argue that because most market players are making bets on future movement, that the moves in price are anticipatory and therefore contain some element of forward-looking information.
However, these moves are not a direct expression of future or implied volatility – that is where the US Volatility Index can be a useful measure of expected equity volatility and can be activity
traded by Pepperstone clients.
While it is not an exact science, given the composition of the options expiries involved, as we highlight in this article - by dividing the current US Volatility index by 15.9 we can understand the
expected daily moves (higher or lower) in the US500 over the coming 30 days. This calculation typically overstates where US equity volatility realises, but it is a good guide on market expectations
around movement and traders can therefore trade the US volatility index based on whether or not they agree with this implied move.
(US volatility index – daily chart)
What is the US volatility index?
The CBoe volatility index (VIX index) incorporates the price of various S&P500 options, which have between 23 and 37 days to expiry – these are weighted to create a blended maturity of 30 days. The
demand for options is a function of whether traders feel the instrument will move above the strike price by the expiration date – the higher the probability of the S&P500 moving ‘in-the-money’, the
greater the demand and subsequently the higher the price of the option - it’s here where we see the US Volatility index rising.
Importantly, the US Volatility index is expressed, quoted and traded in percentage terms as an annualised standard deviation number – this makes it unique in that you’re trading a parentage number
and not an index level per se.
How is the US volatility index priced?
As is the case with most spot prices, the VIX index is a valuation of the options used in the calculation – you can’t directly trade the underlying VIX index and for that, we turn to the VIX futures
– it’s the VIX futures where the US Volatility index takes its price.
The US volatility index is set off a calendar-weighted blend between the front- and second-month VIX futures contracts. In essence, as we get closer to the expiration of the front-month futures
contract the US volatility index will take increased influence from the price of the second-month contract – this allows for an efficient continuous rolling position for clients to trade.
Today is the 18 August 2022.
Front month VIX September futures (expiry date of 21 September) = 24.5%
Second-month VIX October futures (expiry date of 19 October) = 26.2%
For simplicity’s sake let us say today is the 18 August, the day after the VIX August 2022 futures expiry. There is just under a full month to the expiration of the VIX September futures contract,
and therefore very little influence from the October futures contract. As such, we see the US Volatility index trading at 24.8%.
However, as each day passes, and we approach the VIX September futures expiry (on the 21 Sept), the US Volatility index will edge closer to the VIX October futures price.
If today was the 20 September (the day before the September VIX futures expiration), the US volatility index would be almost fully aligned with the October VIX futures price and trading closer to
How to trade the US volatility index?
Pepperstone tries to make position sizing simple for traders and so 1 lot of US volatility index equates to a $1 per point move in the index. Importantly,we price the move off the number before the
decimal point.
For example – I buy 100 lots of the US Volatility index at 24.80% (the quote was 24.64 - 24.80) – on that day, the bid price goes up to 25.80% - my profit is, therefore, $100. Given an outsized daily
move in the US volatility index is around 4 points (or ‘4 volatilities’), we need to consider this in our lot sizing and traders will typically trade a far higher lot size than they would in say in
EURUSD or NAS100.
If a trader believes there will be increased movement in the US equity market – and notably moves lower in the US500 – believing market participants will increase their demand for portfolio hedges
and paying higher prices for S&P index options - then being long the US Volatility index would be a position to consider.
Consider that typically the further out you go in the futures contract the higher the level – this is normal in the VIX futures, and represents holding costs – the ‘steepness’ in the futures curve
represent the cost of carry and its why.
*Clients pay swaps on long positions held past the rollover point.
If a trader felt implied levels of equity market volatility were going lower – perhaps the US Volatility index is at the top of its multi-month range and there could be an event that holds a high
probability of reducing market anxiety (and volatility) – perhaps actions from a central bank, increased corporate buy-backs or a better corporate earnings season, then being short volatility may be
a position to take.
*Clients receive swaps on short positions held past the rollover point.
Trade the US Volatility index with Pepperstone
Simplistically, a rising equity market is synonymous with a falling US volatility index, as traders sell volatility which reduces the price of S&P options. Anticipation of a move lower in the US500
should see demand for the US volitivity index rise, which lifts the US volatility index. Some traders will simply use the index as a guide on the current risk regime – if the US volatility index is
above 30%, they may want to lower position size, reduce leverage and even cut their hold times so they only hold positions when in front of the screens.
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MathFiction: Nearly Gone (Elle Cosimano)
a list compiled by Alex Kasman (College of Charleston)
Home All New Browse Search About
Nearly Gone (2015)
Elle Cosimano
Nearly Boswell has (obviously) a really cool name. She also has a strong interest in her science and math classes. And, for some reason, she also has the ability to taste emotions when she touches
other people's skin. Unsurprisingly, there is a romance with a "bad boy" who turns out not to be what he at first seems. But the serious plot in this YA novel concerns a serial killer who begins
murdering students and leaving cryptic STEM-themed clues for Nearly.
I was sincerely tempted not to include this young adult novel in this database of mathematical fiction. Even though math is discussed and some of the clues have a mathematical component, there really
is very little mathematical content overall. However, I realized that the book is marketing itself as mathematical fiction (e.g. the "E" and "A" in her name on the cover are represented by numerals 3
and 4). And so, I am considering it a public service to review it here just to say that this only barely counts as mathematical fiction. In fact some of the things which might at first glance be
mathematical turn out not to be (like a clue with numbers that actually refers to chemical elements). And, some non-mathematical concepts are incorrectly described in mathematical terms (such as the
Schrödinger's Cat thought experiment which is falsely said to be an example of a mathematical proof by contradiction.)
Perhaps I am selling this book (and its sequels which I have not read) short. If you think so, please use the link below or e-mail me and I'll post your opinion here.
More information about this work can be found at www.amazon.com.
(Note: This is just one work of mathematical fiction from the list. To see the entire list or to see more works of mathematical fiction, return to the Homepage.)
Works Similar to Nearly Gone
According to my `secret formula', the following works of mathematical fiction are similar to this one:
1. The Witch of Agnesi by Robert Spiller
2. Do the Math: Secrets, Lies, and Algebra by Wendy Lichtman
3. Do the Math #2: The Writing on the Wall by Wendy Lichtman
4. The Square Root of Murder by Paul Zindel
5. The Wright 3 by Blue Balliet
6. Chasing Vermeer by Blue Balliet
7. Crimes and Math Demeanors by Leith Hathout
8. The Unknowns: A Mystery by Benedict Carey
9. The Distant Dead by Heather Young
10. The Absolute Value of Mike by Kathryn Erskine
Ratings for Nearly Gone:
Content: Have you seen/read this work of mathematical fiction? Then click here to enter your own votes on its mathematical content and literary quality or send me comments to post on
1/5 (1 votes) this Webpage.
Literary Quality:
2/5 (1 votes)
Genre Mystery, Young Adult,
Motif Math Education,
Medium Novels,
Home All New Browse Search About
Exciting News: The 1,600th entry was recently added to this database of mathematical fiction! Also, for those of you interested in non-fictional math books let me (shamelessly) plug the recent
release of the second edition of my soliton theory textbook.
(Maintained by Alex Kasman, College of Charleston) | {"url":"https://kasmana.people.charleston.edu/MATHFICT/mfview.php?callnumber=mf1552","timestamp":"2024-11-10T09:21:22Z","content_type":"text/html","content_length":"9617","record_id":"<urn:uuid:c00906ce-8ca0-4f4e-9908-fe0635d38a82>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00701.warc.gz"} |
Are we prisoners of reversibility?
(some loose thoughts that are definitely not new, but which I find quite fascinating)
A broken glass will not fix itself; you can either face the fury of your neighbors or forget your ball and run. Whatever you chose, your relationship with these people will change for ever. Some
things are irreversible and we have to accept it.
Or do we?
All of us who work in quantum information surely remember the first lesson that quantum gates are unitary and that the evolution of a qubit can be viewed as a rotation of a vector. We also
immediately find that these rotations are “nice” because they can be inverted. However a few lessons later we learn about Kraus operations and Master equations, and we find that in fact
irreversibility is implicit. I don’t know about you, but my first impression was that quantum irreversibility is a very messy business. Of course, we can refer to “the Church of the Larger Hilbert
space” and purify everything by extending the system and making everything unitary again, but in the end does larger mean simpler?
Correct me if I am wrong, but I think that many people believe that the power of quantum computation, not taking into account the measurement phenomena, is going to be based on unitarity. Perhaps
they are right. Up until now all famous quantum protocols are based on unitary gates. Moreover, since the era of “information is physical” and the works of Landauer and his colleagues, wehave started
to believe that the future of computation will rely on reversibility.
But is it good to fight with irreversibility? First of all, let us clarify that irreversibility does not necessarily mean mess, i.e. it does not imply randomness and unpredictability. The AND gate is
fully deterministic – if you know the input, you know the output. The thing is that it is not the other way around. If you want to trace back your computation, at some point you will have to guess.
This may not be a problem since time flows in one direction so why bother? (well, let’s leave this for the different post…) Everybody has to admit that the AND gate is a very elegant and simple piece
of computational architecture. So irreversibility can be a source of simplicity and elegance.
Still, simplicity and elegance is not enough, so let’s not use Occam’s Razor yet. The fun starts when you realize that actually irreversibility is a very powerful phenomenon. This idea was introduced
by Poincare and Boltzman, and was further developed by Prigogine and others. Reversibility is in a sense boring, since in a finite system you are closed in a loop. Reversibility leads
to reoccurrencesand nothing new and stable can emerge, since you will always go back to the start. On the other hand, irreversibility demands that you have to forget the past – you will
not necessarily come back to the start. Even better, something new can emerge.
Ideas like emergence, self organization and complexity have been around in science for some time. In particular, very simple computer science models, like cellular automata, can nevertheless exhibit
these nontrivial phenomena. On the other hand, the unitary quantum version of cellular automaton does not seem to possess so many intriguing features. Therefore, the natural question is: why stick to
There are “complex” quantum mechanical phenomena that do not work without irreversibility. Laser cannot work if the system follows only a unitary evolution and the dynamics of bosonic condensation
does not seem to be unitary. Therefore, can irreversibility bring something new to quantum information? | {"url":"https://dag.quantumlah.org/are-we-prisoners-of-reversibility/","timestamp":"2024-11-02T14:01:04Z","content_type":"text/html","content_length":"30631","record_id":"<urn:uuid:a04e769b-c3f4-48db-8230-7c5bacdb487e>","cc-path":"CC-MAIN-2024-46/segments/1730477027714.37/warc/CC-MAIN-20241102133748-20241102163748-00383.warc.gz"} |
Q-Learning and SARSA in RL - Similarities and Differences Explained
Q-learning and SARSA are two of the algorithms that one generally encounters early in the journey of learning reinforcement learning. However, despite the high similarity between these two
algorithms, in practice, Q-learning often takes prominence in terms of performance. In this blog post, we’ll discuss the similarities and differences between these two algorithms, as well as the
reason for the strength of one over the other.
Temporal Difference (TD) Learning
Let’s begin by talking about an equation for the value function. We can denote the value of a state as the sum of the reward we get from that state and the value of the next state. Generally, we
multiply the value of the next state by a number less than one. We refer to this value as gamma (γ). All this can be denoted by the following equation:
$V(s_k)=\mathbb{E}(r_k+\gamma V(s_{k+1}))$
Here, the $\mathbb{E}$ symbol is used to show that we don’t yet know whether we will receive the reward rk or whether our value function is correct.
Turns out, we can use this equation to write a new equation that will act as our update function. In other words, this new equation will tell us what we need to do in order to improve our value
function’s estimates.
Value function update equation
This is also known as TD(0) learning and it’s what Q-learning and SARSA are based on.
It’s a big-looking equation, so let me break down what’s going on here. On the left, we have the updated value function: the function we get after updating our old value function. On the right, we do
the update.
You may also notice a familiar set of terms. $r_k + \gamma V_{old}(s_{k+1})$ is the estimate of the value function that we saw in the previous equation (a.k.a. the target estimate). Here, $r_k$ is
the reward we actually received when we went to sk by following some policy. So, the target estimate is likely to be closer to the actual value function than $V_{old}(s_k)$.
But, we don’t want to immediately use that as our new value function. Why?
It’s because it’s still an estimate. We don’t want to accept the result from one experience as the hard truth. Instead, we want to incrementally go towards the correct value function using multiple
So, what TD-learning does is, it takes the difference between the target estimate and the old value estimate, and uses this difference to update the old value function by some amount. This difference
is also known as the TD error. The amount by which we update the function is controlled by $α$.
Let’s look at a simple example. Say the agent did something and it got a better reward than the value function expected. Then, the target estimate is higher than the old estimate. So the TD error
would be positive and the update would give a higher value to that state.
This section is not too important to understand Q-learning or SARSA, but it doesn’t hurt to understand what it is. However, feel free to skip it if you wish.
At the beginning of the previous section, you saw an equation for the value function. It was like looking one time-step into the future. Turns out there is actually a recursive nature to this
equation that we can use to look more timesteps into the future.
Just as we wrote $V_{old}(s_k)$ as $\mathbb{E}(r_k + \gamma V_{old}(s_{k+1}))$, we can replace $V_{old}(s_{k+1})$ with $r_{k+1} + \gamma V_{old}(s_{k+2})$. Note that we don’t have to include
the expectation sign ($\mathbb{E}$) in this substitution because the outer $\mathbb{E}$ already captures the idea. So, now we get the following equation.
$V(s_k)=\mathbb{E}(r_k+\gamma r_{k+1} + \gamma^2 V(s_{k+2}))$
Substituting this back into our update equation gives us the following.
$V^{new}(s_k)=V^{old}(s_k)+\alpha(r_k+\gamma r_{k+1} + \gamma^2 V^{old}(s_{k+2}) - V^{old}(s_k))$
As you might have guessed, we can do this expansion as many times as we want. This would give us the following target estimate.
$\sum_{j=0}^{n} \gamma^j r_{k+j} + \gamma^{n+1} V(s_{k+n+1})$
where n is the number of expansions since the first value function estimate.
Substituting it into the update equation, we get the update equation for TD(n).
$V^{new}(s_k)=V^{old}(s_k)+\alpha(\sum_{j=0}^{n} \gamma^j r_{k+j} + \gamma^{n+1} V(s_{k+n+1})- V^{old}(s_k))$
Substituting back $n=0$, we get our initial update equation from the previous section.
Now that you have an idea of what TD learning is, we can easily transition to what SARSA is. SARSA is the on-policy TD(0) learning of the quality function (Q).
The quality function tells you how good some action is when taken from a given state. All the equations we previously discussed for value functions (V) are also applicable to quality functions (Q).
The only difference is that now we’re taking 2 inputs (state s and action a) instead of just the state s.
$Q^{new}(s_k, a_k)=Q^{old}(s_k, a_k)+\alpha(r_k+\gamma Q^{old}(s_{k+1}, a_{k+1}) - Q^{old}(s_k, a_k))$
Notice how this equation is essentially the same as the TD(0) equation, with the exception that now we’re working with the quality function rather than the value function.
In the SARSA algorithm, we’re working with the current State ($s_k$), Action taken from the current state ($a_k$), Reward observed after taking the action ($r_k$), the next State after taking the
action ($s_{k+1}$), and the Action to take from the next state ($a_{k+1}$).
So, why do we call it on-policy? This is because the policy that resulted in the action ak is the same policy that we’re using to find $a_{k+1}$.
An algorithm is known as off-policy when the behavior policy and the target policy are different, and on-policy when they are the same. The behavior policy is the policy used to perform the actions
and generate experiences, while the target policy is the policy that is being optimized by the algorithm, based on the experiences from the behavior policy.
Q-learning is the off-policy TD(0) learning of the quality function (Q).
Let’s take a look at the Q-learning update equation. First, compare it against the SARSA update equation and identify what the difference between them is.
$Q^{new}(s_k, a_k)=Q^{old}(s_k, a_k)+\alpha(r_k+\gamma \max_{a} Q^{old}(s_{k+1}, a) - Q^{old}(s_k, a_k))$
Look at the TD target estimate. Instead of $r_k+ \gamma Q^{old}(s_{k+1}, a_{k+1})$, we are now taking the maximum Q-value over all the actions we can take from state $s_{k+1}$.
So, what makes the Q-learning algorithm off-policy? Turns out, in Q-learning, the behavior policy (the policy that generated the reward $r_k$) need not be the same as the target policy (the policy
that chooses the action that gives the maximum Q-value). The target policy can try to choose the optimal action (exploitation) without sacrificing exploration because the behavior policy could still
act suboptimally with better exploration.
If you still have doubts about how Q-learning works, take a look at this blog post for a more thorough explanation.
Pros and Cons of Q-Learning and SARSA
Comparison of the safe path (SARSA) vs. optimal path (Q-learning) (source)
• Global Optimum: Given sufficient exploration, Q-Learning will reach the globally optimal policy. SARSA, on the other hand, will require the exploration to be reduced with time in order to reach
the optimal policy. For instance, the epsilon value in epsilon-greedy SARSA would have to be reduced with time. It can be a hassle to tune this hyperparameter.
• Conservative Path: On the other side of the spectrum, SARSA tends to identify the safer path compared to Q-learning. Take a look at the above image (cliff walking). This conservativeness happens
because the exploration steps (random actions) near the cliff also contribute to updating the policy and as a result, the algorithm believes that there is a risk in staying near the cliff.
• Convergence: In Q-learning (and in off-policy algorithms in general), there samples tend to have a higher variance. This is because the action proposed by the behavior policy may be different
from that of the target policy. As a result, Q-learning is likely to take longer to converge.
Final Thoughts
Hope this helps. If you would like to see more explanations for reinforcement learning concepts, check out the RL page of the blog. | {"url":"https://dilithjay.com/blog/q-learning-and-sarsa","timestamp":"2024-11-14T05:22:50Z","content_type":"text/html","content_length":"26159","record_id":"<urn:uuid:1b6572da-d03c-4845-858c-9fd0f4630798>","cc-path":"CC-MAIN-2024-46/segments/1730477028526.56/warc/CC-MAIN-20241114031054-20241114061054-00398.warc.gz"} |
Polynomial Long Division Calculator
Polynomial long division is a formula for dividing a polynomial by other polynomial of the identical or lower degree, the generalised form of the common arithmetic technique is called as long
It is used in manual methods of calculation , because it separates an otherwise complex division issue into smaller ones. | {"url":"https://eguruchela.com/math/Calculator/polynomial-long-division","timestamp":"2024-11-06T07:11:39Z","content_type":"text/html","content_length":"15763","record_id":"<urn:uuid:efd0659f-2e41-45c7-aa9b-8df254f31f4f>","cc-path":"CC-MAIN-2024-46/segments/1730477027910.12/warc/CC-MAIN-20241106065928-20241106095928-00256.warc.gz"} |
Supporting Hyperplane - (Variational Analysis) - Vocab, Definition, Explanations | Fiveable
Supporting Hyperplane
from class:
Variational Analysis
A supporting hyperplane is a flat affine subspace of one dimension less than that of the space containing a convex set, which touches the set at a single point or along a face while separating it
from the outside. This concept is crucial for understanding how convex sets interact with linear functions and plays a key role in separation theorems, as well as in the characterization of
subgradients and optimization problems involving convex functions.
congrats on reading the definition of Supporting Hyperplane. now let's actually learn it.
5 Must Know Facts For Your Next Test
1. A supporting hyperplane exists for every convex set and can provide insights into the structure of the set and its optimization properties.
2. If a point lies on a convex set's boundary, there are infinitely many supporting hyperplanes at that point, reflecting the multiple ways to 'touch' the set without entering it.
3. The existence of supporting hyperplanes is essential in defining optimal solutions in convex optimization problems, where they help determine feasible regions.
4. Supporting hyperplanes play a key role in deriving dual problems in convex optimization, linking primal and dual formulations through the notion of support.
5. In geometric terms, supporting hyperplanes can visualize how convex sets are positioned relative to one another, aiding in solving separation problems.
Review Questions
• How do supporting hyperplanes illustrate the relationship between convex sets and their boundary points?
□ Supporting hyperplanes provide a geometric interpretation of how convex sets behave at their boundaries. When a hyperplane touches the boundary of a convex set at a single point or along a
face, it serves to separate the set from other points outside of it. This interaction highlights the notion that multiple supporting hyperplanes can exist at boundary points, emphasizing the
structural complexity of convex sets.
• Discuss how supporting hyperplanes are used in separation theorems and their significance in optimization.
□ Supporting hyperplanes are pivotal in separation theorems, which assert that two disjoint convex sets can be separated by at least one hyperplane. This has significant implications in
optimization as it allows for identifying feasible regions and determining optimal solutions. The ability to separate constraints effectively can lead to more efficient algorithms for solving
convex optimization problems.
• Evaluate how supporting hyperplanes relate to subgradients and duality in convex optimization theory.
□ Supporting hyperplanes are intimately connected to subgradients and duality in convex optimization. A subgradient at a point not only defines a tangent plane but also serves as an indicator
for optimality conditions in optimization problems. In duality theory, supporting hyperplanes facilitate the transition between primal and dual problems by establishing bounds on solutions,
thereby linking feasibility and optimality through their geometric representation.
"Supporting Hyperplane" also found in:
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Math 20-1
6 years ago
What you have there is an expression for the height of the ball. $h\left(t\right)=-4.9t^2+20t$h(t)=−4.9t^2+20t .
We want to find the time period where the height of the ball is higher than 8m.
If we subsitute 8m into our expression for the height, we obtain a quadratic equation. We can then find the roots of this quadratic equation using the quadratic formula $t=\frac{\left(-b\pm\sqrt{b^
2-4ac}\right)}{2a}$t=(−b±√b^2−4ac)2a or by using a graphing calculator and inspecting the graph. You will find two roots. Between those two roots (or times in this case), the ball will be higher than
If you want to use a graphing calculator, there's two ways. You can either graph the $h\left(t\right)=-4.9t^2+20t$h(t)=−4.9t^2+20t in Y1 and h=8 in Y2 and then search for the intersect. Or you can
just graph $-4.9t^2+20t-8$−4.9t^2+20t−8 and calculate the roots. Both will give you the same result. | {"url":"https://www.tutortag.com/chat/alberta/a-rubber-ball-is-thrown-upwards-with-an-initial-speed-of-20m","timestamp":"2024-11-14T17:06:26Z","content_type":"text/html","content_length":"23183","record_id":"<urn:uuid:ee6c2f65-32a7-4335-89c0-5b9a5badbb09>","cc-path":"CC-MAIN-2024-46/segments/1730477393980.94/warc/CC-MAIN-20241114162350-20241114192350-00413.warc.gz"} |
Performance benefits of linking R to multithreaded math libraries - SmartData Collective
Performance benefits of linking R to multithreaded math libraries
8 Min Read
R wasn’t originally designed as a multithreaded application — multiprocessor systems were still rare when the R Project was first conceived in the mid-90’s — and so, by default, R will only use one
processor of your dual-core laptop or quad-core desktop machine when doing calculations. For calculations that take a long time, like big simulations or modeling of large data sets, it would be nice
to put those other processors to use to speed up the computations. There are several parallel processing libraries for R available that allow you to explicitly run loops in R simultaneously (ideally,
each on a different processor), but using them does require you to rewrite your code accordingly.
But there is a way to make use of all your processing power for many computations in R, without changing a line of code. That’s because R is a statistical computing system, and at the heart of many
of the algorithms you use on a daily basis — data restructuring, regressions, classifications, even some graphics functions — is linear algebra. The data are transformed into vector and matrix
objects, and the internals of R have been cleverly designed to link to a standard “BLAS” API to …
R wasn’t originally designed as a multithreaded application — multiprocessor systems were still rare when the R Project was first conceived in the mid-90’s — and so, by default, R will only use one
processor of your dual-core laptop or quad-core desktop machine when doing calculations. For calculations that take a long time, like big simulations or modeling of large data sets, it would be nice
to put those other processors to use to speed up the computations. There are several parallel processing libraries for R available that allow you to explicitly run loops in R simultaneously (ideally,
each on a different processor), but using them does require you to rewrite your code accordingly.
But there is a way to make use of all your processing power for many computations in R, without changing a line of code. That’s because R is a statistical computing system, and at the heart of many
of the algorithms you use on a daily basis — data restructuring, regressions, classifications, even some graphics functions — is linear algebra. The data are transformed into vector and matrix
objects, and the internals of R have been cleverly designed to link to a standard “BLAS” API to perform calculations on vectors and matrices. The binaries provided by the R Core Group on CRAN (with
one exception, see below) are linked to an “internal BLAS which is well-tested and will be adequate for most uses of R”, but is not multi-threaded and so only uses one core. But the beauty of linking
to the BLAS API is that you can re-compile R to link to a different, multi-threaded BLAS library and, voilà, suddenly many computations are using all cores and therefore run much faster.
The MacOS port of R on CRAN is linked to ATLAS, a “tuned” BLAS that uses multiple cores for computations. As a result, R on a multi-core Mac (as all new Macs are these days) really zooms. But the
Windows binaries on CRAN are not linked to an optimized BLAS. It’s possible to compile and link R yourself, but it can be tricky.
That’s what we do at Revolution for our Windows and Linux distributions of Revolution R. When we compile R, we link it to the Intel Math Kernel Libraries, which includes a high-performance BLAS
implementation tuned to multi-core Intel chips. “Tuning” here means using efficient algorithms, optimized assembly code that exploits features of the chipset, and multi-threaded algorithms that use
all cores simultaneously. As a result, you get some serious speed boosts for many operations in R, especially on a multi-core system. Here are some examples:
Calculation Size Command R 2.9.2 Revolution R Revolution R
(1 core) (4 cores)
Matrix Multiply 10000×5000 B <- crossprod(A) 243 sec 22 sec 5.9 sec
Cholesky Factorization 5000×5000 C <- chol(B) 23 sec 3.8 sec 1.1 sec
Singular Value Decomposition 5000×5000 S <- svd (A,nu=0,nv=0) 62 sec 13 sec 4.9 sec
Principal Components Analysis 10000×5000 P <- prcomp(A) 237 sec 41 sec 15.6 sec
As you can see, using the Intel MKL libraries on a 4-core machine gives some dramatic speedups (about a quarter of the 1-core time, as you might expect). Perhaps more surprisingly, using the Intel
MKL libraries on a 1-core machine is also faster than using R’s standard BLAS library: this is a result of the optimized algorithms, not additional computing power. You even get improvements on
non-Intel chipsets (like AMD).
[A side note: These calculations were actually all run on an 8-core machine, specifically, an Intel Xeon 8-core CPU with 18 GB system RAM running Windows Server 2008 operating system. The complete
benchmark code is available on this page. The results for Revolution R 1-core and 4-core were calculated by restricting the Intel MKL library to use 1 thread and 4 threads, using the
Revolution-R-specific commands setMKLthreads(1) and setMKLthreads(4) respectively. This has the effect of using only the power of the specific number of cores, even when more cores are available.
Note: if you’re using Revolution R and are doing explicit parallel programming with doSMP, it’s a good idea to call setMKLthreads(1) first. Otherwise, your parallel loops and the multi-threaded
linear algebra computations will compete for the same processor and actually degrade performance.]
These results are dramatic, but multi-threaded BLAS libraries aren’t a panacea. Not all R commands ultimately link to BLAS code, even ones you might expect. (For example, lm for regression uses a
non-BLAS QR decomposition by default.) And if your R code ultimately does not involve linear algebra, you can’t expect any improvement at all. (For example, the “Program Control” R benchmarks by
Simon Urbanek show only marginal performance gains in Revolution R.) This is when explicit parallel programming is the route to improved performance. We’re also working on dedicated statistical
routines for Revolution R Enterprise that are explicitly multi-threaded for single-machines and also distributable to multiple machines in a cluster or in the cloud, but that’s a topic for another
Revolution Analytics: Performance Benchmarks | {"url":"https://www.smartdatacollective.com/27747/?amp=1","timestamp":"2024-11-10T15:33:53Z","content_type":"text/html","content_length":"116142","record_id":"<urn:uuid:36c3163b-f36d-4b25-a3aa-1b3568dd4497>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.60/warc/CC-MAIN-20241110134821-20241110164821-00133.warc.gz"} |
CS 124: Binary Search
Binary Search
boolean search(int[] values, int lookingFor) {
return false;
assert search(new int[]{1, 2, 4}, 4);
This brief lesson will cover one of the efficient algorithms enabled by sorting: binary search! This is a fun example of an algorithm that you can approach either recursively or iteratively. We’ll
walk through it visually and then let you tackle it on the next homework problem.
One of the reasons that sorting is such an important algorithmic primitive is that it enables other efficiency algorithms. For example, we’ve already seen linear array search, which has runtime O(n):
boolean search(int[] values, int lookingFor) {
for (int i = 0; i < values.length; i++) {
if (values[i] == lookingFor) {
return true;
return false;
assert search(new int[]{1, 2, 5}, 5);
Given that O(n) represents searching the entire array, it is clearly the worst case runtime for array search. But, it is also the best case if we have no idea where the item could be!
But what if we knew something about the structure of the data in the array. Specifically, what if we knew that it was sorted? Could we do better?
Content Restricted to Current CS 124 Students
A publicly-accessible version of this content is available at learncs.online.
Or, as my old graduate school friend David Malan famously explained it:
Binary search is one recursive algorithm where the O(log n) nature of the algorithm is fairly easy to understand. Start with an array of N items. After one round, you have N / 2 left to consider.
Then N / 4. Then N / 8. Until you either get to 1, or you find the item somewhere along the way. So in the worst case we do O(log n) steps, and in the best case we may do better. Cool!
As promised, a few cool search visualizations culled from the amazing interwebs. This one has sound!
Note that the following visualization has flashing lights. Avoid if you are sensitive.
These are also pretty good, as they show runtime for different inputs as well.
Now it’s your chance to implement this classic algorithm! There are good approaches that use both recursion and iteration. The iterative approach maintains the start and end index of where the item
might be in the array and adjusts these on each iteration. The recursive approach has the base case being an empty or single-item array, and makes the problem smaller by determining whether the item
should be in the left half-array or right half-array. Good luck and have fun!
Problem Restricted to Current CS 124 Students
A publicly-accessible version of this content is available at learncs.online.
Homework: Solve BinarySearcher Array
Let's implement a classic algorithm: binary search on an array.
Implement a class named BinarySearcher that provides one static method named search. search takes a SearchList as its first parameter and a Comparable as its second. If either parameter is null, or
if the SearchList is empty, you should throw an IllegalArgumentException.
SearchList is a provided class. It provides a get(int) method that returns the Comparable at that index, and a size method that returns the size of the SearchList. Those are the only two methods you
should need!
search returns a boolean indicating whether the passed value is located in the sorted SearchList. To search the sorted SearchList efficiently, implement the following algorithm:
• Examine the value in the middle of the current array (index (start + end) / 2)
• If the midpoint value is the value that we are looking for, return true
• If the value that we are looking for is greater than the midpoint value, adjust the current array to start at the midpoint
• if the value that we are looking for is less than the midpoint value, adjust the current array to end at the midpoint
• Continue until you find the value, or until the start reaches the end, at which point you can give up and return false
This is a fun problem! Good luck! Keep in mind that every time you call SearchList.get that counts as one access, so you'll need to reduce unnecessary accesses to pass the test suite.
More Practice
Need more practice? Head over to the practice page. | {"url":"https://www.cs124.org/lessons/Fall2023/java/123_binarysearch","timestamp":"2024-11-08T23:17:35Z","content_type":"text/html","content_length":"84882","record_id":"<urn:uuid:075fca87-966d-4cfc-abdd-13ea879685fe>","cc-path":"CC-MAIN-2024-46/segments/1730477028106.80/warc/CC-MAIN-20241108231327-20241109021327-00047.warc.gz"} |
In many research fields, researchers aim to identify significant associations between a set of explanatory variables and a response while controlling the false discovery rate (FDR). To this aim, we
develop a fully Bayesian generalization of the classical model-X knockoff filter. Knockoff filter introduces controlled noise in the model in the form of cleverly constructed copies of the predictors
as auxiliary variables. In our approach we consider the joint model of the covariates and the response and incorporate the conditional independence structure of the covariates into the prior
distribution of the auxiliary knockoff variables. We further incorporate the estimation of a graphical model among the covariates, which in turn aids knockoffs generation and improves the estimation
of the covariate effects on the response. We use a modified spike-and-slab prior on the regression coefficients, which avoids the increase of the model dimension as typical in the classical knockoff
filter. Our model performs variable selection using an upper bound on the posterior probability of non-inclusion. We show how our model construction leads to valid model-X knockoffs and demonstrate
that the proposed characterization is sufficient for controlling the BFDR at an arbitrary level, in finite samples. We also show that the model selection is robust to the estimation of the precision
matrix. We use simulated data to demonstrate that our proposal increases the stability of the selection with respect to classical knockoff methods, as it relies on the entire posterior distribution
of the knockoff variables instead of a single sample. With respect to Bayesian variable selection methods, we show that our selection procedure achieves comparable or better performances, while
maintaining control over the FDR. Finally, we show the usefulness of the proposed model with an application to real data.
Community detection methods have been extensively studied to recover communities structures in network data. While many models and methods focus on binary data, real-world networks also present the
strength of connections, which could be considered in the network analysis. We propose a probabilistic model for generating weighted networks that allows us to control network sparsity and
incorporates degree corrections for each node. We propose a community detection method based on the Variational Expectation-Maximization (VEM) algorithm. We show that the proposed method works well
in practice for simulated networks. We analyze the Brazilian airport network to compare the community structures before and during the COVID-19 pandemic.
We propose reinterpreting copula density estimation as a discriminative task. Under this novel estimation scheme, we train a classifier to distinguish samples from the joint density from those of the
product of independent marginals, recovering the copula density in the process. We derive equivalences between well-known copula classes and classification problems naturally arising in our
interpretation. Furthermore, we show our estimator achieves theoretical guarantees akin to maximum likelihood estimation. By identifying a connection with density ratio estimation, we benefit from
the rich literature and models available for such problems. Empirically, we demonstrate the applicability of our approach by estimating copulas of real and high-dimensional datasets, outperforming
competing copula estimators in density evaluation as well as sampling.
We investigate experimental design for randomized controlled trials (RCTs) with both equal and unequal treatment-control assignment probabilities. Our work makes progress on the connection between
the distributional discrepancy minimization (DDM) problem introduced by Harshaw et al. (2024) and the design of RCTs. We make two main contributions: First, we prove that approximating the optimal
solution of the DDM problem within even a certain constant error is NP-hard. Second, we introduce a new Multiplicative Weights Update (MWU) algorithm for the DDM problem, which improves the
Gram-Schmidt walk algorithm used by Harshaw et al. (2024) when assignment probabilities are unequal. Building on the framework of Harshaw et al. (2024) and our MWU algorithm, we then develop the MWU
design, which reduces the worst-case mean-squared error in estimating the average treatment effect. Finally, we present a comprehensive simulation study comparing our design with commonly used
In this paper we build a joint model which can accommodate for binary, ordinal and continuous responses, by assuming that the errors of the continuous variables and the errors underlying the ordinal
and binary outcomes follow a multivariate normal distribution. We employ composite likelihood methods to estimate the model parameters and use composite likelihood inference for model comparison and
uncertainty quantification. The complimentary R package mvordnorm implements estimation of this model using composite likelihood methods and is available for download from Github. We present two
use-cases in the area of risk management to illustrate our approach.
Missing data often significantly hamper standard time series analysis, yet in practice they are frequently encountered. In this paper, we introduce temporal Wasserstein imputation, a novel method for
imputing missing data in time series. Unlike existing techniques, our approach is fully nonparametric, circumventing the need for model specification prior to imputation, making it suitable for
potential nonlinear dynamics. Its principled algorithmic implementation can seamlessly handle univariate or multivariate time series with any missing pattern. In addition, the plausible range and
side information of the missing entries (such as box constraints) can easily be incorporated. As a key advantage, our method mitigates the distributional bias typical of many existing approaches,
ensuring more reliable downstream statistical analysis using the imputed series. Leveraging the benign landscape of the optimization formulation, we establish the convergence of an alternating
minimization algorithm to critical points. Furthermore, we provide conditions under which the marginal distributions of the underlying time series can be identified. Our numerical experiments,
including extensive simulations covering linear and nonlinear time series models and an application to a real-world groundwater dataset laden with missing data, corroborate the practical usefulness
of the proposed method.
In causal inference, many estimands of interest can be expressed as a linear functional of the outcome regression function; this includes, for example, average causal effects of static, dynamic and
stochastic interventions. For learning such estimands, in this work, we propose novel debiased machine learning estimators that are doubly robust asymptotically linear, thus providing not only doubly
robust consistency but also facilitating doubly robust inference (e.g., confidence intervals and hypothesis tests). To do so, we first establish a key link between calibration, a machine learning
technique typically used in prediction and classification tasks, and the conditions needed to achieve doubly robust asymptotic linearity. We then introduce calibrated debiased machine learning
(C-DML), a unified framework for doubly robust inference, and propose a specific C-DML estimator that integrates cross-fitting, isotonic calibration, and debiased machine learning estimation. A C-DML
estimator maintains asymptotic linearity when either the outcome regression or the Riesz representer of the linear functional is estimated sufficiently well, allowing the other to be estimated at
arbitrarily slow rates or even inconsistently. We propose a simple bootstrap-assisted approach for constructing doubly robust confidence intervals. Our theoretical and empirical results support the
use of C-DML to mitigate bias arising from the inconsistent or slow estimation of nuisance functions.
If the probability model is correctly specified, then we can estimate the covariance matrix of the asymptotic maximum likelihood estimate distribution using either the first or second derivatives of
the likelihood function. Therefore, if the determinants of these two different covariance matrix estimation formulas differ this indicates model misspecification. This misspecification detection
strategy is the basis of the Determinant Information Matrix Test ($GIMT_{Det}$). To investigate the performance of the $GIMT_{Det}$, a Deterministic Input Noisy And gate (DINA) Cognitive Diagnostic
Model (CDM) was fit to the Fraction-Subtraction dataset. Next, various misspecified versions of the original DINA CDM were fit to bootstrap data sets generated by sampling from the original fitted
DINA CDM. The $GIMT_{Det}$ showed good discrimination performance for larger levels of misspecification. In addition, the $GIMT_{Det}$ did not detect model misspecification when model
misspecification was not present and additionally did not detect model misspecification when the level of misspecification was very low. However, the $GIMT_{Det}$ discrimation performance was highly
variable across different misspecification strategies when the misspecification level was moderately sized. The proposed new misspecification detection methodology is promising but additional
empirical studies are required to further characterize its strengths and limitations.
Nonlinear relations between variables, such as the curvilinear relationship between childhood trauma and resilience in patients with schizophrenia and the moderation relationship between mentalizing,
and internalizing and externalizing symptoms and quality of life in youths, are more prevalent than our current methods have been able to detect. Although there has been a rise in network models,
network construction for the standard Gaussian graphical model depends solely upon linearity. While nonlinear models are an active field of study in psychological methodology, many of these models
require the analyst to specify the functional form of the relation. When performing more exploratory modeling, such as with cross-sectional network psychometrics, specifying the functional form a
nonlinear relation might take becomes infeasible given the number of possible relations modeled. Here, we apply a nonparametric approach to identifying nonlinear relations using partial distance
correlations. We found that partial distance correlations excel overall at identifying nonlinear relations regardless of functional form when compared with Pearson's and Spearman's partial
correlations. Through simulation studies and an empirical example, we show that partial distance correlations can be used to identify possible nonlinear relations in psychometric networks, enabling
researchers to then explore the shape of these relations with more confirmatory models.
We propose a restricted win probability estimand for comparing treatments in a randomized trial with a time-to-event outcome. We also propose Bayesian estimators for this summary measure as well as
the unrestricted win probability. Bayesian estimation is scalable and facilitates seamless handling of censoring mechanisms as compared to related non-parametric pairwise approaches like win ratios.
Unlike the log-rank test, these measures effectuate the estimand framework as they reflect a clearly defined population quantity related to the probability of a later event time with the potential
restriction that event times exceeding a pre-specified time are deemed equivalent. We compare efficacy with established methods using computer simulation and apply the proposed approach to 304
reconstructed datasets from oncology trials. We show that the proposed approach has more power than the log-rank test in early treatment difference scenarios, and at least as much power as the win
ratio in all scenarios considered. We also find that the proposed approach's statistical significance is concordant with the log-rank test for the vast majority of the oncology datasets examined. The
proposed approach offers an interpretable, efficient alternative for trials with time-to-event outcomes that aligns with the estimand framework.
The techniques suggested in Frühwirth-Schnatter et al. (2024) concern sparsity and factor selection and have enormous potential beyond standard factor analysis applications. We show how these
techniques can be applied to Latent Space (LS) models for network data. These models suffer from well-known identification issues of the latent factors due to likelihood invariance to factor
translation, reflection, and rotation (see Hoff et al., 2002). A set of observables can be instrumental in identifying the latent factors via auxiliary equations (see Liu et al., 2021). These, in
turn, share many analogies with the equations used in factor modeling, and we argue that the factor loading restrictions may be beneficial for achieving identification.
Ensuring robust model performance across diverse real-world scenarios requires addressing both transportability across domains with covariate shifts and extrapolation beyond observed data ranges.
However, there is no formal procedure for statistically evaluating generalizability in machine learning algorithms, particularly in causal inference. Existing methods often rely on arbitrary metrics
like AUC or MSE and focus predominantly on toy datasets, providing limited insights into real-world applicability. To address this gap, we propose a systematic and quantitative framework for
evaluating model generalizability under covariate distribution shifts, specifically within causal inference settings. Our approach leverages the frugal parameterization, allowing for flexible
simulations from fully and semi-synthetic benchmarks, offering comprehensive evaluations for both mean and distributional regression methods. By basing simulations on real data, our method ensures
more realistic evaluations, which is often missing in current work relying on simplified datasets. Furthermore, using simulations and statistical testing, our framework is robust and avoids
over-reliance on conventional metrics. Grounded in real-world data, it provides realistic insights into model performance, bridging the gap between synthetic evaluations and practical applications.
Resampling methods are especially well-suited to inference with estimators that provide only "black-box'' access. Jackknife is a form of resampling, widely used for bias correction and variance
estimation, that is well-understood under classical scaling where the sample size $n$ grows for a fixed problem. We study its behavior in application to estimating functionals using high-dimensional
$Z$-estimators, allowing both the sample size $n$ and problem dimension $d$ to diverge. We begin showing that the plug-in estimator based on the $Z$-estimate suffers from a quadratic breakdown: while
it is $\sqrt{n}$-consistent and asymptotically normal whenever $n \gtrsim d^2$, it fails for a broad class of problems whenever $n \lesssim d^2$. We then show that under suitable regularity
conditions, applying a jackknife correction yields an estimate that is $\sqrt{n}$-consistent and asymptotically normal whenever $n\gtrsim d^{3/2}$. This provides strong motivation for the use of
jackknife in high-dimensional problems where the dimension is moderate relative to sample size. We illustrate consequences of our general theory for various specific $Z$-estimators, including
non-linear functionals in linear models; generalized linear models; and the inverse propensity score weighting (IPW) estimate for the average treatment effect, among others.
This paper deals with Elliptical Wishart distributions - which generalize the Wishart distribution - in the context of signal processing and machine learning. Two algorithms to compute the maximum
likelihood estimator (MLE) are proposed: a fixed point algorithm and a Riemannian optimization method based on the derived information geometry of Elliptical Wishart distributions. The existence and
uniqueness of the MLE are characterized as well as the convergence of both estimation algorithms. Statistical properties of the MLE are also investigated such as consistency, asymptotic normality and
an intrinsic version of Fisher efficiency. On the statistical learning side, novel classification and clustering methods are designed. For the $t$-Wishart distribution, the performance of the MLE and
statistical learning algorithms are evaluated on both simulated and real EEG and hyperspectral data, showcasing the interest of our proposed methods.
We examine the challenges in ranking multiple treatments based on their estimated effects when using linear regression or its popular double-machine-learning variant, the Partially Linear Model
(PLM), in the presence of treatment effect heterogeneity. We demonstrate by example that overlap-weighting performed by linear models like PLM can produce Weighted Average Treatment Effects (WATE)
that have rankings that are inconsistent with the rankings of the underlying Average Treatment Effects (ATE). We define this as ranking reversals and derive a necessary and sufficient condition for
ranking reversals under the PLM. We conclude with several simulation studies conditions under which ranking reversals occur. | {"url":"https://papers.cool/arxiv/stat.ME","timestamp":"2024-11-06T05:19:47Z","content_type":"text/html","content_length":"70524","record_id":"<urn:uuid:7192da67-48a6-4cea-b40e-d5bede1614bc>","cc-path":"CC-MAIN-2024-46/segments/1730477027909.44/warc/CC-MAIN-20241106034659-20241106064659-00280.warc.gz"} |
Markov Chains
The document covers both Markov chains in both the discrete and continuous time models.
1. Preamble
A Markov chain is described by,
1. State space $\mathcal{S}$: Set of all possible states $s$ that the chain can be in.
2. Transition Kernel\Matrix $\mathbf{P}$: A two-dimensional matrix that stores the probability of going from state $i$ to another state in the state space $j$. State space $\mathcal{S}$ cardinality
is $|\mathcal{S}| = N$. The probability is denoted by, $$p_{ij} := Pr(s_{t+1} = j | s_t = i), i,j \in \mathcal{S}$$ where $t$ is the timestep counter for a *discrete* Markov chain.
$$ \mathbf{P} = \begin{bmatrix} p_{11} & p_{12} & \cdots & p_{1N}\\ p_{22} & p_{22} & \ddots & p_{2N}\\ p_{N1} & p_{N2} & \cdots & p_{NN} \end{bmatrix} $$
Note that the row vectors give transition probabilities for a state $i$, which combined with the normalization axiom must give $\sum_{j=1}^{N} p_{ij} = 1 \hspace{1em} \forall i \in \mathcal{S}$.
A self-transition $p_{ii}$ gives the probability of remaining in the same state at the next timestep.
Markov Property
The current transition at timestep $t$ is independent from the history $\mathcal{H}$, where it does not depend on previous timestep transitions $t-1, t-2, \cdots, t=0$. Formally, the probability of
transitioning to state $j$ is given by,
$$ p_{ij} = $$ $$ P(s_{t+1} = j | s_t = i_t, s_{t-1} = i_{t-1}, \cdots, s_{t=0} = i_{t=0}) $$ $$ = P(s_{t+1} = j | s_t = i_t) $$
Transition is said to be memoryless for all trajectories' enumerations $i_{t-1}, i_{t-2}, \cdots, i_{t=0}$.
Important note
we are not restricted to only use information given at the current timestep in transitioning. It is possible to perserve the Markov property by designing the state $s_t$ to include past
information up to a window $W_t = [t-1, t-2, \cdots, t-|W_t|]$. One popular example is presented in the DQN paper ^1, where the authors used the past three frames as part of the current state.
The DQN agent made its decision based on a state constructed from fresh and past information.
State Classifications
2. Discrete-Time Markov Chains (DTMCs)
Let $t = 0,1,\cdots, T$ be the timestep with horizon $T$. A Markov chain transition based on $\mathbf{P}$ in each $t$.
3. Continuous-Time Markov Chains (CTMCs)
The continuous-time equivalent builds on top of the discrete version, but where a defined timeslots exists in the discrete version, continuous-time take the limit over timeslot interval. The
continuous-time version is evaluated for integrals over time.
A stochastic process $X(t)$ is a CTMC if:
1. Timesteps are $t \in \mathbb{R}$.
2. Has state spacec $X(t) \in \mathcal{S}$, with $\mathcal{S}$ being a countable set ($|\mathcal{S}|$ either finite or infinite).
3. Holds the Markov property, $$ Pr(X(t+s) | X(u), u \leq s) $$ $$ = Pr(X(t+s) | X(s)) $$
Meaning that the conditional probability depends only on the current state $X(s)$.
Assumption 1
Non-explosivness For a finite time interval $\delta > 0$, the chain transitions to a finite number of states.
Time-homogenous CTMC: if transitions probabilities $Pr(X(t+s) | X(s))$ are independent of time $s$, then the CTMC is time-homogenous.
Transitions in a CTMC are defined as jumps, with the state $Y(k)$ being the state after $k$ jumps. The time interval between the $(k-1)^{th}$ and $k^{th}$ jumps is defined as $T_k$. $T_k$ is an
exponentially distributed random variable that depends only on the $Y(k-1)$ state. We define the time spent in state $i$ at time $t$ as $\gamma_i(t)$, $$\gamma_i(t) := inf\{s > 0 : X(t+s) \neq X(t) \
text{ and } X(t) = i\}$$
$\gamma_i(t)$ is an exponentially distributed random variable if the CTMC is time-homogenous. Denote $\frac{1}{q_i}$ as the mean time spent in state $i \in \mathcal{S}$.
Stationary Distribution $\pi$ of CTMCs
Theorem 1
CTMCs with finite state space $\mathcal{S}$ and is irreducible has a stationary distribution $\pi$ and $\lim\limits_{t \rightarrow \infty} p(t) = \pi \text{ } \forall \text{ } p(0)$. The
stationary distribution may not necessarily be unique.
The irreducability condition is not enough to ensure a stationary distribution for infinite state spaces. A stationary distribution may still exist for infinite state spaces.
In a CTMC, the states can be categorized as recurrent or transient (same as in DTMCs), but using different time intervals. A state $i$ is recurrent if, $$ \lim\limits_{T \rightarrow \infty} Pr \{ \
tau_i < T \} = 1 $$
With the intervals $\tau_i$ and $\gamma_i$ for state $i$ defined as, $$ \tau_i := inf \{ t > \gamma_i : X(t) = i \text{ and } X(0) = i \} $$
$$ \gamma_i := inf \{t > 0 : X(t) \neq i \text{ and } X(0) = i \} $$
If the above condition is not satisfied, then the state $i$ is transient.
Global and Local Balance Equations
Foster-Lyapunov for CTMCs
With the same goal as in DTMCs, of proving positive-recurrency for a Markov chain, the Foster-Lyapunov theorem can be extended to the continuous-time domain. This is another method of providing a
sufficient condition for positive recurrency.
Theorem 2
For an irreducible, non-explosive CTMC, if a function $V : \mathcal{S} \rightarrow \mathbb{R}^{+}$ exists such that:
1. $\sum\limits_{j \neq i} Q_{ij}(V(j) - V(i)) \leq - \epsilon$ if $i \in \beta^{c}$.
2. $\sum\limits_{j \neq i} Q_{ij} (V(j) - V(i)) \leq M$ if $i \in \beta$. | {"url":"https://khalednakhleh.com/posts/stochastic_process/markov_chains/","timestamp":"2024-11-02T08:46:59Z","content_type":"text/html","content_length":"19434","record_id":"<urn:uuid:bf629268-5003-490e-8e44-376dc33769d6>","cc-path":"CC-MAIN-2024-46/segments/1730477027709.8/warc/CC-MAIN-20241102071948-20241102101948-00680.warc.gz"} |
A collection of various methods for accelerating Modulus Sym are presented below. The figures below show a summary of performance improvements using various Modulus Sym features over different
Fig. 51 Speed-up across different Modulus Sym releases on V100 GPUs. (MFD: Meshless Finite Derivatives)
Fig. 52 Speed-up across different Modulus Sym releases on A100 GPUs. (MFD: Meshless Finite Derivatives)
The higher vRAM in A100 GPUs means that we can use twice the batch size/GPU compared to the V100 runs. For comparison purposes, the total batch size is held constant, hence the A100 plots use 2 A100
These figures are only for summary purposes and the runs were performed on the flow part of the example presented in Industrial Heat Sink. For more details on performance gains due to individual
features, please refer to the subsequent sections.
TensorFloat-32 (TF32) is a new math mode available on NVIDIA A100 GPUs for handing matrix math and tensor operations used during the training of a neural network.
On A100 GPUs, the TF32 feature is “ON” by default and you do not need to make any modifications to the regular scripts to use this feature. With this feature, you can obtain up to 1.8x speed-up over
FP32 on A100 GPUs for the FPGA problem. This allows us to achieve same results with dramatically reduced training times (Fig. 53) without change in accuracy and loss convergence (Table 2 and Fig. 54
Fig. 53 Achieved speed-up using the TF32 compute mode on an A100 GPU for the FPGA example
Table 2 Comparison of results with and without TF32 math mode
Case Description \(P_{drop}\) \((Pa)\)
Modulus Sym: Fully Connected Networks with FP32 29.24
Modulus Sym: Fully Connected Networks with TF32 29.13
OpenFOAM Solver 28.03
Commercial Solver 28.38
Fig. 54 Loss convergence plot for FPGA simulation with TF32 feature
JIT compilation is a feature where elements of the computational graph can be compiled from native PyTorch to the TorchScript backend. This allows for optimizations like avoiding python’s Global
Interpreter Lock (GIL) as well as compute optimizations including dead code elimination, common substring elimination and pointwise kernel fusion.
PINNs used in Modulus Sym have many peculiarities including the presence of many pointwise operations. Such operations, while being computationally inexpensive, put a large pressure on the memory
subsystem of a GPU. JIT allows for kernel fusion, so that many of these operations can be computed simultaneously in a single kernel and thereby reducing the number of memory transfers from GPU
memory to the compute units.
JIT is enabled by default in Modulus Sym through the jit option in the config file. You can optionally disable JIT by adding a jit: false option in the config file or add a jit=False command line
Modulus Sym supports CUDA Graph optimization which can accelerate problems that are launch latency bottlenecked and improve parallel performance. Due to the strong scaling of GPU hardware, some
machine learning problems can struggle keeping the GPU saturated resulting in work submission latency. This also impacts scalability due to work getting delayed from these bottlenecks. CUDA Graphs
provides a solution to this problem by allowing the CPU to submit a sequence of jobs to the GPU rather than individually. For problems that are not matrix multiplied bound on the GPU, this can
produce a notable speed up. Regardless of performance gains, it is recommended to use CUDA Graphs when possible, particularly when using multi-GPU and multi-node training. For additional details on
CUDA Graphs in PyTorch, the reader is refered to the PyTorch Blog.
There are three steps to using CUDA Graphs:
1. Warm-up phase where training is executed normally.
2. Recording phase during which the forward and backward kernels during one training iteration are recorded into a graph.
3. Replay of the recorded graph which is used for the rest of training.
Modulus Sym supports this PyTorch utility and is turned on by default. CUDA Graphs can be enabled using Hydra. It is suggested to use at least 20 warm-up steps, which is the default. After 20
training iterations, Modulus Sym will then attempt to record a CUDA Graph and if successful it will replay it for the remainder of training.
cuda_graphs: True
cuda_graph_warmup: 20
CUDA Graphs is presently a beta feature in PyTorch and may change in the future. This feature requires newer NCCL versions and host GPU drivers (R465 or greater). If errors are occurring please
verify your drivers are up to date.
CUDA Graphs do not work for all user guide examples when using multiple GPUs. Some examples requires find_unused_parameters when using DDP, which is not compatible with CUDA Graphs.
NVTX markers do not work inside of CUDA Graphs, thus we suggest shutting this feature off when profiling the code.
Meshless finite derivatives is an alternative approach for calculating derivatives for physics-informed learning. Rather than relying on automatic differentiation to compute analytical gradients,
meshless finite derivatives queries stencil points on the fly to approximate the gradients using finite difference. With autodiff, multiple automatic differentiation calls are needed to calculate the
higher-order derivatives as well as the backward pass for optimization. The trouble is that computational complexity exponentially increases for every additional autodiff pass needed, which can
significantly slow training. Meshless finite derivatives replaces the need for autodiff with additional forward passes. Since the finite difference stencil points are queried on demand, no grid
discretion is needed preserving mesh free training.
For many problems, the additional computation needed for the foward passes in meshless finite derivatives is far less than the autodiff equivalent. This approach can potentially yield anywhere from a
\(2-4\) times speed-up over the autodiff approach with comparable accuracy.
To use meshless finite derivatives, one just needs to define a MeshlessFiniteDerivative node and add it to a constraint that will require gradient quantities. Modulus Sym will prioritize the use of
meshless finite derivatives over autodiff when provided. When creating a MeshlessFiniteDerivative node, the derivatives that will be needed must be explicitly defined. This can be done though just a
list, or accessing needed derivatives from other nodes. Additionally, this node requires a node that has the inputs consist of the independent variables and output being the quantities derivatives
are needed for. For example, the derivative \(\partial f / \partial x\) with require a node with input variables that contain \(x\) and outputs \(f\). Switching to meshless finite derivatives is
straight forward for most problems. As an example, for LDC the following code snippet turns on meshless finite derivative providing a \(3\) times speed-up:
from modulus.sym.eq.derivatives import MeshlessFiniteDerivative
# Make list of nodes to unroll graph on
ns = NavierStokes(nu=0.01, rho=1.0, dim=2, time=False)
flow_net = instantiate_arch(
input_keys=[Key("x"), Key("y")],
output_keys=[Key("u"), Key("v"), Key("p")],
flow_net_node = flow_net.make_node(name="flow_network", jit=cfg.jit)
# Define derivatives needed to be calculated
# Requirements for 2D N-S
derivatives_strs = set(["u__x", "v__x", "p__x", "v__x__x", "u__x__x", "u__y", "v__y", \
"p__y", "u__y__y", "v__y__y"])
derivatives = Key.convert_list(derivatives_strs)
# Or get the derivatives from the N-S node itself
derivatives = []
for node in ns.make_nodes():
for key in node.derivatives:
derivatives.append(Key(key.name, size=key.size, derivatives=key.derivatives))
# Create MFD node
mfd_node = MeshlessFiniteDerivative.make_node(
# Add to node list
nodes = ns.make_nodes() + [flow_net_node, mfd_node]
Meshless Finite Derivatives is a development from the Modulus Sym team and is presently in beta. Use at your own discretion; stability and convergence is not garanteed. API subject to change in
future versions.
• Setting the dx parameter is a very critical part of meshless finite derivatives. While classical numerical methods offer clear guidance on this topic, these do not directly apply here due
additional stability constraints placed by the backwards pass and optimization. For most problems in our user guide a dx close to 0.001 works well and yields good convergence, lower will likely
lead to instability during training with a float32 precision model. Additional details, tools and guidance on the specification of dx will be forthcoming in the near future.
• Meshless finite derivatives can increase the noise during training compared to automatic differentiation due its approximate nature. Thus this feature is currently not suggested for problems that
are exhibit unstable training characteristics for automatic differentiation.
• Meshless finite derivatives can converge to the wrong solution and accuracy is highly dependent on the dx used.
• Performance gains are problem specific and is based on the derivatives needed. Presently the best way to further increase the performance of meshless finite derivatives, users should increase
max_batch_size when creating the meshless finite derivative node.
• Modulus Sym will add automatic differentiation nodes if all required derivatives are not specified to the meshless finite derivative.
To boost performance and to run larger problems, Modulus Sym supports multi-GPU and multi-node scaling. This allows for multiple processes, each targeting a single GPU, to perform independent forward
and backward passes and aggregate the gradients collectively before updating the model weights. The Fig. 55 shows the scaling performance of Modulus Sym on the laminar FPGA test problem (script can
be found at examples/fpga/laminar/fpga_flow.py) up to 1024 A100 GPUs on 128 nodes. The scaling efficiency from 16 to 1024 GPUs is almost 85%.
This data parallel fashion of multi-GPU training keeps the number of points sampled per GPU constant while increasing the total effective batch size. You can use this to your advantage to increase
the number of points sampled by increasing the number of GPUs allowing you to handle much larger problems.
To run a Modulus Sym solution using multiple GPUs on a single compute node, one can first find out the available GPUs using
Once you have found out the available GPUs, you can run the job using mpirun -np #GPUs. Below command shows how to run the job using 2 GPUs.
mpirun -np 2 python fpga_flow.py
Modulus Sym supports running a problem on multiple nodes as well using a SLURM scheduler. Simply launch a job using srun and the appropriate flags and Modulus Sym will set up the multi-node
distributed process group. The command below shows how to launch a 2 node job with 8 GPUs per node (16 GPUs in total):
srun -n 16 --ntasks-per-node 8 --mpi=none python fpga_flow.py
Modulus Sym also supports running on other clusters that do not have a SLURM scheduler as long as the following environment variables are set for each process:
• MASTER_ADDR: IP address of the node with rank 0
• MASTER_PORT: port that can be used for the different processes to communicate on
• RANK: rank of that process
• WORLD_SIZE: total number of participating processes
• LOCAL_RANK (optional): rank of the process on it’s node
For more information, see Environment variable initialization
Fig. 55 Multi-node scaling efficiency for the FPGA example | {"url":"https://docs.nvidia.com/deeplearning/modulus/modulus-sym-v120/user_guide/features/performance.html","timestamp":"2024-11-09T19:55:20Z","content_type":"text/html","content_length":"695351","record_id":"<urn:uuid:83291bbc-67bb-4783-a612-254ce4148712>","cc-path":"CC-MAIN-2024-46/segments/1730477028142.18/warc/CC-MAIN-20241109182954-20241109212954-00883.warc.gz"} |
Why We Test Lactate Threshold - Part 1
Mark Turnbull
Let's look at four cyclists with wildly different lactate thresholds, here measured in watts. Notice that there are wide differences in VO2 max.
Below are four cyclists with different threshold values. What could cause the differences? The highest threshold is 406 w and this cyclist has the highest VO2 max. But Cyclist C with the next highest
VO2 max has a substantially lower threshold than Cyclist B. Why? Cyclist D has a threshold 150 watts less than Cyclist A. His VO2 max is lower but is this difference in threshold due solely to a
lower VO2 max?
If you want to understand what the threshold is and what causes it, read on. We will revisit these four cyclist later on in this discussion.
You will also learn how to train the threshold in the direction you desire.
VO2 Max and Threshold Four Cyclists
Cyclist VO2 Max Lactate Threshold
There is a threshold number for each of the cyclists above but what exactly does it mean? Most know that it somehow affects performance and want to change it to make it higher. But we find that few
actually know what the Lactate Threshold is even though they have heard the term many times. The more interesting thing is that few who claim to understand the concept actually know what causes it.
One reason is that there is no one accepted definition of it. The other is that there is almost no discussion of the causes of the LT in the academic literature.
First, we will define the Lactate Threshold even though we acknowledge that there are several other definitions in the literature. The definition of the lactate threshold we use is the maximum
lactate steady state (often called MAXLASS or MLSS). This is:
• the maximal speed or effort that an athlete can maintain for an extended period and still have little or no increase in lactate. Â This extended period of time can be as much as an hour or
possibly longer.
• At this speed or effort, lactate levels in the blood remain relatively constant which is how the name maximum lactate steady state arose. Â (Originally this phenomenon was called just the Maximum
Steady State but the term lactate was added to emphasize that this marker was key in both the measurement of it and to explain what was causing this effort level to be the maximum steady state.)
• Any increase in effort or speed above this level will cause lactate to increase steadily. Something will also force the athlete to either stop or slow down.
• Something is happening physiologically to cause this to happen and we will see this process affects performance.
• Originally it was thought that lactate accumulation is the problem but we now know that lactate is not the issue but the problem was due to the accumulation of other metabolites that accompany
the increase in lactate.
Lactate Threshold (LT) - This term has many other definitions than the one above. People argue with each other as to what is the correct way to define it. The answer is that there is no one
universally accepted definition. Thus, to make it simple, we use the definition above. We believe this one is best because it represents a physiological phenomenon that is both related to lactate and
is important for endurance performance. These other definitions of the lactate threshold are discussed at various places in this course.
The chart below illustrates the LT: At small effort levels above the LT the athlete's lactate level will rise, and he or she will be forced to stop, sometimes within a few minutes, sometimes a bit
longer. Above this maximal lactate steady state there are no more steady states but an inevitable and frequently rapid progression to exhaustion. This effort level is often called the anaerobic
threshold or the onset of blood lactate accumulation (OBLA) but we will see that these terms also have other definitions. The chart below shows that for this runner, at a pace of 4.2 m/s the lactate
levels remain relatively constant but above 4.2 m/s the lactate is no longer in a steady state and the athlete is forced to stop after a period of time
If you just want to know what we consider the best definition of this concept then one can stop here. If you want to know what causes the lactate threshold and how to train it so that it gets better,
then this discussion will answer these questions. To understand it all takes some time but we believe every coach and athlete will profit from this understanding. A proper understanding will make
one's training more effective and efficient We first start by examining what leads to a good performance and why the lactate threshold is so important for a good performance. Improving one's
performance is what most are interested in. We will then explain why measuring the lactate threshold is not important. If you have read the preceding tutorial chapters you will understand why.
What follows now are some reasons why the lactate threshold is important. For those interested there is a more detailed discussion of thresholds in general and how to train them on this website, and
specifics for triathletes in our triathlon section. But first a key correlation. The lactate threshold highly correlates with performance in an endurance race. MLSS correlates about .92 with running
time in a 8k race. (The validity of the lactate minimum test for determination of the maximal lactate steady state. Jones, A. M.; Doust, J. H., Med. Sci. Sports Exerc., Vol. 30, No. 8, pp. 1304-1313,
1998) The MLSS is very hard to measure and thus sports scientists have developed other measures that correlate highly with the MLSS and are much easier to measure. Below are two other studies showing
measures that correlate with MLSS and which predict performance in an endurance event.
Predicting competition performance in long-distance running by means of a treadmill test  Roecker, K.; Schotte, O.; Niess, A.; Horstmann, T.; Dickhuth, H  Medicine & Science in Sports & Exercise:
October 1998 - Volume 30 - Issue 10 - p 1552-1557 Applied Physiology of Marathon RunningSjödin, Bertil & Svedenhag, Jan  Sports medicine 1985  2. 83-99.
In 2009, a review of the Lactate Threshold, listed 25 different LT concepts that had been found in the literature. Â They also listed 32 studies that investigated the relationship between these LT
concepts and performance in endurance events. (Lactate Threshold Concepts - How Valid are They? Faude, O., Kindermann, W. & Meyer, T. Sports Med (2009) 39: 469) This study had 195 references and has
been cited by over 700 other studies.
The Lactate Threshold has been called the "gold standard" of athletic performance and has been intensely studied as the Faude study indicates. So there is no doubt that it reliably predicts
performance in distance events. Â
One recent study tried to evaluate various threshold measurements to see which one came closest to actually measuring the MLSS. (Jamnick NA, Botella J, Pyne DB, Bishop DJ (2018) Manipulating graded
exercise test variables affects the validity of the lactate threshold and VO2 peak PLoS ONE 13(7))Â
Interesting no one in these studies questions the need to measure the MLSS or estimate it accurately. They assume that knowing it is the sine qua non of lactate testing. But is it? We will argue that
it is not. Â But finding an easy measure that highly correlates with the MLSS is all that is needed. Â
Since lactate levels are the best indicator of potential race performance for endurance events, frequent lactate threshold testing (every 4-6 weeks) is the best way to find out whether the training
program is working or not. For short events such as swimming and rowing the maximal lactate steady state is also correlated with performance but not as much as for distance events. Â Anaerobic
capacity or the ability to produce lactate and speed become more important as the events get shorter.
Each of the terms (lactate threshold, anaerobic threshold, OBLA, V4) are associated with this maximum steady state measure (MLSS). Â There is even a term called the aerobic threshold, which has
appeared in the training literature and is in one of the charts immediately above. See the Roecker study above. The large number of similar terms with different definitions has caused much confusion
with both coaches and athletes.Â
Researchers have identified two other factors besides the lactate threshold that are highly correlated with performance in endurance sports. These three are
• VO2 max or aerobic capacity
• Economy of Movement
• Lactate Threshold
The chart below depicts this model
Those who propose this model says that the Lactate Threshold is the most trainable of the three. We believe that all three are trainable to a certain extent and we actually find that most coaches pay
more attention to aspects of economy of movement than anything else.
But the more important question is this model correct? Does this model distort what is actually happening? Is it missing something important? Â Is it conflating one thing with another. Â First, we
believe this model misrepresents what is happening during a competition. Â Second, we believe there is missing a key factor that determines the effectiveness of training and performance during a
race. Part 2 starts with the conclusion of a extremely good study that hints at what is wrong with this model but does not actually identify what the problem is. It in fact ended up muddying the
situation for years instead of clarifying it. | {"url":"https://www.sparksinto.life/post/why-we-test-lactate-threshold-part-1","timestamp":"2024-11-06T23:48:24Z","content_type":"text/html","content_length":"1050483","record_id":"<urn:uuid:08e33133-d543-4b46-8878-54d4b47708be>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00192.warc.gz"} |
Preprints of the St. Petersburg Mathematical Society
Authors: Filimonenkova, N. V.; Bakusov, P. A.
Title: Analysis of m-convexity of multydimensional paraboloids and hyperboloids
Comments: Russian, LaTeX, 22 pp.
Subj-class: 53A07, Secondary: 53C45
Submitted: 11.02.2017
Abstract, LaTeX, PDF 2017-02
Author: Mekler, A. A.
Title: Regular functions and conditional expectation operators on ordered ideals of $L^1(0,1)$-space
Comments: Russian, LaTeX, 85 pp.
Subj-class: 46E30
Submitted: 09.03.2017; updated: 01.10.2017
Abstract, LaTeX, PDF 2017-03
Author: Meshkova, Yu. M.
Title: On operator estimates for homogenization of hyperbolic systems with periodic coefficients
Comments: Russian, LaTeX, 35 pp.
Subj-class: 35B27, Secondary: 35L52
Submitted: 25.04.2017
Abstract, LaTeX, PDF 2017-04
Author: Rastegaev, N. V.
Title: On spectral asymptotics of the tensor product of operators with almost regular marginal asymptotics
Comments: Russian, LaTeX, 29 pp.
Subj-class: 47A80, Secondary: 47A75, 60G15
Submitted: 28.04.2017
Abstract, LaTeX, PDF 2017-05
Author: Krym, V. R.
Title: Problem: how to find a functional with the given energy-momentum tensor
Comments: Russian, LaTeX, 3 pp.
Subj-class: 83F05, Secondary: 35G05
Submitted: 24.05.2017
Abstract, LaTeX, PDF 2017-06
Author: Rastegaev, N. V.
Title: On the spectrum of the Sturm-Liouville problem with arithmetically self-similar weight
Comments: Russian, LaTeX, 14 pp.
Subj-class: 34B09, Secondary: 34B24
Submitted: 29.05.2017
Abstract, LaTeX, PDF 2017-07
Author: Reinov, O. I.
Title: On $Z_d$-symmetry of spectra of linear operators in Banach spaces
Comments: English, LaTeX, 23 pp.
Subj-class: 47B10
Submitted: 10.09.2017
Abstract, LaTeX, PDF 2017-08
Author: Reinov, O. I.
Title: Approximation properties associated with quasi-normed operator ideals of $(r,p,q)$-nuclear operators
Comments: English, LaTeX, 9 pp.
Subj-class: 46B28
Submitted: 19.09.2017
Abstract, LaTeX, PDF 2017-09
Author: Reinov, O. I.
Title: On $Z_d$-symmetry of spectra of some nuclear operators
Comments: English, LaTeX, 15 pp.
Subj-class: 47B06, Secondary: 47B10
Submitted: 30.09.2017
Abstract, LaTeX, PDF 2017-10
Authors: Apushkinskaya, D. E.; Nazarov, A. I.
Title: Keep the traces of Man in a sand of time!
Comments: Russian, LaTeX, 12 pp.
Subj-class: 01A70
Submitted: 07.12.2017; updated: 16.12.2017
Abstract, LaTeX, PDF
Back to the main page | {"url":"http://www.mathsoc.spb.ru/preprint/2017/index.html","timestamp":"2024-11-02T14:00:27Z","content_type":"text/html","content_length":"5369","record_id":"<urn:uuid:4d4be455-93d7-4927-b5a3-6a489358b0bf>","cc-path":"CC-MAIN-2024-46/segments/1730477027714.37/warc/CC-MAIN-20241102133748-20241102163748-00541.warc.gz"} |
Embeddings of von Neumann algebras into uniform Roe algebras and quasi-local algebras
• We study which von Neumann algebras can be embedded into uniform Roe algebras and quasi-local algebras associated to a uniformly locally finite metric space $X$. Under weak assumptions, these
$mathrm{C}^*$-algebras contain embedded copies of $prod_{k}mathrm{M}_{n_k}(mathbb C)$ for any emph{bounded} countable (possibly finite) collection $(n_k)_k$ of natural numbers; we aim to show
that they cannot contain any other von Neumann algebras. One of our main results shows that $L_infty[0,1]$ does not embed into any of those algebras, even by a not-necessarily-normal
$*$-homomorphism. In particular, it follows from the structure theory of von Neumann algebras that any von Neumann algebra which embeds into such algebra must be of the form $prod_{k}mathrm{M}_
{n_k}(mathbb C)$ for some countable (possibly finite) collection $(n_k)_k$ of natural numbers. Under additional assumptions, we also show that the sequence $(n_k)_k$ has to be bounded: in other
words, the only embedded von Neumann algebras are the ``obvious'' ones. | {"url":"https://vivo.library.tamu.edu/vivo/display/n686296SE","timestamp":"2024-11-03T16:03:57Z","content_type":"text/html","content_length":"18357","record_id":"<urn:uuid:9f380e51-d965-44a8-8303-5429e2990169>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00340.warc.gz"} |
Understanding Mathematical Functions: How To Know If The Function Is O
Understanding the Basics of Mathematical Functions
Mathematical functions are essential components of mathematical analysis and play a crucial role in various fields of science and engineering. They provide a way to describe and understand the
relationship between input and output values. In simple terms, a mathematical function is a rule that assigns to each input value exactly one output value.
A Define what a mathematical function is
A mathematical function is a relation between a set of inputs and a set of possible outputs, where each input is related to exactly one output. It can be represented as an equation, a graph, or a
table of values. The input values are typically represented by the variable 'x', and the corresponding output values are denoted by 'f(x)' or 'y'.
B Explain the significance of identifying odd and even functions
Identifying whether a function is odd or even is important as it helps in understanding its symmetry and behavior. Odd and even functions exhibit specific characteristics that can be used to simplify
mathematical analysis and problem-solving. Understanding these properties can lead to more efficient and accurate solutions to various mathematical problems.
C Outline the importance of symmetry in mathematical analysis
Symmetry plays a crucial role in mathematical analysis as it helps in identifying patterns, making predictions, and simplifying complex problems. In the context of functions, symmetry can help in
determining key properties such as periodicity, range, and behavior in different quadrants. Odd and even functions are specific examples of symmetric functions that exhibit distinct patterns and
Key Takeaways
• Understanding odd and even functions
• Identifying symmetry in mathematical functions
• Key characteristics of odd and even functions
• Testing for odd or even functions
• Applications of odd and even functions
Understanding Mathematical Functions: How to know if the function is odd or even
When dealing with mathematical functions, it is important to understand their properties and characteristics. One key aspect of functions is whether they are odd or even. In this chapter, we will
delve into what makes a function odd or even, including their symmetry properties and algebraic criteria.
Describe the symmetry properties of odd functions (symmetric about the origin)
An odd function is symmetric about the origin. This means that if you were to fold the graph of the function along the y-axis and then along the x-axis, the resulting graph would be identical to the
original. Visually, this symmetry results in a graph that looks the same when rotated 180 degrees about the origin.
Mathematically, the symmetry property of odd functions can be expressed as f(-x) = -f(x). This means that if you replace x with -x in the function, and then negate the result, you will get the same
function back. This property is what gives odd functions their characteristic symmetry.
Explain the symmetry properties of even functions (symmetric about the y-axis)
An even function is symmetric about the y-axis. This means that if you were to fold the graph of the function along the y-axis, the resulting graph would be identical to the original. Visually, this
symmetry results in a graph that looks the same when reflected across the y-axis.
Mathematically, the symmetry property of even functions can be expressed as f(-x) = f(x). This means that if you replace x with -x in the function, you will get the same function back. This property
is what gives even functions their characteristic symmetry.
Provide the algebraic criteria for odd and even functions (f(-x) = -f(x) for odd, f(-x) = f(x) for even)
For a function to be classified as odd, it must satisfy the algebraic criteria f(-x) = -f(x). This means that replacing x with -x in the function and then negating the result should yield the same
function back. If this property holds true, the function is odd.
On the other hand, for a function to be classified as even, it must satisfy the algebraic criteria f(-x) = f(x). This means that replacing x with -x in the function should yield the same function
back. If this property holds true, the function is even.
Understanding the symmetry properties and algebraic criteria for odd and even functions is essential for analyzing and working with mathematical functions. By recognizing these characteristics, one
can gain deeper insights into the behavior and properties of functions.
Graphical Interpretation: Visualizing Odd and Even Functions
Understanding the graphical representation of odd and even functions is essential for gaining insight into their behavior and properties. By visualizing these functions on a graph, we can easily
determine their parity and identify key characteristics.
A. Visualizing Functions on a Graph
When graphing a mathematical function, the x-axis represents the input values (independent variable) while the y-axis represents the output values (dependent variable). The graph of a function is a
visual representation of the relationship between the input and output values.
B. Strategies for Graphically Determining Parity
Determining whether a function is odd or even can be done by examining its graph. One strategy is to look for symmetry. An even function exhibits symmetry with respect to the y-axis, meaning that if
you fold the graph along the y-axis, the two halves will coincide. On the other hand, an odd function exhibits symmetry with respect to the origin, meaning that if you rotate the graph 180 degrees
about the origin, it will look the same.
Another strategy is to consider the behavior of the function for negative input values. For an even function, if f(-x) = f(x) for all x in the domain, it is symmetric with respect to the y-axis. For
an odd function, if f(-x) = -f(x) for all x in the domain, it is symmetric with respect to the origin.
C. Examples of Common Odd and Even Functions and Their Graphs
Common examples of even functions include the quadratic function f(x) = x^2 and the cosine function f(x) = cos(x). When graphed, these functions exhibit symmetry with respect to the y-axis.
On the other hand, common examples of odd functions include the cubic function f(x) = x^3 and the sine function f(x) = sin(x). When graphed, these functions exhibit symmetry with respect to the
By examining the graphs of these functions, we can clearly observe the symmetry properties that define them as odd or even.
Algebraic Techniques: How to Analyze Functions Algebraically
When analyzing mathematical functions to determine whether they are odd or even, algebraic techniques can be incredibly useful. By following a step-by-step method and using examples, it becomes
easier to understand and verify the parity of a function.
A. Outline step-by-step methods to verify if a function is odd or even algebraically
• Step 1: Replace x with -x in the function.
• Step 2: Simplify the function after replacing x with -x.
• Step 3: Determine if the simplified function is equal to the original function.
• Step 4: If the function is equal to its negative, it is odd. If the function is equal to its original, it is even.
Following these steps allows for a systematic approach to analyzing the parity of a function. It is important to carefully perform each step to ensure an accurate determination.
B. Use examples to demonstrate algebraic verification
Let's consider the function f(x) = x^3 - x as an example. To verify if this function is odd or even, we can follow the algebraic steps outlined:
• Step 1: Replace x with -x: f(-x) = (-x)^3 - (-x) = -x^3 + x
• Step 2: Simplify the function after replacing x with -x: f(-x) = -x^3 + x
• Step 3: Determine if the simplified function is equal to the original function: f(-x) = -x^3 + x ≠ f(x) = x^3 - x
From this example, we can see that f(x) = x^3 - x is an odd function because f(-x) = -f(x).
C. Address common mistakes in the algebraic analysis of function parity
One common mistake in the algebraic analysis of function parity is overlooking the simplification step after replacing x with -x. It is crucial to simplify the function correctly to accurately
determine its parity. Additionally, ensuring that the equality or inequality between the original and simplified function is correctly evaluated is essential to avoid errors in the analysis.
By being mindful of these common mistakes and following the outlined algebraic techniques, it becomes easier to analyze functions algebraically and determine whether they are odd or even.
Applications in Real World Problems
Understanding whether a mathematical function is odd or even has significant applications in real-world problems. Let's explore how this concept applies to physical phenomena, mathematical modeling,
and problem-solving.
A. Concept of odd and even functions in physical phenomena
The concept of odd and even functions is crucial in understanding physical phenomena, particularly in the field of physics. For example, in wave functions, the concept of odd and even functions helps
in analyzing the symmetry of wave patterns. In the case of wave functions, odd functions represent asymmetric wave patterns, while even functions represent symmetric wave patterns. This understanding
is essential in various areas of physics, including quantum mechanics and electromagnetism.
B. Role in mathematical modeling and problem-solving
The concept of odd and even functions plays a vital role in mathematical modeling and problem-solving. When creating mathematical models to represent real-world phenomena, determining the parity of a
function helps in simplifying the model and making accurate predictions. It allows mathematicians and scientists to identify patterns and relationships within the data, leading to more effective
problem-solving strategies.
C. Scenarios where determining the parity of a function is crucial
There are numerous scenarios in which determining the parity of a function is crucial. For instance, in signal processing, identifying whether a signal is odd or even can help in filtering out
unwanted noise and extracting meaningful information. In cryptography, understanding the parity of mathematical functions is essential for developing secure encryption algorithms. Additionally, in
financial modeling, the concept of odd and even functions aids in analyzing market trends and making informed investment decisions.
Troubleshooting Common Misconceptions and Challenges
When it comes to understanding mathematical functions, identifying whether a function is odd or even can be a common source of confusion. Let's address some typical misunderstandings and challenges
that arise in this area, and provide some clarification and tips for overcoming them.
Identify typical misunderstandings regarding odd and even functions
One common misunderstanding is that people often confuse the concepts of odd and even functions. They may mistakenly believe that odd functions only contain odd powers of x, and even functions only
contain even powers of x. Another misconception is that all functions must be either odd or even, when in fact, many functions are neither odd nor even.
Provide clarification on these misconceptions
Odd functions: An odd function is one where f(-x) = -f(x) for all x in the domain of the function. This means that if you replace x with -x in the function, the result is the negative of the original
function. It's important to note that odd functions can contain terms with even powers of x, as long as the overall function satisfies the odd function property.
Even functions: An even function is one where f(-x) = f(x) for all x in the domain of the function. This means that if you replace x with -x in the function, the result is the same as the original
function. Similar to odd functions, even functions can also contain terms with odd powers of x, as long as the overall function satisfies the even function property.
Functions that are neither odd nor even: It's important to understand that not all functions fall into the categories of odd or even. Functions that do not satisfy the properties of odd or even
functions are simply classified as neither odd nor even. An example of such a function is f(x) = x^3 + 2x.
Offer tips for overcoming challenges when working with complex functions
When dealing with complex functions, it can be helpful to break down the function into its individual components and analyze each part separately. This can make it easier to determine whether the
function is odd, even, or neither. Additionally, practicing with a variety of functions and seeking guidance from resources such as textbooks, online tutorials, and practice problems can help improve
understanding and proficiency in identifying odd and even functions.
Another tip is to pay close attention to the properties of odd and even functions and how they manifest in different types of functions. Understanding the fundamental properties and characteristics
of odd and even functions can provide a solid foundation for identifying and working with these functions in more complex mathematical contexts.
Conclusion: Best Practices and Summary of Key Takeaways
Understanding mathematical functions and their symmetries is crucial for mastering various mathematical concepts and applications. By knowing how to determine the parity of a function, individuals
can gain valuable insights into its behavior and properties. In this conclusion, we will summarize the importance of understanding functions and their symmetries, recapitulate best practices in
determining the parity of a function, and encourage the application of these concepts to enhance mathematical reasoning and analysis skills.
A Summarize the importance of understanding functions and their symmetries
Understanding functions and their symmetries is essential for gaining a deeper insight into the behavior of mathematical relationships. By recognizing the patterns and properties of functions,
individuals can make informed decisions and predictions in various mathematical and real-world scenarios. Symmetry, in particular, provides a powerful tool for analyzing and interpreting functions,
allowing for the identification of key characteristics and behaviors.
B Recapitulate best practices in determining the parity of a function
When determining the parity of a function, it is important to recapitulate best practices to ensure accuracy and efficiency. This includes examining the behavior of the function under the operations
of reflection and rotation, as well as applying the properties of even and odd functions to identify their symmetry. Additionally, utilizing algebraic techniques such as substitution and manipulation
can aid in determining the parity of a function with precision.
C Encourage applying these concepts to enhance mathematical reasoning and analysis skills
It is encouraged to apply the concepts of function symmetry and parity to enhance mathematical reasoning and analysis skills. By actively engaging with these concepts, individuals can develop a
deeper understanding of mathematical relationships and their properties. Furthermore, the application of these concepts can lead to improved problem-solving abilities and critical thinking skills,
which are essential in various academic and professional pursuits. | {"url":"https://dashboardsexcel.com/blogs/blog/understanding-mathematical-functions-odd-even","timestamp":"2024-11-14T17:54:23Z","content_type":"text/html","content_length":"224536","record_id":"<urn:uuid:5f9d6493-3aff-466a-9be6-bf16365a799a>","cc-path":"CC-MAIN-2024-46/segments/1730477393980.94/warc/CC-MAIN-20241114162350-20241114192350-00143.warc.gz"} |
Factorial Worksheets
Factorial worksheets benefit 8th grade and high school students to test their understanding of factorial concepts like writing factorial notation in product form and vice versa; evaluating factorial,
simplifying factorial expressions, solving factorial equation and more. Additionally, MCQ worksheet pdfs are provided to reinforce the concept. Procure some of these worksheets for free!
This set of printable factorial worksheets are divided into two parts. Part-A requires students to express the factorial in product form and Part-B is vice-versa.
Level 2 raises the bar by introducing variables. Part-A requires students to write an algebraic expression with factorial notation in general form & Part-B is the other way around.
Evaluate the Factorial: Level 1
These factorial pdf worksheets contain basic arithmetic operations and require students to simplify the numerical expression involving factorials.
Express in Specified Factorial Form
Each worksheet is divided into two sections. In Part A, express the given factorial in terms of specified factorial form. In Part B, write the numerals in factorial form.
Evaluate the Factorial - Level 2
Level 2 printable worksheets comprise more complex factorial expressions, including exponents and square roots. Evaluate the expressions.
Solving Factorial Equations - Level 1
Write the factorials in general form, isolate the variable and solve the equations involving factorials. Use the answer key to verify your solutions.
These printable factorial worksheets combine all the aspects of factorials to reinforce the knowledge of grade 8 and high school students on factorials. | {"url":"https://www.mathworksheets4kids.com/factorial.php","timestamp":"2024-11-10T10:44:10Z","content_type":"text/html","content_length":"42672","record_id":"<urn:uuid:85f8b907-d3d6-483e-a759-d1eff805fca5>","cc-path":"CC-MAIN-2024-46/segments/1730477028186.38/warc/CC-MAIN-20241110103354-20241110133354-00708.warc.gz"} |
Notice that the first row of the array was plotted at the top of the image.
This may be counterintuitive if when you think of row #0 you think of y=0, which in a normal x-y coordinate system is on the bottom.
This be changed using the "origin" keyword argument.
The reason for this is that this command was made for displaying CCD image data, and often the pixel (0,0) was considered to be the one in the upper left.
But it also matches the standard print-out of arrays, so that's good as well. | {"url":"https://notebook.community/CUBoulder-ASTR2600/lectures/lecture_15_ndarraysII","timestamp":"2024-11-10T13:58:12Z","content_type":"text/html","content_length":"131191","record_id":"<urn:uuid:02170721-95dc-4f49-b743-c9cdc4b6074e>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.60/warc/CC-MAIN-20241110134821-20241110164821-00340.warc.gz"} |
How to Calculate Date Range in Excel (4 Ideal Methods) - ExcelDemy
Method 1 – Using Arithmetic Formula to Calculate Date Range in Excel
• Enter the formula in Cell D5 and press Enter.
• The formula will give the following output.
• Enter the following formula in Cell C6.
• Press Enter to get the result.
• Copy Cell D5 to Cell D6.
• Select both ranges and drag them down using the Fill Handle tool.
• You will get the following result.
Note: If your start is the present day then use the TODAY function and it will generate this formula.
Read More: How to Filter Date Range in Excel
Method 2 – Creating Date Sequence with Date Range in Excel
• Enter the Invoice and Payment Dates of the first 2 products in the Cell range C5:D6.
• Select both ranges and drag them down.
• You will get the sequential date ranges.
Read More: How to Use Formula for Past Due Date in Excel
Method 3 – Inserting Excel TEXT Function to Calculate Date Range
• Enter the 1st product’s Invoice and Payment Dates in Cell C5 and Cell D5.
• Enter the following formula in Cell D5.
=TEXT(C6,"d mmm yy") & "-" & TEXT(D6,"d mmm yy")
The TEXT function converts values of Cells C5 and D5 to text in a specific number format. The Ampersand (&) operator is used to get the date range value in the custom format (“mmm d”) in a single
• Press Enter to get the following result.
• Change the format of the concatenated date range by using the following formula in Cell D6.
=TEXT(C6,"d mmm yy") & "-" & TEXT(D6,"d mmm yy")
• The final output is as follows.
Method 4 – Combining TEXT & IF Functions to Create a Date Range in Excel
• The Payment Date of the range is missing.
• Enter the following formula.
=TEXT(C5,"mmm d")&IF(D5<>""," - "&TEXT(D5,"mmm d"),"")
The TEXT function returns the value in a number format. The formula checks whether a condition is met and returns one value if TRUE and another value if FALSE.
• The output result will be as shown below.
• Both the start and end dates of the range are missing.
• Enter the following formula.
=IF(C6<>"",TEXT(C6,"mmmm d")&IF(D6<>""," - "&TEXT(D6,"mmm d"),""),"")
• You’ll get the output as Blank (“ “).
Read More: How to Use IF Formula for Date Range in Excel
How to Calculate Interval of Days within a Date Range in Excel
Method 1 – Using Mathematical Operation to Calculate Interval within Date Range
• Calculate the difference between the dates in Cell C5 and Cell D5 with the formula shown in the image below.
• It will give the output as a number for the dates in the specified date range.
• Use the Autofill tool to get all the intervals.
Note: You can also apply the DAYS function to subtract the dates and get intervals. For this, use the following formula based on the above dataset.
=DAYS(D5, C5)
Read More: How to Calculate Average If within Date Range in Excel
Method 2 – Calculating Date Range Interval with DATEDIF Function in Excel
• Calculate the date range difference in Year with the formula shown in the following image.
• Press Enter to get the output.
The DATEDIF function helps to calculate the number of years from Cells C5 and D5.
• Calculate the date difference in a Month. Enter the following formula.
• Calculate the difference in Days with the formula shown in the following image.
Read More: Excel Formula to Add Date Range
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How can I find this formula for the magnetic flux density? (EMagn)
How can I find this formula for the magnetic flux density? (EMagn)
• Engineering
• Thread starter Boltzman Oscillation
• Start date
In summary, the conversation discusses the use of the Biot-Savart formula to find the magnetic flux density at point P, with one person providing guidance and hints on how to correctly use the
formula. The conversation also touches on the integration process and the result for a semi-infinite line.
Homework Statement
A semi-infinite linear conductor extends between z = 0 and z = inf. along the z- axis. If the current in the inductor flows along the positive z-direction find H(vector) at a point in the x-y
plane at a distance r from the conductor.
Relevant Equations
H = I/(4*pi) Integral[( dl x R)/R^2]
I drew an illustration to make this easier:
Point P is where I wish to find the magnetic flux density H.
Given the Biot-Savart formula:
$$d\textbf{H} = \frac{I}{4\pi}\frac{d\textbf{l}\times\textbf{R}}{R^2}$$
I can let
$$d\textbf{l} = \hat{z}dz$$
$$\hat{z}dz\times\textbf{R} = \hat{\phi}sin(\theta_{Rdl})dz$$
Have I done this correctly so far? If so, what should I let R^2 in the Biot-Savart equation be?
Looks ok. I'll give you a hint: What is ## \frac{r}{R} ##? One other hint is you would do well to also express ## z ## in terms of ## \theta ## and ## r ##, and write ## dz ## as a ## d \theta ##
expression, and integrate over ## \theta ##.
Charles Link said:
Looks ok. I'll give you a hint: What is ## \frac{r}{R} ##? One other hint is you would do well to also express ## z ## in terms of ## \theta ## and ## r ##, and write ## dz ## as a ## d \theta ##
expression, and integrate over ## \theta ##.
Ah, I think I see what you mean.
$$R = rcsc(\theta)$$
$$z = rcsc^2(\theta)d\theta$$
$$dz = rcsc^2(\theta)d\theta$$
Thus Biot-Savart's law becomes:
Then doing all the integration from 0 to limiting angle will eventually lead me to:
$$H = \hat{\phi}\frac{I}{4\pi r}$$
Of course this is taking into account that this is a semi-infinite line.
thank you for that clarification.
FAQ: How can I find this formula for the magnetic flux density? (EMagn)
1. Where can I find the formula for magnetic flux density (B)?
The formula for magnetic flux density is typically found in physics or engineering textbooks, or on reputable scientific websites. It is also commonly available in reference materials or databases
related to electromagnetism.
2. Is there only one formula for magnetic flux density?
No, there are different formulas for calculating magnetic flux density depending on the specific situation. For example, the formula may differ depending on the type of magnet or the medium in which
the magnetic field is being measured.
3. How is the formula for magnetic flux density derived?
The formula for magnetic flux density is derived from Maxwell's equations, which describe the relationship between electric and magnetic fields.
4. Can the formula for magnetic flux density be applied to all types of magnets?
The formula for magnetic flux density is applicable to all types of magnets, as long as the magnet is stationary and not in motion. For moving magnets, the formula for magnetic flux density must take
into account the additional effects of velocity and acceleration.
5. How can I use the formula for magnetic flux density in practical applications?
The formula for magnetic flux density is commonly used in calculations and measurements related to electromagnetism, such as in the design of magnetic materials, motors, generators, and other devices
that utilize magnetic fields. | {"url":"https://www.physicsforums.com/threads/how-can-i-find-this-formula-for-the-magnetic-flux-density-emagn.979124/","timestamp":"2024-11-10T20:56:11Z","content_type":"text/html","content_length":"94265","record_id":"<urn:uuid:c2f73182-0948-4574-afcf-829d19110631>","cc-path":"CC-MAIN-2024-46/segments/1730477028191.83/warc/CC-MAIN-20241110201420-20241110231420-00109.warc.gz"} |
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Pay Someone To Take How do I choose the right person to do my statistics homework? Assignment | Pay You To Do Homework
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On The First Day Of Class
Write a function to do this both on the second, and the first, sub-questions. Just as an example: func findCountsByName(firstName: String, lastName: String) -> NSString { return firstName.astypeHow
do I choose the right person to do my statistics homework? I’ve never imagined myself saving my name online and collecting the statistics information at the expense of my money.. Trying to stop
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University of
S3RI Seminar - Data Science Education at School and at University in UK – Key skills and industry expectations, Professor Berthold Lausen (University of Essex) Seminar
14:00 - 15:00
3 May 2018
Room 8031, Lecture Theatre 8C, Building 54, Mathematical Sciences, University of Southampton, Highfield Campus, SO17 1BJ
For more information regarding this seminar, please email Dr Helen Ogden at H.E.Ogden@southampton.ac.uk .
Event details
The landscape for school mathematics in the United Kingdom is very different now compared to even just five years ago, see Lee et al (2016). The study of mathematics is compulsory in England up to
the age of 16 when students complete their GCSE (General Certificate of Secondary Education) examinations. Annually over 550,000 students take GCSE Mathematics(*). Post 16 the most commonly studied
qualifications are A levels (Advanced Levels), where two potential mathematics qualifications are available to students, Mathematics and Further Mathematics. Further Mathematics is an additional A
level only open to students who are already studying a Mathematics A level. In 2005 a total of 52897(**) students completed A level Mathematics, but this number had grown to 95244(**) by the summer
of 2017, making it the most popular A level in the UK. Similarly, the numbers for Further Mathematics have also increased over the same period, raising from 5933(**) in 2005 to 16172(**) in 2017.
Since 2015 there have been many changes in mathematics education with the compulsory 14-16 GCSE examinations being made more difficult with, for example, a greater emphasis on problem solving.
Similarly, the post-compulsory 16+ A level examinations have also been reformed, however this time the examinations have not been made more difficult, rather the curriculum has become more
standardised. For example all students now have to complete a statistics component as part of A level Mathematics, something that was not true under the pre-2017 structure. Another new compulsory
requirement is that students must work with prescribed large data sets during their A level Mathematics studies, and that the use of technology must permeate across their course. The new style A
level has been introduced for the 2017/18 academic year so the first large cohort of students studying under the system will be arriving at Universities for the 2019/20 academic year. These students
will be those who have studied both the new linear GCSE Mathematics and new A level mathematics qualifications.
During the period of recent changes in the UK school system we have seen the emergence of undergraduate (BSc) and postgraduate taught (MSc) qualifications in Data Science, for example in 2014 the
Department of Mathematical Sciences and the School of Computer Science and Electronic Engineering at the University of Essex have introduced a BSc in Data Science and Analytics and an MSc in Data
Science. The curriculum of these courses covers compulsory modules from computer science and mathematical sciences, introducing students to a range of mathematics and statistical topics as well as
computing skills such as programming, software engineering, databases, data mining, web development, and artificial intelligence.
In this talk we will discuss the changes to the school curriculum in the UK and the impact they have on preparing people to study data science at university any beyond. We will also review the
curricula of data science related university degrees, consider how their content matches industry expectations and discuss plans for further developments.
(*) Source: Government Department for Education data
(**) Source: Joint Council for Qualifications Examination Entry data
Lee, S et al (2016) Understanding the UK Mathematics Curriculum Pre-Higher Education – A Guide for Academic Members of Staff (2016 edition) SIGMA. ISBN: 978-1-911217-05 -3
Speaker information
Professor Berthold Lausen , University of Essex. Research interests: biostatistics, classification, clinical research, computational statistics, data analysis, data science, epidemiology, public
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Who Else Is Misleading Us About What Is a Dimension in Math?
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What Is a Dimension in Math Explained
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The Truth About What Is a Dimension in Math
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There are two fundamental techniques for solving nonlinear inequalities in 1 variable. Then The above theorem gives a great idea about what combinatorial geometry is about. This number is known as
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Course Tutorials
Dr. Cyprian Sakwa
Determining Spanning Set of a Vector Space
Scalar triple product and vector triple product of vectors
More Examples on Integration by Algebraic Substitution
Reduced Row Echelon Form (Reduced Echelon Form) of Matrices
Using cross product to find volume of a pyramid
Using cross product to find volume of a parallelepiped
Introduction to Standard Integration
Use of Algebraic Substitution to Integrate Trigonometric Functions
Evaluating Definite Integrals by Substitution
Examples on Determining Cross Product of Vectors
Cross Product of unit vectors i, j, k
Test 2
Determining Spanning Set of a Vector Space
Scalar triple product and vector triple product of vectors
More Examples on Integration by Algebraic Substitution
Reduced Row Echelon Form (Reduced Echelon Form) of Matrices
Using cross product to find volume of a pyramid
Using cross product to find volume of a parallelepiped
Introduction to Standard Integration
Use of Algebraic Substitution to Integrate Trigonometric Functions
Evaluating Definite Integrals by Substitution
Examples on Determining Cross Product of Vectors
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Antonio's is analyzing a project with an initial cost of $31,000
and cash inflows of $27,000...
Antonio's is analyzing a project with an initial cost of $31,000 and cash inflows of $27,000...
Antonio's is analyzing a project with an initial cost of $31,000 and cash inflows of $27,000 a year for 2 years. This project is an extension of the firm's current operations and thus is equally as
risky as the current firm. The firm uses only debt and common stock to finance their operations and maintains a debt-equity ratio of 0.5. The pre-tax cost of debt is 9.6 percent and the cost of
equity is 12.3 percent. The tax rate is 34 percent. What is the projected net present value of this project?
Calculate the NPV as follows:
Therefore, the NPV is $15,664.04. | {"url":"https://justaaa.com/finance/492544-antonios-is-analyzing-a-project-with-an-initial","timestamp":"2024-11-06T01:58:48Z","content_type":"text/html","content_length":"41153","record_id":"<urn:uuid:3b902da8-aa32-402f-8717-29cd1554d22e>","cc-path":"CC-MAIN-2024-46/segments/1730477027906.34/warc/CC-MAIN-20241106003436-20241106033436-00085.warc.gz"} |
Convex Hull - (Tropical Geometry) - Vocab, Definition, Explanations | Fiveable
Convex Hull
from class:
Tropical Geometry
The convex hull of a set of points is the smallest convex set that contains all the points. This geometric concept can be visualized as the shape formed by stretching a rubber band around the
outermost points, ensuring that all points are enclosed within. Understanding the convex hull is essential as it relates to various mathematical constructs, including Newton polygons and tropical
amoebas, where it helps to determine the boundaries and relationships of sets in these contexts.
congrats on reading the definition of Convex Hull. now let's actually learn it.
5 Must Know Facts For Your Next Test
1. The convex hull can be constructed using algorithms like Graham's scan or the QuickHull algorithm, which efficiently find the outermost points in a set.
2. In Newton polygons, the convex hull represents the boundary of the points corresponding to terms of a polynomial, indicating important information about its roots and behavior.
3. Tropical amoebas utilize convex hulls to capture the shape of a variety in tropical geometry, revealing how polynomial roots interact with the surrounding space.
4. The vertices of the convex hull are critical in determining the mixed volume and other combinatorial properties associated with polytopes.
5. Convex hulls provide insight into optimization problems by defining feasible regions for solutions within geometric contexts.
Review Questions
• How does the concept of convex hull relate to Newton polygons and what significance does it have in understanding polynomial behavior?
□ In Newton polygons, the convex hull encapsulates all the points that represent the exponents of polynomial terms. This geometric boundary helps identify critical features of the polynomial,
such as its roots and multiplicities. By analyzing the convex hull, one can better understand how different terms interact and contribute to the overall behavior of the polynomial in various
• Discuss how convex hulls are used in tropical geometry, particularly in relation to tropical amoebas and their properties.
□ In tropical geometry, convex hulls play a vital role by outlining the shape of varieties when transformed through logarithmic maps into amoebas. The convex hull helps visualize how these
varieties project into lower-dimensional spaces and allows for a better understanding of their properties. This connection provides insights into how polynomial roots behave under
tropicalization, affecting aspects like intersection theory and valuation.
• Evaluate the implications of understanding convex hulls in both Newton polygons and tropical amoebas on broader mathematical theories and applications.
□ Understanding convex hulls in both Newton polygons and tropical amoebas has significant implications for broader mathematical theories such as algebraic geometry and optimization. By studying
these concepts, mathematicians can uncover deeper relationships between polynomials and their roots while utilizing tools from combinatorics and geometry. This intersection enriches
mathematical frameworks and has applications in fields like robotics, computer graphics, and data analysis, showcasing how geometric constructs can inform complex problem-solving strategies.
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Sabermetric Research
Home field advantage is naturally higher in a hitter's park
The Rockies have always had a huge home-field advantage (HFA) at Coors. From 1993 to 2001, Colorado has played .545 at home, but only .395 on the road. That's the equivalent of the difference between
going 89-73 and 64-98.
Why such a big difference? I have some ideas I'm working on, but the most obvious one -- although it's not that big, as we will see -- is that higher scoring naturally, mathematically, leads to a
bigger HFA.
When teams play better at home than on the road -- for whatever reason --the manifestation of "better" is in physical performance, not winning percentage as such. The translation from performance to
winning percentage depends on the characteristics of the game.
In MLB, historically, the home team plays around .540. But if the commissioner decreed that now games were going to be 36 innings long instead of 9, the home advantage would roughly double, with the
home team now winning at a .580 pace.
(Why? With the game four times as long, the SD of the score difference by luck would double. But the home team's run advantage would quadruple. So the run differential by talent would double compared
to luck. Since the normal distribution is almost linear at such small differences (roughly, from 0.1 SD to 0.2 SD), HFA would approximately double.)
But it's not *always* that a higher score number increases HFA. If it was decided that all runs now count as 2 points, like in basketball, scoring would double, but, obviously, HFA would stay the
Roughly speaking, increased scoring increases the home advantage only if it also increases the "signal to noise ratio" of performance to luck. Increasing the length of the game does that; doubling
all the scores does not.
In 2000, Coors Field increased scoring by about 40%. If that forty percent was obtained by increasing games from 9 innings to 13 innings, HFA would be around 20% higher. If the forty percent was
obtained by making every run count as 1.4 runs, HFA would be 0% higher. In reality, the increase could be anywhere between 0% and 20%, or beyond.
We probably have the tools available to get a pretty good estimate of the true increase.
Let's start with the overall average HFA. My subscription to Baseball Reference allowed me to obtain home and road batting records, all teams combined, for the 1980-2022 seasons:
AB H 2B 3B HR BB SO
home 3209469 846723 161290 19928 95790 321178 612545
road 3363640 859813 163954 17203 96043 308047 668363
What's the run differential between those two batting lines? We can look at actual runs, or even the difference in run statistics like Runs Created or Extrapolated Runs. But, for better accuracy, I
used Tom Tango's on-line Markov Calculator (the version modified by Bill Skelton, found here). It turns out the home batting line leads to 4.79 runs per nine innings, and the road batting line works
out to 4.36 R/9.
AB H 2B 3B HR BB SO R/9
home 3209469 846723 161290 19928 95790 321178 612545 4.79
road 3363640 859813 163954 17203 96043 308047 668363 4.36
difference 0.43
That's a difference of 0.43 runs per game. Using the rule of thumb that 10 runs equals one win, a rough estimate is that the home team should have a win advantage of 0.043 wins per game, for a
winning percentage of .543.
That's a pretty good estimate -- home teams actually went .539 in that span (51832-44409). But, we'll actually need to be more accurate than that, because the "10 runs per win" figure will change
significantly for higher-scoring environments such as Coors.
So let's calculate an estimate of the actual runs per win for this scoring environment.
The Tango/Skelton Markov calculator includes a feature where, given the batting line, it will show the probability of a team scoring any particular number of runs in a nine-inning game. Here's part
of that output:
home road
2 runs: .1201 .1342
3 runs: .1315 .1404
4 runs: .1282 .1309
From this table, which actually extends from 0 to 30+ runs, we can calculate how many runs it would take for the road team to turn a loss into a win.
Case 1: If the road team is tied after 9 innings, it has about a 50% chance of winning. With one additional run, it turns that into 100%. So an additional run in a tie game is worth half a win.
How often is the game tied? Well, the chance of a 2-2 tie is .1202*.1342, or about 1.6%. The chance of a 3-3 tie is .1315*.1404, or 1.8%. Adding up the 2-2 and the 3-3 and the 0-0 and the 1-1 and the
4-4 and the 5-5, and so on all the way down the line, the overall chance is 9.7%.
Case 2: If the road team is down a run after 9 innings, it loses, which is a 0% chance of winning. With one additional run, it's tied, and turns that into a 50% chance. So, an additional run there is
also worth half a win.
How often is the road team down a run? Well, the chance of a 3-2 result is .1315*.1342, or about 1.8%. The chance of 4-3 is .1282*.1404, another 1.8%. And so on.
The total: a 9.54% chance the road team winds up losing by one run.
What's the chance that the additional run will give the *home* team the extra half win? We can repeat the calculation, but instead of 3-2, we'll calculate 2-3. Instead of 4-3, we'll calculate 3-4.
And so on.
The total: only 8.54%. It makes sense that it's smaller, because the better team is less likely to be behind by a run than ahead by a run.
We'll average the home and road numbers to get 9.04%.
So, we have:
9.7% chance of a tie
9.0% chance of behind one run
18.7% chance that a run will create half a win
Converting that 18.7% chance to R/W:
0.187 half-wins per run
= 5.35 runs per half-win
= 10.7 runs per win
So, we'll use 10.7 runs per win for our calculation.
(Why, by the way, do we get 10.7 runs per win instead of the rule of thumb that it should be 10.0 flat? I think it's becuase the Markov simulation always plays the bottom of the ninth, even when the
home team is already up. It therefore includes a bunch of meaningless runs that don't occur in reality. When some of the run currency is randomly useless, it pushes the price of a win higher.
We'd expect that roughly 1/18 of all runs scored are in the bottom of the ninth with the home team having already won. If we discount those by multiplying 10.7 by 17/18, we get ... 10.1 runs per win.
We saw earlier that the home team had an advantage of 0.43 runs per game. Dividing that by 10.3 runs per win, gives us
Predicted: HFA of .42 wins per game (.542)
Actual: HFA of .39 wins per game (.539)
We're off a bit. The difference is about 2 SD. My guess is that the Markov calculation, which is necessarily simplified, is very slightly off, and we only notice because of the huge sample size of
almost 100,000 actual games.
OK, now let's do the same thing, but this time for Coors Field only.
I could do the same thing I did for MLB as a whole: split the combined Coors batting line into home and road, and calculate those individually. The problem with that is ... well, if I do that, I'll
be getting the Rockies' actual HFA at Coors, which is huge, because it includes all kinds of factors that we're not concerned with, like altitude acclimatization, tailoring of personnel to field,
So, I'm going to try to convert the Coors line into an approximation of what the split would look like if it were similar to MLB as a whole.
Here's that 1980-2022 MLB split from above, except I've added the percentage difference between home and road (on a per-AB basis) below:
AB H 2B 3B HR BB SO
home 3209469 846723 161290 19928 95790 321178 612545
road 3363640 859813 163954 17203 96043 308047 668363
diff +3.2% +3.5% +21.4% +4.5% +9.3% -3.9%
I'll try to create something similar for 2000 Coors. The overall batting line, for both teams, looked like this:
AB H 2B 3B HR BB SO R/9
Coors 5843 1860 359 56 245 633 933 7.43
Here's my arbitrary split, into Rockies vs. road team, in such a way to keep roughly the same percentage differences as in MLB overall, while also keeping the R/9 roughly 7.43. Here's what I came up
AB H 2B 3B HR BB SO
home 5843 1884 362 66 249 672 936
road 5843 1826 350 54 238 615 974
diff +3.2% +3.4% +22.2% +4.6% +9.3% -3.9%
I ran those through Tango's calculator to get runs per 9 innings:
AB H 2B 3B HR BB SO R/9
home 5843 1884 362 66 249 672 936 7.783
road 5843 1826 350 54 238 615 974 7.071
avg 7.427
diff +.712
Next, I ran the runs-per-game distribution calculation to get a runs-per-win estimate. (I won't go through the details here, but it's the same thing as before: calculate the probability of a tie,
then a one-run home win, then a one-run road win, etc.)
The result: 14.37 runs per win.
As expected, that's significantly higher than the 10.7 we calculated for MLB overall. (Adjusting 14.37 for the superfluous bottom-of-the-ninth gives about 13.6, so, if you prefer, you can compare
13.6 Coors to 10.1 overall.)
The difference of .712 runs per game, divided by 14.43 runs per win, gives an HFA of
0.0495 wins per game
Which translates to a home winning percentage of .5495.
Comparing the two results:
.542 home field winning percentage normal
.549 home field winning percentage Coors
.007 difference
The difference of .007 is worth only about half a win per home season. Sure, half a win is half a win, but I'm a little disappointed that's all we wind up with after all this work.
It's certainly not as much of an effect as I thought there would be before I started. Even if you deducted this inherent .007, it would barely make a dent in the Rockies' 150 percentage point
difference between Coors and road. The Rockies would still be in first place on the FanGraphs chart by a sizeable margin -- 42 points instead of 49.
Looked at another way, an additional .007 would move an average team from the middle of the 29-year standings, to about halfway to the top. So maybe it's not that small after all.
Still, our conclusion has to be that the Rockies' huge HFA over the years is maybe 10 percent a mathematical inevitability of all those extra runs, and 90 percent other causes.
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On the 5th of this month, the Office of International Affairs (OIA) Team held an admission information session for Vietnamese students who are enrolled in Korean language classes at our university's
Institute of Language Education.
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The goal of the admission information session event was to expand the opportunities available to Vietnamese students currently studying the Korean language at our university in order to pursue higher
education. The event was hosted by Chung-Ang University¡¯s Institute of Language Education and Office of International Affairs.
±¹Á¦±³·ùÆÀÀº À̳¯ ¼³¸íȸ¸¦ ÅëÇØ ¿ì¸® ´ëÇÐÀÇ ±¹Á¦È ÇöȲ°ú ¿Ü±¹ÀÎ À¯Çлý Àü°ø °ü¸® ÇÁ·Î±×·¥À» ¼Ò°³Çß´Ù. ±¹Á¦Ã³¿¡¼ ±¸ÃàÇÑ ÀÔÇÐ-±³À°-Ãë¾÷À¸·Î À̾îÁö´Â ÀüÁֱ⠰ü¸® ü°è¿¡ ´ëÇÑ ¼³¸íµµ ÇÔ²² ÀÌ·ïÁ³´Ù.
The event began with an introduction of our university¡¯s current international network with other higher education institutions across the globe and plans to expand in the future. Then, the foreign
student major management program was explained in detail. Through the program, the Office of International Affairs guides students through the CAU admissions process, the higher education system, and
the job search to ensure employment after graduation.
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Afterwards, one-on-one consultation sessions were held for the students in Vietnamese. Students were able to learn more in-depth about our university¡¯s departments, majors, and admission procedures
based on their personal academic and professional goals through the consultation sessions.
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One of the Vietnamese students who attended the information session expressed their appreciation, saying, "This was an important opportunity for me to reconsider my career path. At first, I only knew
that it is very difficult to be admitted to Chung-Ang University, but now I've learned that there are many opportunities for Vietnamese students." | {"url":"https://oia.cau.ac.kr/bbs/board.php?tbl=k_bbs21_2&mode=VIEW&num=36&category=&findType=&findWord=&sort1=&sort2=&page=1&mobile_flag=&lang=","timestamp":"2024-11-07T18:51:33Z","content_type":"text/html","content_length":"256438","record_id":"<urn:uuid:b20b560c-c3fe-4820-82b4-655badb9a26c>","cc-path":"CC-MAIN-2024-46/segments/1730477028009.81/warc/CC-MAIN-20241107181317-20241107211317-00404.warc.gz"} |
How Much Coffee Per Cup: The Perfect "Coffee-To-Water" Ratio
How Much Coffee Per Cup – How To Get The Perfect Coffee-To-Water Ratio
If you want to make a delicious cup of coffee at home, all you need to do is follow some basic principles to determine the golden ratio.
One of these principles is to know what amount of coffee is for a cup as well as the coffee to water ratio. When you’ve got this step right then, you can make the perfect cup of coffee each time.
Now, if you’re wondering how much coffee per cup or more cups should be used? Then don’t worry; in this article, you’ll learn how much coffee per cup needs to be used and also the coffee to water
What difference does it make? Firstly, you want to get the most refined flavor you can from the beans you select.
They could have floral, caramel, nutty, or chocolate tones, but using the incorrect amount of coffee in a cup can end up ruining everything.
The second reason is that many take their coffee in cream or milk; therefore, you’ll need enough flavor to pierce through the milk.
Thirdly, you’ve decided to brew your coffee in a particular way, and knowing how to get the best out of your brew is crucial.
What is meant by a cup of coffee?
The discussion on the amount of coffee in a cup must define what a “cup” exactly means? Unfortunately, “cup” is a subjective term and not an exact measurement; therefore, we require a more precise
Let’s begin with this. Do not imagine one cup like you might think of it in baking.
As per US standard, one cup equals 236 milliliters, in other words, 8 ounces of water.
However, all of this doesn’t connect with a real cup or mug, as they are made in different sizes.
The most popular measurement for the term “cup” of coffee is 5 fluid ounces or 150 milliliters.
Your cup of coffee at the beginning of the day could be smaller or larger than this; however, 5 fluid fluid-ounces water volume is how we’ll calculate it. That means eight cups of coffee is
equivalent to 40 fluid ounces.
How do you measure coffee?
Measuring coffee is not as complex as it may seem. You can be extremely specific in your approach, give the issue some thought and avoid going too far.
The idea behind the word “scoop” is meaningless. What is the size, and how big is an actual scoop?
What happens if the ground coffee is small and fine? What happens if the ground is coarse?
When making coffee, it is crucial to be consistent with the quantity of coffee you use. It is possible to have an individual taste. However, there are a few essential factors to take into
The standard coffee to water ratio is 1:18, meaning 1 gram coffee grinds for every 18 millimeters of water.
The most efficient method of achieving this is by using a scale. However, that’s quite a lot of work when you require a quick fix at the beginning of the day.
Remember that everything needs to be measured based on ground coffee beans and not the beans before the grinding.
When we measure the coffee ground, we eliminate the issue of fineness or coarseness of ground coffee beans.
You can measure coffee in tablespoons, grams, or even scoops so the quantities can be identified, and a scale is the most efficient method of figuring out the precise weight of the coffee grounds.
You will always receive the exact amount of coffee in each cup, even if you try various coffee beans, as long as you know the coffee to water ratio.
Every type of bean differs from one another; however, when you ensure that the ratio stays the same, you won’t have trouble.
Measurements of Coffee
From the beginning, it should be noted that measuring coffee in tablespoons is similar to measuring water by gulp.
Tablespoons are a particular measurement that is effective in most situations; however, coffee is different. Coffee’s quantity in one tablespoon will be variable.
The method used to extract the pulp of cherry from the beans affects the amount of moisture left inside the beans.
The degree of coarseness in the beans will also affect the quantity of coffee in one tablespoon. The whole concept is accuracy and consistency.
Scoops or tablespoons can be used if you don’t have a scale; however, we must be aware of what we mean by a tablespoon in relation to coffee.
One tablespoon of coffee has a weight of around 10.6 grams.
Basically, for a coffee cup, you’ll need to make use of approximately 1 ½ -2 tablespoons of coffee grinds.
If you’re using scoops, make sure that the scoop is equivalent to 2 teaspoons.
Earlier, we defined “cup” as 5 ounces. If you use precise measurements, that’s the case. But, If you are using tablespoons or scoops, a cup of coffee is 8 ounces per cup.
To achieve this, you’ll need to make use of approximately 1 ½ -2 tablespoons of coffee. Each tablespoon contains 5.3 grams of ground coffee, and you can determine the appropriate ratio from there.
Here is an easy guideline for you when using tablespoons:
Measurement for 8-ounce cups:
1 cup = 8 oz water + 2 tablespoons coffee
2 cups = 16 oz water + 4 tbsp coffee
3 cups = 24 oz water + 6 tbsp coffee
4 cups = 32 oz water + 8 tbsp coffee
5 cups = 40 oz water + 10 tbsp coffee
Measurement for 10 ounce cups:
1 cup = 10 oz water + 2 1/2 tbsp coffee
2 cups = 20 oz water + 5 tbsp coffee
3 cups = 30 oz water + 7 1/2 tbsp coffee
4 cups = 40 oz water + 10 tbsp coffee
5 cups = 50 oz water + 12 1/2 tbsp coffee
Measurement for 12 ounce cups:
1 cup = 12 oz water + 3 tbsp coffee
2 cups = 24 oz water + 6 tbsp coffee
3 cups = 36 oz water + 9 tbsp coffee
4 cups = 48 oz water + 12 tbsp coffee
Whereas, when scoops are in question, as mentioned above that one scoop equals 2 tablespoons. So, for every 8 oz cup, you want to use one scoop.
Here’s a guideline for using coffee scoops:
4 cups: 20 oz water + 2 1/2 scoops
6 cups: 30 oz water + 3 1/2 scoops
8 cups: 40 oz water + 5 scoops
12 cups: 60 oz water + 7 1/2 scoops
Coffee To Water Ratios
Let’s become more sophisticated and think about the exact measurements of the coffee ratios. You’ll require a scale, and don’t worry; they’re inexpensive if you don’t already have one.
The idea behind “coffee ratio” is relatively easy to understand. It’s the proportion that you get from ground coffee and water. It is the formula that you use to achieve the perfect viscosity,
strength, and flavor.
As previously mentioned, the typical ratio of coffee is 1:18, or one gram of coffee for 18 milliliters of water.
However, you might want to utilize a different ratio based on the taste you like and the beans you’re using. The ratio is what determines the flavor.
These are the general guidelines:
Coffee Ratio Coffee (Grams) Water (Milliliters) Taste
1:15 1 15 Concentrated and Bright
1:16 1 16 Smooth and Bright
1:17 1 17 Smooth and Rounded
1:18 1 18 Lighter and Rounded
Note that these instructions are for brewing coffee with hot water. However, the method employed for extraction can affect the ratio.
The number of tablespoons of coffee, how much coffee, how many milliliters or ounces of water you use for your coffee is up to you at the end.
According to the Specialty Coffee Association of America (SCAA), the coffee to water ratio is between 1:15 to 1:18 (coffee: water), meaning 150ml divided by 18 equals 8.3g in coffee for each cup.
So as per American standard; 8.3g is used for 150ml, 10g for 180ml, 55g for 1000 ml.
The ratios we’ve measured will vary based on the method of brewing. It is essential to consider this and adjust the ratio to suit.
Understanding why ratios are different is crucial to making your perfect drink as it is directly concerned with the type of extraction employed, water temperature, and the extraction duration.
Let’s take a look at how and why different brewing methods make a difference.
Drip Coffee
Drip coffee, often called pour-over coffee, involves pouring coffee grounds into a paper filter, and then water flows through to an insulated carafe under it.
Seems simple right? Take your time. The amount of coffee required differs based on the filters themselves. When you remove the filter, it will be heavier than the quantity of coffee you had used.
What quantity of water do you think the filters can hold back? In general, the filter will hold back twice the amount of coffee consumed.
It’s because a 1:15 ratio is a 1:13 ratio, actually, since 2 grams of water won’t pass through the amount of coffee brewed. Most people believe that pour-over or drip coffee should be made in a 1:20
to 1:177 ratios.
French Press
When making a brew using a French Press, the vessel is filled with hot water and allowed to sit for 4 to 5 minutes.
Once the extraction is completed, the metal filtered plunger moves all grounds down towards the bottom.
Brewing with the French Press is entirely different; in this case, the extraction occurs in the water itself. That means there’s no loss of water in brewing using the French Press.
Making use of the French Press for coffee gives you more control over coffee’s flavor and viscosity. It’s simple to alter the ratio of coffee based on the beans you’re making use of at the time.
The longer you allow your grounds in the steeping process, the better the coffee. As a result, coffee can be prepared to suit your tastes and also the preference of guests.
You might have heard some people say that espresso is too strong. Well, what it indicates is that espresso may be having a different coffee ratio. It’s true.
The difference is that baristas don’t care about the quantity of water extracted but rather the exact weight of the liquid extracted.
In other brewing methods, the ratios depend on the quantity of water required to complete the extraction.
For instance, when you use a French Press, the amount of coffee grounds used directly correlates to the water you pour into the container.
However, espresso brewing does not give you control over the quantity of water used; therefore, it’s all about yield. Hence, for 18 grams of coffee, the yield is 36 grams or a 1:2 ratio.
Baristas can experiment with the amount of coffee used and the yield’s weight to obtain the most flavor from the beans.
Additionally, grounds get tamped to regulate the density. Espresso is the most variable of all other methods, and more variables mean more flexibility.
Cold Brew
Let’s state the obvious that cold brewing affects the coffee ratio. This is because the coffee grounds do not meet hot water; instead, Cold-brew extraction occurs at room temperature.
It is also possible to do this in the fridge; however, it’ll take longer since oil is extracted out of coffee grounds at a slower speed.
Typically, the duration of extraction for cold brew is between 22 to 24 hours.
The long period of cold brew extraction produces concentrated liquid.
When you serve cold brew, the melting ice and since it is diluted with water, they both does the job well.
For Cold brews, a long duration is used for extracting it at room temperature; therefore, a higher coffee ratio is essential.
The standard coffee ratio that is used for cold brewing ranges between 1:10 to 1:13. That is, you’ll use more coffee per every cup of water.
Overview: Key Points To Remember
A standard cup measures 5 fluid ounces
1:18 (1 gram of coffee for 18 milliliters in water) is the ideal coffee ration
To measure accurately, you need the use of a scale
2 tablespoons of ground coffee weight approximately 10.6 grams
Use 2 tablespoons of coffee ground for an 8-ounce cup
If using a scoop, ensure that it’s equivalent to 2 tablespoons of coffee
For Cold brew, 1:10 to 1:15is the ideal ratio
In the end, it all depends on personal taste, so experiment to discover the golden ratio for you.
The equipment you use to brew your coffee will also affect how much coffee you can use. The majority of coffee makers have instruction guides with them.
Single-cup coffee machines usually serve five-ounce cups of coffee in a standard-setting. The longer time spent on brewing, the lesser coffee you’ll need to use for each cup.
We all wait for the initial cup of tea to start the day, and you don’t want to spend your time playing with a scale for coffee and variations in the amount of water.
Finding out how much coffee you’d like to make the perfect cup should not be a matter of deciding before you cook breakfast or hurry your kids to school.
However, determining the correct ratio that suits your taste and quality depends on how you experiment, whether you use scoops, tablespoons, or scale to measure the coffee.
How much coffee do I use per cup?
The amount of coffee per cup means the measurement of coffee in grams. So you can use 14 grams of ground coffee in 8 ounces of water. For a milder cup, you can use 12 grams of coffee.
How much coffee do I use for 4 cups?
For 4 cups, you should use 60 grams of ground coffee, whereas, for a milder cup, use 48 grams of coffee.
How many tablespoons of coffee do you use per cup?
The number of tablespoons of coffee per cup or the golden ratio is 2 tablespoons in 8 ounces of water and 1 ½ tablespoon for a milder cup.
How many tablespoons of coffee do you use for 4 cups?
You can use 8 tablespoons of coffee in 32 ounces of water for 4 cups of coffee. For a milder cup, use 6 ½ tablespoons of ground coffee.
What are coffee ratios for drip coffee makers in tablespoons?
Here is the basic guideline for drip coffee per cup in tablespoons:
4 cups: 20 oz water + 5 tbsp coffee
6 cups: 30 oz water + 7 1/2 tbsp coffee
8 cups: 40 oz water + 10 tbsp coffee
10 cups: 50 oz water + 12 1/2 tbsp coffee
12 cups: 60 oz water + 15 tbsp coffee
How much caffeine per cup of coffee contains?
According to the USDA, a standard cup of brewed coffee contains 11.8 mg caffeine per fluid ounce of brewed coffee, on average.
Fluid Ounces(brewed coffee) Total Caffeine (mg) 5-oz “Cups”
1 11.8 1/5
2 23.6 2/5
3 35.4 3/5
4 47.2 4/5
8 94.4 1.6
12 141.6 2.2
16 188.8 3.2
Note that many coffee machine makers identify the “cups” as 5-oz cups. For example, a 12 cup coffee machine can produce 60 fluid ounces of coffee.
Then multiply 60 fluid ounces of coffee by 11.8 mg to obtain 708 mg of caffeine for each cup of coffee for the specific machine.
To Wrap Up How Much Coffee Per Cup
These are the basic ratios and measurements you can apply to nearly every type of coffee. The key is to select what is the most effective for you.
Feel free to play around with adding some milligrams more or less to determine which one suits your taste most.
Leave a Comment | {"url":"https://coffeegeek.tv/how-much-coffee-per-cup/","timestamp":"2024-11-10T20:41:50Z","content_type":"text/html","content_length":"465442","record_id":"<urn:uuid:ab151ab5-24d4-4cf5-b168-e9ac3a52dfc5>","cc-path":"CC-MAIN-2024-46/segments/1730477028191.83/warc/CC-MAIN-20241110201420-20241110231420-00854.warc.gz"} |
How to Use IF-THEN Statements in Excel
This article explains how to use the IF-THEN function in Excel for Microsoft 365, Excel 2019, 2016, 2013, 2010; Excel for Mac, and Excel Online, as well as a few examples.
Inputting IF-THEN in Excel
The IF-THEN function in Excel is a powerful way to add decision making to your spreadsheets. It tests a condition to see if it’s true or false and then carries out a specific set of instructions
based on the results.
For example, by inputting an IF-THEN in Excel, you can test if a specific cell is greater than 900. If it is, you can make the formula return the text “PERFECT.” If it isn’t, you can make the formula
return “TOO SMALL.”
There are many conditions you can enter into the IF-THEN formula.
The IF-THEN function’s syntax includes the name of the function and the function arguments inside of the parenthesis.
This is the proper syntax of the IF-THEN function:
=IF(logic test,value if true,value if false)
The IF part of the function is the logic test. This is where you use comparison operators to compare two values.
The THEN part of the function comes after the first comma and includes two arguments separated by a comma.
• The first argument tells the function what to do if the comparison is true.
• The second argument tells the function what to do if the comparison is false.
A Simple IF-THEN Function Example
Before moving on to more complex calculations, let’s look at a straightforward example of an IF-THEN statement.
Our spreadsheet is set up with cell B2 as $100. We can input the following formula into C2 to indicate whether the value is larger than $1000.
=IF(B2>1000,"PERFECT","TOO SMALL")
This function has the following arguments:
• B2>1000 tests whether the value in cell B2 is larger than 1000.
• “PERFECT” returns the word PERFECT in cell C2 if B2 is larger than 1000.
• “TOO SMALL” returns the phrase TOO SMALL in cell C2 if B2 is not larger than 1000.
The comparison part of the function can compare only two values. Either of those two values can be:
• Fixed number
• A string of characters (text value)
• Date or time
• Functions that return any of the values above
• A reference to any other cell in the spreadsheet containing any of the above values
The TRUE or FALSE part of the function can also return any of the above. This means that you can make the IF-THEN function very advanced by embedding additional calculations or functions inside of it
(see below).
When inputting true or false conditions of an IF-THEN statement in Excel, you need to use quotation marks around any text you want to return, unless you’re using TRUE and FALSE, which Excel
automatically recognizes. Other values and formulas don’t require quotation marks.
Inputting Calculations Into the IF-THEN Function
You can embed different calculations for the IF-THEN function to perform, depending on the comparison results.
In this example, one calculation is used to calculate the tax owed, depending on the total income in B2.
The logic test compares total income in B2 to see if it’s greater than $50,000.00.
In this example, B2 is not larger than 50,000, so the “value_if_false” condition will calculate and return that result.
In this case, that’s B2*0.10, which is 4000.
The result is placed into cell C2, where the IF-THEN function is inserted, will be 4000.
You can also embed calculations into the comparison side of the function.
For example, if you want to estimate that taxable income will only be 80% of total income, you could change the above IF-THEN function to the following.
This will perform the calculation on B2 before comparing it to 50,000.
Never enter a comma when entering numbers in the thousands. This is because Excel interprets a comma as the end of an argument inside of a function.
Nesting Functions Inside of an IF-THEN Function
You can also embed (or “nest”) a function inside of an IF-THEN function.
This lets you perform advanced calculations and then compare the actual results to the expected results.
In this example, let’s say you have a spreadsheet with five students’ grades in column B. You could average those grades using the AVERAGE function. Depending on the class average results, you could
have cell C2 return either “Excellent!” or “Needs Work.”
This is how you would input that IF-THEN function:
=IF(AVERAGE(B2:B6)>85,"Excellent!","Needs Work")
This function returns the text “Excellent!” in cell C2 if the class average is over 85. Otherwise, it returns “Needs Work.”
As you can see, inputting the IF-THEN function in Excel with embedded calculations or functions allows you to create dynamic and highly functional spreadsheets. | {"url":"https://v3techmedia.online/resources/how-to-use-if-then-statements-in-excel/","timestamp":"2024-11-06T17:59:47Z","content_type":"text/html","content_length":"39354","record_id":"<urn:uuid:aa136093-393b-4aac-a368-239864d19bce>","cc-path":"CC-MAIN-2024-46/segments/1730477027933.5/warc/CC-MAIN-20241106163535-20241106193535-00281.warc.gz"} |
Question 1. Consider a population of perennial plants that breed in th
Question 1. Consider a population of perennial plants that breed in the early spring and
suffer high drought-related mortality late in the summer. Field monitoring experiments
suggest that drought leads to a 50% decline in the population during the late summer (d =
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Scoring: Full credit for providing the correct answer and showing how the answer was
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Suppose that you are monitoring island endemic cricket population that has recently become
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95% C.I. for r = [-0.01, -0.1]
Since the entire confidence interval for your estimate of r is negative, your data imply that the
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How to do operations on cells that contain letters | Microsoft Community Hub
Forum Discussion
How to do operations on cells that contain letters
I need to be able to type a character string like A- into a cell and have excel average that cell along with other cells that contain letters or numbers. I would have conversions for what number each
character string would equal. There is the option of using vlookup but that takes a helper cell for every cell containing a letter and that would take a long time to create since I will end up having
nearly 10000 cells that could contain letters. As context the purpose is to convert letter grades to number grades so they can be averaged. Here is an example, say A- equals 95; B+ equals 93; and B
equals 90. Now have cell A2 contain B, cell A3 contain A-, cell A4 contain B+. Now have cell B2 have this formula =AVERAGE(A2:A4) I want it to come back with a result of 92.6.
• You don't need a helper column (but even if you did it wouldn't be that big a deal and could even be hidden). You can just use your lookup table:
=AVERAGE(XLOOKUP(A2:A4, Grades[letters], Grades[values], ""))
In this case I used "Format as Table" to make the table of grades and values and named the table "Grades", but you could replace Grdae[letters] with the corresponding range.
If a grade entry doesn't match a value in the table I used "" to ignore it, but you might prefer to remove that so you get an error and you know to look into it to find out why.
Notice how the D is ignored because there is no "D" in the table of Grades to the right
One more note: If you really don't like looking at all that inside the formula and want it to look like the simple AVERAGE(A2:A4) you could create a LAMBDA function (inside your named manager)
something like
AVERAGEGRADES =
LAMBDA(grades, AVERAGE(XLOOKUP(grades, Grades[letters], Grades[values], "")))
then you can just enter in you cell:
□ m_tarler Thanks a bunch for the help. That definitely gets me on the right track. I am not a very capable excel user so some of the things you mentioned I am not familiar with but I am sure I
can go figure it out. | {"url":"https://techcommunity.microsoft.com/discussions/excelgeneral/how-to-do-operations-on-cells-that-contain-letters/4190792","timestamp":"2024-11-08T18:12:27Z","content_type":"text/html","content_length":"218267","record_id":"<urn:uuid:4440ba1a-669b-4515-911f-8fc475fd78f0>","cc-path":"CC-MAIN-2024-46/segments/1730477028070.17/warc/CC-MAIN-20241108164844-20241108194844-00566.warc.gz"} |
A note on the quantum collision and set equality problems
A collision for a function f is two distinct inputs x[1] ≠ x[2] such that f outputs the same value on both inputs: f(x[1]) = f(x[2]). The quantum query complexity of finding collisions has been shown
[9, 2, 4, 11] in some settings to be (formula presented)(N^1/3); however, these results do not apply to random functions. The issues are two-fold. First, the Ω(N^1/3) lower bound only applies when
the domain is no larger than the co-domain, which precludes many of the cryptographically interesting applications. Second, most of the results in the literature only apply to r-to-1 functions, which
are quite different from random functions. Understanding the collision problem for random functions is of great importance to cryptography, and we seek to fill the gaps of knowledge for this problem.
To that end, we prove that, as expected, a quantum query complexity of (formula presented)(N^1/3) holds for all interesting domain and co-domain sizes. Our proofs are simple, and combine existing
techniques with several novel tricks to obtain the desired results. Using our techniques, we also give an optimal Ω(M^1/3) lower bound for the set equality problem. This lower bound can be used to
improve the relationship between classical randomized query complexity and quantum query complexity for so-called permutation-symmetric functions.
All Science Journal Classification (ASJC) codes
• Theoretical Computer Science
• Statistical and Nonlinear Physics
• Nuclear and High Energy Physics
• Mathematical Physics
• General Physics and Astronomy
• Computational Theory and Mathematics
• Quantum collision problem
• Random functions
Dive into the research topics of 'A note on the quantum collision and set equality problems'. Together they form a unique fingerprint. | {"url":"https://collaborate.princeton.edu/en/publications/a-note-on-the-quantum-collision-and-set-equality-problems","timestamp":"2024-11-08T09:33:59Z","content_type":"text/html","content_length":"47298","record_id":"<urn:uuid:51b7f016-4994-4489-b3ca-8cd31b9852ad>","cc-path":"CC-MAIN-2024-46/segments/1730477028032.87/warc/CC-MAIN-20241108070606-20241108100606-00655.warc.gz"} |
Journal of Operator Theory
Volume 37, Issue 2, Spring 1997 pp. 223-245.
Multiplication by finite Blaschke factors on de Branges spaces Authors: Dinesh Singh (1) and Virender Thukral (2) Author institution: (1) Department of Mathematics, University of Delhi, Delhi 110007,
INDIA. Current address: Indian Statistical Institute, 7 S.J.S. Sansanwal Marg, New Delhi 110016, INDIA
(2) Department of Mathematics, University of Delhi, Delhi 110007, INDIA Summary: This note characterizes those Hilbert spaces which are algebraically contained in the Hardy space H^2 of scalar valued
analytic functions on the open unit disk $\mathbb D$ and on which multiplication by a finite Blaschke product acts as an isometry. A general inner-outer factorization is deduced and some other
properties of the operator of multiplication by a finite Blaschke product are described. The main theorem generalizes a recent theorem of de Branges as well as a theorem of Peter Lax. Contents
Full-Text PDF | {"url":"http://www.mathjournals.org/jot/1997-037-002/1997-037-002-002.html","timestamp":"2024-11-14T12:10:31Z","content_type":"application/xhtml+xml","content_length":"6410","record_id":"<urn:uuid:3c609f30-b84e-491f-9a32-d45304907426>","cc-path":"CC-MAIN-2024-46/segments/1730477028558.0/warc/CC-MAIN-20241114094851-20241114124851-00250.warc.gz"} |
W dniu 18.05.2021 o 09:34, Jens Thiele via Boost-users pisze:
> I have lots of linestrings and want to find the k nearest linestrings to
> some point.
> Looking at the example
> /libs/geometry/doc/index/src/examples/rtree/polygons_shared_ptr.cpp
> I first thought this should be close to the solution and I just could
> replace the polygons with linestrings. But now I think the nearest query
> in that example only finds the nearest polygon bounding boxes and not
> the nearest polygons. Am I correct?
> If yes, how would one extend that example to find the nearest polygons?
Hi Jens,
Yes, your understanding is correct. Bounding boxes of polygons together
with pointers to polygons are stored in the rtree. This is also what is
returned by the query. So you need to calculate the distances to actual
linestrings by yourself.
I propose you to use query iterators instead of query function. Then you
can iterate over nearest boxes (passing the number of values stored in
the rtree into the nearest predicate). In the loop calculate distances
to linestrings and break when you have enough of them. You should
probably break when the number of linestrings you have is equal to your
K and the distance to the furthest linestring is lesser than the
distance to the current box returned by the rtree query (because then
you know that you will not get any closer linestring). To track the
furthest linestring you can use std::priority_queue.
Boost-users list run by williamkempf at hotmail.com, kalb at libertysoft.com, bjorn.karlsson at readsoft.com, gregod at cs.rpi.edu, wekempf at cox.net | {"url":"https://lists.boost.org/boost-users/2021/05/90913.php","timestamp":"2024-11-13T04:28:36Z","content_type":"text/html","content_length":"12076","record_id":"<urn:uuid:a5ce1ad8-5415-4fbc-8ee2-7a15c3b76eba>","cc-path":"CC-MAIN-2024-46/segments/1730477028326.66/warc/CC-MAIN-20241113040054-20241113070054-00530.warc.gz"} |
numme 29 Let A=[2−132] and f(x)=x2−4x+7. Show that f(A)=0. Us... | Filo
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numme 29 Let and . Show that . Use this rosult to find [NCERT EXEMPLAR] We have,
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Question Text numme 29 Let and . Show that . Use this rosult to find [NCERT EXEMPLAR] We have,
Updated On Jun 21, 2023
Topic Matrices and Determinant
Subject Mathematics
Class Class 12
Answer Type Video solution: 1
Upvotes 98
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Khomenko N. Super-short introduction into Classical TRIZ and OTSM
Submitted by admin on Sat, 10/27/2012 - 19:57
"This material is meant to be the first step for those who desire to reach a more profound understanding of the theoretical foundations of TRIZ and OTSM, as well as to learn to employ their
instruments with ease for the broadest range of purposes. It may be helpful to one studying the modern condition of OTSM-TRIZ on the stages of reflecting upon and systematizing the mastered material"
(from the paper).
Edited by I.Maceralnik
Deposit in CHOUNB 28.10.12 № 3560
Table of Contents
1.1. Theoretical background of Classical TRIZ
1.2. Fundamental instruments of the Classical TRIZ
2.1. Theoretical background of OTSM
2.2. Practical Instruments in OTSM
Within the framework of the OTSM approach, theories are viewed as models created to simplify the process of creating effective instruments to be used in practice. We need scientific theories to
increase the predictability of the results obtained on the basis of a given theory in comparison with a result that can be obtained through the ordinary method of randomly sorting through
Good (effective) theories allow obtaining the best results while spending less time.
For example, there are known cases when specialists in a particular problem area working with OTSM-TRIZ experts managed to obtain, in six days of work, ideas of better quality than the results they
obtained after several years of working without applying OTSM-TRIZ. This happens because the assortment of instruments for practically applying the theories of TRIZ and OTSM helps to clearly identify
the root of a problem and to concentrate, before everything else and with minimal costs, on eliminating the cause of the problem.
This article contains brief information about the classical TRIZ and OTSM. Both theories are described according to one scheme:
1. Theoretical theses.
2. Instruments for using theories in everyday practice.
The theoretical aspects, in their turn, are described as briefly as possible according to the following scheme:
1. The problem, for solving which the theory was created (a problem is often phrased as a question to which the author of the theory would like to find an answer).
2. The driving contradiction of the theory.
From the point of view of the classical TRIZ, a problem is difficult because it has a contradiction to be identified and resolved in order to find a solution for the problem situation. This is why we
give a description of problems which may be solved through TRIZ and OTSM, through the prism of contradiction that drives the evolution of these theories. This class of contradictions emerged in OTSM
and serves to simplify the process of entering the area of information that is new to the problem solver, or to help in identifying the core of the familiar areas of information.
3. Ideal Final Result.
It is the result that must be obtained through resolving a specific contradiction. It is not always attainable and serves as a reference point in the evolution of theory, as a goal that theory
strives to achieve through evolution. While a contradiction, within the framework of TRIZ and OTSM, is a model for describing the Initial Problem Situation (beginning), the IFR is a model for
describing the goal (finishing line) to reach. As they say, there is no favorable wind for ships which don’t know where they are going. The IFR serves precisely to aid one in understanding where to
go from the initially posed problem.
4. Fundamental ideas, assumptions, and models of the Theory.
To cover the distance between the “start” and the “finishing line” of the intellectual journey, some fundamental ideas, assumptions and models are necessary, on which, further on, the entire theory
and its instruments will be based. These ideas, assumptions and models undergo continuous evolution as a result of Theory development. Further on we are going to describe the state in which they are
We imagine the evolution of the Theory as a cycle:
1. Examining and developing the theoretical foundation.
2. Constructing an assortment of instruments to be applied in practice.
3. Applying the instruments in practice.
4. Reflection: Analyzing the effectiveness of applying the theoretical foundation in practice.
5. Examining and revising the theoretical foundation.
6. Revising the assortment of instruments to be applied in practice.
7. Reflection: Analyzing the effectiveness of applying the revised assortment of instruments.
8. Moving to step 5, to the new coil of the evolutionary spiral.
This is in general the manner in which many scientific theories and practical instruments were created. TRIZ and OTSM are not an exception: they came into existence and developed according to the
same scheme.
Further on we will list the most general and fundamental ideas and models of the classical TRIZ and OTSM. These ideas seem, in fact, too general and far removed from practice. This is why their
descriptions may only be rarely found in the “pragmatic” descriptions of the classical TRIZ. At the same time, without this information, it is nearly impossible to fully understand the construction
of instruments that are used in TRIZ for analyzing problems and synthesizing solutions, and there is nothing to do in situations when it seems that instruments do not work and OTSM-TRIZ cannot help.
Being familiar with the essential foundations of any Theory, as a rule, gives one the ease and freedom of possessing its tools for practical application.
This material is meant to be the first step for those who desire to reach a more profound understanding of the theoretical foundations of TRIZ and OTSM, as well as to learn to employ their
instruments with ease for the broadest range of purposes. It may be helpful to one studying the modern condition of OTSM-TRIZ on the stages of reflecting upon and systematizing the mastered material.
This material, also, invites its readers to the conversation about fundamental approaches in the area of problem analysis and solution synthesis, in the context of specific problem situations.
1. Classical TRIZ
1.1. Theoretical background of Classical TRIZ
1.1.1. The key problem to be solved by the classical TRIZ
1.1.2. We need to increase the effectiveness of searching for ideas to solve a specific problem situation, to which we do not know any acceptable professional standard solutions.
How can we significantly increase the effectiveness of arriving at a solution for one or another specific problem situation?
Today, the education for professionals is aimed towards studying standard solutions found by the previous generations of specialists in that area. This is why, when a specialist faces a problem to
which he has no standard solution, he undergoes great difficulties, which often leads to high levels of stress. TRIZ is intended to be helpful in situations such as this one. In many cases, TRIZ may
even help to improve the solutions that are based on standard solutions from a paticular subject area.
1.1.3. The driving contradiction that underlies the Key Problem of the Classical TRIZ.
One of the still-existing stereotypes about problem solving states that one must be able to generate as many ideas for a solution as possible, and then to choose suitable ones. Today, this stereotype
is still the ruling idea in the area of problem solving. However, this approach contains a fundamental contradiction, which cannot be solved through methods oriented only towards the generation of
If we have an infinitely great number of ideas, then, certainly, we have among them the very best idea for solving the problem in question; however, as a result, we cannot identify this idea in the
infinite number of the obtained ideas.
We can easily choose an idea if there is only one idea available; however, then we cannot be sure that it is the best one possible.
G. Altshuller has proposed the three fundamental ideas of the classic TRIZ, which suggest some ways to solving this contradiction (see below).
1.1.4. The Ideal Final Result for the Driving Contradiction which underlies the problem being solved by the Classical TRIZ.
We need an Instrument (Method) that would allow us to come to a single – the very best in the specific situation – solution to the problem, without spending time on generating a great number of ideas
and choosing the best solution from among the generated ideas.
1.1.5. The three fundamental ideas of the Classical TRIZ for overcoming the driving contradiction and approaching the ideal final result.
1.1.5.1. Objective laws, according to which technical systems transform (evolve))
When technical systems evolve, they do it not chaotically, but in accordance with the laws of evolution. Powerful ideas for solving technical problems should correspond to the laws according to which
the technical systems evolve (transform).
When developing a methodology, the solutions to problems must be based on the laws of technical system evolution. After all, essentially, solving a problem means moving to a new evolutionary step of
development for a given system. And the evolution proceeds according to certain laws, which may be identified and used in problem solving.
1.1.5.2. The Postulate of Contradiction
Technical systems develop when contradictions emerge, are intensified and resolved.
Powerful technical solutions always must overcome contradictions. A problem solving methodology must always include mechanisms of identifying, analyzing and resolving contradictions. A problem is
difficult because the relations between the properties of a system are contradictory and do not allow improving every property that needs improvement. To solve a problem, we must find a way to
destroy these relations without harming the system.
1.1.5.3. The Postulate of Specific Situation
The development of systems proceeds under the influence of the specific situation and is determined by the specific resources available to the system for its development (the resources of the system
itself and the resources of the person who is developing the system).
A powerful solution to a problem always arises from the specific character of the situation at hand. A problem solving methodology must include mechanisms for analyzing and utilizing the resources of
a specific problem situation.
1.1.6. The basic model of the classical TRIZ which describes both the thinking process and the mechanism of describing an element of a problem situation
1.1.6.1. Model for describing an Element of a problem situation.
Many-screened scheme of powerful thinking, which uses at least three axes:
· The axis of Hierarchy (Reflects the system-bound[1] interrelations between the components of a problem situation)
· The axis of Time (Reflects various changes in time for the entire hierarchy of the elements of the problem)
· The axis of System/Anti-system (Reflects the development, in time, of the conflicts between the elements of the hierarchy)
Within the framework of approaches of the classical TRIZ and OTSM, the model, according to which technical systems evolve, is based on the three fundamental ideas of resolving the driving
contradiction of the classical TRIZ on all the three axes of the many-screened scheme, as well as on the laws of technical system evolution proposed by G. Altshuller.
Powerful thinking must see the process of evolution through all of these three axes, and every component of the problem must be described and examined as it develops along these three axes.
The problem solving methods (instruments) must be constructed on the basis of the many-screened scheme of powerful thinking.
1.1.6.2. The model of a Problem Solving Process
Within the framework of the classical TRIZ, the process of problem solving is viewed as a series of transformations:
“A problem situation
=> An individual problem[2]
=> A model of an inventive problem
=> An ideal solution
=> A physical solution
=> A technical solution
=> A calculated solution».
1.2. Fundamental instruments of the Classical TRIZ
The fundamental instruments of the classical TRIZ are based on the three key ideas (described above) for resolving the Driving Contradiction of the Classical TRIZ and determining the evolution of
technical systems. All the three ideas are present in every fundamental instrument of the classical TRIZ.
The classical TRIZ contains two types of instruments:
- Instruments for solving problems which can be described by the solver as standard problems that have corresponding general solutions, generally described and therefore simplifying the process of
looking for a necessary instrument (Standards, Principles and other methods used in TRIZ)
- The instrument for working with problems that do not have standard solutions either within the framework of specialized knowledge or among meta-standard solutions, represented by the standard
solutions in TRIZ. This is the fundamental system-forming instrument of the classical TRIZ – the Algorithm of Solving Inventive Problems, ARIZ-85-B, created by Altshuller. It should not be confused
with the numerous modifications, suggested by various authors.
1.2.1. Instruments for solving standard problems in TRIZ
1.2.1.1. The system of laws of technical system evolution and the system of inventive standards or standard solutions
In the mid-70s, Altshuller proposed a system of laws of technical system evolution based on the multi-year research. The existence of technical systems is broken down into three stages:
· Stasis (Reflects the originating of the system);
· Kinematics (Reflects the maturation and development of the system);
· Dynamics (Reflects the change of generations of the system and within the system).
At every stage, a different group of laws is prevalent.
Originally, Altshuller proposed a system of eight laws (G.S. Altshuller. About the laws technical system evolution. 20.01.1977 (Manuscript)):
· The law of completeness of system parts.
· The law of “energy conductivity” in a system.
· The law of rhythm co-ordination of parts in a system.
· The law of increasing degree of ideality in a system.
· The law of irregular development of the system parts.
· The law of transferring into a super-system.
· The law of transferring from a macro-level to a micro-level.
· The law of increasing degree of multi-functionality.
Next, in the process of its own evolution, the system of laws grew into a system of standard inventive solutions. This instrument of the classical TRIZ ensures more effective application of general
laws of system evolution when practically solving specific problems.
ARIZ is also based on the laws of technical system evolution, but, in that case, they manifest themselves in a more covert fashion.
In the process of evolution of TRIZ and as a result of attempts to apply TRIZ in areas other than technical, it was discovered that the system of laws proposed by Altshuller also reflects the
development of many non-technical systems. This opened new perspectives in the development of the classical TRIZ.
1.2.1.2. Indexes of effects for inventors
Indexes of effects, created by TRIZ developers, help to find necessary physical, chemical and geometrical effects which assist in problem solving.
1.2.1.3. Matrix of Technical Contradictions
The matrix of contradictions historically emerged as the first TRIZ-instrument for working with standard problems. The matrix serves to simplify working with a set of 40 inventive methods for
transforming technical systems; these methods were identified by analyzing a great number of inventions.
Starting from the mid-80s of the 20^th century, the matrix has, for all intents and purposes, passed out of use for TRIZ-professionals.
And yet, many beginners in TRIZ use this instrument, since it is the simplest one to master. This instrument can assist in solving problems that pose real difficulties for narrow specialists of some
subject areas, either in engineering or in another field.
1.2.2. TRIZ-Instrument for working with non-standard problems
1.2.2.1. Altshuller’s ARIZ (ARIZ-85-B)
ARIZ is based on laws of system evolution and includes both instruments for carefully analyzing an inventive situation and mechanisms for overcoming psychological inertia. Thus, it allows the solver
to consciously control subconscious creative processes, which are traditionally considered to be uncontrollable.
ARIZ is also based on the model of the inventive problem solving process (see paragraph 1.1.2.5. of this article) as well as on the three fundamental ideas of the classical TRIZ (see paragraph 1.1.4
of this article).
All this allows ARIZ users to effectively analyze a chosen problem, identify and analyze the resources that may be used for solving this problem, posing the goal for the solution and identifying the
contradictions that interfere with reaching the posed goals through using the available resources. ARIZ also offers standard solutions for solving these contradictions and other standard mechanisms
of working with problems.
1.3. Conclusion
1. The theoretical background of the classical TRIZ allows its user to resolve the driving contradiction of the problem-solving process. This background is also helpful in creating and developing the
practical instruments of problem solving.
2. The assortment of instruments based on the theoretical background of the classical TRIZ significantly increases the probability of finding a solution and makes this process more efficient. This
statement is supported by more than fifty years of world experience of applying TRIZ practically.
3. The efficiency of the TRIZ-based process of problem solving increases because the TRIZ instruments use both divergent and convergent types of thinking.
4. In the long run, using both types of thinking allows the problem solver to consciously control subconscious creative processes which are traditionally considered uncontrollable.
5. The TRIZ-OTSM-based process of problem solving and its rules are co-ordinated with the patterns of technical system evolution.
2. OTSM
In the mid-70ies, Altshuller came to a conclusion that TRIZ has a potential for development and must grow into a more general and universal approach, which would allow working with problems while
being dependent on the area in which these problems arise. As TRIZ was developing and spreading through the USSR and Eastern Europe, by the mid-80s this conclusion became quite obvious to many
members of TRIZ-society. At various times before, Altshuller suggested that the society begins developing in this direction. In June of 1997, Altshuller was given a demonstration of the first results
in the field of OTSM. Both the results and the direction of research were approved by the author of TRIZ, and received his personal permission to use the acronym he proposed – OTSM, - on the
conditions that mentioning the acronym will be accompanied by the account of its origin.
2.1. Theoretical background of OTSM
2.1.1. The Key problem to be solved by OTSM
The problem, through solving which the General Theory of Powerful Thinking evolves, may be formulated in the following way:
How is it possible to work on complex non-standard problems, which, essentially, may be represented as networks of numerous interdisciplinary non-standard problems, which, in their turn, develop and
change over time? At that, the speed of these changes is commensurable with the time necessary for solving the problem.
This means that it is necessary to create a universal solving instrument for managing not only the problems that already exist, but also those problems that may emerge in future, as well the areas of
knowledge that are absent today.
The problem formulated in the previous paragraph is obtained by using one of the rules of the classical TRIZ, which states that a problem must be intensified before a solution may be obtained,
formulated in a way that might seem absurd. This rule allows to significantly increase the efficacy of working with a problem when added to the three fundamental principles of the classical TRIZ.
2.1.2. The driving Contradiction that underlies the Key Problem of OTSM.
The above-described problem may be imagined as the driving contradiction of OTSM, with the theory developing as it is attempting to overcome this contradiction:
The rules and methods of problem solving must be as general as possible, so that they can be as universal as possible, regardless of the areas of knowledge required for solving the problem. However,
general rules permit receiving only general solutions, which are not always applicable to a specific problem situation.
This is why – so as to be effective for solving specific problems – rules must be as specific as possible, tightly linked to specific areas of knowledge necessary for solving problems. However, then
these rules, methods and techniques will lose their universality.
2.1.3. Ideal Final Result for the Driving Contradiction
From the above-described system of contradictions, the following Ideal Final Result (IFR) may be derived:
Rules must be as general and universal as possible, ensuring, at the same time, solution of any specific problems that are being solved in specific situations.
2.1.4. General Ideas underlying OTSM
The general idea of solving the above-mentioned driving contradiction of OTSM was obtained in accordance with one of the principles of resolving contradictions in the classical TRIZ: The elements of
a system have one trait value, and the system as a whole has another – opposite – value of the same trait.
A metal watchband may serve as an example. Every link of the band is rigid and inflexible. However, the more links the band has and the smaller the links are the more flexible the band is.
By analogy with this example we can formulate the general model of resolving contradictions in OTSM:
Every rule (method, technique) in OTSM should be as general and abstract as possible – this will ensure the universality of their application.
The overall system into which they are all connected ensures the solution of the specific problem situation in specific circumstances.
Some stipulations should be made here.
1. Neither the classical TRIZ nor the modern OTSM are capable of replacing the knowledge in a specific area of human activity. They can only provide a system of organizing this specialized knowledge
or showing why a problem cannot be solved and which type of information is necessary for solving it, even if this information lies outside of a particular field or is unknown for humanity.
2. Both theories – the classical TRIZ and the modern OTSM – work with information on the qualitative level, offering systems of models for providing specialized qualitative information. They are not
intended for quantitative evaluations, but may help in developing methods of quantitative evaluations that are not yet known in mathematics.
3. The modern OTSM is based on the classical TRIZ and has absorbed the latter into itself as one of its components, developing, coordinating, expanding and clarifying the theoretical theses and
practical methods of the classical TRIZ.
2.1.5. System of OTSM Axioms
The development of OTSM, which has, as its goal, resolving the previously mentioned contradiction and reaching the proposed IFR, is moving within the framework of the following assumptions
(restrictions), represented as the System of Axioms in OTSM.
These axioms, essentially, are rules of thinking, of the highest degree of generalization. This is why, in addition to setting the limits for application of OTSM, they also serve as maximally general
problem-solving instruments used where more specific rules, methods and techniques do not produce results.
It seems that these axioms can be viewed as paradigms on which the modern OTSM is based. As time passes, these paradigms may change and re-form. OTSM is a dynamic and continuously developing theory,
which exists in touch with the practice of solving complex interdisciplinary non-standard problems through the assortment of instruments, developed on the basis of this theory.
In this article, we will not give the detailed wording of the axioms or describe their implications, but only provide the terms for them, as well as some comments.
2.1.5.1. The key Axiom: Axiom of descriptions (models)
This axiom, per se, is the only one. The two groups of axioms given below are, essentially, implications of this one. However, for a number of reasons, as OTSM evolved, these implications had to be
gathered into groups of axioms.
The axiom of descriptions (models) states that a person thinks through descriptions (models) of elements of a problem situation. These models only reflect a certain part of reality and never describe
it completely. Therefore, to increase the efficacy of thinking, and by that – the efficacy of solving problems, it is necessary to be able to construct models that can most effectively ensure the
process of thinking.
It should be noted that, within the framework of the theoretical approach of OTSM, an assumption is made that thinking may be viewed as a process of problem solving. The solver may be clearly aware
of the problem, in which case it is formulated, or, on the other hand, completely unaware of (not reflecting on) it, which significantly complicates the process of solving this problem. From here the
Axiom of Reflection is derived, which belongs to the group of the Axioms of Thinking.
Why are the axioms, described below, sorted into two groups?
As we have just said, thinking, within the framework of OTSM, is viewed as thinking on the subject of a conscious or unconscious problem, the components of which are both material and non-material
elements of our world. In accordance with the axiom of models we should clarify our position both with the models of thinking (the models of the process of problem solving) and with the models of the
world (the image of the world), in which problems to be solved through thinking arise.
2.1.5.2. Group of Axioms about OTSM problem solving process:
2.1.5.2.1. Axiom of the Core (Roots) of problems
2.1.5.2.2. Axiom of Impossibility
2.1.5.2.3. Axiom of Reflection
2.1.5.3. Group of Axioms about OTSM world vision:
2.1.5.3.1. Axiom of Unity
2.1.5.3.2. Axiom of Variety
2.1.5.3.3. Axiom of Connectedness of Unity and Variety
2.1.5.3.4. Axiom of process
2.1.6. Main Models of OTSM for describing the thinking process and components of a problem.
Just like in the case of axioms – the most general rules (theses, assumptions, models), the fundamental models are represented by two models. Both of them are, essentially, systems of models, which
ensure their specific application in a specific problem situation.
The first model is a model that serves to describe those elements of the world that take part, in some way, in the problem situation.
The second model is a model of the process of thinking that takes places as the problem is being solved.
2.1.6.1. OTSM ENV model for describing an Element of the world
OTSM-ENV model (Element-Name of Property-Value of Property) serves to describe those elements of the world that are in some way connected with the problem situation under analysis. The given model
lies in the basis of all other models of OTSM, both theoretical and those intensively used in practice when working with a problem situation.
A model, well-known in Artificial Intelligence work (“Object-Attribute-Value of Attribute”), has served as a prototype for it; however, in comparison with the prototype, the model proposed by OTSM is
significantly revised.
Within the framework of OTSM-approach, every element, both material and non-material, is viewed as a vector in a multi-dimensional space of parameters of infinite dimensions. At that, every axis of
this space can be decomposed into an independent multi-dimensional sub-space of parameters.
Using the ENV-model, among other things, significantly simplifies the fusion of OTSM-approach with many other methods of planning, lowering the costs, and raising the quality of products: Axiomatic
Design, QFD, Six Sigma, Taguchi Methods, etc.
2.1.6.2. OTSM Fractal model of problem-solving process.
This model is based on an earlier model, which Altshuller has proposed within the framework of the classical TRIZ. However, Altshuller’s model has been revised and expanded in accordance with
OTSM-approaches and requirements for working with networks of problems.
In the OTSM Fractal Model of solving complex problems, the problem situation is imagined as a network of problems. Every problem from this network may be potentially solved in some manner, which, in
OTSM-approach, is referred to as a partial solution.
Sets of partial solutions, in their turn, are connected into networks of partial conceptual solutions, which help to arrive to a solution that can be applied in practice. In OTSM, this solution is
referred to as a final conceptual solution.
Therefore, the process of solving a complex non-standard problem is imagined as a network of problems, which gradually grows into a network of partial solutions. The final conceptual solution, which
is suitable for being applied in practice, is constructed out of these partial solutions.
Furthermore, the network that describes the initial problem situation may be, if necessary, viewed as one of the problems from the network of a higher class.
A model is fractal when every problem from the initial network may be viewed as an independent network of problems and partial solutions.
Thus, a problem situation is described as a structure of networks that reminds one of a fractal:[3] every apex may be viewed as a network, identical in structure (a network of problems plus a network
of partial solutions) to the initial network that describes the problem situation.
Taking all this into consideration, as well as the fact that all these networks transform, as the problem is being analyzed, into the final concept solution, this model was given the name of
Self-Organizing Problem Flow Networks, or, for short, Problem Flow.
The word “Self-Organizing” in the title reflects the effect which appears when, while working on a problem, rules and methods of OTSM and classical TRIZ, reworked in accordance with OTSM-approach,
are being systematically applied.
2.2. Practical Instruments in OTSM
Each of the four technologies described below is intended for fulfilling a certain purpose in the course of the process of problem solving.
2.2.1. New Problem Technology
The technology “New Problem” organizes rules and methods of the classical TRIZ and OTSM into a system that ensures a transfer of the description of a problem situation into the form of fractal
network of problems, contradictions and parameters. This allows to identify the specific roots of a specific problem situation and begin working on constructing a solution.
It should be noted that the OTSM-approach to problem solving is characterized by not searching for a solution but, instead, gradually constructing, as analysis goes on, the problem situation on the
basis of OTSM-technologies.
This technology is based on the idea of driving contradictions, in other words, the contradictions that drive and control the process of system evolution.
2.2.2. Typical Solving Technology
The problems posed as a result of applying the technology “New Problem” can be solved through standard solutions and methods developed in TRIZ and OTSM, as well as through methods and techniques
accumulated by specialists in the specific subject areas. Individual solutions for individual problems, as a rule, cannot solve the problem situation on the whole, especially since individual
standard solutions often give rise to new problems.
The technology “Standard Problem” makes it possible to begin accumulating a fund of partial conceptual solutions. Partial solutions may be seen here as a kind of building material for constructing
the final conceptual solution.
This technology is based on standard solutions of the classical TRIZ.
2.2.3. Contradiction Technology
If no standard solutions is known for a problem in question, then the technology “Contradiction” is used, which is based on the fundamental instrument of the classical TRIZ – Altshuller’s ARIZ. In
OTSM-approach the steps of this algorithm have received their own reading and were supplemented with OTSM-recommendations and the rules for carrying them out.
Within the OTSM-approach, an extended, in comparison with the classical TRIZ, system of contradictions is offered, as well as a system of classification for the principles of resolving physical
contradictions (as they are called in the classical TRIZ). The new system of principles of combining opposites, which principles allow to resolve contradictions, is based, in OTSM, on the model ENV
(Name of Element – Name of Property – Value of Property). This significantly simplifies the fusion of OTSM-TRIZ with other methods of design and allows to significantly increase the degree of
formalization in the process of problem solving.
This technology is based on G. Altshuller’s ARIZ.
2.2.4. Self-Organizing Problem Flow Technology (or Problem Flow Technology)
The technology of Problem Flow is intended for constructing the final conceptual solution out of partial conceptual solutions; for evaluating obtained solutions, developing a network of problems on
the basis of new information obtained in the process of working on a problem situation, and constructing a solution.
This technology is based on certain assumptions, which present the process of problem solving as endless evolution of systems that are interacting and forming the evolutionary conditions for one
* * *
All four technologies are closely intertwined and work concurrently, helping one another to carry out their functions. Essentially, here we are dealing with a non-linear technology, the steps of
which are not always carried out in the same order, and the order is determined by the specific problem situation under analysis. This is why the model is referred to as Self-Organizing Flow of
problem networks. Naturally, all this is controlled by the system of general rules in OTSM. As a result, the rules retain their universality, and, at the same time, ensure analysis for specific
problem situations, which are often interdisciplinary.
Both TRIZ and OTSM are, in essence, very simple, but many of their assumptions and methods of work go against some of the modern world’s hardened stereotypes. This creates some difficulties for those
people who are only beginning to master these theories and their instruments. To overcome these difficulties and to help people develop the skills of practically applying the entire assortment of
knowledge of OTSM-TRIZ, we have developed special non-linear technologies of teaching.
1. The classical TRIZ and OTSM are based on the laws of system transformation. This allows proposing a number of alternatives for resolving driving contradictions, which lie within problems of
planning and searching for new solutions for complex interdisciplinary problems, which do not have standard solutions.
2. Models, rules, methods and technologies developed in classical TRIZ and especially in OTSM allow significantly increasing the degree of formalization in the process of problem solving in
comparison with the methods of solving creative problems.
3. Due to the factors previously listed, OTSM-TRIZ is proven to be a favorable foundation for constructing methods of solving various problems that arise in everyday life of various organizations,
such as, for example, developing strategies of product development and organization on the whole; organizing science research; constructing a system of increasing the professional skill level in
personnel; and many others.
4. Neither the classical TRIZ nor OTSM can replace the specialized knowledge, but only can help to organize it more effectively from the point of view of constructing a solution for a specific
problem situation.
5. Thus, OTSM may be viewed as an interdisciplinary language of providing specialized knowledge about a problem situation, with the purpose of analyzing this knowledge and constructing a solution
for the specific problem situation in the specific circumstances.
6. Models, developed in OTSM-approach, contribute to transferring the system of knowledge management in an organization to a qualitatively new level, simplifying the process of solving strategic and
tactical problems faced by the organization.
7. Therefore, OTSM provides an opportunity of viewing the entire assortment of problems faced by the organization, as a single problem in the general context. This, in turn, provides the management
with new instruments that can be used to increase the efficacy of activity in the organization on the whole.
8. Efficacy of TRIZ and OTSM is ensured by the solver consciously controlling his/her own subconscious thinking processes.
9. TRIZ and OTSM use both convergent and divergent thinking. Owing to that, the process of thinking becomes more open to being controlled by the solving, opening before him/her new horizons of
creativity, which, earlier, seemed unreachable.
[1] In this case, the term “system-bound” refers to a number of properties or interrelations that may seem random, but, from a certain point of view, appear to be bound into a system of interest.
[2] In this case, the problem situation is understood as consisting of a network of individual problems, into which it is then separated.
[3] Every level of the system looks identical to every other level and the system on the whole; i.e. every part is identical to the whole.
Recent comments | {"url":"https://otsm-triz.org/en/content/super_short_intr_en?page=8","timestamp":"2024-11-15T03:39:35Z","content_type":"application/xhtml+xml","content_length":"140037","record_id":"<urn:uuid:5b286dde-0565-4805-8a7f-b5b763dfe18f>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00863.warc.gz"} |
On random split of the segment
Applicationes Mathematicae 32 (2005), 243-261 MSC: 60F05, 60F15. DOI: 10.4064/am32-3-1
We consider a partition of the interval $[0,1]$ by two partition procedures. In the first a chosen piece of $[0,1]$ is split into halves, in the second it is split by uniformly distributed points.
Initially, the interval $[0,1]$ is divided either into halves or by a uniformly distributed random variable. Next a piece to be split is chosen either with probability equal to its length or each
piece is chosen with equal probability, and then the chosen piece is split by one of the above procedures. These actions are repeated indefinitely. We investigate the probability distribution of the
lengths of the consecutive pieces after $n$ splits. | {"url":"https://www.impan.pl/en/publishing-house/journals-and-series/applicationes-mathematicae/all/32/3/84391/on-random-split-of-the-segment","timestamp":"2024-11-02T09:10:14Z","content_type":"text/html","content_length":"44682","record_id":"<urn:uuid:4b8f00cb-0534-4d3e-8e39-e4c45e36259f>","cc-path":"CC-MAIN-2024-46/segments/1730477027709.8/warc/CC-MAIN-20241102071948-20241102101948-00338.warc.gz"} |
782 Radian/Square Month to Circle/Square Month
Radian/Square Month [rad/month2] Output
782 radian/square month in degree/square second is equal to 6.4786414218596e-9
782 radian/square month in degree/square millisecond is equal to 6.4786414218596e-15
782 radian/square month in degree/square microsecond is equal to 6.4786414218596e-21
782 radian/square month in degree/square nanosecond is equal to 6.4786414218596e-27
782 radian/square month in degree/square minute is equal to 0.000023323109118694
782 radian/square month in degree/square hour is equal to 0.0839631928273
782 radian/square month in degree/square day is equal to 48.36
782 radian/square month in degree/square week is equal to 2369.78
782 radian/square month in degree/square month is equal to 44805.3
782 radian/square month in degree/square year is equal to 6451963.14
782 radian/square month in radian/square second is equal to 1.1307362386754e-10
782 radian/square month in radian/square millisecond is equal to 1.1307362386754e-16
782 radian/square month in radian/square microsecond is equal to 1.1307362386754e-22
782 radian/square month in radian/square nanosecond is equal to 1.1307362386754e-28
782 radian/square month in radian/square minute is equal to 4.0706504592313e-7
782 radian/square month in radian/square hour is equal to 0.0014654341653233
782 radian/square month in radian/square day is equal to 0.84409007922621
782 radian/square month in radian/square week is equal to 41.36
782 radian/square month in radian/square year is equal to 112608
782 radian/square month in gradian/square second is equal to 7.1984904687329e-9
782 radian/square month in gradian/square millisecond is equal to 7.1984904687329e-15
782 radian/square month in gradian/square microsecond is equal to 7.1984904687329e-21
782 radian/square month in gradian/square nanosecond is equal to 7.1984904687329e-27
782 radian/square month in gradian/square minute is equal to 0.000025914565687438
782 radian/square month in gradian/square hour is equal to 0.093292436474778
782 radian/square month in gradian/square day is equal to 53.74
782 radian/square month in gradian/square week is equal to 2633.09
782 radian/square month in gradian/square month is equal to 49783.67
782 radian/square month in gradian/square year is equal to 7168847.93
782 radian/square month in arcmin/square second is equal to 3.8871848531157e-7
782 radian/square month in arcmin/square millisecond is equal to 3.8871848531157e-13
782 radian/square month in arcmin/square microsecond is equal to 3.8871848531157e-19
782 radian/square month in arcmin/square nanosecond is equal to 3.8871848531157e-25
782 radian/square month in arcmin/square minute is equal to 0.0013993865471217
782 radian/square month in arcmin/square hour is equal to 5.04
782 radian/square month in arcmin/square day is equal to 2901.77
782 radian/square month in arcmin/square week is equal to 142186.63
782 radian/square month in arcmin/square month is equal to 2688317.97
782 radian/square month in arcmin/square year is equal to 387117788.36
782 radian/square month in arcsec/square second is equal to 0.000023323109118694
782 radian/square month in arcsec/square millisecond is equal to 2.3323109118694e-11
782 radian/square month in arcsec/square microsecond is equal to 2.3323109118694e-17
782 radian/square month in arcsec/square nanosecond is equal to 2.3323109118694e-23
782 radian/square month in arcsec/square minute is equal to 0.0839631928273
782 radian/square month in arcsec/square hour is equal to 302.27
782 radian/square month in arcsec/square day is equal to 174106.08
782 radian/square month in arcsec/square week is equal to 8531197.76
782 radian/square month in arcsec/square month is equal to 161299078.49
782 radian/square month in arcsec/square year is equal to 23227067301.87
782 radian/square month in sign/square second is equal to 2.1595471406199e-10
782 radian/square month in sign/square millisecond is equal to 2.1595471406199e-16
782 radian/square month in sign/square microsecond is equal to 2.1595471406199e-22
782 radian/square month in sign/square nanosecond is equal to 2.1595471406199e-28
782 radian/square month in sign/square minute is equal to 7.7743697062315e-7
782 radian/square month in sign/square hour is equal to 0.0027987730942433
782 radian/square month in sign/square day is equal to 1.61
782 radian/square month in sign/square week is equal to 78.99
782 radian/square month in sign/square month is equal to 1493.51
782 radian/square month in sign/square year is equal to 215065.44
782 radian/square month in turn/square second is equal to 1.7996226171832e-11
782 radian/square month in turn/square millisecond is equal to 1.7996226171832e-17
782 radian/square month in turn/square microsecond is equal to 1.7996226171832e-23
782 radian/square month in turn/square nanosecond is equal to 1.7996226171832e-29
782 radian/square month in turn/square minute is equal to 6.4786414218596e-8
782 radian/square month in turn/square hour is equal to 0.00023323109118694
782 radian/square month in turn/square day is equal to 0.13434110852368
782 radian/square month in turn/square week is equal to 6.58
782 radian/square month in turn/square month is equal to 124.46
782 radian/square month in turn/square year is equal to 17922.12
782 radian/square month in circle/square second is equal to 1.7996226171832e-11
782 radian/square month in circle/square millisecond is equal to 1.7996226171832e-17
782 radian/square month in circle/square microsecond is equal to 1.7996226171832e-23
782 radian/square month in circle/square nanosecond is equal to 1.7996226171832e-29
782 radian/square month in circle/square minute is equal to 6.4786414218596e-8
782 radian/square month in circle/square hour is equal to 0.00023323109118694
782 radian/square month in circle/square day is equal to 0.13434110852368
782 radian/square month in circle/square week is equal to 6.58
782 radian/square month in circle/square month is equal to 124.46
782 radian/square month in circle/square year is equal to 17922.12
782 radian/square month in mil/square second is equal to 1.1517584749973e-7
782 radian/square month in mil/square millisecond is equal to 1.1517584749973e-13
782 radian/square month in mil/square microsecond is equal to 1.1517584749973e-19
782 radian/square month in mil/square nanosecond is equal to 1.1517584749973e-25
782 radian/square month in mil/square minute is equal to 0.00041463305099901
782 radian/square month in mil/square hour is equal to 1.49
782 radian/square month in mil/square day is equal to 859.78
782 radian/square month in mil/square week is equal to 42129.37
782 radian/square month in mil/square month is equal to 796538.66
782 radian/square month in mil/square year is equal to 114701566.92
782 radian/square month in revolution/square second is equal to 1.7996226171832e-11
782 radian/square month in revolution/square millisecond is equal to 1.7996226171832e-17
782 radian/square month in revolution/square microsecond is equal to 1.7996226171832e-23
782 radian/square month in revolution/square nanosecond is equal to 1.7996226171832e-29
782 radian/square month in revolution/square minute is equal to 6.4786414218596e-8
782 radian/square month in revolution/square hour is equal to 0.00023323109118694
782 radian/square month in revolution/square day is equal to 0.13434110852368
782 radian/square month in revolution/square week is equal to 6.58
782 radian/square month in revolution/square month is equal to 124.46
782 radian/square month in revolution/square year is equal to 17922.12 | {"url":"https://hextobinary.com/unit/angularacc/from/radpm2/to/circlepm2/782","timestamp":"2024-11-04T11:50:09Z","content_type":"text/html","content_length":"113407","record_id":"<urn:uuid:ce05f592-33a4-40e4-92c5-476478c6fe33>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00595.warc.gz"} |
I have been going through the train accuracy calculation in the last 3.3 section. The code is calculating is the training accuracy between the linear function (Z3) and a categorical variable Y_train.
Will it be alright to compare these two variables to find the accuracy? Don’t we require to apply a softmax function before comparing for accuracy of the model, as we don’t require to calculate for
computation for tf.keras.losses_crossentropy?
Hello Nijaj, welcome to the community!
You’re right! When calculating the training accuracy for a multi-class classification problem, you usually apply a softmax function to the output Z3 to get the predicted probabilities for each class
before comparing them with the true labels (Y_train). The softmax function converts the logits (the unnormalized output of the linear function Z3) into probabilities, which allows you to select the
class with the highest probability as the predicted class.
In TensorFlow, you can bypass the manual application of softmax when calculating accuracy if you’re using certain predefined functions. For example, while you would explicitly apply softmax when
calculating loss (as with cross-entropy), TensorFlow’s built-in accuracy metric functions such as tf.keras.metrics.CategoricalAccuracy are smart enough to handle raw logits by internally applying the
necessary transformations.
If you were to manually code the accuracy calculation without TensorFlow’s built-in metric function, you would need to apply Softmax. However, since you’re using the built-in metric, it works just
fine without it.
I hope this helps!
1 Like
Note that softmax is monotonic, so you can convert the logits output or the softmax output into categorical predictions by simply taking argmax of either input. The answer will be the same because of
monotonicity: the maximum value gives the predicted category. That is probably what the CategoricalAccuracy metric logic is doing to handle both cases.
1 Like
Great observation! This is why tf.keras.metrics.CategoricalAccuracy can operate directly on logits without explicitly applying softmax. The metric function uses argmax internally, effectively
treating the logits as if they had already been processed by softmax. The code works because of this property, which allows TensorFlow to bypass the explicit softmax step for accuracy calculations.
1 Like | {"url":"https://community.deeplearning.ai/t/train-accuracy-update/706145","timestamp":"2024-11-12T10:32:21Z","content_type":"text/html","content_length":"38291","record_id":"<urn:uuid:66d6899a-3266-4437-9d39-0946c35ba132>","cc-path":"CC-MAIN-2024-46/segments/1730477028249.89/warc/CC-MAIN-20241112081532-20241112111532-00191.warc.gz"} |
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A/B Test Statistical Significance Calculator
What is A/B Testing?
A/B testing is a method of comparing two versions of a product or website to determine which one performs better. It is commonly used in the fields of marketing and web design to optimize conversion
rates and improve user experience.
To conduct an A/B test, you would need to create two versions of the product or website, called the "A" version and the "B" version. These two versions should be as similar as possible, except for
the one change that you want to test. For example, you might want to compare two versions of a website's home page, where the only difference is the layout or the color scheme.
Once you have created the two versions, you would then randomly split your target audience into two groups and show each group one of the versions. You would then track how each group interacts with
the product or website, and compare the results to see which version performs better.
A/B testing can be a powerful tool for improving the effectiveness of your product or website, as it allows you to make data-driven decisions about what changes are most likely to be successful. It
is important, however, to carefully design and conduct your A/B test to ensure that the results are reliable and accurately reflect the impact of the changes being tested.
Besides, this process eliminates the doubts associated with website creation and optimization. Instead of judging based on your interests or what you feel is right, you can make data-backed decisions
based on the test result.
What is Statistical Significance?
Statistical significance is a term used to determine the level of certainty that the results of a given test are not due to a sampling error. This test helps researchers and marketers state that it
is unlikely their observations could have occurred under the null hypothesis of a statistical test.
Researchers often denote it by a p-value or probability value, usually set at a threshold of less than 0.05. This value means the data is likely to occur less than 5% of the time under the null
In this situation, when the p-value falls below the chosen alpha value, then we say the result of the test is statistically significant. Statistical significance shows that a relationship between two
or more variables is caused by something other than chance. In contrast, statistical hypothesis testing helps determine whether the result of a data set is statistically significant.
How to Determine Significance in A/B Test?
In A/B testing, the data sets considered are the number of users and the number of conversions for each variation. Statistical significance helps to prove that an A/B test was successful or
unsuccessful. It’s impossible to determine whether a test’s result is due to the change made or sampling error if you only look at the conversion rate difference.
Ideally, the A/B test gets as much as 95% statistical significance, while the very least is at 90%. A value above 90% ensures either a negative or positive impact on your site’s performance. It is
advisable to test pages with a high amount of traffic or a high conversion rate.
At Pixl Labs, our calculator only requires you to input 4 data points to determine your test’s statistical significance. You only need to know data for control visitors and control conversions for
the first test (say A), variant visitors and variant conversions for the second test (say B). To obtain results from the calculator, you need to enter the data into the appropriate space on it. This
calculator will fetch your conversion rate and significant results.
How Many Visitors Do You Need For Your A/B Test?
The number of visitors you need for your A/B test will depend on several factors:
• the size of the effect you are trying to detect
• the level of precision you want to achieve
• the level of statistical significance you are aiming for
In general, the larger the effect size you are trying to detect and the higher the level of precision and statistical significance you want to achieve, the more visitors you will need for your A/B
As a rough rule of thumb, you may need at least several hundred or even several thousand visitors per variation in order to detect small to moderate effect sizes with a reasonable level of precision
and statistical significance.
How to Calculate A/B Testing Sample Size?
Assessing how many visitors will be needed for a test you plan to run in the future is more complicated than evaluating a previous test’s significance. To work out the AB testing sample size you
need, you can use a sample size calculator.
Here are some general guidelines that can help you estimate this number:
Some experts agree it is challenging to get an uplift of more than 10% on a single webpage.
Changing your offers, rebranding, lowering your prices, or restructuring your website are changes you can make to achieving uplift beyond 10%. However, changes such as adjusting a button color,
headline, or image do have less than 7% impact. This kind of change is small and might be insignificant at times.
Some A/B Testing Settings
Below are the vital A/B testing settings you should understand when using a significance calculator.
Hypothesis (of two sides):
It is important to carry out a two-sided hypothesis, which means you are testing if variant B is different from variant A – either better or worse. This form is different from a one-sided hypothesis,
which would only test if B is better than A.
Statistical Power: 80-90%
It is the probability that you will find a difference in conversion rate if any exists. In other words, it is the inverse of the probability of committing a Type 2 error. It depends on your sample
size and the type of results you get. If you want a higher statistical Power in your result, you will need a larger sample size.
Statistical Confidence: 95%
Statistical confidence is the probability that a difference observed in your data is a real effect and not a Type 1 error (an effect observed when no real effect exists). It is common to use a
Confidence of around 95%. That means the chance of seeing a result when one doesn’t exist is 0.05 (or 5%). | {"url":"https://pixllabs.io/tools/ab-test-siginficance-calculator/","timestamp":"2024-11-03T06:54:15Z","content_type":"text/html","content_length":"40816","record_id":"<urn:uuid:55e1c043-a7bb-4c80-acbd-c4f401421b09>","cc-path":"CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00510.warc.gz"} |
Ordinal Numbers Ingles - OrdinalNumbers.com
Ordinal Numbers Ingles
Ordinal Numbers Ingles – There are a myriad of sets that can be easily enumerated using ordinal numbers as a tool. These numbers can be used as a method to generalize ordinal numbers.
The ordinal number is among of the most fundamental concepts in math. It is a number that indicates the location of an object within the list. Ordinal numbers are an integer between 1 and 20.
Although ordinal numbers serve many functions, they’re most often used for indicating the order in which items are placed within an itemized list.
Charts, words, and numbers can all be used to represent ordinal numbers. They can also serve to demonstrate how a collection or pieces are set up.
The vast majority of ordinal numbers can be classified into one of the following two categories. Transfinite ordinal numbers are represented by lowercase Greek letters, while the finite ones are
represented by Arabic numbers.
According to the axioms of selection, every set should contain at least one or more ordinals. For instance, the top possible grade would be awarded to the class’s first member. The winner of the
contest was the student who got the highest grade.
Combinational ordinal numbers
Compounded ordinal numbers are numbers with multiple numbers. They are created when an ordinal value is multiplied by the number of its last digit. They are commonly used to rank and date. They don’t
have an unique ending , like cardinal numbers.
To denote the order in which elements are placed within a collection, ordinal numerals can be used. These numbers also serve to indicate the names of the objects in collections. The two kinds of
regular numbers are regular and flexible.
By prefixing a cardinal numbers with the -u suffix, regular ordinals can be created. Then, the number has to be entered in words followed by a colon added. There are additional suffixes available.For
example the suffix “-nd” is used for numbers that end in 2 while “-th” is used to refer to numbers that end in 9 and 4.
Suppletive ordinals can be created by affixing words with the suffix -u or -e. The suffix is employed to count and is usually bigger than the normal one.
Limit of magnitude ordinal
Limit ordinal values that aren’t zero are ordinal numbers. Limit ordinal numbers have a disadvantage: there isn’t a limit on the number of elements. They can be created by joining sets that are empty
without any maximum elements.
Infinite transfinite-recursion definitions use limited ordinal number. According to the von Neumann model, every infinite cardinal number also has an ordinal limit.
The ordinal numbers that have limits are equal to the sum of all ordinals below it. Limit ordinal quantities can be enumerated using mathematics however they also can be expressed in natural numbers
or a series.
The data is arranged using ordinal numbers. These numbers provide an explanation of the object’s position numerically. They are utilized in the context of set theory and arithmetic. While they share
the same basic form, they aren’t in the same classification as natural number.
In the von Neumann model, a well-ordered set of data is employed. Consider that fyyfy is a subfunction g’ of a function described as a singular operation. Given that g’ is in line with the
requirements, g’ is a limit ordinal of fy if it is the only function (ii).
A limit ordinal that is of the Church-Kleene kind is also referred to as the Church-Kleene ordinal. The Church-Kleene oral defines the term “limit ordinal” as a properly ordered collection of smaller
ordinals, and it has a non-zero ordinal.
Stories that contain normal numbers
Ordinal numbers are commonly used to show the relationship between entities and objects. They are crucial to organize, count, or ranking reasons. They can be utilized to show the order of things and
to show the exact position of objects.
The letter “th” is usually used to indicate the ordinal number. In some instances the letter “nd” is substituted. The titles of books usually contain ordinal numerals.
Ordinary numbers are often expressed as words even though they are often utilized in list format. They can also appear in the form of acronyms and numbers. In comparison, these numbers are much
easier to understand than the traditional ones.
There are three different types of ordinal numbers. You may be able to learn more about them by engaging in games, practice as well as taking part in various other pursuits. You can enhance your
skills in arithmetic by studying more about these concepts. Coloring exercises are a fun easy and fun method to increase your proficiency. You can assess your progress using a marker sheet.
Gallery of Ordinal Numbers Ingles
N meros Ordinais Em Ingl s De 1 A 31 BRAINSTACK
Actividad Interactiva De Ordinal Numbers Para 5th
Leave a Comment | {"url":"https://www.ordinalnumbers.com/ordinal-numbers-ingles/","timestamp":"2024-11-04T22:13:39Z","content_type":"text/html","content_length":"64733","record_id":"<urn:uuid:328e4a58-cdda-4dfa-bdf6-900e5616b88c>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.16/warc/CC-MAIN-20241104194528-20241104224528-00124.warc.gz"} |
Brain Trigger #27 - Sample Number puzzle question - Math Shortcut Tricks
Now here we will discuss few question answers on Number puzzle which are very common in competitive exams. Without these type of Number puzzle question your exam preparation will not complete. Try to
Solve these questions with your math skills. You can use shortcut methods to solve these Number puzzle questions. | {"url":"https://www.math-shortcut-tricks.com/brain-trigger-27-sample-number-puzzle-question/","timestamp":"2024-11-06T16:50:23Z","content_type":"text/html","content_length":"200438","record_id":"<urn:uuid:ae37a651-74c4-4157-bcdb-20936efbc304>","cc-path":"CC-MAIN-2024-46/segments/1730477027933.5/warc/CC-MAIN-20241106163535-20241106193535-00436.warc.gz"} |
A Quick and Dirty Primer on Chaos Theory | Howe & Rusling
“If the flap of a butterfly’s wings can be instrumental in generating a tornado, it can equally well be instrumental in preventing a tornado.”
-Edward Lorenz
Interesting to consider, no? Pop culture is very familiar with the adage about a butterfly’s wings flapping in one part of the world ultimately causing a tornado in a completely different part. We
often interpret this on a kind of karmic scale, considering the far-reaching consequences of seemingly insignificant actions. However, mathematicians, economists, and sometimes meteorologists,
interpret this far differently. Mathematicians and economists think about the butterfly effect in terms of chaos, and naturally set out to quantify it.
Chaos Theory was first discovered in 1961 by meteorologist Edward Lorenz as he was attempting to predict the weather by using a program of twelve recursive equations. This means that each equation
used the information gleaned from the previous equation. While trying to recreate a previous weather pattern, Lorenz started mid-cycle and input his data. To his surprise, the resulting pattern was
very different from the original one (see below). Inquiry into this phenomenon led him to the realization that while his original data had started at .506127, he had tried to recreate the pattern
with the starting point of .506. This led him to the foundational principle of Chaos Theory – these systems are highly sensitive to initial conditions.
There are several layers to the definition of a chaotic system. The system must be dynamical and dynamic – that is, each state of the system is dependent on its previous state and the components of
the system are fluid. The system must also be nonlinear. That means the system’s inputs are not necessarily proportional to its outputs: small changes could affect the system in a big way and big
changes could fail to affect the system at all. Think of a firm whose production and workers are at equilibrium. If we add one worker we could see production rise, but we could also add five workers
and see production stay the same. There are a host of variables that would affect each possible outcome. Chaotic systems can be deterministic or nondeterministic. Deterministic systems are systems
with no random processes and non-deterministic systems do include random processes. Given these parameters, chaos theory seeks to answer several central questions. How sensitive to its initial
conditions is the system? What happens when we impose an iterative process? Can we find the underlying order in a system that looks completely random on the surface?
It’s easier to first think about this on a small scale. Fractals are built on self-similarity and theoretically iterate to infinity. Apply the same function over and over and the fractal will form a
picture. Zoom in on any one portion of the fractal and it will look exactly like the larger picture. This can be done deterministically with software, where the end product is only affected by the
iterated function. If the function is changed, the picture will change. Some of the more famous fractals are the Mandelbrot fractal and the Koch curve:
image from fractal-explorer.com
image from fractal.institute
Nature is also saturated in fractals. Trees, snowflakes, ferns, and cabbage are fractals. Sunflowers, hurricanes, and pinecones are special cases called Fibonacci fractals. It makes sense then that a
fractal found in nature is necessarily non-deterministic as it is subject to any number of random processes that will affect its outcome in any number of ways. Look at any deciduous tree in the
winter and the elements have clearly played a significant role in what it grew to be. The salient question remains, what about Chaos Theory? What happens when we change one thing in the function? Can
we predict the outcome? Well, yes…and no. The underlying functions of our deterministic fractals can be manipulated in the software to create many different patterns. Since these fractals are
essentially grown in a vacuum, we can predict exactly what they will look like for any given number of iterations. It gets more complicated when our fractals are subject to the elements, but we can
add in many different variables and come up with a probability that the result will fall in a given range. That is, if we want to predict how high our tree will grow, we can use information such as
average rainfall, average height of that species, sun exposure, probability of extreme weather events, probability of disease, etc. to come up with a probability that the tree will reach a certain
Things get infinitely more complicated when the system gets more complicated. That is, what if the chaotic system in question is the economy? Currently, it is common thought that economic
fluctuations are due to some exogenous factor. Barnett, Serletis and Serletis (2015) point out that most economic theory assumes that equilibria exist and when there is an absence of exogenous shocks
to the system, the market will tend to a steady state. However, thinking of the economy as a chaotic system means that shocks aren’t exogenous but endogenous. The implication of this is the ability
to assign a probability to what the economy will look like at a given point in time, if one were able to nail down the initial conditions to perfection and assign probabilities of stochastic
variables within an arbitrarily small degree of confidence. The impossibility of that is not the only issue. Economists aren’t yet sure the economy even falls under the definition of a chaotic
system. It’s incredibly difficult to definitively say whether a given economic shock is exogenous and random or endogenous and nonlinear. Further, testing an infinite system with finite data presents
a problem, and Litimi, BenSaida, Belkacem, and Oussama (2019) find there to be too much noise to be of practical use.
Perhaps that’s overcomplicating things – it’s too general to get a handle on. The efficient market hypothesis is widely cited in economic theory. The most salient assumption of the EMH is that asset
prices always reflect all the available information in the market. Thus, most of the players in the market don’t beat it but are instead crowded around the median (this isn’t to say that it isn’t
possible to beat the market under the EMH, just that most don’t. Since investors make up “the market”, if most investors were to beat the market, that would mean the market beat the market.). Another
way of thinking about this is the marginal gain from using information equals the marginal cost of acquiring it. However, there are clear problems with this. The EMH rests on the assumption that
asset prices are rationally set by full information, which is never the case in the real world. Kristoufek (2012) also astutely notes that different groups can interpret the same piece of information
in very different ways. That is, a buy signal at one price to one group can be a sell signal to another. This brings up the question of liquidity, something for which EMH does not account. If one
side of the transaction dominated the market and there were no investors on the other side, the asset would become illiquid and prices could collapse. Enter instead the fractal market hypothesis,
which is based on liquidity and attempts to address the weaknesses of the EMH.
It is a well-established fact that investors are in the market for a myriad of reasons. Each investor has his or her own risk profile, cash needs, industry preferences, and time horizons, among other
things. The FMH zones in on the assumption that investors have investment time horizons that span anywhere from mere minutes to many years. The general idea then is when there is a buyer for every
seller and vice versa, which occurs most of the time in a market where the players have many different timelines, that liquidity works to smooth pricing and stabilize the market. What does this have
to do with Chaos Theory? The FMH works much like a fractal, building itself with the same self-similar structure and feedback loops. A buy to one group is a sell to another and a sell to one group is
a buy to another, and on and on it goes. If there is ever a time in which a group at one spot in the time horizon vacates or investors begin to cluster around a specific time, the fractal will
destabilize and even break down because the underlying function is no longer continuous, or smooth. On the surface, this exists in a vacuum with no exogenous shocks to the system (whether random or
not). FMH does account for these shocks, the discussion of which is outside the scope of this paper.
Where are we now? Economists participate in a lively back-and-forth about the future practicality of Chaos Theory. There isn’t even a consensus when it comes to definitively stating if the financial
markets even fall under the definition of a chaotic structure. If a consensus is ever reached, there are realistically far too many variables to ever accurately predict even one data point long term
within a reasonable confidence interval. Simply put, Chaos Theory isn’t there yet, and it may never be…or it may just join the ranks of the thousands of other ideas whose practical purpose did not
come until much later. At least we have cool pictures to look at while we’re waiting.
Image from Wikipedia
Adrangi, B., Allender, M., & Raffiee, K. 2010, ‘Exchange Rates and Inflation Rates: Exploring Nonlinear Relationships’, Review of Economics and Finance. Available at https://pdfs.semanticscholar.org/
Barnett, W., Serletis, A., & Serletis, D. 2015, ‘Nonlinear and Complex Dynamics in Economics’, Macroeconomic Dynamics, vol. 19, no. 8, pp. 1749-1779.
Cottrell, P. 2016, ‘Chaos Theory and Modern Trading’. Available at SSRN: https://ssrn.com/abstract=2761874.
Kristoufek, L. 2012, ‘Fractal Markets Hypothesis and the Global Financial Crisis: Scaling, Investment Horizons and Liquidity’, Advances in Complex Systems, vol. 15, no. 6, art. 1250065.
Litimi, H., BenSaida, A., Belkacem, L., & Oussama A. 2019, ‘Chaotic Behavior in Financial Market Volatility’, Journal of Risk, vol. 21, no. 3, pp. 27-53.
Peters, E.E. (1994). Fractal Market Analysis: Applying Chaos Theory to Investment and Economics. New York, NY: John Wiley & Sons, Inc.
Tziperman, E., ‘Chaos Theory: A Brief Introduction’. Available at https://courses.seas.harvard.edu/climate/eli/Courses/EPS281r/Sources/Cha… | {"url":"https://www.howeandrusling.com/a-quick-and-dirty-primer-on-chaos-theory/","timestamp":"2024-11-13T16:13:25Z","content_type":"text/html","content_length":"128002","record_id":"<urn:uuid:d3e9127f-85a8-470d-b1f2-c130e6fa7b0a>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00747.warc.gz"} |
What is the Golden Rule of Investment - Knittystash.com
What is the Golden Rule of Investment
The Golden Rule of Investment is a fundamental principle that guides prudent investors. It emphasizes the importance of treating others with fairness and respect during investment activities. This
means adhering to ethical practices, avoiding conflicts of interest, and ensuring that all parties involved in an investment transaction are treated equitably. By following the Golden Rule, investors
can build positive relationships, maintain a reputation for integrity, and ultimately enhance their long-term investment success.
Balancing Risk and Return
Understanding the relationship between risk and return is crucial in investing. Risk refers to the possibility of losing some or all of your invested capital, while return refers to the profit or
gain you can potentially make.
The golden rule of investment is to strike a balance between risk and return. Higher returns typically come with higher risks, while lower risks often yield lower returns. Your risk tolerance,
investment horizon, and financial goals should guide your risk-return preferences.
Risk Tolerance
• Conservative: Low risk tolerance, prefer low-return, low-risk investments.
• Moderate: Open to moderate risks, seek a balance between risk and return.
• Aggressive: High risk tolerance, willing to take on high risks for the potential of higher returns.
Investment Horizon
The length of time you plan to keep your investment. Longer horizons allow for more risk-taking as investments have more time to recover from fluctuations.
Financial Goals
Your long-term financial objectives, such as retirement, education funding, or a house purchase, influence your risk-return preferences.
Table: Risk-Return Relationship
Risk Level Potential Return
Low Low
Moderate Moderate
High High
Remember, there’s no one-size-fits-all approach to balancing risk and return. Consult a financial advisor to assess your individual needs and create a personalized investment strategy.
Time Value of Money
The Golden Rule of Investment dictates that the present value of a future sum of money is worth less than that sum. This is because money can earn interest, so a dollar today is worth more than a
dollar in the future. The time value of money (TVM) is the concept that money has a different value at different points in time.
For example, if you invest $1,000 at a 5% interest rate, it will be worth $1,050 in one year. This is because the interest earned is added to the principal, increasing the value of the investment.
The longer you invest money, the more interest it will earn, and the greater the future value will be. Compound interest helps you grow the money faster as you earn interest on both the principal and
previously earned interest.
• Factors affecting the time value of money:
• Interest rate: Higher interest rates result in a higher future value.
• Investment period: The longer the investment period, the higher the future value.
• Compounding frequency: More frequent compounding results in a higher future value.
Calculating Time Value of Money
The future value (FV) of an investment can be calculated using the following formula:
FV = PV x (1 + r)^n
FV = Future Value
PV = Present Value
r = Interest Rate
n = Number of Years
For example, if you invest $1,000 at a 5% interest rate for 10 years, the future value would be:
FV = 1,000 x (1 + 0.05)^10
FV = $1,628.89
Asset Allocation
Asset allocation is the process of dividing your investment portfolio among different types of assets, such as stocks, bonds, and cash. The goal of asset allocation is to create a portfolio that has
the right combination of risk and return for your individual needs.
There are several factors to consider when determining your asset allocation, including your age, risk tolerance, investment goals, and time horizon.
• Age: Younger investors can typically afford to take on more risk than older investors, as they have more time to recover from any losses.
• Risk tolerance: Your risk tolerance is how much you are comfortable with the potential for losses. Investors with a low risk tolerance should allocate more of their portfolio to less risky
assets, such as bonds.
• Investment goals: Your investment goals will determine how much you need to invest and how much risk you can afford to take. For example, if you are saving for retirement, you will need to invest
more aggressively than if you are saving for a short-term goal, such as a down payment on a house.
• Time horizon: Your time horizon is how long you plan to invest. Investors with a long time horizon can afford to take more risk than investors with a short time horizon.
Once you have considered these factors, you can start to develop an asset allocation strategy. A common approach is to allocate your portfolio based on the following percentages:
Age Stocks Bonds Cash
20-30 70% 20% 10%
30-40 60% 30% 10%
40-50 50% 40% 10%
50-60 40% 50% 10%
60+ 30% 60% 10%
These are just general guidelines, and you may need to adjust your asset allocation based on your individual needs.
The Golden Rule of Investment
The golden rule of investment is a simple but powerful principle that can help you achieve your financial goals. It states that you should always invest for the long term and never sell your
investments out of fear or panic. By following this rule, you can take advantage of the compounding effect, which is the most important factor in building wealth.
Compounding Effect
The compounding effect is the snowball effect of earning interest on your interest. Over time, this can lead to significant growth in your portfolio. For example, if you invest $1,000 at a 10% annual
interest rate, it will grow to $2,594 in 10 years. And if you leave it invested for another 10 years, it will grow to $6,727. That’s over six times your original investment!
Here is a table that shows the power of compounding:
Year Balance
0 1,000
1 1,100
2 1,210
3 1,331
4 1,464
5 1,611
6 1,772
7 1,949
8 2,144
9 2,358
10 2,594
As you can see, the compounding effect can make a big difference in your investment returns. By investing for the long term, you can take advantage of this powerful force and build a substantial nest
Here are some tips for investing for the long term:
• Start saving early
• Invest in a diversified portfolio
• Don’t try to time the market
• Rebalance your portfolio regularly
• Ride out the ups and downs
By following these tips, you can increase your chances of achieving your financial goals.
Well, folks, that’s the Golden Rule of Investment in a nutshell. It’s not rocket science, but it’s a heck of a lot more important than any fancy financial jargon you’ve ever heard. Thanks for
sticking with me through the end. If you found this article helpful, be sure to bookmark it and stop back later for more investment tips and tricks. Remember, the only way to get ahead in this game
is by playing smart and doing your research. So keep learning, keep investing, and keep making that money work for you! | {"url":"https://knittystash.com/what-is-the-golden-rule-of-investment/","timestamp":"2024-11-04T09:00:04Z","content_type":"text/html","content_length":"117213","record_id":"<urn:uuid:82486736-856e-4160-b555-e2ad8ed99c0c>","cc-path":"CC-MAIN-2024-46/segments/1730477027819.53/warc/CC-MAIN-20241104065437-20241104095437-00771.warc.gz"} |
Strange correlations benchmark hadronisation
In high-energy hadronic and heavy-ion collisions, strange quarks are dominantly produced from gluon fusion. In contrast to u and d quarks, they are not present in the colliding particles. Since
strangeness is a conserved quantity in QCD, the net number of strange and anti-strange particles must equal zero, making them prime observable to study the dynamics of these collisions. Various
experimental results from high-multiplicity pp collisions at the LHC demonstrate striking similarities to Pb–Pb collision results. Notably, the fraction of hadrons carrying one or more strange quarks
smoothly increases as a function of particle multiplicity in pp and p–Pb collisions to values consistent with those measured in peripheral Pb–Pb collisions. Multi-particle correlations in pp
collisions also closely resemble those in Pb–Pb collisions.
Explaining such observations requires understanding the hadronisation mechanism, which governs how quarks and gluons rearrange into bound states (hadrons). Since there are no first-principle
calculations of the hadronisation process available, phenomenological models are used, based on either the Lund string fragmentation (Pythia 8, HIJING) or a statistical approach assuming a system of
hadrons and their resonances (HRG) at thermal and chemical equilibrium. Despite having vastly different approaches, both models successfully describe the enhanced production of strange hadrons. This
similarity calls for new observables to decisively discriminate between these two approaches.
In a recently published study, the ALICE collaboration measured correlations between particles arising from the conservation of quantum numbers to further distinguish the two models. In the string
fragmentation model, the quantum numbers are conserved locally through the creation of quark–antiquark pairs from the breaking of colour strings. This leads to a short-range rapidity correlation
between strange and anti-strange hadrons. On the other hand, in the statistical hadronisation approach, quantum numbers are conserved globally over a finite volume, leading to long-range correlations
between both strange–strange and strange–anti-strange hadron pairs. Quantum-number conservation leads to correlated particle production that is probed by measuring the yields of charged kaons (with
one strange quark) and multistrange baryons (Ξ^– and Ξ^+) on an event-by-event basis. In ALICE, charged kaons are directly tracked in the detectors, while Ξ baryons are reconstructed via their weak
decay to a charged pion and a Λ-baryon, which is itself identified via its weak decay into a proton and a charged pion.
In the figure, the first measurement of the normalized variance of the net number (difference between the particle and antiparticle multiplicities) of Ξ baryons and the correlation between net-Ξ and
net-kaon numbers are presented as a function of the average charged-particle multiplicity at midrapidity in pp, p-Pb and Pb-Pb collisions.
The experimental results on the correlations deviate from the uncorrelated baseline (dashed line), and string fragmentation models that mainly correlate strange hadrons with opposite strange quark
content over a small rapidity range fail to describe both observables. At the same time, the measurements agree with the statistical hadronisation model description that includes opposite-sign and
same-sign strangeness correlations over large rapidity intervals. The data indicate a weaker opposite-sign strangeness correlation than that predicted by string fragmentation, suggesting that the
correlation volume for strangeness conservation extends to about three units of rapidity.
The study will be extended using the recently collected data during LHC Run 3. The larger data samples will enable similar measurements for the triply strange Ω baryon and the study of higher
Further reading | {"url":"https://alice.cern/2024_strange_correlation","timestamp":"2024-11-08T14:02:05Z","content_type":"text/html","content_length":"30278","record_id":"<urn:uuid:4c07d94b-4740-4cb4-8d27-8b1bff48607d>","cc-path":"CC-MAIN-2024-46/segments/1730477028067.32/warc/CC-MAIN-20241108133114-20241108163114-00191.warc.gz"} |
critical speed of ball mill in rpm
WEB% Critical speed: Practically, mill speed between 68 and 82% of critical speed. % critical speed is the mill actual speed in RPM divided by nc. Example: meter mill with rotational speed of rpm
then nc =, % critical speed = %. TO DOWNLOAD THE EXCEL SHEET AND ALL THE OTHER USEFUL BOOKS AND RESOURCES . | {"url":"https://lecailleretsaplancha.fr/03/21-7494.html","timestamp":"2024-11-09T06:57:07Z","content_type":"text/html","content_length":"35038","record_id":"<urn:uuid:efa505d2-30e2-4cd3-9ecf-6153566bb2d0>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.30/warc/CC-MAIN-20241109053958-20241109083958-00411.warc.gz"} |
Haleakalā Bike Excursion: Summit & Upcountry Tour
The Ultimate Self-Guided Bike Tour
Touring Haleakala National Park can be a daunting task. There's so much to see and do, how can you make the most of your time? Many people try to tour the park on their own and quickly find out that
it's more than they bargained for. The park is huge and there's a lot of elevation changes, so it can be tough to navigate if you're not familiar with the area.
Our Haleakala Summit and Upcountry Maui Tour is the perfect solution for visitors who want to see as much of the park as possible. We provide an informative guided tour of the park, plus a
self-guided bike ride down the mountain. This allows visitors to explore at their own pace and makes sure they don't miss any of the highlights. It's the ultimate way to explore Maui's dormant
From: $209^.00
Choose Tour Starting From
Choose Tour Starting From
• Haleakala National Park & Upcountry Tour From Maui Hotels
Starting at
per adult
Choose a date to see available times.
• Participant's Information
• Where are you staying?
Having your accommodation information helps us provide directions, travel tips and can more easily contact you if there is a change in plans last minute.
Please arrive by 7:50 AM
Drive time & location: Coming from the South side of Maui Wailea/Makena it should take about an hour to get to Haiku. From Kihei, 45 minutes. On the West side, from Lahaina it’s about a 1 hours
drive. From Kaanapali 1 hour and 15 minutes. From Kapalua 1 hour 30 minutes.
See pickup location on a map
Where are you staying?
Having your accommodation information helps us provide directions, travel tips and can more easily contact you if there is a change in plans last minute.
Where are you staying?
Having your accommodation information helps us provide directions, travel tips and can more easily contact you if there is a change in plans last minute.
Where are you staying?
Having your accommodation information helps us provide directions, travel tips and can more easily contact you if there is a change in plans last minute. | {"url":"https://www.volcanotours.com/tours/haleakala-bike-excursion-summit-upcountry-tour/","timestamp":"2024-11-13T06:09:11Z","content_type":"text/html","content_length":"1049690","record_id":"<urn:uuid:e76af3a6-157b-44f5-b101-aa6ea5da7619>","cc-path":"CC-MAIN-2024-46/segments/1730477028326.66/warc/CC-MAIN-20241113040054-20241113070054-00509.warc.gz"} |
US10330711B2 - Method and device for detecting current of inductor of PFC circuit
- Google Patents
US10330711B2 - Method and device for detecting current of inductor of PFC circuit - Google Patents
Method and device for detecting current of inductor of PFC circuit Download PDF
Publication number
US10330711B2 US10330711B2 US15/305,104 US201415305104A US10330711B2 US 10330711 B2 US10330711 B2 US 10330711B2 US 201415305104 A US201415305104 A US 201415305104A US 10330711 B2 US10330711 B2 US
United States
Prior art keywords
pfc circuit
detection signal
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Tao Liu
Jianping Zhou
Guoxian LIN
Jie Fan
Yong Luo
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ZTE Corp
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ZTE Corp
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☆ G—PHYSICS
☆ G01—MEASURING; TESTING
☆ G01R—MEASURING ELECTRIC VARIABLES; MEASURING MAGNETIC VARIABLES
☆ G01R19/00—Arrangements for measuring currents or voltages or for indicating presence or sign thereof
☆ G01R19/25—Arrangements for measuring currents or voltages or for indicating presence or sign thereof using digital measurement techniques
☆ G01R19/2506—Arrangements for conditioning or analysing measured signals, e.g. for indicating peak values ; Details concerning sampling, digitizing or waveform capturing
☆ G—PHYSICS
☆ G01—MEASURING; TESTING
☆ G01R—MEASURING ELECTRIC VARIABLES; MEASURING MAGNETIC VARIABLES
☆ G01R15/00—Details of measuring arrangements of the types provided for in groups G01R17/00 - G01R29/00, G01R33/00 - G01R33/26 or G01R35/00
☆ G01R15/14—Adaptations providing voltage or current isolation, e.g. for high-voltage or high-current networks
☆ G01R15/18—Adaptations providing voltage or current isolation, e.g. for high-voltage or high-current networks using inductive devices, e.g. transformers
☆ G01R15/183—Adaptations providing voltage or current isolation, e.g. for high-voltage or high-current networks using inductive devices, e.g. transformers using transformers with a magnetic
☆ G—PHYSICS
☆ G01—MEASURING; TESTING
☆ G01R—MEASURING ELECTRIC VARIABLES; MEASURING MAGNETIC VARIABLES
☆ G01R19/00—Arrangements for measuring currents or voltages or for indicating presence or sign thereof
☆ G01R19/0092—Arrangements for measuring currents or voltages or for indicating presence or sign thereof measuring current only
☆ H—ELECTRICITY
☆ H02—GENERATION; CONVERSION OR DISTRIBUTION OF ELECTRIC POWER
☆ H02H—EMERGENCY PROTECTIVE CIRCUIT ARRANGEMENTS
☆ H02H7/00—Emergency protective circuit arrangements specially adapted for specific types of electric machines or apparatus or for sectionalised protection of cable or line systems, and
effecting automatic switching in the event of an undesired change from normal working conditions
☆ H02H7/10—Emergency protective circuit arrangements specially adapted for specific types of electric machines or apparatus or for sectionalised protection of cable or line systems, and
effecting automatic switching in the event of an undesired change from normal working conditions for converters; for rectifiers
☆ H02H7/12—Emergency protective circuit arrangements specially adapted for specific types of electric machines or apparatus or for sectionalised protection of cable or line systems, and
effecting automatic switching in the event of an undesired change from normal working conditions for converters; for rectifiers for static converters or rectifiers
☆ H—ELECTRICITY
☆ H02—GENERATION; CONVERSION OR DISTRIBUTION OF ELECTRIC POWER
☆ H02M—APPARATUS FOR CONVERSION BETWEEN AC AND AC, BETWEEN AC AND DC, OR BETWEEN DC AND DC, AND FOR USE WITH MAINS OR SIMILAR POWER SUPPLY SYSTEMS; CONVERSION OF DC OR AC INPUT POWER INTO
SURGE OUTPUT POWER; CONTROL OR REGULATION THEREOF
☆ H02M1/00—Details of apparatus for conversion
☆ H02M1/42—Circuits or arrangements for compensating for or adjusting power factor in converters or inverters
☆ H02M1/4208—Arrangements for improving power factor of AC input
☆ H—ELECTRICITY
☆ H02—GENERATION; CONVERSION OR DISTRIBUTION OF ELECTRIC POWER
☆ H02M—APPARATUS FOR CONVERSION BETWEEN AC AND AC, BETWEEN AC AND DC, OR BETWEEN DC AND DC, AND FOR USE WITH MAINS OR SIMILAR POWER SUPPLY SYSTEMS; CONVERSION OF DC OR AC INPUT POWER INTO
SURGE OUTPUT POWER; CONTROL OR REGULATION THEREOF
☆ H02M1/00—Details of apparatus for conversion
☆ H02M1/42—Circuits or arrangements for compensating for or adjusting power factor in converters or inverters
☆ H02M1/4208—Arrangements for improving power factor of AC input
☆ H02M1/4225—Arrangements for improving power factor of AC input using a non-isolated boost converter
☆ H—ELECTRICITY
☆ H02—GENERATION; CONVERSION OR DISTRIBUTION OF ELECTRIC POWER
☆ H02M—APPARATUS FOR CONVERSION BETWEEN AC AND AC, BETWEEN AC AND DC, OR BETWEEN DC AND DC, AND FOR USE WITH MAINS OR SIMILAR POWER SUPPLY SYSTEMS; CONVERSION OF DC OR AC INPUT POWER INTO
SURGE OUTPUT POWER; CONTROL OR REGULATION THEREOF
☆ H02M1/00—Details of apparatus for conversion
☆ H02M1/0003—Details of control, feedback or regulation circuits
☆ H02M1/0009—Devices or circuits for detecting current in a converter
☆ H—ELECTRICITY
☆ H02—GENERATION; CONVERSION OR DISTRIBUTION OF ELECTRIC POWER
☆ H02M—APPARATUS FOR CONVERSION BETWEEN AC AND AC, BETWEEN AC AND DC, OR BETWEEN DC AND DC, AND FOR USE WITH MAINS OR SIMILAR POWER SUPPLY SYSTEMS; CONVERSION OF DC OR AC INPUT POWER INTO
SURGE OUTPUT POWER; CONTROL OR REGULATION THEREOF
☆ H02M1/00—Details of apparatus for conversion
☆ H02M1/0083—Converters characterised by their input or output configuration
☆ H02M1/0085—Partially controlled bridges
CROSS-REFERENCE ART COLLECTIONS [XRACs] AND DIGESTS
☆ Y02—TECHNOLOGIES OR APPLICATIONS FOR MITIGATION OR ADAPTATION AGAINST CLIMATE CHANGE
☆ Y02B—CLIMATE CHANGE MITIGATION TECHNOLOGIES RELATED TO BUILDINGS, e.g. HOUSING, HOUSE APPLIANCES OR RELATED END-USER APPLICATIONS
☆ Y02B70/00—Technologies for an efficient end-user side electric power management and consumption
☆ Y02B70/10—Technologies improving the efficiency by using switched-mode power supplies [SMPS], i.e. efficient power electronics conversion e.g. power factor correction or reduction of
losses in power supplies or efficient standby modes
CROSS-REFERENCE ART COLLECTIONS [XRACs] AND DIGESTS
☆ Y02—TECHNOLOGIES OR APPLICATIONS FOR MITIGATION OR ADAPTATION AGAINST CLIMATE CHANGE
☆ Y02P—CLIMATE CHANGE MITIGATION TECHNOLOGIES IN THE PRODUCTION OR PROCESSING OF GOODS
☆ Y02P80/00—Climate change mitigation technologies for sector-wide applications
☆ Y02P80/10—Efficient use of energy, e.g. using compressed air or pressurized fluid as energy carrier
□ the present document relates to the field of power supply technology, and more particularly, to a method and device for detecting inductor current of a critical-conduction mode Power Factor
Correction (PFC) circuit.
□ PFC Power Factor Correction
□ the PFC circuit is widely applied. Meanwhile, the PFC circuit also develops towards a direction of high efficiency and high power density.
□ a Critical-conduction Mode (CRM) PFC circuit is extensively applied.
□ CCM Critical-conduction Mode
□ the topology of the totem-pole-type bridgeless PFC circuit as an example, when working in a critical-conduction mode, it can realize Zero voltage switch (ZVS) or Valley switch (VS) in the
full alternating current input range and full load range, and also can simultaneously meet the requirements of high power density and high efficiency.
□ ZVS Zero voltage switch
□ VS Valley switch
□ FIG. 1 is a structure diagram of a critical-conduction bridge PFC circuit for detecting inductor current of the PFC circuit in the related art.
□ the critical-conduction bridge PFC circuit includes at least two bridge arms connected in parallel between a first connection point A and a second connection point B, herein a first bridge
arm includes two diodes connected in series in the same direction, and a second bridge arm includes two diodes connected in series in the same direction.
□ the critical-conduction bridge PFC circuit also includes one PFC inductor, there is one switch tube S 1 between a third connection point C and the second connection point B, there is one
diode D 5 between the third connection point C and a fourth connection point D, and there are a filter capacitor C 0 and a load R 0 also connected in parallel between the fourth connection
point D and the second connection point B.
□ FIG. 2 is a structure diagram of a totem-pole-type bridgeless PFC circuit for detecting inductor current of the PFC circuit in the related art.
□ the totem-pole-type bridgeless PFC circuit includes at least two bridge arms connected in parallel between a first connection point A and a second connection point B, herein a first bridge
arm includes two switch tubes or diodes connected in series in the same direction, and a second bridge arm includes two switch tubes connected in series in the same direction.
□ the totem-pole-type bridgeless PFC circuit includes one PFC inductor, and a filter capacitor C 0 and a load R 0 also connected in parallel between the first connection point A and the second
connection point B.
□ a diode D 2 When the input voltage is in the positive half cycle, a diode D 2 is always conductive, a switch tube S 2 is closed, a switch tube S 1 is disconnected, and at this point the
current on an inductor L increases from zero to store energy; after the above energy storage process ends, the switch tube S 2 is disconnected, the switch tube S 1 is closed, and at this
point the current on the inductor L decreases from the peak value to release energy.
□ the critical-conduction mode power factor correction circuit is required to timely and accurately obtain an inductor current signal which is used for loop control or implementing the inductor
current protection function, but how to obtain the inductor current signal is a problem required to be urgently solved by the researchers.
□ the object of the embodiments of the present document is to provide a method and device for detecting inductor current of a PFC circuit, which may solve the problem that the inductor current
of the PFC circuit cannot be obtained in a critical-conduction mode.
□ a method for detecting inductor current of a PFC circuit which includes:
□ the inductor voltage detection signal into a voltage signal whose waveform is consistent with a current waveform of the inductor voltage to serve as an inductor current detection signal, to
perform loop protection on the PFC circuit or perform over-current protection on the PFC circuit by using the inductor current detection signal.
□ the step of detecting a voltage on a boost inductor of a critical-conduction mode PFC circuit, and obtaining an inductor voltage detection signal includes:
□ the step of detecting a voltage on a boost inductor of a critical-conduction mode PFC circuit, and obtaining an inductor voltage detection signal includes:
□ obtaining an induced voltage in conformity with the current of the boost inductor of the PFC circuit on the inductor auxiliary winding includes:
□ obtaining an induced voltage in conformity with the current of the boost inductor of the PFC circuit on the inductor auxiliary winding includes:
□ the step of converting the inductor voltage detection signal into a voltage signal whose waveform is consistent with a current waveform of the inductor to serve as an inductor current
detection signal includes:
□ a waveform of the inductor current detection signal is consistent with a waveform of a current signal of the boost inductor of the PFC circuit.
□ the step of performing integral processing on the inductor voltage detection signal includes:
□ the step of performing integral processing on the inductor voltage detection signal includes:
□ a device for detecting inductor current of a PFC circuit which includes:
□ a detection module configured to detect a voltage on a boost inductor of a critical-conduction mode PFC circuit, and obtain an inductor voltage detection signal
□ a conversion module configured to convert the inductor voltage detection signal into a voltage signal whose waveform is consistent with a current waveform the inductor to serve as an inductor
current detection signal, to perform loop protection on the PFC circuit or perform over-current protection on the PFC circuit by using the inductor current detection signal.
□ the conversion module includes:
□ an integral submodule configured to perform integral processing on the inductor voltage detection signal, and take an inductor voltage detection signal on which the integral processing is
performed as the inductor current detection signal, herein a waveform of the inductor current detection signal is consistent with the waveform of a current signal of the boost inductor of the
PFC circuit.
□ the beneficial effects of the embodiments of the present document lie in that: in the embodiments of the present document, through the method for detecting the inductor voltage of the
critical-conduction mode PFC circuit and indirectly obtaining the inductor current, the function of detecting the inductor current of the critical-conduction mode PFC circuit can be realized.
□ FIG. 1 is a structure diagram of a critical-conduction bridge PFC circuit for detecting inductor current of the PFC circuit in the related art
□ FIG. 2 is a structure diagram of a totem-pole-type bridgeless PFC circuit for detecting inductor current of the PFC circuit in the related art
□ FIG. 3 is a principle diagram of a method for detecting inductor current of a PFC circuit provided in the embodiment of the present document;
□ FIG. 4 is a schematic diagram of a structure of a device for detecting inductor current of a PFC circuit provided in the embodiment of the present document;
□ FIG. 5 is a schematic diagram of a device for detecting inductor current of the critical-conduction bridge PFC circuit for detecting the inductor current of the PFC circuit provided in the
embodiment of the present document;
□ FIG. 6 is a circuit principle diagram of detecting the inductor current of the critical-conduction bridge PFC circuit for detecting the inductor current of the PFC circuit provided in the
embodiment of the present document;
□ FIG. 7 is an oscillogram corresponding to various parts of a circuit for detecting the inductor current of the critical-conduction bridge PFC circuit for detecting the inductor current of the
PFC circuit provided in the embodiment of the present document;
□ FIG. 8 is a schematic diagram of a structure of a device for detecting inductor current of the totem-pole-type bridgeless PFC circuit for detecting the inductor current of the PFC circuit
provided in the embodiment of the present document;
□ FIG. 9 is a first circuit principle diagram of detecting the inductor current of the totem-pole-type bridgeless PFC circuit for detecting the inductor current of the PFC circuit provided in
the embodiment of the present document;
□ FIG. 10 is a first oscillogram corresponding to various parts of a circuit for detecting the inductor current of the totem-pole-type bridgeless PFC circuit for detecting the inductor current
of the PFC circuit provided in the embodiment of the present document;
□ FIG. 11 is a second circuit principle diagram of detecting the inductor current of the totem-pole-type bridgeless PFC circuit for detecting the inductor current of the PFC circuit provided in
the embodiment of the present document.
□ FIG. 12 is a second oscillogram corresponding to various parts of a circuit for detecting the inductor current of the totem-pole-type bridgeless PFC circuit for detecting the inductor current
of the PFC circuit provided in the embodiment of the present document.
□ FIG. 3 is a principle diagram of a method for detecting inductor current of a PFC circuit provided in the embodiment of the present document. As shown in FIG. 3 , the specific steps are as
□ step S 1 a device for detecting the inductor current of the PFC circuit detects a voltage on a boost inductor of a critical-conduction mode PFC circuit, and obtains an inductor voltage
detection signal.
□ step S 1 by connecting in series a sense resistor with the boost inductor of the PFC circuit, a voltage drop in conformity with current of the boost inductor of the PFC circuit is obtained
through the sense resistor;
□ the obtained voltage drop is taken as the inductor voltage detection signal.
□ a positive voltage corresponding to a rising waveform of the current of the boost inductor of the PFC circuit and a negative voltage corresponding to a falling waveform of the current of the
boost inductor of the PFC circuit are obtained on an inductor auxiliary winding.
□ the inductor auxiliary winding is enwound on a magnetic core of the boost inductor of the PFC circuit, and an induced voltage in conformity with the current of the boost inductor of the PFC
circuit is obtained on the inductor auxiliary winding;
□ the obtained induced voltage is taken as the inductor voltage detection signal.
□ a positive pulse corresponding to a sawtooth wave rising edge of the current of the boost inductor of the PFC circuit and a negative pulse corresponding to a sawtooth wave falling edge of the
current of the boost inductor of the PFC circuit are obtained on the inductor auxiliary winding.
□ step S 2 the device for detecting the inductor current of the PFC circuit converts the inductor voltage detection signal into a voltage signal whose waveform is consistent with a current
waveform of the inductor to serve as an inductor current detection signal, in order to perform loop protection on the PFC circuit or perform over-current protection on the PFC circuit by
using the inductor current detection signal.
□ step S 2 integral processing is performed on the inductor voltage detection signal, and an inductor voltage detection signal on which the integral processing is performed is taken as the
inductor current detection signal, herein the waveform of the inductor current detection signal is consistent with a waveform of a current signal of the boost inductor of the PFC circuit.
□ the step of performing integral processing on the inductor voltage detection signal includes:
□ integral processing is performed on the positive voltage and the negative voltage obtained on the inductor auxiliary winding, and it is to switch on or switch off a corresponding switch tube
to charge or discharge a capacitor in a connected integral circuit;
□ the step of performing integral processing on the inductor voltage detection signal also includes:
□ a corresponding switch tube is driven to be switched on or switched off to charge or discharge a capacitor in a connected integral circuit
□ FIG. 4 is a schematic diagram of a structure of a device for detecting inductor current of a PFC circuit provided in the embodiment of the present document. As shown in FIG. 4 , the device
includes a critical-conduction PFC circuit, an inductor voltage detection unit and an inductor voltage detection signal processing unit.
□ the critical-conduction PFC circuit includes a related bridge power factor correction circuit and a related power factor correction circuit without a rectifier bridge for which a inductor
current works in a critical continuous mode.
□ the inductor voltage detection unit is used to detect a voltage on the inductor of the PFC circuit and take the detected inductor voltage signal of the PFC circuit as the input of the
inductor voltage detection signal processing unit.
□ the inductor voltage detection unit may perform direct detection by using a divider resistor.
□ the inductor voltage of the PFC circuit is detected by using the divider resistor, the divider resistor is cascaded with the inductor of the PFC circuit, a voltage drop in conformity with the
inductor current of the PFC circuit is detected, and then the detected voltage drop is sent to the inductor voltage detection signal processing unit.
□ the inductor auxiliary winding may also be used to perform detection.
□ the inductor auxiliary winding is preferably a winding coupled on a magnetic core of the inductor of the PFC circuit, voltages at two ends of the inductor of the PFC circuit are respectively
detected, and then the detected voltage signals are respectively sent to the inductor voltage detection signal processing unit.
□ the inductor voltage detection unit implements a function of a detection module, that is, the detection module is used for detecting the voltage on the boost inductor of the
critical-conduction mode PFC circuit, and obtains the inductor voltage detection signal.
□ the inductor voltage detection signal processing unit is used for performing an integral on the detected inductor voltage signal through a certain integral circuit to restore it back to the
inductor current signal.
□ the inductor voltage detection signal processing unit includes one integral circuit, herein, one end of the integral circuit is connected with the inductor voltage detection unit, and the
other end is connected with the earth ground.
□ the voltage on a capacitor in the integral circuit is a detection value of the inductor current, that is, the inductor voltage detection signal processing unit converts the input inductor
voltage signal into the inductor current signal through processing, herein, the output of the inductor voltage detection signal processing unit is the inductor current signal required to be
□ the inductor voltage detection signal processing unit includes some switch tubes used for enabling all inductor currents of positive and negative half cycles of the input voltage to be output
from one output port, and the inductor currents are used for performing loop protection on the PFC circuit or performing over-current protection on the PFC circuit.
□ the inductor voltage detection signal processing unit also includes some other switch tubes used for performing selection on inductor voltage signals of the positive and negative half cycles,
thereby detecting correct inductor current output signals.
□ the inductor voltage detection signal processing unit implements a function of a conversion module, that is, the conversion module is used for converting the inductor voltage detection signal
into a voltage signal whose waveform is consistent with a current waveform of the inductor to serve as an inductor current detection signal, in order to perform loop protection on the PFC
circuit or perform over-current protection on the PFC circuit by using the inductor current detection signal.
□ An integral submodule of the conversion module is used for performing integral processing on the inductor voltage detection signal, and an inductor voltage detection signal on which the
integral processing is performed is taken as the inductor current detection signal, herein the waveform of the inductor current detection signal is consistent with a waveform of a current
signal of the boost inductor of the PFC circuit.
□ the working principle of the inductor voltage detection signal processing unit includes: the inductor current of the critical-conduction mode PFC circuit rises from zero to a maximum value,
and then falls from the maximum value to zero within one switch period. Therefore, the voltage on the inductor of the PFC circuit will have two times of overturn within one switch period, one
is the overturn from the positive voltage to the negative voltage, and the other is the overturn from the negative voltage to the positive voltage.
□ a resistor and capacitor signal processing circuit in the inductor voltage detection signal processing unit will perform a charging integral at the stage of the inductor current rising and
perform discharging at the stage of the inductor current falling, and the voltage signal on the capacitor is the inductor current signal whose waveform is consistent with the inductor current
waveform of the PFC circuit.
□ the inductor voltage detection unit sends the detected inductor voltage of the PFC circuit to the inductor voltage detection signal processing unit to be processed, the inductor voltage
detection signal processing unit makes the received inductor voltage signal of the PFC circuit pass through the signal processing circuit composed of the resistor and capacitor and the switch
tubes, and the inductor voltage signal of the PFC circuit is processed into the inductor current signal of the PFC circuit.
□ FIG. 5 is a schematic diagram of a device for detecting inductor current of the critical-conduction bridge PFC circuit for detecting the inductor current of the PFC circuit provided in the
embodiment of the present document
□ FIG. 6 is a circuit principle diagram of detecting the inductor current of the critical-conduction bridge PFC circuit for detecting the inductor current of the PFC circuit provided in the
embodiment of the present document.
□ one auxiliary winding is added and enwound on a PFC inductor of the related device for detecting the inductor current of the critical-conduction bridge PFC, and the auxiliary winding of the
PFC inductor is coupled on a magnetic core of the PFC inductor.
□ an end A of the inductor auxiliary winding of the PFC circuit and an end 1 of the inductor of the PFC circuit are dotted terminals, an end B of the inductor auxiliary winding of the PFC
circuit is earthed.
□ the end A of the inductor auxiliary winding of the PFC circuit is connected with a resistor R, the other end of the resistor R is connected with a capacitor C, and the other end of the
capacitor C is earthed.
□ the voltage waveform on the capacitor C is obtained in the way that the inductor auxiliary winding of the PFC circuit uses the RC integral according to the inductor voltage of the PFC circuit
obtained through sampling.
□ the voltage waveform on the capacitor C represents a waveform of the inductor current of the PFC circuit. Then, the voltage waveform on the capacitor C enters an Analog-to-Digital Converter
(ADC) sampling port of a Digital Signal Processor (DSP).
□ ADC Analog-to-Digital Converter
□ DSP Digital Signal Processor
□ FIG. 7 is an oscillogram corresponding to various parts of a circuit for detecting the inductor current of the critical-conduction bridge PFC circuit for detecting the inductor current of the
PFC circuit provided in the embodiment of the present document.
□ a first waveform IL is a waveform of the inductor current of the critical-conduction bridge PFC circuit
□ a second waveform VAB is an inductor voltage waveform of the PFC circuit detected on the inductor auxiliary winding of the PFC circuit
□ a third waveform VC is a corresponding waveform on the capacitor C in the RC integral circuit
□ the third waveform VC is similar to the waveform of the inductor current of the PFC circuit, that is, the voltage waveform on the capacitor C is the waveform entering the ADC sampling port of
the DSP.
□ FIG. 8 is a schematic diagram of a structure of a device for detecting inductor current of the totem-pole-type bridgeless PFC circuit for detecting the inductor current of the PFC circuit
provided in the embodiment of the present document
□ FIG. 9 is a first circuit principle diagram of detecting the inductor current of the totem-pole-type bridgeless PFC circuit for detecting the inductor current of the PFC circuit provided in
the embodiment of the present document.
□ the device for detecting the inductor current of the totem-pole-type bridgeless PFC circuit according to the present document will be concretely described through one specific application
example in the embodiments of the present document, certainly the device for detecting the inductor current is not limited to such form of the embodiments of the present document, people
skilled in the art may also choose and adopt other similar forms according to the professional knowledge mastered by them, as long as various functions can be implemented. As shown in FIG. 8
and FIG.
□ two auxiliary windings are added and enwound on the inductor of the PFC circuit of the totem-pole-type bridgeless PFC circuit, and are respectively used for detecting corresponding inductor
voltages when the input alternating voltage is in positive and negative half cycles, and both the two inductor auxiliary windings of the PFC circuit are coupled on a magnetic core of the
inductor of the PFC circuit.
□ the end A 1 of the inductor auxiliary winding 1 of the PFC circuit is connected with a resistor R 1
□ an end B 2 of the inductor auxiliary winding 2 of the PFC circuit is connected with a resistor R 2
□ the other end of the resistor R 1 is connected with a collector of a switch triode VT 1 and the other end of the resistor R 2 is connected with a collector of a switch triode VT 2
□ emitters of the switch triodes VT 1 and VT 2 are earthed
□ a capacitor C 1 is connected in parallel with the collector and emitter of the switch triode VT 1 and a capacitor C 2 is connected in parallel with the collector and emitter of the switch
triode VT 2
□ the collector of the switch triode VT 1 is connected with an anode of a diode D 1
□ the collector of the switch triode VT 2 is connected with an anode of a diode D 2
□ cathodes of the D 1 and D 2 are joined together and connected with the ADC
□ the inductor current of the PFC circuit flows from the end 1 to the end 2 , when the inductor current rises, the end A 1 of the inductor auxiliary winding 1 of the PFC circuit is a positive
voltage, and the end A 2 is a negative voltage, but the end B 1 of the inductor auxiliary winding 2 of the PFC circuit is a negative voltage, and the end B 2 is a positive voltage; when the
inductor current falls, the end A 1 of the inductor auxiliary winding 1 of the PFC circuit is a negative voltage, and the end A 2 is a positive voltage, but the end B 1 of the inductor
auxiliary winding 2 of the PFC circuit is a positive voltage, and the end B 2 is a negative voltage.
□ the switch triode VT 2 is driven, so that the switch triode VT 2 is always in an on-state, and the switch triode VT 1 has no drive voltage, thus the switch triode VT 1 is always in an
□ the voltage waveform on the capacitor C 1 is obtained in a way that the inductor auxiliary winding 1 of the PFC circuit uses the RC integral according to the inductor voltage of the PFC
circuit obtained through sampling, and the voltage waveform on the capacitor C 1 represents a waveform of the inductor current of the PFC circuit, and then enters an ADC sampling port of the
DSP, but since the switch triode VT 2 is always in an on-state, the voltage of the capacitor C 2 is zero, thus the inductor current of the PFC circuit in the power frequency positive half
cycle may be obtained.
□ the inductor current of the PFC circuit flows from the end 2 to the end 1 , when the inductor current rises, the end A 1 of the inductor auxiliary winding 1 of the PFC circuit is a negative
voltage, and the end A 2 is a positive voltage, but the end B 1 of the inductor auxiliary winding 2 of the PFC circuit is a positive voltage, and the end B 2 is a negative voltage; when the
inductor current falls, the end A 1 of the inductor auxiliary winding 1 of the PFC circuit is a positive voltage, and the end A 2 is a negative voltage, but the end B 1 of the inductor
auxiliary winding 2 of the PFC circuit is a negative voltage, and the end B 2 is a positive voltage.
□ the switch triode VT 1 is driven, so that the switch triode VT 1 is always in an on-state, and the switch triode VT 2 has no drive voltage, thus the switch triode VT 2 is always in an
□ the voltage waveform on the capacitor C 2 is obtained in a way that the inductor auxiliary winding 2 of the PFC circuit uses the RC integral according to the inductor voltage of the PFC
circuit obtained through sampling, and the voltage waveform on the capacitor C 2 represents a waveform of the inductor current of the PFC circuit, and then enters an ADC sampling port of the
DSP, but since the switch triode VT 1 is always in an on-state, the voltage of the capacitor C 1 is zero, thus the inductor current of the PFC circuit in the power frequency negative half
cycle may be obtained.
□ a first waveform IL is a waveform of the inductor current of the totem-pole-type bridgeless PFC circuit
□ a second waveform Vg 1 is a drive voltage waveform of the switch triode VT 1
□ a third waveform Vg 2 is a drive voltage waveform of the switch triode VT 2
□ a fourth waveform VC 1 is a voltage waveform of the capacitor C 1
□ a fifth waveform VC 2 is a voltage waveform of the capacitor C 2
□ a sixth waveform VC is the final output inductor current waveform of the PFC circuit.
□ switch tubes S 2 and S 3 are always switched on, switch tubes S 1 and S 4 are always switched off, thus when the inductor current of the PFC circuit rises, the inductor auxiliary winding of
the PFC circuit charges the capacitor C through the resistor R, and when the inductor current of the PFC circuit falls, the capacitor C discharges for the inductor auxiliary winding of the
PFC circuit through the resistor R, thus the voltage waveform obtained on the capacitor C is the inductor current waveform of the PFC circuit in the positive half cycle; during the negative
half cycle of the input alternating voltage, the switch tubes S 1 and S 4 are switched on, the switch tubes S 2 and S 3 are switched off, when the inductor current of the PFC circuit rises,
the inductor auxiliary winding of the PFC circuit charges the capacitor C through the resistor R, and when the inductor current of the PFC circuit falls, the capacitor C discharges for the
inductor auxiliary winding of the PFC circuit through the resistor R,
□ a first waveform IL is a waveform of the inductor current of the totem-pole-type bridgeless PFC circuit
□ a second waveform Vaux is an inductor voltage waveform detected on the inductor auxiliary winding of the PFC circuit
□ a third waveform VC is a corresponding waveform on the capacitor C in the RC integral circuit
□ a shape of the third waveform VC is similar to a waveform of the inductor current of the PFC circuit, that is, the voltage waveform on the capacitor C is the waveform entering the ADC
sampling port of the DSP.
□ the embodiments of the present document have the following technical effects: in the embodiments of the present document, through the method for detecting the inductor voltage of the
critical-conduction mode PFC circuit and indirectly obtaining the inductor current, the function of detecting the inductor current of the critical-conduction mode PFC circuit can be realized,
and loop control on the system or protection control on the inductor current peak value can be achieved by using the detected inductor current.
□ the function of detecting the inductor current of the critical-conduction mode PFC circuit may be realized.
□ Engineering & Computer Science (AREA)
□ Power Engineering (AREA)
□ Physics & Mathematics (AREA)
□ General Physics & Mathematics (AREA)
□ Rectifiers (AREA)
□ Dc-Dc Converters (AREA)
A method and device for detecting inductor current of a PFC circuit are disclosed, which relates to the field of power supply technology. The method includes: detecting a voltage on a boost
inductor of a critical-conduction mode PFC circuit, and obtaining an inductor voltage detection signal (S1); converting the inductor voltage detection signal into a voltage signal whose waveform
is consistent with a current waveform of the inductor to serve as an inductor current detection signal, to perform loop protection on the PFC circuit or perform over-current protection on the PFC
circuit by using the inductor current detection signal (S2).
TECHNICAL FIELD
The present document relates to the field of power supply technology, and more particularly, to a method and device for detecting inductor current of a critical-conduction mode Power Factor
Correction (PFC) circuit.
BACKGROUND OF THE RELATED ART
Nowadays, the application of more and more electric and electronic devices aggravates the harmonic pollution of the power grid, in order to better reduce the harmonic waves, the PFC circuit is
widely applied. Meanwhile, the PFC circuit also develops towards a direction of high efficiency and high power density.
In multitudinous PFC circuits, a Critical-conduction Mode (CRM) PFC circuit is extensively applied. With the topology of the totem-pole-type bridgeless PFC circuit as an example, when working in
a critical-conduction mode, it can realize Zero voltage switch (ZVS) or Valley switch (VS) in the full alternating current input range and full load range, and also can simultaneously meet the
requirements of high power density and high efficiency.
FIG. 1 is a structure diagram of a critical-conduction bridge PFC circuit for detecting inductor current of the PFC circuit in the related art. As shown in FIG. 1, the critical-conduction bridge
PFC circuit includes at least two bridge arms connected in parallel between a first connection point A and a second connection point B, herein a first bridge arm includes two diodes connected in
series in the same direction, and a second bridge arm includes two diodes connected in series in the same direction. The critical-conduction bridge PFC circuit also includes one PFC inductor,
there is one switch tube S1 between a third connection point C and the second connection point B, there is one diode D5 between the third connection point C and a fourth connection point D, and
there are a filter capacitor C0 and a load R0 also connected in parallel between the fourth connection point D and the second connection point B.
FIG. 2 is a structure diagram of a totem-pole-type bridgeless PFC circuit for detecting inductor current of the PFC circuit in the related art. As shown in FIG. 2, the totem-pole-type bridgeless
PFC circuit includes at least two bridge arms connected in parallel between a first connection point A and a second connection point B, herein a first bridge arm includes two switch tubes or
diodes connected in series in the same direction, and a second bridge arm includes two switch tubes connected in series in the same direction. The totem-pole-type bridgeless PFC circuit includes
one PFC inductor, and a filter capacitor C0 and a load R0 also connected in parallel between the first connection point A and the second connection point B. When the input voltage is in the
positive half cycle, a diode D2 is always conductive, a switch tube S2 is closed, a switch tube S1 is disconnected, and at this point the current on an inductor L increases from zero to store
energy; after the above energy storage process ends, the switch tube S2 is disconnected, the switch tube S1 is closed, and at this point the current on the inductor L decreases from the peak
value to release energy. When the input alternating voltage is in the negative half cycle, a diode D1 is always conductive, the switch tube S1 is closed, the switch tube S2 is disconnected, and
at this point the current on the inductor L increases from zero to store energy; after the above energy storage process ends, the switch tube S1 is disconnected, the switch tube S2 is closed, and
at this point the current on the inductor L decreases from the peak value to release energy.
However, in the process of applying the critical-conduction bridge PFC circuit and the totem-pole-type bridgeless PFC circuit in the related art to perform practice and research, the
critical-conduction mode power factor correction circuit is required to timely and accurately obtain an inductor current signal which is used for loop control or implementing the inductor current
protection function, but how to obtain the inductor current signal is a problem required to be urgently solved by the researchers.
The object of the embodiments of the present document is to provide a method and device for detecting inductor current of a PFC circuit, which may solve the problem that the inductor current of
the PFC circuit cannot be obtained in a critical-conduction mode.
According to one aspect of the embodiment of the present document, a method for detecting inductor current of a PFC circuit is provided, which includes:
detecting a voltage on a boost inductor of a critical-conduction mode PFC circuit, and obtaining an inductor voltage detection signal;
converting the inductor voltage detection signal into a voltage signal whose waveform is consistent with a current waveform of the inductor voltage to serve as an inductor current detection
signal, to perform loop protection on the PFC circuit or perform over-current protection on the PFC circuit by using the inductor current detection signal.
Preferably, the step of detecting a voltage on a boost inductor of a critical-conduction mode PFC circuit, and obtaining an inductor voltage detection signal includes:
connecting in series a sense resistor with the boost inductor of the PFC circuit, and obtaining a voltage drop in conformity with a current of the boost inductor of the PFC circuit through the
sense resistor; and
taking the obtained voltage drop as the inductor voltage detection signal.
Preferably, the step of detecting a voltage on a boost inductor of a critical-conduction mode PFC circuit, and obtaining an inductor voltage detection signal includes:
enwinding an inductor auxiliary winding on a magnetic core of the boost inductor of the PFC circuit, and obtaining an induced voltage in conformity with the current of the boost inductor of the
PFC circuit on the inductor auxiliary winding; and
taking the obtained induced voltage as the inductor voltage detection signal.
Preferably, obtaining an induced voltage in conformity with the current of the boost inductor of the PFC circuit on the inductor auxiliary winding includes:
through electromagnetic coupling, obtaining a positive voltage corresponding to a rising waveform of the current of the boost inductor of the PFC circuit and a negative voltage corresponding to a
falling waveform of the current of the boost inductor of the PFC circuit on the inductor auxiliary winding.
Preferably, obtaining an induced voltage in conformity with the current of the boost inductor of the PFC circuit on the inductor auxiliary winding includes:
through electromagnetic coupling, obtaining a positive pulse corresponding to a sawtooth wave rising edge of the current of the boost inductor of the PFC circuit and a negative pulse
corresponding to a sawtooth wave falling edge of the current of the boost inductor of the PFC circuit on the inductor auxiliary winding.
Preferably, the step of converting the inductor voltage detection signal into a voltage signal whose waveform is consistent with a current waveform of the inductor to serve as an inductor current
detection signal includes:
performing integral processing on the inductor voltage detection signal, and taking an inductor voltage detection signal on which the integral processing is performed as the inductor current
detection signal, herein a waveform of the inductor current detection signal is consistent with a waveform of a current signal of the boost inductor of the PFC circuit.
Preferably, the step of performing integral processing on the inductor voltage detection signal includes:
performing integral processing on the positive voltage and the negative voltage obtained on the inductor auxiliary winding, and switching on or switching off a corresponding switch tube to charge
or discharge a capacitor in a connected integral circuit;
detecting a voltage across two ends of the capacitor in the integral circuit, and obtaining a voltage detection signal in conformity with the inductor current waveform after the integral
Preferably, the step of performing integral processing on the inductor voltage detection signal includes:
by performing integral processing on the positive pulse and the negative pulse obtained on the inductor auxiliary winding, driving to switch on or switch off a corresponding switch tube to charge
or discharge a capacitor in a connected integral circuit;
detecting a voltage across two ends of the capacitor in the integral circuit, and obtaining a voltage detection signal in conformity with the inductor current waveform after the integral
According to another aspect of the embodiment of the present document, a device for detecting inductor current of a PFC circuit is provided, which includes:
a detection module, configured to detect a voltage on a boost inductor of a critical-conduction mode PFC circuit, and obtain an inductor voltage detection signal; and
a conversion module, configured to convert the inductor voltage detection signal into a voltage signal whose waveform is consistent with a current waveform the inductor to serve as an inductor
current detection signal, to perform loop protection on the PFC circuit or perform over-current protection on the PFC circuit by using the inductor current detection signal.
Preferably, the conversion module includes:
an integral submodule, configured to perform integral processing on the inductor voltage detection signal, and take an inductor voltage detection signal on which the integral processing is
performed as the inductor current detection signal, herein a waveform of the inductor current detection signal is consistent with the waveform of a current signal of the boost inductor of the PFC
Compared with the related art, the beneficial effects of the embodiments of the present document lie in that: in the embodiments of the present document, through the method for detecting the
inductor voltage of the critical-conduction mode PFC circuit and indirectly obtaining the inductor current, the function of detecting the inductor current of the critical-conduction mode PFC
circuit can be realized.
BRIEF DESCRIPTION OF DRAWINGS
FIG. 1 is a structure diagram of a critical-conduction bridge PFC circuit for detecting inductor current of the PFC circuit in the related art;
FIG. 2 is a structure diagram of a totem-pole-type bridgeless PFC circuit for detecting inductor current of the PFC circuit in the related art;
FIG. 3 is a principle diagram of a method for detecting inductor current of a PFC circuit provided in the embodiment of the present document;
FIG. 4 is a schematic diagram of a structure of a device for detecting inductor current of a PFC circuit provided in the embodiment of the present document;
FIG. 5 is a schematic diagram of a device for detecting inductor current of the critical-conduction bridge PFC circuit for detecting the inductor current of the PFC circuit provided in the
embodiment of the present document;
FIG. 6 is a circuit principle diagram of detecting the inductor current of the critical-conduction bridge PFC circuit for detecting the inductor current of the PFC circuit provided in the
embodiment of the present document;
FIG. 7 is an oscillogram corresponding to various parts of a circuit for detecting the inductor current of the critical-conduction bridge PFC circuit for detecting the inductor current of the PFC
circuit provided in the embodiment of the present document;
FIG. 8 is a schematic diagram of a structure of a device for detecting inductor current of the totem-pole-type bridgeless PFC circuit for detecting the inductor current of the PFC circuit
provided in the embodiment of the present document;
FIG. 9 is a first circuit principle diagram of detecting the inductor current of the totem-pole-type bridgeless PFC circuit for detecting the inductor current of the PFC circuit provided in the
embodiment of the present document;
FIG. 10 is a first oscillogram corresponding to various parts of a circuit for detecting the inductor current of the totem-pole-type bridgeless PFC circuit for detecting the inductor current of
the PFC circuit provided in the embodiment of the present document;
FIG. 11 is a second circuit principle diagram of detecting the inductor current of the totem-pole-type bridgeless PFC circuit for detecting the inductor current of the PFC circuit provided in the
embodiment of the present document; and
FIG. 12 is a second oscillogram corresponding to various parts of a circuit for detecting the inductor current of the totem-pole-type bridgeless PFC circuit for detecting the inductor current of
the PFC circuit provided in the embodiment of the present document.
The preferred embodiments of the present document will be described in detail in combination with the accompanying drawings below. It should be understood that the preferred embodiments described
below are only used to describe and explain the present document, which is not used to limit the present document. The embodiments in the present document and the characteristics in the
embodiments can be arbitrarily combined in the case of no conflict.
FIG. 3 is a principle diagram of a method for detecting inductor current of a PFC circuit provided in the embodiment of the present document. As shown in FIG. 3, the specific steps are as
In step S1, a device for detecting the inductor current of the PFC circuit detects a voltage on a boost inductor of a critical-conduction mode PFC circuit, and obtains an inductor voltage
detection signal.
In the step S1, by connecting in series a sense resistor with the boost inductor of the PFC circuit, a voltage drop in conformity with current of the boost inductor of the PFC circuit is obtained
through the sense resistor; and
the obtained voltage drop is taken as the inductor voltage detection signal.
Alternatively, through electromagnetic coupling, a positive voltage corresponding to a rising waveform of the current of the boost inductor of the PFC circuit and a negative voltage corresponding
to a falling waveform of the current of the boost inductor of the PFC circuit are obtained on an inductor auxiliary winding.
Or, the inductor auxiliary winding is enwound on a magnetic core of the boost inductor of the PFC circuit, and an induced voltage in conformity with the current of the boost inductor of the PFC
circuit is obtained on the inductor auxiliary winding; and
the obtained induced voltage is taken as the inductor voltage detection signal.
Alternatively, through electromagnetic coupling, a positive pulse corresponding to a sawtooth wave rising edge of the current of the boost inductor of the PFC circuit and a negative pulse
corresponding to a sawtooth wave falling edge of the current of the boost inductor of the PFC circuit are obtained on the inductor auxiliary winding.
In step S2, the device for detecting the inductor current of the PFC circuit converts the inductor voltage detection signal into a voltage signal whose waveform is consistent with a current
waveform of the inductor to serve as an inductor current detection signal, in order to perform loop protection on the PFC circuit or perform over-current protection on the PFC circuit by using
the inductor current detection signal.
In the step S2, integral processing is performed on the inductor voltage detection signal, and an inductor voltage detection signal on which the integral processing is performed is taken as the
inductor current detection signal, herein the waveform of the inductor current detection signal is consistent with a waveform of a current signal of the boost inductor of the PFC circuit.
Alternatively, the step of performing integral processing on the inductor voltage detection signal includes:
integral processing is performed on the positive voltage and the negative voltage obtained on the inductor auxiliary winding, and it is to switch on or switch off a corresponding switch tube to
charge or discharge a capacitor in a connected integral circuit;
a voltage across two ends of the capacitor in the integral circuit is detected, and a voltage detection signal in conformity with the inductor current waveform after the integral processing is
Alternatively, the step of performing integral processing on the inductor voltage detection signal also includes:
by performing integral processing on the positive pulse and the negative pulse obtained on the inductor auxiliary winding, a corresponding switch tube is driven to be switched on or switched off
to charge or discharge a capacitor in a connected integral circuit;
a voltage across two ends of the capacitor in the integral circuit is detected, and a voltage detection signal in conformity with the inductor current waveform after the integral processing is
FIG. 4 is a schematic diagram of a structure of a device for detecting inductor current of a PFC circuit provided in the embodiment of the present document. As shown in FIG. 4, the device
includes a critical-conduction PFC circuit, an inductor voltage detection unit and an inductor voltage detection signal processing unit.
The critical-conduction PFC circuit includes a related bridge power factor correction circuit and a related power factor correction circuit without a rectifier bridge for which a inductor current
works in a critical continuous mode.
The inductor voltage detection unit is used to detect a voltage on the inductor of the PFC circuit and take the detected inductor voltage signal of the PFC circuit as the input of the inductor
voltage detection signal processing unit. Herein, the inductor voltage detection unit may perform direct detection by using a divider resistor. For example, the inductor voltage of the PFC
circuit is detected by using the divider resistor, the divider resistor is cascaded with the inductor of the PFC circuit, a voltage drop in conformity with the inductor current of the PFC circuit
is detected, and then the detected voltage drop is sent to the inductor voltage detection signal processing unit. Moreover, the inductor auxiliary winding may also be used to perform detection.
For example, the inductor auxiliary winding is preferably a winding coupled on a magnetic core of the inductor of the PFC circuit, voltages at two ends of the inductor of the PFC circuit are
respectively detected, and then the detected voltage signals are respectively sent to the inductor voltage detection signal processing unit.
Herein, the inductor voltage detection unit implements a function of a detection module, that is, the detection module is used for detecting the voltage on the boost inductor of the
critical-conduction mode PFC circuit, and obtains the inductor voltage detection signal.
The inductor voltage detection signal processing unit is used for performing an integral on the detected inductor voltage signal through a certain integral circuit to restore it back to the
inductor current signal. Alternatively, the inductor voltage detection signal processing unit includes one integral circuit, herein, one end of the integral circuit is connected with the inductor
voltage detection unit, and the other end is connected with the earth ground. The voltage on a capacitor in the integral circuit is a detection value of the inductor current, that is, the
inductor voltage detection signal processing unit converts the input inductor voltage signal into the inductor current signal through processing, herein, the output of the inductor voltage
detection signal processing unit is the inductor current signal required to be obtained. In addition, the inductor voltage detection signal processing unit includes some switch tubes used for
enabling all inductor currents of positive and negative half cycles of the input voltage to be output from one output port, and the inductor currents are used for performing loop protection on
the PFC circuit or performing over-current protection on the PFC circuit. The inductor voltage detection signal processing unit also includes some other switch tubes used for performing selection
on inductor voltage signals of the positive and negative half cycles, thereby detecting correct inductor current output signals.
Herein, the inductor voltage detection signal processing unit implements a function of a conversion module, that is, the conversion module is used for converting the inductor voltage detection
signal into a voltage signal whose waveform is consistent with a current waveform of the inductor to serve as an inductor current detection signal, in order to perform loop protection on the PFC
circuit or perform over-current protection on the PFC circuit by using the inductor current detection signal. An integral submodule of the conversion module is used for performing integral
processing on the inductor voltage detection signal, and an inductor voltage detection signal on which the integral processing is performed is taken as the inductor current detection signal,
herein the waveform of the inductor current detection signal is consistent with a waveform of a current signal of the boost inductor of the PFC circuit.
The working principle of the inductor voltage detection signal processing unit includes: the inductor current of the critical-conduction mode PFC circuit rises from zero to a maximum value, and
then falls from the maximum value to zero within one switch period. Therefore, the voltage on the inductor of the PFC circuit will have two times of overturn within one switch period, one is the
overturn from the positive voltage to the negative voltage, and the other is the overturn from the negative voltage to the positive voltage. Since the inductor voltage of the PFC circuit will be
overturned at the maximum value of the inductor current, a resistor and capacitor signal processing circuit in the inductor voltage detection signal processing unit will perform a charging
integral at the stage of the inductor current rising and perform discharging at the stage of the inductor current falling, and the voltage signal on the capacitor is the inductor current signal
whose waveform is consistent with the inductor current waveform of the PFC circuit.
Alternatively, the inductor voltage detection unit sends the detected inductor voltage of the PFC circuit to the inductor voltage detection signal processing unit to be processed, the inductor
voltage detection signal processing unit makes the received inductor voltage signal of the PFC circuit pass through the signal processing circuit composed of the resistor and capacitor and the
switch tubes, and the inductor voltage signal of the PFC circuit is processed into the inductor current signal of the PFC circuit.
FIG. 5 is a schematic diagram of a device for detecting inductor current of the critical-conduction bridge PFC circuit for detecting the inductor current of the PFC circuit provided in the
embodiment of the present document, and FIG. 6 is a circuit principle diagram of detecting the inductor current of the critical-conduction bridge PFC circuit for detecting the inductor current of
the PFC circuit provided in the embodiment of the present document. As shown in FIG. 5 and FIG. 6, one auxiliary winding is added and enwound on a PFC inductor of the related device for detecting
the inductor current of the critical-conduction bridge PFC, and the auxiliary winding of the PFC inductor is coupled on a magnetic core of the PFC inductor. Herein, an end A of the inductor
auxiliary winding of the PFC circuit and an end 1 of the inductor of the PFC circuit are dotted terminals, an end B of the inductor auxiliary winding of the PFC circuit is earthed. The end A of
the inductor auxiliary winding of the PFC circuit is connected with a resistor R, the other end of the resistor R is connected with a capacitor C, and the other end of the capacitor C is earthed.
When the inductor current of the PFC circuit rises, the end A and end B of the auxiliary winding are positive voltages; and when the inductor current falls, the end A and end B of the auxiliary
winding are negative voltages. Therefore, the voltage waveform on the capacitor C is obtained in the way that the inductor auxiliary winding of the PFC circuit uses the RC integral according to
the inductor voltage of the PFC circuit obtained through sampling. The voltage waveform on the capacitor C represents a waveform of the inductor current of the PFC circuit. Then, the voltage
waveform on the capacitor C enters an Analog-to-Digital Converter (ADC) sampling port of a Digital Signal Processor (DSP).
FIG. 7 is an oscillogram corresponding to various parts of a circuit for detecting the inductor current of the critical-conduction bridge PFC circuit for detecting the inductor current of the PFC
circuit provided in the embodiment of the present document. As shown in FIG. 7, a first waveform IL is a waveform of the inductor current of the critical-conduction bridge PFC circuit, a second
waveform VAB is an inductor voltage waveform of the PFC circuit detected on the inductor auxiliary winding of the PFC circuit, and a third waveform VC is a corresponding waveform on the capacitor
C in the RC integral circuit, and the third waveform VC is similar to the waveform of the inductor current of the PFC circuit, that is, the voltage waveform on the capacitor C is the waveform
entering the ADC sampling port of the DSP.
FIG. 8 is a schematic diagram of a structure of a device for detecting inductor current of the totem-pole-type bridgeless PFC circuit for detecting the inductor current of the PFC circuit
provided in the embodiment of the present document, and FIG. 9 is a first circuit principle diagram of detecting the inductor current of the totem-pole-type bridgeless PFC circuit for detecting
the inductor current of the PFC circuit provided in the embodiment of the present document. The device for detecting the inductor current of the totem-pole-type bridgeless PFC circuit according
to the present document will be concretely described through one specific application example in the embodiments of the present document, certainly the device for detecting the inductor current
is not limited to such form of the embodiments of the present document, people skilled in the art may also choose and adopt other similar forms according to the professional knowledge mastered by
them, as long as various functions can be implemented. As shown in FIG. 8 and FIG. 9, two auxiliary windings are added and enwound on the inductor of the PFC circuit of the totem-pole-type
bridgeless PFC circuit, and are respectively used for detecting corresponding inductor voltages when the input alternating voltage is in positive and negative half cycles, and both the two
inductor auxiliary windings of the PFC circuit are coupled on a magnetic core of the inductor of the PFC circuit. Herein, an end A1 of the inductor auxiliary winding 1 of the PFC circuit, an end
B1 of the inductor auxiliary winding 2 of the PFC circuit and an end 1 of the inductor of the PFC circuit are dotted terminals, an end A2 of the inductor auxiliary winding 1 of the PFC circuit
connecting with the end B1 of the inductor auxiliary winding 2 of the PFC circuit is earthed. The end A1 of the inductor auxiliary winding 1 of the PFC circuit is connected with a resistor R1, an
end B2 of the inductor auxiliary winding 2 of the PFC circuit is connected with a resistor R2, the other end of the resistor R1 is connected with a collector of a switch triode VT1 and the other
end of the resistor R2 is connected with a collector of a switch triode VT2, emitters of the switch triodes VT1 and VT2 are earthed, a capacitor C1 is connected in parallel with the collector and
emitter of the switch triode VT1 and a capacitor C2 is connected in parallel with the collector and emitter of the switch triode VT2, the collector of the switch triode VT1 is connected with an
anode of a diode D1 and the collector of the switch triode VT2 is connected with an anode of a diode D2, cathodes of the D1 and D2 are joined together and connected with the ADC sampling port of
the DSP. During the power frequency positive half cycle, the inductor current of the PFC circuit flows from the end 1 to the end 2, when the inductor current rises, the end A1 of the inductor
auxiliary winding 1 of the PFC circuit is a positive voltage, and the end A2 is a negative voltage, but the end B1 of the inductor auxiliary winding 2 of the PFC circuit is a negative voltage,
and the end B2 is a positive voltage; when the inductor current falls, the end A1 of the inductor auxiliary winding 1 of the PFC circuit is a negative voltage, and the end A2 is a positive
voltage, but the end B1 of the inductor auxiliary winding 2 of the PFC circuit is a positive voltage, and the end B2 is a negative voltage. Therefore, during the power frequency positive half
cycle, the switch triode VT2 is driven, so that the switch triode VT2 is always in an on-state, and the switch triode VT1 has no drive voltage, thus the switch triode VT1 is always in an
off-state. Thus, the voltage waveform on the capacitor C1 is obtained in a way that the inductor auxiliary winding 1 of the PFC circuit uses the RC integral according to the inductor voltage of
the PFC circuit obtained through sampling, and the voltage waveform on the capacitor C1 represents a waveform of the inductor current of the PFC circuit, and then enters an ADC sampling port of
the DSP, but since the switch triode VT2 is always in an on-state, the voltage of the capacitor C2 is zero, thus the inductor current of the PFC circuit in the power frequency positive half cycle
may be obtained. During the power frequency negative half cycle, the inductor current of the PFC circuit flows from the end 2 to the end 1, when the inductor current rises, the end A1 of the
inductor auxiliary winding 1 of the PFC circuit is a negative voltage, and the end A2 is a positive voltage, but the end B1 of the inductor auxiliary winding 2 of the PFC circuit is a positive
voltage, and the end B2 is a negative voltage; when the inductor current falls, the end A1 of the inductor auxiliary winding 1 of the PFC circuit is a positive voltage, and the end A2 is a
negative voltage, but the end B1 of the inductor auxiliary winding 2 of the PFC circuit is a negative voltage, and the end B2 is a positive voltage. Therefore, during the power frequency negative
half cycle, the switch triode VT1 is driven, so that the switch triode VT1 is always in an on-state, and the switch triode VT2 has no drive voltage, thus the switch triode VT2 is always in an
off-state. Thus, the voltage waveform on the capacitor C2 is obtained in a way that the inductor auxiliary winding 2 of the PFC circuit uses the RC integral according to the inductor voltage of
the PFC circuit obtained through sampling, and the voltage waveform on the capacitor C2 represents a waveform of the inductor current of the PFC circuit, and then enters an ADC sampling port of
the DSP, but since the switch triode VT1 is always in an on-state, the voltage of the capacitor C1 is zero, thus the inductor current of the PFC circuit in the power frequency negative half cycle
may be obtained.
FIG. 10 is a first oscillogram corresponding to various parts of a circuit for detecting the inductor current of the totem-pole-type bridgeless PFC circuit for detecting the inductor current of
the PFC circuit provided in the embodiment of the present document. As shown in FIG. 10, a first waveform IL is a waveform of the inductor current of the totem-pole-type bridgeless PFC circuit, a
second waveform Vg1 is a drive voltage waveform of the switch triode VT1, a third waveform Vg2 is a drive voltage waveform of the switch triode VT2, a fourth waveform VC1 is a voltage waveform of
the capacitor C1, a fifth waveform VC2 is a voltage waveform of the capacitor C2, and a sixth waveform VC is the final output inductor current waveform of the PFC circuit.
FIG. 11 is a second circuit principle diagram of detecting the inductor current of the totem-pole-type bridgeless PFC circuit for detecting the inductor current of the PFC circuit provided in the
embodiment of the present document. As shown in FIG. 8 and FIG. 11, a single inductor auxiliary winding of the PFC circuit, four switch tubes, one resistor and one capacitor are used to complete
the inductor current sampling of the PFC circuit. During the positive half cycle of the input alternating voltage, switch tubes S2 and S3 are always switched on, switch tubes S1 and S4 are always
switched off, thus when the inductor current of the PFC circuit rises, the inductor auxiliary winding of the PFC circuit charges the capacitor C through the resistor R, and when the inductor
current of the PFC circuit falls, the capacitor C discharges for the inductor auxiliary winding of the PFC circuit through the resistor R, thus the voltage waveform obtained on the capacitor C is
the inductor current waveform of the PFC circuit in the positive half cycle; during the negative half cycle of the input alternating voltage, the switch tubes S1 and S4 are switched on, the
switch tubes S2 and S3 are switched off, when the inductor current of the PFC circuit rises, the inductor auxiliary winding of the PFC circuit charges the capacitor C through the resistor R, and
when the inductor current of the PFC circuit falls, the capacitor C discharges for the inductor auxiliary winding of the PFC circuit through the resistor R, thus the voltage waveform obtained on
the capacitor C is the inductor current waveform of the PFC circuit in the negative half cycle.
FIG. 12 is a second oscillogram corresponding to various parts of a circuit for detecting the inductor current of the totem-pole-type bridgeless PFC circuit for detecting the inductor current of
the PFC circuit provided in the embodiment of the present document. As shown in FIG. 12, a first waveform IL is a waveform of the inductor current of the totem-pole-type bridgeless PFC circuit, a
second waveform Vaux is an inductor voltage waveform detected on the inductor auxiliary winding of the PFC circuit, and a third waveform VC is a corresponding waveform on the capacitor C in the
RC integral circuit, and a shape of the third waveform VC is similar to a waveform of the inductor current of the PFC circuit, that is, the voltage waveform on the capacitor C is the waveform
entering the ADC sampling port of the DSP.
In conclusion, the embodiments of the present document have the following technical effects: in the embodiments of the present document, through the method for detecting the inductor voltage of
the critical-conduction mode PFC circuit and indirectly obtaining the inductor current, the function of detecting the inductor current of the critical-conduction mode PFC circuit can be realized,
and loop control on the system or protection control on the inductor current peak value can be achieved by using the detected inductor current.
Though the embodiments of the present document have been described in detail above, the present document is not limited to this, and people skilled in the art can make various modifications
according to the principle of the present document. Therefore, it should be understood that all the modifications made according to the principle of the resent document fall into the protection
scope of the present document.
In the above technical scheme, through the method for detecting the inductor voltage of the critical-conduction mode PFC circuit and indirectly obtaining the inductor current, the function of
detecting the inductor current of the critical-conduction mode PFC circuit may be realized.
Claims (9)
What is claimed is:
1. A method for detecting inductor current of a power factor correction (PFC) circuit, comprising:
detecting a voltage on a boost inductor of a critical-conduction mode PFC circuit, and obtaining an inductor voltage detection signal;
converting the inductor voltage detection signal into a voltage signal of which a waveform is consistent with a current waveform of the inductor to serve as an inductor current detection signal,
to perform loop protection on the PFC circuit or perform over-current protection on the PFC circuit by using the inductor current detection signal;
wherein, the step of detecting a voltage on a boost inductor of a critical-conduction mode PFC circuit, and obtaining an inductor voltage detection signal comprises:
enwinding two inductor auxiliary windings on a magnetic core of the boost inductor of the PFC circuit for respectively detecting corresponding inductor voltages when an input alternating voltage
is in positive and negative half cycles, and obtaining induced voltages in conformity with the current of the boost inductor of the PFC circuit on the inductor auxiliary windings; and
taking the obtained induced voltages as the inductor voltage detection signal;
wherein an end of a first inductor auxiliary winding of the PFC circuit is connected with a resistor R1, an end of a second inductor auxiliary winding of the PFC circuit is connected with a
resistor R2, the other end of the resistor R1 is connected with a collector of a switch triode VT1 and the other end of the resistor R2 is connected with a collector of a switch triode VT2,
emitters of the switch triodes VT1 and VT2 are earthed, a capacitor C1 is connected in parallel with the collector and emitter of the switch triode VT1 and a capacitor C2 is connected in parallel
with the collector and emitter of the switch triode VT2, the collector of the switch triode VT1 is connected with an anode of a diode D1 and the collector of the switch triode VT2 is connected
with an anode of a diode D2, cathodes of the D1 and D2 are joined together and connected with an Analog-to-Digital Converter (ADC) sampling port of a Digital Signal Processor (DSP).
2. The method according to
claim 1
, wherein, said obtaining induced voltages in conformity with the current of the boost inductor of the PFC circuit on the inductor auxiliary windings comprises:
through electromagnetic coupling, obtaining a positive voltage corresponding to a rising waveform of the current of the boost inductor of the PFC circuit and a negative voltage corresponding to a
falling waveform of the current of the boost inductor of the PFC circuit on the two inductor auxiliary windings respectively.
3. The method according to
claim 2
, wherein, the step of converting the inductor voltage detection signal into a voltage signal of which a waveform is consistent with a current waveform of the inductor to serve as an inductor
current detection signal comprises:
performing integral processing on the inductor voltage detection signal, and taking the inductor voltage detection signal on which the integral processing is performed as the inductor current
detection signal, wherein a waveform of the inductor current detection signal is consistent with a waveform of a current signal of the boost inductor of the PFC circuit.
4. The method according to
claim 3
, wherein, the step of performing integral processing on the inductor voltage detection signal comprises:
performing integral processing on the positive voltage and the negative voltage obtained on the two inductor auxiliary windings respectively, and switching on or switching off a corresponding
switch tube to charge or discharge a capacitor in a connected integral circuit;
detecting a voltage across two ends of the capacitor in the integral circuit, and obtaining the inductor current detection signal in conformity with the inductor current waveform after the
integral processing.
5. The method according to
claim 1
, wherein, said obtaining induced voltages in conformity with the current of the boost inductor of the PFC circuit on the inductor auxiliary windings comprises:
through electromagnetic coupling, obtaining a positive pulse corresponding to a sawtooth wave rising edge of the current of the boost inductor of the PFC circuit and a negative pulse
corresponding to a sawtooth wave falling edge of the current of the boost inductor of the PFC circuit on the two inductor auxiliary windings respectively.
6. The method according to
claim 5
, wherein, the step of converting the inductor voltage detection signal into a voltage signal whose waveform is consistent with a current waveform of the inductor to serve as an inductor current
detection signal comprises:
performing integral processing on the inductor voltage detection signal, and taking the inductor voltage detection signal on which the integral processing is performed as the inductor current
detection signal, wherein a waveform of the inductor current detection signal is consistent with a waveform of a current signal of the boost inductor of the PFC circuit.
7. The method according to
claim 6
, wherein, the step of performing integral processing on the inductor voltage detection signal comprises:
by performing integral processing on the positive pulse and the negative pulse obtained on the two inductor auxiliary windings respectively, driving to switch on or switch off a corresponding
switch tube to charge or discharge a capacitor in a connected integral circuit;
detecting a voltage across two ends of the capacitor in the integral circuit, and obtaining the inductor current detection signal in conformity with the inductor current waveform after the
integral processing.
8. A device for detecting inductor current of a power factor correction (PFC) circuit, comprising:
a detection circuitry, configured to detect a voltage on a boost inductor of a critical-conduction mode PFC circuit, and obtain an inductor voltage detection signal, wherein two inductor
auxiliary windings are enwound on a magnetic core of the boost inductor of the PFC circuit for respectively detecting corresponding inductor voltages when an input alternating voltage is in
positive and negative half cycles, and induced voltages in conformity with the current of the boost inductor of the PFC circuit on the inductor auxiliary windings are obtained and taken as the
inductor voltage detection signal; and
a conversion circuitry, configured to convert the inductor voltage detection signal into a voltage signal of which a waveform is consistent with a current waveform of the inductor to serve as an
inductor current detection signal, to perform loop protection on the PFC circuit or perform over-current protection on the PFC circuit by using the inductor current detection signal;
wherein an end of a first inductor auxiliary winding of the PFC circuit is connected with a resistor R1, an end of a second inductor auxiliary winding of the PFC circuit is connected with a
resistor R2, the other end of the resistor R1 is connected with a collector of a switch triode VT1 and the other end of the resistor R2 is connected with a collector of a switch triode VT2,
emitters of the switch triodes VT1 and VT2 are earthed, a capacitor C1 is connected in parallel with the collector and emitter of the switch triode VT1 and a capacitor C2 is connected in parallel
with the collector and emitter of the switch triode VT2, the collector of the switch triode VT1 is connected with an anode of a diode D1 and the collector of the switch triode VT2 is connected
with an anode of a diode D2, cathodes of the D1 and D2 are joined together and connected with an Analog-to-Digital Converter (ADC) sampling port of a Digital Signal Processor (DSP).
9. The device according to
claim 8
, wherein, the conversion circuitry comprises:
an integral circuitry, configured to perform integral processing on the inductor voltage detection signal, and take the inductor voltage detection signal on which the integral processing is
performed as the inductor current detection signal, wherein a waveform of the inductor current detection signal is consistent with a waveform of a current signal of the boost inductor of the PFC
US15/305,104 2014-04-22 2014-10-28 Method and device for detecting current of inductor of PFC circuit Active US10330711B2 (en)
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