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1070 - Large party
Irena and Sirup are organizing their engagement party next weekend. They want to invite almost everybody. They have just bought a very big round table for this occasion. But they are now
wondering how should they distribute people around the table. Irena claimed that when there are more than K women next to each other, this group will chat together for the whole night and won’t
talk to anybody else.
Sirup had no other choice but to agree with her. However, being a mathematician, he quickly became fascinated by all the possible patterns of men and women around the table.
Problem specification
There will be N people sitting at the round table. Some of them will be men and the rest will be women.
Your task is to count in how many ways it is possible to assign the places to men and women in such a way that there will not be more than K women sitting next to each other.
If one assignment can be made from another one by rotating all the people around the table, we consider them equal (and thus count this assignment only once).
The first line of the input file contains an integer T specifying the number of test cases. Each test case is preceded by a blank line.
The input for each test case consists of a single line that contains the two integers N and K (N,K<=1000).
For each test case output a single line with one integer – the number of ways how to distribute people around the table, modulo 100000007.
sample input
sample output
In the first test case there are two possibilities: MMM or MMW (M is a man, W is a woman). In the second test case there are two more possibilities: MWW and WWW. In the third test case the three
possibilities are: MMMM, MMMW, and MWMW. | {"url":"http://hustoj.org/problem/1070","timestamp":"2024-11-13T16:28:33Z","content_type":"text/html","content_length":"8938","record_id":"<urn:uuid:5cbc656d-6ff7-441c-acbb-6567b82121a8>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00539.warc.gz"} |
The Exotic Vacuum Object (EVO) as the cause of the vacuum reaction. - LENR Forum
We've been through this before. A tachyon field has an IMAGINARY mass, not negative mass. That is a very different thing. Complex numbers are an extremely basic math concept, and one which it is
very obvious you do not understand. Not understanding the difference between negative and imaginary numbers disqualifies you from making up new physics theories.
Besides the fact that these tachyonic fields are simply a math concept with no evidence at all that tachyons exist. BEC's DO NOT have negative mass. That is a ridiculous statement. There is
nothing at all known to have negative mass. | {"url":"https://www.lenr-forum.com/forum/thread/6810-the-exotic-vacuum-object-evo-as-the-cause-of-the-vacuum-reaction/?postID=202057","timestamp":"2024-11-14T13:15:14Z","content_type":"text/html","content_length":"201960","record_id":"<urn:uuid:1f0c486a-eccb-4c11-bfae-8e518c9c5265>","cc-path":"CC-MAIN-2024-46/segments/1730477028657.76/warc/CC-MAIN-20241114130448-20241114160448-00061.warc.gz"} |
Optimal spinneret layout in Von Koch curves of fractal theory based needleless electrospinning process
Needleless electrospinning technology is considered as a better avenue to produce nanofibrous materials at large scale, and electric field intensity and its distribution play an important role in
controlling nanofiber diameter and quality of the nanofibrous web during electrospinning. In the current study, a novel needleless electrospinning method was proposed based on Von Koch curves of
Fractal configuration, simulation and analysis on electric field intensity and distribution in the new electrospinning process were performed with Finite element analysis software, Comsol
Multiphysics 4.4, based on linear and nonlinear Von Koch fractal curves (hereafter called fractal models). The result of simulation and analysis indicated that Second level fractal structure is the
optimal linear electrospinning spinneret in terms of field intensity and uniformity. Further simulation and analysis showed that the circular type of Fractal spinneret has better field intensity and
distribution compared to spiral type of Fractal spinneret in the nonlinear Fractal electrospinning technology. The electrospinning apparatus with the optimal Von Koch fractal spinneret was set up to
verify the theoretical analysis results from Comsol simulation, achieving more uniform electric field distribution and lower energy cost, compared to the current needle and needleless electrospinning
A. Current issues in large-scale electrospinning
Electrospinning technology which was first patented by Formhals in 1934^1 has been widely used to produce nanofibers for diverse applications such as filter media,^2 sensors,^3,4 polymer batteries
and separators,^5–7 drug release,^8,9 tissue scaffolds,^10,11 wound dressing^12,13 and so on. Currently, the electrospinning technique which enables mass production of nanofibers has been classified
as two categories, i.e., multiple needle (or channel/nozzle/capillary) type^14–18 and needleless type. Multineedle electrospinning, in spite of the advantage of higher productivity than single
needle, its disadvantages such as spinning channel clogging and difficulty in cleaning, especially the “End effect”^19 has severely hindered its industrialization due to its generation of
non-uniformity in electric field intensity. Liu^14,15,20–24 et al has made a lot of efforts to minimize or eradicate the End effect phenomenon in multineedle electrospinning towards massive
electrospinning and high productivity of nanofibers, however, the channel clogging remains the barrier towards massive and nonstop electrospinning process.
In recent years, a lot of published works has been devoted to improve the electrospinning productivity, and various needleless geometries, such as roller and wire commercialized by Elmarco with the
brand name Nanospider™,^25 as well as ball, disk, coil, ring^26–29 have been used to produce nanofibers at large scale. Although a series of needleless geometrical spinnerets without tips have been
designed and commercially applied successfully, tipped needleless electro spinnerets are expected to provide with higher electric field intensity and better distribution. In previous work, the
authors designed a sawtooth shaped needleless electrospinneret,^18 with which finer nanofibers were prepared at lab scale at relatively low voltage.
B. Proposed needleless electrospinneret based on Von Koch Fractal Curves
In the current study, a novel tipped needleless electrospinneret was proposed by the authors based on Fractal Theory. The rudiment of Fractal Theory was established on the basis of Koch snowflake,
which is a mathematical curve and one of the earliest fractal curves that has been described, and appeared in a paper titled ‘On a continuous curve without tangents, constructible from elementary
geometry’ in 1904, by the Swedish mathematician Helge von Koch,^30–32 and was developed by B. B. Mandelbrot in 1975,^33 named based on some phenomena of physical relevance, such as the threshold of
percolating, the oceanic coastlines, represented by self-resembling fractal. Fractal structure could be constructed in steps starting from a given shape, and rescaling it continuously down to a
microscopic length scale. Von Koch curve is the simplest geometric figure (Fig. 1) of Fractal structure, formed by an iterative procedure, with the dimension given by:
where N is the number of subdivision at each step and s is the scaling factor.^32
The Koch curve is created by initiating with an equilateral triangle, then recursively altering each line segment according to the following steps as shown in Fig. 1: (0) draw a line segment; (1)
divide the line segment into three parts with equal length, draw an equilateral triangle which has the middle segment as its base and points upward, with 1/3 length of the line segment in step 0 as
its side length, and then erase the base of the regular triangle in the middle of the line in step 0. The geometric structure obtained this way is referred to as the first level Koch curve, as shown
in step 1, Fig. 1; (2) the second level Koch curve is achieved by repeating the same steps based on step 1, as shown in step 2, Fig. 1; and (3) the third level Koch curve could be reached by
repeating the same steps based on step 2, as shown in step 3, Fig. 1.
The Koch curve has an infinite length because each time the steps above are performed on each line segment of the figure there are four times as many line segments, the length of each being one-third
the length of the segments in the previous step. Hence the total length increases by one third and thus the length at step n will be (4/3) n of the original triangle perimeter; therefore, the fractal
dimension is log4/log3≈1.26.^34 The Koch is continuous everywhere but differentiable nowhere.^35 Taking s as the side length, the original triangle area is $s234$. The side length of each successive
small triangle is 1/3 of those in the previous iteration; because the area of the added triangles is proportional to the square of its side length, the area of each triangle added in the nth step is
(1/9)th of that in the (n-1)th step. In each iteration after the first, 4 times as many triangles are added as in the previous iteration; because the first iteration adds 3 triangles, the nth
iteration will add 3 × 4^n−1 triangles. Combining these two formulae gives the iteration formula (Equation (2)):
where A[0] is area of the original triangle. Substituting in $A1‘=43A0$ and expanding yields:
As n goes to infinity, the limit of the sum of the powers of 4/9 is 4/5, and then
Therefore the area of a Koch snowflake is 8/5 of the area of the original triangle, and thus the A[n+1] > A[n] > A[n−1], i.e., the area under the Koch curve of Level 3 is larger than the area under
the Koch curve of Level 2, and the area of the Koch curve of Level 2 is larger than the area under the Koch curve of Level 1, when they have the same structure units.
C. Simulation process and steps using Comsol Multiphysics
The AC/DC module in Comsol Multiphysics software is employed to simulate the electric field intensity during electrospinning process. The electric field intensity simulation via Comsol software is
comprised of the following steps, i.e., (1) set the physical field as electrostatic field; (2) establish the geometric model; (3) set the model parameters, including attribution of materials and
boundary conditions; (4) differentiating grids; (5) define the physics; (5) solving the physics; (6) post treatment (visualization).
D. Objective of the current study
In the current study, linear and nonlinear geometric configurations based on three different levels, including von Koch curves of Levels 1, 2 and 3 will be used as needleless electrostatic
spinnerets, seeking for a new avenue to improve the electric field intensity and distribution of needleless electrospinning heads and hence modify nanofiber and web quality meanwhile reduce energy
consumption and product cost. The finite element analysis software, Comsol Multiphysics is employed to simulate the electric field intensity. UG software was used for spinneret modeling, and Origin
or MS Excel software is used for statistical analysis on the simulated data. The optimal linear Fractal structure will be selected firstly, and then the layout of electrospinnerets constructed with
nonlinear Fractal structure including circular and spiral types will be further discussed based on the results from Comsol software simulation and analysis.
UG 8.0 (Unigraphics NX, Siemens PLM Software) was employed to model the Fractal based electrospinning process. For conveniently modeling the Fractal structure spinnerets and easily understanding the
simulation results, all tips on the second level and third level Fractal structure spinnerets (as Figure 3 shown) are divided into three layers as shown in Figure 2 (tips through red line are defined
as layer 1, tips through yellow line named layer 2, tips through green line called layer 3), with the first level Fractal structure spinneret only having one layer of tips, and then the electric
field intensity and distribution will be simulated with the finite element analysis software after entering all the relevant parameters required into this software. During the process modeling, the
air boundaries were set as infinite boundaries, the receiving plate was set as zero potential, and the spinneret was set as a certain voltage upon the simulation requirements. Then, process meshing
and solving could be performed and the electric field intensity and distribution were obtained.
A. Modeling of linear spinneret
As shown in Figure 3, the linear Fractal structure is used as the spinneret to model the Fractal electrospinning process, taking the second level Fractal structure for instance to address the
modeling process. A metallic sheet having linearly aligned five-spinning unit of second level Fractal structure is used as the spinneret, high voltage DC power source is used in this Fractal based
electrospinning process, which could be positive or negative in polarity.
The detailed information regarding the parameters used in the process of the linear spinneret modeling, such as the dimensions of the Fractal spinneret and the receiving plate, as well as the
receiving distance, is shown in Table I and Figure 4 respectively. The relative dielectric constants for stainless steel (the metallic materials for Fractal spinnerets and receiving plate) and the
surrounding air used in the modeling process are 1.5 F/m^36 and 1.00059 F/m^37 respectively.
TABLE I.
Model . Length of Fractal unit/mm . Base height/mm . Thickness/mm . Receiving distance/mm . Collector dimension/mm3 . Air space dimension/mm3 .
First level 45 22 2 200 700×700×2 1200×1200×1200
Second level 15 22 2 200 700×700×2 1200×1200×1200
Third level 5 22 2 200 700×700×2 1200×1200×1200
Model . Length of Fractal unit/mm . Base height/mm . Thickness/mm . Receiving distance/mm . Collector dimension/mm3 . Air space dimension/mm3 .
First level 45 22 2 200 700×700×2 1200×1200×1200
Second level 15 22 2 200 700×700×2 1200×1200×1200
Third level 5 22 2 200 700×700×2 1200×1200×1200
B. Modeling of nonlinear Fractal spinneret
The circular and spiral types of second-level Fractal spinnerets having five Fractal structure units were modeled and shown in Figure 4. Considering the convenience in practical electrospinning, the
inner radius is designed as 69 mm, external radius 79 mm, and the location numbers of the two nonlinear Fractal spinneret models are also defined and shown in Figure 5(a) and 5(b), respectively.
COMSOL Multiphysics^38 is a numerical simulation software package that is based on PDE (partial differential equation) to simulate by adopting finite element method in research and engineering. The
simulation for electrostatic field formed during electrospinning process follows the Poisson’s Equation (Equation (5)), a partial differential equation (PDE):
where ξ[0] is the relative permittivity of vacuum, ξ[r] is the relative permittivity of the medium, ρ is the space charge density, V is the potential. The PDE (Equation (5)) is the basis for setting
each subdomain.
Apart from the numerical values set in subdomain, determining the boundary conditions is another important step to complete and every boundary is constrained by Equation (6):
where n is the normal vector of interface, D is the dielectric flux density, ρ[s] is the density of surface charge. In the current study, ambient space is confined to a given area in 2D or space in
3D during simulation process, and the electrospinning model is placed in an open space. Therefore, four boundaries linked together as the limited atmosphere are set as the condition of zero charge/
symmetry, aiming to reach the purpose of the infinity of surroundings corresponding to the Equation (7):
Boundaries such as between spinneret and atmosphere without any charges (ρ[s] = 0) are constrained by the continuity condition that is expressed as shown in Equation (8):
The other basic principle for simulating the field intensity is the Field Superposition Theory in which the whole electric field intensity at a random position in the electrostatic field is
equivalent to the vector sum of field intensity generated by all independent point charge. In physics and systems theory, the superposition principle^39 also known as superposition property, states
that, for all linear systems, the net response at a given place and time caused by two or more stimuli is the sum of the responses which would have been caused by each stimulus individually. If the
system is additive and homogeneous, the superposition principle can be applied. If a homogeneous system F(ax) = aF(x), and an additive system satisfies F(x[1] + x[2]) = F(x[1]) + F(x[2]), where a is
a scalar, then a system that simultaneously has the property of homogeneity and additivity satisfies F(a[1]x[1] + a[2]x[2]) = a[1]F(x[1]) + a[2]F(x[2]), with a[1] and a[2] being the scalars.
Therefore, the electrostatic field to be discussed in the current study, which is of the property of homogeneity and additivity at the same time, abides by the operation principle addressed above.
In the current study, Comsol Multiphysics 4.4 software is employed to simulate the field intensity during Fractal electrospinning, and the parameters used for modeling electrospinning process are
listed in Table I, where 30kV voltage is applied to the spinneret models, the values of relative dielectric permittivity for surrounding air and the materials of Fractal spinneret are defined as
1.00059 F/m and 1.5 F/m, respectively.
The Fractal model established with UG 8.0 software based on the structural/dimensional parameters addressed above is transferred to Comsol software, and the attribution of materials and boundary
conditions are also introduced to Comsol software via a stream of commends. The simulation results are obtained after running for a while as short as a few minutes or as long as several hours.
In order to obtain the optimal Koch fractal spinneret which may generate uniformly distributed nanofibers and even spinning jets at low cost, the electric field intensity and distribution of linear
spinnerets were simulated and analyzed with Comsol software based on three layers, including Layer 1, Layer 2, and Layer 3. The definitions for different layers are indicated in Fig. 2, where the
highest tips on the horizontal red line are defined as Layer 1 tips, composed of three tips from Koch curves of Level 1, Level 2 and Level 3 respectively; the Layer 2 tips stand for the tips located
on the horizontal yellow line, including two tips from lever 2 Koch curve, and two tips from Level 3 Koch curve, with each pair of tips symmetrically located around its geometric center on Levels 2
and 3 Koch curves respectively; Layer 3 tips refer to the four tips on the horizontal green line, including two pairs of tips from Levels 2 and 3 Koch curves respectively. There is equal distance
between the tips on Layers 1 and 2, and between the tips on Layers 2 and 3, which is convenient for the subsequent simulating, analyzing and comparing the field intensity and distribution of the Koch
fractal spinneret.
The tipped Koch curves could be made into needleless electrospinning spinnerets in linear or nonlinear type for large-scale nanofiber production, and the latter could include two kinds of spinnerets,
which are circular and spiral types of spinnerets, respectively, as shown in Fig 5(a) and 5(b) respectively. As easily seen from Fig. 5(a) and Fig. 11(c), only the upper half circumferences of the
nonlinear Koch curve fractal spinnerets could generate nanofibers when located in the electric field, which, however, will generate different electric field intensity values at different tip
locations on the upper half circumferences of the nonlinear spinnerets due to different receiving distances. Therefore, the simulation and analysis will be performed based on different tip positions
on the upper side of the nonlinear spinnerets, i.e., the upper side of the circumferences of the two nonlinear spinnerets would be divided into several different sections based on the radius angles
to understand the field intensity at different tip locations of the spinnerets.
Following the similar strategy used in linear spinnerets, the nonlinear spinnerets will be divided into five layers for easy simulation and analysis, i.e., totally 11 lines were drawn from all the
tips towards the center point of the circles (the side views of both spiral spinneret and circular spinneret would be the same round shape, as shown in Fig. 11(c)). The upper half circles of the two
nonlinear spinnerets are divided into 10 parts with unequal areas or radius angles for each part, and the angles being 0°, 13°, 36°, 59°, 72° and 85° respectively. The angles on the right side of the
longitudinal axis are imparted positive values and negative values are given to the angles on the left side of the longitudinal axis, which is the line with radius angle of 0°.
A. Field intensity and distribution of linear Fractal spinneret model
The electric field intensity of the tips at the three linear Fractal structure spinneret models including the first, second and third levels of Koch curves was simulated and calculated using Comsol
Multiphysics 4.4, the resultant electric field profiles were shown in Figure 6, where (a), (b) and (c) stand for the field intensity distribution on the first, second and third levels of Koch curve
Fractal structure spinnerets respectively, (d), (e) and (f) represent the zoomed-in field intensity distribution on single Fractal structure units of the first, second and third levels of Koch curve
Fractal structure spinnerets respectively. The color bar on the right side of Figure 6 indicates the field intensity against color, with red color indicating high field intensity and dark color
representing low field intensity.
1. Overall field intensity distribution
The overall distribution of electric field intensity of electro spinnerets with Fractal structure of Levels 1, 2 and 3 is depicted in Figure 7, in which the spinneret with Level 1 Koch Fractal
structure shows highest field intensity 41.98 kV/cm, and the lowest coefficient of variation (CV) 13.89%, which is because that only 5 tips exist on Level 1 Fractal spinneret, and all the five tips
are located at the same layer (the first layer), resulting in high field intensity and low CV value. The other spinnerets showed lower field intensity and higher CV values due to they have more tips
located at different layers, leading to broader distribution in field intensity and hence lower field intensity but higher CV values.
It could be found by further observation on Figure 7 that the spinneret having Fractal structure of Level 2 exhibits significantly higher field intensity 35.06 kV/cm and slightly higher CV value of
15.94% than the spinneret with Fractal structure of Level 3. As the Fractal spinnerets with Levels 2 and 3 Fractal structure have similar values of CV, and Level 2 Fractal structure shows relatively
higher field intensity, which facilitates producing finer nanofibers at large scale with lower energy consumption, therefore, the spinneret with Level 2 Fractal structure is chosen as the optimal
Fractal spinneret for future practical electrospinning process.
Although Level 1 Fractal spinneret shows the highest field intensity and the lowest CV, fewer spinning jets would be expected during the practical electrospinning because fewer tips exist on the
Level 1 Fractal spinneret, which is not beneficial for nanofiber production at large scale. If including the missing tip locations into Level 1 Fractal spinneret at the corresponding tip positions on
Levels 2 and 3 Fractal spinnerets, Level 1 Fractal structure is not the optimal spinneret any more, as a result, the Level 2 Fractal structure will become the best spinneret instead, which returns
lowest averaged field intensity(9.99kV/cm) and highest CV values(185%) to Level 1 Fractal structure spinneret, turning out to be a fair way to compare the field intensity distribution among the three
levels of Fractal spinnerets.
As there are many more tips existing on the spinnerets having Levels 2 and 3 Fractal structure spinnerets, the simulation and analysis on field intensity are performed based on different layers on
the Fractal structure spinnerets. The definition about the layers on the Fractal structure spinnerets has been shown in Figure 2.
2. Analysis on field intensity distribution of tips in Layer 1
As shown in Fig. 8, the second level structured Fractal spinneret displays highest field intensity (42.09kV/cm) and the lowest CV (8.46%) at tips of Layer 1 among all the three spinnerets. Based on
the layer definition indicated in Figures 5 and 6, and the area calculation method shown in Equation (4), the Level 1 Fractal spinneret should have the least area under its Level 1 Koch curve, and
hence it should have the highest density of charge on its surface area, therefore it should have the highest field intensity. Also, there are no smaller tips beside the both sides of the 5 main tips
on the Level 1 Fractal spinneret to balance the Columbic force from the neighboring main tips, leading to strongest ‘End effect’ at the two edges of Level 1 Fractal spinneret. However, there are
smaller tips located on both sides of the 5 main tips on Levels 2 and 3 Fractal spinnerets, which can balance the Columbic repellence force and hence enhance the field intensity of the main tips
(i.e., the first layer of tips) on their spinnerets, therefore, Level 1 Fractal spinneret does not exhibit the highest field intensity but exhibits the highest CV value, and the field intensity
distribution of which follows the quadratic equation, y = 0.0383x^2-1.8953x+59.236 (R^2 = 0.9904).
The tips in Layer 1 of the Level 2 Fractal spinneret show the highest field intensity and lowest coefficient of deviation due to a pair of tips located on both sides of the 5 main tips which balance
the Columbic force and enhance the electric field force and field intensity, resulting in smallest End effect and CV value of the field intensity of the tips in Layer 1, Level 2 Fractal spinneret,
and the field intensity of which follows the quadratic equation, y = 0.022x^2-1.0895x+52.009 (R^2 = 0.8776). In the case of the Level 3 Fractal spinneret, more tips are located on both sides of its 5
main tips, leading to higher Columbic force among the tips and hence the most severe End effect generated at the two edges of the spinneret, this spinneret shows the highest CV value but the lowest
field intensity of field intensity of the tips in Layer 1, Level 2 Fractal structure, due to its largest surface area and hence smallest areal charge density among the three levels of spinnerets,
with its field intensity distribution following the quadratic equation, y = 0.0213x^2-1.052x+45.253 (R^2 = 0.9831). Therefore, the spinneret with Level 2 Fractal structure is again selected as the
optimal spinneret based on the analysis on field intensity of Layer 1.
3. Analysis on field intensity distribution of Layer 2
Since the second layer does not exist on the spinneret with Level 1 Fractal structure, the current analysis will be conducted between the spinnerets with Level 2 and 3 Fractal structure. The
comparison of field intensity of the two spinnerets is shown in Figure 9, where the spinneret with Level 2 Fractal structure displays significant high field intensity and almost the same CV value
than the spinneret with Level 3 Fractal structure, which is mainly because that there is less surface area and hence higher charge density under the former Fractal curve than the later. Thus, the
spinneret with Level 2 Fractal structure is again considered as the optimal spinneret based on the analysis on field intensity of Layer 2.
4. Analysis on field intensity distribution of Layer 3
Since the third layer does not exist on the spinneret with Level 1 Fractal structure, the current analysis will be limited between the spinnerets with Level 2 and 3 Fractal structure. The comparison
of field intensity of the two spinnerets is shown in Figure 10, where the spinneret with Level 2 Fractal structure displays much higher field intensity and significantly lower CV value than the
spinneret with Level 3 Fractal structure, which is mainly because that there is less surface area and hence higher charge density under the former Fractal curve than the later, and less number of
tips in Level 2 spinneret than Level 3 spinneret results in weaker Columbic repellence among the third layer of tips and hence narrower distribution of its field intensity. Thus, the spinneret with
Level 2 Fractal structure is again considered as the optimal spinneret based on the analysis on field intensity of Layer 3.
After comprehensive analysis on the overall and layer based field intensity and its distribution of the linear Fractal spinnerets, the optimal structure is selected as the spinneret with Level 2
Fractal structure. However, linear type spinneret will generate non-continuous solution feeding issue during the real electrospinning process which should be taken for consideration during the
large-scale electrospinning process. Thus, nonlinear needleless electrospinnerets based on Fractal Theory are proposed to solve the feeding problem, and two typical types of nonlinear Fractal
spinnerets are further analyzed in terms of field intensity, its distribution and the energy efficiency, using Comsol software.
B. Field intensity and distribution of nonlinear Fractal spinneret model
1. Results from Spiral Model
Fig. 11(a) and 11(b) illustrate the electric field distribution on spiral Fractal structure spinnerets. The high intensity electric field is mainly distributed on the semicircle closed to the
collector, and the tips nearest to the collector have the highest electric field intensity, i.e., nearer tips to the collector have the higher electric field intensity. The similar strategy is
employed in analyzing the field intensity along the circular Fractal spinneret, and between the different spinnerets. For easy analysis on the simulation results from Comsol Multiphysics, the tips on
each of the five circular Fractal spinnerets are classified into different layers based on the angles formed between the lines through the tips and the center of the circular Fractal spinnerets, and
the longitudinal axis, where the 0° line is located at, as shown in Fig. 11(c).
The spiral Fractal array structure involved in the discussion include totally five spiral spinnerets marked as Numbers 1, 2, 3, 4 and 5 respectively from the left to the right, and the five spiral
spinnerets are located at symmetric positions with No.3 spiral spinneret as the center, as shown in Fig. 12, and thus No.1 and No. 5 spinnerets share the same averaged field intensity and
distribution, No. 2 and No. 4 spinnerets have the same averaged field intensity and CV values, respectively. Due to the existence of End effect, No.3 spinneret has the lowest field intensity of 19.12
kV/cm, No.2 and No.4 have the medium field intensity of 32.07 kV/cm, and No.1 and No.5 spinnerets have the highest field intensity of 30.90 kV/cm.
2. Results from Circular Model
Fig. 13(a) and 13(b) illustrate the electric field distribution on circular Fractal structure spinnerets. The high intensity electric field is mainly distributed on the semicircle closed to the
collector, and the tips nearest to the collector have the highest electric field intensity, i.e., nearer tips to the collector have the higher electric field intensity. The similar strategy is
employed to analyze the field intensity along the circular Fractal spinneret, and between the different spinnerets. For easy analysis on the simulation results from Comsol Multiphysics, the tips on
each of the five circular Fractal spinnerets are classified into different layers based on the angles formed between the lines through the tips and the center of the circular Fractal spinnerets, and
the longitudinal axis, where the 0° line is located at, as shown in Fig. 11(c).
The circular Fractal array structure involved in the analysis include totally five circular spinnerets with the order numbers of 1, 2, 3, 4 and 5 respectively from the left to the right, and the five
circular spinnerets are located at symmetric positions with No.3 spiral spinneret as the center, as shown in Fig. 14, and thus No.1 and No. 5 spinnerets share the same averaged field intensity and
distribution, No. 2 and No. 4 spinnerets have the same averaged field intensity and CV values, respectively. Due to the existence of End effect, No.3 spinneret has the lowest field intensity of
21.36kV/cm, No.2 and No.4 have the medium field intensity of 22.57kV/cm, and No.1 and No.5 spinnerets have the highest field intensity of 29.97kV/cm.
3. Comparison of circular and spiral Fractal spinnerets
As shown in Figure 15, the circular and spiral Fractal spinneret arrays have the similar field intensity distribution within the angles between -85° through 85°, with the former having higher field
intensity than the later. Additionally, the circular Fractal spinneret shows lower and more regular CV distribution in the angles ranging from -85° to 85°. Therefore, circular Von Kouch Fractal
structure is considered as the better type of mass electrospinning as this type of nonlinear Fractal spinneret is capable of providing higher electric field intensity at narrower distribution range,
which indicates finer and more uniform nanofiber product could be massively produced at lower energy consumption.
In order to verify the feasibility of the modeling Comsol analysis, the needleless electrospinning apparatus composed of the optimal Level 2 Koch fractal spinneret was set up (as shown in Figure 16).
The electrospinning solution containing 8% wt/wt thermoplastic polyurethanes (TPU) was prepared on the magnetic stirrer for 5h at the temperature of 60°C. The nanofiber web was obtained after the
electrospinning experiment (shown in Figure 17(a)) at 20 kV voltage, 20 cm fiber receiving distance and 30rpm rotating rate of spinneret, and the scanning electron microscope (SEM, Hitach S-4800,
Japan) and Image Pro 6.0 (Media Cybernetics Company) were used to measure the surface micromorphology of the TPU nanofiber web and fiber diameter respectively.
It could be easily seen from Figures 17(b) and 17(c) that the surfaces of the nanofibers are regular and smooth, most fibers having diameter of 657nm with the CV value of 16.2% (without using the
traverse mechanism), which have the similar dimension and distribution to the fibers generated from the needle spinnerets, but similar productivity (∼1kg/h) to the currently used needleless
electrospinning technology such as the Nanospider™ (100-120kV)from Elmarco, the spiral coil spinneret (50-65kV) from Deakin University, etc., at lower applied voltage, due to the high field intensity
generated from the tips on the fractal spinneret.
In the current study, a new type of needleless electrospinning technology was proposed towards mass production of nanofibers and their materials products, based on the linear von Koch curves of
Fractal Theory. The feasibility of three levels of Koch curve Fractal structure as a new type of needleless electrospinnerets was investigated and discussed based on the simulation results via the
finite element analysis software, Comsol Multiphysics. The results indicate that the spinneret having the second level Fractal structure is capable of providing the optimal electric field intensity
and its distribution based on the overall analysis and layer based analysis using Comsol Multiphysics. Furthermore, two types of nonlinear Fractal spinneret arrays were constructed using spiral model
and circular model, and their field intensity and distribution profiles were investigated comparatively and systematically. The results show that the circular Fractal structure is the optimal
nonlinear Fractal spinneret. The setup with the optimal fractal spinneret was prepared to testify its feasibility. The result indicated that the new setup could be used in mass electrospinning
process to produce finer and more uniform nanofiber product at much lower voltage and hence lower cost.
Although the second level Koch curve based Fractal structure was selected as the optimal needleless electrospinneret based on the simulation results via Comsol Multiphysics, and the circular Fractal
spinneret array has the potential of producing nanofibers at large scale, the End effect, which leads to higher electric field intensity at the two edges and lower electric field intensity in the
middle of the Fractal spinneret array, remains an issue during the electrospinning process with the Fractal structure as spinnerets. In the future work, End effect problem is about to be solved by
means of several methods including the adjustment of (1) spacing between the circular Fractal spinnerets, (2) the diameter of the circular Fractal spinnerets at two edges, and (3) the applied
voltages at the circular Fractal spinnerets at two edges, and so on. In addition, the effect of the electrospinning parameters such as revolving rate of the spinneret, voltage and collector distance
will be study in order to further improve the quality of the nanofiber web.
This work was supported by National Science Foundation of China, with the project approval No. of 51373121.
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Multiphysics Modeling Using Comsol: A First Principles Approach
Jones & Bartlett Learning | {"url":"https://pubs.aip.org/aip/adv/article/6/6/065223/22580/Optimal-spinneret-layout-in-Von-Koch-curves-of","timestamp":"2024-11-04T20:54:07Z","content_type":"text/html","content_length":"341628","record_id":"<urn:uuid:c01f2075-0f83-4767-96f3-e9eb722f9a0f>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.16/warc/CC-MAIN-20241104194528-20241104224528-00231.warc.gz"} |
Given a day of the week encoded as 0=Sun, 1=Mon, 2=Tue, ...6=Sat, and a boolean indicating if we are on vacation, return a string of the form "7:00" indicating when the alarm clock should ring.
Weekdays, the alarm should be "7:00" and on the weekend it should be "10:00". Unless we are on vacation -- then on weekdays it should be "10:00" and weekends it should be "off".
alarm_clock(1, False) → '7:00'
alarm_clock(5, False) → '7:00'
alarm_clock(0, False) → '10:00'
...Save, Compile, Run (ctrl-enter) | {"url":"https://codingbat.com/prob/p119867","timestamp":"2024-11-05T16:14:41Z","content_type":"text/html","content_length":"4766","record_id":"<urn:uuid:6d4e5e05-855d-40f5-9b44-5a02ee83a775>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00395.warc.gz"} |
University of Pittsburgh
Thursday, November 30, 2023 - 12:00 to 13:00
Abstract or Additional Information
The Fundamental Lemma is a conjecture about identities of p-adic integrals that appear in the trace formula for a reductive group. The conjecture was made by Langlands and Shelstad in the 1980s and
was solved in 2008 by Ngo Bao Chau. For this result, he was awarded a Fields Medal. This talk will describe some of the general context and history of the fundamental lemma, including the work of
Forey, Loeser, and Wyss, who proved a motivic version of the fundamental lemma in August 2023.
Research Area | {"url":"https://www.mathematics.pitt.edu/content/notes-fundamental-lemma","timestamp":"2024-11-08T04:50:09Z","content_type":"text/html","content_length":"92781","record_id":"<urn:uuid:32500cc9-0c00-4ad8-b1ee-9e421444ed0a>","cc-path":"CC-MAIN-2024-46/segments/1730477028025.14/warc/CC-MAIN-20241108035242-20241108065242-00831.warc.gz"} |
Thermal Resistance of Resistors in context of resistor voltage
31 Aug 2024
Title: An Exploration of the Thermal Resistance of Resistors in Relation to Resistor Voltage
Abstract: This study delves into the thermal resistance characteristics of resistors, a crucial aspect in understanding their behavior under varying voltage conditions. The relationship between
resistor voltage and thermal resistance is examined, providing insights into the underlying physics.
Resistors are fundamental components in electronic circuits, playing a pivotal role in controlling current flow. However, as voltage across them increases, so does the heat generated due to Joule
heating (I^2R). This phenomenon necessitates an understanding of thermal resistance, which is the opposition to heat flow through a material.
The thermal resistance (Rth) of a resistor can be described by the following formula:
Rth = ΔT / P
where ΔT is the temperature difference across the resistor and P is the power dissipated in it. The power dissipation (P) itself is given by:
P = V^2 / R
where V is the voltage applied across the resistor and R is its resistance.
Relationship between Resistor Voltage and Thermal Resistance:
As the voltage (V) across a resistor increases, so does the power dissipated in it. Consequently, the temperature difference (ΔT) across the resistor also increases, leading to higher thermal
resistance (Rth). This relationship can be expressed as:
Rth ∝ V^2
The direct proportionality between resistor voltage and thermal resistance underscores the importance of considering thermal effects when designing electronic circuits. As voltage levels increase, so
does the heat generated by resistors, potentially leading to overheating and component failure.
This study has provided an in-depth examination of the thermal resistance characteristics of resistors in relation to resistor voltage. The findings highlight the need for careful consideration of
thermal effects when designing electronic circuits, particularly at high voltage levels.
• [1] “Thermal Resistance of Resistors” by J. Smith et al.
• [2] “Joule Heating and Thermal Resistance” by K. Johnson et al.
Note: The references provided are fictional and for demonstration purposes only.
Related articles for ‘resistor voltage’ :
• Reading: Thermal Resistance of Resistors in context of resistor voltage
Calculators for ‘resistor voltage’ | {"url":"https://blog.truegeometry.com/tutorials/education/71eed169f9912c0d72b32668d3be382d/JSON_TO_ARTCL_Thermal_Resistance_of_Resistors_in_context_of_resistor_voltage.html","timestamp":"2024-11-02T08:05:06Z","content_type":"text/html","content_length":"15819","record_id":"<urn:uuid:16de6a07-caa4-4fe0-8dd5-0ec447772347>","cc-path":"CC-MAIN-2024-46/segments/1730477027709.8/warc/CC-MAIN-20241102071948-20241102101948-00607.warc.gz"} |
Partial differential equation - Wikiwand
In mathematics, a partial differential equation (PDE) is an equation which computes a function between various partial derivatives of a multivariable function.
A visualisation of a solution to the two-dimensional heat equation with temperature represented by the vertical direction and color.
The function is often thought of as an "unknown" to be solved for, similar to how x is thought of as an unknown number to be solved for in an algebraic equation like x^2 − 3x + 2 = 0. However, it is
usually impossible to write down explicit formulae for solutions of partial differential equations. There is correspondingly a vast amount of modern mathematical and scientific research on methods to
numerically approximate solutions of certain partial differential equations using computers. Partial differential equations also occupy a large sector of pure mathematical research, in which the
usual questions are, broadly speaking, on the identification of general qualitative features of solutions of various partial differential equations, such as existence, uniqueness, regularity and
stability.^[1] Among the many open questions are the existence and smoothness of solutions to the Navier–Stokes equations, named as one of the Millennium Prize Problems in 2000.
Partial differential equations are ubiquitous in mathematically oriented scientific fields, such as physics and engineering. For instance, they are foundational in the modern scientific understanding
of sound, heat, diffusion, electrostatics, electrodynamics, thermodynamics, fluid dynamics, elasticity, general relativity, and quantum mechanics (Schrödinger equation, Pauli equation etc.). They
also arise from many purely mathematical considerations, such as differential geometry and the calculus of variations; among other notable applications, they are the fundamental tool in the proof of
the Poincaré conjecture from geometric topology.
Partly due to this variety of sources, there is a wide spectrum of different types of partial differential equations, and methods have been developed for dealing with many of the individual equations
which arise. As such, it is usually acknowledged that there is no "general theory" of partial differential equations, with specialist knowledge being somewhat divided between several essentially
distinct subfields.^[2]
Ordinary differential equations can be viewed as a subclass of partial differential equations, corresponding to functions of a single variable. Stochastic partial differential equations and nonlocal
equations are, as of 2020, particularly widely studied extensions of the "PDE" notion. More classical topics, on which there is still much active research, include elliptic and parabolic partial
differential equations, fluid mechanics, Boltzmann equations, and dispersive partial differential equations.^[3]
A function u(x, y, z) of three variables is "harmonic" or "a solution of the Laplace equation" if it satisfies the condition ${\displaystyle {\frac {\partial ^{2}u}{\partial x^{2}}}+{\frac {\partial
^{2}u}{\partial y^{2}}}+{\frac {\partial ^{2}u}{\partial z^{2}}}=0.}$ Such functions were widely studied in the 19th century due to their relevance for classical mechanics, for example the
equilibrium temperature distribution of a homogeneous solid is a harmonic function. If explicitly given a function, it is usually a matter of straightforward computation to check whether or not it is
harmonic. For instance ${\displaystyle u(x,y,z)={\frac {1}{\sqrt {x^{2}-2x+y^{2}+z^{2}+1}}}}$ and ${\displaystyle u(x,y,z)=2x^{2}-y^{2}-z^{2}}$ are both harmonic while ${\displaystyle u(x,y,z)=\sin
(xy)+z}$ is not. It may be surprising that the two examples of harmonic functions are of such strikingly different form. This is a reflection of the fact that they are not, in any immediate way,
special cases of a "general solution formula" of the Laplace equation. This is in striking contrast to the case of ordinary differential equations (ODEs) roughly similar to the Laplace equation, with
the aim of many introductory textbooks being to find algorithms leading to general solution formulas. For the Laplace equation, as for a large number of partial differential equations, such solution
formulas fail to exist.
The nature of this failure can be seen more concretely in the case of the following PDE: for a function v(x, y) of two variables, consider the equation ${\displaystyle {\frac {\partial ^{2}v}{\
partial x\partial y}}=0.}$ It can be directly checked that any function v of the form v(x, y) = f(x) + g(y), for any single-variable functions f and g whatsoever, will satisfy this condition. This is
far beyond the choices available in ODE solution formulas, which typically allow the free choice of some numbers. In the study of PDEs, one generally has the free choice of functions.
The nature of this choice varies from PDE to PDE. To understand it for any given equation, existence and uniqueness theorems are usually important organizational principles. In many introductory
textbooks, the role of existence and uniqueness theorems for ODE can be somewhat opaque; the existence half is usually unnecessary, since one can directly check any proposed solution formula, while
the uniqueness half is often only present in the background in order to ensure that a proposed solution formula is as general as possible. By contrast, for PDE, existence and uniqueness theorems are
often the only means by which one can navigate through the plethora of different solutions at hand. For this reason, they are also fundamental when carrying out a purely numerical simulation, as one
must have an understanding of what data is to be prescribed by the user and what is to be left to the computer to calculate.
To discuss such existence and uniqueness theorems, it is necessary to be precise about the domain of the "unknown function". Otherwise, speaking only in terms such as "a function of two variables",
it is impossible to meaningfully formulate the results. That is, the domain of the unknown function must be regarded as part of the structure of the PDE itself.
The following provides two classic examples of such existence and uniqueness theorems. Even though the two PDE in question are so similar, there is a striking difference in behavior: for the first
PDE, one has the free prescription of a single function, while for the second PDE, one has the free prescription of two functions.
• Let B denote the unit-radius disk around the origin in the plane. For any continuous function U on the unit circle, there is exactly one function u on B such that ${\displaystyle {\frac {\partial
^{2}u}{\partial x^{2}}}+{\frac {\partial ^{2}u}{\partial y^{2}}}=0}$ and whose restriction to the unit circle is given by U.
• For any functions f and g on the real line R, there is exactly one function u on R × (−1, 1) such that ${\displaystyle {\frac {\partial ^{2}u}{\partial x^{2}}}-{\frac {\partial ^{2}u}{\partial y^
{2}}}=0}$ and with u(x, 0) = f(x) and ∂u/∂y(x, 0) = g(x) for all values of x.
Even more phenomena are possible. For instance, the following PDE, arising naturally in the field of differential geometry, illustrates an example where there is a simple and completely explicit
solution formula, but with the free choice of only three numbers and not even one function.
• If u is a function on R^2 with ${\displaystyle {\frac {\partial }{\partial x}}{\frac {\frac {\partial u}{\partial x}}{\sqrt {1+\left({\frac {\partial u}{\partial x}}\right)^{2}+\left({\frac {\
partial u}{\partial y}}\right)^{2}}}}+{\frac {\partial }{\partial y}}{\frac {\frac {\partial u}{\partial y}}{\sqrt {1+\left({\frac {\partial u}{\partial x}}\right)^{2}+\left({\frac {\partial u}{\
partial y}}\right)^{2}}}}=0,}$ then there are numbers a, b, and c with u(x, y) = ax + by + c.
In contrast to the earlier examples, this PDE is nonlinear, owing to the square roots and the squares. A linear PDE is one such that, if it is homogeneous, the sum of any two solutions is also a
solution, and any constant multiple of any solution is also a solution.
A partial differential equation is an equation that involves an unknown function of ${\displaystyle n\geq 2}$ variables and (some of) its partial derivatives. That is, for the unknown function ${\
displaystyle u:U\rightarrow \mathbb {R} ,}$ of variables ${\displaystyle x=(x_{1},\dots ,x_{n})}$ belonging to the open subset ${\displaystyle U}$ of ${\displaystyle \mathbb {R} ^{n}}$, the ${\
displaystyle k^{th}}$-order partial differential equation is defined as ${\displaystyle F[D^{k}u,D^{k-1}u,\dots ,Du,u,x]=0,}$ where ${\displaystyle F:\mathbb {R} ^{n^{k}}\times \mathbb {R} ^{n^{k-1}}
\dots \times \mathbb {R} ^{n}\times \mathbb {R} \times U\rightarrow \mathbb {R} ,}$ and ${\displaystyle D}$ is the partial derivative operator.
When writing PDEs, it is common to denote partial derivatives using subscripts. For example: ${\displaystyle u_{x}={\frac {\partial u}{\partial x}},\quad u_{xx}={\frac {\partial ^{2}u}{\partial x^
{2}}},\quad u_{xy}={\frac {\partial ^{2}u}{\partial y\,\partial x}}={\frac {\partial }{\partial y}}\left({\frac {\partial u}{\partial x}}\right).}$ In the general situation that u is a function of n
variables, then u[i] denotes the first partial derivative relative to the i-th input, u[ij] denotes the second partial derivative relative to the i-th and j-th inputs, and so on.
The Greek letter Δ denotes the Laplace operator; if u is a function of n variables, then ${\displaystyle \Delta u=u_{11}+u_{22}+\cdots +u_{nn}.}$ In the physics literature, the Laplace operator is
often denoted by ∇^2; in the mathematics literature, ∇^2u may also denote the Hessian matrix of u.
Linear and nonlinear equations
A PDE is called linear if it is linear in the unknown and its derivatives. For example, for a function u of x and y, a second order linear PDE is of the form ${\displaystyle a_{1}(x,y)u_{xx}+a_{2}
(x,y)u_{xy}+a_{3}(x,y)u_{yx}+a_{4}(x,y)u_{yy}+a_{5}(x,y)u_{x}+a_{6}(x,y)u_{y}+a_{7}(x,y)u=f(x,y)}$ where a[i] and f are functions of the independent variables x and y only. (Often the mixed-partial
derivatives u[xy] and u[yx] will be equated, but this is not required for the discussion of linearity.) If the a[i] are constants (independent of x and y) then the PDE is called linear with constant
coefficients. If f is zero everywhere then the linear PDE is homogeneous, otherwise it is inhomogeneous. (This is separate from asymptotic homogenization, which studies the effects of high-frequency
oscillations in the coefficients upon solutions to PDEs.)
Nearest to linear PDEs are semi-linear PDEs, where only the highest order derivatives appear as linear terms, with coefficients that are functions of the independent variables. The lower order
derivatives and the unknown function may appear arbitrarily. For example, a general second order semi-linear PDE in two variables is ${\displaystyle a_{1}(x,y)u_{xx}+a_{2}(x,y)u_{xy}+a_{3}(x,y)u_{yx}
In a quasilinear PDE the highest order derivatives likewise appear only as linear terms, but with coefficients possibly functions of the unknown and lower-order derivatives: ${\displaystyle a_{1}(u_
{x},u_{y},u,x,y)u_{xx}+a_{2}(u_{x},u_{y},u,x,y)u_{xy}+a_{3}(u_{x},u_{y},u,x,y)u_{yx}+a_{4}(u_{x},u_{y},u,x,y)u_{yy}+f(u_{x},u_{y},u,x,y)=0}$ Many of the fundamental PDEs in physics are quasilinear,
such as the Einstein equations of general relativity and the Navier–Stokes equations describing fluid motion.
A PDE without any linearity properties is called fully nonlinear, and possesses nonlinearities on one or more of the highest-order derivatives. An example is the Monge–Ampère equation, which arises
in differential geometry.^[5]
Second order equations
The elliptic/parabolic/hyperbolic classification provides a guide to appropriate initial- and boundary conditions and to the smoothness of the solutions. Assuming u[xy] = u[yx], the general linear
second-order PDE in two independent variables has the form ${\displaystyle Au_{xx}+2Bu_{xy}+Cu_{yy}+\cdots {\mbox{(lower order terms)}}=0,}$ where the coefficients A, B, C... may depend upon x and y.
If A^2 + B^2 + C^2 > 0 over a region of the xy-plane, the PDE is second-order in that region. This form is analogous to the equation for a conic section: ${\displaystyle Ax^{2}+2Bxy+Cy^{2}+\cdots =
More precisely, replacing ∂[x] by X, and likewise for other variables (formally this is done by a Fourier transform), converts a constant-coefficient PDE into a polynomial of the same degree, with
the terms of the highest degree (a homogeneous polynomial, here a quadratic form) being most significant for the classification.
Just as one classifies conic sections and quadratic forms into parabolic, hyperbolic, and elliptic based on the discriminant B^2 − 4AC, the same can be done for a second-order PDE at a given point.
However, the discriminant in a PDE is given by B^2 − AC due to the convention of the xy term being 2B rather than B; formally, the discriminant (of the associated quadratic form) is (2B)^2 − 4AC = 4(
B^2 − AC), with the factor of 4 dropped for simplicity.
1. B^2 − AC < 0 (elliptic partial differential equation): Solutions of elliptic PDEs are as smooth as the coefficients allow, within the interior of the region where the equation and solutions are
defined. For example, solutions of Laplace's equation are analytic within the domain where they are defined, but solutions may assume boundary values that are not smooth. The motion of a fluid at
subsonic speeds can be approximated with elliptic PDEs, and the Euler–Tricomi equation is elliptic where x < 0. By change of variables, the equation can always be expressed in the form: ${\
displaystyle u_{xx}+u_{yy}+\cdots =0,}$where x and y correspond to changed variables. Thus justifies Laplace equation as an example of this type.^[6]
2. B^2 − AC = 0 (parabolic partial differential equation): Equations that are parabolic at every point can be transformed into a form analogous to the heat equation by a change of independent
variables. Solutions smooth out as the transformed time variable increases. The Euler–Tricomi equation has parabolic type on the line where x = 0. By change of variables, the equation can always
be expressed in the form: ${\displaystyle u_{xx}+\cdots =0,}$where x correspond to changed variables. Thus justifies heat equation, which are of form ${\textstyle u_{t}-u_{xx}+\cdots =0}$, as an
example of this type.^[6]
3. B^2 − AC > 0 (hyperbolic partial differential equation): hyperbolic equations retain any discontinuities of functions or derivatives in the initial data. An example is the wave equation. The
motion of a fluid at supersonic speeds can be approximated with hyperbolic PDEs, and the Euler–Tricomi equation is hyperbolic where x > 0. By change of variables, the equation can always be
expressed in the form: ${\displaystyle u_{xx}-u_{yy}+\cdots =0,}$where x and y correspond to changed variables. Thus justifies wave equation as an example of this type.^[6]
If there are n independent variables x[1], x[2 ], …, x[n], a general linear partial differential equation of second order has the form ${\displaystyle Lu=\sum _{i=1}^{n}\sum _{j=1}^{n}a_{i,j}{\frac
{\partial ^{2}u}{\partial x_{i}\partial x_{j}}}\quad +{\text{lower-order terms}}=0.}$
The classification depends upon the signature of the eigenvalues of the coefficient matrix a[i,j].
1. Elliptic: the eigenvalues are all positive or all negative.
2. Parabolic: the eigenvalues are all positive or all negative, except one that is zero.
3. Hyperbolic: there is only one negative eigenvalue and all the rest are positive, or there is only one positive eigenvalue and all the rest are negative.
4. Ultrahyperbolic: there is more than one positive eigenvalue and more than one negative eigenvalue, and there are no zero eigenvalues.^[7]
The theory of elliptic, parabolic, and hyperbolic equations have been studied for centuries, largely centered around or based upon the standard examples of the Laplace equation, the heat equation,
and the wave equation.
However, the classification only depends on linearity of the second-order terms and is therefore applicable to semi- and quasilinear PDEs as well. The basic types also extend to hybrids such as the
Euler–Tricomi equation; varying from elliptic to hyperbolic for different regions of the domain, as well as higher-order PDEs, but such knowledge is more specialized.
Systems of first-order equations and characteristic surfaces
The classification of partial differential equations can be extended to systems of first-order equations, where the unknown u is now a vector with m components, and the coefficient matrices A[ν] are
m by m matrices for ν = 1, 2, …, n. The partial differential equation takes the form ${\displaystyle Lu=\sum _{u =1}^{n}A_{u }{\frac {\partial u}{\partial x_{u }}}+B=0,}$ where the coefficient
matrices A[ν] and the vector B may depend upon x and u. If a hypersurface S is given in the implicit form ${\displaystyle \varphi (x_{1},x_{2},\ldots ,x_{n})=0,}$ where φ has a non-zero gradient,
then S is a characteristic surface for the operator L at a given point if the characteristic form vanishes: ${\displaystyle Q\left({\frac {\partial \varphi }{\partial x_{1}}},\ldots ,{\frac {\partial
\varphi }{\partial x_{n}}}\right)=\det \left[\sum _{u =1}^{n}A_{u }{\frac {\partial \varphi }{\partial x_{u }}}\right]=0.}$
The geometric interpretation of this condition is as follows: if data for u are prescribed on the surface S, then it may be possible to determine the normal derivative of u on S from the differential
equation. If the data on S and the differential equation determine the normal derivative of u on S, then S is non-characteristic. If the data on S and the differential equation do not determine the
normal derivative of u on S, then the surface is characteristic, and the differential equation restricts the data on S: the differential equation is internal to S.
1. A first-order system Lu = 0 is elliptic if no surface is characteristic for L: the values of u on S and the differential equation always determine the normal derivative of u on S.
2. A first-order system is hyperbolic at a point if there is a spacelike surface S with normal ξ at that point. This means that, given any non-trivial vector η orthogonal to ξ, and a scalar
multiplier λ, the equation Q(λξ + η) = 0 has m real roots λ[1], λ[2], …, λ[m]. The system is strictly hyperbolic if these roots are always distinct. The geometrical interpretation of this
condition is as follows: the characteristic form Q(ζ) = 0 defines a cone (the normal cone) with homogeneous coordinates ζ. In the hyperbolic case, this cone has nm sheets, and the axis ζ = λξ
runs inside these sheets: it does not intersect any of them. But when displaced from the origin by η, this axis intersects every sheet. In the elliptic case, the normal cone has no real sheets.
Separation of variables
Linear PDEs can be reduced to systems of ordinary differential equations by the important technique of separation of variables. This technique rests on a feature of solutions to differential
equations: if one can find any solution that solves the equation and satisfies the boundary conditions, then it is the solution (this also applies to ODEs). We assume as an ansatz that the dependence
of a solution on the parameters space and time can be written as a product of terms that each depend on a single parameter, and then see if this can be made to solve the problem.^[8]
In the method of separation of variables, one reduces a PDE to a PDE in fewer variables, which is an ordinary differential equation if in one variable – these are in turn easier to solve.
This is possible for simple PDEs, which are called separable partial differential equations, and the domain is generally a rectangle (a product of intervals). Separable PDEs correspond to diagonal
matrices – thinking of "the value for fixed x" as a coordinate, each coordinate can be understood separately.
This generalizes to the method of characteristics, and is also used in integral transforms.
Method of characteristics
The characteristic surface in n = 2-dimensional space is called a characteristic curve. In special cases, one can find characteristic curves on which the first-order PDE reduces to an ODE – changing
coordinates in the domain to straighten these curves allows separation of variables, and is called the method of characteristics.
More generally, applying the method to first-order PDEs in higher dimensions, one may find characteristic surfaces.
Integral transform
An integral transform may transform the PDE to a simpler one, in particular, a separable PDE. This corresponds to diagonalizing an operator.
An important example of this is Fourier analysis, which diagonalizes the heat equation using the eigenbasis of sinusoidal waves.
If the domain is finite or periodic, an infinite sum of solutions such as a Fourier series is appropriate, but an integral of solutions such as a Fourier integral is generally required for infinite
domains. The solution for a point source for the heat equation given above is an example of the use of a Fourier integral.
Change of variables
Often a PDE can be reduced to a simpler form with a known solution by a suitable change of variables. For example, the Black–Scholes equation ${\displaystyle {\frac {\partial V}{\partial t}}+{\tfrac
{1}{2}}\sigma ^{2}S^{2}{\frac {\partial ^{2}V}{\partial S^{2}}}+rS{\frac {\partial V}{\partial S}}-rV=0}$ is reducible to the heat equation ${\displaystyle {\frac {\partial u}{\partial \tau }}={\frac
{\partial ^{2}u}{\partial x^{2}}}}$ by the change of variables^[10] {\displaystyle {\begin{aligned}V(S,t)&=v(x,\tau ),\\[5px]x&=\ln \left(S\right),\\[5px]\tau &={\tfrac {1}{2}}\sigma ^{2}(T-t),\\
[5px]v(x,\tau )&=e^{-\alpha x-\beta \tau }u(x,\tau ).\end{aligned}}}
Fundamental solution
Inhomogeneous equations can often be solved (for constant coefficient PDEs, always be solved) by finding the fundamental solution (the solution for a point source ${\displaystyle P(D)u=\delta }$),
then taking the convolution with the boundary conditions to get the solution.
This is analogous in signal processing to understanding a filter by its impulse response.
Superposition principle
The superposition principle applies to any linear system, including linear systems of PDEs. A common visualization of this concept is the interaction of two waves in phase being combined to result in
a greater amplitude, for example sin x + sin x = 2 sin x. The same principle can be observed in PDEs where the solutions may be real or complex and additive. If u[1] and u[2] are solutions of linear
PDE in some function space R, then u = c[1]u[1] + c[2]u[2] with any constants c[1] and c[2] are also a solution of that PDE in the same function space.
Methods for non-linear equations
There are no generally applicable methods to solve nonlinear PDEs. Still, existence and uniqueness results (such as the Cauchy–Kowalevski theorem) are often possible, as are proofs of important
qualitative and quantitative properties of solutions (getting these results is a major part of analysis). Computational solution to the nonlinear PDEs, the split-step method, exist for specific
equations like nonlinear Schrödinger equation.
Nevertheless, some techniques can be used for several types of equations. The h-principle is the most powerful method to solve underdetermined equations. The Riquier–Janet theory is an effective
method for obtaining information about many analytic overdetermined systems.
The method of characteristics can be used in some very special cases to solve nonlinear partial differential equations.^[11]
In some cases, a PDE can be solved via perturbation analysis in which the solution is considered to be a correction to an equation with a known solution. Alternatives are numerical analysis
techniques from simple finite difference schemes to the more mature multigrid and finite element methods. Many interesting problems in science and engineering are solved in this way using computers,
sometimes high performance supercomputers.
Lie group method
From 1870 Sophus Lie's work put the theory of differential equations on a more satisfactory foundation. He showed that the integration theories of the older mathematicians can, by the introduction of
what are now called Lie groups, be referred, to a common source; and that ordinary differential equations which admit the same infinitesimal transformations present comparable difficulties of
integration. He also emphasized the subject of transformations of contact.
A general approach to solving PDEs uses the symmetry property of differential equations, the continuous infinitesimal transformations of solutions to solutions (Lie theory). Continuous group theory,
Lie algebras and differential geometry are used to understand the structure of linear and nonlinear partial differential equations for generating integrable equations, to find its Lax pairs,
recursion operators, Bäcklund transform and finally finding exact analytic solutions to the PDE.
Symmetry methods have been recognized to study differential equations arising in mathematics, physics, engineering, and many other disciplines.
The three most widely used numerical methods to solve PDEs are the finite element method (FEM), finite volume methods (FVM) and finite difference methods (FDM), as well other kind of methods called
meshfree methods, which were made to solve problems where the aforementioned methods are limited. The FEM has a prominent position among these methods and especially its exceptionally efficient
higher-order version hp-FEM. Other hybrid versions of FEM and Meshfree methods include the generalized finite element method (GFEM), extended finite element method (XFEM), spectral finite element
method (SFEM), meshfree finite element method, discontinuous Galerkin finite element method (DGFEM), element-free Galerkin method (EFGM), interpolating element-free Galerkin method (IEFGM), etc.
Finite element method
The finite element method (FEM) (its practical application often known as finite element analysis (FEA)) is a numerical technique for finding approximate solutions of partial differential equations
(PDE) as well as of integral equations.^[14]^[15] The solution approach is based either on eliminating the differential equation completely (steady state problems), or rendering the PDE into an
approximating system of ordinary differential equations, which are then numerically integrated using standard techniques such as Euler's method, Runge–Kutta, etc.
Finite difference method
Finite-difference methods are numerical methods for approximating the solutions to differential equations using finite difference equations to approximate derivatives.
Finite volume method
Similar to the finite difference method or finite element method, values are calculated at discrete places on a meshed geometry. "Finite volume" refers to the small volume surrounding each node point
on a mesh. In the finite volume method, surface integrals in a partial differential equation that contain a divergence term are converted to volume integrals, using the divergence theorem. These
terms are then evaluated as fluxes at the surfaces of each finite volume. Because the flux entering a given volume is identical to that leaving the adjacent volume, these methods conserve mass by
Neural networks
Physics informed neural networks have been used to solve partial differential equations in both forward and inverse problems in a data driven manner.
One example is the reconstructing fluid flow governed by the
Navier-Stokes equations
. Using physics informed neural networks does not require the often expensive mesh generation that conventional
methods rely on.
Weak solutions are functions that satisfy the PDE, yet in other meanings than regular sense. The meaning for this term may differ with context, and one of the most commonly used definitions is based
on the notion of distributions.
An example for the definition of a weak solution is as follows:
Consider the boundary-value problem given by: {\displaystyle {\begin{aligned}Lu&=f\quad {\text{in }}U,\\u&=0\quad {\text{on }}\partial U,\end{aligned}}} where ${\displaystyle Lu=-\sum _{i,j}\partial
_{j}(a^{ij}\partial _{i}u)+\sum _{i}b^{i}\partial _{i}u+cu}$ denotes a second-order partial differential operator in divergence form.
We say a ${\displaystyle u\in H_{0}^{1}(U)}$ is a weak solution if ${\displaystyle \int _{U}[\sum _{i,j}a^{ij}(\partial _{i}u)(\partial _{j}v)+\sum _{i}b^{i}(\partial _{i}u)v+cuv]dx=\int _{U}fvdx}$
for every ${\displaystyle v\in H_{0}^{1}(U)}$, which can be derived by a formal integral by parts.
An example for a weak solution is as follows: ${\displaystyle \phi (x)={\frac {1}{4\pi }}{\frac {1}{|x|}}}$ is a weak solution satisfying ${\displaystyle abla ^{2}\phi =\delta {\text{ in }}R^{3}}$ in
distributional sense, as formally, ${\displaystyle \int _{R^{3}}abla ^{2}\phi (x)\psi (x)dx=\int _{R^{3}}\phi (x)abla ^{2}\psi (x)dx=\psi (0){\text{ for }}\psi \in C_{c}^{\infty }(R^{3}).}$
Well-posedness refers to a common schematic package of information about a PDE. To say that a PDE is well-posed, one must have:
• an existence and uniqueness theorem, asserting that by the prescription of some freely chosen functions, one can single out one specific solution of the PDE
• by continuously changing the free choices, one continuously changes the corresponding solution
This is, by the necessity of being applicable to several different PDE, somewhat vague. The requirement of "continuity", in particular, is ambiguous, since there are usually many inequivalent means
by which it can be rigorously defined. It is, however, somewhat unusual to study a PDE without specifying a way in which it is well-posed.
The energy method
The energy method is a mathematical procedure that can be used to verify well-posedness of initial-boundary-value-problems (IBVP).^[20] In the following example the energy method is used to decide
where and which boundary conditions should be imposed such that the resulting IBVP is well-posed. Consider the one-dimensional hyperbolic PDE given by ${\displaystyle {\frac {\partial u}{\partial t}}
+\alpha {\frac {\partial u}{\partial x}}=0,\quad x\in [a,b],t>0,}$
where ${\displaystyle \alpha eq 0}$ is a constant and ${\displaystyle u(x,t)}$ is an unknown function with initial condition ${\displaystyle u(x,0)=f(x)}$. Multiplying with ${\displaystyle u}$ and
integrating over the domain gives ${\displaystyle \int _{a}^{b}u{\frac {\partial u}{\partial t}}\mathrm {d} x+\alpha \int _{a}^{b}u{\frac {\partial u}{\partial x}}\mathrm {d} x=0.}$
Using that ${\displaystyle \int _{a}^{b}u{\frac {\partial u}{\partial t}}\mathrm {d} x={\frac {1}{2}}{\frac {\partial }{\partial t}}\|u\|^{2}\quad {\text{and}}\quad \int _{a}^{b}u{\frac {\partial u}
{\partial x}}\mathrm {d} x={\frac {1}{2}}u(b,t)^{2}-{\frac {1}{2}}u(a,t)^{2},}$ where integration by parts has been used for the first relationship, we get ${\displaystyle {\frac {\partial }{\partial
t}}\|u\|^{2}+\alpha u(b,t)^{2}-\alpha u(a,t)^{2}=0.}$
Here ${\displaystyle \|\cdot \|}$ denotes the standard ${\displaystyle L^{2}}$ norm. For well-posedness we require that the energy of the solution is non-increasing, i.e. that ${\textstyle {\frac {\
partial }{\partial t}}\|u\|^{2}\leq 0}$, which is achieved by specifying ${\displaystyle u}$ at ${\displaystyle x=a}$ if ${\displaystyle \alpha >0}$ and at ${\displaystyle x=b}$ if ${\displaystyle \
alpha <0}$. This corresponds to only imposing boundary conditions at the inflow. Well-posedness allows for growth in terms of data (initial and boundary) and thus it is sufficient to show that ${\
textstyle {\frac {\partial }{\partial t}}\|u\|^{2}\leq 0}$ holds when all data are set to zero.
Existence of local solutions
The Cauchy–Kowalevski theorem for Cauchy initial value problems essentially states that if the terms in a partial differential equation are all made up of analytic functions and a certain
transversality condition is satisfied (the hyperplane or more generally hypersurface where the initial data are posed must be non-characteristic with respect to the partial differential operator),
then on certain regions, there necessarily exist solutions which are as well analytic functions. This is a fundamental result in the study of analytic partial differential equations. Surprisingly,
the theorem does not hold in the setting of smooth functions; an example discovered by Hans Lewy in 1957 consists of a linear partial differential equation whose coefficients are smooth (i.e., have
derivatives of all orders) but not analytic for which no solution exists. So the Cauchy-Kowalevski theorem is necessarily limited in its scope to analytic functions.
Some common PDEs
Types of boundary conditions
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1.1: To c or Not to c
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Einstein’s famous special theory of relativity has gained unquestioned acceptance in the scientific world. It has been proved in countless ways and is the foundation upon which gravitation and
high-energy quantum physics are based. Is it something that a “normal” person can understand? Relativity forces us to abandon our ideas about time, which is a hard thing to do, but the basic
mathematics of it are relatively simple—just a picture away.
The picture depends on extending something you know about on an intuitive level. Suppose I tell you that you have an hour to drive and that you must go south at 50 miles/hour. Can you tell me where
you will end up?* Suppose I tell you to go south at 100 miles/ hour: Where will you find yourself after one hour?^† Suppose I tell you to go south for two hours at 50 miles/hour, where do you end up?
^‡ So you see, you know the distance traveled if you know how fast you are going and how much time is allowed. That is,
\[\text { distance traveled is velocity } \times \text { time. } \nonumber\]
Let’s check this against our intuition for the last case:
\[100 \text { miles }=50 \text { miles/hour } \times 2 \text { hours. }^§ \nonumber\]
But I’m a slow typist so I want to abbreviate this understanding by using the first letter of each word:
\[d=v t. \nonumber\]
Any heart attacks yet? Well that one relation, which you already understand enough to do the calculation in your head, is all the math you need to do much of relativity!
* Fifty miles south of your start. In my case, Salem, Oregon.
† On the way to jail.
‡ One hundred miles south of your start. In my case, Eugene, Oregon.
§ Of course, the 50 × 2 on the right-hand side gives 100, the numerical part of the result on the left-hand side, but also notice that the hour in “2 hours” cancels the hour in “miles/hour” (or
“miles per hour”) leaving just “miles” as the unit of the result—correctly a unit of distance. Another way to say this is “hour/hour = 1,” and 1 times anything just gives that thing back.
It’s About Time!
You are at the airport trying to catch a plane you are late for. As you run onto Concourse E, the announcer says the jetway door will be closing in 1 minute, and you see the door 300 meters (328
yards) away. With your luggage you can only run at 3 meters per second (3 m/s is 6 miles/hour), which means that it will take you 100 seconds to get to the plane, but you only have 60 seconds before
the door closes. You notice a sliding walkway (slidewalk) ahead with people standing still relative to it, yet moving at the same velocity you are (3 m/s) relative to the hallway. What would you do?
If you run onto the slidewalk, past the people standing still relative to the slidewalk, how fast would you be moving relative to the hallway? Would this solve your problem?*
You know that these sorts of gadgets are irresistible to children. What is the first thing a child will do on a slidewalk (or on an escalator)? Run in the “wrong” direction at 3 m/s. If she does so,
what would her velocity be relative to the hallway?^† She is playing with the idea that in our everyday experience, velocities add and subtract. (This is why we must use the term velocity rather than
speed whenever we wish to be mindful of directions, since the latter is only the size of the velocity with no indication of direction. If direction is immaterial, speed and velocity are often used
During the late 1800s, physicists were trying to find how fast the Earth was moving through a cosmic substance that they called ether.^‡ They thought that they could determine the Earth’s speed (like
the slidewalk’s) by comparing the speed relative to the ether (akin to the hallway) of one beam of light cast one way from the Earth to another beam of light cast sideways from that direction. But
every time they tried this, they found no difference for the two cases.
This was a big mystery. Albert Einstein sorted it out it in 1905 by asking what the consequences would be for our experiences if the speed of light were the same no matter what the state of motion of
the object that projects the beam is.^§ He also said that there is no absolute state of motion (or absolute reference frame) to which we can compare our motion and thus no need to pose the existence
of the “ether.” He showed that the major consequence of the speed of light being the same in all frames of reference is that the passage of time depends on the motion of the viewer.
To show the general idea, consider one of Einstein’s thought experiments. Suppose you were reading this book while you sat on a train moving at the speed of light away from a clock fixed on a tower.
If you passed the tower precisely at twelve o’clock, what time would the clock face show 10 minutes later as you look back at the light coming from it?^¶ Does that match the 12:10 showing on your
wristwatch (or smartphone)? What time would it show three hours later? Not three o’clock, like on your watch? What about three days or three months later? Indeed, since you are moving at the same
speed that the light is, it would never overtake you with new information. It would always read twelve o’clock.
It turns out that matter like you and the train cannot ever reach the speed of light for reasons we will show a bit later. So we have to modify the thought experiment a bit and say you are reading
this book while you sit on a train moving at the 99.86% of the speed of light away from a clock fixed on a tower. If you passed the tower precisely at twelve o’clock, what time would the clock face
show 10 minutes later as you look back at the light coming from it? Well, it would be slowing overtaking you and would read something like 12:03, but it would still not match the 12:10 showing on
your wristwatch. The picture we develop in the next section will allow us to determine whether the clock shows 12:01 or 12:00.32 or 12:03.
We think of light as the colors red through violet in the color spectrum. But consider, why is it that you wear sunglasses that block out ultraviolet radiation? UV is a higher-energy form of light
that the dyes in our eyes do not register but that will interact with (burn) our bodies. X-rays, gamma rays, and so on are even higher-energy forms of light. On the other side of the visible spectrum
are the lower-energy forms of light starting with infrared, microwave, radio waves, and radar. They all travel at the same speed. The speed of light is always represented by the letter \(c\) and has
been measured to be \(c\) = 186,000 miles/second.
Once we know the value of \(c\), we can measure the distance to a spaceship by counting how many seconds pass between when we send it a radio wave and when it returns the wave to us. Likewise, when
we bounce a laser beam off the Moon, we notice that there is a 1.28-second delay for each leg of the trip. That means that the distance between Earth and the Moon is
\[d=c t \nonumber\]
\[d=186,000 \text { miles } / \text { second } \times 1.28 \text { seconds }=238,000 \text { miles. } \nonumber\]
* Your velocity of 3 m/s would add to that of the slidewalk, 3 m/s, to give 6 m/s. At this velocity, you need only 50 seconds to travel 300 m and have 10 extra seconds to casually walk onboard the
† It would be 0 m/s; she would make no progress, and that is why it is fun.
‡ This term was a holdover from the Ptolemaic model of the Solar System, where ether was a substance said to fill the celestial spheres.
§ A. Einstein, Ann. Phys. (Leipzig) 17, 891 (1905); 20, 371 (1906).
¶ Twelve o’clock.
Did You Ever Wish You Had More Time?
Consider a beam of light bouncing between the mirrors of a spaceship moving, perpendicular to the light beam, at speed \(v\) relative to the ground, as seen in Figure \(\PageIndex{1}\).
To someone sitting in the ship, the distance the beam must travel is simply the height H of the ship. The distance is the velocity \(c\) times the time of travel \(\tau\) between bounces as measured
in the ship or
\[H=c \tau. \nonumber\]
Note that we use the Greek letter tau (\(\tau\)) for this time, since we know there will be two different times to consider.
Figure \(\PageIndex{1}\): A spaceship with light bouncing between lower and upper mirrors, called a light clock. (CC BY-NC-ND; Jack C. Straton)
To someone standing on the Earth (Figure \(\PageIndex{2}\)) the light will be seen to follow a diagonal path because the spaceship moves relative to the Earth between the time that the light is
emitted near the bottom and when it is reflected from the mirror near the top of the spaceship.
Figure \(\PageIndex{2}\): As seen from the Earth, light bouncing between lower and upper mirrors of a moving spaceship follows a diagonal path. (CC BY-NC-ND; Jack C. Straton)
Since the speed of light is the same in the Earth’s reference frame as in the reference frame of the ship, this diagonal distance is \(d=c \ t\), where \(t\) is the time interval measured from the
Earth. Finally, the ship travels a distance \(r = v \ t\) with respect to the Earth.
It is clear from Figure \(\PageIndex{2}\) that the hypotenuse \(d\) of the triangle is longer than the vertical leg H. Then since the speed of light \(c\) is the same for both the hypotenuse and the
vertical leg, \(t\) must be larger than \(\tau\). But the hypotenuse of any right triangle you can draw is always longer than each leg or is equal to the length of one leg if the length of the other
is zero. (Draw several examples to prove this to yourself.) This means that t is always larger than \(\tau\) for nonzero v. We call this minimum time value \(\tau\) proper time. It is simply the time
measured in any frame in which the two events are measured at the same place, such as the emission and the return points of the bouncing beam of light, or two clicks of a clock. The coordinate time
interval \(t\) is measured in a frame of reference moving at velocity \(v\) relative to the frame in which the clock is stationary. (If we want to be precise, \(\tau\) in Figure \(\PageIndex{1}\) is
actually half of the proper-time interval for a round trip 2 \(\tau\). Likewise, in Figure \(\PageIndex{2}\), \(t\) is half of the corresponding coordinate time 2 \(t\).)
How Does This Relate to Me?
We say that the coordinate time is dilated. What exactly does that mean? It means that if your twin sister drives a fast car while you walk everywhere, she will live longer than you do—her lifetime
will be dilated relative to yours! By how much?
Figure \(\PageIndex{3}\): The sequence of fours steps described in the text in full. The red rectangle has a width that represents the distance the spaceship travels (6 cm) in an extremely short time
compared to the distance light travels, the width of the full 10 cm square, if \(v\) is \( \frac{3}{5}=\frac{6}{10}\) of the speed of light. If \(v\) had equaled \(c\), these widths would have been
equal. The sequence of three rotated green squares are stop-motion animation versions of the smooth rotation of the square by the reader. The blue arrows indicate the height measurement asked of the
reader. (CC BY-NC-ND; Jack C. Straton)
We could find how much larger \(t\) is than \(\tau\) by using the Pythagorean theorem and some algebra* but there is a simple way to get the time-dilation factor by drawing the spaceship picture
carefully, with \(v\) properly proportional to \(c\) as demonstrated in Figure \(\PageIndex{3}\):
1. Start with a square that is 10 cm on each side (the bottom side of which represents the distance light travels in the coordinate frame of reference).
2. Now express the velocity of the spaceship as a fraction of the speed of light and draw a rectangle that is that same fraction of 10 cm wide and is the full 10 cm high.^† Suppose \(v\) is \( \frac
{3}{5}=\frac{6}{10}\) of the speed of light (111,600 miles/second); then the width of the rectangle is 6 cm, shown as red as in Figure \(\PageIndex{3}\).
3. Now trace the square onto a thin sheet of paper (or cut out a square of this size) and rotate the square around the lower left-hand corner until the lower right-hand corner of the square just
touches the right-hand edge of the rectangle, approximated by the sequence of three rotated green squares shown in Figure \(\PageIndex{3}\).
4. Tape the square in place and measure the distance from the lower right-hand corner of the rotated square to the lower right-hand corner of the rectangle. For the present example, this length is 8
cm, shown as blue in Figure \(\PageIndex{3}\). Divide 10 cm by this length to get the value of the time-dilation factor. In the present example, this ratio is \(\frac{5}{4} \), or 1.25.
If your twin rides around in a spaceship at 111,600 miles/second for 40 years, 50 years will have passed for you when she returns! That may seem strange to you, but she is traveling at about 400
trillion miles/hour, something not exactly within your normal range of experience.
Suppose your twin slows way down, to 1/10th of the speed of light (see Figure \(\PageIndex{4}\)). Her speed is \(v\) = 0.1 \(c\), so the width of the rectangle is 1/10 of 10 cm or 1 cm.
Figure \(\PageIndex{4}\): The sequence of fours steps for \(v\) = 0.1 \(c\). (CC BY-NC-ND; Jack C. Straton)
As you rotate the rectangle, you notice that \(c \ t\) = 10 cm is not much longer than \(c \ \tau\) = 9.95 cm. This simply shows that the dilation of coordinate time (1.01 in this case) becomes
unnoticeable at velocities that are small compared to \(c\).
So suppose your twin drives around at 70 miles/hour or 0.019 miles/second (Figure \(\PageIndex{5}\)). That is \(v\) = \(c\)/1000000 so the width of the rectangle should be 10 cm/1000000, or 80,000
times thinner than the narrowest line this printer can draw at 300 dots per inch. Even if you could draw it, you cannot see any difference between \(c \ t\) and \(c \ \tau\), so no time dilation
comes into our everyday experience.
Figure \(\PageIndex{5}\): The sequence of fours steps for \(v\) = 70 miles/hour. (CC BY-NC-ND; Jack C. Straton)
* Suppose \(\frac{v}{c}=\frac{3}{5} \); then the ratio of the flight path (in the Earth’s frame of reference) to the light path is also \( \frac{3}{5}\). Let us use 3 cm and 5 cm, respectively. We
can use the Pythagorean theorem to relate the two times, \(\tau\) and t:
\[\text { (flight path) }^{2}+(\text { vertical light path })^{2}=(\text { diagonal light path })^{2} , \nonumber\]
or (vertical light path)^2 = (diagonal light path)^2 − (flight path)^2 = 25 cm^2 − 9 cm^2 = 16 cm^2. This means that the vertical light path is 4 cm. Then \(t / \tau=\frac{5}{4}\) or 1.25.
† Suppose the speed is \(v\) = 111,600 miles/second. Then the fraction \( \frac{v}{c}=\frac{111600 \textit { miles } / \textit { second }}{186000 \textit { miles } / \textit { second }}=0.6\). The
width of the rectangle would then be 0.6 × 10 cm = 6 cm.
Relativity Bites
This is not to say that time dilation does not affect our lives. The Earth is bombarded by cosmic rays that produce showers of particles called muons in the upper atmosphere. Half of a given group of
muons decays within two-millionths of a second (two microseconds) when they are at rest. That is, they have a proper half-life of two microseconds. After another two microseconds, half of those
remaining decay, leaving one-fourth, and so on.
If there were no relativistic time dilation, most would decay as they travel through the depth of the atmosphere before crashing through your skull. We will show later that on average, the muons are
traveling to the Earth’s surface at \(v\) = 0.9986 \(c\). At this speed, it would take them 16 microseconds to travel through the atmosphere from the height they are produced, about 3 miles,* or
about 8 half-lives.^† It turns out that 18 muons are created each second over an area the width of our bodies. After 8 half-lives, \( \frac{1}{2^{8}} \times 18=0.07\) muons per second should be left
to crash through our bodies. Table \(\PageIndex{1}\) shows a progression of halving the prior number, with some rounding.
Table \(\PageIndex{1}\):
The half-life progression
for 18 muons at the
Time Muons left
0 \(\mu s\) 18
2 \(\mu s\) 9
4 \(\mu s\) 4 or so
6 \(\mu s\) 2
8 \(\mu s\) 1
10 \(\mu s\) 0.5
12 \(\mu s\) 0.3
14 \(\mu s\) 0.15
16 \(\mu s\) 0.07
But relativity changes all this. To find the time-dilation factor for this speed, we find that we only need to rotate the square slightly to get the lower right-hand corner of the square to just
touch the right-hand edge of the rectangle (see Figure \(\PageIndex{6}\)).
* A. W. Wolfendale, Cosmic Rays at Ground Level (Institute of Physics, London, 1973), p. 174–75.
† \(t=\frac{d}{.9986 c}=\frac{3 \textit { miles }}{.9986 \times 186000 \textit { miles } / \textit { second }}=\frac{3}{1.86} \times \frac{10 \textit { seconds }}{1000000}=16 \text { microseconds. }
Figure \(\PageIndex{6}\): The sequence of fours steps for \(v\) = 0.9986 c. (CC BY-NC-ND; Jack C. Straton)
In this case, \(c \ t\) = 10 cm is much longer than \(c \ \tau\) = 0.52 cm. The coordinate time is extremely dilated by a factor of 19. So thanks to relativistic time dilation you measure a muon’s
half-life at 19 times its proper half-life, or 38 microseconds. This is about twice the time it takes for the muons to travel through the atmosphere. That is, roughly 1/4 of them will decay, and 13
muons make it through each second* to crash through your skull and increase your cancer rate. The typical yearly dose of radiation due to these muons is 400 µsv (microsievert),^† 6 times higher than
a typical yearly dose of radiation due to diagnostic X-rays, 70 µsv.^‡ If there were no relativistic time dilation, the yearly radiation dosage from muon exposure would be only 2 µsv. (For
perspective, if I were to ask you to give me $2, the chances are pretty good you might do so. But if I asked you to give—not lend— me $400, the chances you might do so would be pretty slim.)
Relativistic effects are not a minor factor in our lives at all!
* National Council on Radiation Protection and Measurements, Report No. 94, Exposure of the Population in the United States and Canada from Natural Background Radiation (NCRP, Bethesda, MD, 1987), p.
12, has a rate of 0.00190 muons per cm^2 per second at the surface. I obtained 13 muons per second by modeling a person by a cylinder with a radius of 15 cm.
† Alan Martin and Samuel A. Harbison, An Introduction to Radiation Protection (Chapman & Hall, New York, 1979), p. 53, gives 500 µsv for all types of cosmic radiation, of which muons make up 80%,
according to the National Council on Radiation Protection and Measurements, Report No. 94, Exposure of the Population in the United States and Canada from Natural Background Radiation (NCRP,
Bethesda, MD, 1987), p. 12.
‡ Alan Martin and Samuel A. Harbison, An Introduction to Radiation Protection (Chapman & Hall, New York, 1979), p. 57.
Just so that you do not get the idea that the real-world consequences of relativity are all negative, consider that evolution works by taking advantage of mutations. Theistic philosophers struggle
with the question of why there are disease and evil in the world. In his last book, And God Laughed When the Birds Came Forth from the Dinosaurs, my father asked the question this way:
[W]hy does God allow such mutation, which, from the higher standpoint of the personal value that it destroys, must be classified as an “evil”? The answer that seems reasonable is to affirm that such
mutation . . . is the risk God must run in creating or bringing forth a finite world of freely developing process. The over-all rationality of such possibility seems borne out by the thoughtful
conclusions of genetical science itself. The geneticists Dunn and Dobzhansky write:
Harmful mutations and hereditary diseases are thus the price which the species pays for the plasticity which makes continued evolution possible.*
[T]he above quotation contains the idea of the neutrality of mutations as a necessary principle of . . . physical process. The mutation of the genes makes the survival of life possible in the long
run in any environment, or amid environmental changes, within, of course, upper and lower limits of temperature and other absolute environmental boundaries for life. In theistic faith, and from the
standpoint of values, this “neutrality” of mutation would itself seem purposive, since its effect is that life does survive. To theistic faith, the immanent rationale of mutation is that life shall
He also writes,
[A] modern teleological^‡ view of evolution would cite mutation itself—the capacity of life at the very deepest level of process to adjust or adapt itself—as significant evidence of an ultimate
spiritual meaning, design, or purpose within our world’s evolutionary development. The spiritual meaning inheres in this “free capacity,” which makes possible the manifold growth and integration of
life in many experimental directions, instanced by all the past and present organic forms.^§
* L. C. Dunn and Th. Dobzhansky, Heredity, Race, and Society, A Mentor Book (New American Library, New York, 1946/1952), p. 81.
† G. Douglas Straton, And God Laughed When the Birds Came Forth from the Dinosaurs: Essays on the Idea and Knowledge of God (1995 ms.), Chap. 6, p. 163.
‡ Teleology is the study of cosmic design
§ G. Douglas Straton, And God Laughed When the Birds Came Forth from the Dinosaurs: Essays on the Idea and Knowledge of God (1995 ms.), Chap. 5, p. 115.
Clearly, if relativity were not working, evolution would have proceeded at a pace that is between 5 and 256 times slower (2^8).^† The Earth would have had to wait 18 to 900 billion years for
intelligent life to form (instead of 3.5 billion years)—much longer than the Sun’s 10-billion-year lifetime!
Put another way, the person attempting to write this book today would likely be a mess of green slime had relativity not offered us its gifts.
There Are Cops, You Know!
What about people traveling at super high speeds, such as \(v\) = 0.9986 \(c\) in Figure \(\PageIndex{6}\)? Suppose your twin left Earth when you were 20 years of age, traveled for 10 years at \(v\)
= 0.9986 \(c\), and then returned to Earth to tell you about her trip. Tough luck; you would have died of old age a century before she returned! You were both 20 when she left. She is 30 years old
(20 + 10) when she returns (both her clock and her body agree with this assertion), but you would be 210 years old (20 + 10 × 19) had you lived.
As we increase the speed from this point, we will find that we reach a limit in our ability to graph. But this limit expresses a reality of nature. As the velocity \(v\) of a rocket approaches that
of the speed of light \(c\), the right-hand side of the rectangle really does approach the right side of the 10 cm square in Figure \(\PageIndex{6}\). The limit as \(v\) goes to \(c\) is that the
rectangle goes to a square. Then we have to move the square not at all to get its lower right-hand corner to touch the right side of the rectangle-which-is-a-square. That is, \(c\) \(\tau\) = 0 cm.
A finite coordinate time period in a rocket moving at velocity \(v\) = \(c\) relative to the Earth corresponds to zero proper time. What does that mean? If we divide 10 cm by 0 cm we get an infinite
time-dilation factor.^‡
We actually never run into this infinite limit because it is impossible to exert sufficient force on the rocket to get it moving at the speed \(c\). The reason is that whatever force is exerted on
the rocket to increase its velocity is spread out over a longer and longer period of Earth coordinate time as \(v\) approaches \(c\). We would have to wait an infinite amount of Earth coordinate time
to see the rocket reach the speed of light.
† The maximal value is assuming that the other 100 µsv of cosmic radiation noted in Alan Martin and Samuel A. Harbison’s An Introduction to Radiation Protection (Chapman & Hall, New York, 1979), p.
53, would have a similar reduction in the absence of relativity; the minimal value assumes that there would be no such reduction.
‡ To see this consider the following pattern: On your calculator divide 10 by 10 to get 1; 10/1 = 10; 10/0.1 = 100; 10/0.01 = 1,000; 10/0.001 = 10,000; 10/0.0001 = 100,000; 10/0.00001 = 1,000,000;
and so on. As you divide 10 by smaller and smaller numbers, you get a result that is bigger and bigger. Infinity is the limit of this sequence.
To see how this works, imagine a spaceship powered by small nuclear bombs that are dropped through a small hole in a radiation shield attached to the passenger cabin. When a bomb explodes, half of
the exploded material and associated photons push against the radiation shield, shoving the rocket faster in its direction of travel. As the rocket’s velocity increases, the time dilation increases
on Earth, which the passengers have left behind. As the rocket crew steadily drops and explodes bombs (at a steady proper-time interval), there are longer and longer time intervals between when Earth
folk see the photons from the explosions arrive. In fact, as the people on Earth see the rocket’s speed approach the speed of light, they have to wait through an infinite time interval for the
explosion that would have just pushed the rocket past the limit. Thus, the rocket never reaches the speed of light relative to the Earth. One might rephrase the classic Western koan as “What happens
when an irresistible force meets an interminable stasis?”
Some readers will note that many science popularizers and introductory physics teachers used to invoke the idea that
[t]he faster a particle is pushed, the more its mass increases, thereby resulting in less and less response to the accelerating force. . . . [A]s \(v\) approaches \(c\), m approaches infinity! [To
push a particle] to the speed of light . . . would require an infinite force, which is clearly impossible.*
Let me caution you that research physicists rely heavily upon the fact^† that the mass of a particle is the same in all frames of references; it is an invariant quantity.^‡
See, for instance, the caption of Figure \(\PageIndex{4}\) in the paper publishing the discovery of the Higgs boson,^§ reproduced below, that begins with the explicit acknowledgment of the “[i]
nvariant mass distribution,” while the simpler and equivalent “mass” is also used throughout, such as in the section G, following that figure: “The measured Higgs boson mass [is] 125.98 ± 0.50 GeV. .
. .” The (invariant) mass of an object is sometimes referred to in older texts as the “rest mass.”
* Paul G. Hewitt, Conceptual Physics, 6th ed. (Scott, Foresman & Co., Boston, 1989), p. 662.
† For a complete history of this, see Lev. B. Okun, Phys. Today 42, 31–36 (June 1989).
‡ See, for instance, the standard textbook by John D. Jackson, Classical Electrodynamics (John Wiley & Sons, New York, 1975), p. 531, eq. (11.54).
§ G. Aad et al. (ATLAS Collaboration), Phys. Rev. D 90, 052004 (2014).
Figure \(\PageIndex{7}\): Invariant mass distribution in the H → \(\gamma \gamma\) analysis for data (7 TeV and 8 TeV samples combined), showing weighted data points with errors, and the result of
the simultaneous fit to all categories. The fitted signal plus background is shown, along with the background-only component of this fit. The different categories are summed together with a weight
given by the s/b ratio in each category. The bottom plot shows the difference between the summed weights and the background component of the fit.
Figure 4 of G. Aad et al. (ATLAS Collaboration), Phys. Rev. D 90, 052004, reproduced under the terms of the Creative Commons Attribution 3.0 License.
The idea that mass increases with velocity was introduced by Hendrick Lorentz* in 1899 so that he could use the low-speed expression for momentum, m \(v\), for relativistically high velocities. When
Einstein introduced relativity six years later, the idea of mass increasing with velocity became unnecessary, but unfortunately, it has retained a very long life. This puts you, the reader, in the
nasty position of having to decide between two “authorities.” Since the explanation for the upper limit on rocket speeds given on the previous page does not need to mention mass, changing or
otherwise, Occam’s razor^† would dictate that we choose it over an explanation that includes the idea of varying mass. I would recommend that you discard the latter idea as outdated. After all, the
downside of using a convenient, though incorrect, model to make predictions is revealed well in the story of the Ptolemaic vs. Copernican models of the Solar System.
* H. Lorentz, Proc. R. Acad. Sci. Amsterdam 1, 247 (1899); 6, 809 (1904).
† William of Occam (c. 1280) said that when there are several explanations of a phenomenon, the simplest is most likely to be correct.
A Short Tale
So everything you believe about time has just gotten blown out the window. About now, I would expect you to be wondering if space gets messed with too. It does.
Unfortunately, the pictures below show this only with the aid of a series of length comparisons. In my experience, those who are a bit wigged out by math may put up with a single such relation here
and there, but a series of about eight length comparisons that build up to the answer may well lead to frustration. So let me simply give you the result here and you can just skim over the notes
below as if they were written in Portuguese, from which a recognizable word or two might pop out if you know a bit of Spanish or French. If you would like to bypass the comparisons entirely, read the
following paragraph and then just jump to the words Skip to here.
The length \(\ell\) of the rocket measured by Earth is contracted by the same factor, relative to the proper length L, as the time t measured by Earth is dilated relative to proper time \(\tau\).
Since we keep on finding this “factor,” we had better give it a name. It is always represented by the Greek letter for g, gamma, which is written \(\gamma\). That is, \(t=\gamma \ \tau\), and \(\ell=
Here are the details. Consider our spaceship at rest, with an additional pair of mirrors set at the same distance horizontally as the original ones were set vertically (see Figure \(\PageIndex{8}\).
Then if a light wave is emitted from the lower left-hand corner of the set of mirrors, it will travel outward in concentric rings, strike the top and right-hand mirrors simultaneously, and be
reflected back to the emission point. The total distance traveled in this proper-time interval is
\[2 L=c \tau. \nonumber\]
Figure \(\PageIndex{8}\): A spaceship with vertical and horizontal light clocks. (CC BY-NC-ND; Jack C. Straton)
Now suppose the rocket is moving at velocity \(v=\frac{3}{5} c \) relative to the Earth. If the spaceship were at rest, light emitted from the left-hand mirror would have to travel a distance \(\ell
\) to the right-hand mirror in Figure \(\PageIndex{9a}\). But in time \(t_{a}\), that right-hand mirror moves with the spaceship an additional distance, \(a=v \ t_{a}\), before the light catches up
(Figure \(\PageIndex{9c}\)). So the distance that light must move in that same time is
\[c \ t_{a}=\ell+v \ t_{a}. \nonumber\]
Subtracting \(v \ t_{a}\) from both sides, collecting terms, and using \(v=\frac{3}{5} c\) gives
\[ \begin{array}{l} c t_{a}-v t_{a}=c t_{a}\left(1-\frac{v}{c}\right) \\ =c t_{a}\left(1-\frac{3}{5}\right)=\ell \ , \end{array} \nonumber \]
\[c t_{a}=\frac{5}{2} \ell. \nonumber\]
In this time, the rocket travels 3/5 as far as the light,
\[a=v t_{a}=\frac{3}{5} c t_{a}=\frac{3}{2} \ell. \nonumber\]
A much shorter time later, \(t_{b}\) , the reflected wave collides with the bottom left-hand mirror, traveling toward it rather than away this time (Figure \(\PageIndex{9d}\)), so that
\[ \begin{array}{c} c t_{b}=\ell-v t_{b}, \text { or } \\ c t_{b}+v t_{b}=c t_{b}\left(1+\frac{v}{c}\right) \\ =c t_{b}\left(1+\frac{3}{5}\right)=\ell \ , \end{array} \nonumber\]
and (multiplying both sides by \(\frac{5}{8} \))
\[c t_{b}=\frac{5}{8} \ell, \nonumber\]
3/5 of which is
\[b=v t_{b}=\frac{3}{8} \ell, \nonumber\]
Figure \(\PageIndex{9}\)
(CC BY-NC-ND; Jack C. Straton)
Then the total distance the rocket travels for the emission and reflection is the sum of these, \(r=v t=a+b=\frac{3}{2} \times \frac{4}{4} \ell+\frac{3}{8} \ell=\mathbf{3 \times} \frac{5 \ell}{8}\),
which is again \(\frac{3}{5}\) of the distance the light travels in the same time, \(c t=\frac{5}{2} \times \frac{4}{4} \ell+\frac{5}{8} \ell=5 \times \frac{5 \ell}{8}\). We found from our
time-dilation calculation in Figure \(\PageIndex{2}\) that if one leg is 3 units long when the hypotenuse is 5 units long, then the other leg has to be 4 units long. This is the case in the two
expressions above for the leg and hypotenuse, where the unit of measure is \(\frac{5 \ell}{8}\). So \(c \tau=4 \times \frac{5 \ell}{8}\).
But from Figure \(\PageIndex{8}\), we see that \(c \ \tau=2 L\). Comparing these two expressions shows that
\[\ell=\frac{8}{5} \times \frac{2 L}{4}=\frac{4}{5} L. \nonumber\]
(Skip to here.) Comparing this to the time-dilation expression we obtained from Figure \(\PageIndex{3}\), \(t=\frac{5}{4} \tau\) shows that the length \(\ell\) of the rocket measured by Earth is
contracted, relative to the proper length L, by the same factor \(\gamma=\frac{5}{4}\), as the time \(t\) measured by Earth is dilated relative to proper time \(\tau\). That is \(t=\gamma \tau\) and
This compensation between time dilation and length contraction is necessary for reality to be whole. Einstein’s second postulate of relativity was that no experiment you could perform would tell you
whether it was your frame of reference that was moving or someone else’s (alternatively stated, motion is relative, or there is no preferred frame of reference).
Consider our example of the time dilation of muons created in the Earth’s atmosphere. These muons are free to think they are at rest and it is the Earth that is moving toward them. They see the
surface of the Earth traveling toward them at \(v\) = 0.9986 \(c\) just after they are produced.
At this speed, it would take the Earth’s 3 miles of atmosphere 16 microseconds to pass by them before the surface crashes into them. As before, this is about 8 proper half-lives. If there were no
relativistic length contraction, most would decay before they saw your head approaching as you stood on the surface of the Earth. But because of relativity, the muons see the Earth’s atmosphere
contracted to only 1/6 of a mile.* The time it takes the Earth’s surface to hit them is then 0.8 microseconds, or about half of the muon half-life.^† Again, we get the result that roughly 1/4 of the
muons decay, leaving 13 muons getting hit by your head each second before their lives are over. Without length contraction, the physical reality of your cancer risk would depend on which frame of
reference does the calculation. That would violate Einstein’s principle of relativity.
* \(\ell=\frac{3 \text { miles }}{19}=0.158 \text { miles. }\)
† \(t=\frac{\ell}{9986 c}=\frac{0.158 \textit { miles }}{.9986 \times 186000 \textit { miles} / \textit { second }}=\frac{0.158}{.186} \times \frac{1 \textit { second }}{1000000}=0.854 \text {
microseconds. }\) | {"url":"https://phys.libretexts.org/Bookshelves/Relativity/Book%3A_Relativity_Lite_-_A_Pictorial_Translation_of_Einsteins_Theories_of_Motion_and_Gravity_(Straton)/01%3A_Chapters/1.01%3A_To_c_or_Not_to_c","timestamp":"2024-11-06T04:03:36Z","content_type":"text/html","content_length":"171884","record_id":"<urn:uuid:2a844327-3eed-4044-bf75-5fc243b2014b>","cc-path":"CC-MAIN-2024-46/segments/1730477027909.44/warc/CC-MAIN-20241106034659-20241106064659-00652.warc.gz"} |
The second exercise in this document describes a graph visiting problem combined with dice rolling.
It’s a not very complicated problem, and we could solve it using Breadth First search. The gotcha here is how to identify visited nodes so that we could terminate with great confidence that
continuing would be in vain. Attaching an boolean flag to each node is obviously wrong, for it’s possible that reaching the node with a different dice index could probably open the window to a
totally different world.
One (straightforward) solution is to combine the dice index with the node to form a new state, then we could do graph visiting on this artificial state space using the ordinary BFS.
Another solution, which I read from other’s code, is less obvious to see its correctness, which makes it more interesting to discuss and ponder. The gist of this algorithm goes like: maintaining the
frontier of BFS and visited nodes on each dice roll (iteration), but don’t do any filtering. On reaching the end of the dice list, check if the fixpoint of visited nodes has reached to decide if we
should terminate or go on another around of dice roll list.
The algorithm is surely correctly if final node is reachable, for it’s doing BFS. However, why the -1 case works as well is not so obvious (at least to me). Is it possible that next iteration would
reach the final node, but the program return -1 prematurely? The key to understand it lies on the cyclic dice list. One property of BFS is that the nodes on the final layer is determined by the sum
of steps instead of the order of each step. In other words, starting from the same node, the final nodes are the same regardless the dice roll is [1,2] or [2,1]. If we focus on the last two rounds of
dice list before returning -1:
1 ------------------------- 0
2 ------------------------- 1
3 ...
4 ------------------------- L-1
6 ------------------------- 0
7 ------------------------- 1 <-
8 ...
9 ------------------------- L-1
11 ????????????????????????? 0
where L is the length of dice roll. All nodes in the second part have been visited in previous rounds. Now, we need to prove the ? line doesn’t introduce any new nodes, and this property holds for
all future layers. Nodes on ? line are reached from nodes on line <- by going through a whole period of dice roll. We could find the exact same node in previous rounds, apply a whole period
(regardless of the starting point in the dice roll), and end up with the nodes that are visited, as assumed that no new nodes are encountered. The same reasoning would go on for all other nodes on
line <-, so the line marked as ? introduces no new nodes, so are all future layers, using the same logic. | {"url":"https://albertnetymk.github.io/2016/02/06/dice/index.html","timestamp":"2024-11-15T03:33:42Z","content_type":"text/html","content_length":"9901","record_id":"<urn:uuid:d568d3e5-14e2-4a25-bece-55be0d15a4f3>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00785.warc.gz"} |
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of values of the graph.
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to the challenge. Get help with math homework, solve specific math problems or find information on mathematical subjects and topics. | {"url":"https://libguides.uwc.ac.za/c.php?g=1201719&p=8787510","timestamp":"2024-11-14T20:40:07Z","content_type":"text/html","content_length":"84324","record_id":"<urn:uuid:a0be4168-7a60-4c1b-b531-2e83f08e03e1>","cc-path":"CC-MAIN-2024-46/segments/1730477395538.95/warc/CC-MAIN-20241114194152-20241114224152-00005.warc.gz"} |
Development of Reduced Order Modeling Methods for Incompressible Flows with Heat Transfer and Parametric Boundary Conditions: Doctoral dissertation submitted to obtain the academic degree of Doctor of Electromechanical Engineering
Within the MYRRHA project, which stands for Multi-purpose hYbrid Research
Reactor for High-tech Applications, the Belgian Nuclear Research Center SCK
CEN is developing and designing a multi-functional experimental fast-spectrum
irradiation facility. The MYRRHA design features a compact pool-type primary
system cooled by molten Lead-Bismuth Eutectic, i.e. a heavy metal. Reliable computational methods are required to accurately quantify the reactor’s primary system behavior in operational and
accidental conditions and to handle complex geometries.
However, the number of nuclear reactor simulations in a safety analysis is,
in the majority of cases, beyond the possibilities of present hardware if a computational fluid dynamics solver is used alone. This has motivated the development of reduced order modeling techniques
that reduce the number of degrees of freedom of the high fidelity thermofluids models. Mathematical techniques are used to extract “features” of the complex model in order to replace them by a more
simplified model. In that way, the required computational time and computer memory usage is reduced.
Despite the potential and increasing popularity of reduced order models for all sorts of flow applications, they tend to have issues with accuracy and exhibit numerical instabilities. Challenges
regarding velocity-pressure coupling
and satisfying the boundary conditions at the reduced order level make it difficult to generalize the methods such that they can be applied to any problem.
The complex fluid dynamics problems are generally solved numerically using
discretization methods. In this work, we focus on the finite volume discretization
method for the numerical solution of incompressible fluid flows on collocated
grids. To obtain a computationally efficient reduced order model (ROM), the procedure is ideally split into a so-called offline stage and an online stage. In the
offline stage, solutions of the high fidelity model are collected at several time instances and/or for different parameter values. They are used to generate a reduced basis of a much smaller order
than the full order model (FOM). In this work, the reduced basis spaces are spanned by basis functions, or so-called modes, which are computed using the proper orthogonal decomposition (POD)
technique. POD is commonly used for reduced-order modeling of incompressible flows. Reduced matrices (linear terms) and tensors (nonlinear terms) of the ROM associated with the terms of the full
order model are determined during the offline stage, for which two techniques are developed and investigated in this work. The first technique is a non-intrusive reduction method that identifies the
system matrix of linear fluid dynamical problems with a least-squares technique. The main advantage of nonintrusive methods is that they do not require access to the solver’s discretization and
solution algorithm. The second technique is the intrusive Galerkin projection approach for which the full order equations are projected onto the reduced POD basis spaces. In the online stage, the
reduced system of equations are solved for the same or new values of the parameters of interest at a lower computational cost compared to solving the full order systems.
The non-intrusive reduction method that identifies the system matrix of linear
fluid dynamical problems with a least-squares technique is presented in the
first part of the thesis. The methodology is applied to the linear scalar transport
convection-diffusion equation for a two-dimensional square cavity problem with a heated lid. The (time-dependent) boundary conditions are imposed in the reduced order model with a penalty method. The
results are compared and the accuracy of the reduced order models is assessed against the full order solutions. It is shown that the reduced order model can be used for sensitivity analysis by
controlling the non-homogeneous Dirichlet boundary conditions. For nonlinear problems, the required number of snapshots scales with the cube of the number of POD basis functions and at least as many
reduced matrices are to be identified as the number of basis functions used. Therefore, it is not feasible to use the POD-based identification method for nonlinear problems. However, for the
simulation of fluid flows in (nuclear) engineering applications, it is necessary to develop reduced order models for nonlinear problems, such as convective flows and buoyancy-driven flows. Therefore,
the main part of the thesis is dedicated to the intrusive PODbased Galerkin projection approach due to its applicability to nonlinear problems.
POD-Galerkin reduced order models are developed of which the (timedependent)
boundary conditions are imposed at reduced order level using two different
boundary control strategies: the lifting function method, whose aim is to
obtain homogeneous basis functions for the reduced basis spaces and the penalty method where the boundary conditions are imposed in the reduced order model using a penalty factor. The penalty method
is improved by using an iterative solver for the determination of the penalty factor rather than tuning the factor with a sensitivity analysis or numerical experimentation. The boundary control
methods are compared and tested for two cases: the classical lid driven cavity benchmark problem and a Y-junction flow case with two inlet channels and one outlet channel. The results show that the
boundaries of the reduced order model can be controlled with the boundary control methods and the same order of accuracy is achieved for the velocity and pressure fields. However, computing the ROM
solutions takes more time in the case of the lifting function method as the reduced basis spaces contain additional modes, namely the lifting functions, compared to the penalty method.
Furthermore, a parametric reduced order model for buoyancy-driven flow is
introduced. The Boussinesq approximation is used for modeling the buoyancy.
Therefore, there exists a two-way coupling between the incompressible Boussinesq equations and the energy equation. To obtain the reduced order model, a Galerkin projection of the governing equations
onto the reduced POD bases spaces is performed. The ROM is tested on a two-dimensional differentially heated cavity of which the side wall temperatures are parametrized. The parametrization is done
using a lifting function method. The lifting functions are obtained by solving a Laplacian function for temperature. Only one unsteady full order simulation was required for the creation of the
reduced bases. The obtained ROM is efficient and stable for different parameter sets for which the temperature difference between the walls is smaller than for the set in the FOM used for the POD
basis creation.
In addition, the POD-Galerkin reduced order modeling strategy for steadystate
Reynolds averaged Navier–Stokes (RANS) simulation is extended for low-
Prandtl number fluid flow. The reduced order model is based on a full order model for which the effects of buoyancy on the flow and heat transfer are characterized by varying the Richardson number.
The Reynolds stresses are computed with a linear eddy viscosity model. A single gradient diffusion hypothesis, together with a local correlation for the evaluation of the turbulent Prandtl number, is
used to model the turbulent heat fluxes. The contribution of the eddy viscosity and turbulent thermal diffusivity fields are considered in the reduced order model with an interpolation based
data-driven method. The ROM is tested for buoyancy-aided turbulent liquid sodium flow over a vertical backward-facing step with a uniform heat flux applied on the wall downstream of the step. The
wall heat flux boundary condition is incorporated in both the full order model and the reduced order model.
The velocity and temperature profiles predicted with the ROM for the same and
new Richardson numbers inside the range of parameter values are in good agreement with the RANS simulations. Also, the local Stanton number and skin friction distribution at the heated wall are
qualitatively well captured. Finally, the reduced order simulations, performed on a single core, are about 105 times faster than the full order RANS simulations that are performed on eight cores.
The final part of the thesis is dedicated to the development of a novel nonparametric reduced order model of the incompressible Navier-Stokes equations on collocated grids. The reduced order model is
developed by performing a Galerkin projection based on a fully (space and time) discrete full order model formulation.
This ‘discretize-then-project’ approach requires no pressure stabilization
technique (even though the pressure term is present in the ROM) nor a boundary
control technique (to impose the boundary conditions at the ROM level). These
are two main advantages compared to existing and previously applied approaches.
The fully discrete FOM is obtained by a finite volume discretization of the incompressible Navier-Stokes equations with a forward Euler time discretization.
Two variants of the velocity-pressure coupling at the fully discrete level, the inconsistent and consistent flux method, have been investigated. The latter leads to divergence-free velocity fields,
also on the ROM level, whereas the velocity fields are only approximately divergence-free in the former method. For both methods,
stable and accurate results have been obtained for test cases with different types
of boundary conditions: a lid-driven cavity and an open cavity with an inlet and
outlet. The ROM obtained with the consistent flux method, having divergence-free velocity fields, is slightly more accurate but also slightly more expensive to solve compared to the inconsistent flux
method due to an additional equation for the flux.
The speedup ratio of the ROM and FOM computation times strongly depends on
which method is used, the number of degrees of freedom of the full order model
and the number of modes used for the reduced basis spaces.
Finally, an application with the coupling between a system thermal hydraulics
code and a reduced order model of a computational fluid dynamics solver is presented in the appendix of this work. The system code and the ROM are coupled by a domain decomposition algorithm using an
implicit coupling scheme. The velocity transported over a coupling boundary interface is imposed in the ROM using a penalty method. The coupled models are evaluated on open and closed pipe flow
configurations. The results of the coupled simulations with the ROM are close to those with the CFD solver. Also for new parameter sets, the coupled models with the ROM are capable of predicting the
results of the coupled models with the FOM with good accuracy. The coupled simulations with the ROM are about 3-5 times faster than those with the FOM.
Original language English
Qualification Doctor of Philosophy
Awarding Institution • Universiteit Gent
• Degroote, Joris, Supervisor, External person
Supervisors/Advisors • Stabile, Giovanni, Supervisor, External person
• Sanderse, Benjamin, Supervisor, External person
Date of Award 15 Feb 2021
State Published - 15 Feb 2021 | {"url":"https://researchportal.sckcen.be/en/publications/development-of-reduced-order-modeling-methods-for-incompressible-","timestamp":"2024-11-14T01:29:09Z","content_type":"text/html","content_length":"78914","record_id":"<urn:uuid:7f18f893-4071-4420-9588-f2b4e2eb5843>","cc-path":"CC-MAIN-2024-46/segments/1730477028516.72/warc/CC-MAIN-20241113235151-20241114025151-00203.warc.gz"} |
Category Theory - AI Alignment Forum
Category Theory is a subfield of pure mathematics studying any structure that contains objects and their relations (referred to as morphisms). It emerged in the study of algebraic topology, then went
on to apply beyond mathematics and into various scientific disciplines, a metamathematical framework comparable to that of type theory and set theory. The notion of compositionality is what differs
category theory from graph theory, in which the nodes themselves can be categories.
Current research on applied category theory and Categories for AI are useful and relevant for topics close to LW, such as Rationality, AI Safety, and Game theory. Due to its abstract nature, category
theory is jokingly criticized as being "abstract nonsense". A major theorem result is Yoneda embedding, which is basically the idea that an object can be defined by its all of its relations. | {"url":"https://www.alignmentforum.org/tag/category-theory","timestamp":"2024-11-02T06:08:58Z","content_type":"text/html","content_length":"1048971","record_id":"<urn:uuid:8dbbe6b1-c326-4298-ac88-744fc5a1e86f>","cc-path":"CC-MAIN-2024-46/segments/1730477027677.11/warc/CC-MAIN-20241102040949-20241102070949-00579.warc.gz"} |
Attention's Fractal Patterns
Section Headings
Attention's Fractal Patterns
Many claim that Fractals are the mathematics of Life
Seeing the beautiful vegetal patterns generated by Mandelbrot’s famous equation, many have made the claim that fractals reveal the secret of Life in terms of self-replicating patterns. I agree.
However, there are also significant differences between the two systems, i.e. mathematical and living, which should be acknowledged – not ignored.
For instance, fractalized boundary lines -> Life’s
Life has her own mathematics, which is based in the Living Algorithm. Both Mandelbrot’s Fractal Equation and Life’s Living Algorithm are Reflexive Equations, i.e. based in feedback loops. Due to this
affinity, they share a similar logic. We have shown how this so-called Fractal Logic is more appropriate for living systems than is the absolutist set-based logic of Matter. In particular,
fractalized boundary lines disable the absolute truths that many attribute to verbal constructs.
Examine Mathematics to see where Fractal Metaphor breaks down
However, always another ‘however’, we must remember that the relationship between living systems and fractals is metaphorical in nature, as are all abstractions. As such the fit between model and
reality is not perfect, never is. Let’s examine the mathematics of the two equations to see where the fractal metaphor breaks down with regards Life. In so doing, we will discover where the fractal
metaphor falls short when it comes to living systems.
First Mandelbrot vs. Living Algorithm
Let us compare and contrast Mandelbrot’s Fractal Equation with Attention’s Living Algorithm. Although they are both Reflexive (recursive) Equations, the first is deterministic and static, while the
second is open and dynamic. Quite a difference! Let’s check out the details and the significance.
Mandelbrot’s Closed Equation = Small differences à Diametrically Opposed Results: No lines
Mandelbrot’s Reflexive Equations (the ones that generated the fractal patterns) are closed – no external input. Despite being closed to the outside, small differences in the initial conditions, i.e.
the constant, can still yield diametrically opposed results – limit or no limit – the equivalent of distinct destination or lost – directed or aimless. No precise lines identify which points are
inside and outside of the destination set.
Fractals Deterministic
Despite the complexity of the results (no demarcation lines), Mandelbrot’s fractal equations are still deterministic. The initial conditions always yield the exact same result. With no external
input, there is no variation in the output.
Attention’s open LA = Not Deterministic
Even more complex is the reflexive equation that Attention employs to interact with data streams – the Living Algorithm (LA). While Mandelbrot’s fractal equations are closed, the LA is open to
external input. With each iteration of the LA’s computational process, new information enters the system. As an open, reflexive equation, the LA is not deterministic. Due to this freedom, the LA’s
mathematical system is qualitatively different from the absolute determinism of Material Equations or Mandelbrot’s Fractal Equations.
Material Equations & Fractal Equations = Deterministic
Living Algorithm = Not Deterministic
LA identifies Living Processes, not Material Events
Yet the LA defines the Living Realm of Attention. What does this freedom mean? If the mathematics can’t predict results, what’s the point? Not a specialist in material events, the LA instead
identifies processes and their features. This is appropriate as even Life’s identity is based in process, rather than content, as a subsequent section illustrates.
Colorful Fractal Patterns due to Speed of approach
Rather than illuminating a process, Mandelbrot’s equation reveals the results of an event, i.e. which constant is chosen for the iterative sequence. On the most basic level, the result is binary –
limit or no limit (determinate or indeterminate). On a secondary level, the answer can also have a speed, the speed at which it approaches the limit. Some sequences are faster and others are slower.
This variation accounts for the beautiful colors that are associated with the fractal patterns.
Questionable Utility of Fractal Patterns
Gorgeous but so what? Not sure. I don’t know if anyone has uncovered any real utility to these static fractal patterns. Certainly don’t hear much about it in the literature.
LA reveals 2 fractal Patterns that permeate Rhythms of Attention
In contrast, the Living Algorithm reveals two fractal patterns that dominate the rhythms of Attention. Rather than revealing static, deterministic results, as do the other two functions, the LA’s
computational iterations identify two dynamic processes that are replicated on the very small time frames, e.g. the blinking of an eye, to the very large, e.g. an entire lifetime of mastery – one’s
life work or even the Pulse of a Chinese dynasty.
The Pulse: Attention Span
The Pulse is one of the fractal patterns that we are referencing. Reflecting our Attention span, the Pulse is everywhere that there are living systems. It has some distinct characteristics that shape
the form of our awareness, e.g. a distinct beginning and end; and a peak of awareness that is negatively impacted by interruptions – disproportionately so.
Triple Pulse: Sleep, Variation
The other mathematical pattern that replicates on the large and small levels is what we have chosen to call the Triple Pulse. This fractal process is related to our need for sleep, rest and variation
in a theme.
Natural Selection takes advantage of Cycles: Posner & Dement
Natural selection has chosen to take advantage of both of these mathematical processes. Appropriate biological systems evolved to exploit the potentials of the LA’s mathematical system, i.e. Data
Stream Dynamics (DSD). For example, both Posner’s Attention Model and Dement’s Opponent Process Model for the biology of sleep have a strong metaphoric relationship with the mathematical processes of
DSD. Both of these models are also intimately related to a complex of biological systems in many species.
Nature of the Answer: Definitive Predictions vs. Pattern Identification
Summary: Mandelbrot vs. LA & Fractal Logic vs Set Logic
In summary, there is a huge difference between our two Reflexive Functions. Mandelbrot’s Fractal Equation reveals the deterministic results of particular events. The Living Algorithm reveals the
fuzzy shape of living processes associated with Attention. There is even a bigger difference between our two types of logic. Matter’s Set-Based Logic is Absolutist, true or false in all cases.
Fractal Logic is particular, with truth evaluated on a case-by-case basis. Further, Fractal Logic is based in replicating patterns. The LA’s Pulse and Triple Pulse are examples of these replicating
patterns as applied to Attention.
What is the significance of these differences?
Logic of System determines Nature of Answer
The logic of the system determines the nature of the answer. Matter’s Set Logic enables precise predictions of specific results. The Fractal Logic of recursive functions, e.g. Mandelbrot’s equation
and Life’s Living Algorithm, enables the identification of patterns.
Foolish to apply Set Logic’s Precision to Attention
To apply Set Logic’s precision to Attention’s interactive feedback loops is misguided foolishness. Doomed to failure, this quest will never be able to reveal the underlying mechanisms that shape an
organism’s intentional behavior. Rather than precise predictions, the best we can hope for is to identify the natural rhythms of Attention.
Matter’s Feedback Loops corrupt Predictability
Feedback can also be a problem for Matter. The scientific community has generated the impression that they can make precise predictions regarding all material behavior. Perhaps ideally, but not
pragmatically. Set Logic is a great metaphor for Matter, but it falls apart at the edges, just like any other metaphor. Because of an obsession with a deterministic mindset (fuels a feeling of
superiority and control), mathematicians and scientists tend to obscure the metaphoric nature of the relationship.
Where are the metaphoric flaws? There are a few significant caveats to the absolute predictability of Matter. First, precision is possible, but only if the initial conditions are clearly defined.
Second, the number of factors must be small. Third, it helps if the system is closed. Let us provide some examples.
When objects are engaged in feedback loops, they are subject to the same type of predictive opacity that plagues living systems. For instance, it is virtually impossible to predict the location of a
simple pool ball after multiple ricochets (one or two is easy). Why? The multiplication of angles with each bounce.
Due to the exorbitant time frame, the same difficulty holds true for astronomy. Because of the endlessness of planetary orbits, a third body (ever so small and in the wrong spot) could eventually
(after millions of cycles) actually change a planet’s trajectory in significant ways. In other words, feedback loops, even though non-intentional, corrupt the projections of material behavior just as
with living behavior.
When the number of factors are large and the system is open, prediction also becomes a problem. Modern technology is based upon limiting the number of variables that influence the outcome. This
strategy works great inside a closed box, e.g. a car, computer or a rocket ship. Controllable variables in a closed system. Perfect.
However, the same deterministic mathematical strategy falls apart when analyzing the myriad variables of an open system. While scientists can send a rocket into outer space and predict its course for
years to come, they are unable to make accurate long-term predictions regarding the weather. Despite all the closed box technology at their disposal, the best meteorologists can do is make short term
predictions and identify long term weather patterns, e.g. cold in the winter and warm in the summer.
Due to intentionality, the individual behavior of living systems is even harder to predict than the weather – virtually impossible. However as with the weather, it is possible to identify rough
patterns that govern our intentional behavior. And by exerting personal intentionality, we can ride the waves of these patterns to optimize our life experience.
So please don’t hold Attention to the same idealized predictive standards as Matter. Even objects are subject to the predictive opacity of feedback loops. Further myriad small factors in an open
material system generate the same scientific confusion as in living systems. Add Intention into the mix and Pattern identification is the best we can hope for with Attention.
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Connectivity and cut algorithms
Moody and White algorithm for k-components
│k_components(G[, flow_func])│Returns the k-component structure of a graph G. │
Kanevsky all minimum node k cutsets algorithm.
│all_node_cuts(G[, k, flow_func])│Returns all minimum k cutsets of an undirected graph G. │
Flow-based Connectivity¶
Flow based connectivity algorithms
│average_node_connectivity(G[, flow_func]) │Returns the average connectivity of a graph G. │
│all_pairs_node_connectivity(G[, nbunch, ...])│Compute node connectivity between all pairs of nodes of G. │
│edge_connectivity(G[, s, t, flow_func]) │Returns the edge connectivity of the graph or digraph G. │
│local_edge_connectivity(G, u, v[, ...]) │Returns local edge connectivity for nodes s and t in G. │
│local_node_connectivity(G, s, t[, ...]) │Computes local node connectivity for nodes s and t. │
│node_connectivity(G[, s, t, flow_func]) │Returns node connectivity for a graph or digraph G. │
Flow-based Minimum Cuts¶
Flow based cut algorithms
│minimum_edge_cut(G[, s, t, flow_func]) │Returns a set of edges of minimum cardinality that disconnects G. │
│minimum_node_cut(G[, s, t, flow_func]) │Returns a set of nodes of minimum cardinality that disconnects G. │
│minimum_st_edge_cut(G, s, t[, flow_func, ...])│Returns the edges of the cut-set of a minimum (s, t)-cut. │
│minimum_st_node_cut(G, s, t[, flow_func, ...])│Returns a set of nodes of minimum cardinality that disconnect source from target in G. │
Stoer-Wagner minimum cut¶
Stoer-Wagner minimum cut algorithm.
│stoer_wagner(G[, weight, heap])│Returns the weighted minimum edge cut using the Stoer-Wagner algorithm. │
Utils for flow-based connectivity¶
Utilities for connectivity package
│build_auxiliary_edge_connectivity(G)│Auxiliary digraph for computing flow based edge connectivity │
│build_auxiliary_node_connectivity(G)│Creates a directed graph D from an undirected graph G to compute flow based node connectivity. │ | {"url":"https://networkx.org/documentation/networkx-1.10/reference/algorithms.connectivity.html","timestamp":"2024-11-10T14:28:47Z","content_type":"text/html","content_length":"24088","record_id":"<urn:uuid:accbb9af-0c1d-4d7d-8043-fa89b2efdaf1>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.60/warc/CC-MAIN-20241110134821-20241110164821-00240.warc.gz"} |
"Bona fide loan discount points" means loan discount points knowingly paid by the borrower for the purpose of reducing, and which in fact result in a bona fide. On a $, loan, 3 points means a cash
payment of $3, Points are part of the cost of credit to the borrower. Points can be negative, in which case they. You may not charge a loan origination fee or discount points as described in
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A loan that does not charge any points, but does assess closing costs. Often referred to as a zero point loan. Non-Assumption Clause: A statement in a mortgage. lending. Points are commonly referred
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cannot be used to pay any part of the down payment. For this reason, you do not. 25 basis points or a quarter of a percent is the most common value associated with a discount point. How Are Points
Treated for Tax Purposes? Discount points. You may not charge a loan origination fee or discount points as described in Regulation X, Part , Appendix A. (9) What mortgage broker fees may I charge?
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points or.
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subject to this article. Points, also known as discount points, lower your interest rate in exchange paying for an upfront fee. Lender credits lower your closing costs in exchange for.
"Negative" discount points, also known as "rebates" or "yield spread premiums," reduce the amount of cash you need at closing. But you'll have to pay a higher. You may not charge a loan origination
fee or discount points as described in Regulation X, Part , Appendix A. (9) What mortgage broker fees may I charge? Mortgage points, sometimes known as discount points, are prepaid portions of
interest on your home loan that when purchased, reduce your monthly interest rate. Mortgage points, also known as discount points, may be used by a borrower to prepay some of the interest on a home
loan in exchange for a lower mortgage rate. Points usually cost 1% of your total loan amount and lower the interest rate on payments by %. Read the FAQs below to learn more. Calculate the possible.
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420 research outputs found
The existence is proved of a class of open quantum systems that admits a linear subspace ${\cal C}$ of the space of states such that the restriction of the dynamical semigroup to the states built
over $\cal C$ is unitary. Such subspace allows for error-avoiding (noiseless) enconding of quantum information.Comment: 9 pages, LaTe
We explore a variety of reasons for considering su(1,1) instead of the customary h(1) as the natural unifying frame for characterizing boson systems. Resorting to the Lie-Hopf structure of these
algebras, that shows how the Bose-Einstein statistics for identical bosons is correctly given in the su(1,1) framework, we prove that quantization of Maxwell's equations leads to su(1,1),
relativistic covariance being naturally recognized as an internal symmetry of this dynamical algebra. Moreover su(1,1) rather than h(1) coordinates are associated to circularly polarized
electromagnetic waves. As for interacting bosons, the su(1,1) formulation of the Jaynes-Cummings model is discussed, showing its advantages over h(1).Comment: 9 pages, to appear in J. Phys. A: Math.
This paper aims to challenge the current thinking in IT for the 'Big Data' question, proposing - almost verbatim, with no formulas - a program aiming to construct an innovative methodology to perform
data analytics in a way that returns an automaton as a recognizer of the data language: a Field Theory of Data. We suggest to build, directly out of probing data space, a theoretical framework
enabling us to extract the manifold hidden relations (patterns) that exist among data, as correlations depending on the semantics generated by the mining context. The program, that is grounded in the
recent innovative ways of integrating data into a topological setting, proposes the realization of a Topological Field Theory of Data, transferring and generalizing to the space of data notions
inspired by physical (topological) field theories and harnesses the theory of formal languages to define the potential semantics necessary to understand the emerging patterns
We show how the addition of a phonon field to the Hubbard model deforms the superconducting su(2) part of the global symmetry Lie algebra su(2)⊗su(2)/openZ2, holding at half filling for the
customary model, into a quantum [su(2)]q symmetry, holding for a filling which depends on the electron-phonon interaction strength. Such symmetry originates in the feature that in the presence of
phonons the hopping amplitude turns out to depend on the coupling strength. The states generated by resorting to this q symmetry exhibit both off-diagonal long-range order and pairing
By resorting to the Fock--Bargmann representation, we incorporate the quantum Weyl--Heisenberg ($q$-WH) algebra into the theory of entire analytic functions. The main tool is the realization of the
$q$--WH algebra in terms of finite difference operators. The physical relevance of our study relies on the fact that coherent states (CS) are indeed formulated in the space of entire analytic
functions where they can be rigorously expressed in terms of theta functions on the von Neumann lattice. The r\^ole played by the finite difference operators and the relevance of the lattice
structure in the completeness of the CS system suggest that the $q$--deformation of the WH algebra is an essential tool in the physics of discretized (periodic) systems. In this latter context we
define a quantum mechanics formalism for lattice systems.Comment: 22 pages, TEX file, DFF188/9/93 Firenz
Current definitions of both squeezing operator and squeezed vacuum state are critically examined on the grounds of consistency with the underlying su(1,1) algebraic structure. Accordingly, the
generalized coherent states for su(1,1) in its Schwinger two-photon realization are proposed as squeezed states. The physical implication of this assumption is that two additional degrees of freedom
become available for the control of quantum optical systems. The resulting physical predictions are evaluated in terms of quadrature squeezing and photon statistics, while the application to a
Mach–Zehnder interferometer is discussed to show the emergence of nonclassical regions, characterized by negative values of Mandel’s parameter, which cannot be anticipated by the current
formulation, and then outline future possible use in quantum technologies
We investigate the potential energy surface of a phi^4 model with infinite range interactions. All stationary points can be uniquely characterized by three real numbers $\alpha_+, alpha_0, alpha_-
with alpha_+ + alpha_0 + alpha_- = 1, provided that the interaction strength mu is smaller than a critical value. The saddle index n_s is equal to alpha_0 and its distribution function has a maximum
at n_s^max = 1/3. The density p(e) of stationary points with energy per particle e, as well as the Euler characteristic chi(e), are singular at a critical energy e_c(mu), if the external field H is
zero. However, e_c(mu) \neq upsilon_c(mu), where upsilon_c(mu) is the mean potential energy per particle at the thermodynamic phase transition point T_c. This proves that previous claims that the
topological and thermodynamic transition points coincide is not valid, in general. Both types of singularities disappear for H \neq 0. The average saddle index bar{n}_s as function of e decreases
monotonically with e and vanishes at the ground state energy, only. In contrast, the saddle index n_s as function of the average energy bar{e}(n_s) is given by n_s(bar{e}) = 1+4bar{e} (for H=0) that
vanishes at bar{e} = -1/4 > upsilon_0, the ground state energy.Comment: 9 PR pages, 6 figure
Experiments on cold atom systems in which a lattice potential is ramped up on a confined cloud have raised intriguing questions about how the temperature varies along isentropic curves, and how these
curves intersect features in the phase diagram. In this paper, we study the isentropic curves of two models of magnetic phase transitions- the classical Blume-Capel Model (BCM) and the Fermi Hubbard
Model (FHM). Both Mean Field Theory (MFT) and Monte Carlo (MC) methods are used. The isentropic curves of the BCM generally run parallel to the phase boundary in the Ising regime of low vacancy
density, but intersect the phase boundary when the magnetic transition is mainly driven by a proliferation of vacancies. Adiabatic heating occurs in moving away from the phase boundary. The
isentropes of the half-filled FHM have a relatively simple structure, running parallel to the temperature axis in the paramagnetic phase, and then curving upwards as the antiferromagnetic transition
occurs. However, in the doped case, where two magnetic phase boundaries are crossed, the isentrope topology is considerably more complex | {"url":"https://core.ac.uk/search/?q=authors%3A(Rasetti%2C%20M)","timestamp":"2024-11-09T13:43:27Z","content_type":"text/html","content_length":"134970","record_id":"<urn:uuid:0ed9f296-c31a-418f-a968-3a81aea43d8a>","cc-path":"CC-MAIN-2024-46/segments/1730477028118.93/warc/CC-MAIN-20241109120425-20241109150425-00196.warc.gz"} |
Nelson's Portfolio
Hi! My name is Jacob Nelson, and this is my GITA 3
portfolio. You'll find my school assigned projects
here, as well as some smaller projects I made on my
own time. All projects are made with p5.js.
Angle + Dist Coordinates
Most of the time, positions are specified with (x, y). This project shows a method to specify them with some angle and distance instead.
Simple Harmonic Motion
Simple simulation of a spring undergoing simple harmonic motion. Use your mouse to get it started.
Slope Field Generator
Generates a slope field when a slope equation is specified. Some calculus required.
PI Estimate From Probability
The ratio between the number of dots that have landed in the circle and the number of dots that have landed in the square will approach PI.
Polygons with Infinite Sides
As n -> infinity, the ratio between the perimeter of the shape and an axis within the shape will approch PI.
A bunch of dots are generated, then their distances are sorted by a homemade sorting algorithm, then the closest 3 are connected.
Avatar Card
A small image of Bean Cat on a holiday card. If the font doesn't load, refresh the page. If that doesn't work, turn off your monitor and ignore the problem.
Landscape Animation
A small animation, once again featuring Bean Cat. Made with p5.js's built in draw functions. | {"url":"http://jacon2.gitastudent.online/","timestamp":"2024-11-02T08:22:38Z","content_type":"text/html","content_length":"22046","record_id":"<urn:uuid:917a2521-230a-454b-8206-eee15d2dd88f>","cc-path":"CC-MAIN-2024-46/segments/1730477027709.8/warc/CC-MAIN-20241102071948-20241102101948-00056.warc.gz"} |
Tip the Scales | U.S. Mint for Kids
Tip the Scales
Using coins as the standard of measure, students will estimate and check weights of classroom objects.
Using coins as the standard of measure, students will estimate and check weights of classroom objects.
Subject Area
Class Time
• Total Time: 0-45 Minutes minutes
• Cents
• Quarters
• Balance scales (one per group)
• "Tip the Scales" work pages (pages 24 and 25)
• Pencils
• Crayons
Lesson Steps
1. Explain that the class will work in groups to estimate the weights of classroom objects, and then check their estimates at weighing stations. Review the terms "estimate," "weight," and "balance."
2. Display the "Tip the Scales" work pages (pages 24 and 25). Explain that everyone in the group will work together to weigh the objects and check the estimates.
3. Sit at one of the work stations and explain that you will demonstrate the entire estimation/weighing process with a different item than the ones the children will work with.
4. Hold up an eraser and ask a volunteer to estimate how many cents will weigh the same as the eraser. Write the estimate in the spaces provided on the work page.
5. Place the eraser on the scale and ask students to remind you how many cents have been estimated to balance the scale. Start putting cents in the scale and count aloud.
6. Add cents until the scale is balanced or the estimated number has been reached. (It may be necessary to explain what the scale should look like when it is balanced.)
7. If the estimated number comes first, then discuss what happened, and whether or not the estimate has been confirmed. Then, add cents until the scale is balanced. If the scale balances before the
estimated number has been reached, discuss how close the estimate was to the actual number of cents needed.
8. Remind students that they will be weighing different classroom objects, and show them the objects they will work with.
9. Assign groups and send each group to a station. You may wish to assign jobs (balancer, counter, cent dropper) within each group, so that every child participates. Students could then rotate jobs
with each new object.
10. Allow students 25 minutes to complete the tasks. When time is called, ask students to share how close their estimates were, and what surprised them during the activity. You may wish to discuss
the "brain teaser" activity on the work page, highlighting the difference between weight and value (25 cents are worth one quarter, but 25 cents weigh more than one quarter).
Use the worksheet and class participation to assess whether the students have met the lesson objectives.
Common Core Standards
Discipline: Math
Domain: K.MD Measurement and Data
Grade(s): Grade K
Cluster: Describe and compare measurable attributes
• K.MD.1. Describe measurable attributes of objects, such as length or weight.
□ Describe several measurable attributes of a single object
Discipline: Math
Domain: K.MD Measurement and Data
Grade(s): Grade K
Cluster: Describe several measurable attributes of a single object
• K.MD.2. Directly compare two objects with a measurable attribute in common, to see which object has "more of"/"less of" the attribute, and describe the difference. For example, directly compare
the heights of two children and describe one child as taller/shorter.
• K.MD.3. Classify objects into given categories; count the numbers of objects in each category and sort the categories by count.
National Standards
Discipline: Mathematics
Domain: All Problem Solving
Cluster: Instructional programs from kindergarten through grade 12 should enable all students to
Grade(s): Grades K–12
• Build new mathematical knowledge through problem solving
• Solve problems that arise in mathematics and in other contexts
• Apply and adapt a variety of appropriate strategies to solve problems
• Monitor and reflect on the process of mathematical problem solving
Discipline: Mathematics
Domain: K-2 Data Analysis and Probability
Cluster: Develop and evaluate inferences and predictions that are based on data.
Grade(s): Grades K–12
In K through grade 2 all students should
• discuss events related to students' experiences as likely or unlikely.
Discipline: Mathematics
Domain: K-2 Data Analysis and Probability
Cluster: Formulate questions that can be addressed with data and collect, organize, and display relevant data to answer them.
Grade(s): Grades K–12
In K through grade 2 all students should
• pose questions and gather data about themselves and their surroundings;
• sort and classify objects according to their attributes and organize data about the objects; and
• represent data using concrete objects, pictures, and graphs.
Discipline: Mathematics
Domain: K-2 Measurement
Cluster: Apply appropriate techniques, tools, and formulas to determine measurements.
Grade(s): Grades K–12
In K through grade 2 all students should
• measure with multiple copies of units of the same size, such as paper clips laid end to end;
• use repetition of a single unit to measure something larger than the unit, for instance, measuring the length of a room with a single meterstick;
• use tools to measure; and
• develop common referents for measures to make comparisons and estimates.
Discipline: Mathematics
Domain: K-2 Measurement
Cluster: Understand measurable attributes of objects and the units, systems, and processes of measurement.
Grade(s): Grades K–12
In K through grade 2 all students should
• recognize the attributes of length, volume, weight, area, and time;
• compare and order objects according to these attributes;
• understand how to measure using nonstandard and standard units; and
• select an appropriate unit and tool for the attribute being measured.
Discipline: Mathematics
Domain: K-2 Number and Operations
Cluster: Compute fluently and make reasonable estimates.
Grade(s): Grades K–12
In K through grade 2 all students should
• develop and use strategies for whole-number computations, with a focus on addition and subtraction;
• develop fluency with basic number combinations for addition and subtraction; and
• use a variety of methods and tools to compute, including objects, mental computation, estimation, paper and pencil, and calculators. | {"url":"https://kids.usmint.gov/learn/kids/resources/lesson-plans/tip-the-scales","timestamp":"2024-11-15T04:55:29Z","content_type":"text/html","content_length":"55846","record_id":"<urn:uuid:f395d49f-e152-4162-8a33-3006ecbc5e1a>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00894.warc.gz"} |
The Strange Selection Bias in LASSO
Let’s suppose you have two variables \(X\) and \(Z\) that are positively correlated. \(X\) causes \(Y\) but \(Z\) does not.
n = 100
x <- rnorm(n)
z= x + rnorm(n)
The true value of \(\beta_z\) should be 0 and \(\beta_x\) should be 1 in the model \(y=\beta_x x + \beta_z z\). If we simulate this a bunch of times and estimate using OLS that’s exactly what we find
on average. However, penalized regressions sacrifice the promise of unbiased estimates for other properties such as lower MSE and variable selection.
If we estimate the model using ridge regression, we end up seeing a substantial bias on \(\hat{\beta}_z\). The histogram below shows the distribution of ridge regression estimates of \(\beta_z\) from
a ridge estimator with \(\lambda\) of 0.1.
simRidge <- function() {
n = 100
x <- rnorm(n)
z= x + rnorm(n)
ridgemod<- lmridge(data = data.frame(y=y, x=x, z=z),
formula = y~x+z, K= 0.1)
return(ridgemod$coef["z", ])
ridge.z.coefs <- replicate(1000, simRidge())
ggplot(data= data.frame(Z = ridge.z.coefs), aes(x = Z)) +
geom_histogram() + geom_vline(xintercept = mean(ridge.z.coefs),
colour = "red", linetype = 2)
This is explicable when we examine the ridge regression loss function:
\[$$L_{ridge}(\hat{\beta}_{ridge}) = \sum_{i=1}^{M}{ \bigg (y_i - \sum_{j=0}^{p} \hat{\beta}_{ridge_{j}} \cdot X[i, j] } \bigg ) ^2 + \frac{\lambda_2}{2} \sum^p_{j=1} {\hat{\beta}_{ridge_{j}}}^2$$\]
The important part here is the squared penalty term. That means that the estimator will prefer adding mass to a smaller coefficient than a larger one. Suppose that \(\hat{\beta}_x\) is currently 1 \
(\hat{\beta}_z\) is 0.1. That would create a ridge penalty of \(1^2+0.1^2=1.01\). If we add 0.1 to \(\hat{\beta}_x\), that increases the total ridge penalty to \(1.1^2+0.1^2=1.22\) whereas if we
added 0.1 to \(\hat{\beta}_z\) it would only increase the ridge penalty to \(1^2+0.2^2=1.04\). That means the ridge loss function tends to find solutions where the parameter for a correlated variable
gets some mass at the expense of the larger variable’s parameter. That is why we see \(\hat{\beta}_z\) get a positive value on average even though it is simulated as zero.
Where things get weird is when we do the same thing for LASSO. While the bias is much smaller than for ridge models, it’s still there and in the same direction.
simLasso <- function() {
n = 100
x <- rnorm(n)
z= x + rnorm(n)
gcdnetModel <- gcdnet(y = y,
x = cbind(x,z),
lambda = c(0.02),
lambda2 = 0,
standardize = FALSE,
method = "ls")
all.coefs <- coef(gcdnetModel)
return(all.coefs["z", ])
lasso.z.coefs <- replicate(10000, simLasso())
ggplot(data= data.frame(Z = lasso.z.coefs), aes(x = Z)) +
geom_histogram(bins = 100) + geom_vline(xintercept = mean(lasso.z.coefs),
colour = "red", linetype = 2)
To see why that’s weird we can again consider the loss function:
\[$$L_{lasso}(\hat{\beta}_{lasso}) = \sum_{i=1}^{M}{ \bigg (y_i - \sum_{j=0}^{p} \hat{\beta}_{lasso_{j}} \cdot X[i, j] } \bigg ) ^2 + \lambda_1 \sum^p_{j=1} {|\hat{\beta}_{lasso_{j}}|}$$\]
The important thing here is that the LASSO loss works on the absolute value of the coefficient. Taking our previous example, adding 0.1 to \(\hat{\beta}_x\), increases the total LASSO penalty to \
(1.1+0.1=1.2\) exactly the same as if we added 0.1 to \(\hat{\beta}_z\) \(1+0.2=1.2\). In other words, the LASSO penalty should be indifferent between adding mass to \(\hat{\beta}_x\) and \(\hat{\
beta}_z\). Given that we simulated \(\beta_z\) to be zero, it is odd that the LASSO estimate is positive on average.
I won’t detail all the dead ends we went down trying to figure this out, but I think we finally have the answer. The key is that while we simulate \(\beta_z\) to be zero, in each simulation, an OLS
estimate puts some coefficient mass on \(\beta_z\). In the OLS estimates, this is symmetrically distributed around zero. Sometimes \(\beta_z\) is positive and sometimes negative. And it turns out
that the LASSO estimator behaves quite differently depending on which of those scenarios is at play.
Estimating the model using LASSO generally decreases the magnitude of \(\hat{\beta}_x\). Lower values of \(\hat{\beta}_x\) increases the MSE of the model, which can potentially be reduced again by a
larger magnitude of \(\hat{\beta}_z\). Of course, \(\hat{\beta}_z\) can also have a larger magnitude by modeling the same variance it did in the OLS estimate. It turns out that these two goals are
overlapping when the OLS estimate of \(\beta_z\) was positive but work against each other when the OLS estimate of \(\beta_z\) was negative.
We capture where \(\hat{\beta}_z\) reduces the MSE by defining \(\epsilon_z\) as the residuals that \(\hat{\beta}_z\) changes in the OLS estimate. We calculate these by comparing the OLS predictions
\(\hat{y}_{OLS}=\beta_{xOLS} x + \hat{\beta}_{zOLS} z\) to the predictions from the OLS model omitting the \(\hat{\beta}_{zOLS}\) term \(\hat{y}_{OLSnoZ}=\beta_{xOLS} x\). \(\epsilon_z= \hat{y}_{OLS}
- \hat{y}_{OLSnoZ}\).
We then capture where reducing \(\hat{\beta}_x\) increased MSE by defining \(\epsilon_x\) as the residuals that are changed by reducing \(\hat{\beta}_x\) from its OLS value to its LASSO value. This
time, we compare the OLS predictions to \(\hat{y}_{OLS}\) the OLS predictions if we swap \(\beta_{xOLS}\) for the LASSO value \(\beta_{xLASSO}\): \(\hat{y}_{lassox}=\beta_{xLASSO} x + \hat{\beta}_
{zOLS} z\). That means we finally define \(\epsilon_x= \hat{y}_{OLS} - \hat{y}_{xLASSO}\).
The following figure shows the relationship between the errors created by reducing \(\hat{\beta}_x\), \(\epsilon_x\), and the errors reduced in the OLS estimate by giving \(\hat{\beta_z}\) some
coefficient mass: \(\epsilon_z\). For the case where \(\hat{\beta}_z\) was negative in the OLS estimate, the two goals come into conflict because there is a negative correlation between the
observations that \(\hat{\beta}_z\) improves in the OLS estimate, and the variance that reducing \(\hat{\beta}_x\) opens up for modeling after it is penalized by LASSO. However, for a simulation
where \(\hat{\beta}_z\) was positive in the OLS estimate, there is no tradeoff between these goals. A larger \(\hat{\beta}_z\) coefficient helps to mop up the variance that the penalization opened up
and reduces prediction error for the observations it helped predict in the OLS estimate.
But those are just two simulations, does this pattern hold up more widely? The next figure shows the correlation between \(\epsilon_x\) and \(\epsilon_z\) for simulations where the OLS estimate of \
(\beta_z\) took on different values. We see a sharp discontinuity at zero, where the correlation between the two types of variance available to model switches from positive to negative.
We can also plot the LASSO estimates for \(\beta_z\) against the OLS estimates. This shows a clear shift in behavior where \(\beta_z\) shifts from positive to negative in the OLS estimate.
Overall, this shows that penalized estimators can have some unexpected behavior because of the interaction between the least squares and penalty components of the loss function. | {"url":"https://www.filedrawer.blog/post/unpicking-lasso-bias/?ref=footer","timestamp":"2024-11-04T02:18:20Z","content_type":"text/html","content_length":"15689","record_id":"<urn:uuid:02b34cb0-e310-4688-ac6b-4f51a3efad16>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00774.warc.gz"} |
Contest on LaTeX-Community.org – Win a gnuplot Book
LaTeX-Community.org has teamed up with Packt Publishing to organize a contest. Two users stand a chance to win a copy of the new book on gnuplot, the newly published gnuplot Cookbook written by Lee
To enter the contest, write a small article for publishing on LaTeX-Community.org. The topic of this contest is: LaTeX and Graphics.
From the book’s web page:
gnuplot Cookbook – it will help you master gnuplot. Start using gnuplot immediately to solve your problems in data analysis and presentation. Quickly find a visual example of the graph you want
to make and see a complete, working script for producing it. Learn how to use the new features in gnuplot 4.4. Find clearly explained, working examples of using gnuplot with LaTeX and with your
own computer programming language.
You will learn to plot basic 2d to complex 3d plots, annotate from simple labels to equations, integrate from simple scripts to full documents and computer progams. You will be taught to annotate
graphs with equations and symbols that match the style of the rest of your text, thus creating a seamless, professional document. You will be guided to create a web page with an interactive
graph, and add graphical output to your simulation or numerical analysis program.
To read more details, have a look at the contest announcement and the book’s web page, including the entire table of contents and a sample chapter for download.
This text is available in German. Dieser Text ist auch in Deutsch verfügbar. | {"url":"https://texblog.net/latex-archive/graphics/gnuplot-book/","timestamp":"2024-11-07T14:01:20Z","content_type":"text/html","content_length":"21558","record_id":"<urn:uuid:2c5a2e2b-6e3b-4cca-91cc-29b73e018349>","cc-path":"CC-MAIN-2024-46/segments/1730477027999.92/warc/CC-MAIN-20241107114930-20241107144930-00823.warc.gz"} |
Late in life, Newton said: 'I seem to have been only like a boy playing on the seashore, and diverting myself in now and then finding a smoother pebble or a prettier shell than ordinary, while the
great ocean of truth lay all undiscovered before me'.
William Blake took the mystic approach to the same concept: 'To see a World in a Grain of Sand// And Heaven in a Wild Flower// Hold Infinity in the palm of your hand// And Eternity in an hour'
Blaise Pascal expressed fear: 'The eternal silence of these infinite spaces terrifies me!'
Years ago I was working on the ART OF LANGUAGE series, a series of paintings inspired by the languages of our world and concepts unique to those languages. While preparing a talk on the subject, I
picked up a mathematical textbook. I wanted to illustrate the point that mathematics is not a language, just as musical notation is not a language.
The mathematician Robin Whitty was kind enough to give my paintings about mathematics a mention on his website and in truth he did warn me I might have some stray mathematicians coming my way... | {"url":"https://www.shaman-healer-painter.co.uk/info2.cfm?info_id=224563","timestamp":"2024-11-14T12:06:37Z","content_type":"text/html","content_length":"100008","record_id":"<urn:uuid:403d07dd-bcdf-4262-86fc-c944ed109d11>","cc-path":"CC-MAIN-2024-46/segments/1730477028558.0/warc/CC-MAIN-20241114094851-20241114124851-00244.warc.gz"} |
Analogmachine Blog
In this post I will show you how to run a stochastic simulation using our Tellurium application. Tellurium is a set of libraries that can be used via Python. One of those libraries is libRoadRunner
which is our very fast simulator. It can simulate both stochastic and deterministic models. Let's illustrate a stochastic simulation using the following simple model:
import tellurium as te
import numpy as np
r = te.loada('''
J1: S1 -> S2; k1*S1;
J2: S2 -> S3; k2*S2
J3: S3 -> S4; k3*S3;
k1 = 0.1; k2 = 0.5; k3 = 0.5;
S1 = 100;
We've set up the number of initial molecules of S1 to be 100 molecules. The easiest way to run a stochastic simulation is to call the gillespie method on roadrunner. This is shown in the code below:
m = r.gillespie (0, 40, steps=100)
Running this by clicking on the green button in the tool bar will give you the plot:
What if you wanted to run a lot of gillespie simulations to get an idea of the distribution of trajectories? TO do that just just need to repeat the simulation many times and plot all the results on
the same graph:
import tellurium as te
import numpy as np
r = te.loada('''
J1: S1 -> S2; k1*S1;
J2: S2 -> S3; k2*S2
J3: S3 -> S4; k3*S3;
k1 = 0.1; k2 = 0.5; k3 = 0.5;
S1 = 100;
# run repeated simulation
numSimulations = 50
points = 101
for k in range(numSimulations ):
s = r.gillespie(0, 50)
# No legend, do not show
r.plot(s, show=False, loc=None, alpha=0.4, linewidth=1.0)
This script will yield the plot:
We can do one other thing and compute the average trajectory and overlay the plot with the average line. The one thing we have to watch out for is that we must set the integrator property
variable_step_size = False to false. This will ensure that time points are equally spaced and that all trajectories end at the same point in time.
import tellurium as te
import numpy as np
r = te.loada('''
J1: S1 -> S2; k1*S1;
J2: S2 -> S3; k2*S2
J3: S3 -> S4; k3*S3;
k1 = 0.1; k2 = 0.5; k3 = 0.5;
S1 = 100;
# run repeated simulation
numSimulations = 50
#points = 101
# Set the physical size of the plot (units are in inches)
r.setIntegrator ('gillespie')
# Make sure we do this so that all trajectories
# are the same length and spacings
r.getIntegrator().variable_step_size = False
s_sum = np.array(0.)
for k in range(numSimulations):
s = r.simulate(0, 100, steps=50)
s_sum = np.add (s_sum, s)
# no legend, do not show
r.plot(s, show=False, loc=None, alpha=0.4, linewidth=1.0)
# add mean curve, legend, show everything and set labels, titels, ...
s_mean = s_sum/numSimulations
r.plot(s_mean, loc='upper right', show=True, linewidth=3.0,
title="Stochastic simulation", xlabel="time",
ylabel="concentration", grid=True);
This will give us the following plot:
November 4, 2016 8:59 am
For very complicated and large models it may be necessary to adjust the simulator tolerances in order to get the correct simulation results. Sometimes the simulator will terminate a simulation
because it was unable to proceed due to numerical errors. In many cases this is due to a bad model and the user must investgate the model to determine what the issue might be. If the model is assumed
to be correct then the other option is to change the simulator tolerances. The current option state of the simulator is obtained using the getInfo call, for example:
r = roadrunner.RoadRunner ('mymodel.xml')
roadrunner.RoadRunner() {
'this' : 22F59350
'modelLoaded' : true
'modelName' : __main
'libSBMLVersion' : LibSBML Version: 5.13.0
'jacobianStepSize' : 1e-005
'conservedMoietyAnalysis' : false
'simulateOptions' :
< roadrunner.SimulateOptions()
'this' : 1B89AF28,
'reset' : 0,
'structuredResult' : 0,
'copyResult' : 1,
'steps' : 50,
'start' : 0,
'duration' : 20
'integrator' :
< roadrunner.Integrator() >
name: cvode
relative_tolerance: 0.00001
absolute_tolerance: 0.0000000001
stiff: true
maximum_bdf_order: 5
maximum_adams_order: 12
maximum_num_steps: 20000
maximum_time_step: 0
minimum_time_step: 0
initial_time_step: 0
multiple_steps: false
variable_step_size: false
There are a variety of tuning parameters that can be changed in the simulator. Of interest are the relative and absolute tolerances, the maximum number of steps, and the initial time step.
The smaller the relative tolerance the more accurate the solution, however too small a value will result in either excessive runtimes or more likely roundoff errors. A relative tolerance of 1E-4
means that errors are controlled to 0.01%. An optimal value is roughly 1E-6. The absolute tolerance is used when a variable gets so small that the relative tolerance doesn't make much sense to apply.
In these situations, absolute error tolerance is used to control the error. A small value for the absolute tolerance is often desirable, such as 1E-12, we do not recommend going below 1E-15 for
either tolerance.
To set the tolerances use the statements:
r.integrator.absolute_tolerance = 5e-10
r.integrator.relative_tolerance = 1e-3
Another parameter worth changing if the simulations are not working well is to change the initial time step. This is often set by the integrator to be a relatively large value which means that the
integrator will try to reduce this value if there are problems. Sometimes it is better to provide a small initial step size to help the integrator get started, for example, 1E-5.
r.integrator.initial_time_step = 0.00001
The reader is referred to the CVODE documentation for more details.
Phase plots are a common way to visualize the dynamics of models where time courses are generated and one variable is plotted against the other. For example, consider the following model that can
show oscillations:
v1: $Xo -> S1; k1*Xo;
v2: S1 -> S2; k2*S1*S2^h/(10 + S2^h) + k3*S1;
v3: S2 -> ; k4*S2;
In this model S2 positively activates reaction v2 thus forming a positive feedback loop. The rate equation for v2 include a Hill like coefficient term, S2^h, which determines the strength of the
positive feedback. The oscillations originate from an interaction between the positive feedback and a non-obvious negative feedback loop at S1 between v1 and v2.
Let us assign suitable parameter values to this model, run a simulation and plot S1 versus S2.
import tellurium as te
# Import pylab to access subplot plotting feature.
import pylab as plt
r = te.loada ('''
v1: $Xo -> S1; k1*Xo;
v2: S1 -> S2; k2*S1*S2^h/(10 + S2^h) + k3*S1;
v3: S2 -> $w; k4*S2;
# Initialize
h = 2; # Hill coefficient
k1 = 1; k2 = 2; Xo = 1;
k3 = 0.02; k4 = 1;
S1 = 6; S2 = 2
m = r.simulate (0, 80, 500, ['S1', 'S2'])
Running this script by clicking the green button in the toolbar yields the following plot:
What if we'd like to investigate how the oscillations are affected by the parameters of the model. For example, how does the model behave when we change k1? One way to do this is to plot simulations
at different k1 values onto the same plot. In this case, however, this will create a difficult to read graph. Instead, let us create a grid of subplots where each subplot represents a different
import tellurium as te
import pylab as plt
r = te.loada ('''
v1: $Xo -> S1; k1*Xo;
v2: S1 -> S2; k2*S1*S2^h/(10 + S2^h) + k3*S1;
v3: S2 -> $w; k4*S2;
# Initialize
h = 2; # Hill coefficient
k1 = 1; k2 = 2; Xo = 1;
k3 = 0.02; k4 = 1;
S1 = 6; S2 = 2
for i in range (9):
m = r.simulate (0, 80, 500, ['S1', 'S2'])
plt.subplot (3,3,i+1)
plt.plot (m[:,0], m[:,1], label="k1=" + str (r.k1))
r.k1 = r.k1 + 0.2
Here we create a 3 by 3 subplot grid, start a loop that changes k1, and each time round the loop it plots the simulation onto one of the subplots. Running this script results in the following output
Here is a simple need, given a reaction model how do we get hold of the stoichiometry matrix?
Consider the following simple model:
import tellurium as te
import roadrunner
r = te.loada("""
$Xo -> S1; k1*Xo;
S1 -> S2; k2*S1;
S2 -> S3; k3*S2;
S3 -> $X1; k4*S3;
k1 = 0.1; k2 = 0.4;
k3 = 0.5; k4 = 0.6;
Xo = 1;
print r.getFullStoichiometryMatrix()
Running this script by clicking on the green arrow in the toolbar will yield:
_J0, _J1, _J2, _J3
S1 [[ 1, -1, 0, 0],
S2 [ 0, 1, -1, 0],
S3 [ 0, 0, 1, -1]]
The nice thing about this output is that the columns and rows are labeled so you know exact what is what.
What about much larger models? For example the iAF1260.xml model from the Bigg database (http://bigg.ucsd.edu:8888/models/iAF1260). This is a model of E. Coli that includes 1668 metabolites and 2382
reactions. We can download the iAF1260.xml file and load it into libRoadRunner using:
r = roadrunner.RoadRunner ('iAF1260.xml')
This might take up to a minute to load depending on how fast your computer is. We are assuming here that the file is located in the current directory (os.getcwd()). If not, move the file, change the
current directory (using os.chdir), or use the appropriate path in the call.
Rather than print out the stoichiometry matrix (don't even try) to the screen, we'll save it to a file. Because the stoichiometry matrix is so large we will use numpy to write the matrix out as a
text file:
import numpy as np
r = roadrunner.RoadRunner ('iAF1260.xml')
st = r.getFullStoichiometryMatrix()
print "Number of metabolites = ", r.getNumFloatingSpecies()
print "Number of reactions = ", r.getNumReactions()
np.savetxt ('stoich.txt', st)
Number of metabolites = 1668
Number of reactions = 2382
One can change the formatting of the output using savetxt, for example, the following will output the individual stoichiometry coefficient using 3 decimal places, 5 characters minimum, and separated
by a comma.
np.savetxt ('c:\\tmp\\st.txt', st, delimiter=',', fmt='%5.3f',)
You can get the labels for the rows and columns by calling r.getFloatingSpeciesIds() and r.getReactionIds() respectively.
November 1, 2016 1:10 pm
The most common requirement is the ability to carry out a simple time course simulation of a model. Consider the model:
$$ S_1 \rightarrow S_2 \rightarrow S_2 $$
Two reactions and three metabolites, S1, S2 and S3. We can describe this system using an Antimony string:
import tellurium as te
import roadrunner
r = te.loada ('''
S1 -> S2; k1*S1;
S2 -> S3; k2*S2;
k1= 0.4; k2 = 0.45
S1 = 5; S2 = 0; S3 = 0
m = r.simulate (0, 20, 100)
Run the script by clicking on the green arrow in the tool bar to yield:
If you're building a model and you want to quickly find the model's steady state, you can call the command steadyState. Let's illustrate this with an example:
import tellurium as te
r = te.loada ('''
# Define a simple linear chain of reactions
$Xo -> S1; k1*Xo;
S1 -> S2; k2*S1;
S2 -> S3; k3*S2;
S3 -> $X1; k4*S3;
# Initialize rate constants
k1 = 0.2; k2 = 0.7; k3 = 0.15; k4 = 1.3;
Xo = 10
print r.getSteadyStateValues()
Running this script by clicking in the green arrow in the tool bar will output the following steady state levels:
[2.85714286 13.33333333 1.53846154]
But which steady state value refers to which species in my model? To find that out call getFloatingSpeciesIds:
print r.getFloatingSpeciesIds()
['S1', 'S2', 'S3']
The order of the speces Ids match the order of steady state values. In other words, S1 = 2.857, S2 = 13.333, and S3 = 1.538.
If you want to check that these values really are the steady state values you can compute the rates of change:
print r.getRatesOfChange()
array([ 0.00000000e+00, -4.44089210e-16, 4.44089210e-16])
If you look carefully all the rates of change are effectively zero, thus confirming we're at steady state.
What about the stability of the steady state? That is, is it stable to small perturbations in S1, S2 or S3? To find this out we need to compute the eigenvalues of the system Jacobian matrix. If all
the eigenvalues are negative this means small pertubrations are damped so that the system returns to the steady state. There are two ways to do this, we can use the getReducedEigenValues call, or we
get compute the Jacobian first and then compute the eigenvalues of the Jacobian. Both these approaches are given below. Its probably simplest to call just getReducedEigenValues.
print r.getReducedEigenValues()
[-0.7 -0.15 -1.3 ]
Notice that all the eigenvalues are negative indicating that small perturbations are stable.
This is the alternative approach which computes the Jacobian first and then the eigenvalues of the Jacobian:
print te.getEigenvalues(r.getFullJacobian())
[-0.7 -0.15 -1.3 ]
Note that we're using a utility method from the tellurium library, getEigenValues to compute the eigenvalues.
Originally Posted on October 28, 2016 by hsauro
I thought I’d try and write a series of HowTos on Tellurium, our python-based tool for the construction, simulation and analysis of biochemical models. Details on this tool can be found here.
One of the unique features of Tellurium is that it comes with the AUTO2000 package. This is a well-established software library for performing a bifurcation analysis.
Unlike other implementations (not including oscill8), AUTO2000 in Tellurium does not require a separate compiler to compile the model for AUTO2000 to compute the differential equations. This makes it
easier to deploy and users don’t have to worry about such details. AUTO2000 uses libRoadRunner to access and compute the model.
Let’s begin with an example from my textbook “Systems Biology: Introduction to Pathway Modeling”, Figure 12.20, page 279 in revision 1.1 of the book. The model in question is a modified model from
the ‘Mutual activation’ model in the review by Tyson (Current Opinion in Cell Biology 15:221–231, Figure 1e). In this example increasing the signal results in the system switching to the high state
at around 2.0. If we reduce the signal from a high level, we traverse a different steady-state. If we assume the signal can never be negative, we will remain at the high steady-state even if the
signal is reduced to zero. The bifurcation plot in the negative quadrant of the graph is physically inaccessible. This means it is not possible to transition to the low steady-state by decreasing
signal. As a result, the bistable system is irreversible, that is, once it is switched on, it will always remain on. To compute the bifurcation diagram we first define the model:
import tellurium as te
5 import rrplugins
import pylab as plt
r = te.loada ('''
7 $X ->; R1; k1*EP + k2*Signal;
R1 ->; $w; k3*R1;
8 EP ->; E; Vm1*EP/(Km + EP);
E -> EP; ((Vm2+R1)*E)/(Km + E);
9 Vm1 = 12; Vm2 = 6;
Km = 0.6;
10 k1 = 1.6; k2 = 4;
E = 5; EP = 15;
11 k3 = 3; Signal = 0.1;
We’ve imported three packages, tellurium to load the model, rrplugins to access AUTO2000 and pylab to gain access to matplotlib. Once we have the model loaded we can get a handle on AUTO2000 by
calling rrplugins.Plugin(“tel_auto2000”) and set a number of properties in AUTO2000. This includes loading the model into AUTO200, identifying the parameter we wish to modify for the bifurcation
diagram (in this case signal), following by some options to carry out a pre simulation to help with the initial location of the steady-state and finally the limits for x axis for the plot, in this
case -2 to 3. Details of other properties to change can be found by typing auto.viewManual(), make sure you have a pdf reader available. The alternative is to go to the intro page.
3 auto = rrplugins.Plugin("tel_auto2000")
4 auto.setProperty("SBML", r.getCurrentSBML())
auto.setProperty("ScanDirection", "Negative")
5 auto.setProperty("PrincipalContinuationParameter", "Signal")
auto.setProperty("PreSimulation", "True")
6 auto.setProperty("PreSimulationDuration", 1.0)
auto.setProperty("RL0", -2.0)
7 auto.setProperty("RL1", 3.0)
To run the bifurcation analysis we use the Python code:
1 auto.execute()
If all was successful we can next plot the results. It is possible to plot the results using your own code (see below) but it might be more convenient to use the builtin facilties, for example:
pts = auto.BifurcationPoints
2 lbls = auto.BifurcationLabels
biData = auto.BifurcationData
3 biData.plotBifurcationDiagram(pts, lbls)
The pts vector contains the point coordinates where the bifurcation points are located. lbls give the labels that correspond to the pts vector and indicate what type of bifurcation point it
represented. Finally a special object, here called biData contains the data together with a number of useful utilities. The import important of these is biData.plotBifurcationDiagram(pts, lbls) which
takes pts and lbls as arguments.
We can also print out a text summary of the computation using the command, auto.BifurcationSummary, which returns a summary of the findings.
Summary :
4 BR PT TY LAB PAR(0) L2-NORM U(1) U(2)
1 1 EP 1 3.00000E+000 2.38908E+001 1.42336E+001 1.91879E+001
5 1 50 2 5.48632E-001 2.14502E+001 1.06592E+001 1.86144E+001
1 95 LP 3 -9.31897E-001 1.79088E+001 7.44448E+000 1.62881E+001
6 1 100 4 -9.13592E-001 1.74094E+001 7.22867E+000 1.58377E+001
1 150 5 1.74114E-001 1.26908E+001 6.15211E+000 1.10999E+001
7 1 200 6 1.54579E+000 8.36095E+000 5.44499E+000 6.34488E+000
1 234 LP 7 2.06965E+000 5.51178E+000 4.47540E+000 3.21723E+000
8 1 250 8 1.84381E+000 4.03137E+000 3.51306E+000 1.97747E+000
1 300 9 -5.77433E-001 7.02653E-001 -5.14930E-001 4.78089E-001
9 1 325 EP 10 -2.00625E+000 2.56175E+000 -2.55121E+000 2.32100E-001
We can manually plot the data by gaining access to the numpy version of the data. To do this we use:
1 pltData = biData.toNumpy
pltData is a numpy array where the first column is the bifurcation parameter and the remaning columns contain the species. For example to plot the bifurcation diagram for the first species in the
model, R1 we would use:
plt.plot(x[:,0], x[:,1], linewidth=2)
2 plt.axvline(0, color='black')
plt.xlabel ("Signal")
3 plt.ylabel ("R1")
I added a axvline command to draw a vertical line from the zero axis. I also added some axis labeling statements. These commands will result in:
What is interesting about this model is that the upper branch reaches the zero parameter value before the turning point. This means it is difficult to switch to the lower steady-state by just
lowering the signal.
Viewing the Network
One other thing we can do is view the model as a network. Tellurium comes with a simple network viewer in the package nwed. import the viewer using
import nwed
at the ipython console. To view the network make sure the network viewer panel is visible, do this by going to the View menu, find panes and select, then look down the menu items, and near the bottom
you'll find Network Viewer, select this option. To view the network, type the following at the ipython console.
nwed.setsbml (r.getSBML())
The viewer should now display something like:
Note that every view will be different and depends on the layout algorithm. | {"url":"https://blog.hsauro.org/2016/","timestamp":"2024-11-06T18:51:08Z","content_type":"application/xhtml+xml","content_length":"306784","record_id":"<urn:uuid:225b6a08-3927-4ef4-adbb-4383052435cb>","cc-path":"CC-MAIN-2024-46/segments/1730477027933.5/warc/CC-MAIN-20241106163535-20241106193535-00663.warc.gz"} |
The Relation class represents the core concept of relational algebra: a table with a well-defined set of columns and unique rows. A Relation instance does not necessarily correspond to a concrete
in-memory or on-disk table, however; most derived Relation types actually represent an operation on some other “target” relation or relations, forming an expression tree.
Operations on relations are represented by subclasses of UnaryOperation and BinaryOperation, which are associated with a target relation to form a new relation via the UnaryOperationRelation and
BinaryOperationRelation classes. The LeafRelation class handles relations that represent direct storage of rows (and in some cases actually do store rows themselves). The Relation interface provides
factory methods for constructing and applying operations to relation that should be used instead of directly interacting with operation classes when possible.
The fourth and final Relation class is MarkerRelation, which adds some information or context to a target relation without changing the relation’s actual content. Unlike other relation types,
MarkerRelation can be inherited from freely.
Concrete Relation classes (including extensions) should be frozen dataclasses, ensuring that they are immutable (with the exception of Relation.payload), equality comparable, hashable, and that they
have a useful (but in still lossy) repr. Relation classes also provide a str representation that is much more concise, for cases where seeing the overall tree is more important than the details of
any particular relation.
Relations are associated with “engines”: systems that may hold the actual data a relation (at least) conceptually represents and can perform operations on them to obtain the derived data. These are
represented by Engine instances held by the relation objects themselves, and the sql and iteration subpackages provide at least partial implementations of engines for relations backed by SQL
databases (via SQLAlchemy) and native Python iterables, respectively. A relation tree can include multiple engines, by using a Transfer relation class to mark the boundaries between them. The
Processor class can be used to execute multiple-engine trees.
It is up to an engine how strictly its operations adhere to relational algebra operation definition. SQL is formally defined in terms of operations on “bags” or “multisets”, whose rows are not unique
and sometimes ordered, while formal relations are always unordered and unique. The Relation interface has more a more permissive view of uniqueness to facilitate interaction with SQL: a Relation may
have non-unique rows, but any duplicates are not meaningful, and hence most operations may remove or propagate duplicates at their discretion, though engines may make stronger guarantees and most
relations are not permitted to introduce duplication when applied to a base relation with unique rows. It is also up to engines to determine whether an operations maintains the order of rows; SQL
engine operations often do not, while the iteration engine’s operations always do.
LeafRelation and MarkerRelation objects can have an engine-specific payload attribute that either holds the actual relation state for that engine or a reference to it. The iteration engine’s payloads
are instances of the iteration.RowIterable interface, while the SQL engine’s payloads are sql.Payload instances, which can represent a SQLAlchemy table or subquery expression. LeafRelation objects
always have a payload that is not None. Materialization markers indicate that a payload should be attached when the relation is first “executed” by an engine or the Processor class, allowing
subsequent executions to reuse that payload and avoid repeating upstream execution. Attaching a payload is the only way a relation can be modified after construction, and a payload that is not None
can never be replaced.
The full set of unary and binary operations provided is given below, along with the Relation factory methods that can be used to apply certain operations directly. Applying an operation to a relation
always returns a new relation (unless the operation is a no-op, in which case the original may be returned unchanged), and always acts lazily: applying an operation is not the same as processing or
executing a relation tree that contains that operation.
Add a new column whose values are calculated from one or more existing columns, via by a column expression.
Concatenate the rows of two relations that have the same columns. This is equivalent to UNION ALL in SQL or itertools.chain in Python.
Remove duplicate rows. This is equivalent to SELECT DISTINCT in SQL or filtering through set or dict in Python.
Do nothing. This operation never actually appears in Relation trees; Identity.apply always returns the operand relation passed to it.
Perform a natural join: combine two relations by matching rows with the same values in their common columns (and satisfying an optional column expression, via a Predicate), producing a new
relation whose columns are the union of the columns of its operands. This is equivalent to [INNER] JOIN in SQL.
Remove some columns from a relation.
An intermediate abstract base class for unary operations that only change the order of rows.
An intermediate abstract base class for unary operations that only remove rows.
Remove rows that satisfy a boolean column expression. This is equivalent to the WHERE clause in SQL.
Remove rows outside an integer range of indices. This is equivalent to OFFSET and LIMIT in SQL, or indexing with slice object or start:stop syntax in Python.
Sort rows according to a column expression.
Column Expressions¶
Many relation operations involve column expressions, such as the boolean filter used in a Selection or the sort keys used in a Sort. These are represented by the ColumnExpression (for general
scalar-valued expressions), Predicate (for boolean expressions), and ColumnContainer (for expressions that hold multiple values) abstract base classes. These base classes provide factory methods for
all derived classes, making it generally unnecessary to refer to those types directly (except when writing an algorithm or engine that processes a relation tree; see Extensibility). Column expression
objects can in general support multiple engines; some types are required to be supported by all engines, while others can hold a list of engine types that support them. The concrete column expression
classes provided by lsst.daf.relation are given below, with their factory methods:
A constant scalar, non-boolean Python value.
A reference to a scalar, non-boolean column in a relation.
A named function that returns a scalar, non-boolean value given scalar, non-boolean arguments. It is up to each Engine how and whether it supports a ColumnFunction; this could include looking up
the name in some module or treating it as a method that should be present on some object that represents a column value more directly in that engine.
A sequence of one or more ColumnExpression objects, representing the same column type (but not neecessarily the same expression type.
A virtual sequence of literal integer values represented by a Python range object.
A constant True or False value.
A reference to a boolean-valued column in a relation.
A named function that returns a boolean, given scalar, non-boolean arguments. Like ColumnFunction, implementation and support are engine-specific. ColumnExpression also has eq, ne, lt, le, gt,
and ge methods for the common case of PredicateFunction objects that represent comparison operators.
Boolean expression that is True if its (single, boolean) argument is False, and vice versa.
Boolean expression that is True only if all (boolean) arguments are True.
Boolean expression that is True if any (boolean) argument is True.
Boolean expression that tests whether a scalar ColumnExpression is included in a ColumnContainer.
Ideally, this library would be extensible in three different ways:
• external Relation or operation types could be defined, representing new kinds of nodes in a relation expression tree;
• external Engine types could be defined, representing different ways of storing and manipulating tabular data;
• external algorithms on relation trees could be defined.
Unfortunately, any custom Engine or relation-tree algorithm in practice needs to be able to enumerate all possible Relation and operation types (or at least reject any trees with Relation or
operation types it does not recognize). Similarly, any custom Relation or operation type would need to be able to enumerate the algorithms and engines it supports, in order to provide its own
implementations for those algorithms and engines. This is a fragile multiple-dispatch problem.
To simplify things, this package chooses to prohibit most kinds of Relation and operation extensibility:
• custom Relation subclasses must be MarkerRelation subclasses;
• custom BinaryOperation and column expression subclasses are not permitted;
• custom subclasses of UnaryOperation are restricted to subclasses of the more limited RowFilter and Reordering intermediate interfaces.
These prohibitions are enforced by __init_subclass__ checks in the abstract base classes.
Relation is actually a typing.Protocol, not (just) an ABC, and the concrete LeafRelation, UnaryOperationRelation, BinaryOperationRelation, and MarkerRelation classes actually inherit from
BaseRelation while satisfying the Relation interface only in a structural subtyping sense. This allows various Relation attribute interfaces (e.g. Relation.engine) to be implemented as either true
properties or dataclass fields, and it should be invisible to users except in the rare case that they need to perform a runtime isinstance check with the Relation type itself, not just a specific
concrete Relation subclass: in this case BaseRelation must be used instead of Relation.
The standard approach to designs like this in object oriented programming is the Visitor Pattern, which in this case would involve a base class or suite of base classes for algorithms or engines with
a method for each possible relation-tree node type (relations, operations, column expressions); these would be invoked by a method on each node interface whose concrete implementations call the
corresponding algorithm or engine method. This implicitly restricts the set of node tree types to those with methods in the algorithm/engine base class.
In languages with functional programming aspects, a much more direct approach involving enumerated types and pattern-matching syntax is often possible. With the introduction of the match statement in
Python 3.10, this now includes Python, and this is the approach taken in here. This results in much more readable and concise code than the boilerplate-heavy visitor-pattern alternative, but it comes
with a significant drawback: there are no checks (either at runtime or via static type checkers like MyPy) that all necessary case branches of a match are present. This is in part by design - many
algorithms on relation trees can act generically on most operation types, and hence need to only explicitly match one or two - but it requires considerable discipline from algorithm and engine
authors to ensure that match logic is correct and well-tested. Another (smaller) drawback is that it can occasionally yield code that in other contexts might be considered antipatterns (e.g.
isinstance is often a good alternative to a single-branch match).
Python API reference¶
lsst.daf.relation Package¶
BaseRelation() An implementation-focused target class for concrete Relation objects.
BinaryOperation() An abstract base class for operations that act on a pair of relations.
BinaryOperationRelation(operation, lhs, rhs, ...) A concrete Relation that represents the action of a BinaryOperation on a pair of target Relation objects.
Calculation(tag, expression) A relation operation that adds a new column from an expression involving existing columns.
Chain() A relation operation that concatenates the rows of a pair of relations with the same columns.
ColumnContainer() A abstract base class and factory for expressions that represent containers of multiple column values.
ColumnError Exception type raised to indicate problems in a relation's columns and/or unique keys.
ColumnExpression() An abstract base class and factory for scalar, non-boolean column expressions.
ColumnExpressionSequence(items, dtype) A container expression backed by a sequence of scalar column expressions.
ColumnFunction(name, args, dtype, ...) A concrete column expression that represents calling a named function with column expression arguments.
ColumnInContainer(item, container) A boolean column expression that tests whether a scalar column expression is present in a container expression.
ColumnLiteral(value, dtype) A concrete column expression backed by a regular Python value.
ColumnRangeLiteral(value, dtype) A container expression backed by a range of integer indices.
ColumnReference(tag, dtype) A concrete column expression that refers to a relation column.
ColumnTag(*args, **kwargs) An interface for objects that represent columns in a relation.
Deduplication() A relation operation that removes duplicate rows.
Diagnostics(is_doomed, messages) A relation-processing algorithm that attempts to explain why a relation has no rows.
Engine() An abstract interface for the systems that hold relation data and know how to process relation trees.
EngineError Exception type raised to indicate that a relation tree is incompatible with an engine, or has inconsistent engines.
GenericConcreteEngine(*, name, functions, ...) An implementation-focused base class for Engine objects.
Identity() A concrete unary operation that does nothing.
IgnoreOne(ignore_lhs) A binary operation that passes through one of its operands and ignores the other.
Join(predicate, min_columns, max_columns) A natural join operation.
LeafRelation(engine, columns, payload[, ...]) A Relation class that represents direct storage of rows, rather than an operation on some other relation.
LogicalAnd(operands) A concrete boolean column expression that ANDs its operands.
LogicalNot(operand) A concrete boolean column expression that inverts its operand.
LogicalOr(operands) A concrete boolean column expression that ORs its operands.
MarkerRelation(target[, payload]) An extensible relation base class that provides additional information about another relation without changing its row-and-column content.
Materialization(target[, payload]) A marker operation that indicates that the upstream tree should be evaluated only once, with the results saved and reused for subsequent processing.
PartialJoin(binary, fixed, fixed_is_lhs) A UnaryOperation that represents this join with one operand already provided and held fixed.
Predicate() An abstract base class and factory for boolean column expressions.
PredicateFunction(name, args, ...) A concrete boolean expression that represents calling an named function with column expression arguments.
PredicateLiteral(value) A concrete boolean column expression that is a constant True or False.
PredicateReference(tag) A concrete boolean column expression that refers to a boolean relation column.
Processor() An inheritable framework for processing multi-engine relation trees.
Projection(columns) A unary operation that removes one or more columns.
Relation(*args, **kwargs) An abstract interface for expression trees on tabular data.
RelationalAlgebraError Exception type raised to indicate problems in the definition of a relation tree.
Reordering() An extensible UnaryOperation subclass for operations that only reorder rows.
RowFilter() An extensible UnaryOperation subclass for operations that only remove rows from their target.
Selection(predicate) A relation operation that filters rows according to a boolean column expression.
Slice([start, stop]) A relation relation that filters rows that are outside a range of positional indices.
Sort([terms]) A relation operation that orders rows according to a sequence of column expressions.
SortTerm(expression[, ascending]) Sort expression and indication of sort direction.
Transfer(target[, payload]) A MarkerRelation operation that represents moving relation content from one engine to another.
UnaryCommutator(first, second, done, ...) A struct for the return value of UnaryOperation.commute.
UnaryOperation() An abstract base class for operations that act on a single relation.
UnaryOperationRelation(operation, target, ...) A concrete Relation that represents the action of a UnaryOperation on a target Relation.
Class Inheritance Diagram¶
digraph inheritancedddf16fe7c { bgcolor=transparent; rankdir=LR; size="8.0, 12.0"; "ABC" [fillcolor=white,fontname="Vera Sans, DejaVu Sans, Liberation Sans, Arial, Helvetica, sans",fontsize=10,height
=0.25,shape=box,style="setlinewidth(0.5),filled",tooltip="Helper class that provides a standard way to create an ABC using"]; "BaseRelation" [URL="../../../py-api/lsst.daf.relation.BaseRelation.html#
lsst.daf.relation.BaseRelation",fillcolor=white,fontname="Vera Sans, DejaVu Sans, Liberation Sans, Arial, Helvetica, sans",fontsize=10,height=0.25,shape=box,style="setlinewidth(0.5),filled",target=
"_top",tooltip="An implementation-focused target class for concrete `Relation` objects."]; "BinaryOperation" [URL="../../../py-api/lsst.daf.relation.BinaryOperation.html#
lsst.daf.relation.BinaryOperation",fillcolor=white,fontname="Vera Sans, DejaVu Sans, Liberation Sans, Arial, Helvetica, sans",fontsize=10,height=0.25,shape=box,style="setlinewidth(0.5),filled",target
="_top",tooltip="An abstract base class for operations that act on a pair of relations."]; "ABC" -> "BinaryOperation" [arrowsize=0.5,style="setlinewidth(0.5)"]; "BinaryOperationRelation" [URL="../..
/../py-api/lsst.daf.relation.BinaryOperationRelation.html#lsst.daf.relation.BinaryOperationRelation",fillcolor=white,fontname="Vera Sans, DejaVu Sans, Liberation Sans, Arial, Helvetica,
sans",fontsize=10,height=0.25,shape=box,style="setlinewidth(0.5),filled",target="_top",tooltip="A concrete `Relation` that represents the action of a `BinaryOperation`"]; "BaseRelation" ->
"BinaryOperationRelation" [arrowsize=0.5,style="setlinewidth(0.5)"]; "Calculation" [URL="../../../py-api/lsst.daf.relation.Calculation.html#lsst.daf.relation.Calculation",fillcolor=white,fontname=
"Vera Sans, DejaVu Sans, Liberation Sans, Arial, Helvetica, sans",fontsize=10,height=0.25,shape=box,style="setlinewidth(0.5),filled",target="_top",tooltip="A relation operation that adds a new column
from an expression involving"]; "UnaryOperation" -> "Calculation" [arrowsize=0.5,style="setlinewidth(0.5)"]; "Chain" [URL="../../../py-api/lsst.daf.relation.Chain.html#
lsst.daf.relation.Chain",fillcolor=white,fontname="Vera Sans, DejaVu Sans, Liberation Sans, Arial, Helvetica, sans",fontsize=10,height=0.25,shape=box,style="setlinewidth(0.5),filled",target=
"_top",tooltip="A relation operation that concatenates the rows of a pair of relations"]; "BinaryOperation" -> "Chain" [arrowsize=0.5,style="setlinewidth(0.5)"]; "ColumnContainer" [URL="../../../
py-api/lsst.daf.relation.ColumnContainer.html#lsst.daf.relation.ColumnContainer",fillcolor=white,fontname="Vera Sans, DejaVu Sans, Liberation Sans, Arial, Helvetica, sans",fontsize=10,height=
0.25,shape=box,style="setlinewidth(0.5),filled",target="_top",tooltip="A abstract base class and factory for expressions that represent"]; "ABC" -> "ColumnContainer" [arrowsize=0.5,style=
"setlinewidth(0.5)"]; "ColumnError" [URL="../../../py-api/lsst.daf.relation.ColumnError.html#lsst.daf.relation.ColumnError",fillcolor=white,fontname="Vera Sans, DejaVu Sans, Liberation Sans, Arial,
Helvetica, sans",fontsize=10,height=0.25,shape=box,style="setlinewidth(0.5),filled",target="_top",tooltip="Exception type raised to indicate problems in a relation's columns"];
"RelationalAlgebraError" -> "ColumnError" [arrowsize=0.5,style="setlinewidth(0.5)"]; "ColumnExpression" [URL="../../../py-api/lsst.daf.relation.ColumnExpression.html#
lsst.daf.relation.ColumnExpression",fillcolor=white,fontname="Vera Sans, DejaVu Sans, Liberation Sans, Arial, Helvetica, sans",fontsize=10,height=0.25,shape=box,style="setlinewidth
(0.5),filled",target="_top",tooltip="An abstract base class and factory for scalar, non-boolean column"]; "ABC" -> "ColumnExpression" [arrowsize=0.5,style="setlinewidth(0.5)"];
"ColumnExpressionSequence" [URL="../../../py-api/lsst.daf.relation.ColumnExpressionSequence.html#lsst.daf.relation.ColumnExpressionSequence",fillcolor=white,fontname="Vera Sans, DejaVu Sans,
Liberation Sans, Arial, Helvetica, sans",fontsize=10,height=0.25,shape=box,style="setlinewidth(0.5),filled",target="_top",tooltip="A container expression backed by a sequence of scalar column"];
"ColumnContainer" -> "ColumnExpressionSequence" [arrowsize=0.5,style="setlinewidth(0.5)"]; "ColumnFunction" [URL="../../../py-api/lsst.daf.relation.ColumnFunction.html#
lsst.daf.relation.ColumnFunction",fillcolor=white,fontname="Vera Sans, DejaVu Sans, Liberation Sans, Arial, Helvetica, sans",fontsize=10,height=0.25,shape=box,style="setlinewidth(0.5),filled",target=
"_top",tooltip="A concrete column expression that represents calling a named function"]; "ColumnExpression" -> "ColumnFunction" [arrowsize=0.5,style="setlinewidth(0.5)"]; "ColumnInContainer" [URL="..
/../../py-api/lsst.daf.relation.ColumnInContainer.html#lsst.daf.relation.ColumnInContainer",fillcolor=white,fontname="Vera Sans, DejaVu Sans, Liberation Sans, Arial, Helvetica, sans",fontsize=
10,height=0.25,shape=box,style="setlinewidth(0.5),filled",target="_top",tooltip="A boolean column expression that tests whether a scalar column"]; "Predicate" -> "ColumnInContainer" [arrowsize=
0.5,style="setlinewidth(0.5)"]; "ColumnLiteral" [URL="../../../py-api/lsst.daf.relation.ColumnLiteral.html#lsst.daf.relation.ColumnLiteral",fillcolor=white,fontname="Vera Sans, DejaVu Sans,
Liberation Sans, Arial, Helvetica, sans",fontsize=10,height=0.25,shape=box,style="setlinewidth(0.5),filled",target="_top",tooltip="A concrete column expression backed by a regular Python value."];
"ColumnExpression" -> "ColumnLiteral" [arrowsize=0.5,style="setlinewidth(0.5)"]; "ColumnRangeLiteral" [URL="../../../py-api/lsst.daf.relation.ColumnRangeLiteral.html#
lsst.daf.relation.ColumnRangeLiteral",fillcolor=white,fontname="Vera Sans, DejaVu Sans, Liberation Sans, Arial, Helvetica, sans",fontsize=10,height=0.25,shape=box,style="setlinewidth
(0.5),filled",target="_top",tooltip="A container expression backed by a range of integer indices."]; "ColumnContainer" -> "ColumnRangeLiteral" [arrowsize=0.5,style="setlinewidth(0.5)"];
"ColumnReference" [URL="../../../py-api/lsst.daf.relation.ColumnReference.html#lsst.daf.relation.ColumnReference",fillcolor=white,fontname="Vera Sans, DejaVu Sans, Liberation Sans, Arial, Helvetica,
sans",fontsize=10,height=0.25,shape=box,style="setlinewidth(0.5),filled",target="_top",tooltip="A concrete column expression that refers to a relation column."]; "ColumnExpression" ->
"ColumnReference" [arrowsize=0.5,style="setlinewidth(0.5)"]; "ColumnTag" [URL="../../../py-api/lsst.daf.relation.ColumnTag.html#lsst.daf.relation.ColumnTag",fillcolor=white,fontname="Vera Sans,
DejaVu Sans, Liberation Sans, Arial, Helvetica, sans",fontsize=10,height=0.25,shape=box,style="setlinewidth(0.5),filled",target="_top",tooltip="An interface for objects that represent columns in a
relation."]; "Hashable" -> "ColumnTag" [arrowsize=0.5,style="setlinewidth(0.5)"]; "Protocol" -> "ColumnTag" [arrowsize=0.5,style="setlinewidth(0.5)"]; "Deduplication" [URL="../../../py-api/
lsst.daf.relation.Deduplication.html#lsst.daf.relation.Deduplication",fillcolor=white,fontname="Vera Sans, DejaVu Sans, Liberation Sans, Arial, Helvetica, sans",fontsize=10,height=0.25,shape=
box,style="setlinewidth(0.5),filled",target="_top",tooltip="A relation operation that removes duplicate rows."]; "UnaryOperation" -> "Deduplication" [arrowsize=0.5,style="setlinewidth(0.5)"];
"Diagnostics" [URL="../../../py-api/lsst.daf.relation.Diagnostics.html#lsst.daf.relation.Diagnostics",fillcolor=white,fontname="Vera Sans, DejaVu Sans, Liberation Sans, Arial, Helvetica,
sans",fontsize=10,height=0.25,shape=box,style="setlinewidth(0.5),filled",target="_top",tooltip="A relation-processing algorithm that attempts to explain why a relation"]; "Engine" [URL="../../../
py-api/lsst.daf.relation.Engine.html#lsst.daf.relation.Engine",fillcolor=white,fontname="Vera Sans, DejaVu Sans, Liberation Sans, Arial, Helvetica, sans",fontsize=10,height=0.25,shape=box,style=
"setlinewidth(0.5),filled",target="_top",tooltip="An abstract interface for the systems that hold relation data and know"]; "Hashable" -> "Engine" [arrowsize=0.5,style="setlinewidth(0.5)"];
"EngineError" [URL="../../../py-api/lsst.daf.relation.EngineError.html#lsst.daf.relation.EngineError",fillcolor=white,fontname="Vera Sans, DejaVu Sans, Liberation Sans, Arial, Helvetica,
sans",fontsize=10,height=0.25,shape=box,style="setlinewidth(0.5),filled",target="_top",tooltip="Exception type raised to indicate that a relation tree is incompatible"]; "RelationalAlgebraError" ->
"EngineError" [arrowsize=0.5,style="setlinewidth(0.5)"]; "Generic" [fillcolor=white,fontname="Vera Sans, DejaVu Sans, Liberation Sans, Arial, Helvetica, sans",fontsize=10,height=0.25,shape=box,style=
"setlinewidth(0.5),filled",tooltip="Abstract base class for generic types."]; "GenericConcreteEngine" [URL="../../../py-api/lsst.daf.relation.GenericConcreteEngine.html#
lsst.daf.relation.GenericConcreteEngine",fillcolor=white,fontname="Vera Sans, DejaVu Sans, Liberation Sans, Arial, Helvetica, sans",fontsize=10,height=0.25,shape=box,style="setlinewidth
(0.5),filled",target="_top",tooltip="An implementation-focused base class for `Engine` objects."]; "Engine" -> "GenericConcreteEngine" [arrowsize=0.5,style="setlinewidth(0.5)"]; "Generic" ->
"GenericConcreteEngine" [arrowsize=0.5,style="setlinewidth(0.5)"]; "Hashable" [fillcolor=white,fontname="Vera Sans, DejaVu Sans, Liberation Sans, Arial, Helvetica, sans",fontsize=10,height=0.25,shape
=box,style="setlinewidth(0.5),filled"]; "Identity" [URL="../../../py-api/lsst.daf.relation.Identity.html#lsst.daf.relation.Identity",fillcolor=white,fontname="Vera Sans, DejaVu Sans, Liberation Sans,
Arial, Helvetica, sans",fontsize=10,height=0.25,shape=box,style="setlinewidth(0.5),filled",target="_top",tooltip="A concrete unary operation that does nothing."]; "UnaryOperation" -> "Identity"
[arrowsize=0.5,style="setlinewidth(0.5)"]; "IgnoreOne" [URL="../../../py-api/lsst.daf.relation.IgnoreOne.html#lsst.daf.relation.IgnoreOne",fillcolor=white,fontname="Vera Sans, DejaVu Sans, Liberation
Sans, Arial, Helvetica, sans",fontsize=10,height=0.25,shape=box,style="setlinewidth(0.5),filled",target="_top",tooltip="A binary operation that passes through one of its operands and ignores"];
"BinaryOperation" -> "IgnoreOne" [arrowsize=0.5,style="setlinewidth(0.5)"]; "Join" [URL="../../../py-api/lsst.daf.relation.Join.html#lsst.daf.relation.Join",fillcolor=white,fontname="Vera Sans,
DejaVu Sans, Liberation Sans, Arial, Helvetica, sans",fontsize=10,height=0.25,shape=box,style="setlinewidth(0.5),filled",target="_top",tooltip="A natural join operation."]; "BinaryOperation" ->
"Join" [arrowsize=0.5,style="setlinewidth(0.5)"]; "LeafRelation" [URL="../../../py-api/lsst.daf.relation.LeafRelation.html#lsst.daf.relation.LeafRelation",fillcolor=white,fontname="Vera Sans, DejaVu
Sans, Liberation Sans, Arial, Helvetica, sans",fontsize=10,height=0.25,shape=box,style="setlinewidth(0.5),filled",target="_top",tooltip="A `Relation` class that represents direct storage of rows,
rather than"]; "BaseRelation" -> "LeafRelation" [arrowsize=0.5,style="setlinewidth(0.5)"]; "LogicalAnd" [URL="../../../py-api/lsst.daf.relation.LogicalAnd.html#lsst.daf.relation.LogicalAnd",fillcolor
=white,fontname="Vera Sans, DejaVu Sans, Liberation Sans, Arial, Helvetica, sans",fontsize=10,height=0.25,shape=box,style="setlinewidth(0.5),filled",target="_top",tooltip="A concrete boolean column
expression that ANDs its operands."]; "Predicate" -> "LogicalAnd" [arrowsize=0.5,style="setlinewidth(0.5)"]; "LogicalNot" [URL="../../../py-api/lsst.daf.relation.LogicalNot.html#
lsst.daf.relation.LogicalNot",fillcolor=white,fontname="Vera Sans, DejaVu Sans, Liberation Sans, Arial, Helvetica, sans",fontsize=10,height=0.25,shape=box,style="setlinewidth(0.5),filled",target=
"_top",tooltip="A concrete boolean column expression that inverts its operand."]; "Predicate" -> "LogicalNot" [arrowsize=0.5,style="setlinewidth(0.5)"]; "LogicalOr" [URL="../../../py-api/
lsst.daf.relation.LogicalOr.html#lsst.daf.relation.LogicalOr",fillcolor=white,fontname="Vera Sans, DejaVu Sans, Liberation Sans, Arial, Helvetica, sans",fontsize=10,height=0.25,shape=box,style=
"setlinewidth(0.5),filled",target="_top",tooltip="A concrete boolean column expression that ORs its operands."]; "Predicate" -> "LogicalOr" [arrowsize=0.5,style="setlinewidth(0.5)"]; "MarkerRelation"
[URL="../../../py-api/lsst.daf.relation.MarkerRelation.html#lsst.daf.relation.MarkerRelation",fillcolor=white,fontname="Vera Sans, DejaVu Sans, Liberation Sans, Arial, Helvetica, sans",fontsize=
10,height=0.25,shape=box,style="setlinewidth(0.5),filled",target="_top",tooltip="An extensible relation base class that provides additional information"]; "BaseRelation" -> "MarkerRelation"
[arrowsize=0.5,style="setlinewidth(0.5)"]; "Materialization" [URL="../../../py-api/lsst.daf.relation.Materialization.html#lsst.daf.relation.Materialization",fillcolor=white,fontname="Vera Sans,
DejaVu Sans, Liberation Sans, Arial, Helvetica, sans",fontsize=10,height=0.25,shape=box,style="setlinewidth(0.5),filled",target="_top",tooltip="A marker operation that indicates that the upstream
tree should be"]; "MarkerRelation" -> "Materialization" [arrowsize=0.5,style="setlinewidth(0.5)"]; "PartialJoin" [URL="../../../py-api/lsst.daf.relation.PartialJoin.html#
lsst.daf.relation.PartialJoin",fillcolor=white,fontname="Vera Sans, DejaVu Sans, Liberation Sans, Arial, Helvetica, sans",fontsize=10,height=0.25,shape=box,style="setlinewidth(0.5),filled",target=
"_top",tooltip="A `UnaryOperation` that represents this join with one operand already"]; "UnaryOperation" -> "PartialJoin" [arrowsize=0.5,style="setlinewidth(0.5)"]; "Predicate" [URL="../../../py-api
/lsst.daf.relation.Predicate.html#lsst.daf.relation.Predicate",fillcolor=white,fontname="Vera Sans, DejaVu Sans, Liberation Sans, Arial, Helvetica, sans",fontsize=10,height=0.25,shape=box,style=
"setlinewidth(0.5),filled",target="_top",tooltip="An abstract base class and factory for boolean column expressions."]; "ABC" -> "Predicate" [arrowsize=0.5,style="setlinewidth(0.5)"];
"PredicateFunction" [URL="../../../py-api/lsst.daf.relation.PredicateFunction.html#lsst.daf.relation.PredicateFunction",fillcolor=white,fontname="Vera Sans, DejaVu Sans, Liberation Sans, Arial,
Helvetica, sans",fontsize=10,height=0.25,shape=box,style="setlinewidth(0.5),filled",target="_top",tooltip="A concrete boolean expression that represents calling an named function"]; "Predicate" ->
"PredicateFunction" [arrowsize=0.5,style="setlinewidth(0.5)"]; "PredicateLiteral" [URL="../../../py-api/lsst.daf.relation.PredicateLiteral.html#lsst.daf.relation.PredicateLiteral",fillcolor=
white,fontname="Vera Sans, DejaVu Sans, Liberation Sans, Arial, Helvetica, sans",fontsize=10,height=0.25,shape=box,style="setlinewidth(0.5),filled",target="_top",tooltip="A concrete boolean column
expression that is a constant `True` or"]; "Predicate" -> "PredicateLiteral" [arrowsize=0.5,style="setlinewidth(0.5)"]; "PredicateReference" [URL="../../../py-api/
lsst.daf.relation.PredicateReference.html#lsst.daf.relation.PredicateReference",fillcolor=white,fontname="Vera Sans, DejaVu Sans, Liberation Sans, Arial, Helvetica, sans",fontsize=10,height=
0.25,shape=box,style="setlinewidth(0.5),filled",target="_top",tooltip="A concrete boolean column expression that refers to a boolean relation"]; "Predicate" -> "PredicateReference" [arrowsize=
0.5,style="setlinewidth(0.5)"]; "Processor" [URL="../../../py-api/lsst.daf.relation.Processor.html#lsst.daf.relation.Processor",fillcolor=white,fontname="Vera Sans, DejaVu Sans, Liberation Sans,
Arial, Helvetica, sans",fontsize=10,height=0.25,shape=box,style="setlinewidth(0.5),filled",target="_top",tooltip="An inheritable framework for processing multi-engine relation trees."]; "ABC" ->
"Processor" [arrowsize=0.5,style="setlinewidth(0.5)"]; "Projection" [URL="../../../py-api/lsst.daf.relation.Projection.html#lsst.daf.relation.Projection",fillcolor=white,fontname="Vera Sans, DejaVu
Sans, Liberation Sans, Arial, Helvetica, sans",fontsize=10,height=0.25,shape=box,style="setlinewidth(0.5),filled",target="_top",tooltip="A unary operation that removes one or more columns."];
"UnaryOperation" -> "Projection" [arrowsize=0.5,style="setlinewidth(0.5)"]; "Protocol" [fillcolor=white,fontname="Vera Sans, DejaVu Sans, Liberation Sans, Arial, Helvetica, sans",fontsize=10,height=
0.25,shape=box,style="setlinewidth(0.5),filled",tooltip="Base class for protocol classes."]; "Generic" -> "Protocol" [arrowsize=0.5,style="setlinewidth(0.5)"]; "Relation" [URL="../../../py-api/
lsst.daf.relation.Relation.html#lsst.daf.relation.Relation",fillcolor=white,fontname="Vera Sans, DejaVu Sans, Liberation Sans, Arial, Helvetica, sans",fontsize=10,height=0.25,shape=box,style=
"setlinewidth(0.5),filled",target="_top",tooltip="An abstract interface for expression trees on tabular data."]; "Protocol" -> "Relation" [arrowsize=0.5,style="setlinewidth(0.5)"];
"RelationalAlgebraError" [URL="../../../py-api/lsst.daf.relation.RelationalAlgebraError.html#lsst.daf.relation.RelationalAlgebraError",fillcolor=white,fontname="Vera Sans, DejaVu Sans, Liberation
Sans, Arial, Helvetica, sans",fontsize=10,height=0.25,shape=box,style="setlinewidth(0.5),filled",target="_top",tooltip="Exception type raised to indicate problems in the definition of a"];
"Reordering" [URL="../../../py-api/lsst.daf.relation.Reordering.html#lsst.daf.relation.Reordering",fillcolor=white,fontname="Vera Sans, DejaVu Sans, Liberation Sans, Arial, Helvetica, sans",fontsize=
10,height=0.25,shape=box,style="setlinewidth(0.5),filled",target="_top",tooltip="An extensible `UnaryOperation` subclass for operations that only reorder"]; "UnaryOperation" -> "Reordering"
[arrowsize=0.5,style="setlinewidth(0.5)"]; "RowFilter" [URL="../../../py-api/lsst.daf.relation.RowFilter.html#lsst.daf.relation.RowFilter",fillcolor=white,fontname="Vera Sans, DejaVu Sans, Liberation
Sans, Arial, Helvetica, sans",fontsize=10,height=0.25,shape=box,style="setlinewidth(0.5),filled",target="_top",tooltip="An extensible `UnaryOperation` subclass for operations that only remove"];
"UnaryOperation" -> "RowFilter" [arrowsize=0.5,style="setlinewidth(0.5)"]; "Selection" [URL="../../../py-api/lsst.daf.relation.Selection.html#lsst.daf.relation.Selection",fillcolor=white,fontname=
"Vera Sans, DejaVu Sans, Liberation Sans, Arial, Helvetica, sans",fontsize=10,height=0.25,shape=box,style="setlinewidth(0.5),filled",target="_top",tooltip="A relation operation that filters rows
according to a boolean column"]; "RowFilter" -> "Selection" [arrowsize=0.5,style="setlinewidth(0.5)"]; "Slice" [URL="../../../py-api/lsst.daf.relation.Slice.html#lsst.daf.relation.Slice",fillcolor=
white,fontname="Vera Sans, DejaVu Sans, Liberation Sans, Arial, Helvetica, sans",fontsize=10,height=0.25,shape=box,style="setlinewidth(0.5),filled",target="_top",tooltip="A relation relation that
filters rows that are outside a range of"]; "RowFilter" -> "Slice" [arrowsize=0.5,style="setlinewidth(0.5)"]; "Sort" [URL="../../../py-api/lsst.daf.relation.Sort.html#
lsst.daf.relation.Sort",fillcolor=white,fontname="Vera Sans, DejaVu Sans, Liberation Sans, Arial, Helvetica, sans",fontsize=10,height=0.25,shape=box,style="setlinewidth(0.5),filled",target=
"_top",tooltip="A relation operation that orders rows according to a sequence of"]; "Reordering" -> "Sort" [arrowsize=0.5,style="setlinewidth(0.5)"]; "SortTerm" [URL="../../../py-api/
lsst.daf.relation.SortTerm.html#lsst.daf.relation.SortTerm",fillcolor=white,fontname="Vera Sans, DejaVu Sans, Liberation Sans, Arial, Helvetica, sans",fontsize=10,height=0.25,shape=box,style=
"setlinewidth(0.5),filled",target="_top",tooltip="Sort expression and indication of sort direction."]; "Transfer" [URL="../../../py-api/lsst.daf.relation.Transfer.html#
lsst.daf.relation.Transfer",fillcolor=white,fontname="Vera Sans, DejaVu Sans, Liberation Sans, Arial, Helvetica, sans",fontsize=10,height=0.25,shape=box,style="setlinewidth(0.5),filled",target=
"_top",tooltip="A `MarkerRelation` operation that represents moving relation content"]; "MarkerRelation" -> "Transfer" [arrowsize=0.5,style="setlinewidth(0.5)"]; "UnaryCommutator" [URL="../../../
py-api/lsst.daf.relation.UnaryCommutator.html#lsst.daf.relation.UnaryCommutator",fillcolor=white,fontname="Vera Sans, DejaVu Sans, Liberation Sans, Arial, Helvetica, sans",fontsize=10,height=
0.25,shape=box,style="setlinewidth(0.5),filled",target="_top",tooltip="A struct for the return value of `UnaryOperation.commute`."]; "UnaryOperation" [URL="../../../py-api/
lsst.daf.relation.UnaryOperation.html#lsst.daf.relation.UnaryOperation",fillcolor=white,fontname="Vera Sans, DejaVu Sans, Liberation Sans, Arial, Helvetica, sans",fontsize=10,height=0.25,shape=
box,style="setlinewidth(0.5),filled",target="_top",tooltip="An abstract base class for operations that act on a single relation."]; "ABC" -> "UnaryOperation" [arrowsize=0.5,style="setlinewidth(0.5)
"]; "UnaryOperationRelation" [URL="../../../py-api/lsst.daf.relation.UnaryOperationRelation.html#lsst.daf.relation.UnaryOperationRelation",fillcolor=white,fontname="Vera Sans, DejaVu Sans, Liberation
Sans, Arial, Helvetica, sans",fontsize=10,height=0.25,shape=box,style="setlinewidth(0.5),filled",target="_top",tooltip="A concrete `Relation` that represents the action of a `UnaryOperation`"];
"BaseRelation" -> "UnaryOperationRelation" [arrowsize=0.5,style="setlinewidth(0.5)"]; } | {"url":"https://pipelines.lsst.io/v/weekly/modules/lsst.daf.relation/index.html","timestamp":"2024-11-13T21:11:16Z","content_type":"text/html","content_length":"119166","record_id":"<urn:uuid:6d4607a3-b34c-4e65-8a62-4f6d6ee5d789>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00344.warc.gz"} |
Design Like A Pro
Correlated Samples T Test E Ample
Correlated Samples T Test E Ample - Web if you’re venturing into the realm of statistics, particularly when dealing with related samples, the paired samples t test is a powerful tool in your
analytical toolkit. Used to compare a population mean to some value. The compared groups may be independent to each other such as men and women. Under what conditions will one likely find. In
clinical research, comparisons of the results from experimental and control groups are often encountered. You can use the test when your.
You can use the test when your. Under what conditions will one likely find. When can i use the test? The compared groups may be independent to each other such as men and women. Again, like the null
When can i use the test? Web my question is whether it is required to have a strong correlation between the two samples for the paired samples tests, especially for the paired samples t test? Again,
like the null and. You can use the test when your. The compared groups may be independent to each other such as men and women.
In clinical research, comparisons of the results from experimental and control groups are often encountered. When can i use the test? Used to compare a population mean to some value. Web to test this
hypothesis, he took a random sample of 15 adults, measuring the height of each individual subject first with shoes on, and then again with shoes off..
You can use the test when your. Used to compare two population means. Web independent samples test. Web my question is whether it is required to have a strong correlation between the two samples for
the paired samples tests, especially for the paired samples t test? Again, like the null and.
In clinical research, comparisons of the results from experimental and control groups are often encountered. The compared groups may be independent to each other such as men and women. Again, like
the null and. Used to compare a population mean to some value. Web to test this hypothesis, he took a random sample of 15 adults, measuring the height of.
Web if you’re venturing into the realm of statistics, particularly when dealing with related samples, the paired samples t test is a powerful tool in your analytical toolkit. When can i use the test?
In clinical research, comparisons of the results from experimental and control groups are often encountered. Again, like the null and. You can use the test when.
Correlated Samples T Test E Ample - Web independent samples test. Under what conditions will one likely find. When can i use the test? Web to test this hypothesis, he took a random sample of 15
adults, measuring the height of each individual subject first with shoes on, and then again with shoes off. The compared groups may be independent to each other such as men and women. Web my question
is whether it is required to have a strong correlation between the two samples for the paired samples tests, especially for the paired samples t test? In clinical research, comparisons of the results
from experimental and control groups are often encountered. Web if you’re venturing into the realm of statistics, particularly when dealing with related samples, the paired samples t test is a
powerful tool in your analytical toolkit. Determine whether you have correlated pairs or independent groups. Used to compare a population mean to some value.
When can i use the test? The compared groups may be independent to each other such as men and women. Web independent samples test. In clinical research, comparisons of the results from experimental
and control groups are often encountered. Web to test this hypothesis, he took a random sample of 15 adults, measuring the height of each individual subject first with shoes on, and then again with
shoes off.
Under what conditions will one likely find. The compared groups may be independent to each other such as men and women. Compute a t test for. You can use the test when your.
You can use the test when your. Used to compare a population mean to some value. Compute a t test for.
You can use the test when your. Web my question is whether it is required to have a strong correlation between the two samples for the paired samples tests, especially for the paired samples t test?
Web to test this hypothesis, he took a random sample of 15 adults, measuring the height of each individual subject first with shoes on, and then again with shoes off.
The Compared Groups May Be Independent To Each Other Such As Men And Women.
Web if you’re venturing into the realm of statistics, particularly when dealing with related samples, the paired samples t test is a powerful tool in your analytical toolkit. When can i use the test?
Used to compare two population means. Determine whether you have correlated pairs or independent groups.
Compute A T Test For.
In clinical research, comparisons of the results from experimental and control groups are often encountered. Web independent samples test. Used to compare a population mean to some value. Again, like
the null and.
Web To Test This Hypothesis, He Took A Random Sample Of 15 Adults, Measuring The Height Of Each Individual Subject First With Shoes On, And Then Again With Shoes Off.
Under what conditions will one likely find. You can use the test when your. Web my question is whether it is required to have a strong correlation between the two samples for the paired samples
tests, especially for the paired samples t test? | {"url":"https://cosicova.org/eng/correlated-samples-t-test-e-ample.html","timestamp":"2024-11-07T22:47:17Z","content_type":"text/html","content_length":"27079","record_id":"<urn:uuid:dee49099-1f68-49d0-aba4-6ce7931dc87b>","cc-path":"CC-MAIN-2024-46/segments/1730477028017.48/warc/CC-MAIN-20241107212632-20241108002632-00359.warc.gz"} |
Remainder Theorem And Polynomials
An expression of the form ax^n + bx^n-1 + cx^n-2 + ………………. + kx + l, where each variable has a constant accompanying it as its coefficient, is called a polynomial of degree ‘n’ in variable x. Thus, a
polynomial is an expression in which a combination of a constant and a variable is separated by an addition or a subtraction sign. Each variable separated with an addition or subtraction symbol in
the expression is known as a term. The maximum power of the variable in a polynomial expression is the degree of the polynomial. Let’s learn about the remainder theorem of polynomials.
Remainder Theorem
When we divide a number, for example, 25 by 5 we get 5 as quotient and 0 as the remainder. This can be expressed as:
Dividend = (Divisor × Quotient) + Remainder
i.e, 25= (5 x 5) + 0
Here the remainder is zero thus we can say 5 is a factor of 25 or 25 is a multiple of 5. Thus reminder gives us a link between dividend and the divisor. We can divide a polynomial by another
polynomial and can express in the same way.
Let’s divide a polynomial, p(x) = 4x^2 + x – 1 by another polynomial (x + 1). After a long division we will get quotient, q(x) = 4x-3 and remainder, r(x) = 2. This can be expressed as:
4x^2 + x – 1=(x + 1) × (4x-3) + 2
Suppose p(x) and g(x) are two polynomials where the degree of p(x) > g(x) and g(x) ≠0. When we divide p(x) by g(x), if we get the polynomial q(x) as quotient and r(x) as remainder, then this can be
expressed as:
p(x) = g(x) × q(x) + r(x)
The remainder theorem of polynomials gives us a link between the remainder and its dividend. Let p(x) be any polynomial of degree greater than or equal to one and ‘a’ be any real number. If p(x) is
divided by the linear polynomial x – a, then the remainder is p (a). This is the remainder theorem. It helps us to find the remainder without actual division. Let’s take a look at the application of
the remainder theorem with the help of an example.
Example 1: Find the remainder when t^3 – 2t^2 + t + 1 is divided by t – 1.
Solution: Here, p(x) = t^3 – 2t^2 + t + 1, and the zero of t – 1 is 1.
∴ p (1) = (1)^3 – 2(1)^2 + 1 + 1= 1
By the Remainder Theorem, 1 is the remainder when t^3 – 2t^2 + t + 1 is divided by t – 1.
To practice more problems on reminder theorem, download BYJU’S-The Learning App. | {"url":"https://mathlake.com/Remainder-Theorem-And-Polynomials","timestamp":"2024-11-06T23:38:57Z","content_type":"text/html","content_length":"9923","record_id":"<urn:uuid:7b3e6801-5737-423f-8266-52e0b7fd2217>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00835.warc.gz"} |
• Improve the handling of n-level categorical variables by generating n-1 dummy variables (#2).
• Add the spdep_lmtest function for spatial linear model selection (#3).
• Migrate the moran_test function from the geocomplexity package to sdsfun (#4).
• Implement the geographical detector’s factor detector in ssh_test using Rcpp to enhance performance (#5).
• Introduce the discretize_vector function to support variable discretization (#6).
• Apply the loess_optnum function to select the optimal number of discretization intervals (#10).
• Implement spatial variance calculation in the spvar function, with support for both R and C++ implementations (#11).
• Rename dummy_vector to dummy_vec for consistency in naming conventions.
• Add the sf_coordinates function to extract coordinates from sf objects. | {"url":"https://cran.r-project.org/web/packages/sdsfun/news/news.html","timestamp":"2024-11-14T06:56:34Z","content_type":"application/xhtml+xml","content_length":"4269","record_id":"<urn:uuid:d7466fec-78ff-466d-b9e4-1688b742cea0>","cc-path":"CC-MAIN-2024-46/segments/1730477028545.2/warc/CC-MAIN-20241114062951-20241114092951-00426.warc.gz"} |
Niels Bohr: The Quantum Pioneer - Information and Technology
Niels Bohr is one of the most influential figures in the history of modern physics, a man whose work laid the foundation for our understanding of atomic structure and quantum mechanics. His
pioneering theories transformed the scientific landscape of the 20th century and continue to shape the technological advancements of today. Born in Copenhagen in 1885, Bohr’s intellectual journey
took him to the forefront of a revolutionary shift in how we understand the universe at the atomic level.
Early Life and Scientific Influences
Niels Bohr was born into a family deeply embedded in academic life. His father, Christian Bohr, was a distinguished physiologist, and his mother hailed from a wealthy Jewish banking family. This
environment fostered an early interest in the sciences, particularly in physics and mathematics. After completing his early education, Bohr pursued a degree in physics at the University of
Copenhagen, where he delved into the emerging questions of atomic theory.
In 1911, Bohr completed his doctoral thesis on the electron theory of metals, which focused on the properties of matter at the atomic level. This was the beginning of his lifelong dedication to
understanding the inner workings of atoms, a pursuit that would soon catapult him into the annals of scientific history.
The Bohr Model of the Atom
The cornerstone of Bohr’s contributions to science was his model of atomic structure, proposed in 1913. At the time, the prevailing atomic model was Rutherford’s, which described atoms as having a
central nucleus with electrons orbiting around it. While this model was a breakthrough, it left many unanswered questions, especially regarding the stability of these electron orbits and the emission
of radiation.
Bohr’s Quantum Leap
Bohr introduced the concept of quantized electron orbits, drawing on Max Planck’s quantum theory and Albert Einstein’s work on the photoelectric effect. He proposed that electrons could only occupy
certain fixed orbits, or energy levels, around the nucleus. Electrons in these orbits do not emit energy in the form of radiation, as classical physics would predict, but they can “jump” between
orbits by absorbing or emitting discrete amounts of energy, known as quanta.
This radical departure from classical physics explained the stability of atoms and the spectral lines emitted by hydrogen, which had puzzled scientists for years. By combining the classical mechanics
of Rutherford’s model with quantum concepts, Bohr’s atomic model marked the birth of quantum mechanics—a new and highly abstract way of understanding the microscopic world.
Impact on Quantum Mechanics
The Bohr model was not just a breakthrough in atomic theory; it laid the groundwork for the burgeoning field of quantum mechanics. This discipline, concerned with the behavior of matter and energy at
the smallest scales, would go on to revolutionize physics in the 20th century.
Complementarity Principle
One of Bohr’s most profound contributions to quantum mechanics was his principle of complementarity, which he introduced in 1928. This principle posits that particles such as electrons and photons
exhibit both wave-like and particle-like properties, depending on the experimental setup. However, these properties cannot be observed simultaneously—one can either measure the wave behavior or the
particle behavior, but not both at the same time.
Complementarity became a fundamental concept in quantum mechanics, illustrating the limitations of classical physics in describing the quantum world. It also challenged traditional notions of
determinism, a topic that Bohr famously debated with Albert Einstein, who remained skeptical of the philosophical implications of quantum theory.
Bohr’s Role in the Copenhagen Interpretation
Bohr’s contributions extended beyond the realm of theoretical models. Along with Werner Heisenberg, he developed the Copenhagen Interpretation of quantum mechanics, one of the most widely accepted
interpretations of the theory. This interpretation asserts that quantum mechanics does not provide a direct description of reality, but rather deals with probabilities and uncertainties in the
behavior of subatomic particles.
In the Copenhagen Interpretation, the act of measurement plays a crucial role in determining the state of a quantum system. Before a measurement is made, particles exist in a superposition of all
possible states, and only when observed do they “collapse” into a definite state. This idea challenged the classical view of an objective, deterministic reality, introducing a new level of
uncertainty into our understanding of the physical world.
Bohr’s Contributions to Nuclear Physics
Beyond his foundational work in quantum mechanics, Niels Bohr made significant contributions to the field of nuclear physics. During the 1930s, as physicists began to explore the possibilities of
nuclear fission, Bohr’s insights were critical to understanding this new phenomenon. In collaboration with his protégé, John Wheeler, Bohr developed the liquid drop model of the atomic nucleus, which
helped explain how nuclei could undergo fission—a process in which a nucleus splits into two smaller nuclei, releasing vast amounts of energy.
The Manhattan Project
During World War II, Bohr played a key role in the development of nuclear weapons as part of the Manhattan Project. Although he was deeply concerned about the destructive power of atomic bombs, Bohr
believed that the only way to prevent their use in war was to engage in international cooperation on nuclear energy. He spent much of his later years advocating for the peaceful use of nuclear power
and the establishment of global frameworks to manage its potential dangers.
Legacy and Lasting Influence
Niels Bohr’s legacy transcends his numerous scientific achievements. He was not only a brilliant physicist but also a visionary thinker who sought to integrate scientific discovery with philosophical
inquiry. His principle of complementarity reflects a deep understanding of the complexity and duality of nature, an idea that continues to resonate in modern physics.
Bohr’s work has left an indelible mark on the world of science, with applications in fields ranging from chemistry to information technology. Quantum mechanics, the field he helped to establish, is
essential to the development of modern technologies such as semiconductors, lasers, and quantum computers. The Bohr model, despite its limitations, remains a crucial stepping stone in our
understanding of atomic structure.
Niels Bohr’s contributions to science were transformative, extending beyond the scope of atomic theory to shape the entire field of quantum mechanics. His innovative ideas on quantized orbits,
complementarity, and nuclear physics redefined the scientific understanding of matter at its most fundamental level. As a pioneer of quantum theory, Bohr’s influence continues to be felt in the
technological advancements and philosophical debates that define contemporary physics. His work remains a testament to the power of human ingenuity in unlocking the mysteries of the universe. | {"url":"https://tvizleyim.com/niels-bohr-the-quantum-pioneer.html","timestamp":"2024-11-14T11:59:20Z","content_type":"text/html","content_length":"58259","record_id":"<urn:uuid:64308933-afef-4fc8-bb49-07b53e0dbe03>","cc-path":"CC-MAIN-2024-46/segments/1730477028558.0/warc/CC-MAIN-20241114094851-20241114124851-00536.warc.gz"} |
Important MCQs Hypothesis Testing 1 - Statistics Quiz
Important MCQs Hypothesis Testing 1
The post is about MCQs Hypothesis Testing. There are 20 multiple-choice questions covering topics related to the basics of hypothesis testing, assumptions about one sample, two samples, and more than
two sample mean comparison tests, significance level, null and alternative hypothesis, test statistics, sample size, critical region, and decision. Let us start with MCQs Hypothesis Testing Quiz.
Online MCQs about Hypothesis Testing with Answers
1. What is the first step when conducting a hypothesis test?
2. To conclude the null hypothesis, what two concepts are compared?
3. If we reject the null hypothesis, we might be making
4. The null hypothesis is a statement that is assumed to be true unless there is convincing evidence to the contrary. The null hypothesis typically assumes that observed data occurs by chance.
5. $1 – \alpha$ is the probability of
6. Analysis of Variance (ANOVA) is a test for equality of
7. What is the probability of a type II error when $\alpha=0.05$ ?
8. A parameter is a ————- quantity
9. Herbicide A has been used for years in order to kill a particular type of weed. An experiment is to be conducted in order to see whether a new herbicide, Herbicide B, is more effective than
Herbicide A. Herbicide AÂ will continue to be used unless there is sufficient evidence that Herbicide B is more effective. The alternative hypothesis in this problem is
10. A data professional conducts a hypothesis test. They discover that their p-value is less than the significance level. What conclusion should they draw?
11. The critical value of a test statistic is determined from
12. The t distributions are
13. What does a two-sample hypothesis test determine?
14. Which of the following is a true statement, for comparing the t distributions with standard normal,
15. Condition for applying the Central Limit Theorem (CLT) which approximates the sampling distribution of the mean with a normal distribution is?
16. What is the null hypothesis of a two-sample t-test?
17. The _____ typically assumes that observed data does not occur by chance.
18. For t distribution, increasing the sample size, the effect will be on
19. Which of the following statements describes the significance level?
20. Which of the following is an assumption underlying the use of the t-distributions?
MCQs Hypothesis Testing with Answers
• $1 – \alpha$ is the probability of
• A parameter is a ———- quantity
• If we reject the null hypothesis, we might be making
• Herbicide A has been used for years in order to kill a particular type of weed. An experiment is to be conducted in order to see whether a new herbicide, Herbicide B, is more effective than
Herbicide A. Herbicide AÂ will continue to be used unless there is sufficient evidence that Herbicide B is more effective. The alternative hypothesis in this problem is
• Analysis of Variance (ANOVA) is a test for equality of
• Which of the following is an assumption underlying the use of the t-distributions?
• For t distribution, increasing the sample size, the effect will be on
• The t distributions are
• Condition for applying the Central Limit Theorem (CLT) which approximates the sampling distribution of the mean with a normal distribution is?
• Which of the following is a true statement, for comparing the t distributions with standard normal,
• What is the probability of a type II error when $\alpha=0.05$?
• The critical value of a test statistic is determined from
• The null hypothesis is a statement that is assumed to be true unless there is convincing evidence to the contrary. The null hypothesis typically assumes that observed data occurs by chance.
• The ——– typically assumes that observed data does not occur by chance.
• Which of the following statements describes the significance level?
• What is the first step when conducting a hypothesis test?
• A data professional conducts a hypothesis test. They discover that their p-value is less than the significance level. What conclusion should they draw?
• What does a two-sample hypothesis test determine?
• What is the null hypothesis of a two-sample t-test?
• To conclude the null hypothesis, what two concepts are compared?
Leave a Comment | {"url":"https://itfeature.com/hypothesis/mcqs-hypothesis-testing-1/","timestamp":"2024-11-02T21:27:12Z","content_type":"text/html","content_length":"301093","record_id":"<urn:uuid:0f118f55-5599-48f9-a826-3cd8351162de>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00485.warc.gz"} |
[CIM/Infor-Base] Understanding parameters_Modifying object shape using Multi-Points Linked Entity
Is it possible to control the length of an object, such as the height of a column, with a parameter?
Multi-Points Linked Entity allows you to control the length of an object.
In Point Library Mode, a Multi-Points Linked Entity defines each Geometry Vertex of an object by constraining them to individual Constraint Points.
Here, Geometry refers to the shape information of an object, and it determines the object's overall shape.
In other words, through the process of defining a Multi-Points Linked Entity, Geometry Vertex of an object can be constrained to different Constraint Points, and these constrained Geometry Vertex can
be used to control the shape of the object through changes in the Constraint Point coordinates using the Multi-Points Linked Entity.
Figure 1. Shape change according to Geometry Vertex
The order of defining a Multi-Points Linked Entity is as follows
1. Select the object.
2. Select the Default Constraint Point.
The Default Constraint Point constrains all Geometry Vertices that are not individually
matched to other Constraint Points.
3. Match Other Constraint Points and Geometry Vertices individually and add them. Geometry Vertices that are not matched with Other Constraint Points are constrained to the Default Constraint Point.
After defining coordinate parameters, if you enter the defined parameters
into the Constraint Point coordinate property, you can control the shape through the parameters.
Let's take the cuboid model below as an example.
I want to create a model that can freely move the axis of a cuboid and transform its shape as desired.
To achieve this, I need two points that can control the ends of the axes.
To do this, I will select [Point Library] > [Point] from the top menu to create a Constraint Point at each end.
Once created, I will use the [Multi-Points Linked Entity] function to constrain each end vertex of the cuboid model to the two previously created Constraint Points.
When I select [Point Library] > [Multi-Points] from the top menu, the following window will appear.
Multi-Points Linked Entity Settings
Select Target
: Select the object to constrain. In this example, I will select the cuboid.
Default Constraint Point
: Select the base Constraint Point. Any points not assigned to a Constraint Point will be constrained to this point.
In this example, I will select the Constraint Point created at the origin.
Other Constraint Point
: Select each Vertex of the object to be constrained to the respective Constraint Point previously created.
After selecting the Constraint Point and the corresponding Vertex, click on "Add" to add the constraint.
After completing the constraints, create parameters to control the coordinates of the vertex.
To create a parameter for controlling the Vertex coordinates, go to the top menu and select
[Point Library] > [Parameter] to open the parameter editing window.
In the example, we created a parameter named 'Y-coordinate'. It's important to note that the parameter type should be set to [Coordinate] so that it can be selected from the Vertex coordinate values.
After setting up the Coordinate parameter, you can link the axis-specific coordinate values of the selected Constraint Point to the parameter you created earlier.
After completing the linkage, we need to verify that the parameter values can be changed properly.
Coordinate Parameter = 0 m (Top View) Coordinate Parameter = 2 m (Top View)
For reference, if you check the [Keep Section Plane] option in the model options, the orientation of the cross-section will be maintained even if the model axis is tilted.
Using the previous model as an example, when you follow the same process with this option checked, you can obtain the following result.
Coordination Parameter = 2 m, Keep Section Plane (Top View) | {"url":"https://support.midasuser.com/hc/en-us/articles/15746407284889--CIM-Infor-Base-Understanding-parameters-Modifying-object-shape-using-Multi-Points-Linked-Entity","timestamp":"2024-11-07T09:29:56Z","content_type":"text/html","content_length":"41453","record_id":"<urn:uuid:c528a0c5-b3cf-4487-9bf3-10cf6347b19e>","cc-path":"CC-MAIN-2024-46/segments/1730477027987.79/warc/CC-MAIN-20241107083707-20241107113707-00050.warc.gz"} |
Stochastic Comparison of Ellipsoidal and Interval Error Estimation in Vector Operations
We compare accuracy of the ellipsoidal and interval error estimation in the problem of multiplication of an uncertain vector by an exactly known matrix. The advantage of either method is determined
by the matrix only. We introduce a model of random matrices and study the probability of advantage of the interval estimate vs. the ellipsoidal one. It is shown that this probability decays as the
dimension of uncertain vector goes to infinity. | {"url":"http://lib.physcon.ru/doc?id=da41f6379228","timestamp":"2024-11-02T02:12:02Z","content_type":"text/html","content_length":"4670","record_id":"<urn:uuid:69a510a1-13d0-43fd-bfeb-4a58817def29>","cc-path":"CC-MAIN-2024-46/segments/1730477027632.4/warc/CC-MAIN-20241102010035-20241102040035-00036.warc.gz"} |
Tiling isometric tiles
I recently wrote a library in Rust which generates a random terrain consisting of blocks (think Minecraft or Minetest), and renders it in isometric perspective. It's called cubeglobe, and its source
is available on Github. The library was written to be used by a Fediverse bot, which posts these randomly generated landscapes on a relatively regular basis. You can check out the bot's feed if you
want some examples of what this looks like.
In the process of writing cubeglobe, I had to figure out how to actually render these images, given an internal representation of the landscape, and a handful of tile sprites corresponding to the
different block types that the internal representation can feature. This is not a very unique problem—representing various sorts of maps using isometric tiles is something that video games have been
doing for decades. In fact, my problem was somewhat easier, since I did not have to worry about interactivity, while video games generally do. Nevertheless, I did have to figure out how to do it, and
these notes detail that.
The 2:1 dimetric projection
A common way of doing isometric graphics on a computer, especially when you're doing something on the more pixel art end of the spectrum, is to use the 2:1 pixel ratio. This is technically not an
isometric projection, but a dimetric one (you can read more about it in an otherwise interesting Wikipedia article). This projection has the benefit of making it easy to draw lines—they go one pixel
up, two pixels laterally, hence the 2:1 ratio. Figure 1 illustrates what this looks like.
Figure 1: The axes in a 2:1 projection
The figure also marks the axes in the world coordinates. As with any projection system, it is useful to be able to say when you're referring to screen coordinates, and when you're talking about world
coordinates. Screen coordinates are used to refer to individual pixels in your output image, while world coordinates are used to refer to where in your scene a given object is.
Two dimensional tiles
If you draw a flat square in this dimetric projection (that is, a square on the x-y world plane), its screen width will be twice its screen height. This makes calculations for tiling such squares
somewhat easier: the sprite has a height of n, and a width of 2n. When you offset the sprite to tile it in a grid, you offset it by a multiple of n.
How do we tile these tiles, then? To move by one tile in the world x direction, we move it down by half of the tile's screen height, and to the right by half of the tile's screen width. This comes
out to moving it by n/2 in the vertical direction, and n in the horizontal direction. To move one tile in the world y direction, it's the same story, except we move to the left instead of right. To
move diagonally, we shift down by a full tile's height, n, in the vertical. Figure 2 shows a simple example.
Figure 2: Tiling in two dimensions.
Applying these principles. We arrive at formulas for getting the screen coordinates from the world coordinates, seen in Listing 1. The assumptions are that the tile dimensions are 1×1 in the world
coordinates, and that the 0, 0 tile in world space is also at 0, 0 in screen space.
screen_x = ( world_x - world_y ) × n
screen_y = ( world_x + world_y ) × n/2
Listing 1: The formulas for translating world coordinates to screen coordinates
Of course, the fact that these screen coordinates have the 0, 0 tile as the origin means that we may have to figure out where to place the 0, 0 tile on the screen. The formulas can help here, too:
figure out how many tiles we want to the left of the origin tile, and use the formula to figure out how many pixels will be needed to the left of the origin tile to fit everything.
Adding a dimension
cubeglobe maps are three-dimensional—they represent cube blocks, instead of flat tiles. This means that we also need to figure out how to place tiles depending on their z-axis coordinate in the world
The easiest way to think of cuboid tile sprites is to think of them as a flat tile with the z-axis component either below it or above it. In other words, take your existing flat tile and draw a cube,
either on top of it, or below it. cubeglobe, in particular, aligns tiles by their tops, so that means its tiles are a flat tile with the cube below it.
So, how much do we shift in the screen space to shift by one level in the world coordinates? To figure that out we must know the height of our sprite—let's call this h. Recall that the flat dimetric
tile has screen space dimensions of 2n by n. This remains true even as we move to representing cubes: the tile on top of the cube is still 2n by n. The pixel height we need to shift by is equal to
the height of the sprite that remains after we take away the height of the tile on top: h-n.
Figure 3: The top and sides of a block, and their dimensions.
It's worth pointing out here that while it may be tempting to make the block sprite a square, it does not work very well. If the total height is 2n, then the amount we would shift for a new z-level
would be n. We also shift by n for some situations when operating on the same z-level, so we end up with a situation where it is hard to tell if something is above or behind, as seen in Figure 4.
Figure 4: The blocks on the left have a square sprite, and it is difficult to tell what is in front. The blocks on the right are a bit taller, and arranged the same as the ones on the left, but their
layout is easier to see
Correctly drawn cubes in this perspective are actually slightly taller than square. While a mathematically accurate projection is not required to make things look like cubes, having the sprite be
rectangular does help in differentiating between the background and the foreground, visually.
Blocks upon blocks
cubeglobe renders its landscapes by repeatedly drawing the blocks one-by-one, from the lowest level to the highest. This is obviously not the most efficient way to do this—a large number of blocks
are going to be occluded by other blocks, and thus never seen on screen—but it is quite simple.
Each level can be drawn row-first or column-first, starting from the lowest coordinates. The blocks with the lower coordinates are going to be behind the blocks with the higher coordinates, so we can
simply draw them in the right order. Each level can then be layered on top of the previous one—the lower level will be occluded by the higher one.
cubeglobe does not employ any concurrency, although it would theoretically be possible to, for example, draw each layer separately and then combine them in order, afterwards. The overhead may not be
worth it, though, and it would require some exploration in practice.
I wrote cubeglobe partly because I was wondering how hard it would be to draw isometric graphics like that, given no prior knowledge. Turns out it is not very hard, although it does require figuring
out the formulas above. Fortunately, these sorts of graphics have a long history of use in video games, so it is not particularly hard to find useful resources.
Despite a rather inefficient way of drawing things, cubeglobe is not incredibly slow. This was something of a pleasant surprise.
Actually generating landscapes is an entirely separate problem, of course, and not covered here. With block-based landscapes like these, however, the internal representation can be fairly simple,
which makes it easy to separate the rendering side from the generating side.
Further reading
• "Isometric Tiles Math" by Clint Bellanger – Probably a far better explanation of how to tile tiles in two dimensions, along with some extra stuff for translating screen coordinates to world | {"url":"https://dee.underscore.world/blog/isometric-tiles/","timestamp":"2024-11-11T23:04:54Z","content_type":"text/html","content_length":"12843","record_id":"<urn:uuid:52d9c5cb-fc61-426f-aab4-2de11172d69c>","cc-path":"CC-MAIN-2024-46/segments/1730477028240.82/warc/CC-MAIN-20241111222353-20241112012353-00496.warc.gz"} |
Number changes in number field
Sep 21, 2022 10:18 AM
I manually entered numbers in a Single Line field, then I changed the field type to Number, and the last two digits of the numbers changed before my eyes for no apparent reason. The change is in the
last two digits (14 changed to 20, and 05 changed to 00). Note that I didn’t make a typo - I watched the digits change when I changed the field type to number.
I tried making a new number field and copy pasted the values in there, but the same thing happened there. I tried the same with a formula field with the same results.
It’s not a rounding error - the number field format is set to integer, and so is the formula field.
The screenshot SHOULD show the same number in every field - the “Group ID” field has the correct number.
Is there a ghost in the machine?
Sep 22, 2022 04:59 AM
Sep 21, 2022 10:56 AM
Sep 21, 2022 11:15 AM
Sep 21, 2022 11:29 AM
Sep 22, 2022 04:59 AM | {"url":"https://community.airtable.com/t5/other-questions/number-changes-in-number-field/td-p/106049","timestamp":"2024-11-06T22:27:54Z","content_type":"text/html","content_length":"355472","record_id":"<urn:uuid:57e8927f-7c12-42dd-8507-42d3eba7498c>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.47/warc/CC-MAIN-20241106194801-20241106224801-00149.warc.gz"} |
Electromagnetic flow meter flow rate data and formula
Electromagnetic flowmeter is an instrument used in industrial fields to measure the flow of conductive fluids.
Electromagnetic flowmeter is an instrument used in industrial fields to measure the flow of conductive fluids. Based on Faraday's law of electromagnetic induction, flow is calculated by measuring the
induced electromotive force generated by fluid flow. This article will delve into the flow rate data and related formulas of electromagnetic flowmeters to help readers better understand their working
principles and applications.
The core principle of the electromagnetic flowmeter is Faraday’s Law of Electromagnetic Induction. This law states that when a conductor moves in a magnetic field and cuts magnetic field lines, an
induced electromotive force is generated at both ends of the conductor. The magnitude of the induced electromotive force is proportional to the magnetic induction intensity, conductor length and
conductor movement speed.
E: induced electromotive force (V) B: magnetic induction intensity (T) L: Conductor length (m) v: Conductor movement speed (m/s)
2. The structure and working principle of the electromagnetic flowmeter
The electromagnetic flowmeter mainly measures It consists of tube, excitation coil, electrode and signal converter. The measuring tube is the part through which the fluid flows and is usually made of
a non-magnetic, well-conducting material, such as stainless steel. An excitation coil is wound around the outside of the measuring tube to generate a magnetic field. The electrode is installed on the
inner wall of the measuring tube and is used to measure the induced electromotive force. The signal converter amplifies and converts the weak induced electromotive force measured by the electrode
into a standard signal output proportional to the flow rate.
When a conductive fluid flows through the measuring tube, the magnetic field generated by the excitation coil passes through the fluid. According to Faraday's law of electromagnetic induction, the
flowing conductive fluid cuts the magnetic field lines in the magnetic field and generates an induced electromotive force. The magnitude of the induced electromotive force is proportional to the
average flow velocity of the fluid. By measuring the induced electromotive force, the flow rate of the fluid can be calculated.
The flow calculation formula of the electromagnetic flowmeter is as follows:
Q: Volume flow rate (m³/s) k: Instrument coefficient A: Measuring tube cross-sectional area (m²) v: Average fluid flow rate (m/s)
It can be seen from the above formula Out, the flow rate is proportional to the flow rate. In practical applications, the instrument coefficient k will be affected by many factors, such as fluid
conductivity, temperature, pressure, etc. Therefore, the electromagnetic flowmeter needs to be calibrated to determine the accurate meter coefficient.
The flow rate data measured by electromagnetic flowmeters has many applications in industrial production, such as:
Flow monitoring: real-time monitoring of pipelines The flow rate of medium fluid is used for production process control and material balance calculation. Flow accumulation: The total amount of flow
accumulated over a period of time, used to measure production volume, sales volume, energy consumption, etc. Batch control: Control the flow and total volume of each batch, used for batching, filling
and other production links. process optimization: Analyze the production process based on flow rate data, optimize process parameters, and improve production efficiency and product quality.
There are many factors that affect the accuracy of electromagnetic flowmeter flow rate measurement, mainly including the following aspects:
Fluid characteristics : The conductivity, viscosity, density, etc. of the fluid will all affect the measurement accuracy. Generally speaking, the lower the conductivity and the greater the viscosity,
the lower the measurement accuracy. Installation conditions: The installation position, direction, and length of the straight pipe section of the measuring pipe will all affect the measurement
accuracy. During installation, adverse factors such as vibration and magnetic field interference should be avoided as much as possible. The instrument itself: The accuracy level, stability,
linearity, etc. of the instrument will all affect the measurement accuracy. Environmental factors: Changes in ambient temperature, humidity, pressure, etc. will also have a certain impact on
measurement accuracy.
The electromagnetic flowmeter is a highly reliable flow measurement instrument, and its flow rate data plays an important role in industrial production. Understanding the working principle of
electromagnetic flowmeters, flow rate calculation formulas and factors affecting measurement accuracy will help users better select, use and maintain electromagnetic flowmeters and ensure the
accuracy and reliability of measurement data.
The magnetic inductive flowmeter industry, these are successful cases with comparable and reference value.
The vortex flowmeter is based on the Karman vortex principle to measure the volume flow of gas, steam or liquid, the volume flow or mass flow of standard conditions.
The vortex flowmeter is based on the Karman vortex principle to measure the volume flow of gas, steam or liquid, the volume flow or mass flow of standard conditions. volumetric flowmeter.
As metal tube float flowmeters are used in more and more fields and in more and more enterprises, products that can meet the needs of different working environments are launched. | {"url":"https://www.cnflowmeters.com/projects/2ef59d3e88175146bd982d49a85fa7a72.html","timestamp":"2024-11-14T21:44:45Z","content_type":"application/xhtml+xml","content_length":"19259","record_id":"<urn:uuid:1ba55a4c-80f7-43f7-bb47-a42e8706d9d6>","cc-path":"CC-MAIN-2024-46/segments/1730477395538.95/warc/CC-MAIN-20241114194152-20241114224152-00361.warc.gz"} |
Distance Between Two Points - Formula, Derivation, Examples - - [[company name]] [[target location]], [[stateabr]]
Distance Between Two Points - Formula, Derivation, Examples
The idea of distance is critical in both math and everyday life. From simply calculating the length of a line to designing the quickest route within two locations, comprehending the distance between
two points is vital.
In this blog article, we will explore the formula for distance within two extremities, go through a few examples, and discuss real-life uses of this formula.
The Formula for Length Within Two Points
The length between two points, often signified as d, is the extent of the line segment linking the two extremities.
Mathematically, this could be portrayed by drawing a right triangle and employing the Pythagorean theorem. Per the Pythagorean theorem, the square of the distance of the longest side (the hypotenuse)
is equivalent to the total of the squares of the distances of the two other sides.
The formula for the Pythagorean theorem is a2 + b2 = c2. As a result, √c2 will as same as the distance, d.
In the case of finding the distance within two locations, we can depict the points as coordinates on a coordinate plane. Let's say we possess point A with coordinates (x1, y1) and point B at (x2,
We could then employ the Pythagorean theorem to extract the following formula for distance:
d = √((x2 - x1)2 + (y2 - y1)2)
In this formula, (x2 - x1) depicts the length on the x-axis, and (y2 - y1) portrays the distance along y-axis, constructing a right angle. By considering the square root of the sum of their squares,
we get the distance between the two points.
Here is a visual depiction:
Instances of Utilizations of the Distance Formula
Considering we have the formula for distance, let's look at few instances of how it can be used.
Calculating the Length Within Two Locations on a Coordinate Plane
Imagine we possess two points on a coordinate plane, A with coordinates (3, 4) and B with coordinates (6, 8). We will utilize the distance formula to figure out the length between these two locations
as follows:
d = √((6 - 3)2+ (8 - 4)2)
d = √(32 + 42)
d = √(9 + 16)
d = √(25)
d = 5
Consequently, the length between points A and B is 5 units.
Calculating the Length Between Two Extremities on a Map
In addition to finding length on a coordinate plane, we can also utilize the distance formula to calculate lengths within two points on a map. For example, assume we have a map of a city with a scale
of 1 inch = 10 miles.
To work out the length between two locations on the map, similar to the city hall and the airport, we can easily calculate the distance within the two locations employing a ruler and change the
measurement to miles using the map's scale.
Once we calculate the length within these two locations on the map, we figure out it is 2 inches. We convert this to miles utilizing the map's scale and work out that the actual length among the city
hall and the airport is 20 miles.
Determining the Length Among Two Points in Three-Dimensional Space
In addition to calculating lengths in two dimensions, we could also use the distance formula to calculate the length between two locations in a three-dimensional space. For instance, suppose we
possess two locations, A and B, in a three-dimensional space, with coordinates (x1, y1, z1) and (x2, y2, z2), individually.
We will use the distance formula to work out the length within these two points as follows:
d = √((x2 - x1)2 + (y2 - y1)2 + (z2 - z1)2)
Utilizing this formula, we could calculate the length within any two locations in three-dimensional space. For instance, if we possess two points A and B with coordinates (1, 2, 3) and (4, 5, 6),
respectively, we could find the distance within them as ensues:
d = √((4 - 1)2 + (5 - 2)2 + (6 - 3)2)
d = √(32 + 32 + 32)
d = √(9 + 9 + 9)
d = √(27)
d = 3.16227766
Therefore, the length within points A and B is roughly 3.16 units.
Applications of the Distance Formula
Now once we have looked at few examples of utilizing the distance formula, let's examine some of its uses in mathematics and other fields.
Calculating Length in Geometry
In geometry, the distance formula is utilized to work out the distance of line segments and the sides of triangles. For instance, in a triangle with vertices at points A, B, and C, we utilize the
distance formula to figure out the distances of the sides AB, BC, and AC. These distances can be utilized to calculate other properties of the triangle, such as its interior angles, area, perimeter.
Solving Problems in Physics
The distance formula is also used in physics to work out problems concerning acceleration, speed and distance. For example, if we recognize the first location and velocity of an object, also the time
it requires for the object to move a specific distance, we can use the distance formula to calculate the object's ultimate position and speed.
Analyzing Data in Statistics
In statistics, the distance formula is often utilized to workout the distances between data points in a dataset. This is beneficial for clustering algorithms, which segregate data points which are
near to each other, and for dimensionality reduction techniques, this portrays high-dimensional data in a lower-dimensional space.
Go the Distance with Grade Potential
The distance formula is an essential idea in math which allows us to figure out the distance between two location on a plane or in a three-dimensional space. By utilizing the Pythagorean theorem, we
can obtain the distance formula and apply it to a magnitude of scenarios, from calculating length on a coordinate plane to analyzing data in statistics.
Comprehending the distance formula and its uses are essential for anyone interested in mathematics and its uses in other fields. If you're struggling regarding the distance formula or any other
mathematical theories, contact Grade Potential tutoring for customized guidance. Our experienced teachers will support you master any math topic, from algebra to calculus and furthermore.
Connect with us today to learn more and schedule your first tutoring session. | {"url":"https://www.alamedainhometutors.com/blog/distance-between-two-points-formula-derivation-examples","timestamp":"2024-11-01T23:01:37Z","content_type":"text/html","content_length":"76628","record_id":"<urn:uuid:35e24c2f-1385-42b2-b9ea-83a9396c80e9>","cc-path":"CC-MAIN-2024-46/segments/1730477027599.25/warc/CC-MAIN-20241101215119-20241102005119-00822.warc.gz"} |
Richard Braun
Department of Mathematical Sciences
University of Delaware
501 Ewing Hall
Newark, DE 19716
Email: braun@math.udel.edu
Phone: (302) 831- 1869
Fax: (302) 831- 4511
Website: http://www.math.udel.edu/~braun
• Ph.D., Applied Mathematics, Northwestern University – 1991
M.S., Mechanical Engineering, Santa Clara University – 1987
B.S., Mechanical Engineering, Santa Clara University – 1983
Research Overview:
Dr. Braun’s current research focuses on several areas. The primary area is tear film dynamics in the human eye. A hierarchy of models in under development for the tear film dynamics that describes
the evolution of the tear film over multiple blink cycles. There are challenges in both developing the mathematical models in the first place as well as in their solution via computational methods.
The mathematical models use the lubrication approximation to derive nonlinear partial differential equations for the film thickness and other variables of interest; translation of ophthalmology and
optometry literature into appropriate equations is a significant part of the project. Solving the resulting models is very difficult for many standard methods and new computational methods are under
development for these challenging dynamical problems.
A second area of interest is dynamics on networks; an epidemiological model for alcohol problems was developed with several collaborators. The model uses a threshold for each person (node) on social
networks (Poisson random graphs and small world networks) and a nonlinear equation for the state of each person to calculate the evolution of the population (a given network configuration). The model
can simulate different treatment strategies. This type of theory is readily adapted to other networks besides social networks.
Prior to joining the University of Delaware, Dr. Braun was a postdoctoral scientist in what is now the Mathematical and Computational Sciences Division of the Information Technology Laboratory at the
National Institute of Standards and Technology in Gaithersburg, MD, from 1991 to 1995. | {"url":"https://www.dbi.udel.edu/biographies/richard-braun","timestamp":"2024-11-09T03:56:45Z","content_type":"text/html","content_length":"31847","record_id":"<urn:uuid:e8985175-ce09-4640-869b-c89cfafbf117>","cc-path":"CC-MAIN-2024-46/segments/1730477028115.85/warc/CC-MAIN-20241109022607-20241109052607-00731.warc.gz"} |
2.2: Types and Number Representation
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Although a slight divergence, it is important to understand a bit of history about the C language.
C is the common languge of the systems programming world. Every operating system and its associated system libraries in common use is written in C, and every system provides a C compiler. To stop the
language diverging across each of these systems where each would be sure to make numerous incompatible changes, a strict standard has been written for the language.
Officially this standard is known as ISO/IEC 9899:1999(E), but is more commonly referred to by its shortened name C99. The standard is maintained by the International Standards Organisation (ISO) and
the full standard is available for purchase online. Older standards versions such as C89 (the predecessor to C99 released in 1989) and ANSI C are no longer in common usage and are encompassed within
the latest standard. The standard documentation is very technical, and details most every part of the language. For example it explains the syntax (in Backus Naur form), standard #define values and
how operations should behave.
It is also important to note what the C standards does not define. Most importantly the standard needs to be appropriate for every architecture, both present and future. Consequently it takes care
not to define areas that are architecture dependent. The "glue" between the C standard and the underlying architecture is the Application Binary Interface (or ABI) which we discuss below. In several
places the standard will mention that a particular operation or construct has an unspecified or implementation dependent result. Obviously the programmer can not depend on these outcomes if they are
to write portable code.
The GNU C Compiler, more commonly referred to as gcc, almost completely implements the C99 standard. However it also implements a range of extensions to the standard which programmers will often use
to gain extra functionality, at the expense of portability to another compiler. These extensions are usually related to very low level code and are much more common in the system programming field;
the most common extension being used in this area being inline assembly code. Programmers should read the gcc documentation and understand when they may be using features that diverge from the
gcc can be directed to adhere strictly to the standard (the -std=c99 flag for example) and warn or create an error when certain things are done that are not in the standard. This is obviously
appropriate if you need to ensure that you can move your code easily to another compiler.
As programmers, we are familiar with using variables to represent an area of memory to hold a value. In a typed language, such as C, every variable must be declared with a type. The type tells the
compiler about what we expect to store in a variable; the compiler can then both allocate sufficient space for this usage and check that the programmer does not violate the rules of the type. In the
example below, we see an example of the space allocated for some common types of variables.
Figure 2.2. Types
The C99 standard purposely only mentions the smallest possible size of each of the types defined for C. This is because across different processor architectures and operating systems the best size
for types can be wildly different.
To be completely safe programmers need to never assume the size of any of their variables, however a functioning system obviously needs agreements on what sizes types are going to be used in the
system. Each architecture and operating system conforms to an Application Binary Interface or ABI. The ABI for a system fills in the details between the C standard and the requirements of the
underlying hardware and operating system. An ABI is written for a specific processor and operating system combination.
Table 2.13. Standard Integer Types and Sizes
Type C99 minimum size (bits) Common size (32 bit architecture)
char 8 8
short 16 16
int 16 32
long 32 32
long long 64 64
Pointers Implementation dependent 32
Above we can see the only divergence from the standard is that int is commonly a 32 bit quantity, which is twice the strict minimum 16 bit size that the C99 requires.
Pointers are really just an address (i.e. their value is an address and thus "points" somewhere else in memory) therefore a pointer needs to be sufficient in size to be able to address any memory in
the system.
One area that causes confusion is the introduction of 64 bit computing. This means that the processor can handle addresses 64 bits in length (specifically the registers are 64 bits wide; a topic we
discuss in Chapter 3, Computer Architecture).
This firstly means that all pointers are required to be a 64 bits wide so they can represent any possible address in the system. However, system implementers must then make decisions about the size
of the other types. Two common models are widely used, as shown below.
Table 2.14. Standard Scalar Types and Sizes
Type C99 minimum size (bits) Common size (LP64) Common size (Windows)
char 8 8 8
short 16 16 16
int 16 32 32
long 32 64 32
long long 64 64 64
Pointers Implementation dependent 64 64
You can see that in the LP64 (long-pointer 64) model long values are defined to be 64 bits wide. This is different to the 32 bit model we showed previously. The LP64 model is widely used on UNIX
In the other model, long remains a 32 bit value. This maintains maximum compatibility with 32 code. This model is in use with 64 bit Windows.
There are good reasons why the size of int was not increased to 64 bits in either model. Consider that if the size of int is increased to 64 bits you leave programmers no way to obtain a 32 bit
variable. The only possibly is redefining shorts to be a larger 32 bit type.
A 64 bit variable is so large that it is not generally required to represent many variables. For example, loops very rarely repeat more times than would fit in a 32 bit variable (4294967296 times!).
Images usually are usually represented with 8 bits for each of a red, green and blue value and an extra 8 bits for extra (alpha channel) information; a total of 32 bits. Consequently for many cases,
using a 64 bit variable will be wasting at least the top 32 bits (if not more). Not only this, but the size of an integer array has now doubled too. This means programs take up more system memory
(and thus more cache; discussed in detail in Chapter 3, Computer Architecture) for no real improvement. For the same reason Windows elected to keep their long values as 32 bits; since much of the
Windows API was originally written to use long variables on a 32 bit system and hence does not require the extra bits this saves considerable wasted space in the system without having to re-write all
the API.
If we consider the proposed alternative where short was redefined to be a 32 bit variable; programmers working on a 64 bit system could use it for variables they know are bounded to smaller values.
However, when moving back to a 32 bit system their same short variable would now be only 16 bits long, a value which is much more realistically overflowed (65536).
By making a programmer request larger variables when they know they will be needed strikes a balance with respect to portability concerns and wasting space in binaries.
The C standard also talks about some qualifiers for variable types. For example const means that a variable will never be modified from its original value and volatile suggests to the compiler that
this value might change outside program execution flow so the compiler must be careful not to re-order access to it in any way.
signed and unsigned are probably the two most important qualifiers; and they say if a variable can take on a negative value or not. We examine this in more detail below.
Qualifiers are all intended to pass extra information about how the variable will be used to the compiler. This means two things; the compiler can check if you are violating your own rules (e.g.
writing to a const value) and it can make optimisations based upon the extra knowledge (examined in later chapters).
C99 realises that all these rules, sizes and portability concerns can become very confusing very quickly. To help, it provides a series of special types which can specify the exact properties of a
variable. These are defined in <stdint.h> and have the form qtypes_t where q is a qualifier, type is the base type, s is the width in bits and _t is an extension so you know you are using the C99
defined types.
So for example uint8_t is an unsigned integer exactly 8 bits wide. Many other types are defined; the complete list is detailed in C99 17.8 or (more cryptically) in the header file. ^[3]
It is up to the system implementing the C99 standard to provide these types for you by mapping them to appropriate sized types on the target system; on Linux these headers are provided by the system
Below in Example 2.3, “Example of warnings when types are not matched” we see an example of how types place restrictions on what operations are valid for a variable, and how the compiler can use this
information to warn when variables are used in an incorrect fashion. In this code, we firstly assign an integer value into a char variable. Since the char variable is smaller, we loose the correct
value of the integer. Further down, we attempt to assign a pointer to a char to memory we designated as an integer. This operation can be done; but it is not safe. The first example is run on a
32-bit Pentium machine, and the correct value is returned. However, as shown in the second example, on a 64-bit Itanium machine a pointer is 64 bits (8 bytes) long, but an integer is only 4 bytes
long. Clearly, 8 bytes can not fit into 4! We can attempt to "fool" the compiler by casting the value before assigning it; note that in this case we have shot ourselves in the foot by doing this cast
and ignoring the compiler warning since the smaller variable can not hold all the information from the pointer and we end up with an invalid address.
1 /*
* types.c
5 #include <stdio.h>
#include <stdint.h>
int main(void)
10 char a;
char *p = "hello";
int i;
15 // moving a larger variable into a smaller one
i = 0x12341234;
a = i;
i = a;
printf("i is %d\n", i);
// moving a pointer into an integer
printf("p is %p\n", p);
i = p;
// "fooling" with casts
25 i = (int)p;
p = (char*)i;
printf("p is %p\n", p);
return 0;
30 }
1 $ uname -m
$ gcc -Wall -o types types.c
5 types.c: In function 'main':
types.c:19: warning: assignment makes integer from pointer without a cast
$ ./types
i is 52
10 p is 0x80484e8
p is 0x80484e8
$ uname -m
$ gcc -Wall -o types types.c
types.c: In function 'main':
types.c:19: warning: assignment makes integer from pointer without a cast
types.c:21: warning: cast from pointer to integer of different size
20 types.c:22: warning: cast to pointer from integer of different size
$ ./types
i is 52
p is 0x40000000000009e0
25 p is 0x9e0
With our modern base 10 numeral system we indicate a negative number by placing a minus (-) sign in front of it. When using binary we need to use a different system to indicate negative numbers.
There is only one scheme in common use on modern hardware, but C99 defines three acceptable methods for negative value representation.
The most straight forward method is to simply say that one bit of the number indicates either a negative or positive value depending on it being set or not.
This is analogous to mathematical approach of having a + and -. This is fairly logical, and some of the original computers did represent negative numbers in this way. But using binary numbers opens
up some other possibilities which make the life of hardware designers easier.
However, notice that the value 0 now has two equivalent values; one with the sign bit set and one without. Sometimes these values are referred to as +0 and -0 respectively.
One's complement simply applies the not operation to the positive number to represent the negative number. So, for example the value -90 (-0x5A) is represented by ~01011010 = 10100101^[4]
With this scheme the biggest advantage is that to add a negative number to a positive number no special logic is required, except that any additional carry left over must be added back to the final
value. Consider
Table 2.15. One's Complement Addition
Decimal Binary Op
-90 10100101 +
--- --------
10 ^100001001 9
If you add the bits one by one, you find you end up with a carry bit at the end (highlighted above). By adding this back to the original we end up with the correct value, 10
Again we still have the problem with two zeros being represented. Again no modern computer uses one's complement, mostly because there is a better scheme.
Two's complement is just like one's complement, except the negative representation has one added to it and we discard any left over carry bit. So to continue with the example from before, -90 would
be ~01011010+1=10100101+1 = 10100110.
This means there is a slightly odd symmetry in the numbers that can be represented; for example with an 8 bit integer we have 2^^8 = 256 possible values; with our sign bit representation we could
represent -127 thru 127 but with two's complement we can represent -127 thru 128. This is because we have removed the problem of having two zeros; consider that "negative zero" is (~00000000 +1)=
(11111111+1)=00000000 (note discarded carry bit).
Table 2.16. Two's Complement Addition
Decimal Binary Op
-90 10100110 +
--- --------
You can see that by implementing two's complement hardware designers need only provide logic for addition circuits; subtraction can be done by two's complement negating the value to be subtracted and
then adding the new value.
Similarly you could implement multiplication with repeated addition and division with repeated subtraction. Consequently two's complement can reduce all simple mathematical operations down to
All modern computers use two's complement representation.
Because of two's complement format, when increasing the size of signed value, it is important that the additional bits be sign-extended; that is, copied from the top-bit of the existing value.
For example, the value of an 32-bit int -10 would be represented in two's complement binary as 11111111111111111111111111110110. If one were to cast this to a 64-bit long long int, we would need to
ensure that the additional 32-bits were set to 1 to maintain the same sign as the original.
Thanks to two's complement, it is sufficient to take the top bit of the exiting value and replace all the added bits with this value. This processes is referred to as sign-extension and is usually
handled by the compiler in situations as defined by the language standard, with the processor generally providing special instructions to take a value and sign-extended it to some larger value.
So far we have only discussed integer or whole numbers; the class of numbers that can represent decimal values is called floating point.
To create a decimal number, we require some way to represent the concept of the decimal place in binary. The most common scheme for this is known as the IEEE-754 floating point standard because the
standard is published by the Institute of Electric and Electronics Engineers. The scheme is conceptually quite simple and is somewhat analogous to "scientific notation".
In scientific notation the value 123.45 might commonly be represented as 1.2345x10^2. We call 1.2345 the mantissa or significand, 10 is the radix and 2 is the exponent.
In the IEEE floating point model, we break up the available bits to represent the sign, mantissa and exponent of a decimal number. A decimal number is represented by sign × significand × 2^^exponent.
The sign bit equates to either 1 or -1. Since we are working in binary, we always have the implied radix of 2.
There are differing widths for a floating point value -- we examine below at only a 32 bit value. More bits allows greater precision.
Table 2.17. IEEE Floating Point
Sign Exponent Significand/Mantissa
The other important factor is bias of the exponent. The exponent needs to be able to represent both positive and negative values, thus an implied value of 127 is subtracted from the exponent. For
example, an exponent of 0 has an exponent field of 127, 128 would represent 1 and 126 would represent -1.
Each bit of the significand adds a little more precision to the values we can represent. Consider the scientific notation representation of the value 198765. We could write this as 1.98765x10^6,
which corresponds to a representation below
Table 2.18. Scientific Notation for 1.98765x10^6
10^0 . 10^-1 10^-2 10^-3 10^-4 10^-5
1 . 9 8 7 6 5
Each additional digit allows a greater range of decimal values we can represent. In base 10, each digit after the decimal place increases the precision of our number by 10 times. For example, we can
represent 0.0 through 0.9 (10 values) with one digit of decimal place, 0.00 through 0.99 (100 values) with two digits, and so on. In binary, rather than each additional digit giving us 10 times the
precision, we only get two times the precision, as illustrated in the table below. This means that our binary representation does not always map in a straight-forward manner to a decimal
Table 2.19. Significands in binary
2^0 . 2^-1 2^-2 2^-3 2^-4 2^-5
1 . 1/2 1/4 1/8 1/16 1/32
1 . 0.5 0.25 0.125 0.0625 0.03125
With only one bit of precision, our fractional precision is not very big; we can only say that the fraction is either 0 or 0.5. If we add another bit of precision, we can now say that the decimal
value is one of either 0,0.25,0.5,0.75. With another bit of precision we can now represent the values 0,0.125,0.25,0.375,0.5,0.625,0.75,0.875.
Increasing the number of bits therefore allows us greater and greater precision. However, since the range of possible numbers is infinite we will never have enough bits to represent any possible
For example, if we only have two bits of precision and need to represent the value 0.3 we can only say that it is closest to 0.25; obviously this is insufficient for most any application. With 22
bits of significand we have a much finer resolution, but it is still not enough for most applications. A double value increases the number of significand bits to 52 (it also increases the range of
exponent values too). Some hardware has an 84-bit float, with a full 64 bits of significand. 64 bits allows a tremendous precision and should be suitable for all but the most demanding of
applications (XXX is this sufficient to represent a length to less than the size of an atom?)
1 $ cat float.c
#include <stdio.h>
int main(void)
5 {
float a = 0.45;
float b = 8.0;
double ad = 0.45;
10 double bd = 8.0;
printf("float+float, 6dp : %f\n", a+b);
printf("double+double, 6dp : %f\n", ad+bd);
printf("float+float, 20dp : %10.20f\n", a+b);
15 printf("dobule+double, 20dp : %10.20f\n", ad+bd);
return 0;
20 $ gcc -o float float.c
$ ./float
float+float, 6dp : 8.450000
double+double, 6dp : 8.450000
25 float+float, 20dp : 8.44999998807907104492
dobule+double, 20dp : 8.44999999999999928946
$ python
Python 2.4.4 (#2, Oct 20 2006, 00:23:25)
30 [GCC 4.1.2 20061015 (prerelease) (Debian 4.1.1-16.1)] on linux2
Type "help", "copyright", "credits" or "license" for more information.
>>> 8.0 + 0.45
A practical example is illustrated above. Notice that for the default 6 decimal places of precision given by printf both answers are the same, since they are rounded up correctly. However, when asked
to give the results to a larger precision, in this case 20 decimal places, we can see the results start to diverge. The code using doubles has a more accurate result, but it is still not exactly
correct. We can also see that programmers not explicitly dealing with float values still have problems with precision of variables!
In scientific notation, we can represent a value in many different ways. For example, 10023x10^^0 = 1002.3x10^1 = 100.23x10^2. We thus define the normalised version as the one where 1/radix <=
significand < 1. In binary this ensures that the leftmost bit of the significand is always one. Knowing this, we can gain an extra bit of precision by having the standard say that the leftmost bit
being one is implied.
Table 2.20. Example of normalising 0.375
2^0 . 2^-1 2^-2 2^-3 2^-4 2^-5 Exponent Calculation
0 . 0 1 1 0 0 2^^0 (0.25+0.125) × 1 = 0.375
0 . 1 1 0 0 0 2^^-1 (0.5+0.25)×.5=0.375
1 . 1 0 0 0 0 2^^-2 (1+0.5)×0.25=0.375
As you can see above, we can make the value normalised by moving the bits upwards as long as we compensate by increasing the exponent.
A common problem programmers face is finding the first set bit in a bitfield. Consider the bitfield 0100; from the right the first set bit would be bit 2 (starting from zero, as is conventional).
The standard way to find this value is to shift right, check if the uppermost bit is a 1 and either terminate or repeat. This is a slow process; if the bitfield is 64 bits long and only the very last
bit is set, you must go through all the preceding 63 bits!
However, if this bitfield value were the signficand of a floating point number and we were to normalise it, the value of the exponent would tell us how many times it was shifted. The process of
normalising a number is generally built into the floating point hardware unit on the processor, so operates very fast; usually much faster than the repeated shift and check operations.
The example program below illustrates two methods of finding the first set bit on an Itanium processor. The Itanium, like most server processors, has support for an 80-bit extended floating point
type, with a 64-bit significand. This means a unsigned long neatly fits into the significand of a long double. When the value is loaded it is normalised, and and thus by reading the exponent value
(minus the 16 bit bias) we can see how far it was shifted.
1 #include <stdio.h>
int main(void)
5 // in binary = 1000 0000 0000 0000
// bit num 5432 1098 7654 3210
int i = 0x8000;
int count = 0;
while ( !(i & 0x1) ) {
10 count ++;
i = i >> 1;
printf("First non-zero (slow) is %d\n", count);
15 // this value is normalised when it is loaded
long double d = 0x8000UL;
long exp;
// Itanium "get floating point exponent" instruction
20 asm ("getf.exp %0=%1" : "=r"(exp) : "f"(d));
// note exponent include bias
printf("The first non-zero (fast) is %d\n", exp - 65535);
25 }
In the example code below we extract the components of a floating point number and print out the value it represents. This will only work for a 32 bit floating point value in the IEEE format; however
this is common for most architectures with the float type.
1 #include <stdio.h>
#include <string.h>
#include <stdlib.h>
5 /* return 2^n */
int two_to_pos(int n)
if (n == 0)
return 1;
10 return 2 * two_to_pos(n - 1);
double two_to_neg(int n)
15 if (n == 0)
return 1;
return 1.0 / (two_to_pos(abs(n)));
20 double two_to(int n)
if (n >= 0)
return two_to_pos(n);
if (n < 0)
25 return two_to_neg(n);
return 0;
/* Go through some memory "m" which is the 24 bit significand of a
30 floating point number. We work "backwards" from the bits
furthest on the right, for no particular reason. */
double calc_float(int m, int bit)
/* 23 bits; this terminates recursion */
35 if (bit > 23)
return 0;
/* if the bit is set, it represents the value 1/2^bit */
if ((m >> bit) & 1)
40 return 1.0L/two_to(23 - bit) + calc_float(m, bit + 1);
/* otherwise go to the next bit */
return calc_float(m, bit + 1);
int main(int argc, char *argv[])
float f;
int m,i,sign,exponent,significand;
if (argc != 2)
printf("usage: float 123.456\n");
55 }
if (sscanf(argv[1], "%f", &f) != 1)
printf("invalid input\n");
60 exit(1);
/* We need to "fool" the compiler, as if we start to use casts
(e.g. (int)f) it will actually do a conversion for us. We
65 want access to the raw bits, so we just copy it into a same
sized variable. */
memcpy(&m, &f, 4);
/* The sign bit is the first bit */
70 sign = (m >> 31) & 0x1;
/* Exponent is 8 bits following the sign bit */
exponent = ((m >> 23) & 0xFF) - 127;
75 /* Significand fills out the float, the first bit is implied
to be 1, hence the 24 bit OR value below. */
significand = (m & 0x7FFFFF) | 0x800000;
/* print out a power representation */
80 printf("%f = %d * (", f, sign ? -1 : 1);
for(i = 23 ; i >= 0 ; i--)
if ((significand >> i) & 1)
printf("%s1/2^%d", (i == 23) ? "" : " + ",
85 23-i);
printf(") * 2^%d\n", exponent);
/* print out a fractional representation */
90 printf("%f = %d * (", f, sign ? -1 : 1);
for(i = 23 ; i >= 0 ; i--)
if ((significand >> i) & 1)
printf("%s1/%d", (i == 23) ? "" : " + ",
95 (int)two_to(23-i));
printf(") * 2^%d\n", exponent);
/* convert this into decimal and print it out */
100 printf("%f = %d * %.12g * %f\n",
(sign ? -1 : 1),
calc_float(significand, 0),
/* do the math this time */
printf("%f = %.12g\n",
(sign ? -1 : 1) *
110 calc_float(significand, 0) *
return 0;
115 }
Sample output of the value 8.45, which we previously examined, is shown below.
$ ./float 8.45
8.450000 = 1 * (1/2^0 + 1/2^5 + 1/2^6 + 1/2^7 + 1/2^10 + 1/2^11 + 1/2^14 + 1/2^15 + 1/2^18 + 1/2^19 + 1/2^22 + 1/2^23) * 2^3
8.450000 = 1 * (1/1 + 1/32 + 1/64 + 1/128 + 1/1024 + 1/2048 + 1/16384 + 1/32768 + 1/262144 + 1/524288 + 1/4194304 + 1/8388608) * 2^3
8.450000 = 1 * 1.05624997616 * 8.000000
8.450000 = 8.44999980927
From this example, we get some idea of how the inaccuracies creep into our floating point numbers.
^[3] Note that C99 also has portability helpers for printf. The PRI macros in <inttypes.h> can be used as specifiers for types of specified sizes. Again see the standard or pull apart the headers for
full information.
^[4] The ~ operator is the C language operator to apply NOT to the value. It is also occasionally called the one's complement operator, for obvious reasons now! | {"url":"https://eng.libretexts.org/Bookshelves/Computer_Science/Programming_and_Computation_Fundamentals/Computer_Science_from_the_Bottom_Up_(Wienand)/02%3A_Binary_and_Number_Representation/2.02%3A_Types_and_Number_Representation","timestamp":"2024-11-15T03:24:09Z","content_type":"text/html","content_length":"186374","record_id":"<urn:uuid:cf539a69-55b9-4321-b5c6-a56c0107b853>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00264.warc.gz"} |
How do you use the slope of a line to assist in graphing with alegebra?
how do you use the slope of a line to assist in graphing with alegebra? Related topics: calculate algebra
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Author Message Author Message
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Are comparative reports used in horizontal analysis?
Horizontal analysis can either use absolute comparisons or percentage comparisons, where the numbers in each succeeding period are expressed as a percentage of the amount in the baseline year, with
the baseline amount being listed as 100%. This is also known as base-year analysis.
What is comparative horizontal analysis?
Horizontal analysis is the comparison of historical financial information over various reporting periods. It helps determine a companies’ growth and financial position versus competitors. The
horizontal analysis technique uses a base year and a comparison year to determine a company’s growth.
What does horizontal analysis of comparative financial statements include?
Horizontal analysis of financial statements involves comparison of a financial ratio, a benchmark, or a line item over a number of accounting periods. For example, this analysis can be performed on
revenues, cost of sales, expenses, assets, cash, equity and liabilities.
How do you find the comparative horizontal analysis?
Horizontal Analysis (%) = [(Amount in Comparison Year – Amount in Base Year) / Amount in Base Year] * 100
1. The overall growth has been relatively higher in the year 2018 compared to that of the year 2017.
2. Further, it is also noticed that the operating income moves in tandem with the revenue growth, which is a good sign.
What is the main difference between horizontal and vertical analysis?
Given these descriptions, the main difference between vertical analysis and horizontal analysis is that vertical analysis is focused on the relationships between the numbers in a single reporting
period, while horizontal analysis spans multiple reporting periods.
What is an example of horizontal analysis?
Horizontal analysis compares account balances and ratios over different time periods. For example, you compare a company’s sales in 2014 to its sales in 2015. The analysis computes the percentage
change in each income statement account at the far right.
What is the difference between horizontal and vertical analysis?
Horizontal analysis is performed horizontally across time periods, while vertical analysis is performed vertically inside of a column. Horizontal analysis represents changes over years or periods,
while vertical analysis represents amounts as percentages of a base figure.
What is an example of vertical analysis?
In accounting, a vertical analysis is used to show the relative sizes of the different accounts on a financial statement. For example, when a vertical analysis is done on an income statement, it will
show the top-line sales number as 100%, and every other account will show as a percentage of the total sales number.
What is the purpose of horizontal and vertical analysis?
Horizontal analysis usually examines many reporting periods, while vertical analysis typically focuses on one reporting period. Horizontal analysis can help you compare a company’s current financial
status to its past status, while vertical analysis can help you compare one company’s financial status to another’s.
How do you interpret a horizontal and vertical analysis?
For a horizontal analysis, you compare like accounts to each other over periods of time — for example, accounts receivable (A/R) in 2014 to A/R in 2015. To prepare a vertical analysis, you select an
account of interest (comparable to total revenue) and express other balance sheet accounts as a percentage.
How do you calculate Horizontal analysis?
To do a horizontal analysis, you will need the condensed balance sheets for the company that cover the years in question. Start with the first two years you have balance sheets for. Go to the first
item, current assets. Subtract the value for the first year from the second. Negative values are usually denoted by parentheses rather than minus signs.
How to calculate Horizontal analysis?
note the line item’s amount in the base year from the financial statement.
note the amount of the line item in the comparison year.
What is an example of Horizontal analysis?
Example of Horizontal Analysis. Horizontal analysis typically shows the changes from the base period in dollar and percentage. For example, when someone says that revenues have increased by 10% this
past quarter, that person is using horizontal analysis.
How do you calculate vertical analysis percentage?
In vertical analysis each line item is calculated as percentage of a common base line item. The vertical analysis formula used to calculate the line item percentages is as follows: Line item % = Line
item amount / Base line item amount. | {"url":"https://elegant-question.com/are-comparative-reports-used-in-horizontal-analysis/","timestamp":"2024-11-13T13:13:38Z","content_type":"text/html","content_length":"71722","record_id":"<urn:uuid:9a06a20a-e864-45d7-a6fc-6a9151fb6c64>","cc-path":"CC-MAIN-2024-46/segments/1730477028347.28/warc/CC-MAIN-20241113103539-20241113133539-00012.warc.gz"} |
781 Radian/Square Week to Circle/Square Microsecond
Radian/Square Week [rad/week2] Output
781 radian/square week in degree/square second is equal to 1.2233482394295e-7
781 radian/square week in degree/square millisecond is equal to 1.2233482394295e-13
781 radian/square week in degree/square microsecond is equal to 1.2233482394295e-19
781 radian/square week in degree/square nanosecond is equal to 1.2233482394295e-25
781 radian/square week in degree/square minute is equal to 0.00044040536619462
781 radian/square week in degree/square hour is equal to 1.59
781 radian/square week in degree/square day is equal to 913.22
781 radian/square week in degree/square week is equal to 44748
781 radian/square week in degree/square month is equal to 846049.05
781 radian/square week in degree/square year is equal to 121831063.54
781 radian/square week in radian/square second is equal to 2.1351454676521e-9
781 radian/square week in radian/square millisecond is equal to 2.1351454676521e-15
781 radian/square week in radian/square microsecond is equal to 2.1351454676521e-21
781 radian/square week in radian/square nanosecond is equal to 2.1351454676521e-27
781 radian/square week in radian/square minute is equal to 0.0000076865236835475
781 radian/square week in radian/square hour is equal to 0.027671485260771
781 radian/square week in radian/square day is equal to 15.94
781 radian/square week in radian/square month is equal to 14766.34
781 radian/square week in radian/square year is equal to 2126353.19
781 radian/square week in gradian/square second is equal to 1.3592758215883e-7
781 radian/square week in gradian/square millisecond is equal to 1.3592758215883e-13
781 radian/square week in gradian/square microsecond is equal to 1.3592758215883e-19
781 radian/square week in gradian/square nanosecond is equal to 1.3592758215883e-25
781 radian/square week in gradian/square minute is equal to 0.0004893392957718
781 radian/square week in gradian/square hour is equal to 1.76
781 radian/square week in gradian/square day is equal to 1014.69
781 radian/square week in gradian/square week is equal to 49720
781 radian/square week in gradian/square month is equal to 940054.5
781 radian/square week in gradian/square year is equal to 135367848.38
781 radian/square week in arcmin/square second is equal to 0.000007340089436577
781 radian/square week in arcmin/square millisecond is equal to 7.340089436577e-12
781 radian/square week in arcmin/square microsecond is equal to 7.340089436577e-18
781 radian/square week in arcmin/square nanosecond is equal to 7.340089436577e-24
781 radian/square week in arcmin/square minute is equal to 0.026424321971677
781 radian/square week in arcmin/square hour is equal to 95.13
781 radian/square week in arcmin/square day is equal to 54793.47
781 radian/square week in arcmin/square week is equal to 2684880.23
781 radian/square week in arcmin/square month is equal to 50762943.14
781 radian/square week in arcmin/square year is equal to 7309863812.65
781 radian/square week in arcsec/square second is equal to 0.00044040536619462
781 radian/square week in arcsec/square millisecond is equal to 4.4040536619462e-10
781 radian/square week in arcsec/square microsecond is equal to 4.4040536619462e-16
781 radian/square week in arcsec/square nanosecond is equal to 4.4040536619462e-22
781 radian/square week in arcsec/square minute is equal to 1.59
781 radian/square week in arcsec/square hour is equal to 5707.65
781 radian/square week in arcsec/square day is equal to 3287608.44
781 radian/square week in arcsec/square week is equal to 161092813.68
781 radian/square week in arcsec/square month is equal to 3045776588.6
781 radian/square week in arcsec/square year is equal to 438591828758.77
781 radian/square week in sign/square second is equal to 4.077827464765e-9
781 radian/square week in sign/square millisecond is equal to 4.077827464765e-15
781 radian/square week in sign/square microsecond is equal to 4.077827464765e-21
781 radian/square week in sign/square nanosecond is equal to 4.077827464765e-27
781 radian/square week in sign/square minute is equal to 0.000014680178873154
781 radian/square week in sign/square hour is equal to 0.052848643943355
781 radian/square week in sign/square day is equal to 30.44
781 radian/square week in sign/square week is equal to 1491.6
781 radian/square week in sign/square month is equal to 28201.64
781 radian/square week in sign/square year is equal to 4061035.45
781 radian/square week in turn/square second is equal to 3.3981895539709e-10
781 radian/square week in turn/square millisecond is equal to 3.3981895539709e-16
781 radian/square week in turn/square microsecond is equal to 3.3981895539709e-22
781 radian/square week in turn/square nanosecond is equal to 3.3981895539709e-28
781 radian/square week in turn/square minute is equal to 0.0000012233482394295
781 radian/square week in turn/square hour is equal to 0.0044040536619462
781 radian/square week in turn/square day is equal to 2.54
781 radian/square week in turn/square week is equal to 124.3
781 radian/square week in turn/square month is equal to 2350.14
781 radian/square week in turn/square year is equal to 338419.62
781 radian/square week in circle/square second is equal to 3.3981895539709e-10
781 radian/square week in circle/square millisecond is equal to 3.3981895539709e-16
781 radian/square week in circle/square microsecond is equal to 3.3981895539709e-22
781 radian/square week in circle/square nanosecond is equal to 3.3981895539709e-28
781 radian/square week in circle/square minute is equal to 0.0000012233482394295
781 radian/square week in circle/square hour is equal to 0.0044040536619462
781 radian/square week in circle/square day is equal to 2.54
781 radian/square week in circle/square week is equal to 124.3
781 radian/square week in circle/square month is equal to 2350.14
781 radian/square week in circle/square year is equal to 338419.62
781 radian/square week in mil/square second is equal to 0.0000021748413145413
781 radian/square week in mil/square millisecond is equal to 2.1748413145413e-12
781 radian/square week in mil/square microsecond is equal to 2.1748413145413e-18
781 radian/square week in mil/square nanosecond is equal to 2.1748413145413e-24
781 radian/square week in mil/square minute is equal to 0.0078294287323488
781 radian/square week in mil/square hour is equal to 28.19
781 radian/square week in mil/square day is equal to 16235.1
781 radian/square week in mil/square week is equal to 795520.07
781 radian/square week in mil/square month is equal to 15040872.04
781 radian/square week in mil/square year is equal to 2165885574.12
781 radian/square week in revolution/square second is equal to 3.3981895539709e-10
781 radian/square week in revolution/square millisecond is equal to 3.3981895539709e-16
781 radian/square week in revolution/square microsecond is equal to 3.3981895539709e-22
781 radian/square week in revolution/square nanosecond is equal to 3.3981895539709e-28
781 radian/square week in revolution/square minute is equal to 0.0000012233482394295
781 radian/square week in revolution/square hour is equal to 0.0044040536619462
781 radian/square week in revolution/square day is equal to 2.54
781 radian/square week in revolution/square week is equal to 124.3
781 radian/square week in revolution/square month is equal to 2350.14
781 radian/square week in revolution/square year is equal to 338419.62 | {"url":"https://hextobinary.com/unit/angularacc/from/radpw2/to/circlepmicros2/781","timestamp":"2024-11-13T06:46:08Z","content_type":"text/html","content_length":"113803","record_id":"<urn:uuid:b8be00a8-53de-4b10-a656-c878501f7ebd>","cc-path":"CC-MAIN-2024-46/segments/1730477028326.66/warc/CC-MAIN-20241113040054-20241113070054-00626.warc.gz"} |
Chemistry I Laboratory Manual
Learning Objectives
By the end of this section, you will be able to:
• Calculate the density of a sugar solution.
• Evaluate lab sources of error and their effect on an experiment.
The density of an object is defined as the ratio of its mass to its volume. We write this mathematically by using the equations:
Equation 1
[latex]\displaystyle\text{density}=\frac{\text{mass}}{\text{volume}};\text{ d}=\frac{\text{m}}{\text{V}}\\[/latex]
For an example of density, consider the following: Imagine a brick that is made of Styrofoam. Imagine a second brick that is made of lead. Note that even though the bricks take up the same amount of
space – that is, they have the same volume – there is a major difference in their mass. We would say that the lead is denser, that is it has more mass in the same volume.
It is important to note that water has a density of 1.0 g/mL. Objects that have a density less than water, that is, less than 1.0 g/mL, will float on the surface of the water. Those that have a
density greater than 1.0 g/mL will sink. Consider our two bricks again. The brick of Styrofoam will float if we toss it into water. The lead will quickly sink.
Modern ship manufacturers make use of density when designing the ships they build. They use materials that are denser than water but shape the materials so that they take up enough space to float.
Although the ships weigh several thousand tons, that mass takes up a lot of space. Overall, the ship has a density less than water and therefore floats.
Two factors have an effect on the density of water:
1) Temperature will have a small effect on the density. For water, density increases as temperature decreases. See Table 1 for the density of water at different temperatures.
2) If more dense materials are dissolved in the water, the solution density will increase. We will see this effect in today’s lab when we measure the effect of dissolving sucrose on the density of
Table 1. Density of water at different temperatures
Temp (°C) dH2O (g/mL) Temp (°C) dH2O (g/mL)
18.0 0.99860 22.0 0.99777
18.5 0.99850 22.5 0.99765
19.0 0.99841 23.0 0.99754
19.5 0.99830 23.5 0.99742
20.0 0.99820 24.0 0.99730
20.5 0.99809 24.5 0.99716
21.0 0.99799 25.0 0.99704
21.5 0.99788 25.5 0.99690
In this experiment you will test your laboratory technique by calibrating a 10 mL graduated cylinder, making up an aqueous sucrose solution of a particular mass percent in solute, and measuring the
density of the solution with the calibrated graduated cylinder. The density result will be evaluated by students for accuracy and precision. Since the correct density will depend on a correctly
prepared sugar solution, careful sample preparation will be critical.
There are many ways of describing the concentration of a solution. The mass percent of solute in a solution is given by the symbol and the equation:
[latex]\displaystyle\text{Mass }\%=\frac{\text{Mass of Solute}}{\text{Total Mass of Solution}}\times100\\[/latex]
The advantage of this type of concentration unit is that it depends only on the mass, which is accurately measured with an analytical balance. It is not dependent on the temperature.
Note: Volumes are dependent on temperature. For example, a 10.000 mL volume of water will increase by 0.016 mL when the temperature is raised from 18°C to 25°C. Table 1 gives the density of water at
different temperatures.
Another useful property is using the percent error to determine the amount a measurement is off from the theoretical value. The equation for finding percent error is:
[latex]\displaystyle\text{Percent Error}=\frac{\text{Experimental Value}-\text{Theoretical Value}}{\text{Theoretical Value}}\times100\\[/latex]
This allows us a more reasonable comparison of numbers than looking at the difference only because the magnitude of the theoretical value is considered. A table of the theoretical values of density
for sucrose solutions of various (w/w)% is included in Table 2 below.
Table 2: Theoretical Density Values of Sucrose Solutions with Known Mass Percent
Mass % Density (g/mL) Mass % Density (g/mL)
0.00 1.000 12.50 1.051
2.50 1.011 15.00 1.062
5.00 1.021 17.50 1.073
7.50 1.030 20.00 1.084
10.00 1.042 22.50 1.102
Graphing Data
It is imperative students learn to properly organize and graph data. Students may wish to review graphing data and calculating the slope prior to coming to lab this week if it has been a few years
since you have had a math course. A brief review is included here but may not be sufficient for some students.
Manual graphs should always:
• Be drawn on graph paper (included within the lab handout).
• Include data points (and possibly the labels as well).
• Have labels for the graph itself (named Y vs. X), the axes (with both name and units), and (if applicable) the legend.
• Be drawn large enough to visually see all components.
• Include axis scales that are appropriate (they may not start at 0, depending on the data).
• Contain a line of best-fit.
Graphs done in Microsoft Excel should always
• Include all of the components of manual graphs.
• Be in the “scatter” chart type unless otherwise specified.
• Include the equation for the line of best fit. | {"url":"https://courses.lumenlearning.com/chemistry1labs/chapter/lab-2-introduction/","timestamp":"2024-11-03T09:38:33Z","content_type":"text/html","content_length":"53438","record_id":"<urn:uuid:e5db7739-0094-47cc-b1e1-f4edbfad5b1b>","cc-path":"CC-MAIN-2024-46/segments/1730477027774.6/warc/CC-MAIN-20241103083929-20241103113929-00019.warc.gz"} |
Bounded cohomology of subgroups of mapping class groups.
Bounded cohomology of subgroups of mapping class groups.
Bestvina, Mladen, and Fujiwara, Koji. "Bounded cohomology of subgroups of mapping class groups.." Geometry & Topology 6 (2002): 69-89. <http://eudml.org/doc/122903>.
author = {Bestvina, Mladen, Fujiwara, Koji},
journal = {Geometry & Topology},
keywords = {bounded cohomology; mapping class groups; hyperbolic groups},
language = {eng},
pages = {69-89},
publisher = {University of Warwick, Mathematics Institute, Coventry; Mathematical Sciences Publishers, Berkeley},
title = {Bounded cohomology of subgroups of mapping class groups.},
url = {http://eudml.org/doc/122903},
volume = {6},
year = {2002},
TY - JOUR
AU - Bestvina, Mladen
AU - Fujiwara, Koji
TI - Bounded cohomology of subgroups of mapping class groups.
JO - Geometry & Topology
PY - 2002
PB - University of Warwick, Mathematics Institute, Coventry; Mathematical Sciences Publishers, Berkeley
VL - 6
SP - 69
EP - 89
LA - eng
KW - bounded cohomology; mapping class groups; hyperbolic groups
UR - http://eudml.org/doc/122903
ER -
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Application of Fractal Theory in Brick-Concrete Structural Health Monitoring
Engineering Vol.08 No.09(2016), Article ID:71000,11 pages
Application of Fractal Theory in Brick-Concrete Structural Health Monitoring
Changmin Yang^1, Xia Zhao^2, Yanfang Yao^2, Zhongqiang Zhang^1^
^1Architectural Engineering Institute, Hebei University, Baoding, China
^2Hebei College of Science and Technology, Baoding, China
Copyright © 2016 by authors and Scientific Research Publishing Inc.
This work is licensed under the Creative Commons Attribution International License (CC BY 4.0).
Received: August 4, 2016; Accepted: September 26, 2016; Published: September 29, 2016
In order to monitor and forecast the deformation of the brick-concrete building, by taking a brick-concrete building as research object, fiber grating sensors were used to collect the monitoring data
and double logarithmic curve of limit value characteristic and monitoring data were obtained based on the fractal theory. Constant dimension fractal method cannot be used to analyze the data
directly. With the method of variable dimension fractal, we accumulate data, and the double logarithmic curve is smooth. Piecewise fractal dimensions are close. The outer interpolation method is used
to calculate the fractal dimension of the next point and then back calculate the vertical displacement. The relative errors are calculated by comparing the forecast values and monitoring values, and
the maximum relative error is 5.76%. The result shows that the fractal theory is suitable to use in the forecast of the deformation and the accuracy is good.
Brick-Concrete Building, Real-Time Monitoring, Fiber Grating Sensors, Constant Dimension Fractal, Variable Dimension Fractal, Log-Log Line, Prediction
1. Introduction
In China, the structure is generally divided into brick-concrete building, post and panel structure, and reinforced concrete structure, and these buildings during service are bound to produce
cumulative damage affected by corrosion, fatigue, aging and other factors, so it is particularly important to monitor the buildings which are on active service.
Health monitoring is an important means to understand the situation of the buildings in use and to layout reasonable monitoring position; analyzing and processing monitoring data will be a key step
to master the state of buildings in use; reasonable data processing and effective monitoring models can help staffs to discover abnormities and they can take appropriate measures to ensure the
security of the persons and property within the building [1] - [3] . With the progress and mutual integration of modern testing, analysis technics, computer technology, mathematical theory and
wireless communication technology, and the traditional monitoring methods are changing to an online, dynamic, real-time direction [4] [5] . Fiber grating technology applying on-line monitoring is a
major breakthrough in the field of civil engineering structures or buildings, it has aroused widespread attention [6] - [9] .
Fractal geometry developed by the Mandelbrot [10] in the 1970s is a new branch of mathematics, it focuses on the similarity between the parts and the whole, starting directly from the complex
non-linear system, and it recognizes the inherent regularity [11] [12] by the no simplified and abstract study itself. The fractal theory applied to fault diagnosis and monitoring of structural
damage is only appeared recently; fault diagnosis and damage identification of fractal theory has been widely developed in the field of machinery, aerospace, ships, vehicles [13] - [16] , but in the
field of civil engineering [17] seems less. Yongqiang Jin [18] makes a more reliable prediction to dam uplift pressure with the application of fractal theory. Displacement-time curve variability of
buildings under various loads also satisfies the self-similarity of fractal theory in the process of service, therefore, the fractal theory applied to health monitoring and early warning is
theoretically feasible. The traditional health monitoring methods are time series, neural network, etc., the theory is applied to analyze the structure by the observational data, and analyze the
characteristics of a specific observed quantity in particular period of time and then predict its future variation by extrapolation methods, but it requires a large amount of observation data. When
the observed quantities and the range of time change, the model also need to do a great change, and we cannot find any common between observed quantities of building systems. Fractal theory is a very
important part in the modern non-linear science research, it can describe some kinds of scientific complexity of nature and various non-linear systems encountered, and it requires only small amount
of data (15 - 30 values) to predict accurately, so the calculation is faster than traditional forecasting methods; it is a new method of processing prototype observation data of buildings.
Fractal theory means the parts in some way are similar to the whole, under normal circumstances it can be regarded as the state of gathering of fragments. It generally has the following
characteristics: 1) fractal sets have a ratio of the details of any small scales, or have a fine structure; 2) fractal sets have some self-similar forms, they may be approximate self-similarity or
statistical self-similarity; 3) the fractal dimension of fractal sets is strictly greater than its corresponding topological dimension.
The remote monitoring and warning system used in the experiment includes: fiber sensing systems, signal acquisition and transmission systems, data processing and monitoring and warning system. Fiber
grating sensor system includes: types of fiber grating sensors, modulation system and installation of fiber grating sensors. The system of signal transmission and collection includes a correction of
fiber grating sensors, application of module, storage structure and methods of vast amounts of real-time data. Data processing, monitoring and warning system is a key part of this experiment,
including visualization system of data analyzing and structure running status, and the function of disaster early warning. This study is based on the fractal theory, using the methods of constant
dimension fractal and variable dimension fractal to deal with the data collected by fiber grating sensors, and we can predict the vertical displacement of the next point in time, and then to achieve
the goal of monitoring and early warning.
2. Constant Dimension Fractal and Variable Dimension Fractal
At present, the application of constant dimension fractal is described by Equation (1)
Here: r is the characteristic linearity; N is the function associated with r; C is undetermined constant; D is the dimension. Because D is a constant, Equation (1) in log-log line is a straight line,
so any two data points
If there is a negative number in the logarithm operation, all the values of this sequence plus a constant to eliminate the impact of negative number. However, in the curve of log-log line, if there
is a nonlinear function, this constant dimension fractal cannot be dealt with. Variable dimension fractal can be introduced to solve the problem, specifically as follows:
Because the fractal dimension D is a function of the characteristic linearity r:
Then a Function relationship
available variable dimension fractal form,
That is a form of variable dimension fractal.
We can know from the Equation (5), any functions the same as
Then use the basic sequence to construct other cumulative sequence. If you construct a first order accumulation sequence
3. Monitoring General Situation about Brick-Concrete Structure
3.1. Monitoring Purpose and the Arrangement of Measuring Points
The test takes the telecommunication building of Hebei University as research object. The telecommunication building belongs to brick-concrete building, 6 layers, 5 layers of main building, built in
1973.12-1976.12, parts of beams, columns exists aging, corrosion and other phenomena. Damage of the aging, corrosion place can easily occur in the future, the main building appears minute vibrations,
deformation under the environmental loads, then the vibrations and deformation may lead to crack development, even worse, the whole building may collapse. To make sure the safety of the whole
building, teachers and students, we analyze the status of the telecommunication building, then make sure the physical types and the appropriate sensor type and the sensor location.
In the outer wall surface of each floor of typical aging parts, we put FBG surface crack meters to monitor the development of main cracks. In the southeast, northeast, southwest and northwest corners
of the main building roof, we install a FBG fiber level to monitor the vertical deformation of main building under outside interference. This is because the top of the building is the most sensitive
to external factors, and it seems inconspicuous to artificial disturbance. In the typical beams, columns of the building, we arrange FBG strain gauge to monitor the strain of the beams and columns
under outside interference, monitoring point arrangement is as shown in Figure 1, each monitoring instrument parameters are as shown in Table 1 (instrument parameters), the monitor-
Figure 1. Structure model and arrangement of measuring points.
ing devices are as shown in Figure 2 (Measuring device).
3.2. Data Analysis
Due to the huge amount of test data, the test is in order to deal with the level monitoring data, we cut out 30 s data of vertical displacement from the northeast corner of the telecommunication
building roof, we cut out a data point every second, a total of 30 data points, we take these points as the research object, then we numbers them in chronological order, we use the former 20 points
to build prediction model, then use the latter
Figure 2. Measuring device. (a) roof level; (b) speed fiber grating demodulator; (c) strain gage; (d) crack meter.
10 points to check the correctness of the prediction model.
Table 2 shows that: this part of the vertical displacement includes positive number and negative number, however, the negative number will be unable to operate in the logarithmic coordinates, then we
plus 0.2 for them all to eliminate the impact, the processed data is shown in Table 3 (after processing).
The data after processing can obtain piecewise fractal dimension of monitoring data according to Equation (2), Table 4 (Piecewise fractal dimension of monitoring data):
We build log-log coordinate according to the data in Table 4 (Piecewise fractal dimension of monitoring data), then draw the Figure 3 (The log-log curve of the original monitoring data): log-log
curve has a big fluctuation, piecewise fractal dimension (D) includes negative number and positive number, so it is difficult to predict.
So we deal with the original monitoring data by first-order accumulation, depending on Equation (7), then we obtain the piecewise fractal dimension (Table 5 the monitoring data of first-order
accumulative piecewise fractal dimension). Draw Figure 4, the monitoring data of first-order accumulative double logarithmic curve),we can see in the Table 4 (Piecewise fractal dimension of
monitoring data): the original complex curve change into a relatively smooth curve, each of the piecewise fractal dimension seems similar, so we can predict unknown piecewise fractal dimension
according to the known piecewise fractal dimension, then we can anti-derived displacement, this will achieve the purpose of early warning.
Table 2. The original data of northeast.
3.3. The Prediction on of Monitoring Data
By comparing Figure 3 (The log-log curve of the original monitoring data) and 4 (the monitoring data of first-order accumulative double logarithmic curve), a cumulative fist-order log-log curve
(fractal dimension curve) can act as a predictive model to predict the data of the telecommunication building, and using equations to predict the 10 values in the next time. Specific methods are as
1. Figure out the total increment Table 4 (Piecewise fractal dimension of monitoring data), so the average of the neighboring piecewise fractal dimension is.
2. Figure out the piecewise fractal dimension after the19th subsection.
Table 4. Piecewise fractal dimension of monitoring data.
Figure 3. The log-log curve of the original monitoring data.
Table 5. The monitoring data of first-order accumulative piecewise fractal dimension.
Figure 4. The monitoring data of first-order accumulative double logarithmic curve.
3. depending on the equation we can obtain:
We can predict the displacement of the sequence (21 ~ 30), substituting D of Equation (8) into Equation (9). According to the result, we can discover that the relative error is between −5.74% and
+5.76% (Table 6, the prediction results of variable dimension fractal). So the method of variable dimension fractal can predict the structural deformation.
4. Conclusion
The result of the test shows that FBG sensing technology can achieve the goal of the remote real-time dynamic prediction to the deformation and displacement of the brick- concrete buildings. Through
processing the displacement-time graph, we can visually monitor the state of buildings. Health monitoring data usually presents self-similarity and satisfies the conditions of application of fractal
theory. Fractal theory can make a reasonable assessment quickly for the health status of the buildings which are in use after analyzing the dimension changes of displacement curve. When the range of
the measurements and time changes, there is no need to change the prediction model, and the similarity of systematic measurements of the brick-concrete structures can be reflected. Displacement data
which FBG sensors collect meet the fractal characteristics, but D, piecewise fractal dimension, has a big fluctuation, then variable dimension fractal can be a predictive model to monitor
brick-concrete buildings, and the relative error between predictive value and true value ranges from −5.74% to +5.76%, so accuracy of the prediction is better than others. By setting the alarm value
of the building, fractal theory provides a new type of monitoring, early warning methods for the practical engineering. Related conclusions have yet to be studied further.
Table 6. The prediction results of variable dimension fractal.
Cite this paper
Yang, C.M. Zhao, X., Yao, Y.F. and Zhang, Z.Q. (2016) Application of Fractal Theory in Brick-Concrete Structural Health Monitoring. Engineering, 8, 646-656. http://dx.doi.org/10.4236/eng.2016.89058 | {"url":"https://file.scirp.org/Html/7-8102670_71000.htm","timestamp":"2024-11-13T08:37:55Z","content_type":"application/xhtml+xml","content_length":"49925","record_id":"<urn:uuid:3e42e08a-7d17-431c-a9a8-c73f4c5dd5df>","cc-path":"CC-MAIN-2024-46/segments/1730477028342.51/warc/CC-MAIN-20241113071746-20241113101746-00320.warc.gz"} |
Pre-service Secondary Mathematics Teachers Making Sense of Definitions of Functions
Pre-service Secondary Mathematics Teachers Making Sense of Definitions of Functions
Definitions play an essential role in mathematics. As such, mathematics teachers and students need to flexibly and productively interact with mathematical definitions in the classroom. However, there
has been little research about mathematics teachers’ understanding of definitions. At an even more basic level, there is little clarity about what teachers must know about mathematical definitions in
order to support the development of mathematically proficient students. This paper reports on a qualitative study of pre-service secondary mathematics teachers choosing, using, evaluating, and
interpreting definitions. In an undergraduate capstone course for mathematics majors, these future teachers were assigned three tasks which required them to (1) choose and apply definitions of
functions, (2) evaluate the equivalence of definitions of functions, and (3) interpret and critique a secondary school textbook’s definition of a specific type of function. Their performances
indicated that many of these pre-service mathematics teachers had deficiencies reasoning with and about mathematical definitions. The implications of these deficiencies are discussed and suggestions
for teacher educators are proposed. | {"url":"https://mted.merga.net.au/index.php/mted/article/view/17","timestamp":"2024-11-14T00:27:16Z","content_type":"text/html","content_length":"14722","record_id":"<urn:uuid:3cb65e22-7749-4ce5-9b6a-7dfefc346444>","cc-path":"CC-MAIN-2024-46/segments/1730477028516.72/warc/CC-MAIN-20241113235151-20241114025151-00737.warc.gz"} |
Succeed with maths: part 2
2 Scientific notation and large numbers
So how can you use powers of ten to write numbers in scientific notation? Let’s look at the example of six million to start with. Written in full this number is six followed by six zeroes: 6 000 000.
This can also be written as: . Now 6 million is shown like this it can be written using a power of ten by noting that:
10 multiplied by itself 6 times is the same as 10^6
So, this now means that 6 million can be written as:
You would therefore write 6 million using scientific notation as: .
Similarly, if the example had been six and a half million (6 500 000), this can be written in scientific notation as:
or in scientific notation.
So, a number written in scientific notation takes the form of: a number between 1 and 10 multiplied by a whole number power of 10. This can be shown mathematically as:
Thus, there are two steps to writing a number using scientific notation, as follows:
1. Work out what the number between 1 and 10 will be.
2. From this, decide on the power of 10 required.
So, taking the example of 130 000, the number between 1 and 10 must be 1.3, as it cannot be 0.13 or 13. 0.13 is less than 1, and 13 is greater than 1.
So, 130 000 written in scientific notation is . Now, it’s your turn to try some examples.
Activity 2 Understanding and writing numbers in scientific notation
Timing: Allow approximately 10 minutes
Write the following numbers without using powers of 10.
Write the following numbers in scientific notation.
1 billion is 1 followed by 9 zeroes.
1 trillion is a million million.
• g.Which of these numbers is the biggest?
Compare the powers of ten.
• g.The highest power of the three numbers in this activity is 14, so 400 trillion is the biggest number here.
Now you’ve found out how to write large numbers using scientific notation, in the next section you’ll turn your attention back to the problem posed at the beginning of this week: how to work out the
width of the observable universe in kilometres. | {"url":"https://www.open.edu/openlearn/mod/oucontent/view.php?id=19187§ion=2","timestamp":"2024-11-07T00:43:07Z","content_type":"text/html","content_length":"254539","record_id":"<urn:uuid:68892969-3f3b-42ed-9e5a-6d460488ec57>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00010.warc.gz"} |
To resubmit or Not to resubmit - Codeforces
Hello CodeForces!
Recently, I am thinking about the resubmission strategy in CodeForces. Last week I participated in CodeForces Round #905 (Div.1), and I successfully solved first 4 problem with ~10 minutes remaining.
However, after that, I saw the pretest result again and I noticed that execution time of problem C was 2979/3000ms.
Then I decided to speed up the solution of problem C and finally resubmitted just a few seconds before the contest ends. The execution time in pretest was shortened to 2464ms, and all of my solution
passed in system test.
But the biggest blunder in this contest was resubmission — the number of testcases in pretest and system test was almost the same, and even my solution before resubmission passed (I confirmed after
the contest). Since the score is mainly determined by submission time, my final rank in the contest dropped from ~60th to ~100th, and this made a difference between rating [S:increase:S] almost
increase (UPD: miscalculated) and decrease.
So now I would like to ask everyone about resubmission strategy. There are some cases which may require resubmission in CodeForces, like as follows:
• My initial solution was not proved yet, and I found another proved solution
• I found a counter-example of my solution
• My solution is very close to time limit
But when will you decide the resubmission? I am very appreciate for sharing your opinion.
Thank you for reading this blog.
12 months ago, # |
» +35
chromate00 For the TLE case, I would prefer not resubmitting unless I have a clear counterexample that makes the solution definitely TLE. This is because codeforces rechecks the solution up to 3
times when the verdict is TLE, to check if the cause is really the solution or just a noise in the measurement. You can [S:abuse:S]utilize this to your advantage however you want.
• 12 months ago, # ^ |
← Rev. 2 → +18
» I'd say, figure out the reason why it's taking so long to run and take a decision based on that. And the multiple execution doesn't always help. Sharing a fun experience. Spoilered since
it's kinda long.
Fun Experience
Takeaway: If I figured out during the contest that I made the mistake, I'd submit it again.
□ » 12 months ago, # ^ |
» ← Rev. 5 → 0
---deleted since the typo was fixed and Noone like my little joke--
☆ »
» 12 months ago, # ^ |
» 0
Or I might be bad at math lol. Fixed, thanks!
12 months ago, # |
In my opinion, it is only necessary to resubmit if you are sure that your solution is wrong (there exists a counter-example). Contest problems have to be very strict with their
» testcases, and generally the pretests should account for extreme cases (large input size) and tricky edge cases.
awesomeguy856 For the issue of slow solutions, I think if you are worried about TLE in main tests then that's something you should watch for immediately after your submission when you can quickly
optimize your code, but it's not really worth it if it's been a long time since your submission.
I don't think solutions being unproven is something to worry about, oftentimes the proof might be difficult and a waste time to find, and if you think your solution might not be correct
then quickly think for possible counter examples then move on.
12 months ago, # |
» +40
ko_osaga I remember doing the same things as what you described. I considered it as a mistake and decided to never do the same again.
Anyway, for the cases you mentioned, I guess I will resubmit for #1 and #3 only if I have a very cheap fix, and resubmit for #2 if I know how to fix it.
12 months ago, # |
We can use a simple mathematical model to help answer this question. Assume that your objective is to maximize your expected rating. I'm going to assume for simplicity that all aspects of
the contest results are known except whether your original submission and updated submission will pass systests. Let $$$p_1$$$ and $$$p_2$$$ be the probabilities that your old and new
submissions, respectively, get AC.
Consider three cases.
1: You don't resubmit your C. In this case, by my calculations your score would have been 2760 if you got AC and 2082 otherwise. According to cfviz, this would have led to -7 if you got AC
and -58 otherwise. Thus, your expected delta is $$$-7p_1 - 58 (1 - p_1) = 51p_1 - 58.$$$
2: You resubmit your C after solving D (this is what you did in contest). In this case, your delta would be -58 as before if you FST and -35 (what you received in-contest) with AC. Thus,
your expected delta is $$$-35p_2 - 58(1-p_2) = 23 p_2 - 58.$$$
3: You resubmit C before solving D. This option is a bit less attractive in practice because in practice it might stop you from solving D. In this case, assuming resubmitting C takes you
Geothermal 12 minutes (the amount of time it took after D), you would have solved C at 1:20 with two penalties, so the value of C would have been about 580. You would have solved D at 1:59, for a
score of about 655. This would give you a total score of 2602 with an AC and 2022 with an FST on C. Your resulting delta would have been -29 with an AC and -68 with an FST, for an expected
delta of $$$-29 p_2 - 68 (1 - p_2) = 39p_2 - 68.$$$
First, let's figure out when (2) is preferable to (3). This holds when $$$23 p_2 - 58 \ge 39 p_2 - 68$$$, i.e., $$$10 \ge 16 p_2$$$, i.e., $$$p_2 \le \frac{5}{8}.$$$ Thus, if your new
submission has at least a $$$5/8$$$ chance of AC, you should resubmit immediately, while otherwise you should resubmit after solving D.
To compare (1) and (2) we have $$$23 p_2 - 58 \ge 51 p_1 - 58$$$ when $$$p_2 \ge \frac{51}{23} p_1 \approx 2.52 p_1.$$$ Thus, resubmitting after solving D is only optimal if it multiplies
your probability of AC by at least $$$2.5$$$, but the resulting AC probability is at most $$$5/8.$$$
To compare (1) and (3), we have $$$39 p_2 - 68 \ge 51 p_1 - 58$$$ when $$$39 p_2 \ge 51 p_1 + 10,$$$ i.e., $$$p_2 \ge 1.31 p_1 + 0.32$$$. If this condition holds and $$$p_2 \ge 5/8$$$,
then resubmitting C immediately after solving is optimal. As a simple example, if $$$p_2 = 1$$$, then this is optimal if $$$p_1 \le 0.52.$$$
My intuitive assessment of this situation is that (2) is unlikely to be preferable to (3) unless there's a high probability that resubmitting costs you the opportunity to solve another
problem. The decision to resubmit or not immediately after solving C is closer and depends largely on your assessment of whether pretests are likely to be strong or if systests are likely
to have a test on which your solution slows down substantially.
» 12 months ago, # |
Just never resubmit and write a comment complaining about tight TL if you FST
• »
» 12 months ago, # ^ | | {"url":"https://mirror.codeforces.com/blog/entry/121849","timestamp":"2024-11-10T18:35:13Z","content_type":"text/html","content_length":"122269","record_id":"<urn:uuid:8ccb6c9a-ae4c-4071-9cfc-3fb418a9e8d3>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.61/warc/CC-MAIN-20241110170046-20241110200046-00663.warc.gz"} |
Job Scheduling Problem with DEAP
Problem Description:
In job scheduling, we aim to assign jobs to machines or workers in such a way that the overall completion time (makespan) is minimized.
This is a common problem in manufacturing, project management, and computer systems where tasks need to be allocated efficiently to minimize delays and optimize resources.
We have multiple jobs that need to be scheduled on different machines.
Each machine can only handle one job at a time, and jobs have different processing times.
The goal is to minimize the maximum time it takes to complete all jobs (i.e., the makespan).
We will solve this using the $DEAP$ (Distributed Evolutionary Algorithms in $Python$) library, leveraging a $Genetic$ $Algorithm$ ($GA$).
Steps for Solving Using DEAP
1. Define the Problem:
□ We have N jobs and M machines.
□ Each job has a specific processing time.
□ We want to assign jobs to machines such that the overall makespan is minimized.
2. Genetic Algorithm Setup:
□ Individuals: Each individual in the population represents a possible solution, i.e., a specific assignment of jobs to machines.
□ Fitness Function: The fitness function will evaluate the makespan of the solution. We aim to minimize this value.
□ Mutation: Swapping jobs between machines.
□ Crossover: Combining two individuals (solutions) by taking part of one solution and merging it with another.
□ Selection: Using tournament selection to choose the best solutions from the population.
DEAP Implementation
1 import random
2 from deap import base, creator, tools, algorithms
3 import numpy as np
5 # Job scheduling problem parameters
6 N_JOBS = 10 # Number of jobs
7 M_MACHINES = 3 # Number of machines
8 PROCESSING_TIMES = [random.randint(1, 20) for _ in range(N_JOBS)] # Random processing times for each job
10 # DEAP Setup
11 creator.create("FitnessMin", base.Fitness, weights=(-1.0,)) # Minimize makespan
12 creator.create("Individual", list, fitness=creator.FitnessMin)
14 def create_individual():
15 """Create an individual where jobs are randomly assigned to machines."""
16 return [random.randint(0, M_MACHINES - 1) for _ in range(N_JOBS)]
18 def evaluate(individual):
19 """Evaluate the makespan of the current job allocation to machines."""
20 machine_times = [0] * M_MACHINES # Track processing time for each machine
21 for i, machine in enumerate(individual):
22 machine_times[machine] += PROCESSING_TIMES[i]
23 return max(machine_times), # Minimize the maximum machine time (makespan)
25 # DEAP tools setup
26 toolbox = base.Toolbox()
27 toolbox.register("individual", tools.initIterate, creator.Individual, create_individual)
28 toolbox.register("population", tools.initRepeat, list, toolbox.individual)
29 toolbox.register("mate", tools.cxUniform, indpb=0.5)
30 toolbox.register("mutate", tools.mutUniformInt, low=0, up=M_MACHINES - 1, indpb=0.1)
31 toolbox.register("select", tools.selTournament, tournsize=3)
32 toolbox.register("evaluate", evaluate)
34 # Genetic Algorithm Parameters
35 POPULATION_SIZE = 100
36 CXPB = 0.7 # Crossover probability
37 MUTPB = 0.2 # Mutation probability
38 NGEN = 50 # Number of generations
40 # Initialize population
41 population = toolbox.population(n=POPULATION_SIZE)
43 # Run the Genetic Algorithm
44 algorithms.eaSimple(population, toolbox, cxpb=CXPB, mutpb=MUTPB, ngen=NGEN, verbose=True)
46 # Find the best solution
47 best_individual = tools.selBest(population, k=1)[0]
48 best_makespan = evaluate(best_individual)[0]
50 print(f"Best job allocation: {best_individual}")
51 print(f"Best makespan: {best_makespan}")
1. Problem Setup:
□ We defined N_JOBS as $10$ jobs and M_MACHINES as $3$ machines.
Each job has a randomly assigned processing time.
□ The create_individual function generates a random assignment of jobs to machines.
2. Fitness Function:
□ The fitness function (evaluate) calculates the makespan, which is the maximum time it takes any machine to complete its assigned jobs.
We aim to minimize this makespan.
3. Evolutionary Process:
□ We use genetic operators such as crossover and mutation to evolve better solutions over generations.
□ Crossover: This combines parts of two individuals (job assignments).
□ Mutation: Randomly alters a small part of the individual (a job is moved to a different machine).
□ Selection: The algorithm uses tournament selection to choose the best individuals to evolve in the next generation.
4. Result:
□ After running for $50$ generations, the algorithm returns the best job assignment and the minimum makespan.
Sample Output:
1 Best job allocation: [2, 1, 0, 0, 2, 0, 1, 2, 1, 0]
2 Best makespan: 32
Interpretation of Results:
• The best solution indicates which jobs should be assigned to which machines, represented by a list (e.g., [2, 1, 0, 0, 2, ...]).
• The makespan is the total time taken by the machine that finishes last, which in this example is $32$ units of time.
This approach demonstrates how we can use $DEAP$ to optimize a complex scheduling problem by evolving solutions to find the best possible job assignments. | {"url":"https://ailogsite.netlify.app/2024/10/11/2024/20241011/","timestamp":"2024-11-06T07:34:51Z","content_type":"text/html","content_length":"24111","record_id":"<urn:uuid:0c1c1f2f-5c32-4ef9-979f-b3be9723806a>","cc-path":"CC-MAIN-2024-46/segments/1730477027910.12/warc/CC-MAIN-20241106065928-20241106095928-00589.warc.gz"} |
Dalculator Download For PC 2022 [New] — Lourenço CargasDalculator Download For PC 2022 [New]
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We’ve got a game for that! It’s a fun way to review basic human anatomy and physiology for kids. Let’s learn something together!
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Some secrets of HAL 9000 are revealed in an excerpt from “ HAL 9001: A ComputerTake…
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Dalculator is a simple, easy-to-use calculator. But can Dalculator also do some magic? I say yes and yes. Dalculator can write formulas to your clipboard and other things which you can use
afterwards. If you want to know more about this text-utility you can read further: Dalculator – My First Step
Dalculator is the default calculator app from Dropbox. I’ve noticed that over the past years, I’ve used alot of these tools. But, I don’t think I’ve ever used the pretty one before. Maybe it’s
because I’m new to the iPhone. But one of the greatest features of this calculator is that you can create your own functions. Now, even if you’re the type that has an abridged version of a math
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Dalculator is a nice and easy-to-use calculator for Windows 8 that is available in the Microsoft Store for free. The calculator gives you the option to quickly evaluate mathematical expressions. It
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Deixar um comentário | {"url":"https://lourencocargas.com/dalculator-download-for-pc-2022-new/","timestamp":"2024-11-05T23:13:26Z","content_type":"text/html","content_length":"83150","record_id":"<urn:uuid:be400d91-65ba-4610-bf89-1cfe2c651e52>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00211.warc.gz"} |
Folded rouble / Etudes // Mathematical Etudes
So can we or not increase the initial perimeter of the rectangle? In the years 1991 and 1993, yet the banknotes were changed, and that of the rouble of 1961 came out of circulation. But the Arnold
problem still remained unsolved.
Since that time a Russian rouble worth, unfortunately, so little that there are no more notes, but only coins, of that value.
At the beginning of the twenty-first century, however, the problem was solved. The first mathematically rigorous solution was provided by a student of N.P. Dolbilin, Alexei Tarasov. He invented
an algorithm for folding a square so that in total you get a planar polygon with a greater perimeter.
Who only wishes to enjoy the movie, can skip the next part, which has been added for those who want to understand well how the sheet has to be folded. | {"url":"https://en.etudes.ru/etudes/napkin-folding-problem/1/","timestamp":"2024-11-09T19:55:49Z","content_type":"application/xhtml+xml","content_length":"46188","record_id":"<urn:uuid:3473da4d-1f24-46c5-a0be-a4431677e6b4>","cc-path":"CC-MAIN-2024-46/segments/1730477028142.18/warc/CC-MAIN-20241109182954-20241109212954-00872.warc.gz"} |
Rich McLaughlin (UNC-CH), Colloquium - Department of Mathematics
Rich McLaughlin (UNC-CH), Colloquium
October 13, 2016 @ 4:00 pm - 5:00 pm
Tea at 3:30 pm in 330 Phillips Hall
Title: Tailoring the Tails in Taylor Dispersion
Abstract: The interaction of the physics of advection and diffusion is a well studied problem in turbulent transport, and gives rise to quantified understandings of things like wind chill effects.
Diffusive effects tend to be strongly dependent upon a contaminants exposed surface area to the fluid flow. The interaction of fluid flow with diffusion can be quickly understood by thinking about
how a pollutant, say a dye, can be stretched by a non-spatially constant fluid flow with to have greater surface area exposed to the fluid over which diffusivity effects can be greatly enhanced.
This is the mechanism first described mathematically by Sir G. I. Taylor in 1953, where he explicitly computed this effect for laminar flow in a straight, circular pipe, computing an effective,
“renormalized”, diffusivity which is seen to be tremendously boosted over the usual molecular diffusivity, which typically is only on the order of 10^(-6) cm^2/s. This phenomena sets in after
waiting a sufficient amount of time for molecular diffusion to have time to act, and as such is a long time asymptotic theory. Very, very interesting effects occur on shorter timescales, which is
what we describe in this lecture.
The pure diffusion equation preserves initial symmetries. But we will show how the interaction of fluid advection with diffusion can break up-stream/downstream initial symmetries, and lead to, for
example, chemical deliveries which are front-loaded, arriving with a sharper front, then depart with a much gentler tail, or the reverse of this depending upon the shape of the cross-section as we
show in this lecture. The lowest order statistic which captures these differences is the skewness, which is the centered, normalized third moment of the up-stream/downstream distribution of tracer.
The sign of this quantity, under suitable conditions, distinguishes the front-loaded versus back-loaded distributions. These different effects are of interest in micro-scale fluidics, and in drug
We explore the role different geometries (amongst rectangular and elliptical domains of arbitrary aspect ratios) play in controlling emerging up-stream/downstream asymmetries in the cross-sectionally
averaged distribution of diffusing passive scalars advected by laminar, pressure driven shear flows. We show using a combination of rigorous analysis, asymptotic expansions, and Monte-Carlo
simulations, that on short time scales relative to the shortest diffusion times, elliptical domains preserve initial uptream/downstream symmetric distributions, while rectangular ducts break this
symmetry. Skinny ducts produce distributions with negative skewness, while fat ducts produce positive skewness for symmetric initial data which is uniformly distributed in the cross-section. There
is a special aspect ratio of approximately 2-1 ratio for which symmetry is preserved. In turn, long-time (relative to the longest diffusion timescale) exact analysis and simulation shows that all
geometries generically break symmetry before ultimately symmetrizing in infinite time. Laboratory experiments are presented which confirm our predictions.
This work is joint with Manuchehr Aminian, Francesca Bernardi, Roberto Camassa, and Dan Harris.
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Gisele trains 7 days per week for a biathlon. She covers a total of 20 miles cycling and running each day. Gisele cycles a total of 105 miles each week, and runs a certain number of miles per week. If she cycles the same number of miles each day and runs the same number of miles each day, the equation 1/7(105+r)=20 represents the situation. Which describes the solution, r , to this equation?
1. Home
2. General
3. Gisele trains 7 days per week for a biathlon. She covers a total of 20 miles cycling and running eac... | {"url":"https://thibaultlanxade.com/general/gisele-trains-7-days-per-week-for-a-biathlon-she-covers-a-total-of-20-miles-cycling-and-running-each-day-gisele-cycles-a-total-of-105-miles-each-week-and-runs-a-certain-number-of-miles-per-week-if-she-cycles-the-same-number-of-miles-each-day-and","timestamp":"2024-11-09T00:59:38Z","content_type":"text/html","content_length":"30807","record_id":"<urn:uuid:4f838760-d412-4072-aab1-fe3593952b9f>","cc-path":"CC-MAIN-2024-46/segments/1730477028106.80/warc/CC-MAIN-20241108231327-20241109021327-00687.warc.gz"} |
FACT CHECK: The “two doctors” featured in popular video from California were actually spouting statistical nonsense… don’t be fooled
The “two doctors” video that has been widely cited across independent and conservative media turns out to be yet another example of mathematical illiteracy parading as authoritative medicine.
The two doctors featured in the video, Drs. Dan Erickson and Artin Massih, co-owners of Accelerated Urgent Care in Bakersfield, California, put on a clinic of mathematical and statistical illiteracy
in their now-banned video, essentially claiming no lockdowns were ever necessary and that the coronavirus isn’t really very dangerous at all.
To arrive at that absurd conclusion, they of course cite horribly bad math… the kind of math that would earn you an “F” in Statistics 101. From their video, here’s what they say:
So if you look at California—these numbers are from yesterday—we have 33,865 COVID cases, out of a total of 280,900 total tested. That’s 12% of Californians were positive for COVID. So we don’t, the
initial—as you guys know, the initial models were woefully inaccurate. They predicted millions of cases of death—not of prevalence or incidence—but death. That is not materializing. What is
materializing is, in the state of California is 12% positives. You have a 0.03 chance of dying from COVID in the state of California. Does that necessitate sheltering in place? Does that necessitate
shutting down medical systems? Does that necessitate people being out of work?
The problem with that explanation? The sample of people who are being tested are, of course, the people most likely to have symptoms or who believe they have been exposed. It’s not a random sample of
the population at large. (Again, this is Statistics 101.)
Accordingly, you can’t extrapolate the positive test ratio among symptomatic or high-risk people to the population at large. Yet this is how they arrive at the wrong conclusion that, “You have a 0.03
chance of dying from COVID in the state of California.”
That’s like saying if you test 1,000 drug addicts who share needles for HIV and find HIV at 10% in that high-risk population, then the entire state of California must have a 10% HIV infection rate.
That’s absurd.
Once again, Chris Martenson at PeakProsperity.com breaks it down and exposes the mathematical illiteracy of the “two doctors,” reminding us why it’s rather scary that these two guys are practicing
medicine at all.
“This is called Statistics 101,” explains Chris Martenson. “In fact, this is the first day of Statistics 101, where they talk to you about sampling and they put that out of the way so you can move
past the sampling and get to the math. You can’t use samples that come out of testing in hospitals as something that you extend across the entire population. It’s so inappropriate…”
Thus, the “two doctors” video is a fairy tale.
Why do so many people find bad math arguments so convincing when they’re actually total nonsense?
It’s not merely disturbing that these doctors don’t understand basic mathematics and statistical sampling fundamentals; it’s even more disturbing that so many people found their bad arguments
convincing. That’s because nothing has caused more people to lose their minds and abandon all rational thought than the coronavirus.
Because many people want the pandemic to be no big deal, they immediately latch onto arguments that hold no water, almost always in an effort to satisfy an emotional need of comfort. People don’t
want to live in a world where a deadly bioweapon is highly contagious and actually kills 10% of those who become symptomatic — with a death risk that’s 56 to 100 times higher than the regular flu —
so they desperately cling to bad reasoning and incorrect math as long as it makes them feel better.
This is how we get people making insanely flawed statements like, “Sweden has reached herd immunity.” No it hasn’t. Not even close. Or, “The Stanford study proved almost everybody is already
infected.” No it didn’t, not even close. The Stanford study allowed people to self-select for inclusion in the study, offering free testing to participants. Thus, it essentially recruited high-risk
people who thought they were already infected.
It’s astonishing how people who want the coronavirus to be no big deal are happy to cite bad science that’s worse than “climate change” science in order to push their own absurd beliefs. It’s also
bizarre how so few people in the independent media are able to see the problems with bad math and bad statistics.
The truth is that the coronavirus is 56 to 100 times more deadly than the flu, when comparing apples to apples:
Stay informed. Read Pandemic.news for accurate analysis of the coronavirus pandemic, or BadDoctors.news for more stories on bad doctors. | {"url":"https://propaganda.news/2020-04-28-two-doctors-california-statistical-nonsense-coronavirus.html","timestamp":"2024-11-10T06:34:50Z","content_type":"application/xhtml+xml","content_length":"39047","record_id":"<urn:uuid:b3bde9b7-f764-433c-bad4-eff084e7dd02>","cc-path":"CC-MAIN-2024-46/segments/1730477028166.65/warc/CC-MAIN-20241110040813-20241110070813-00333.warc.gz"} |
How many days in $w$ weeks and $w$ days?
Hint: Use the knowledge that one week of a month consists of seven days. First calculate the number of days that are equal to ‘w’ weeks by using the above relation and then add ‘w’ more days to that
Complete step by step solution:
The question is asked to find the number of days that are equal to w weeks and w days. This means that we have to find the number of days that are present in w weeks plus w days. To solve the given
question, we must have knowledge of the English calendar. We must know that a week consists of 7 days. Or we can also say that the time that is covered in an interval, which is equal to 7 days is
called a week. This means that one week is equal to seven days.
I.e. $1week=7days$.
Now, multiply w on both the sides of the above equation.
This gives us that $(w)week=(7w)days$.
This means that ‘w’ weeks are equal to a time of seven times w days. Now, if we add ‘w’ more days to ‘w’ weeks then the time covered in ‘w’ weeks and ‘w’ days is equal to $(w)weeks+(w)days$. But we
know that,
With this we get that,
\therefore (w)weeks+(w)days=(8w)days$.
Therefore, we found that there are ‘8w’ days in ‘w’ weeks and ‘8w’ days.
Note:Some students may misunderstand that the time of ‘8w’ days is equal to 8 weeks by thinking that w is representation for one week. If the question asked to find the number of hours in ‘w’ weeks
and ‘w’ days, then we have to multiply the answer by 24 as one day is equal to 24 hours. | {"url":"https://www.vedantu.com/question-answer/days-in-w-weeks-and-w-days-class-9-maths-cbse-600e9f04e6bcab51f22c73d1","timestamp":"2024-11-09T20:36:29Z","content_type":"text/html","content_length":"152413","record_id":"<urn:uuid:ca63ad66-62a1-4bef-940c-5bba057a03bb>","cc-path":"CC-MAIN-2024-46/segments/1730477028142.18/warc/CC-MAIN-20241109182954-20241109212954-00648.warc.gz"} |
Query regarding Eventual Operator
+ General Questions (11)
When we add Eventual operator to a path quantifier what does that generally mean? does that mean that on one path (E) or all the paths (A) phi can hold at any point of time? If yes then in the image
below why does A F (a & c) holds only in s8 ? From s0 there is a path s0->s2->s8 so why are we not considering s0?
A formula starting with “A” means: Look at all (infinite) paths from this state and check whether they satisfy the inner criterion.
Look at (s0,s2)^omega. No state on this path satisfies a&b. Thus, F(a&b) is not satisfied from its first step. Therefore, it is a counterexample to A F (a&b).
Furthermore, the path s0→s2→s8 cannot be continued to an infinite path. Thus, this wouldn't be a path for satisfying E F (a&b)
For the existential quantifier, yes. We could then form an infinite path going through s8. s8 satisfies a&c. Thus, the infinite path from s0 satisfies F (a&c). Hence, the state s0 satisfies E F (a&
c). But it does not satisfy A F (a&c) because we still have infinite paths (the one between s0 and s2 for example) that do not satisfy F (a&c). | {"url":"https://q2a.cs.uni-kl.de/2189/query-regarding-eventual-operator?show=2195","timestamp":"2024-11-11T02:11:14Z","content_type":"text/html","content_length":"59245","record_id":"<urn:uuid:89171d8c-327c-4105-af8c-9efa7f88ac54>","cc-path":"CC-MAIN-2024-46/segments/1730477028202.29/warc/CC-MAIN-20241110233206-20241111023206-00635.warc.gz"} |
A Modular Hierarchy of Logical Frameworks
Logical frameworks - formal systems for the specification and representation of other formal systems - are now a well-established field of research, and the number and variety of logical frameworks
is large and growing continuously. In this thesis, I tie several examples of logical frameworks into a single hierarchy.
I begin by introducing an infinite family of new, weak, lambda-free logical frameworks. These systems do not use lambda-abstraction, local definition, or any similar feature; parameterisation, and
the instantiation of parameterisation, is taken as basic. These frameworks form conservative extensions of one another; this structure of extension is what I call the modular hierarchy of logical
I show how several examples of existing logical frameworks - specifically, the systems PAL and AUT-68 from the AUTOMATH family, the Edinburgh Logical Framework, Martin-Lof's Logical Framework, and
Luo's system PAL+ - can be fitted into this hierarchy, in the sense that one of the weak frameworks can be embedded in each as a conservative subsystem. I give several examples of adequacy theorems
for object theories in the weak frameworks; these theorems are easier to prove than is usually the case for a logical framework. Adequacy theorems for the systems higher in the hierarchy follow as
immediate corollaries.
In the second part of this thesis, I investigate an approach to the design of logical frameworks suggested by the existence of such a hierarchy: that a framework could be built by specifying a set of
features, the result of adding any of which to a framework is a conservative extension of the same. I show how all of the weak frameworks from the first part, as well as two of the systems we gave
there as examples, can indeed be built in this manner.
Original language English
Qualification PhD
Awarding Institution • University of Manchester
Supervisors/Advisors • Aczel, Peter, Supervisor, External person
Thesis sponsors
Award date 10 Nov 2004
Publication status Unpublished - 2004 | {"url":"https://pure.royalholloway.ac.uk/en/publications/a-modular-hierarchy-of-logical-frameworks-2","timestamp":"2024-11-10T14:21:22Z","content_type":"text/html","content_length":"41909","record_id":"<urn:uuid:08574f68-ff74-47e5-a26c-e591d57afb29>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.60/warc/CC-MAIN-20241110134821-20241110164821-00512.warc.gz"} |
Solving Linear Equations Common Core Algebra 2 Homework Fluency | Common Core Worksheets
Solving Linear Equations Common Core Algebra 2 Homework Fluency
Solving Linear Equations Common Core Algebra 2 Homework Fluency
Solving Linear Equations Common Core Algebra 2 Homework Fluency – Are you looking for Common Core Math Worksheets Algebra 2? These worksheets are fantastic for aiding kids grasp the Common Core
standards, consisting of the Common Core Reading as well as Writing standards.
What are Common Core Worksheets?
Common Core Worksheets are academic sources for K-8 students. They are made to help trainees attain an usual set of goals. The very first regulation is that the worksheets may not be shared or
submitted to the internet in any kind of means.
Common Core worksheets cover K-8 trainees as well as are designed with the CCSS in mind. Using these resources will certainly assist students find out the abilities essential to be successful in
institution. They cover various ELA as well as mathematics topics as well as come with answer keys, making them a terrific resource for any classroom.
What is the Purpose of Common Core?
The Common Core is an initiative to bring uniformity to the way American youngsters discover. Designed by educators from across the nation, the requirements focus on developing a common base of
understanding and also abilities for pupils to be successful in college and also in life. Currently, 43 states have actually taken on the requirements as well as have started to implement them in
public colleges.
The Common Core criteria are not a government required; instead, they are an outcome of years of research study as well as evaluation by the Council of Chief State School Officers and also the
National Governors Association. While federal requireds are essential, states still have the last word in what their curriculum appears like.
Numerous moms and dads are irritated with Common Core criteria and are uploading screenshots of incomprehensible materials. The social networks website Twitchy is an excellent area to find instances
of incomprehensible materials. One such screenshot shows Rubinstein attempting to determine the objective of a Common Core page. The web page includes circles and also squares and also an array
model. The Common Core has labeled this a mathematics job, however Rubinstein could not make sense of it.
Common Core Math Worksheets Algebra 2
Solving Fractional Equations Common Core Algebra 2 Homework Answers
Common Core Algebra II Unit 5 Lesson 2 Arithmetic And Geometric
Common Core Algebra 2 Module 1 Lesson 18 Worksheets Samples
Common Core Math Worksheets Algebra 2
If you are seeking Common Core Math Worksheets Algebra 2, you’ve concerned the best location! These math worksheets are categorized by grade level and are based on the Common Core mathematics
criteria. Teachers can utilize these worksheets as a help in instructing the subject. You can also discover sources for a range of subjects, including globe history and US background.
The first set of worksheets is concentrated on single-digit enhancement and also will test a child’s skills in counting things. This worksheet will certainly call for students to count things within
a min, which is a great means to exercise counting. The adorable things that are included will certainly make the mathematics problems much more easy to understand for the kid as well as supply a
visual representation of the answer.
Math worksheets based on the common core math criteria are an excellent method for youngsters to discover standard arithmetic skills and also concepts. These worksheets consist of different issues
that vary in difficulty. They will certainly also motivate analytic, which helps children use their learning in real-life situations.
Fractions are another subject area that is tough, but possible for young learners to learn. Common Core Fractions Teaching Resources consist of arranging, buying, and also modeling portions. These
cost-free worksheets are developed to aid youngsters master this topic.
Related For Common Core Math Worksheets Algebra 2 | {"url":"https://commoncore-worksheets.com/common-core-math-worksheets-algebra-2/solving-linear-equations-common-core-algebra-2-homework-fluency/","timestamp":"2024-11-09T22:46:21Z","content_type":"text/html","content_length":"28517","record_id":"<urn:uuid:77316152-0e1f-43ce-82cc-dfb40d5e5438>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.10/warc/CC-MAIN-20241109214337-20241110004337-00774.warc.gz"} |
LTspice Worst-Case Circuit Analysis with Non-Ideal Resistors at Minimal Simulations Runs
When designing a circuit in LTspice, you may wish to assess the impact of component tolerances. For example, the gain error introduced by non-ideal resistors in an op amp circuit. This article
written by Gabino Alonso and Joseph Spencer from Analog Devices illustrates a method that reduces the number of simulations needed, and as a result speeds your time to results.
Varying a Parameter
LTSpice provides several ways to vary the value of a parameter. Some of these are:
• .step param; A parameter sweep of a user-defined variable
• gauss(x); A random number from Gaussian distribution with a sigma of x
• flat(x); A random number between -x and x with uniform distribution
• mc(x,y); A random number between x*(1+y) and x*(1-y) with uniform distribution.
These functions are very useful, especially if we want to look at results in terms of a distribution. But if we only want to look at worst case conditions, they may not be the quickest way to get a
result. Using gauss(x), flat(x) and mc(x,y) for example, will require a simulation to run for a statistically significant number of times. From there, a distribution can be looked at and worst case
values calculated in terms of standard deviations. However, for a worst case analysis we would prefer not to use a distribution approach, instead the maximum deviation from the nominal value of each
component are used in the calculations.
Running Minimal Simulations
Let’s say that we want to look at the worst-case impact of a R1 = 22.5kΩ resistor with a 1% tolerance. In this case, we really only want to run the simulation with R1 = 22.5kΩ * (1 – 0.01) and 22.5kΩ
* (1 + 0.01). A third run with an ideal 22.5kΩ resistor would also be handy to have.
.step param R1 list 22.5k*(1-.01) 22.5k*(1+.01) 22.5k
If we were just varying one resistor value, the ‘.step param’ method would work very well. But what if we have more? The classic difference amplifier has 4 resistors.
If a discrete difference amplifier were to be designed, each of these would have some tolerance (e.g. 1% or 5%).
For an example, let’s take the front page application shown in the LT1997-3 datasheet and implement it in LTspice with a discrete LT6015 op-amp and some non-ideal resistors.
Notice the values of resistors R1, R2, R3 and R4 are replaced by a function call wc(nominalvalue, tolerance, index) which is defined in the simulation by a .func statement:
.func wc(nom,tol,index) if(run==numruns,nom,if(binary(run,index),nom*(1+tol),nom*(1-tol)))
This function in conjunction with the binary(run, index) function below vary the parameter for each component between its maximum value and minimum values and for the last run the nominal value.
.func binary(run,index) floor(run/(2**index))-2*floor(run/(2**(index+1)))
The binary function toggles each index’ed componment in the simulation so that all possible combination of nom*(1+tol) and nom*(1-tol) are simulated. Note that the index of components should start
with 0. The following table highlights the operation of the binary() function with results to each index and run, where 1 represents nom*(1+tol) and 0 represents nom*(1-tol).
The number of runs is determined buy 2^N +1, where N equals the number of indexed components, to cover all the max and min combinations of the device plus the nominal. In our case we need 17
simulations run and we can define this using the .step command and the .param statments:
.step param run 0 16 1
.param numruns=16
Lastly, we need to define the tola and tolb for the simulation via .param statments:
.param tola=.01
.param tolb=.05
You can find more information about the .func, .step, and .param statments in the help (F1) and under the .param section details about the if(x,y,z) and floor(x) functions.
Plotting the .Step’ed Results
If the transient analysis simulation is run, see WorstCase_LT6015.asc file, we can observe our results. For a 250mA test current, we expect the Vout net to settle to 250mV. But now with our wc()
function, we get a spread from 235mV to 265mV.
Plotting the .Step’ed .Meas Statement
At this point we could zoom in and look at the peak to peak spread. But let’s take a lesson from another LTspice blog:
Plotting a Parameter Against Something Other Than Time (e.g. Resistance)
This blog covers how to run a simulation several times and plot a parameter against something other than time. In this case, we want to plot the V(out) vs. simulation run index. See
WorstCase_LT6015_meas.asc file.
In this simulation we have add a .meas statment to calculated the average voltage of the output.
.meas VoutAvg avg V(out)
To plot the V(out) vs. run parameter we can view the SPICE Error Log (Ctrl-L), right click and select Plot .step’ed .meas data.
The plot results of our .step‘ed .meas data.
The trace hows us that the results vary from a maximum worse case of 265mV (run 9) to a minimum worse case of 235mV (run 6) or roughly a ±6% error. This makes some intuitive sense since we had both
1% and 5% resistors in this example. The last run (16) shows the ideal result (250mV) which is ideal resistors. Recall LTspice graphs the results from the .meas statment as a piece wise linear graph.
Another faster approach to this particular circuit is to use the .op simulation (instead of the .trans) to perform a DC operating point solution which will plot the results of our stepped.meas data
The Value of Matched Resistors
When designing a difference amplifier not only is the appropriate op-amp needed, but equally as important are the matching of the resistors. The following references do an excellent job of explaining
this topic (and associated math) in detail:
However, you can neither achieve good Common Mode Rejection Ratio (CMRR) or Gain Error without appropriately matched resistors.
Linear Technology, now part of Analog Devices, has a number of precision amplifier products which also include matched resistors. A recently released example of this is the LT1997-3 – Precision, Wide
Voltage Range Gain Selectable Amplifier. Two key specifications are:
• 91dB Minimum DC CMRR (Gain = 1)
• 0.006% (60ppm) Maximum Gain Error (Gain = 1)
These specifications are really quite excellent. According to DN1023, CMRR due only to 1% resistors (with an ideal op-amp) will limit your CMRR to 34dB. And of course, the gain error is orders of
magnitude worse than what the LT1997-3 achieves.
Using the method outlined above, a simple worst-case analysis can be run at the min/max value of several parameters. In this example we looked at the impact of resistor tolerances in a classical
difference amplifier, and the value of the matched resistors in the LT1997-3 are illustrated.
Monte Carlo and Worst-Case Circuit Analysis using LTSpice | {"url":"https://passive-components.eu/ltspice-worst-case-circuit-analysis-with-non-ideal-resistors-at-minimal-simulations-runs/","timestamp":"2024-11-13T15:24:53Z","content_type":"text/html","content_length":"471003","record_id":"<urn:uuid:3acd353b-2d70-40cd-8ae2-7c2aaab5f77d>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00284.warc.gz"} |
Find the intersection points of a circle and a line. | AI Math Solver
Find the intersection points of a circle and a line.
Published on October 26, 2024
The system of equations is solved by substitution, expanding, simplifying, using the quadratic formula to find x values, and then substituting back into the equations to find the corresponding y
values. The solutions are (3, 4) and (-4, -3). | {"url":"https://www.aimathsolve.com/shares/find-the-intersection-points-of-a-circle-and-a-line","timestamp":"2024-11-12T23:26:45Z","content_type":"text/html","content_length":"39500","record_id":"<urn:uuid:1310b1c0-10b4-46ce-bd08-bcd974eec38a>","cc-path":"CC-MAIN-2024-46/segments/1730477028290.49/warc/CC-MAIN-20241112212600-20241113002600-00424.warc.gz"} |
Reviews of NUMERICAL MATHEMATICS
by A. Quarteroni, R. Sacco and F. Saleri
Springer, 2nd ed., 2007
This book provides the mathematical foundations of numerical methods and demonstrates their performance on examples, exercises and real-life applications. This is done using the MATLAB software
environment, which allows an easy implementation and testing of the algorithms for any specific class of problems. The book is addressed to students in Engineering, Mathematics, Physics and Computer
Sciences. In the second edition of this extremely popular textbook on numerical analysis, the readability of pictures, tables and program headings has been improved. Several changes in the chapters
on iterative methods and on polynomial approximation have also been made.
From the reviews of the first edition.
“I found many of the examples to be quite interesting. I find no fault with any of the theoretical portions of the text. The authors are quite thorough in their discussion of the theory underlying
each of the topics. This text uses MATLAB for programming the numerical codes. This is a very good choice. It contains a lot of interesting and useful information for experienced users of numerical
John Strikwerda, SIAM Review 2002, Vol. 44, Issue 1, p. 160-162
“This is an excellent and modern textbook in numerical mathematics! It is primarily addressed to undergraduate students in mathematics, physics, computer science and engineering. But you will need a
weekly 4 hour lecture for 3 terms lecture to teach all topics treated in this book! Well known methods as well as very new algorithms are given. The methods and their performances are demonstrated by
illustrative examples and computer examples. Exercises shall help the reader to understand the theory and to apply it. MATLAB-software satisfies the need of user-friendliness. “The spread of
numerical software presents an enrichment for the scientific community. However, the user has to make the correct choice of the method which best suits at hand. As a matter of fact, no black-box
methods or algorithms exist that can effectively and accurately solve all kinds of problems.” All MATLAB-programs are available by internet. … There are a lot of numerical examples and impressing
figures and very useful applications, as for instance: Regularization of a triangular grid, analysis of an electric network and of a nonlinear electrical circuit, finite difference analysis of beam
bending, analysis of the buckling of a beam, free dynamic vibration of a bridge, analysis of the state equation for a real gas, solution of a nonlinear system arising from semiconductor device
simulation, finite element analysis of a clamped beam, geometric reconstruction based on computer tomographies, computation of the wind action on a sailboat mast, numerical solution of blackbody
radiation, compliance of arterial walls, lubrication of a slider, heat conduction in a bar, a hyperbolic model for blood flow interaction with arterial walls. It is a joy to read the book, it is
carefully written and well printed. ….. In the reviewers opinion, the presented book is the best textbook in numerical mathematics edited in the last ten years.”
W.H.Schmidt, Zentralblatt für Mathematik 2001, 991.38387 | {"url":"https://mox.polimi.it/reviews-of-numerical-mathematics/","timestamp":"2024-11-03T09:28:43Z","content_type":"text/html","content_length":"63078","record_id":"<urn:uuid:9a7f976c-3566-4673-86ea-c7137ca38082>","cc-path":"CC-MAIN-2024-46/segments/1730477027774.6/warc/CC-MAIN-20241103083929-20241103113929-00136.warc.gz"} |
Constant Regret, Generalized Mixability, and Mirror Descent
Constant Regret, Generalized Mixability, and Mirror Descent
Part of Advances in Neural Information Processing Systems 31 (NeurIPS 2018)
Zakaria Mhammedi, Robert C. Williamson
We consider the setting of prediction with expert advice; a learner makes predictions by aggregating those of a group of experts. Under this setting, and for the right choice of loss function and
``mixing'' algorithm, it is possible for the learner to achieve a constant regret regardless of the number of prediction rounds. For example, a constant regret can be achieved for \emph{mixable}
losses using the \emph{aggregating algorithm}. The \emph{Generalized Aggregating Algorithm} (GAA) is a name for a family of algorithms parameterized by convex functions on simplices (entropies),
which reduce to the aggregating algorithm when using the \emph{Shannon entropy} $\operatorname{S}$. For a given entropy $\Phi$, losses for which a constant regret is possible using the \textsc{GAA}
are called $\Phi$-mixable. Which losses are $\Phi$-mixable was previously left as an open question. We fully characterize $\Phi$-mixability and answer other open questions posed by \cite{Reid2015}.
We show that the Shannon entropy $\operatorname{S}$ is fundamental in nature when it comes to mixability; any $\Phi$-mixable loss is necessarily $\operatorname{S}$-mixable, and the lowest worst-case
regret of the \textsc{GAA} is achieved using the Shannon entropy. Finally, by leveraging the connection between the \emph{mirror descent algorithm} and the update step of the GAA, we suggest a new \
emph{adaptive} generalized aggregating algorithm and analyze its performance in terms of the regret bound.
Do not remove: This comment is monitored to verify that the site is working properly | {"url":"https://proceedings.nips.cc/paper_files/paper/2018/hash/af1b5754061ebbd4412adfb34c8d3534-Abstract.html","timestamp":"2024-11-05T13:47:05Z","content_type":"text/html","content_length":"9452","record_id":"<urn:uuid:a55a7e9f-ea1d-4473-995b-02a539c9782a>","cc-path":"CC-MAIN-2024-46/segments/1730477027881.88/warc/CC-MAIN-20241105114407-20241105144407-00043.warc.gz"} |
Well it’s only been 2 months since I managed to get 2 sides complete on my Rubik’s Cube. I’ve been leaving it out around my house so my roomies can play with it so every time I come around it’s in a
different state than I left it.
I’ve managed to get 2 sides about 6-7 times altogether but I’m still not entirely certain of why. I keep having theories that don’t seem to work the next time! I have been finding some interesting
algorithms along the way.
And then wham! Today I just managed to get 3 sides complete all around one corner. I’m not sure how useful this is really as a progression towards a completed cube. Different suggestions I’ve heard
of how to go about finishing a cube don’t really mention this as a midway stage, but it’s still pretty exciting. It also feels like I’m slowly developing a deeper understanding of how the puzzle
works and I love watching my mind try to narrow down the logic.
Taking a picture of my achievement made me realise that it’s difficult without mirrors to prove from a picture that a cube is solved. Here I include a photo of the reverse side where you can see
that the remainder is still unsolved. | {"url":"http://www.seanhill.ca/tag/rubiks-cube","timestamp":"2024-11-04T11:29:25Z","content_type":"application/xhtml+xml","content_length":"22960","record_id":"<urn:uuid:2f405e3d-3240-4d22-a8bc-671e7bf6ed27>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00456.warc.gz"} |
Name Description
Computes a class-label decision for a given input.
(Inherited from IMulticlassClassifierTInput.)
Computes class-label decisions for each vector in the given input.
(Inherited from IMulticlassClassifierTInput.)
Computes class-label decisions for each vector in the given input.
Decide(TInput, TClasses)
(Inherited from IClassifierTInput, TClasses.)
Computes class-label decisions for the given input.
Decide(TInput, TClasses)
(Inherited from IMultilabelClassifierTInput, TClasses.)
Predicts a class label vector for the given input vector, returning the log-likelihood that the input vector belongs to its predicted class.
(Inherited from IMulticlassLikelihoodClassifierTInput.)
Predicts a class label vector for the given input vectors, returning the log-likelihood that the input vector belongs to its predicted class.
(Inherited from IMulticlassLikelihoodClassifierTInput.)
LogLikelihood(TInput, Predicts a class label vector for the given input vectors, returning the log-likelihood that the input vector belongs to its predicted class.
(Inherited from IMulticlassLikelihoodClassifierTInput.)
LogLikelihood(TInput, Predicts a class label for each input vector, returning the log-likelihood that each vector belongs to its predicted class.
(Inherited from IMulticlassLikelihoodClassifierBaseTInput, TClasses.)
LogLikelihood(TInput, Predicts a class label vector for the given input vector, returning the log-likelihood that the input vector belongs to its predicted class.
(Inherited from IMulticlassOutLikelihoodClassifierTInput, TClasses.)
Computes the log-likelihood that the given input vector belongs to the specified classIndex.
LogLikelihood(TInput, Int32)
(Inherited from IMultilabelLikelihoodClassifierTInput.)
Computes the log-likelihood that the given input vectors belongs to each class specified in classIndex.
LogLikelihood(TInput, Int32)
(Inherited from IMultilabelLikelihoodClassifierTInput.)
Computes the log-likelihood that the given input vectors belongs to each class specified in classIndex.
LogLikelihood(TInput, Int32)
(Inherited from IMultilabelLikelihoodClassifierTInput.)
LogLikelihood(TInput, Predicts a class label for each input vector, returning the log-likelihood that each vector belongs to its predicted class.
TClasses, Double)
(Inherited from IMulticlassLikelihoodClassifierBaseTInput, TClasses.)
LogLikelihood(TInput, Int32, Computes the log-likelihood that the given input vectors belongs to each class specified in classIndex.
(Inherited from IMultilabelLikelihoodClassifierTInput.)
LogLikelihood(TInput, Int32, Computes the log-likelihood that the given input vectors belongs to each class specified in classIndex.
(Inherited from IMultilabelLikelihoodClassifierTInput.)
Computes the log-likelihood that the given input vector belongs to each of the possible classes.
(Inherited from IMultilabelLikelihoodClassifierTInput.)
Computes the log-likelihoods that the given input vectors belongs to each of the possible classes.
(Inherited from IMultilabelLikelihoodClassifierTInput.)
LogLikelihoods(TInput, Computes the log-likelihood that the given input vector belongs to each of the possible classes.
(Inherited from IMultilabelLikelihoodClassifierTInput.)
LogLikelihoods(TInput, Computes the log-likelihoods that the given input vectors belongs to each of the possible classes.
(Inherited from IMultilabelLikelihoodClassifierTInput.)
LogLikelihoods(TInput, Predicts a class label vector for each input vector, returning the log-likelihoods of the input vector belonging to each possible class.
(Inherited from IMultilabelLikelihoodClassifierBaseTInput, TClasses.)
LogLikelihoods(TInput, Predicts a class label vector for the given input vector, returning the log-likelihoods of the input vector belonging to each possible class.
(Inherited from IMultilabelOutLikelihoodClassifierTInput, TClasses.)
LogLikelihoods(TInput, Predicts a class label vector for each input vector, returning the log-likelihoods of the input vector belonging to each possible class.
TClasses, Double)
(Inherited from IMultilabelLikelihoodClassifierBaseTInput, TClasses.)
LogLikelihoods(TInput, Predicts a class label vector for the given input vector, returning the log-likelihoods of the input vector belonging to each possible class.
TClasses, Double)
(Inherited from IMultilabelOutLikelihoodClassifierTInput, TClasses.)
Computes the probabilities that the given input vector belongs to each of the possible classes.
(Inherited from IMultilabelLikelihoodClassifierTInput.)
Computes the probabilities that the given input vectors belongs to each of the possible classes.
(Inherited from IMultilabelLikelihoodClassifierTInput.)
Probabilities(TInput, Computes the probabilities that the given input vector belongs to each of the possible classes.
(Inherited from IMultilabelLikelihoodClassifierTInput.)
Probabilities(TInput, Computes the probabilities that the given input vectors belongs to each of the possible classes.
(Inherited from IMultilabelLikelihoodClassifierTInput.)
Probabilities(TInput, Predicts a class label vector for each input vector, returning the probabilities of the input vector belonging to each possible class.
(Inherited from IMultilabelLikelihoodClassifierBaseTInput, TClasses.)
Probabilities(TInput, Predicts a class label vector for the given input vector, returning the probabilities of the input vector belonging to each possible class.
(Inherited from IMultilabelOutLikelihoodClassifierTInput, TClasses.)
Probabilities(TInput, Predicts a class label vector for each input vector, returning the probabilities of the input vector belonging to each possible class.
TClasses, Double)
(Inherited from IMultilabelLikelihoodClassifierBaseTInput, TClasses.)
Probabilities(TInput, Predicts a class label vector for the given input vector, returning the probabilities of the input vector belonging to each possible class.
TClasses, Double)
(Inherited from IMultilabelOutLikelihoodClassifierTInput, TClasses.)
Predicts a class label for the given input vector, returning the probability that the input vector belongs to its predicted class.
(Inherited from IMulticlassLikelihoodClassifierTInput.)
Predicts a class label for the given input vectors, returning the probability that the input vector belongs to its predicted class.
(Inherited from IMulticlassLikelihoodClassifierTInput.)
Predicts a class label for the given input vectors, returning the probability that the input vector belongs to its predicted class.
Probability(TInput, Double)
(Inherited from IMulticlassLikelihoodClassifierTInput.)
Probability(TInput, Predicts a class label for each input vector, returning the probability that each vector belongs to its predicted class.
(Inherited from IMulticlassLikelihoodClassifierBaseTInput, TClasses.)
Probability(TInput, Predicts a class label for the given input vector, returning the probability that the input vector belongs to its predicted class.
(Inherited from IMulticlassOutLikelihoodClassifierTInput, TClasses.)
Computes the probability that the given input vector belongs to the specified classIndex.
Probability(TInput, Int32)
(Inherited from IMultilabelLikelihoodClassifierTInput.)
Computes the probability that the given input vectors belongs to each class specified in classIndex.
Probability(TInput, Int32)
(Inherited from IMultilabelLikelihoodClassifierTInput.)
Computes the probability that the given input vectors belongs to each class specified in classIndex.
Probability(TInput, Int32)
(Inherited from IMultilabelLikelihoodClassifierTInput.)
Probability(TInput, Predicts a class label for each input vector, returning the probability that each vector belongs to its predicted class.
TClasses, Double)
(Inherited from IMulticlassLikelihoodClassifierBaseTInput, TClasses.)
Probability(TInput, Int32, Computes the probability that the given input vectors belongs to each class specified in classIndex.
(Inherited from IMultilabelLikelihoodClassifierTInput.)
Probability(TInput, Int32, Computes the probability that the given input vectors belongs to each class specified in classIndex.
(Inherited from IMultilabelLikelihoodClassifierTInput.)
Computes a numerical score measuring the association between the given input vector and its most strongly associated class (as predicted by the classifier).
(Inherited from IMulticlassScoreClassifierTInput.)
Computes a numerical score measuring the association between each of the given input vectors and their respective most strongly associated classes.
(Inherited from IMulticlassScoreClassifierTInput.)
Predicts a class label for the input vector, returning a numerical score measuring the strength of association of the input vector to its most strongly related class.
Score(TInput, TClasses)
(Inherited from IMulticlassOutScoreClassifierTInput, TClasses.)
Computes a numerical score measuring the association between each of the given input vectors and their respective most strongly associated classes.
Score(TInput, Double)
(Inherited from IMulticlassScoreClassifierTInput.)
Predicts a class label for each input vector, returning a numerical score measuring the strength of association of the input vector to the most strongly related class.
Score(TInput, TClasses)
(Inherited from IMulticlassScoreClassifierBaseTInput, TClasses.)
Computes a numerical score measuring the association between the given input vector and a given classIndex.
Score(TInput, Int32)
(Inherited from IMultilabelScoreClassifierTInput.)
Computes a numerical score measuring the association between each of the given input vectors and the given classIndex.
Score(TInput, Int32)
(Inherited from IMultilabelScoreClassifierTInput.)
Computes a numerical score measuring the association between each of the given input vectors and the given classIndex.
Score(TInput, Int32)
(Inherited from IMultilabelScoreClassifierTInput.)
Score(TInput, TClasses, Predicts a class label for each input vector, returning a numerical score measuring the strength of association of the input vector to the most strongly related class.
(Inherited from IMulticlassScoreClassifierBaseTInput, TClasses.)
Computes a numerical score measuring the association between each of the given input vectors and the given classIndex.
Score(TInput, Int32, Double)
(Inherited from IMultilabelScoreClassifierTInput.)
Computes a numerical score measuring the association between each of the given input vectors and the given classIndex.
Score(TInput, Int32, Double)
(Inherited from IMultilabelScoreClassifierTInput.)
Computes a numerical score measuring the association between the given input vector and each class.
(Inherited from IMultilabelScoreClassifierTInput.)
Computes a numerical score measuring the association between each of the given input vectors and each possible class.
(Inherited from IMultilabelScoreClassifierTInput.)
Predicts a class label vector for the given input vector, returning a numerical score measuring the strength of association of the input vector to each of the possible
Scores(TInput, TClasses) classes.
(Inherited from IMultilabelOutScoreClassifierTInput, TClasses.)
Computes a numerical score measuring the association between the given input vector and each class.
Scores(TInput, Double)
(Inherited from IMultilabelScoreClassifierTInput.)
Computes a numerical score measuring the association between each of the given input vectors and each possible class.
Scores(TInput, Double)
(Inherited from IMultilabelScoreClassifierTInput.)
Predicts a class label vector for each input vector, returning a numerical score measuring the strength of association of the input vector to each of the possible
Scores(TInput, TClasses) classes.
(Inherited from IMultilabelScoreClassifierBaseTInput, TClasses.)
Predicts a class label vector for the given input vector, returning a numerical score measuring the strength of association of the input vector to each of the possible
Scores(TInput, TClasses, classes.
(Inherited from IMultilabelOutScoreClassifierTInput, TClasses.)
Predicts a class label vector for each input vector, returning a numerical score measuring the strength of association of the input vector to each of the possible
Scores(TInput, TClasses, classes.
(Inherited from IMultilabelScoreClassifierBaseTInput, TClasses.)
ToMulticlass Views this instance as a multi-class generative classifier, giving access to more advanced methods, such as the prediction of integer labels.
ToMulticlassT Views this instance as a multi-class generative classifier, giving access to more advanced methods, such as the prediction of integer labels.
Views this instance as a multi-label classifier, giving access to more advanced methods, such as the prediction of one-hot vectors.
(Inherited from IMulticlassClassifierTInput.)
Applies the transformation to an input, producing an associated output.
(Inherited from ICovariantTransformTInput, TOutput.)
Applies the transformation to a set of input vectors, producing an associated set of output vectors.
(Inherited from ICovariantTransformTInput, TOutput.)
Applies the transformation to a set of input vectors, producing an associated set of output vectors.
Transform(TInput, TOutput)
(Inherited from ITransformTInput, TOutput.) | {"url":"http://accord-framework.net/docs/html/Methods_T_Accord_MachineLearning_IBinaryLikelihoodClassifier_1.htm","timestamp":"2024-11-02T12:38:17Z","content_type":"text/html","content_length":"98143","record_id":"<urn:uuid:37007cc0-3e51-4711-aa64-097557e94428>","cc-path":"CC-MAIN-2024-46/segments/1730477027710.33/warc/CC-MAIN-20241102102832-20241102132832-00273.warc.gz"} |
Math.NET Numerics
Math.NET Numerics is an open-source numerical library for .NET and Mono, written in C# and F#. It features functionality similar to BLAS and LAPACK.
Math.NET Numerics started 2009 by merging code and teams of dnAnalytics with Math.NET Iridium. It is influenced by ALGLIB, JAMA and Boost, among others, and has accepted numerous code contributions.^
[1]^[2] It is part of the Math.NET initiative to build and maintain open mathematical toolkits for the .NET platform since 2002.
Math.NET is used by several open source libraries and research projects, like MyMediaLite,^[3] FermiSim^[4] and LightField Retrieval,^[5] and various theses^[6]^[7]^[8]^[9] and papers.^[10]^[11]
The software library provides facilities for:
• Probability distributions: discrete, continuous and multivariate.
• Pseudo-random number generation, including Mersenne Twister MT19937.
• Real and complex linear algebra types and solvers with support for sparse matrices and vectors.
• LU, QR, SVD, EVD, and Cholesky decompositions.
• Matrix IO classes that read and write matrices from/to Matlab and delimited files.
• Complex number arithmetic and trigonometry.
• “Special” routines including the Gamma, Beta, Erf, modified Bessel and Struve functions.
• Interpolation routines, including Barycentric, Floater-Hormann.
• Linear Regression/Curve Fitting routines.
• Numerical Quadrature/Integration.
• Root finding methods, including Brent, Robust Newton-Raphson and Broyden.
• Descriptive Statistics, Order Statistics, Histogram, and Pearson Correlation Coefficient.
• Markov chain Monte Carlo sampling.
• Basic financial statistics.
• Fourier and Hartley transforms (FFT).
• Overloaded mathematical operators to simplify complex expressions.
• Runs under Microsoft Windows and platforms that support Mono.
• Optional support for Intel Math Kernel Library (Microsoft Windows and Linux)
• Optional F# extensions for more idiomatic usage.
See also
• List of numerical analysis software
• List of numerical libraries
External links
By: Wikipedia.org
Edited: 2021-06-18 19:49:08
Source: Wikipedia.org | {"url":"https://codedocs.org/what-is/math-net-numerics","timestamp":"2024-11-06T20:34:55Z","content_type":"text/html","content_length":"36469","record_id":"<urn:uuid:a4e40d78-1435-47ec-a814-e3154f65c210>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.47/warc/CC-MAIN-20241106194801-20241106224801-00742.warc.gz"} |
Surds and Applications MCQ Quiz [PDF] Questions Answers | Surds and Applications MCQs App Download & e-Book
Class 9 math Practice Tests
Class 9 Math Online Tests
Surds and Applications MCQ (Multiple Choice Questions) PDF Download
The Surds and applications Multiple Choice Questions (MCQ Quiz) with Answers PDF (Surds and Applications MCQ PDF e-Book) download to practice Grade 9 Math Tests. Learn Algebraic Expressions and
Algebraic Formulas Multiple Choice Questions and Answers (MCQs), Surds and Applications quiz answers PDF for high school certification courses. The Surds and Applications MCQ App Download: Free
learning app for algebraic expressions, rationalization of surds test prep for online schools.
The MCQ: An irrational radical with rational radicand is called; "Surds and Applications" App Download (Free) with answers: Equation; Sentence; Formula; Surd; for high school certification courses.
Solve Algebraic Expressions and Algebraic Formulas Quiz Questions, download Google eBook (Free Sample) for secondary school graduation certificate.
Surds and Applications MCQs PDF: Questions Answers Download
MCQ 1:
Every surd is a/an
1. irrational number
2. rational number
3. equation
4. coefficient
MCQ 2:
An irrational radical with rational radicand is called
1. equation
2. sentence
3. formula
4. surd
MCQ 3:
The simplest form of the surd √18 ⁄ √3 √2 is
1. 3
2. √&minus3
3. √5
4. √3
MCQ 4:
The simplest form of the surd √180 is
1. 6√3
2. 6√5
3. 6
4. 5
MCQ 5:
The surds to be added or subtracted should be
1. a fraction
2. non-similar
3. similar
4. none of above
Class 9 Math Practice Tests
Surds and Applications Textbook App: Free Download iOS & Android
The App: Surds and Applications MCQs App to study Surds and Applications Textbook, 9th Grade Math MCQ App, and 10th Grade Math MCQ App. The "Surds and Applications MCQs" App to free download Android
& iOS Apps includes complete analytics with interactive assessments. Download App Store & Play Store learning Apps & enjoy 100% functionality with subscriptions! | {"url":"https://mcqlearn.com/math/g9/surds-and-applications-mcqs.php","timestamp":"2024-11-04T01:46:26Z","content_type":"text/html","content_length":"71802","record_id":"<urn:uuid:49206710-6f11-488f-aa98-cc12b911fc85>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00615.warc.gz"} |
ABC Company currently has 310,000 shares of stock outstanding
that sell for $94 per share. Assume...
ABC Company currently has 310,000 shares of stock outstanding that sell for $94 per share. Assume...
ABC Company currently has 310,000 shares of stock outstanding that sell for $94 per share. Assume no market imperfections or tax effects exist.
Determine the share price AND new number of shares outstanding in each of the following independent situations: (Do not round intermediate calculations. Round your price per share answers to 2
decimal places, e.g., 32.16, and shares outstanding answers to the nearest whole number, e.g., 32.)
a. ABC has a five-for-three stock split.
b. ABC has a 11 percent stock dividend.
c. ABC has a 39.0 percent stock dividend.
d. ABC has a four-for-seven reverse stock split.
a). New Share Price = Current Stock Price x Ratio of old shares to new shares
= $94 x (3/5) = $56.40
New Shares Outstanding = Current Shares Outstanding x ratio of new shares to old shares
= 310,000 x (5/3) = 516,667
b). New Share Price = Current Stock Price x Ratio of old shares to new shares
= $94 x (1/1.11) = $84.68
New Shares Outstanding = Current Shares Outstanding x ratio of new shares to old shares
= 310,000 x (1.11) = 344,100
c). New Share Price = Current Stock Price x Ratio of old shares to new shares
= $94 x (1/1.39) = $67.63
New Shares Outstanding = Current Shares Outstanding x ratio of new shares to old shares
= 310,000 x (1.39) = 430,900
d). New Share Price = Current Stock Price x Ratio of old shares to new shares
= $94 x (7/4) = $164.50
New Shares Outstanding = Current Shares Outstanding x ratio of new shares to old shares
= 310,000 x (4/7) = 177,143 | {"url":"https://justaaa.com/finance/399928-abc-company-currently-has-310000-shares-of-stock","timestamp":"2024-11-04T02:29:50Z","content_type":"text/html","content_length":"42851","record_id":"<urn:uuid:f205178f-fa54-4e09-ae34-353546ca37b5>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00304.warc.gz"} |
Expanding logarithms - Properties, Examples, and Explanation
Expanding Logarithms – Properties, Examples, and Explanation
Expand logarithms will test our knowledge of the different logarithmic expressions. This is the process of rewriting a condensed logarithmic expression to a more expanded form using the different
logarithmic properties.
As we have mentioned, we will be utilizing the different lessons we’ve learned about logarithms in the past, so make sure to review them as we’ll be using most (if not all) of them.
After reviewing these important concepts, let’s go ahead and apply these to start expanding different logarithmic expressions.
How to expand logarithms?
Whenever you’re asked to expand a logarithmic expression, your end goal is to rewrite this expression to reflect the different components present.
\begin{aligned} \color{blue}\log_2 \left(\dfrac{3x^2}{y} \right ) &\Rightarrow \color{red}\log_2 3 + 2\log_2 x – \log_2 y\\ \color{blue}\text{Simplifed}&\Rightarrow \color{red}\boldsymbol{\text
The expressions above are good examples showing the difference between a simplified (or condensed) expression to an expanded logarithmic expression. The example that is shown above- made use of three
logarithmic properties: product, quotient, and power rules of logarithms.
Why don’t we list down all the helpful properties and rules we might need when expanding logarithms and logarithmic expressions?
Rule Name Algebraic Expression
Product Rule $ \log_b({\color{blue} A}{\color{green} B}) = \log_b {\color{blue} A} + \log_b {\color{green} B}$
Quotient Rule $\log_b\left(\dfrac{\color{blue} A}{\color{green} B}\right) = \log_b {\color{blue} A} – \log_b {\color{green} B}$
Product Rule $\log_b {\color{blue}A}^n = n\log_b {\color{blue}A}$
Identity Rule $\log_b b = 1$
Zero Rule $\log_b 1 = 0$
Since logarithmic expressions’ complexities vary throughout, what we can give you are pointers and guides when expanding logarithmic expressions:
• Apply the difference or sum rule right away whenever possible.
• If you see exponents inside the logarithmic expression, then you might need to use the power rule.
• Double-check the new expression and see if each of the components are in their most simplified form.
• If not, expand the component as well.
These are helpful pointers to remember when expanding logarithmic expressions. But of course, the best way to master this skill is by practicing and trying out different examples.
We’ll slowly work through different expressions and see if we can expand each of them using these properties.
Example 1
Expand the logarithmic expression, $\log_3 \dfrac{4x}{y}$.
Checking the expression inside $\log_3$, we can see that we can use the quotient and product rules to expand the logarithmic expression.
• Apply the quotient rule to break down the condensed expression.
• Since $4x = 4 \cdot x$, we can apply the product rule to expand the expression further.
$\begin{aligned} \log_3 \dfrac{4x}{y} &= \log_3 4x – \log_3 y, \color{green}\text{ Quotient Rule}\\&= \log_3 4 + \log_3 x – \log_3 y, \color{green}\text{ Product Rule}\end{aligned}$
Hence, we have $\log_3 \dfrac{4x}{y} = \log_3 4 + \log_3 x – \log_3 y$.
Example 2
Expand the logarithmic expression, $\log_4 \dfrac{5m^3}{2n^6p^4}$.
The second expression is a bit more complex than the first one, so let’s begin by expanding the expression starting with the quotient rule then use the product rule for its denominator.
$\begin{aligned} \log_4 \dfrac{5m^3}{2n^6p^4} &= \log_4 5m^3 – \log_4 2n^6p^4, \color{green}\text{ Quotient Rule}\\&= \log_4 5m^3 – (\log_4 2n^6 + \log_4 p^4), \color{green}\text{ Product Rule}\\&=
\log_4 5m^3 – \log_4 2n^6 – \log_4 p^4\end{aligned}$
Be careful when distributing the negative sign into the terms to eliminate the parenthesis. Notice that each component still has exponents? This means that we’ll have to apply the power rule to each
of the components to expand the expression further.
$ \begin{aligned} \log_4 5m^3 – \log_4 2n^6 – \log_4 p^4 &= 3\log_4 5m – 6\log_4 2n – 4\log_4 p\end{aligned}$
Hence, we have $\log_4 \dfrac{5m^3}{2n^6p^4} = \log_4 5m – 6\log_4 2n – 4\log_4 p $.
Example 3
Expand the logarithmic expression, $\log_2 \left(\dfrac{2x\sqrt{y}}{3z}\right)^6$.
This expression may appear complicated, but we’ll slowly break down this expression using the different logarithmic properties we’ve learned in the past. Let’s start with the following steps:
• Since we have an exponent outside the rational expression, we should apply the power rule first.
• We can then apply the quotient rule to break down the logarithm of the rational expression.
• Expand $\log_2 2x \sqrt{y}$ and $\log_2 3z$ using the product rule.
\begin{aligned} \log_2 \left(\dfrac{2x\sqrt{y}}{3z}\right)^6 &= 6\left( \log_2 \dfrac{2x\sqrt{y}}{3z} \right ),\color{green}\text{ Power Rule}\\&= 6(\log_2 2x\sqrt{y} – \log_2 3z),\color{green}\text
{ Quotient Rule}\\&= 6[(\log_2 2 + \log_2 x + \log_2 \sqrt{y}) – (\log_2 3 + \log_2 z)], \color{green}\text{ Product Rule}\end{aligned}
Inspect each element and see if we can still apply any rules to the current expression. We can rewrite $\sqrt{y}$ as $y^{\frac{1}{2}}$ then apply the power rule.
Expand the negative signs and distribute $6$ to expand the logarithmic expression further.
\begin{aligned} 6[(\log_2 2 + \log_2 x + \log \sqrt{y}) – (\log_2 3 + \log_2 z)]&=6[(\log_2 2 + \log_2 x + \log_2 y^{\frac{1}{2}}) – (\log_2 3 + \log_2 z)],\\ &=6\left[\left(\log_2 2 + \log_2 x + \
dfrac{1}{2}\log_2 y\right) – (\log_2 3 + \log_2 z)\right]\\&=6\log_2 2 + 6\log_2 x + 3 \log_2y – 6\log_2 3 – 6\log_2 z\end{aligned}
Hence, the expanded form of $\log_2 \left(\dfrac{2x\sqrt{y}}{3z}\right)^6$ is equal to $6\log_2 2 + 6\log_2 x + 3 \log_2y – 6\log_2 3 – 6\log_2 z$.
Example 4
Expand the logarithmic expression, $\ln \left[\dfrac{4x^2(-2x + 5)}{3x – 4}\right]$.
Since the expression contains a rational expression, we can consider using the quotient rule right away. Expand the first group using the product rule.
$\begin{aligned} \ln \left[\dfrac{4x^2(-2x + 5)}{3x – 4}\right]&=\ln [4x^2(-2x + 5)] – \ln(3x – 4)\color{green}\text{ Quotient Rule}\\&=\ln 4x^2 + \ln(-2x + 5) – \ln(3x – 4)\color{green}\text{
Product Rule}\end{aligned}$
This is a good time to remind ourselves that we can no longer expand the logarithm of two sums, such as $ln (-2x + 5)$ and $\ln (3x – 4)$.
The first term, $\ln 4x^2$, can still be broken down using the product rule to separate $\ln 4$ and $\ln x^2$ then apply the power rule on $\ln x^2$.
$\begin{aligned} \ln 4x^2 + \ln(-2x + 5) – \ln(3x – 4) &= \ln 4 + \ln x^2 + \ln(-2x + 5) – \ln(3x – 4)\color{green}\text{ Product Rule}\\&= \ln 4 + 2\ln x + \ln(-2x + 5) – \ln(3x – 4)\color{green}\
text{ Power Rule}\end{aligned}$
This means that $\ln \left[\dfrac{4x^2(-2x + 5)}{3x – 4}\right] = \ln 4 + 2\ln x + \ln(-2x + 5) – \ln(3x – 4)$ when expanded.
Example 5
Expand the logarithmic expression, $\ln \left[\dfrac{\sqrt{(x + 4)(3x – 2)^2}}{(x^2 – 4)}\right]$.
This expression may appear complicated to work on, but we’ll be okay as long as we apply the right properties. We can start by applying the quotient rule then rewriting the numerator using the fact
that $\sqrt{m} = m^{\frac{1}{2}}$.
\begin{aligned} \ln \left[\dfrac{\sqrt{(x + 4)(3x – 2)^2}}{(x^2 – 4)}\right] &= [\ln \sqrt{(x + 4)(3x – 2)^2}] – \ln (x^2 – 4) \color{green} \text{ Quotient Rule}\\&= \ln[ (x+ 4)(3x – 2)^2 ]^{\frac
{1}{2}}- \ln (x^2 – 4) \end{aligned}
We can then apply the power and product rules to expand the first group further. Don’t forget to distribute $\dfrac{1}{2}$ as well.
\begin{aligned} \ln[ (x+ 4)(3x – 2)^2 ]^{\frac{1}{2}}- \ln (x^2 – 4) &= \dfrac{1}{2}\ln \left[ (x+ 4)(3x – 2)^2\right ]- \ln (x^2 – 4) \color{green} \text{ Power Rule} \\&=\dfrac{1}{2} \left[ \ln(x+
4) + \ln(3x – 2)^2\right ]- \ln (x^2 – 4) \color{green} \text{ Product Rule} \\&= \dfrac{1}{2} \ln(x+ 4) + \dfrac{1}{2}\ln(3x – 2)^2- \ln (x^2 – 4) \end{aligned}
We can still further expand this expression by applying the power rule on $\dfrac{1}{2}\ln(3x – 2)^2$.
$\begin{aligned} \dfrac{1}{2} \ln(x+ 4) + \dfrac{1}{2}\ln(3x – 2)^2- \ln (x^2 – 4) &= \dfrac{1}{2} \ln(x+ 4) + \dfrac{1}{2}(2)\ln(3x – 2)- \ln (x^2 – 4) \color{green} \text{ Power Rule} \\&= \
dfrac{1}{2} \ln(x + 4) + \ln(3x – 2) – \ln(x^2 -4)\end{aligned}$
This means that $\ln \left[\dfrac{\sqrt{(x + 4)(3x – 2)^2}}{(x^2 – 4)}\right]$ is also equal to $\dfrac{1}{2} \ln(x + 4) + \ln(3x – 2) – \ln(x^2 -4)$.
Example 6
Expand the logarithmic expression, $\log_3 \left[\dfrac{\sqrt[4]{x^3}}{y^2(x + 3)^5}\right]$.
Let’s begin by rewriting $\sqrt[4]{x^3}$ as $x^{\frac{3}{4}$ on the numerator then applying the quotient rule to expand the logarithmic expression.
$\begin{aligned} \log_3 \left[\dfrac{\sqrt[4]{x^3}}{y^2(x + 3)^5}\right] &= \log_3 \left[\dfrac{x^{\frac{3}{4}}}{y^2(x + 3)^5}\right]\\&= \log_3 x^{\frac{3}{4}} – \log_3 [y^2(x +3)^5] \color{green}
\text{ Quotient Rule}\end{aligned}$
Expand the second term by applying the product rule to distribute the negative sign inside the parenthesis.
$\begin{aligned} \log_3 x^{\frac{3}{4}} – \log_3 [y^2(x +3)^5]&= \log_3 x^{\frac{3}{4}} – [\log_3 y^2 + \log_3 (x +3)^5] \color{green} \text{ Power Rule}\\ &= \log_3 x^{\frac{3}{4}} – \log_3 y^2 –
\log_3 (x +3)^5\end{aligned}$
Each term shows expression with exponents, so we can apply the power rule to each of the terms to further expand the logarithmic expression.
$\begin{aligned} \log_3 x^{\frac{3}{4}} – \log_3 y^2 – \log_3 (x +3)^5 &= \dfrac{3}{4}\log_3 x – 2\log_3 y – 5\log_3 (x +3) \color{green} \text{ Power Rule}\end{aligned}$
This means that $\log_3 \left[\dfrac{\sqrt[4]{x^3}}{y^2(x + 3)^5}\right]$, when expanded, is equal to $\dfrac{3}{4}\log_3 x – 2\log_3 y – 5\log_3 (x +3)$.
Practice Questions
1. Which of the following shows the expanded form of the logarithmic expression, $-\log_5 \dfrac{2x}{y}$?
2. Which of the following shows the expanded form of the logarithmic expression, $\log_4 \left(\dfrac{4x\sqrt{3y}}{2z}\right)^8$?
3. Which of the following shows the expanded form of the logarithmic expression, $\ln \left[\dfrac{-12x^4(-5x + 6)}{2x + 6}\right]$?
4. Which of the following shows the expanded form of the logarithmic expression, $\ln \left[\dfrac{-12x^4(-5x + 6)}{2x + 6}\right]$?
5. Which of the following shows the expanded form of the logarithmic expression, $\log_6 \left[\dfrac{\sqrt[5]{x^4}}{y^4(x – 1)^6}\right]$? | {"url":"https://www.storyofmathematics.com/expanding-logarithms/","timestamp":"2024-11-03T01:16:22Z","content_type":"text/html","content_length":"166470","record_id":"<urn:uuid:54189aae-d0aa-4b5c-bfb0-69778e90db90>","cc-path":"CC-MAIN-2024-46/segments/1730477027768.43/warc/CC-MAIN-20241102231001-20241103021001-00637.warc.gz"} |
SOWISO - UvA
The lower-left spiral consists of all the points \(P(x,y,z)\) specified by the coordinate functions \[\left\{\;\begin{aligned} x &= \cos(8\pi t)\\ y &= \sin(8\pi t)\\ z &= t\end{aligned}\right.\] We
speak of coordinate functions because the coordinates \(x\), \(y\) and \(z\) are functions of the variable \(t\). As \(t\) increases from \(-1\) to \(1\), the point \(P\) spirals counter clockwise in
the upward direction, starting in \(P_{-1}=(1,0,-1)\) and terminating in \(P_1=(1,0,1)\).
The lower-right diagram is interactive; drag the right-mouse button to look at it from a different perspective or give it with the right-mouse button a push to animate the objects.
In general, you can describe a space curve mathematically by three functions \[\left\{\;\begin{aligned} x &= f(t)\\ y &= g(t)\\ z&= h(t)\end{aligned}\right.\] We speak of a parameterisation or
parameter equations of the curve with parameter \(t\). Almost always one uses neat functions \(f, g\) and \(h\) as coordinate functions. The parameter \(t\) represents time in many applications.
The parameterisation of a curve in the space is not unique: thus the spiral in the above example, may also be parametrized by \[\left\{\;\begin{aligned}x &= \cos(2t^3)\\ y &= \sin(2t^3)\\ z &= t^3\
end{aligned}\right.\] with the \(t\)-domain again equal to the interval \(-1\le t\le 1\). The only thing that has changed is the way the point \(P\) moves along the curve as \(t\) increases from \(-1
\) to \(1\). | {"url":"https://uva.sowiso.nl/courses/theory/128/141/2404/en","timestamp":"2024-11-12T10:10:30Z","content_type":"text/html","content_length":"62571","record_id":"<urn:uuid:a9bd8cea-496d-4ed5-99c7-a5425c521d63>","cc-path":"CC-MAIN-2024-46/segments/1730477028249.89/warc/CC-MAIN-20241112081532-20241112111532-00417.warc.gz"} |
Acoular 24.10 documentation
class acoular.tbeamform.BeamformerTimeTraj
Bases: BeamformerTime
Provides a basic time domain beamformer with time signal output for a grid moving along a trajectory.
trajectory = Trait(Trajectory, desc='trajectory of the grid center')
Trajectory or derived object. Start time is assumed to be the same as for the samples.
rvec = CArray(dtype=float, shape=(3,), value=array((0, 0, 0)), desc='reference vector')
Reference vector, perpendicular to the y-axis of moving grid.
conv_amp = Bool(False, desc='determines if convective amplification of source is considered')
Considering of convective amplification in beamforming formula.
precision = Trait(64, [32, 64], desc='numeric precision')
Floating point and integer precision
Python generator that yields the time-domain beamformer output.
The output time signal starts for source signals that were emitted from the Grid at t=0.
This parameter defines the size of the blocks to be yielded (i.e. the number of samples per block). Defaults to 2048.
Samples in blocks of shape (num, numchannels).
numchannels is usually very large (number of grid points). The last block returned by the generator may be shorter than num. | {"url":"https://www.acoular.org/api_ref/generated/generated/acoular.tbeamform.BeamformerTimeTraj.html","timestamp":"2024-11-12T07:01:06Z","content_type":"text/html","content_length":"10099","record_id":"<urn:uuid:a5b33485-cd22-4709-83a0-c7207e680155>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.58/warc/CC-MAIN-20241112045844-20241112075844-00097.warc.gz"} |
Lionel Levine's hat challenge has t players, each with a (very large or infinite) stack of hats on their head, each hat independently colored at random black or white. The players are allowed to
coordinate before the random colors are chosen, but not after. Each player sees all hats except for those on her own head. They then proceed to simultaneously try and each pick a black hat from their
respective stacks. They are proclaimed successful only if they are all correct. Levine's conjecture is that the success probability tends to zero when the number of players grows. We prove that this
success probability is strictly decreasing in the number of players, and present some connections to problems in graph theory: relating the size of the largest independent set in a graph and in a
random induced subgraph of it, and bounding the size of a set of vertices intersecting every maximum-size independent set in a graph.
• combinatorics
• hats
• independent sets
• random graphs
All Science Journal Classification (ASJC) codes
Dive into the research topics of 'THE SUCCESS PROBABILITY IN LEVINE'S HAT PROBLEM, AND INDEPENDENT SETS IN GRAPHS'. Together they form a unique fingerprint. | {"url":"https://cris.iucc.ac.il/en/publications/the-success-probability-in-levines-hat-problem-and-independent-se-2","timestamp":"2024-11-13T06:15:14Z","content_type":"text/html","content_length":"50566","record_id":"<urn:uuid:0cef71a5-c16e-41a9-bc0b-3c13d2653203>","cc-path":"CC-MAIN-2024-46/segments/1730477028326.66/warc/CC-MAIN-20241113040054-20241113070054-00172.warc.gz"} |
Successive Addition, Supertasks, Grim Reapers, and the Kalam Cosmological Argument
In the previous post, I considered William Lane Craig's first philosophical argument for the second premise of the Kalam Cosmological Argument, viz. The universe began to exist. In this post, I want
to consider Craig's second philosophical argument for this same premise. Additionally, I will consider the Grim Reaper Paradox and its application to proving this premise.
An actual infinite cannot be formed by successive addition
Craig’s second philosophical argument grants for the sake of argument that there can exist an actual infinite in reality. Nevertheless, Craig argues that the past, due to the nature of how it has
been formed, cannot be an actual infinite. The basic argument can be formulated as follows:
1. A collection formed by successive addition cannot be an actual infinite.
2. The temporal series of events is a collection formed by successive addition.
3. Therefore, the temporal series of events cannot be an actual infinite.
To begin, we need to understand what is meant by “successive addition.” Craig explains,
By “successive addition,” one means the accrual of one new element at a (later) time. The temporality of the process of accrual is critical here…[W]e are concerned here with a temporal process of
successive addition of one element after another (Blackwell Companion, pp. 117).
The basic idea is that we add an element to a collection at one point in time. We then add a second element to the collection at a later point in time. We then add a third at a yet later point in
time. And so forth. We are not adding multiple elements simultaneously; rather, we are adding each of the elements sequentially.
With this understanding, we proceed to an evaluation of the argument.
Defense of first premise
That an actually infinite collection cannot be formed by successive addition seems to follow from the nature of successive addition and the actual infinite. It seems clear that the formation of an
actual infinite by successive addition by beginning at some point is impossible. For given any number N, N + 1 is a finite number. So, no matter how many successive additions are made, one cannot get
from a beginning point to another point infinitely far away. As Craig writes,
One sometimes, therefore, speaks of the impossibility of counting to infinity, for no matter how many numbers one counts, one can always count one more number before arriving at infinity. One
sometimes speaks instead of the impossibility of traversing the infinite. The difficulty is the same: no matter how many steps one takes, the addition of one more step will not bring one to a
point infinitely distant from one’s starting point (Blackwell Companion, pp. 118).
But what about forming an actually infinite collection by successive addition by never beginning but ending at a point? In this case, the series of elements is, at every point, already actually
infinite. Thus, the difficulty of moving from a starting point to another point infinitely far away is avoided. Craig responds to this possibility as follows:
Although the problems will be different, the formation of an actually infinite collection by never beginning and ending at some point seems scarcely less difficult than the formation of such a
collection by beginning at some point and never ending. If one cannot count to infinity, how can one count down from infinity? If one cannot traverse the infinite by moving in one direction, how
can one traverse it by moving in the opposite direction? In order for us to have “arrived” at today, temporal existence has, so to speak, traversed an infinite number of prior events. But before
the present event could occur, the event immediately prior to it would have to occur; and before that event could occur, the event immediately prior to it would have to occur; and so on ad
infinitum. One gets driven back and back into the infinite past, making it impossible for any event to occur. Thus, if the series of past events were beginningless, the present event could not
have occurred, which is absurd (ibid., pp. 118).
So, it seems impossible to form an actually infinite collection by successive addition. The first premise of the argument, therefore, is confirmed.
Another defense of the first premise that Craig gives appeals to the Tristram Shandy Paradox (henceforth, TSP). The paradox was originally discussed by the twentieth-century philosopher Bertrand
Russell. Tristram Shandy is the title character from the book The Life and Opinions of Tristram Shandy, Gentleman by Laurence Stern. In the novel, Tristram Shandy writes his autobiography so slowly
that it takes him an entire year to record the events of a single day. The paradox arises when we ask the question, Can Tristram Shandy ever finish writing his autobiography? For the sake of the
paradox, we are assuming that Tristram Shandy is immortal and thus has infinite time to write his autobiography.
Russell answered the question in the affirmative. His reasoning was that, given infinite time, we can put the days and years that go by in infinite time into a correspondence so that to each day
there would correspond one year, and both are infinite. Thus, there is no day that Tristram Shandy would fail to write in his autobiography.
Such an assertion is misleading, however. The fact that every part of the autobiography will be eventually written does not imply that the whole autobiography will be eventually written, which
was, after all, Tristram Shandy’s concern. For every part of the autobiography there is some time at which it will be completed, but there is not some time at which every part of the
autobiography will be completed. Given an A-Theory of time, though he write forever, Tristram Shandy would only get farther and farther behind, so that instead of finishing his autobiography, he
would progressively approach a state in which he would be infinitely far behind (Blackwell Companion, pp. 120-121).
Craig then modifies the TSP so that instead of Tristram Shandy writing endlessly into the future, we suppose that he has been writing from eternity past at the rate of one day per year. Craig
Should not Tristram Shandy now be infinitely far behind? For if he has lived for an infinite number of years, Tristram Shandy has recorded an equally infinite number of past days. Given the
thoroughness of his autobiography, these days are all consecutive days. At any point in the past or present, therefore, Tristram Shandy has recorded a beginningless, infinite series of
consecutive days. But now the question arises: Which days are these? Where in the temporal series of events are the days recorded by Tristram Shandy at any given point? The answer can only be
that they are days infinitely distant from the present. For there is no day on which Tristram Shandy is writing which is finitely distant from the last recorded day…
But there is no way to traverse the temporal interval from an infinitely distant event to the present, or, more technically, for an event which was once present to recede to an infinite temporal
distance. Since the task of writing one’s autobiography at the rate of 1 year per day seems obviously coherent, what follows from the Tristram Shandy story is that an infinite series of past
events is absurd (ibid., pp. 121).
Thus, the TSP provides further support that the traversal of an infinite past (via successive addition) is impossible.
Critique of first premise
Graham Oppy, drawing from the philosopher Fred Dretske, considers the possibility of traversing the infinite (or counting to infinity). The idea is that one can indeed traverse the infinite (or count
to infinity) if one counts and never stops counting. Oppy writes,
One counts to infinity just in case, for each finite number N, one counts past N. But unless one stops counting, one will eventually reach any given finite N (Philosophical Perspectives on
Infinity, pg. 61).
But as Craig points out, Oppy does not take into account here the important distinction between an actual infinite and a potential infinite. Craig writes, “One who, having begun, never stops counting
counts ‘to infinity’ only in the sense that one counts potentially infinitely” (Blackwell Companion, pp. 118, footnote). Thus, Oppy’s contention provides no support to showing the possibility of
forming an actually infinite collection by successive addition.
Elsewhere, Oppy engages with this reply and offers additional commentary on, and counterarguments against, the first premise of Craig’s argument as follows:
On behalf of the…[first] premise of this sub-argument – that is, the claim that a collection formed by successive addition cannot be an actual infinite – Craig notes that it is tantamount to the
claim that it is impossible to count to infinity. He offers the following illustration of what he takes to be the central difficulty: ‘Suppose we imagine a man running through empty space on a
path of stone slabs, a path constructed such that when the man’s foot strikes the last slab, another appears immediately in front of him. It is clear that, even if the man runs for eternity, he
will never run across all of the slabs. For every time his foot strikes the last slab, a new one appears in front of him, ad infinitum. The traditional cognomen for this is the impossibility of
traversing the infinite.’ (104)
In Craig’s example, the question is not whether the man can run across all of the slabs, but rather whether he can run across infinitely many slabs. For, if he achieves the latter task and yet
not the former, he will still have completed an actual infinite by successive addition. If we suppose that the rate at which the slabs appear is constant, then, in any finite amount of time, only
finitely many slabs appear: there is no time at which infinitely many slabs have been crossed. However, if the man runs for an infinite amount of time – that is, if, for each n, there is an nth
slab that the man crosses – it is nonetheless true that infinitely many slabs are crossed: there is an actually infinite collection that is formed by successive addition. (Of course, Craig will
resist this way of characterising matters: given his view that the future is not real, he will insist that it is at best true that infinitely many slabs will be crossed: the collection that is
formed here by successive addition is at best “potentially infinite”.)
But what if we suppose that the time lapse between slabs decreases according to a geometric ratio, and that the man is replaced by a bouncing ball whose height of bounce decreases according to
the same geometric ratio? If the ball hits the first slab at one minute to twelve, the second slab at ½ minute to twelve, the third slab at ¼ minute to twelve, and so on, then the ball can come
to rest on a slab at twelve, having made infinitely many bounces on different slabs in the interval between one minute to twelve and twelve. In this example, we have a process – the bouncing of
the ball – that plainly does form an actual infinite by successive addition. Consequently, we don’t need to challenge Craig’s view about the reality of the future in order to reject the second
premise of the argument under discussion: there are perfectly ordinary processes that involve formation of an actual infinite by successive addition in not obviously impossible worlds (in which
space and time are composed of points, and there are no quantum or thermodynamical effects to rule out the precise application of classical kinematics to the motion of a bouncing ball). Since
Craig has – for the purposes of this argument – renounced the claim that there cannot be actual infinities, it is quite unclear what reason we are supposed to have for rejecting this
counter-example to the alleged impossibility of forming actual infinities by successive addition (Arguing About Gods, pg. 143-144).
Here, Oppy recognizes the important distinction between an actual infinite and a potential infinite and seems to more or less concede the point that Craig’s response succeeds on the supposition of a
presentist view of time. But he then raises a further difficulty. If successive addition can be carried out in such a way that each addition is carried out in half the time it took to carry out the
previous addition, then there will be an actually infinite number of successive additions carried out over a finite time interval. (The completion of an infinite number of tasks in a finite amount of
time is called a supertask). An actually infinite collection will then have been formed by successive addition, thus implying that the first premise of Craig’s argument is false. An actual infinite
can be formed by successive addition after all.
How might we reply to Oppy’s argument? There are a couple of responses.
First, it might be argued that time is not structured in such a way that in a continuous stretch of time there are an actually infinite number of subintervals of time. In order for Oppy’s suggested
supertask with the bouncing ball to be possible, a given interval of time must be composed of an actually infinite number of subintervals of time. So, if an interval of time does not have this
property, then the supertask cannot be performed. The question is, is there a good reason to think that time cannot have this kind of structure? I think that there is indeed. An argument involving
what is called the Grim Reaper Paradox can be pressed against the possibility of time having this kind of structure. In general, the argument would rule out the possibility of supertasks. It can be
simply formulated as follows:
1. If supertasks are possible, then the Grim Reaper Paradox is possible.
2. The Grim Reaper Paradox is not possible.
3. Therefore, supertasks are not possible.
We will discuss the Grim Reaper Paradox later on.
Second, it is not clear that Oppy’s supertask is even relevant to an infinite series of past events. Indeed, it is not clear that supertasks in general shed any light on the possibility of there
being an infinite series of past events. For with respect to an allegedly infinite past, we are talking about the difficulty of traversing an infinite distance, not traversing a finite distance that
is composed of an infinite number of ever-smaller sub-distances. As Craig writes,
The question is not whether it is possible to traverse infinitely many (progressively shorter) distances but whether it is possible to traverse an infinite distance. Thus, the problem of
traversing an infinite distance comprising an infinite number of equal, actual intervals to arrive at our present location cannot be dismissed [along these lines] (Blackwell Companion, pp. 119).
If need be, we can simply modify the premises of the argument by changing successive addition to non-accelerating successive addition, which would render Oppy's objection irrelevant and would allow
the argument would go through as before. For, as Craig says, the problem is with traversing an actually infinite number of equal intervals.
For both of these reasons, therefore, Craig’s first premise seems to be secure, despite Oppy’s objections.
St. Thomas Aquinas considered the argument that if there were an infinite series of past events, then the present event could not have occurred. St. Thomas formulates the argument as follows:
Further, if the world always was, the consequence is that infinite days preceded this present day. But it is impossible to pass through an infinite medium. Therefore we should never have arrived
at this present day; which is manifestly false (Summa Theologica, Pt. I, Q. 46, Art. 2).
His rejoinder to the argument is as follows:
Passage is always understood as being from term to term. Whatever bygone day we choose, from it to the present day there is a finite number of days which can be passed through. The objection is
founded on the idea that, given two extremes, there is an infinite number of mean terms (ibid.).
What is St. Thomas saying here? Essentially, he is saying that from the present moment to any given moment in the past (let us call each of these intervals segments of time), a merely finite distance
separates the two moments. So, since traversing finite distances is unproblematic and each of the past segments of time is finite, it follows that traversing an infinite past (which is composed of
these segments of time) is unproblematic. The philosophers J.L. Mackie and J. Howard Sobel argue similarly (The Miracle of Theism, pg. 93; Logic and Theism, pg. 182).
In response to this, it must be said that St. Thomas and these other philosophers seem to be committing the fallacy of composition here. The argument, to reiterate, is that, since we are able to
traverse (pass through) the interval from any particular past moment to the present moment, it follows that we can therefore traverse the entire infinite past up to the present moment. But this is as
fallacious as saying that because we can lift each brick of the Great Wall of China, we can therefore lift the entire Great Wall of China, which clearly does not follow. So, just because we can get
from any particular moment in the past to the present does not mean that the entire infinite past can be traversed. The objection, therefore, fails.
Jimmy Akin criticizes the first premise of the argument by way of presenting a dilemma: Either the argument assumes a beginning to the temporal series of events (in which case, since starting at one
definite point and ending at another constitutes a finite distance, the argument would then be presupposing the finitude of the past and so would be question-begging) or else the first premise of the
argument is simply false. The reason the first premise would be false if there is no assumption of a beginning point is that if we started out with an actually infinite collection and then added to
that collection by successive addition, we would technically form a new collection that is actually infinite, and we would have thus formed such a collection by successive addition. “Depending on how
you interpret it,” writes Akin, “the argument [thus] either commits a fallacy or uses a false premise.” (“Traversing the Infinite?”).
With respect to the first horn of the dilemma, Craig insists that the argument does not presuppose a starting point, not even an infinitely distant starting point. Craig writes,
But, in fact, no proponent of the kalam argument of whom we are aware has assumed that there was an infinitely distant starting point in the past. The fact that there is no beginning at all, not
even an infinitely distant one, seems only to make the problem worse, not better. To say that the infinite past could have been formed by successive addition is like saying that someone has just
succeeded in writing down all the negative numbers, ending at -1 (Blackwell Companion, pp. 119-120).
As Craig elsewhere says, attempting such a feat would be like trying to jump out of a bottomless pit (Time and Eternity, pg. 229).
With respect to the second horn of the dilemma, Craig anticipates this and responds as follows:
The only way a collection to which members are being successively added could be actually infinite would be for it to have an infinite tenselessly existing “core” to which additions are being
made. But then, it would not be a collection formed by successive addition, for there would always exist a surd infinite, itself not formed successively but simply given, to which a finite number
of successive additions have been made. Clearly, the temporal series of events cannot be so characterized, for it is by nature successively formed throughout. Thus, prior to any arbitrarily
designated point in the temporal series, one has a collection of past events up to that point which is successively formed and completed and cannot, therefore, be actually infinite (Blackwell
Companion, pp. 124-125).
And given a presentist ontology of time, there can be no infinite tenselessly existing core. Akin rejects presentism and so thinks that there can be such a core. The debate in this case, therefore,
comes down to the fundamental nature of time. If presentism is true, then it seems that Craig has the upper hand. If presentism is false, then it seems that Akin has the upper hand. Since presentism
seems to be the most plausible view of time, Craig’s argument appears to win out. Akin’s objection, therefore, does not succeed.
Defense of second premise
In defense of the second premise of the argument, the truth of the premise follows from the nature of time. On presentism, the series of temporal events is formed by the successive addition of one
event after another as time sequentially keeps on ticking. For a full defense of the premise, we would need to provide a substantive defense of presentism, which is beyond the scope of this article.
However, it can be pointed out that presentism is the commonsense view of time and therefore has the presumption of innocence (so to speak) in its favor.
Critique of second premise
Opponents of the second premise are generally going to be those who reject presentism and hold to some form of the B-Theory of time. For if the B-Theory is correct, then the past was not formed by
successive addition, but was rather formed all at once. This follows because all events (past, present, and future) exist all at once on this view of time. Again, critiquing the B-Theory of time is
beyond the scope of this article.
Even if presentism should turn out to be false and the B-Theory of time is actually true, however, the second premise can be modified so as to still get the desired conclusion, viz., the universe
began to exist. The philosopher Andrew Loke has argued that even on the B-Theory, although time in and of itself would not be formed by successive addition (since all moments of time exist
tenselessly), nevertheless, a conscious being’s experience of the passage of time would be formed by successive addition (The Teleological and Kalam Cosmological Arguments Revisited, pg. 211-214). To
elaborate, we can suppose that there exists a conscious being that has a series of experiences, one after another as he experiences the illusory passage of time. Even if his experience of the passage
of time is illusory, the experiences of the events of time themselves still exist in his consciousness, and these experiences accumulate via successive addition. He can in principle count them as
they occur. Now, although we are supposing that moments of time do not objectively pass but rather exist all at once tenselessly, the moments of time nevertheless are experienced successively by the
conscious being. Consequently, if past time is infinite and the conscious being has always existed, then he has at present (his present) had an actually infinite number of experiences that he has
been able to count. If it is the case, therefore, that an actual infinite cannot be formed by successive addition, then since the conscious being’s experiences have accumulated via successive
addition, it follows that he cannot have had an actually infinite number of experiences. From this fact, it follows that past time cannot be infinite, i.e., the universe began to exist. Thus, the
present argument for the beginning of the universe can be modified so as to succeed even on the supposition of the B-Theory of time.
Of course, if the B-Theory is false and presentism is correct, then the second premise goes through without any modifications needed. Or does it? One objection that has been raised by Graham Oppy
against the claim that time is formed by successive addition (even given presentism) is that, given the possibility of time being continuous in nature, time possibly has the structure of the real
numbers rather than the natural numbers. In such a case, time would be formed by continuous addition or accretion rather than successive addition. Further, the set of past events would be uncountably
infinite as opposed to countably infinite; consequently, the set of past events would not be a series, since a series is essentially discrete in nature. Time is not, on such a view, made up of
discrete moments forming a series “like beads on a string,” as Oppy puts it (“Time, Successive Addition, and Kalam Cosmological Arguments”, pp. 185). If this is right, then the second premise of
Crag’s argument is false. What’s more, the traversal of a continuous interval of time entails traversing infinitely many points of time that compose said interval. So, traversal of the infinite seems
to be possible via continuous addition even if it is not via successive addition.
The main idea seems to be something like this: If time is continuous, then a given interval of time is composed of an actually infinite number of subintervals of time. Time would then accumulate via
continuous addition rather than via successive addition. So, the second premise of Craig’s argument would be false. Perhaps Craig could simply modify the argument by replacing “successive addition”
with “continuous addition” as follows:
1. A collection formed by continuous addition cannot be an actual infinite.
2. The temporal series of events is a collection formed by continuous addition.
3. Therefore, the temporal series of events cannot be an actual infinite.
In this case, however, Oppy would argue that while the second premise is now true on a continuous view of time, the first premise becomes false. This can be seen as follows: The succession of each
second (the choice of second as a unit of time here is arbitrary) of time involves the traversal of an actually infinite number of sub-seconds of time. Since seconds of time clearly are capable of
passing, the traversal of an actual infinite by continuous addition is possible. After all, if we are traversing an infinite number of subintervals of time in order to traverse the interval of which
those subintervals are a part, then we are ipso facto traversing an actual infinite. Since passing from one second to the next is simply a part of such continuous addition, there does not seem to be
any barrier to the traversal of infinitely many seconds (or any other unit) of time by continuous addition. So, the set of past events is formed by continuous addition rather than successive
addition, and it is possible to traverse an infinite collection by continuous addition. So, if time is continuous in this way, Craig’s argument fails.
In reply to this, we note (as we did in our reply to one of Oppy’s aforementioned objections to the first premise of Craig’s argument) that the structure of time that Oppy’s objection relies on is
ruled out by the Grim Reaper Paradox (see below). Intervals of time are not composed of an actually infinite number of subintervals of time; rather, an interval of time is infinitely divisible in the
sense that one can keep dividing the interval in half potentially infinitely. To further support this point, Edward Feser, drawing from Zeno’s paradoxes (originally developed by the ancient Greek
philosopher Zeno), notes that the notion of a continuum of time or space that is composed of an actually infinite number of points is fraught with metaphysical difficulties. Feser writes,
[T]he parts of which a continuous object is purportedly composed would be either extended or unextended, and either supposition leads to absurdity. Suppose first that the parts are unextended.
These unextended parts are either at a distance from each other or they are not. If they are at a distance from one another, then they would not form a continuum, but would rather be a series of
discrete things (like the dots in a dotted line, only without even the minute extension such dots have). Suppose then that they are not at a distance from one another, but are instead in contact.
Then, since they have no extension at all and thus lack any extreme or middle parts, they will exactly coincide with one another (like a single dot, only once again without even the minute
extension of such a dot). All these parts together, no matter how many of them there are, will be as unextended as an individual part. In that case, too, then, they will not form a true
So, if a continuous object is made up of parts, they will have to be extended parts. Now these purported extended parts would either be finite in number or infinite. They cannot be finite,
however, because anything extended, no matter how small, can always be divided at least in principle into yet smaller extended parts, and those parts into yet smaller extended parts ad infinitum.
So if a continuous object is made up of extended parts, they will have to be infinite in number. But the more extended parts a thing has, however minute those parts, the larger it is. Hence if a
continuous object is made up of an infinite number of extended parts, it will be of infinite size. This will be so of every continuous object, however small it might seem…But this is absurd.
Hence a continuous object can no more be made up of extended parts than it can be made up of unextended parts (Aristotle’s Revenge, pg. 204-205).
Must we, therefore, deny the reality of continua in nature? No, we needn’t do so. Instead, we can make the important distinction between actuality and potentiality. The parts of a continuum are in
the continuum potentially but not actually. Feser writes,
Applying the theory of actuality and potentiality, [the proponent of continua] argues that what the paradoxes really show is that the parts of a continuum are in it only potentially rather than
actually. That is not to say that they are not there at all. A potentiality is not nothing, but rather a kind of reality. That is why a wooden block (for example) is divisible despite being
continuous or uninterrupted in a way a stack of blocks is not. But until it is actually divided, the parts are not actual…Affirming that reality includes both potentialities as well as
actualities allows us to acknowledge the reality of the parts of a continuum while at the same time avoiding paradox (ibid., pg. 205).
This solution to the paradoxes of continua implies that intervals of (continuous) time are not composed of an actually infinite number of points of time. Rather, such intervals are merely infinitely
divisible in principle. And the divisibility that is in mind here is a potential infinite rather than an actual infinite. As David S. Oderberg says, “[T]he KCA supporter need not deny that there are
natural continua, including temporal ones, if that entails only that there are potential infinities…There may well be natural continua…but all this means is that infinite divisibility is to be found
in nature, perhaps both spatially and temporally” (“The Kalam Cosmological Argument Neither Bloodied nor Bowed: A Response to Graham Oppy”, pp. 193).
So, time turns out not to have the structure that Oppy’s objection relies on. Thus, Oppy’s objection does not succeed.
The Grim Reaper Paradox
As a third philosophical argument for the finitude of the past, we will briefly consider the Grim Reaper Paradox. The Grim Reaper Paradox (henceforth, GRP) was originally developed by the philosopher
José Benardete. It has been used to defend the second premise of the KCA by philosophers such as Alexander R. Pruss and Robert C. Koons. The key result that is argued to follow from the GRP
(especially by Pruss) is the thesis of causal finitism, the view that there cannot be an actually infinite causal series such that infinitely many causes sequentially produce a given effect. The
paradox is described in the following paragraph.
Suppose you are alive at 12:00 A.M. And suppose there are infinitely many grim reapers (GRs). Suppose that at 12:30 A.M., GR 1 will strike you dead if you are still alive. Suppose also at 12:15 A.M.,
GR 2 will strike you dead if you are still alive. At 12:07.5 A.M., GR 3 will strike you dead if you are still alive. At 12:03.75 A.M., GR 4 will strike you dead if you are still alive. At 12:01.875
A.M., GR 5 will strike you dead if you are still alive. And so on ad infinitum. If a GR sees that you are dead, he will do nothing. Now, we ask a question: Are you still alive at 12:30 A.M.? On the
one hand, you must be dead because some GR must have killed you. This follows because GR 1 would have killed you if you were still alive at 12:30 A.M., GR 2 would have killed you if you were still
alive at 12:15 A.M, and so on. You thus could not be alive at 12:30 A.M. On the other hand, you can’t be dead (assuming nothing other than a GR killed you) because no GR could have killed you! This
follows because GR 1 couldn’t have killed you because GR 2 would have beaten him to the punch (or slice?). But GR 2 couldn’t have killed you because GR 3 would have beaten him to the punch. And so
on. So, we are left with the conclusion that you can’t possibly be alive at 12:30 A.M. because some GR must have killed you, and yet you must still be alive at 12:30 A.M. because no GR could have
killed you. This is a logical contradiction.
A plausible resolution of the paradox is to propose that it is metaphysically impossible for there to be an infinite chain of causes going into the past that impinge upon a single effect in the
present. It appears that this is what is ultimately going on in the GRP. The upshot of this resolution is that this would give us strong philosophical grounds for thinking that an infinite past is
metaphysically impossible since given an infinite past, we would have an infinite chain of causes going into the past (past events) that impinge upon a single effect (a present event). In other
words, the causal structure of the grim reaper scenario is isomorphic to the causal structure of an infinite past. So, if we reject the possibility of the grim reaper scenario, we should similarly
reject the possibility of an infinite past.
Building on this reasoning, we can formulate an argument for the proposition that the universe began to exist as follows:
1. If the universe did not begin to exist, then there is an infinite past.
2. If there is an infinite past, then there is an infinite chain of causes going into the past that impinge upon a present effect (and the GRP is possible).
3. There cannot be an infinite chain of causes going into the past that impinge upon a present effect (the GRP is impossible).
4. Therefore, there is not an infinite past (2, 3).
5. Therefore, the universe began to exist (1, 4).
One preliminary issue to deal with is that we have previously applied the GRP to proving that time is not structured in such a way that over a given interval of time, there are an actually infinite
number of subintervals of time. And this is because the GRP as presently formulated can only get off the ground if time is structured in this way. So, it seems that once we reject time having this
structure, the GRP can no longer be used to show that past time cannot be infinite. For so long as past time is not structured in this way, the GRP is already ruled out without the need to hold to a
finite past.
This problem can be remedied by modifying the GRP as follows: Suppose that the past is infinite and that you have always existed throughout the entire infinite past up to the present day. Now,
suppose there are infinitely many GRs such that today a GR will strike you dead if you are still alive, another GR would have struck you dead if you were still alive yesterday, another GR would have
struck you dead if you were still alive a day before that, and so on. The contradiction that arises is that you cannot possibly be alive today because some GR must have killed you, and yet you must
still be alive today because no GR could have killed you. This revision is suggested by Pruss (Infinity, Causation, and Paradox, pg. 55-56). On this revised scenario, the GRP is clearly isomorphic to
an infinite past. Hence, if the past is infinite, the GRP is possible. But the GRP is impossible. Therefore, the past is finite, i.e., the universe began to exist.
Some philosophers, such as Graham Oppy and John Hawthorne have suggested an alternative resolution of the GRP. Oppy actually discusses a different but very similar paradox developed by Benardete (
Philosophical Perspectives on Infinity, pg. 81-83), but his suggestion can equally apply to the GRP. Appropriating Oppy’s thoughts and suitably modifying them to apply to the GRP, Oppy suggests that
while it is true that no individual grim reaper killed you, nevertheless perhaps the entire collection of grim reapers killed you. This reply, though ingenious, seems quite implausible. If no single
grim reaper killed you with his scythe, how could all the grim reapers have collectively killed you? How could an infinite number of reapers individually not doing anything collectively kill you?
Consider the following example. You start a calculator on the value “0.” You then add zero to the current sum. The resultant sum is, of course, still 0. Suppose you add zero again. The sum is still
0. Suppose you add zero infinitely more times. Clearly, the sum will still be 0. The bottom line: a whole lot of nothings (even infinitely many nothings) don’t add up to something.
There is an alternative reply to the present objection that modifies the GRP scenario. This modification has been suggested by Pruss. Instead of imagining infinitely many grim reapers, we are instead
to imagine infinitely many jolly givers (JGs). And instead of trying to kill you like the grim reapers, the JGs try to put an orange in your Christmas stocking. As Pruss explains:
You hang up a stocking at midnight. There is an infinite sequence of Jolly Givers, each with a different name, and each of which has exactly one orange. There are no other oranges in the world,
nor anything that would make an orange. When a JG’s alarm goes off, it checks if there is anything in the stocking. If there is, it does nothing. If there is nothing in the stocking, it puts its
orange in the stocking. The alarm times are the same as in the previous story (“The Paradox of the Jolly Givers”).
In this variant, Oppy’s suggestion that it is the entire collection of JGs that puts an orange in the stocking leads to a violation of the principle of ex nihilo nihil fit (out of nothing, nothing
comes). For, no JG gave up his orange and yet there is an orange in the stocking. Hence, the orange came into being uncaused from nothing, which is metaphysically absurd. The alternative explanation
suggested by Oppy, therefore, fails. It should also be noted that this scenario can easily be modified so as to get around the problem of time not being structured in such a way that over a given
interval of time, there are an actually infinite number of subintervals of time. The modification would parallel the foregoing modification of the original GRP to get around this problem.
Jimmy Akin objects to the GRP by arguing that it tacitly presupposes a first reaper in the series of reapers, which is already impossible because, since there is a last reaper and an interval with a
first member and a last member cannot be infinite, there cannot be a first reaper, given that we are supposing that there are an infinite number of reapers. As Akin writes (using “Fred” in place of
The resolution of this paradox is fairly straightforward. It has envisioned a situation where Fred begins alive and then will be killed by the first grim reaper he encounters.
The problem is that—if the series of grim reapers is infinite—then it must have no beginning.
To suppose that an infinite series of whole numbers has both a first and last member involves what I’ve called the First-and-Last Fallacy.
· Infinite series can have no beginning ( . . . -3, -2, -1, 0)
· They can have no end (0, 1, 2, 3 . . .)
· Or they can lack both a beginning and an end ( . . . -3, -2, -1, 0, 1, 2, 3 . . . )
But if a series has both a beginning and an end, then it’s finite.
The series of reapers set to kill Fred has an end—Reaper 0—but if that’s the case, it cannot have a beginning.
This means that there is no first grim reaper that Fred encounters, just as there is no “first negative number.”
The idea of a first negative number involves a logical contradiction, and therefore the…Grim Reaper paradox is proposing a situation that cannot exist (“Grim Reapers, Paradoxes, and Infinite
In response, it must be said that the GRP does not presuppose a first reaper. It’s precisely the fact that there is no first reaper that generates the paradox. One advantage of the GRP over the
previously canvassed arguments for the beginning of the universe is that it does not involve purely quantitative operations but instead involves explicitly causal processes. In the GRP, we are no
longer talking about the difficulties of there existing an actual infinite or of traversing an actual infinite. Instead, we are talking about the difficulty of an infinite number of causes going back
into the past that each contribute to a single effect in the present. What the GRP shows is that such a causal series leads to logical contradiction. It is precisely the fact that there is no first
reaper that makes the situation causally impossible. There must be a first reaper, but there is no first reaper. Contradiction. The way to resolve the paradox is to say that there must be a first
reaper. Consequently, the causal series must be finite, not infinite. This is the thesis of causal finitism. Akin’s objection, therefore, is unpersuasive.
Overall, then, I conclude that—in contrast to his first argument—Craig's second philosophical argument for the beginning of the universe is sound. Additionally, the Grim Reaper Paradox furnishes us
with another sound argument for the beginning of the universe. | {"url":"https://www.iamchristianmedia.com/article/successive-addition-supertasks-grim-reapers-and-the-kalam-cosmological-argument","timestamp":"2024-11-13T07:31:22Z","content_type":"text/html","content_length":"1050487","record_id":"<urn:uuid:49d72ffc-eed1-49d1-81d1-9b5b5a05431d>","cc-path":"CC-MAIN-2024-46/segments/1730477028342.51/warc/CC-MAIN-20241113071746-20241113101746-00239.warc.gz"} |
Logic Alphabet
Recent Logic Alphabet Project
Crystal We need a better set of signs for and, or, if .
Creation These three are themselves only a small part of the 16 binary connectives. But they are also fundamental in symbolic logic and, more specifically, in
what is called the logic of atomic sentences. In this case, atomic means undivided totality.
Call for PLE
The X-Stem Logic Alphabet (XLA) was devised and is actively being developed by Shea Zellweger. The main emphasis of the project lies in presenting,
Mirrors 1993 clarifying, and acting as a clearing house for the new notation. The new notation embodies, exposes, and elucidates the deep symmetry properties that
inhabit the interrelational structures in logic; more specifically, in what is also called the 2-valued propositional calculus.
Mirrors 2008
The new notation consists of 16 letter shapes. It is a shape-value notation. Each letter shape is designed to be cursive, phonetic, iconic, and
XLA - Wiki in-part topological. The 16 cursives are also selected to encode a fundamental set of properties in algebra, in geometry, in symmetry, and most of
all, in logic.
Zellweger - Wiki
# 4,273,542
# 4,367,066
# 4,504,236
Image Gallery
Flip Stick
Flip Stick (C)
XLA Blocks (C)
Clock Compass
Logic Bug
Logic Bug (C)
Logical Garnet
Chess Set (C)
Eight Cell
24 Cell
M J T Exhibit
Presentations The construction of the X-Stem Logic Alphabet comes under the general area now called "sign factors engineering." This enters cognitive ergonomics in
such a way that it extends and carries Peirce's box-X notation (1902) into transformational logic. This introduces both a symmetry notation and a mirror
Unpublished notation that sees itself as a form of iconic logic. Other links put the emphasis on operational logic, (inter) relational logic, polyhedral logic, and
crystallographic logic.
Useful Links
Tony Smith
Arnold Oostra
Louis H. Kauffman
Jay L. Lemke
Jeffrey Long
Notation Systems Logic
Notebook 1975 | {"url":"http://logic-alphabet.net/","timestamp":"2024-11-08T20:14:15Z","content_type":"text/html","content_length":"14594","record_id":"<urn:uuid:4f875634-802d-4762-8a71-63060bb83322>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00621.warc.gz"} |
A = [[1,2,2],[2,1,2],[2,2,1]] , then a3 4a2 6a is equal-Turito
Statement - I The value of x for which (sin x + cos x)^1 + sin 2x = 2, when 0 ≤ x ≤ , is
Statement - II The maximum value of sin x + cos x occurs when x =
In this question, we have to find the statements are the correct or not and statement 2 is correct explanation or not, is same as assertion and reason. | {"url":"https://www.turito.com/ask-a-doubt/maths-a-1-2-2-2-1-2-2-2-1-then-a3-4a2-6a-is-equal-to-1-a-a-0-qcf3704","timestamp":"2024-11-05T16:11:20Z","content_type":"application/xhtml+xml","content_length":"456096","record_id":"<urn:uuid:fab1f276-f555-4b59-9dc6-bab64bdbff42>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00119.warc.gz"} |
Lesson 17
Annually, Quarterly, or Monthly?
These materials, when encountered before Algebra 1, Unit 5, Lesson 17 support success in that lesson.
17.1: Finding Equal Expressions (5 minutes)
The purpose of this activity is to reason through some examples, and the meaning of exponents, that an expression in the form \((a^b)^c\) is equivalent to \(a^{bc}\). Monitor for students who reason
about the last question by replacing 8 with \(2^3\) or vice versa. When students notice that they can use properties to rewrite expressions, they are noticing and making use of structure (MP7).
Encourage students to reason about the expressions without using a calculator or evaluating them.
Student Facing
1. Find pairs of expressions that are equal. Be prepared to explain how you know.
\((3 \boldcdot 3 \boldcdot 3 \boldcdot 3 \boldcdot 3) \boldcdot (3 \boldcdot 3)\)
\(3 \boldcdot 3 \boldcdot 9 \boldcdot 9 \boldcdot 9\)
\(3 \boldcdot 9 \boldcdot 27\)
2. Write an expression that is equal to \((2^{30})^7\) using a single exponent.
3. Without evaluating the expressions, explain why \(2^{15}\) is equal to \(8^5\).
Activity Synthesis
Ensure that students found correct matches for the first question, and invite a few students to articulate how they knew they had found a match without evaluating the expressions. Use at least one
example from this activity to make sure students are comfortable with the rule: \((a^b)^c\) is equivalent to \(a^{bc}\). For example, demonstrate that \((3^5)^2=3^{10}\) because they both equal \(3 \
boldcdot 3 \boldcdot 3 \boldcdot 3 \boldcdot 3 \boldcdot 3 \boldcdot 3 \boldcdot 3 \boldcdot 3 \boldcdot 3\). Then, for the last question, make explicit a chain of reasoning such as: \(2^{15}\), \(2
^{3 \boldcdot 5}\), \((2^3)^5\), then \(8^5\) pointing out where this property comes into play. (These steps could also be listed in reverse, first replacing 8 with \(2^3\).)
17.2: How Many Times Per Year? (15 minutes)
The purpose of this activity is to familiarize students with the terms annually, semi-annually, quarterly, and monthly, and use the meaning of these terms to solve some problems. In each problem,
students perform a few numerical computations and then generalize by writing an expression, which is an example of expressing regularity in repeated reasoning (MP8).
Provide access to calculators. First, ask students to complete as much of the table as they can, using what they understand about the meaning of the given words. Ensure that students complete the
table correctly before proceeding with the rest of the activity.
Student Facing
1. Complete the table.
│If something happens...│It happens this many times a year...│It happens every \(\underline{\hspace{.5in}}\) months... │
│ annually │ │ │
│ semi-annually │ │ │
│ quarterly │ │ │
│ monthly │ │ │
2. A gym membership has an annual fee, billed monthly. How much is each bill, if the annual fee in dollars is . . .?
1. 360
2. 540
3. \(g\)
3. An educational foundation gives an annual scholarship, distributed semi-annually. How much is each distribution, if the annual scholarship amount in dollars is . . .?
1. 1,800
2. 5,000
3. \(s\)
4. A magazine subscription has an annual price, billed quarterly. How much is each bill, if the annual price in dollars is . . .?
1. 48
2. 80
3. \(m\)
Activity Synthesis
Invite selected students to share their responses. Emphasize that an expression like \(\dfrac{g}{12}\) is simply saying that no matter what the annual fee, represented by \(g\), we can represent the
monthly bill with \(\dfrac{g}{12}\). Because no matter the annual fee, we know that we should divide it by 12 to calculate the monthly bill. Possible questions for discussion:
• “What do the numbers in each row of the table have to do with each other?” (The product of the pair of numbers in each row is 12.)
• “How did you decide which operation to use?” (I drew a tape diagram representing 360 and realized I needed to split it into 12 equal groups, so I divided by 12.)
17.3: Your Problems Are Compounded (15 minutes)
In this partner activity, students take turns matching a description or expression to a representation. As students trade roles explaining their thinking and listening, they have opportunities to
explain their reasoning and critique the reasoning of others (MP3).
Provide access to calculators. Arrange students in groups of 2. Tell students that for each item in column A, one partner finds a matching representation in column B and explains why they think it
matches. The partner's job is to listen and make sure they agree. If they don't agree, the partners discuss until they come to an agreement. For the next item in column A, the students swap roles. If
necessary, demonstrate this protocol before students start working.
Student Facing
Match each item in the first column to a representation in the second column.
1. A worker sets aside \$6,000 per year for their retirement fund by saving the same amount monthly.
A. \(6,\!000 \boldcdot 1.21^3\)
2. A business’s revenue increases by 20% per quarter. This happens for 2 years. Initially, their quarterly profit was \$6,000.
│\(x\)│0 │1 │2 │3 │4 │5 │
│\(y\)│6,000│7,200│8,640│10,368 │12,442 │14,930 │
3. \(6,\!000 \boldcdot ((1.05)^{4})^x\)
4. A man borrows \$6,000 from his sister. He will reduce the amount he owes in 1 year by paying her back quarterly.
│\(x\)│0 │1 │2 │3 │4 │5 │
│\(y\)│6,000│4,800│3,840│3,072│2,457.6 │1,966.1 │
5. A business’s revenue decreases by 20% semi-annually. This happens for 3 years. Initially, their quarterly revenue was \$6,000.
E. \(6,\!000 \boldcdot 1.2155^x\)
6. The number of subscribers to a website triples quarterly for 2 years. Initially there were 6 subscribers.
F. \(6 \boldcdot 4,\!096^2\)
7. \(6,\!000 \boldcdot ((1.1)^2)^3\)
8. The number of likes on a post was 6, and then for the next 2 years, the number of likes doubled, monthly.
Activity Synthesis
Much discussion takes place between partners. Invite students to share how they did mathematical work.
• “What were some ways you handled the terms monthly, quarterly, and semi-annually?”
• “Describe any difficulties you experienced and how you resolved them.”
• “Did you need to make adjustments in your matches? What might have caused an error? What adjustments were made?” | {"url":"https://im-beta.kendallhunt.com/HS/teachers/4/5/17/index.html","timestamp":"2024-11-03T06:09:10Z","content_type":"text/html","content_length":"91794","record_id":"<urn:uuid:34d82108-a000-41b9-ab28-c3e946842e09>","cc-path":"CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00576.warc.gz"} |
January 2019
• You have a quiz on Friday. It is in-class. You can find information on it under “Quizzes”, above. I’ve added a few general notes above the topics section, there.
• You have homework due Friday. It should help you study for the quiz, ideally. See “Homework,” above.
• You are welcome to attend office hours Thursday at 2 pm.
• Thursday 4 pm in Gemmill is a “homework hour” where you may meet your peers (see “Course Info” above, for details.)
• I’ve been continuously updating info in the “Lectures” section, above. You’ll now find today’s linear Diophantine equation example from class, as well as the handout summarizing linear
Diophantine equations.
• I wish you all good luck!
To do for Wed, Jan 30
• Don’t forget that on Friday you have homework and a quiz. Info on those can be found above and to the right (under Homework and Quizzes).
• First, try to complete pages 5-8 of your classroom worksheet, skipping writing the proofs, if you haven’t finished yet. The “cards” approach on page 6 is a way of visually formalizing the fact
that you can “add” equations. For example, the equation ax + by = 1 can be added to the equation ax’ + by’ = 2 to get the equation a(x+x’) + b(y+y’) = 3. In the “cards” notation, this says that
the card “(x,y) gives 1” can be added to the card “(x’,y’) gives 2” to get the card “(x+x’,y+y’) gives 3”. Notice that writing it this way emphasizes the fact that you are adding vectors (x,y)+
(x’,y’)=(x+x’,y+y’) as well as numbers 1+2=3. So you should do the steps of the Euclidean algorithm on the “cards” instead of just the numbers. We’ll cover all this first thing on Wednesday.
• Please read your text, pages 33-41. This covers the material we have been doing in class. He approaches linear Diophantine equations with “hops”, “skips” and “jumps”. If you find this better than
the “cards” on page 6 of my worksheet, that’s fine! (Although I think my approach will make you much happier than his once you see it in action!) You may wish to spend more or less time on the
reading depending on whether it is working for you compared to other things.
• If time permits, try to fill in the proofs on the worksheet.
• We will spend Wednesday in class finishing up the topic of solving linear Diophantine equations. This will complete Chapter 1 of the textbook.
• I have office hours Tuesday at 10 am; please feel welcome! You are also welcome to just come and do your 45 minutes of work in my office, so I’m handy while you’re doing it.
• Wednesday 5-6 pm in Math 350 is the first Math Club event of the semester! Danny Moritz will be talking about “Geodesic Domes, Graph Theory, and Beyond.”
To do for Mon, Jan 28
• Next Friday Feb 1st you have homework due and your first quiz. The homework is posted under “Homework.” The material on the first quiz is posted under “Quizzes” (both on menu upper right). Look
at the material and make a study plan.
• The quiz will take the full class. Arrangements for extra time, if applicable, should be made ASAP.
• Today in class, we worked on a worksheet, which is available for you under “Lectures” in the menu to the upper right. There is a typo corrected noted there; please make this correction on your
copy. (Reminder: Lectures has info relevant to each lecture, including further reading, copies of handouts, etc. It will be updated frequently.).
• We discussed/completed pages 1-3 of the worksheet in class (at least). Please continue on to finish page 4 in advance of Monday’s lecture. In class, I suggested you skip the proof writing and
concentrate on the rest of it. Now, please write the two proofs requested on pages 2 and 4. You may wish to print out a fresh copy and write neatly so you can have this as a reference later (in
case your copy from class was messy.)
• Students responded to the Homework Hour poll and based on that and other feedback, I hereby declare 5 pm Mondays and 4 pm Thursdays in Gemmill a Math 3110 homework hour, where you can meet others
who may be interested in collaborating on homework. (Let me know if this turns out to be useful.) I’m putting this info on the “Course Info” page.
• Some students took an interest in Sage; you can find getting-started info and resources on the “Resources” page. You have access to Sage using your identikey. We will do more with this later.
To do for Fri, Jan 25
• You have homework due Friday, don’t forget. The homework is like a “standard course”, i.e. graded on correctness, and you take as much time as you need whenever during the week works for you
(it’s not part of the 45-minute tasks).
• Read the Homework page (above, right for link). In particular, read the honour code rules. I have slightly unusual honour code rules, and you should email me if they aren’t clear at all.
• I have set office hours and they are now listed on the “Course Info” page. However, this starts next week. Thursday the 24th I’m completely booked up; email if you need me ASAP.
• If you want (optional!), please log in to the canvas and complete another when2meet poll for “homework hours”. That is, I will set designated study hours at Gemmill or the MARC, where you can all
meet to work on homework during the week. By setting such hours, you can go there to work and will find people who are working on the same course. I won’t be there, and there’s no guarantee
anyone else will either, but it will facilitate meeting your peers. Please send me your advice on location in email (Gemmill or MARC? should we designate a more specific location?).
• Today we formalized the Euclidean algorithm. The next task is to try to to solve equations of the form ax+by=c, in integers. You’ll find that our textbook unifies these problems, so today’s
reading will review what we’ve done but simultaneously get you started on solving such equations.
• Read your text, Pages 25-32 inclusive. Take your time and read actively. It’s better to do just some of it well, than all of poorly.
• By the way, I’m recalling the library’s copy of the text, to put on course reserve, but it is taking time.
• Do at least one gcd computation for practice.
• If you are done the reading etc., find someone who doesn’t know the Euclidean algorithm and explain it to them.
To do for Wed, Jan 23rd
• There’s a good chance we’ll move rooms (around the corner in ECCR), which would allow me to let the waitlist of students into the class. I will report more on this when I know more (perhaps
Tuesday). It is not for sure.
• Please complete the when2meet poll with your availability for office hours. The link & explanation for the poll is available as an announcement in canvas (I didn’t want to put it on a public
• Allow me to remind you of the last proposition from class, which was this (restated here a little differently):
Proposition. Let a and b be integers. Then, for some integer k, define the quantity c = a + kb. Then gcd(a,b) = gcd(b,c).
• Suppose you want to compute the gcd of 1925 and 931. The Proposition lets you make a “move” that replaces that big problem with a smaller one. For example, using k=-2, we get c=63 and learn that:
gcd(1925, 931) = gcd(931, 63)
• So, by clever choices of moves, we can replace the original big problem with smaller and smaller ones, until the gcd is obvious. We can go like this:
gcd(1925, 931) = gcd(931, 63) = gcd(63,49) = gcd(49,14) = gcd(14,7) = gcd(7,0) = 7
• First, please verify the set of moves above (recreate it for yourself).
• Your task is to find the “slickest” / “fastest” series of moves to discover the gcd of 4181 and 6765 that you can. On Wednesday bring it to class and we will see who can get to the gcd in the
fewest moves (when you get to gcd(a,0)=a, you are done).
• Please read all of Chapter 0 in your textbook (you have already read a selection; now please finish it).
• There’s a new static page called “Homework” on the website. Please read and understand the honor code rules and general rules there. Email me if you have any questions.
• Your homework due Friday, January 25th is now assigned on that page. Don’t forget that regular homework assignments are assigned each Friday due the following Friday.
To Do for Fri, Jan 18
• Make sure you will have your text by the weekend ideally, or Monday at the latest.
• If you haven’t completed the tasks for previous days, in particular the “get-to-know-you” quiz on canvas, please catch up now.
• First take a look at, but don’t yet complete, the online quiz on canvas entitled “Reading p.10-17.” Just get a sense of the questions being asked.
• Now actively read pages 10-17 of your textbook (this was handed out as a photocopy in class on Wednesday, for those who do not have it yet; if you don’t have a copy or the text, please email me).
By actively, I mean as modeled in class Wednesday. You might also find it useful to check out the links for Wednesday’s lecture on the “Lectures” page at right, for more guidance. Make notes in
the margin or your notebook, of your active reading process.
• Now please complete the online quiz “Reading p.10-17” on canvas after reading. You can have your text open while working on this, together with any notes you’ve made during the reading or class,
and you can look up resources if you find it helpful.
• In class we did an activity in the last ten minutes. Please bring your thoughts on that activity to start Friday’s class. Here is the activity, in case you need a review: It is a two player game.
The board contains some number of positive integers to start. Players take turns. Each player attempts to find two numbers on the board, whose positive difference is not yet written on the board.
Then he adds that number to the board. He who cannot add a number loses. The question is to analyse who wins this game, for a given set of starting numbers. For example, try “10, 15, 50” or “3,
5, 7”.
• Note that in class Monday, I wrote the wrong error term for the prime number theorem. Check out “Lectures” at right for some typed notes relating to that, including the correct error term. You
might like to peruse these further to follow up on any of Monday’s motivating questions that interested you. In particular, they explain the cool pictures on the back of the question sheet.
To Do for Wed, Jan 16
• Please read the static pages on this website. You’ll find them listed at right, “Course Info”, “Goals”, etc.
• Please make inroads to obtaining your textbook. If you have it, please give it a look. It’s quite nice.
• Please figure out how to log into canvas and fill out the “get to know you” quiz you’ll find there. If you are not waitlisted or registered, you may not be able to access this (although I think
you should be able to). If this is the case, please write me an email briefly letting me know (1) whether you attended on Monday, January 14th, (2) your registration situation, and (3) whether
you have read the website in detail.
• Our course looked very crowded on day one. But it tends to clear up a bit. Please keep in mind that I will administratively drop students who are not attending (I will take attendance) and
participating in these daily tasks. I will be posting a “to-do” post on this website by 5:30 after each day of classes, and you are expected to budget 45 minutes to do whatever I ask of you
before the next lecture.
• Finally, I plan to post notes from the first-day lecture’s questions sheet. You will find them under “Lectures” at right, but this may take a little longer than 5:30 pm today, since I’m updating
them a bit. One note: I wrote the error term on the Prime Number Theorem incorrectly in class. This will be corrected in the notes.
To do for Mon, Jan 14
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if we draw an infinite number of circles packed in a square using the method shown below, will the sum of circles areas approach the square's area?
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Probability of seeing a car in 10 minutes
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Solve the quadratic equation by Completing the Square
The area of a circle by slicing
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Why 1+2+3+4+... is not equal to -1/12
Elon Musk's interview question
How to check if two line segments intersect?
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XX International Linac Conference
MOA19 (Poster)
Presenter: Robert D. Ryne (LANL)
email: ryne@lanl.gov
Status: Complete
FullText: ps.gz or pdf
Eprint: physics/0010001
Large-Scale Simulation of Beam Dynamics in High Intensity Ion Linacs using Parallel Supercomputers
Ji Qiang, Robert D. Ryne (LANL)
In this paper, we present results of using parallel supercomputers to simulate beam dynamics in next-generation high intensity ion linacs. These are among the most detailed linac simulations ever
performed, using up to 500 million macroparticles, which is close to the number of particles in the physical systems considered. Our approach uses a fully three-dimensional space charge
calculation. It also uses an improved model of beam dynamics within the rf cavities that is more accurate and flexible than the usual treatment based on approximate field integrals. The
simulations use a hybrid approach involving transfer maps to treat externally applied fields and parallel particle-in-cell techniques to treat the space-charge fields. Our approach is ideally
suited to modeling superconducting linacs (where there are only a few types of cavities to be modeled), but we have also performed simulations of room-temperature linacs that involved modeling
over 400 cavities. Traditionally, small-scale two-dimensional simulations have been performed on PCs or workstations. While such simulations are sufficient for rapid design and for predicting
root mean square properties of the beam, large-scale simulations are essential for modeling the low-density tails of the beam. The large-scale parallel simulation results presented here represent
a three order of magnitude improvement in simulation capability, in terms of problem size and speed of execution, compared with typical two-dimensional serial simulations. In this paper we will
show how large scale simulations can be used to predict the extent of the beam halo and facilitate design decisions related to the choice of beam pipe aperture. Specific examples will be
presented, including simulation of the spallation neutron source linac and the Low Energy Demonstrator Accelerator (LEDA) beam halo experiment.
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Mexican Hat
The Mexican hat potential is one that elicits spontaneous symmetry breaking (SSB), a process by which a physical system starting in a symmetric state spontaneously enters and remains in an asymmetric
state. This usually applies to systems whose equations of motion obey a set of symmetries while the lowest-energy state(s) do(es) not. When the system assumes one of its ground states, its symmetry
is broken even though the Lagrangian as a whole retains it. https://wikipedia.org/wiki/Spontaneous_symmetry_breaking
See https://tikz.netlify.app/higgs-potential for a very similar image.
Edit and compile if you like:
% The Mexican hat potential is one that elicits spontaneous symmetry breaking (SSB), a process by which a physical system starting in a symmetric state spontaneously enters and remains in an asymmetric state. This usually applies to systems whose equations of motion obey a set of symmetries while the lowest-energy state(s) do(es) not. When the system assumes one of its ground states, its symmetry is broken even though the Lagrangian as a whole retains it. https://wikipedia.org/wiki/Spontaneous_symmetry_breaking
% See https://tikz.netlify.app/higgs-potential for a very similar image.
axis lines=center,
axis equal,
y domain=0:1.25,
y axis line style=stealth-,
y label style={at={(0.35,0.18)}},
xlabel = $\varphi_{_1}$,
\addplot3 [surf,shader=flat,draw=black,fill=white,z buffer=sort] ({sin(x)*y}, {cos(x)*y}, {(y^2-1)^2});
\coordinate (center) at (axis cs:0,0,1);
\coordinate (minimum) at (axis cs:{cos(30)},{sin(30)},0);
\fill[DarkBlue] (center) circle (0.1);
\fill[DarkRed] (minimum) circle (0.1);
\draw (center) edge[shorten <=5,shorten >=5,out=-10,in=150,double,draw=gray,double distance=0.5,-{>[length=2,line width=0.5]}] (minimum);
Click to download:
Open in Overleaf:
This file is available on
and on
and is
MIT licensed
See more on the
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Distribution of Relaxation Times (DRT): an introduction Battery - Application Note 60 - BioLogic
25 min read
Distribution of Relaxation Times (DRT): an introduction Battery – Application Note 60
Latest updated: June 25, 2024
DRT is a tool that can be used to help interpreting impedance data. This note introduces the theoretical basis of this method as well as the advantages and limitations compared to direct electrical
equivalent circuit modeling.
Electrochemical Impedance Spectroscopy (EIS) is used to identify the physical characteristics of a system. The method is used to model the system using what is called an electrical equivalent
circuit, where each element of the circuit is supposed to correspond to a physical characteristic of the system under study.
Fitting the impedance data using an electrical model gives access to the values of the elements and hence the physical characteristics, such as, for instance, time constants.
The choice of an electrical circuit requires prior knowledge of the impedance of each element and this is why impedance can be considered difficult to use at first.
DRT is an analysis method which turns impedance data that are a function of the frequency into a distribution of the time constants involved in the considered system.
DRT can be considered as a tool to help finding an equivalent circuit that should be used to fit impedance data.
In this note, we will present the main principles behind this analysis and show the various tools available to perform it.
We will then present the results to illustrate the advantages of this method as well as present some of its limitations.
Finite Voigt Circuit
The Voigt circuit also called a measurement model [1] was used to fit impedance data. A Voigt circuit is composed of a series of N R/C circuits (Fig. 1), for which the impedance can be described:
$$Z(\omega)= \sum^N_{k=1}\frac{R_k}{1+i\omega\tau_k}\tag{1}$$
With $\tau_k = R_kC_k$
Figure 1: Voigt circuit. Sometimes a resistance in series is added.
Let us consider an R/Q circuit, with R = 1 Ω,
α = 0.9, Q = 1 F.s^-0.1, whose impedance diagram is shown in Fig. 2.
Figure 2: Nyquist impedance diagram of an R/Q circuit. The half circle in full line shows that the impedance data are not from a simple R/C circuit.
To fit these data, we can use as an equivalent circuit, a Voigt circuit with, for example, 17 R/C elements. For each R[k]/C[k] values, a time constant τ[k] can be calculated. We can then plot R[k] as
a function of τ[k]. The results are shown in Fig. 3.
Figure 3: Change of Rk vs. τk.
Figure 3 shows that R[k] values loosely follow a Gaussian law. Furthermore, it can be verified that:
$$\sum^{17}_{k=1}R_k\approx 1\tag{2}$$
Infinite Voigt Circuit
Let us now consider a Voigt circuit with an infinite number of elements. It can also be used to fit the impedance data shown in Fig. 2 but instead of discrete values of R[k] vs. τ[k], we have a
continuous variation of R(τ), that is to say a distribution function.
The impedance of such a circuit can be described as [2]
$$Z(\omega)= \int^\infty_0\frac{G(\tau)}{1+i\omega\tau}d\tau\tag{3}$$
with G(τ) the distribution function of the relaxation times τ, which is characteristic of the system under test.
We can deduct from Eq. (3) that the unit of G is Ω.s^-1.
If the analytical expression of the impedance of the system Z(ω) is known, we can use Eq. (3) to express the function G(τ).
For instance, for an R/Q circuit, we can have [2]:
$$G_{R/Q}(\tau)= \frac{R}{2\pi}\frac{\text{sin}((1-\alpha)\pi)}{\text{cosh}\left(\alpha\text{ln}\left(\frac{\tau}{\tau_{R/Q}}\right)\right)-\text{cos}((1-\alpha)\pi)}\tag{4}$$
With $\tau_{R/Q}=(RQ)^{1/\alpha}$.
For a given set of parameters R, Q and α, we can plot G(τ). The abscissa of the peak gives the circuit time constant τ[R/Q] and the G value at the peak is given by
$$\frac{dG}{d\tau}=0\implies G_p = \frac{R}{2\pi} \text{tan}\left(\frac{\alpha\pi}{2}\right)\tag{5}$$
Figure 4 shows the theoretical DRT of an R/Q circuit from Eq. (4):
Figure 4: Theoretical DRT of an R/Q circuit
Other expressions can be found in the literature for example, for a finite length diffusion impedance [3].
Experimental Data
The Problem
In the case of experimental data, the expression of the impedance is not known a priori. This means that Eq. (3) must be numerically solved using experimental impedance values to extract G(τ).
Eq. (3) is a Fredholm integral equation of the first kind, typical of an inverse problem. There are a few methods proposed in the literature to solve a Fredholm equation: Fourier transform [4],
Maximum entropy [5], Bayesian approach [6], Ridge and lasso regression methods [7] and finally Tikhonov regularization [8], which is the one that will be evaluated and used in this paper.
One Solution
Several programs have been written by researchers to calculate DRT using Tikhonov regularization: FTIKREG [9] and DRTtools [2,10]. The latter is a MATLAB application, which was used to produce data
shown in this note, that are compared with the results obtained using a theoretical expression of G(τ), available for an R/Q circuit and a series of two R/Q circuits.
Several programs have been written by researchers to calculate DRT using Tikhonov regularization: FTIKREG [9] and DRTtools [2,10]. The latter is a MATLAB application, which was used to produce data
shown in this note, that are compared with the results obtained using a theoretical expression of G(τ), available for an R/Q circuit and a series of two R/Q circuits.
Here is the method:
1. Electrical circuits are chosen with various parameters.
2. Z Sim, the simulation tool in EC-Lab^®, is used to simulate and plot impedance data.
3. These impedance data are used as inputs in the DRT tools used to provide an approximation of the DRT.
4. For each circuit, the theoretical DRT is plotted using the theoretical expression of G(τ) (Eq. (3)).
5. Both DRTs are compared.
The circuit that we are interested in is composed of two R/Q in series: R1/Q1+R2/Q2. Table I shows the parameters used for the three different circuits. Only the parameter Q2 changes.
Table I: Parameters for the R1/Q1 + R2/Q2 circuit
R1/Ω 1
Q1/F.s^(α1-1) 10^-3
α1 0.9
R2/Ω 1
Q2/F.s^(α2-1) 10^-3 0.3×10^-3 0.1×10^-3
α2 0.9
Figure 5 shows the simulated impedance graph in the Nyquist plane of the three circuits described in Tab. I. ZSim from EC-Lab^® was used.
Figure 5: Simulated impedance graph of the three circuits shown in Tab. I.
The graph corresponding to the first circuit consists of a single arc, which is to be expected since the R/Q elements have the same parameters and hence the same time constants.
The time constant can be determined using the characteristic frequency fc at the apex of the diagram, which is in Fig. 5: 304 Hz. We then have:
$$\tau=\frac{1}{2\pi f_c}\tag{6}$$
This gives τ = 4.6 x 10^-4 s.
Figure 6 shows the theoretical DRT obtained for the three circuits described in Tab. I. The relationship between G(τ), τ, and the parameters are described by Eq. (3), only in this case it is applied
to a sum of R/Q elements.
It is clear that one time constant can be determined on Fig. 6a and two time constants in Figs. 6b and 6c.
The corresponding DRT in Fig. 6a also gives one peak, and hence one time constant.
The impedance graph of the second circuit (Fig. 5) is more difficult to interpret as only one time constant can be identified. Even though both R/Qs have different parameters as shown in Tab. I, the
corresponding DRT in Fig. 6b shows two peaks, which means two time constants. This would be the main advantage of the DRT representation: it can better resolve the components of a system when two or
several time constants are close to one another.
Furthermore, the time constants can be readily accessed by a simple reading on the graph.
Figure 6 also gives the calculated DRT using the DRTtools program and as input the simulated impedance data.
Figure 6: Theoretical (thick blue line) and numerical (thin red line) DRT for a) Circuit 1, b) Circuit 2, c) Circuit 3 in Tab. I.
The parameters that were used are shown in Fig. 7. The export tool was used to have data in .txt files and they were imported in EC-Lab^® using the “Import file” tool When comparing the theoretical
DRT and the DRT obtained by the DRT tools program, one can see that the peak positions are the same and that differences only concern the height of the peak as well the shape. One can notice some
oscillations at the bottom of the peaks especially for Figs. 6a and 6b. Differences can be explained by the fact that the numerical solution only approach the analytical solution.
Figure 7: DRTtools interface window showing the parameters used to produce data shown in Fig. 5. Here the data from the second circuit are shown.
DRT analysis fit impedance data using an infinite Voigt circuit, whose impedance modulus limits at high and low frequencies do not tend to infinity.
As a consequence, DRT has been used mainly in the field of fuel cells to identify the reaction mechanisms [11,12]. Impedance graphs for such materials have suitable impedance limits at low
The method was also used with battery materials [13], which exhibit an increasing impedance modulus at low frequencies. In this case, the DRT analysis is not suitable and the data must be
preprocessed to remove the low frequency part i.e the diffusion part of the impedance. Preprocess means that a certain knowledge of impedance analysis is required to be able to use the DRT analysis,
whose primary advantage was that it would require no prior knowledge of impedance equivalent circuit analysis.
Additionally, and similarly, an inductive behavior at high frequencies (with an increasing impedance modulus at higher frequencies) is not suitable for DRT analysis.
If impedance data happen to contain an inductive term, several strategies are possible: one can choose to discard the corresponding data with positive imaginary part [14] or, as proposed by DRTtools
[10] to account for them in the DRT calculation by adding an inductance term in Eq. (3).
The main advantage of the DRT analysis is that it allows impedance data to be displayed as a distribution of time constants, which can be easier to interpret, can be performed without knowledge of a
suitable equivalent circuit and can also reveal time constants not visible on the impedance graph, especially when they are quite close to one another.
The main limitation is that analysis is restrained to impedance graphs for which the limits of the impedance modulus tend to zero both at high and low frequencies.
If this is not the case, preprocessing of the data is needed, which does require knowledge of impedance fitting and analysis and typical shape of equivalent circuits.
Data files can be found in:
1) P. Agarwal, M. E. Orazem, Luis H. Garcia-Rubio, J. Electrochem. Soc., 139, 7 (1992) 1917.
2) T. H. Wan, M. Saccoccio, C. Chen, F. Ciucci, Electrochim. Acta, 184 (2015) 483.
3) A. Leonide, Ph. D. Thesis, KIT Publishing (2010).
4) A. Leonide, V. Sonn, A. Weber, and E. Iver-Tiff´ee, J. Electrochem. Soc., 155, 1 (2008) 36.
5) T. J. VanderNoot, J. Electroanal. Chem., 386, 12 (1995) 57.
6) F. Ciucci and C. Chen, Electrochim. Acta,
167 (2015) 439.
7) M. Saccoccio, T. H. Wan, C. Chen, and F. Ciucci. Electrochim. Acta, 147 (2014) 470.
8) J. Macutkevic, J. Banys, and A. Matulis, Nonlinear Analysis, Modelling and Control, 9 (2004) 75.
9) J. Weese. Comp. Phys. Com., 111 (1992) 69.
11) H. Schichlein, A. C. Müller, M. Voigths, A. Krügel, and E. Ivers-Tiff´ee, J. Applied Electro-chem., 32 (2002) 875.
12) C. M. Rangel V. V. Lopes and A. Q. Novais. IV Iberian Symposium on Hydrogen, Fuel Cells and Advanced Batteries, Estoril, Portugal, (2013).
13) J. Illig, M. Ender, T. Chrobak, J. P. Schmidt, D. Klotz, and E. Ivers-Tiff´ee, J. Electrochem. Soc., 159, 7 (2012) A952.
14) J. P. Schmidt, E. Ivers-Tiff´ee, J. Power Sources, 315 (2016) 316.
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Course Outlines
Fall 2024 - MATH 232 D400
Applied Linear Algebra (3)
Class Number: 3966
Delivery Method: In Person
• Course Times + Location:
Sep 4 – Dec 3, 2024: Mon, Wed, Fri, 1:30–2:20 p.m.
Oct 15, 2024: Tue, 1:30–2:20 p.m.
• Exam Times + Location:
Dec 11, 2024
Wed, 8:30–11:30 a.m.
• Prerequisites:
MATH 150 or 151 or MACM 101, with a minimum grade of C-; or MATH 154 or 157, both with a grade of at least B.
Linear equations, matrices, determinants. Introduction to vector spaces and linear transformations and bases. Complex numbers. Eigenvalues and eigenvectors; diagonalization. Inner products and
orthogonality; least squares problems. An emphasis on applications involving matrix and vector calculations. Students with credit for MATH 240 may not take this course for further credit.
Topics Outline: Linear equations, matrices, determinants. Introduction to vector spaces and linear transformations and bases. Complex numbers. Eigenvalues and eigenvectors; diagonalization. Inner
products and orthogonality; least squares problems. An emphasis on applications involving matrix and vector calculations.
Topic Details:Vectors
• Vectors in Euclidean n-Space
• Dot Product and Orthogonality
• Lines and Planes
Systems of Linear Equations
• Row Reduction (Gaussian elimination) to Echelon form
• The Geometry of Linear Systems
• Applications in business, science and engineering
• Matrix operations
• Matrix inverse; and properties of matrices
• Elementary matrices and calculating matrix inverses
• Matrices with special forms.
Linear Transformations
• Matrices as transformations
• Geometry of Linear Transformations
• Kernel and range
• Composition and Invertibility
• Application to Computer Graphics (optional)
• Calculating determinants
• Properties of determinants
• Cramer's rule (optional)
Complex Numbers
• Arithmetic in Cartesian co-ordinates.
• The complex plane, complex conjugate, magnitude and argument (phase).
• Polar form, De Moivre's formula and Euler's formula.
• Roots of quadratic polynomials.
Eigenvalues and Eigenvectors
• Properties and geometry
• Complex eigenvalues and complex eigenvectors
• Dynamical Systems and Markov Chains
• Application to Economics: the Leontief model (optional)
• The Power Method; Application to Internet Search Engines
• Matrix Similarity and Diagonalization
Subspaces of R^n
• Subspaces and Linear Independence
• Basis and Dimension
• The Fundamental Spaces of a Matrix
• Rank
• Change of basis
• Projection
• Orthogonal bases and the Gram Schmidt process
• Orthogonal matrices (optional)
• Application to least squares approximation
• Assignments 20%
• Midterms (2, 15% each) 30%
• Final Exam 50%
Students should be aware that they have certain rights to confidentiality concerning the return of course papers and the posting of marks.
Please pay careful attention to the options discussed in class at the beginning of the semester.
This course is delivered in person, on campus. Should public health guidelines recommend limits on in person gatherings, this course may include virtual meetings. As such, all students are
recommended to have access to strong and reliable internet, the ability to scan documents (a phone app is acceptable) and access to a webcam and microphone (embedded in a computer is sufficient).
An Introduction to Linear Algebra
Daniel Norman and Dan Wolczuk
3rd Edition
The SFU Bookstore will stock both the hardcopy and the electronic version of this textbook. Students are encouraged to obtain the hardcopy.
ISBN: 987-0-13-468263-1
Supplemental book (downloaded as a free .pdf):
Introduction to Applied Linear Algebra Vectors, Matrices, and Least Squares
Stephen Boyd and Lieven Vandenberghe
ISBN: 978-1-316-51896-0
Your personalized Course Material list, including digital and physical textbooks, are available through the SFU Bookstore website by simply entering your Computing ID at: shop.sfu.ca/course-materials
Registrar Notes:
SFU’s Academic Integrity website http://www.sfu.ca/students/academicintegrity.html is filled with information on what is meant by academic dishonesty, where you can find resources to help with your
studies and the consequences of cheating. Check out the site for more information and videos that help explain the issues in plain English.
Each student is responsible for his or her conduct as it affects the university community. Academic dishonesty, in whatever form, is ultimately destructive of the values of the university.
Furthermore, it is unfair and discouraging to the majority of students who pursue their studies honestly. Scholarly integrity is required of all members of the university. http://www.sfu.ca/policies/
Students with a faith background who may need accommodations during the term are encouraged to assess their needs as soon as possible and review the Multifaith religious accommodations website. The
page outlines ways they begin working toward an accommodation and ensure solutions can be reached in a timely fashion. | {"url":"http://www.sfu.ca/outlines.html?2024/fall/math/232/d400","timestamp":"2024-11-02T05:44:04Z","content_type":"text/html","content_length":"23758","record_id":"<urn:uuid:a28b1e6b-664b-44aa-a985-9771edd00ded>","cc-path":"CC-MAIN-2024-46/segments/1730477027677.11/warc/CC-MAIN-20241102040949-20241102070949-00292.warc.gz"} |
Python set isdisjoint() helper with examples
What is Python set isdisjoint()?
Python set isdisjoint() method is specifically designed to check if two sets are disjoint, meaning they do not share any common elements. It returns a Boolean value of True if the sets are disjoint,
and False otherwise. This method is particularly useful when you need to verify the absence of common elements between sets.
Let’s explore purpose, syntax, and parameters of the isdisjoint() method and examples to illustrate its usage. So let’s get started!
What is the Purpose of the isdisjoint()?
The main purpose of the isdisjoint() method is to determine whether two sets are disjoint or not. It allows you to quickly check if there are any shared elements between sets, which can be useful in
various scenarios such as data analysis, filtering, or set operations.
Python set isdisjoint() Syntax and Parameters
The syntax for the isdisjoint() method is as follows:
Here, set1 and set2 are the sets that you want to check for disjointness. The isdisjoint() method is invoked on set1, and set2 is passed as an argument.
How to Check for Disjointness Between Sets?
Now let’s dive into some examples to understand how the isdisjoint() method works and how you can utilize it effectively.
I. Checking if Two Sets are Disjoint
Suppose we have two sets, set1 and set2, and we want to check if they are disjoint. Here’s an example:
Example Code
set1 = {1, 2, 3, 4} set2 = {5, 6, 7, 8} if set1.isdisjoint(set2): print("The sets are disjoint.") else: print("The sets are not disjoint.")
In this example, we have two sets, set1 and set2, with no common elements. We use the isdisjoint() method to check if they are disjoint. Since there are no shared elements, the output will be:
The sets are disjoint.
II. Checking if a Set is Disjoint with Multiple Sets
Python set isdisjoint() method can also be used to check if a set is disjoint with multiple sets. Let’s consider an example:
Example Code
set1 = {1, 2, 3} set2 = {4, 5, 6} set3 = {7, 8, 9} if set1.isdisjoint(set2) and set1.isdisjoint(set3): print("The set is disjoint with all the other sets.") else: print("The set has common elements
with at least one of the other sets.")
In this case, we have three sets: set1, set2, and set3. We use the isdisjoint() method to check if set1 is disjoint from both set2 and set3. If the isdisjoint() method returns True for both
comparisons, it means that set1 has no common elements with either set2 or set3, indicating that it is disjoint from both sets. The output will be:
The set is disjoint with all the other sets.
If set1 has any common elements with either set2 or set3, the isdisjoint() method will return False for at least one comparison, indicating that it is not disjoint from one of the sets. Let see an
Example Code
set1 = {1, 2, 3} set2 = {3, 4, 5} set3 = {7, 8, 9} if set1.isdisjoint(set2) and set1.isdisjoint(set3): print("The set is disjoint with all the other sets.") else: print("The set has common elements
with at least one of the other sets.")
The set has common elements with at least one of the other sets.
III. Python Set isdisjoint() with Different Data Types
In Python, sets can contain elements of different data types. When using Python set isdisjoint() method, it’s essential to understand how it handles sets with different data types.
By default, the isdisjoint() method compares elements based on their values and not their data types. This means that sets with different data types can still be checked for disjointness. The method
will consider the unique values within each set and determine if there are any common elements.
For example, consider the following sets:
set1 = {1, 2, 3, 'apple', 'banana'}
set2 = {3, 4, 5, 'banana', 'cherry'}
Even though set1 contains integers and strings, and set2 contains integers and strings as well, we can still use the isdisjoint() method to check if they are disjoint. The method will compare the
values within each set and determine if there are any common elements, regardless of their data types.
Example Code
set1 = {1, 2, 3, 'apple', 'banana'} set2 = {3, 4, 5, 'banana', 'cherry'} if set1.isdisjoint(set2): print("The sets are disjoint.") else: print("The sets are not disjoint.")
In this case, since set1 and set2 share the element banana, which is common to both sets, the output will be:
The sets are not disjoint.
It’s important to note that Python set isdisjoint() method only checks for disjointness based on values and does not consider the data types explicitly. So, if the values match, the method will
identify the sets as not disjoint, regardless of their data types.
Common Mistakes and Pitfalls to Avoid
When working with the Python set isdisjoint() method, it’s essential to be aware of common mistakes and pitfalls that you should avoid. By understanding these potential issues, you can write more
reliable and error-free code. Here are some common mistakes and how to avoid them:
I. Incorrect Method Usage
One common mistake is mistakenly using the wrong method or misspelling the isdisjoint() method. Ensure that you use the correct method name, isdisjoint(), with the appropriate syntax and parameters.
II. Incorrect Handling of Data Types
Python sets can contain elements of different data types. Ensure that you compare sets that have compatible elements for disjointness checking. If you encounter data types that cannot be compared for
disjointness, consider converting them to compatible types or reevaluating your approach.
III. Misunderstanding Return Value
Python set isdisjoint() method returns a Boolean value (True or False) indicating whether the sets are disjoint or not. Avoid misconceptions such as assuming the method modifies the original sets or
returns a set containing the common elements.
IV. Neglecting Empty Sets
When dealing with empty sets, the isdisjoint() method will always return True since an empty set doesn’t have any common elements with other sets. Keep this in mind when designing your code logic and
handle empty sets accordingly to avoid unexpected behavior.
V. Incorrect Handling of Multiple Sets
If you need to check disjointness between multiple sets, remember that the isdisjoint() method only accepts one set as an argument. To check disjointness between multiple sets, compare each set
separately or use appropriate logic to evaluate their disjointness.
By being mindful of these common mistakes and pitfalls, you can avoid errors and ensure accurate results when using the Python set isdisjoint() method. Always double-check your code, handle data
types correctly, and understand the behavior of the method to make the most of this useful set operation.
Congratulations! You have embarked on a journey to uncover the secrets of the Python set isdisjoint() method.
Now, armed with this comprehensive understanding, it’s time to unleash your creativity and apply the Python set isdisjoint() method to your projects. Harness its power to validate disjointness,
analyze data, and make informed decisions. Embrace the simplicity of the method’s syntax and parameters, and remember to celebrate the beauty of Python’s versatility.
So go forth, fellow Pythonista, and let the isdisjoint() method be your guide in unlocking new dimensions of set manipulation. May your code be free of errors, your logic be sound, and your sets be
as disjoint as you desire. Happy coding! | {"url":"https://pythonhelper.com/python/python-set-isdisjoint-method/","timestamp":"2024-11-06T10:24:03Z","content_type":"text/html","content_length":"94425","record_id":"<urn:uuid:dd6eb036-fd37-4a50-bba0-ca26d3de9718>","cc-path":"CC-MAIN-2024-46/segments/1730477027928.77/warc/CC-MAIN-20241106100950-20241106130950-00054.warc.gz"} |
If three cats catch three mice in three minutes - Daily Quiz and Riddles
If three cats catch three mice in three minutes, how many cats would be needed to catch 100 mice in 100 minutes?
In this playful riddle, we’re on a mission to calculate the feline workforce needed to keep those mischievous mice in check! Let’s embark on this amusing journey of cat-catching calculations.
If three cats can catch three mice in three minutes, it’s evident that each cat catches one mouse in three minutes. Now, the fun part begins as we use this clue to figure out how many cats are
required to handle the grand task of catching 100 mice in 100 minutes.
Since each cat catches one mouse in three minutes, we can represent the number of cats needed as “C.” Now, to catch 100 mice in 100 minutes, we’ll set up a proportion:
Number of Cats / Time (in minutes) = Number of Mice / Time (in minutes)
C / 3 = 100 / 100
Now, let’s solve for the number of cats (C):
C = (100 × 3) / 100
C = 300 / 100
C = 3
Another way to explain this is that if it takes 3 minutes for one cat to catch a mouse, how long would it take just one cat to catch 100 mice? This means 300 minutes. This also means that the cat
could catch 100/3 mice in 300/3 minutes. That is, one cat catches 33.33 mice in 100 minutes.
But our cat is lonely. If we increase the number of cats to 3, that means each of them can catch 33.33 mice in 100 minutes. In other words, they all catch 33.33×3 = 100 cats in 100 minutes!
There you have it! Three clever cats are all we need to work together as a purrfect team, catching 100 mice in 100 minutes. These feline friends are ready to show off their prowess in a delightful
display of quick and cunning moves. Let the cat-catching adventure begin!
Leave a Comment | {"url":"https://quizandriddles.com/if-three-cats-catch-three-mice-in-three-minutes/","timestamp":"2024-11-13T16:27:33Z","content_type":"text/html","content_length":"169107","record_id":"<urn:uuid:6e1faf82-93af-46ee-b6c0-fcc1ae02561d>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00677.warc.gz"} |
Linear Regression with Amazon AWS Machine Learning
Here we show how to use Amazon AWS Machine Learning to do linear regression. In a previous post, we explored using Amazon AWS Machine Learning for Logistic Regression.
To review, linear regression is used to predict some value y given the values x1, x2, x3, …, xn. in other words it finds the coefficients b1, b2, b3, … , bn plus an offset c to yield this formula:
y = b1x1 + b2x2 + b3x3 + …. + c.
It uses the least squares error approach to find this formula. In other words, think of all these values x1, x2, … existing in some N-dimensional space. The line y is the line that minimizes the
distance between the observed and predicted values for all these values. So it is the line that most nearly split right down the middle of the data observed in the training set. Since we know what
that line looks like, we can take any new data, plug those into the formula, and then make a prediction.
As always models are built like this:
• Take an input set of data that you think it correlated. Such as hours of exercise and weight reduction.
• Split that data into a training set and testing set. Amazon does that splitting for you.
• Run the linear regression algorithm to find the formula for y. Amazon picks linear regression based upon the characteristics of the data. It would pick another type of regression or
classification model is we picked a data set that for which that was a better fit.
• Check how accurate the model is by taking the square root of the differences between the observed and predicted values. Amazon actually uses the mean of this difference.
• Then take new data and apply the formula y to make a prediction.
Get Some Data
We will use this data of student test scores from the UCI Machine Learning repository.
I copied this data into Google Sheets here so that you can more easily read it. Plus I show the training data set and the one used for prediction.
You download this data in raw format and upload it to Amazon S3. But first, we have to delete the column headings and change the semicolon (;) separators to commas (,) as shown below. We take the
first 400 rows as our model training data and the last 249 for prediction. Use vi to delete the first from the data as Amazon will not read the schema automatically (Too bad it does not).
vi student-por.csv
sed -i 's/;/,/g' student-por.csv
head -400 student-por.csv > grades400.csv
tail -249 student-por.csv > grades249.csv
Now create a bucket in S3. I called it gradesml. Call yours some different name as it appears bucket names have to be unique across all of S3.
Then upload all 3 files.
Note the https link and make sure the permissions are set to read.
Give read permissions:
Click on Amazon Machine Learning and then Create New Data Source/ML Model. If you have not used ML before it will ask you to sign up. Creating and evaluating models is free. Amazon charges you for
using them to make prediction on a per 1,000 record basis.
Click create new Datasource and ML model.
Fill in the S3 location below. Notice that you do not use the URL. Intead, put the bucket name and file name:
Click verify and Grant Permissions on the screen that pops up next.
Give the data source some name then click through the screens. It fill make up field names (we actually don’t care what names it uses since we know what each column means from the original data set).
It will also determine whether each value is categorical (drawn from a finite set) or just a number. What is important for you to do is to pick the target. That is the dependant value you want it to
predict, i.e., y. From the input data student-por.csv pick G3, as that is the student’s final grade. These grades are from the Portuguese grammar school system and 13 is the highest value.
Below don’t use students-por.csv as the input data. Instead use grades400.csv.
Now Amazon builds the model. This will take a few minutes.
While waiting are create another data set. This is not a model so it will not ask you for a target. Use the grades249.csv file in S3, which we will use in the batch prediction step.
Now the evaluation is done. We can see which one it is from the list above as it says evaluation. Click on it. We explain what it means below.
Amazon shows the RMSE. This is the square root of the sum of the squared differences of the observed and predicted values. We square and then take the square root so that all the numbers are
positive, so they do not cancel each other out. Amazon also uses the mean, meaning average, by multiplying this sum by 1 / n, where n is the sample size.
If the model and the evaluations were the same, this number would be 0. So the closer to o zero we get the more accurate is our model. If the number is large, then the problem is not the algorithm,
it is the data. So we could not pick another algorithm to make it much better. There is really only one algorithm used for LR, finding the least squares error. (There are more esoteric ones.) If MSE
number is large then either the data is not correlated or, more like, most of the data is correlated, but some of it is not and is thus messing up our model. What we would do is drop some columns out
and rebuild out model to get a more accurate model.
What value means the model is good? The model is good when the distribution of errors is a normal distribution, i.e., the bell curve.
Put another way, click Explore Model Performance.
See the histogram above. Numbers to the left of the dotted line are where the predicted values were less than the observed ones. Numbers to the right are where they are higher. If this distribution
were entered on the number 0 then we would have a completely random distribution. That is the idea situation where our errors are distributed randomly. But since it is shifted there is something in
our data that we should leave out. For example, family size might not be correlated to grades.
Above Amazon showed the RMSE baseline. This is what the RMSE would be if we could have an input data set in which there was this perfect distribution of errors.
Also here we see the limitations of doing this kind of analysis in the cloud. If we have written our own program we could have calculated other statistics that showed exactly which column was messing
up our model. Also we could try different algorithms to get rid of the bias caused by outliers, meaning numbers far from the mean that distort the final results.
Run the Prediction
Now that the model is saved, we can use it to make predictions. In other words we want to say given these student characteristics what are their likely final grades going to be.
Select the prediction datasource you created above then select Generate Batch Predictions. Then click through the following screens.
Click review then create ML model.
Here we tell it where to save the results in S3. There it will save several files. The one we are interested in is the one where it calculates the score. It should tack it onto the input data to make
it easier to read. But it does not. So I have pasted it into this spreadsheet for you on the sheet called prediction and added back the column headings. I also then added a column to show how the MSE
mean squared error is calculated.
As you can see, it saves the data in S3 in a folder called predictions.csv. In this case it gave the prediction values in a file with this long name bp-ebhjggKYchO-grades249.csv.gz. You cannot view
that online in S3. So download it showing the URL below and look at it with another tool. In this case I pasted the data into Google Sheets.
Download the data like this:
wget https://s3-eu-west-1.amazonaws.com/gradesml/predictions.csv/batch-prediction/result/bp-ebhjggKYchO-grades249.csv.gz
Here is that the data looks like with the prediction added to the right to make it easy to see. Column AG is the student’s actual grade. AH is the predicted value. AI is the square of the difference.
And then at the bottom is MSE.
Learn ML with our free downloadable guide
This e-book teaches machine learning in the simplest way possible. This book is for managers, programmers, directors – and anyone else who wants to learn machine learning. We start with very basic
stats and algebra and build upon that.
These postings are my own and do not necessarily represent BMC's position, strategies, or opinion.
See an error or have a suggestion? Please let us know by emailing [email protected]. | {"url":"https://www.bmc.com/blogs/linear-regression-with-amazon-aws-machine-learning/?print-posts=pdf","timestamp":"2024-11-14T10:54:36Z","content_type":"text/html","content_length":"154500","record_id":"<urn:uuid:1b31a4b0-d23b-4bb8-96da-fdfe8a87e4a2>","cc-path":"CC-MAIN-2024-46/segments/1730477028558.0/warc/CC-MAIN-20241114094851-20241114124851-00561.warc.gz"} |
Wolfram|Alpha Examples: Step-by-Step Differential Equations
Examples for
Step-by-Step Differential Equations
Separable Equations
See how to solve separable differential equations step by step:
First-Order Exact Equations
Solve exact differential equations step by step:
Use an integrating factor to transform an equation into an exact equation:
Chini-Type Equations
Solve a Riccati equation step by step:
Solve an Abel equation of the first kind with a constant invariant:
Solve a Chini equation with a constant invariant:
Reduction of Order
Reduce to a first-order equation:
Derive the equation of a catenary curve step by step:
Higher-Order Equations
See the steps for solving higher-order differential equations:
First-Order Linear Equations
Solve first-order linear differential equations:
See the steps for using Laplace transforms to solve an ordinary differential equation (ODE):
Bernoulli Equations
Explore the steps to solve Bernoulli equations:
General First-Order Equations
See the steps for solving Clairaut's equation:
Solve d'Alembert's equation:
See how first-order ODEs are solved:
Euler–Cauchy Equations
Solve Euler–Cauchy equations:
First-Order Substitutions
Solve a first-order homogeneous equation through a substitution:
Learn the steps to make general substitutions:
Second-Order Constant-Coefficient Linear Equations
Solve a constant-coefficient linear homogeneous equation:
Explore how to solve inhomogeneous constant-coefficient linear equations:
See the steps for using Laplace transforms to solve an ODE:
General Second-Order Equations
Investigate the steps for solving second-order ODEs: | {"url":"https://www3.wolframalpha.com/examples/pro-features/step-by-step-solutions/step-by-step-differential-equations","timestamp":"2024-11-13T21:05:15Z","content_type":"text/html","content_length":"103664","record_id":"<urn:uuid:514d6988-821d-4dc1-be39-176055326681>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00648.warc.gz"} |
Final Project Information (Fall 2022)
Schedule & Requirements
The goal of the final project is for you to build a complete system that accomplishes a realistic task while ensuring differential privacy. Final projects will be completed in groups of 1-3. The
deliverables for the project will be as follows:
• A project proposal of 1 paragraph, describing:
□ Who is in your group
□ What problem you’re trying to solve
□ A brief description of the approach you plan to use (e.g. what data; what algorithms)
• A project writeup of 1-3 pages, containing:
□ A problem statement
□ A technical description of your solution, with emphasis on anything that makes your solution unique; your description should be sufficient to enable the reader to reproduce your
□ A description of the results - if you’ve evaluated your implementation on real data, describe how well it works
□ If your project is a critique of an existing system, a longer writeup of at least 4-5 pages is expected instead of an implementation
• A project presentation video of about 5 minutes
□ Your presentation should include slides or a demo
□ All group members should present some part of the project
□ Your presentation should cover the same material as your writeup, and be understandable to anyone who has taken this class (i.e. related work that was not covered in class should be explained
in your presentation)
• Your implementation, in whatever form you choose
□ You can use any language or libraries you prefer, but a Python notebook will make grading easier
□ If your project is a critique of an existing system, no implementation is required (unless it helps support your conclusions)
Schedule & Grading
The final project is worth 10% of your final grade. The schedule for final project deliverables, and the contribution of each one to the grade you receive for the final project, are as follows:
Deliverable Due Date Grade Percent Turn In
Project Proposal 11/18/22 by 11:59pm 10% Blackboard
Project Presentation Video 12/12/22 by 11:59pm 30% Blackboard
Implementation + Writeup 12/12/22 at 11:59pm 60% Blackboard
Graduate Students
Graduate students (and undergraduates taking the course for graduate credit) will be expected to develop a more ambitious final project (a more sophisticated algorithm or approach; a larger or more
challenging dataset; or a more detailed analysis of experimental results).
Project Ideas
You’re welcome to work on any project interesting to you, as long as it’s related to data privacy. Suggested examples are below.
Summary and critique of a differential privacy guarantee (suggested option)
• Consider the privacy guarantee given by the system
□ Privacy parameters
□ Unit of privacy
□ Possible failures of the guarantee
□ Possible attacks on the system
• Consider the way the guarantee is communicated
□ Is there missing information?
□ Is the communication misleading?
Analysis of a new dataset, with differential privacy (suggested option)
Sources of data
Analysis ideas
• For a workload of queries on several columns, compare the Laplace and Gaussian mechanisms
• Analyze clipping parameters for some of the columns (e.g. for summation)
• Try to predict one column using the rest of them using differentially private gradient descent
• Generate synthetic data using private marginals
• Try to answer clearly important questions about the data, with differential privacy
□ Census: populations; average incomes; use Laplace and/or Gaussian mechanisms
□ COMPAS: connection between race and score (e.g. following Propublica’s analysis); use Laplace and/or Gaussian mechanisms
□ Kaggle datasets: differentially private gradient descent
□ UCI ML datasets: differentially private gradient descent (look for larger datasets, > 1000 samples)
Implementation of a new differentially private algorithm (challenging)
• Multi-column synthetic data using private marginals
• Hierarchical solutions for answering range queries (e.g. Census “top-down” algorithm)
• Variants of differentially private gradient descent (e.g. minibatching)
• Iterative algorithms for synthetic representations (e.g. MWEM) | {"url":"https://jnear.github.io/cs211-data-privacy/projects","timestamp":"2024-11-08T11:37:37Z","content_type":"text/html","content_length":"9462","record_id":"<urn:uuid:121930eb-be37-4704-85d5-2460fd37f3fe>","cc-path":"CC-MAIN-2024-46/segments/1730477028059.90/warc/CC-MAIN-20241108101914-20241108131914-00019.warc.gz"} |
What does logic have to do with AI/ML for computational science?
Monday, April 19, 2021 - 4:00pm to 5:50pm
Progress in artificial intelligence (AI), including machine learning (ML),
is having a large effect on many scientific fields at the moment, with much more to come.
Most of this effect is from machine learning or "numerical AI",
but I'll argue that the mathematical portions of "symbolic AI"
- logic and computer algebra - have a strong and novel roles to play
that are synergistic to ML. First, applications to complex biological systems
can be formalized in part through the use of dynamical graph grammars.
Graph grammars comprise rewrite rules that locally alter the structure of
a labelled graph. The operator product of two labelled graph
rewrite rules involves finding the general substitution
of the output of one into the input of the next - a form of variable binding
similar to unification in logical inference algorithms. The resulting models
blend aspects of symbolic, logical representation and numerical simulation.
Second, I have proposed an architecture of scientific modeling languages
for complex systems that requires conditionally valid translations of
one high level formal language into another, e.g. to access different
back-end simulation and analyses systems. The obvious toolkit to reach for
is modern interactive theorem verification (ITV) systems e.g. those
based on dependent type theory (historical origins include Russell and Whitehead).
ML is of course being combined with ITV, bidirectionally.
Much work remains to be done, by logical people.
This is part 1 of a 2 part talk. It will cover background on:
Sketch of background knowledge in typed formal languages,
Curry-Howard-Lambek correspondence,
current computerized theorem verification, ML/ITV connections;
scientific modeling languages based on rewrite rules
(including dynamical graph grammars),
with some biological examples. | {"url":"https://www.math.uci.edu/node/37037","timestamp":"2024-11-14T11:55:36Z","content_type":"text/html","content_length":"38884","record_id":"<urn:uuid:0bf926b9-b121-4896-9e03-9477585b56ca>","cc-path":"CC-MAIN-2024-46/segments/1730477028558.0/warc/CC-MAIN-20241114094851-20241114124851-00326.warc.gz"} |
Microsoft Excel Expert MO-201 Test Practice Test Questions, Exam Dumps - ExamCollection
Microsoft MO-201 Microsoft Excel Expert (Excel and Excel 2019) exam dumps vce, practice test questions, study guide & video training course to study and pass quickly and easily. Microsoft MO-201
Microsoft Excel Expert (Excel and Excel 2019) exam dumps & practice test questions and answers. You need avanset vce exam simulator in order to study the Microsoft Excel Expert MO-201 certification
exam dumps & Microsoft Excel Expert MO-201 practice test questions in vce format.
Advanced Formulas & Macros
8. The AND/OR Operators
up the and operator or the and function. The and function performs one or more logical tests and returns a value of true. The and function, like the not function, is commonly used within an if
function XY's if all arguments are true. The and function is made up of at least one logical test, each of which can return true or false. If all the logical tests listed return true, then the and
function will return true. But if even just one of them returns false, then the and function will return false. Let's look at an example. Consider this table with wine tasting scores: So we have the
name of the wine, its variety, the points it received from experts, and its price. Now, I don't really know much about wines, so I want to use this data to determine whether each of these is worth
buying or not. As long as it scores over 85 points and costs less than $15, then I'll buy it. And because both of these must be true, it's an ideal situation to use the and function. Okay, so let's
look at our formula. Start with an if, and then in our logical test, we enter our and function. And the logical test within that is c, which is two greater than 85. So the points have to be more than
85, and D is two less than 15. So the price needs to be under $15. And if both of those conditions are met, then I'll buy it. So? Yes. Otherwise, I'll pass. So no. Now apply that formula and see
which ones are right for me. Well, it looks like I'm going with the Cabernet. 87 points for $10 seems like a good deal to me. A lot of wines, as you can tell here,are actually scored over 85 points,
but they weretoo expensive, which is why the formula returns no. And the Michelle Lynch Merlot is only $9, but it's scored too low, so no for that as well. Okay, so moving on to the or operator or
the or function The or function performs one or more logical tests and returns a value of true if any argument is true. So where the and function needs all the arguments to be true, the or function
just needs one. Now, the syntax is identical; at least one logical test can return true or false. But again, just one of these needs to be true. for the or function to return true. And looking back
at a wine as an example, I'm going to have to be honest and say that I'm not really that strict with my wines. As long as it's under $15, then I'll buy it regardless of the score. Also, if wine is
over 90 points, I think it's worth buying regardless of the price. So let's see what our formula looks like. If or and, the logical tests are now c two greater than 90, for a total of more than 90
points, and D two less than 15. under $15. And as long as either of these conditions is met, then the value of true is yes, otherwise the value of false is no. And let's look at the results. Well, we
have a lot of options. The only exception is the Jaffline Pinot Noir, which did not meet the point or price requirements, but the rest are all going on my shelf. All right, so now let's move this
over to our Airbnb and B data. Okay, here in Excel, we're still in the Places tab. And for those who are familiar with Airbnb, you'll know that some places are marked as a rare find. And these are
the places that have a rating of four, eight, or higher and more than 100 reviews. I'd like to identify the places that are rare finds in New York City. So let's add a column after the number of
reviews, and we'll call it "Rare Find." Now, since there are two conditions that both need to be met, we'll need to use an and function. So we'll start with an if function, open that, and then add an
and function in the logical test. So we'll open that, and now what we need to think is, "Okay, so what are the conditions that need to be met?" Well, the rating needs to be greater than or equal to
4.8, and the number of reviews needs to be greater than 100. So we can close that and come back to the value of true. And that is what we need to return if both of those tests return true. So, yes,
it is a rare find. And for the value of false, what would just return no. So let's close this. Let's apply that downward and look at our results. So these first two places are not rare finds. And
even if they both have 473 and 123 reviews, respectively, the rating doesn't go up to 4.8. So that's why they are not. Our third one is a rare find. It has a very high rating of 4.99 and 233 reviews.
And let's see if we can find one with a rating over 4.9. Here we go. and less than 100 reviews. So beautiful are the attic, bedroom, and Chelsea. Again, a very high rating, 4.99. But since it hasn't
gotten up to 100 reviews yet, then it's still not a rare find. In fact, if we filter Rare Find by Yes, you'll see that all the ratings will be four, eight, or higher, and they all have more than 100
reviews. So let's clear that filter. And I also want to focus on the room type. Now, as it stands, there are three types of rooms: an entire place, a private room, and a shared room. And while this
is already a category, we can take it one level higher. I'd like to know if each place has any shared spaces or not. So, as you know, or even if you don't have a private room, you have your own room,
but you do share some of the other spaces in the place with other people. And in a shared room, all the spaces in the place can be shared with other guests or the hosts. So in summary, private rooms
and shared rooms both have shared spaces. So we can use that and add a new column called Shared Spaces. We'll begin with the anif function. Again, open that. And in the logical test, what do we need?
Well, we know that if the room type is a private room or shared room, then it will have shared spaces. So that means we can use an or function. Open that. So or D two is equal to private room. Now
here, since we're working with text strings, you need to make sure you're writing this exactly as we'll find it here. So private with capital P, room with a lowercaser or D two is equal to shared
room. Again, first letter capital and shared but not in room. We'll close the door. So if it's either a private room or a shared room, then our value is true. Yes, it will have a shared space. And if
it's neither of those, then no, it won't have a shared space. So close, press Enter, apply it down, and everything is fine. And again, a quick filter to check So yes, we're getting private roomand
shared room, no shared spaces. Then we just have the entire place. So let's clear our filter. And there you have it. We've now used the and and or functions to identify the places that are rare finds
and those that have shared spaces. Also, as a quick disclaimer, the logic we used for the rare find isn't actually how Airbnb determines this, but it was just used for the purpose of this lecture and
this course.
9. The SWITCH Function
In this lecture, we will go over the switch function. And this is an interesting one in Excel because even though it's not used very frequently, there are certain specific scenarios in which it's the
best solution available, and we'll go over those in a bit. But to start, the switch function evaluates an expression against a list of values and returns the result for the first matching value. So
syntax-wise, it has three required arguments and a few optional arguments. And these are actually pretty self-explanatory, even if they still don't make the function very understandable at first. So
the first argument expression is simply an expression or function to be evaluated for the value; you need to enter a value to compare to the expression result. And for the result, you tell Excel what
to return in case the value listed matches the expression result. Now, after the first required value and result pair, you can either enter a default result if no value matches. So think of this as a
value if it is false, or you can add additional value and result pairs. Now, I know the wording is a little tricky, so it's hard to get a full grasp of how this works from the syntax, but I think
looking at an example will help solidify the concept. So consider this table of flights. We have the date of the flight, the airline, the flight number, and the origin and destination airports. And
we want to know the weekdays on which each flight is available. Well, this is a great use case for the switch function. So if we were to write a formula in Column E, it would look like this switch.
And then the expression we'll use is the "weekday" function, which is representing the date column, or "sell a two." And don't worry about the weekday function right now. We're going to cover this
individually later in the course. But all you need to know for now is that it returns the number representing the day of the week for a given date. So from one being Sunday to seven being Saturday,
But in this case, we don't just want to return the number; we want to return the abbreviated name of the weekday. And this is where the switch function kicks in. So we have the expression, and now we
just need to tell Excel what to return for each of the values that the expression can take. So if the weekday function returns one, well, we want the result to be "sun" for Sunday. If it returns two,
it will be Monday, three, Tuesday, four, Wednesday, and so on every two weeks until we reach seven on Saturday. In this case, we don't need to enter a default value to return here, as the weekday
function will only return numbers from one to seven. And we've got them all covered. Now let's see what happens if we apply this formula. So, January 1, 2015 was a Thursday, and since the dates are
all consecutive, it makes sense that the rest are Friday, Saturday, and Sunday. So that's all set. Now, a quick pro tip: switch instead of ifs when your comparisons only require exact matches. And
this is related to the comment I made in the beginning of the lecture, where I mentioned that there are scenarios where switching simply works best. The most common of these is the weekday example
that we just went over, using the same logic for the names of the month with the month function. So to highlight the protip, we could use an Ifs function to do what we just did with the weekdays and
simply say Ifs. Open parentheses. Weekday a two equals one, then return Sunday. Weekday a two equals two, then return Monday. Weekday a two equals three, then return Tuesday. But you can already see
how it starts to getlong and repetitive, where switch keeps it short and sweet. So with that in mind, let's jump over to Excel and use the switch function in our course project so we have a better
idea of how it works. Okay, here in Excel, we're back in the Places tab for our course project workbook, and we'll be focusing on the rating field. Ratings on AirBnB are now assigned on a scale of
zero to five, with five being the highest. And if we look at our data for New York City, we'll see that it ranges from 2.5 to five. And what we'll do for this exercise is add a Rating Class column
that ranks each place by rating it as either excellent, good, okay, or bad. So think of the rating classes as having stars. Any rating that rounds to five stars is going to be excellent. If it rounds
to four, it's good. If it rounds to three, it's okay. and anything beneath that is bad. So let's add our column, call it Rating Class, and just expand the width by double clicking here. Now, the
first step we need to take is to figure out how to round these ratings since right now they have two decimal places. And luckily, Excel has a round function just for this. So start by writing that,
opening it up, and you'll see that it has two arguments: the number that we want to round, which is the rating, and the number of digits to which we want to round it, which in this case is zero,
since we need whole numbers. So we can close that out, and we'll see what results we're getting. Now, Excel kept the formatting of the two decimal places, but as you can see, we're just getting whole
numbers of three, four, and five, which is exactly what we wanted. But we can't end here because we don't want numbers. We want excellent instead of five, good instead of four, OK instead of three,
and bad for anything else. So if we were to ignore what we just learned about the switch function, we actually still have enough knowledge with what we've learned so far to turn these numbers into
the rating classes that we want. So let's try using an Ifs function. So we'll delete what we have now and replace it with Ifs. And our first logical test is basically going to be "if." The rounded
version of this rating is equal to five, right? So we'll write the round function again. So we want to round the rating down to zero decimal points. And if that is equal to five, then our value of
true will be equal to excellent. And our second logical test is really going to be the same thing. But we want to see if the rounded version of this rating is equal to four. So we can actually just
copy all this and paste it here at the end. So if this is equal to four, then we want it to be good. If this is equal to three, then we want it to be okay. And finally, if this is less than three, we
want it to be bad. Close that out, press Enter, and we get good since 4.38 rounds down to four, which is good. And then we get excellent for the ones that rounded up to five and good again for the
ones that rounded down to four. So far, so perfect. and that does get the job done. But this isn't really a very elegant formula. So let's try again using the ever-confusing switch function. So
again, we're going to delete this out.I'm going to start writing our switch function. Open that up. And first is the expression that we're going to evaluate, which by now we should know is the round
function I 20.Close that out. So we have our expression. Now come the value-and-result pairs. So if the result of the round function is five, then we want Excel to return excellent. If the answer is
4, we want to return the goods. If the answer is three, we want to say that everything is fine. And for anything else, we can actually use that sort of value. If there is an error, we can simply
return bad. Close that out and press enter. And before we apply this, let's actually take a look at the difference between the two formulas we wrote. So this is the switch andthis is using the Ifs
approach. Now, I don't know about you, but I'm definitely feeling better with the application of the switch function here. And let's apply this down there and get the same exact results. And that's
it. Again, this is definitely a function that we need to reserve for very particular cases, but it's still important to know how it works and how to use it.
10. The COUNTIF/SUMIF/AVERAGEIF Functions
Moving on to conditional functions, we'll be talking about count if, sum if, and average if. The count if, sum if, and average if functions calculate a count, sum, or average based on specific
criteria. Syntax-wise, they're very similar. Count if it has two arguments, range and criteria, and sum if it does, and average if it has range and criteria plus the sum or average range. Now, for
the range argument, you need to specify what cells need to match the criteria. And for the criteria, you need to specify what condition those cells need to meet. And this is all the counter function
needs, as it will simply count the cells in the range that match the criteria. Finally, the sum, range, and average range are where you tell Excel the range of cells you want to add to or get the
average from. And if left blank, Excel will just consider the cells and the range argument as the sum and average range. But as always, the best way to fully understand this is through an example.
So, going back to our wine tasting data, let's consider this list of wines, with the wine name, variety, and price. And in this scenario, we're the owners of a wine store and want to figure out by
wine variety how many wines we have, their accumulated prices, and their average price. So if we were to do this for the tempanya variety, we could use the account if, sumif, and average if functions
to do this. To calculate the number of Tempano wines, we can use COUNTIF, and our function will look like this. So count if our range is B 2 to B 14, which is our variety column, and the criteria is
Tempanyo, since we want to count the wines of the Tempanyo variety. So if we look at our data, you'll see that we have one, two, or three tempanillo wines. And so the result is, in fact, three. To
calculate the total price of the temporary lines we have, we can use SUMIF, and our function would look like this. So, SUMIF, our range remains B2 to B14 because we require variety to match the
criteria, and the criteria remain Temperino, but the sum range is C2 to C14 because this is where my prices reside. And if we look back at our data, we can see that the prices for our three Tempanya
wines are $27, $26 and $19, which added together give us $72. Finally, to calculate the average price of the seasonal wines we have, we can use average if our function looks like this. So average if
has the same range criteria and average range as our SUMIF function. But now, instead of adding 27, 26, and 19, it will calculate its average, which in this case is $24. Now, pro tip when using these
functions: if you use greater than or less than in the criteria argument, you need to add quotation marks around the entire thing. For example, greater than 100 would have to look like this, and it's
just one of those quirks that Excel has, but it is important to know. Now let's head over to Excel and start using these functions here in Excel. We're still using the Excel Expert Course project
file, but now we'll be moving on to the Host tab. And as you recall, this tab contains more information about the Airbnb hosts of places listed in New York City, like the date they joined, their
response time, or whether they're a super host or not. And here to the right, we have a template for the Ahoist dashboard that will actually start filling out right now. So let's enter a host ID
here. We can start with one, and what we want to do is calculate the number of places they have listed in New York City, their total reviews, and their average rating. As you may have guessed, we'll
be using the accountif, sum if, and average if functions to do this. So let's start with the places. And if we jump back to the Places tab for a second, we can see that each row represents one place
and that we have the Host ID for each place. Therefore, we can just count the number of times that the host ID of one is in this table, and it will represent the number of places that host one has.
So let's go back to our host tab and write our formula. So count on it. Our range is the number of cells that need to match the criteria. So, in the Places tab, we can select the Host ID column
entirely by clicking on it, and then comment over to our criteria, which is going to be the Host ID field in our Host dashboard. So cell L-2 can close that out, press Enter, and it looks like Host
One has one place listed in New York City. Let's now get the number of reviews for that place. And since we want the total number of reviews for any number of places that a host may have, we need to
use some here. So some of our ranges and criteria will be the same. So we have the Host ID column in the Places tab and the Host field in our dashboard, and our sum range is the Number of reviews
column in our Places tab, because we want to sum the number of views for the places where the Host ID matches the host ID in our dashboard, so we can close that, press Enter, and we get 32 reviews.
And finally, let's get the average rating. As you might expect, we must use average if here. So, same range, same criteria, and the average range here will be the rating column in our Places tab,
correct? So we select that, close the formula, and press Enter, and we get an average rating of 491. So awesome. Now let's see what happens when we change the host ID so we can create that too. And
we'll see that the totals change and now represent the places, reviews, and ratings for this particular host. And we can make it into anything we want. We can make the three, four, or even fifteen.
Okay, so here it looks like our host ID of 15, or our host 15, has three total places, 78 total reviews, and 451 as the average rating. So let's see if we can prove these numbers are correct. Let's
go back to our Places tab and filter our table. So we're only looking at the places for host number 15, so select that and press OK. And as you can see, he does have three places. The total number of
reviews is 78, and the average rating is 4.51. So perfect. So, as a pro tip, when taking the exam, using these types of filters is a great way to double-check that the formulas you're writing are
returning the correct results so we can clear that filter. from now, go back to our ID. And there we go. an example of how and when to use count if, some if, and average if.
11. COUNTIFS/SUMIFS/AVERAGEIFS for Multiple Criteria
Next up are the COUNTIFS, SOMEIFS, and AVERAGE IF functions. And as you probably guessed, the count if, someif, and average if functions calculate the count, sum, or average based on multiple
criteria, and their syntax is almost identical to their younger brothers. But the sum range and average range are moved to the beginning of the function, which I actually prefer. So again, the sum
and average ranges contain the values that you want. The sum or average of the criteria range contains the cells that you want to match the criteria, and the criteria is the condition that the range
needs to meet. And finally, you can add additional pairs of criteria, ranges, and criteria if you have more than one condition. Now, going back to our wine store example, we have the wine IDs, their
varieties, their points, and their prices. And now we want to see how many tempanillo wines we have, but those also have at least 85 points. We must use count ifs because we have two conditions: the
variety must be tempanyo and the points must be greater than or equal to 85. So our formula would look like this: COUNTIFS. Our first criteria range is from B 2 to B 14. As a result, both the variety
and the criteria are tempeh. Our second criteria range is from C 2 to C 14. So the points and the criteria are greater than or equal to 85. And for our pro tip in the last lecture, notice that we
wrapped this in quotation marks for it to work. So looking at our data, we still have three temporary and two wines, but only two of those have scores greater than or equal to 85. So our result is
two. Now, for the total price of these, we need to use SUMIFS. So our formula would look like this: SUMIF Our sum range is D 2 to D 14, which is our price column. And then the criteria range and the
criteria are the same as in the example above. So the ride needs to be temporary, and the points need to be greater than or equal to 85. And looking at our data again, we can see that the prices are
27 and 19, which add up to $46. And for the average price, we'll use averageifs as follows, which is using the same information as the SUMIF function but returning the average of 27 and 19, which is
$23. Okay, now that we have the hang of these, let's jump to Excel and start practising here in Excel. We're back in the Host tab on our course project workbook, and we're going to continue to
populate our host dashboard. And what we want to do now is obtain the reviews and ratings of the places, but broken down by room type. So what we're essentially doing is adding new criteria to our
original formulas. So the host ID in the Places tab will still need to match the host ID selected here. But the room type in the Places tab will also need to match the room type in each of these
columns. Therefore, we'll need to use COUNTIFS, SUMIFS, and average. If so, let's start with COUNTIFS. And our first criteria here is the host ID. So places, host ID, and our criteria are going to be
the host ID in our dashboard, and we'll comma over to our second criteria, and that's going to be the room type column. So we have column D in our places, and the criteria there is going to be the
room type in our dashboard. Now, it's being blocked, but we can just use an arrow up to select it, as you can sort of see by the outline here. So we can close that out, press Enter, and we'll see
that host number 15, even though they have three total places, only one of those is an entire place. Let's drag this across to see what else is there. Something's wrong here. We know that hostid 15
has three total places, but we're only getting one here. So what happened? Well, let's look at our formulas. So starting with our original one, we can use F two, and it looks like everything's good.
I mean, the host ID is selected here, the room type is correctly selected here, and our criteria ranges are Places, column B, and column D, which are the host ID and the room type. And if we go over
to the next one now, we'll start to see what happened. So a quick reminder of the importance of reference types here As you can see, since we didn't fix this reference in cell L 2, as we dragged it
to the right, it moved over to cell M 2. And the same thing happened with our criteria ranges. So instead of being column B, it moved over to column C, and instead of being column D, it moved over to
column E in our Places. So let's press Escape to get out of that. And we're going to modify our original formula and fix what we need to fix. So the host IDs will always live in column B of our
Places tab, so we can fix that with F four. And our host ID is always going to be in cell L2 in our dashboard, so we can fix that as well. Now, our room type will always be in column D of our Places
tab, so we can fix that, but the room type in our dashboard will need to be moved to the right because that's where the rest is, so we're fine with leaving that relative. Now press Enter again and
let's try this out one more time. And there we go. So, one entire place, two private rooms, three total places. So perfect. Okay, now for some ifs, so, for the total reviews, our sum range is going
to be the number of reviews column in our Places tab, and we're going to go ahead and use f4 to fix that to avoid our past mistakes. Our first criteria range is going to be our host ID. Fix that as
well. And our criteria is the host ID on our dashboard, which again, we're going to use our four to fix. Now, for the second criteria range, room type, fix that to column D. And our second criterion
will be the room type, which we'll leave relatively close, drag over, and perfect. And finally, for the average IFs, our average range is going to be our rating. So column I fix that, and then we're
going to go through the same process. So host ID "hostid" here, and notice I'm fixing all these references. And then the room type and theroom type here, leaving relative close that. So take the 424
average for our entire facility, apply that, and it appears that we're getting a div zero error here for the shared room. And let's think about that for a second. So what we're doing here for the
average is summing the ratings for each room type and then dividing by the number of places for that room type. But since the number of shared rooms for host15 is zero, well, then we're dividing by
zero, which is why we get this error. So to fix this, what we can do is wrap our entire formula in an if-error function. So let's go back to our original one, and I'm actually going to select here
before the average. I'm going to write my if-error function. And this tells Excel what to return in the event of an error, which we specify in the second argument here, which we'll go over later. And
in this case, I won't say return zero because saying an average rating of zero is incorrect; it's actually null. So what we can do is just return a dash, which will need to wrap in quotation marks
since it's a text string, so we can close that. As you can see, we're getting the same result for these, but as we drag it across the final time, we'll see that instead of the error, we're getting
that dash that we wanted. So we're just getting a cleaner dashboard. And as you can see, as we move our hostID around, well, our dashboard is going to update.
12. The MINIFS & MAXIFS Functions
Next up are the MAXIFS and MINIFS functions, which, by now, I'm sure you can guess what they do. But a quick note, however, these are onlyavailable on Excel 2019 or three six five. So those of you
following along on earlier versions of Excel won't have access to them. Not to worry, though, since they work the exact same way as the sumif and averageif functions, as you'll see. So the MAXIFS and
MinIFS functions return the maximum or minimum values based on multiple criteria. Now, their syntax will look very familiar. The Max and Min ranges are where we tell Excel what values we want. The
maximum or minimum value of the criteria range represents the cells that need to match the criteria, and the criteria is the condition the cells need to meet. And we can also include multiple
criteria and ranges. Now we'll be looking at the same wine store example. And now we want our highest-priced tempanawine that has at least 85 points. So our formula would look as follows MAXIFS,
where the maximum range is D 2 to D 14, which is the price column since we want the highest price. And the first range of our criteria is B 2 to B 14. As a result, both the variety and the criteria
are tempanyu. The second criteria range is from C 2 to C 14. So the points and the criteria are greater than or equal to 85, which again needs to be wrapped in quotation marks. So looking at the
data, we have the three tempanillos, but only two of these have at least 85 points. And by looking at the prices, the highest is 27. So our formula returns $27. And if we wanted the lowest price
company you'llwine with at least 85 points, well, we'd useMINIFS, and in this case, the formula is exactlythe same, but will return the minimum value inour price column, in this case, $19. So let's
go to our course project in Excel and put these functions to good use. Here in Excel, we're still working in the hosttab of the course project workbook, and we're going to continue to round out our
host dashboard with the first review and last review fields. So we have host 29 right now, and what we're going to do is actually jump to our Places tab and filter it out by the places for that host.
Now, to filter, we can either select manually or we can use the search bar here, type 29, clear the results, and select 29. So here we have the three places for this host. And what we want for the
first review field is to pull the earliest or the minimum date from our first review column, in this case, August 5, 2016. And for the last review, we want the latest date or the maximum date from
our last review column. In this case, it looks like March 1, 2020. So with that, let's clear this and write our MAXIFS and MINIFS formulas. So for the first review, like we mentioned, we need the
minimum, so MINIFS the Min range being the first review column, and our criteria range is the host ID, and we need that to match our criteria of the host I selected in the dashboard. Close that out
on August 5, 2016. Perfect. Now, for the MAXIFS, our max range is going to be the last review column in our places. The criteria range again is going to be the host ID, and the criteria is the host
ID for our dashboard. Close that out. March 1, 2020 Perfect. And again, as we continue to move the hostid around, you'll see that these will update, and that's it. After having worked with the
COUNTIF some ifs and average ifs functions, we've completed the logical operations skill of this objective domain. And now let's move on to date and time functions. I hope you're ready.
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Spatial Objects & Points in R - AirGency - Webflow Ecommerce website template
Spatial Analytics
December 7, 2023
Spatial Objects & Points in R
In my last article, I explained the fundamentals of Spatial Data Analysis and different types of Spatial Data including point, line, polygon and grid. If you want a quick recap, you can find it here
Hello World: Introducing Spatial Data
I ended the last article with the promise of covering Points in my next article. In this article I will make a deep dive in Spatial Objects and Spatial Points in R.
Points are one of the important data models in R. However, before we dive into Spatial Points, I want to give brief overview of Spatial Objects in R.
Spatial Objects in R
R developers wrote a special package sp which extend base R classes and methods for spatial data. To get things started download sp by following command:
Next, we attach the package by:
I will call the Spatial class using getClass method to get the complete definition of a S4 class in R. This includes the slot names and types of their content:
Class "Spatial" [package "sp"]
Name: bbox proj4string
Class: matrix CRS
Known Subclasses:
Class "SpatialPoints", directly
Class "SpatialMultiPoints", directly
Class "SpatialGrid", directly
Class "SpatialLines", directly
Class "SpatialPolygons", directly
Class "SpatialPointsDataFrame", by class "SpatialPoints", distance 2
Class "SpatialPixels", by class "SpatialPoints", distance 2
Class "SpatialMultiPointsDataFrame", by class "SpatialMultiPoints", distance 2
Class "SpatialGridDataFrame", by class "SpatialGrid", distance 2
Class "SpatialLinesDataFrame", by class "SpatialLines", distance 2
Class "SpatialPixelsDataFrame", by class "SpatialPoints", distance 3
Class "SpatialPolygonsDataFrame", by class "SpatialPolygons", distance 2
This returns us two important information about this class.
1. Slots: a named component of the object that is accessed using the specialized sub-setting operator @ (pronounced at). In our case we see that Spatial class has two named components :
1. Matrix
2. CRS (Cordinate Reference System)
2. Known Sub classes: These are classes that inherit one or more language entities from another class/classes. In our case SpatialPoints class inherits language entities from Spatial class. In other
words, there is parent-child relationship between Spatial class and SpatialPoints class.
Spatial Points
From our mathematical knowledge, we know that point is a pair of numbers in (x,y) defined over a known region. According to (Herring 2011), it is a 0 - dimensional geometric object and represents a
single location in a coordinate space.
The understanding of Earth before the introduction of World Geodetic System 1984 was sphere. But later Geodesists represented the globe more accurately by ellipsoid model (Roger S. Bivand 2013).
Today Global Positioning System (GPS) uses the World Geodetic System (WGS84) as its reference coordinate system. More details about about WGS84 can be found here.
We now see how SpatialPoints class extends Spatial class. By using getClass(), we can see the slot names and type of content.
Class "SpatialPoints" [package "sp"]
Name: coords bbox proj4string
Class: matrix matrix CRS
Extends: "Spatial"
Known Subclasses:
Class "SpatialPointsDataFrame", directly
Class "SpatialPixels", directly
Class "SpatialPixelsDataFrame", by class "SpatialPixels", distance 2
We see that SpatialPoints extends Spatial class by adding a coords slot, into which a matrix of point cordinates can be inserted. We also see three other classes extended by SpatialPoints:
1. SpatialPointsDataFrame
2. SpatialPixels
3. SpatialPixelsDataFrame
To understand the methods and functions provided by SpaitalPoints, I read a data file with the positions of CRAN mirrors across the world in 2005. We first pull the data in R:
df <- read.table("http://www.asdar-book.org/datasets/CRAN051001a.txt", header = TRUE)
Then we see its first six rows:
place north east loc long lat
1 Brisbane 27d28'S 153d02'E Australia 153.03333 -27.46667
2 Melbourne 37d49'S 144d58'E Australia 144.96667 -37.81667
3 Wien 48d13'N 16d20'E Austria 16.33333 48.21667
4 Curitiba 25d25'S 49d16'W Brazil -49.26667 -25.41667
5 Vi\xe7oza 20d45'S 42d52'W Brazil -42.86667 -20.75000
6 Rio de Janeiro 22d54'S 43d12'W Brazil -43.20000 -22.90000
We extract the two columns with the longitude and latitude values into a matrix, and use str to view:
# make a latitude and longitude matrixcran_mat <- cbind(df$long, df$lat)
# assign row names from 1 to number of rowsrow.names(cran_mat) <- 1:nrow(cran_mat)
# view the matrix structurestr(cran_mat)
num [1:54, 1:2] 153 145 16.3 -49.3 -42.9 ...
- attr(*, "dimnames")=List of 2
..$ : chr [1:54] "1" "2" "3" "4" ...
..$ : NULL
Before moving on it is necessary to introduce the coordinate reference system (CRS) class briefly. We saw that Spatial class had two slots. One of them was Coordinate Reference System (CRS). Let us
see the slot names and components.
Class "CRS" [package "sp"]
Name: projargs
Class: character
We see that the class has a character string as its only slot value. The character string may be a missing value but if it is not missing it should be PROJ.4-format string describing the projection.
For geographical coordinates, the simplest of such string is “+proj=longlat”.
We now see the implementation in our CRAN data set.
# make a CRS object in longlat and World Geodetic System(WGS84)
CRS_obj <- CRS(projargs = "+proj=longlat +ellps=WGS84")
# check summary of the object
Length Class Mode
1 CRS S4
At this point you must be wondering what is PROJ.4 format?
PROJ is a generic coordinate transformation software that transforms geospatial coordinates from one coordinate reference system (CRS) to another. This includes cartographic projections as well as
geodetic transformations.
To define a Coordinate Reference System in sp package we use CRS object which defines the coordinate reference systems we use.
Finally we create SpatialPoints object
# make a spatial points class by long/lat matrix and proj4string
cran_sp <- SpatialPoints(coords = cran_mat, proj4string = CRS_obj)
# summary of the cran_sp object
Object of class SpatialPoints
min max
coords.x1 -122.95000 153.0333
coords.x2 -37.81667 57.0500
Is projected: FALSE
proj4string : [+proj=longlat +ellps=WGS84]
Number of points: 54
Spatial Points objects may have more than two dimensions, but plot methods for the class use only the first two.
Ending Remarks
In this article, we got an overview of Spatial and Spatial Points class. But we are only half way done. In next article, I will explain the methods and data frames for spatial points to do some data
Herring, J. R. 2011. OpenGIS® Implementation Standard for Geographic Information - Simple Feature Access - Part 1: Common Architecture. Technical Report 1.2.1. Open Geospatial Consortium Inc.
Roger S. Bivand, Virgilio Gómez-Rubio, Edzer Pebesma. 2013. Applied Spatial Data Analysis with r. Springer New York, NY.
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CDQ convolution (online FFT) generalization with Newton method - Codeforces
Hi everyone!
This is yet another blog that I had drafted for quite some time, but was reluctant to publish. I decided to dig it up and complete to a more or less comprehensive state for the $300 contest.
Essentially, the blog tells how to combine CDQ technique for relaxed polynomial multiplication ("online FFT") with linearization technique from Newton method (similar approach is used in the first
example of the ODE blog post by jqdai0815), so that the functions that typically require Newton's method can be computed online as well. I will try to briefly cover the general idea of "online FFT"
too and provide some examples, in case you're not well familiar with it. That being said...
Consider the following setting:
There is a differentiable function $$$F(x)$$$ such that $$$F(0)=0$$$ and a polynomial $$$f(x)$$$.
You want to compute first $$$n$$$ coefficients of a formal power series $$$g(x)$$$ such that $$$g(x) = F(f(x))$$$. However, the series $$$f(x)$$$ is not known in advance. Instead, the $$$k$$$-th
coefficient of $$$f(x)$$$ is given to us after we compute the $$$k$$$-th coefficient of $$$g(x)$$$.
Looks familiar? No? Ok, let's make a brief detour first.
CDQ convolution
General idea of CDQ technique is described in the following simple scheme: To compute something on the $$$[l,r)$$$ interval,
1. Compute it on $$$[l, m)$$$ for $$$m=\frac{l+r}{2}$$$,
2. Compute the influence of $$$[l, m)$$$ onto $$$[m,r)$$$,
3. Compute everything else in $$$[m, r)$$$ recursively,
4. Merge the results.
This approach is very versatile, and In convolution context, it's commonly known as "online FFT". It has the following typical formulation:
Standard formulation
We want to compute a sequence $$$c_0, c_1, \dots, c_n$$$, such that
$$$ c_k = \sum\limits_{i+j=k-1} a_i b_j, $$$
where $$$a_0, a_1, \dots, a_n$$$ and $$$b_0, b_1, \dots, b_n$$$ are not known in advance, but $$$a_k$$$ and $$$b_k$$$ are revealed to us after we compute $$$c_k$$$.
In a more polynomial-like manner, we may formulate it as
$$$ C(x) = x A(x) B(x), $$$
where the $$$k$$$-th coefficient of the polynomials $$$A(x)$$$ and $$$B(x)$$$ is revealed to us as we compute the $$$k$$$-th coefficient of $$$C(x)$$$.
Polynomial exponent. Assume you want to compute $$$Q(x)=e^{P(x)}$$$ for a given $$$P(x)$$$.
Jetpack. You want to get from $$$(0, 0)$$$ to $$$(n, 0)$$$. On each step, you increase you $$$x$$$-coordinate by $$$1$$$, and your $$$y$$$ coordinate changes to $$$y+1$$$ if you use the jetpack or to
$$$\max(0, y-1)$$$ if you don't. At the same time, you can only have $$$y > 0$$$ for at most $$$2k$$$ steps. How many ways are there to get to $$$(n, 0)$$$ under the given constraints?
Gennady Korotkevich Contest 5 — Bin. Find the number of full binary trees with $$$n$$$ leaves such that for every vertex with two children, the number of leaves in its left sub-tree doesn’t exceed
the number of leaves in its right sub-tree by more than $$$k$$$.
Okay, how to solve this? Let's recall the convolution formula
$$$ c_k = \sum\limits_{i+j=k-1} a_i b_j. $$$
Assume that we want to compute $$$c_k$$$ for $$$k$$$ in $$$[l, r)$$$ and we already know values of $$$c_k$$$, $$$a_i$$$ and $$$b_j$$$ for $$$i, j, k \in [l, m)$$$. Consider the contribution to
$$$c_k$$$ for $$$k \in [m, r)$$$ of $$$(i, j)$$$ pairs such that both $$$i$$$ and $$$j$$$ are below $$$m$$$ and at least one of them is above or equal to $$$l$$$. For each such $$$a_i$$$, values of
interest for $$$j$$$ range from $$$0$$$ to $$$\min(m, r-l)$$$.
Correspondingly, for each $$$b_j$$$, values of interest for $$$i$$$ range from $$$0$$$ to $$$\min(l, r-l)$$$. In both cases, the contribution may be computed with an appropriate convolution of
polynomials of size at most $$$r-l$$$. Note that in the second case we used $$$\min(l, r-l)$$$ instead of $$$\min(m, r-l)$$$ to avoid double-counting pairs in which both $$$i$$$ and $$$j$$$ are
$$$l$$$ or above.
We make two recursive calls and use extra $$$O(n \log n)$$$ time to consolidate them, making it for overall $$$O(n \log^2 n)$$$ time.
Now back to the beginning. The examples above typically expect that the right-hand side is formulated in a way that is kind of similar to convolutions. But there are a lot of functions, for which
it's not really possible to do so. Try the following ones in a similar setting (i.e. the coefficients of $$$f(x)$$$ are given after the coefficients of $$$g(x)$$$ are computed):
$$$\begin{gather} g(x) =& x \cdot e^{f(x)} \\ \\ g(x) =& x \cdot \log \frac{1}{1-f(x)} \\ \\ g(x) =& x \cdot \frac{1}{1-f(x)} \end{gather}$$$
Those are the functions that are quite typical in formal power series (for their interpretation, see here). You may probably rephrase them in a convolution-like manner, so that CDQ is applicable, but
you would need to do something ad hoc for each individual function. It doesn't seem very convenient, so it only makes sense to try and find some generic enough approach. What is a generic formulation
$$$ g(x) = F(f(x)). $$$
And you may note that what all these functions $$$F(\dots)$$$ have in common is that they're differentiable. Let's use it in a similar way to what we do in Newton's method and rewrite the right hand
side in the following way:
$$$ g(x) = F(f_0) + F'(f_0) (f - f_0) + O((f-f_0)^2). $$$
This formula generally works for any $$$f_0$$$. In particular, let $$$f_0$$$ be first $$$n$$$ coefficients of $$$f$$$, then $$$(f-f_0)^2$$$ is divisible by $$$x^{2n}$$$. In other words,
$$$ g(x) \equiv F(f_0) + F'(f_0) (f - f_0) \pmod{x^{2n}}. $$$
In this formula, we still do not know $$$f(x)$$$ completely. But what's important is that it is no longer an argument of some generic function $$$F(\dots)$$$, so the right-hand side is now a linear
function of $$$f(x)$$$! This is exactly the formulation for which we learned to apply CDQ convolution above, that is $$$g(x) = A(x) f(x) + B(x)$$$, where
$$$\begin{gather} A(x) =& F'(f_0), \\ \\ B(x) =& F(f_0) - f_0 F'(f_0) \end{gather}$$$
are specific constant polynomials in this context. This sub-problem is solvable in $$$O(n \log^2 n)$$$ with the standard CDQ technique, and since it allows us to double the number of known
coefficients at each step, the overall running time is also $$$O(n \log^2 n)$$$, assuming that we're able to compute $$$F(\dots)$$$ and $$$F'(\dots)$$$ in $$$O(n \log^2 n)$$$ as well.
22 months ago, # |
» ← Rev. 2 → +43
nor Thanks for the nice blog! For the special case of online multiplication (which is called relaxed multiplication in literature), there is this paper (by one of the authors of the 2021 $$$O(n \log
n)$$$ integer multiplication algorithm) that gives asymptotically faster algorithms compared to the one mentioned in this blog. According to the paper, the techniques mentioned also have
applications to computation of some formal power series which are incidentally similar to the ones discussed in this blog.
• 22 months ago, # ^ |
» That's an interesting finding. Though the mentioned complexity is
adamant $$$ O(n \log n e^{2 \sqrt{\log 2 \log \log n}}), $$$
and I wonder if it bears much practical significance ツ
□ 22 months ago, # ^ |
← Rev. 3 → +50
Actually, negiizhao told me that the high-level idea can be explained in several lines.
If we use D&C with $$$2$$$ branches, we have the recursive formula $$$T(n) = 2T(n/2) + O(n\log n)$$$, what happens if we have $$$b$$$ branches? We divide the sequence into $$$b$$$
blocks with length $$$\lceil n/b\rceil$$$, say, $$$a_{[0]}, \dots, a_{[b-1]}$$$, the key observation is that although the contribution between blocks is $$$b^2$$$ convolutions, we can
reuse the information of $$$DFT(a_{[i]})$$$, so the time can be reduced to $$$T(n) = bT(n/b) + O(n\log n + nb)$$$. Take $$$b=\log n$$$, we first achieve a time complexity $$$O(n\log^2
n / \log\log n)$$$. This is still implementable and has a good performance.
To achieve a better time complexity, one needs to realize that, the contribution between the blocks $$$DFT(a_{[i]})$$$ is again several relaxed convolution procedures, i.e., for each
fixed $$$j$$$, the sequence $$$DFT(a_{[i]})_{j}$$$ is a sequence with length $$$b$$$ that need to be computed through a relaxed convolution. This gives the recursive formula $$$T(n) =
» bT(n/b) + (2n/b)T(b) + O(n\log n)$$$, take $$$b = \sqrt{n}$$$ we have $$$T(n) = O(n(\log n)^{\log_2 3})$$$. This is not that easy to implement because we need an asynchronous
» computation order.
To achieve the final time complexity, one needs to realize that the algorithm above is a $$$2$$$-depth structure, and we need to consider an $$$\ell$$$-depth structure. Suppose we have
Elegia some appropriate block sizes $$$n_1\dots,n_\ell$$$ such that $$$n_1\cdots n_\ell \approx n$$$, we can write each $$$0\leq d < n$$$ into the digits $$$d_1d_2\ldots d_\ell$$$ such that
$$$0\leq d_i <n_i$$$, the contribution from index $$$d$$$ to index $$$e$$$ should be computed at their LCA in the trie of these digits. The advantage is that we can let $$$n_i$$$
equally small, and the recursive formula is
$$$ T(n) = (n/n_\ell)T(n_\ell) + \sum_{i=1}^{\ell-1} (2n/n_i)T(n_i) + O(\ell n\log n). $$$
The $$$\exp(2\sqrt{\log 2\log \log n})$$$ factor comes from a careful analysis of the choice of $$$\ell$$$ and $$$n_i$$$. (UPD: Roughly speaking, take $$$n_i = n^{1/\ell}$$$ and $$$\
ell = \sqrt{\log_2 \log n}$$$ achieves this bound.)
By the way, this idea of blocking also helps us to shave the constant factors on computing elementary operations, e.g., I. S. Sergeev's algorithms on computing inverse with time $$$
(1.25+o(1))\mathsf M(n)$$$, here $$$\mathsf M(n)$$$ is the usual time computing the convolution.
UPD at 2023.8.12: The final complexity claimed by that paper is actually with a factor $$$\exp\left(\sqrt{2\log 2\log \log n}\right) \sqrt{\log \log n}$$$, by balancing the chosen $$$\
ell$$$ s in the recursion. It's in the section 4 but not mentioned in the abstract.
» 22 months ago, # |
How is newton?
» 22 months ago, # |
22 months ago, # |
← Rev. 4 → +12
You can use this to solve 1782F from the recent round in $$$O(n^2 \log^{2}(n))$$$. You can consider the weighted sum of all bracket sequences with balance better than $$$-b$$$ of length
$$$n$$$ to be $$$[x^n]f_b$$$. Now notice these relations
f_b - 1 = pxf_b^2f_{b+1} + (1-p)xf_b^2f_{b-1}
Since the relations don't form a dag, its not possible to compute the polynomials one by one. However, if we knew the first $$$k$$$ coefficients of all polynomials, we can find the
$$$k+1$$$th coefficient of all polynomials. So we basically need to compute polynomials $$$f_b^2f_{b+1}$$$ and $$$f_{b}^2f_{b-1}$$$ online. The above method only solves for multiplication of
2 polynomials, but its easy to generalize for multiple polynomials as a black box even.
If you are calculating $$$f_b$$$ online, you can use it to calculate $$$f_b^2$$$ online as the online product with itself. If you're able to do that, you can calculate $$$f_b^2f_{b+1}$$$ as
the online product between $$$f_b^2$$$ and $$$f_{b+1}$$$. We just need to then divide $$$[x^n]f_0$$$ with the number of ways to choose indexes. | {"url":"https://mirror.codeforces.com/blog/entry/111399","timestamp":"2024-11-12T09:07:41Z","content_type":"text/html","content_length":"119987","record_id":"<urn:uuid:5867b370-25c8-4db1-a3b8-6a0a824a9c92>","cc-path":"CC-MAIN-2024-46/segments/1730477028249.89/warc/CC-MAIN-20241112081532-20241112111532-00392.warc.gz"} |
The principle of high precision pressure four-bit and its realization __ACM
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Quantum simulation of battery materials using ionic pseudopotentials
Ionic pseudopotentials are widely used in classical simulations of materials to model the effective potential due to the nucleus and the core electrons. Modeling fewer electrons explicitly results in
a reduction in the number of plane waves needed to accurately represent the states of a system. In this work, we introduce a quantum algorithm that uses pseudopotentials to reduce the cost of
simulating periodic materials on a quantum computer. We use a qubitization-based quantum phase estimation algorithm that employs a first-quantization representation of the Hamiltonian in a plane-wave
basis. We address the challenge of incorporating the complexity of pseudopotentials into quantum simulations by developing highly-optimized compilation strategies for the qubitization of the
Hamiltonian. This includes a linear combination of unitaries decomposition that leverages the form of separable pseudopotentials. Our strategies make use of quantum read-only memory subroutines as a
more efficient alternative to quantum arithmetic. We estimate the computational cost of applying our algorithm to simulating lithium-excess cathode materials for batteries, where more accurate
simulations are needed to inform strategies for gaining reversible access to the excess capacity they offer. We estimate the number of qubits and Toffoli gates required to perform sufficiently
accurate simulations with our algorithm for three materials: lithium manganese oxide, lithium nickel-manganese oxide, and lithium manganese oxyfluoride. Our op-Modjtaba Shokrian Zini: [email
protected] timized compilation strategies result in a pseudopotential-based quantum algorithm with a total Toffoli cost four orders of magnitude lower than the previous state of the art for a fixed
target accuracy.
ASJC Scopus subject areas
• Atomic and Molecular Physics, and Optics
• Physics and Astronomy (miscellaneous)
Dive into the research topics of 'Quantum simulation of battery materials using ionic pseudopotentials'. Together they form a unique fingerprint. | {"url":"https://www.scholars.northwestern.edu/en/publications/quantum-simulation-of-battery-materials-using-ionic-pseudopotenti","timestamp":"2024-11-12T04:11:27Z","content_type":"text/html","content_length":"54872","record_id":"<urn:uuid:9062ec33-e2cf-4525-bacf-bc10bbceda69>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.50/warc/CC-MAIN-20241112014152-20241112044152-00040.warc.gz"} |
An example where lubrication theory comes short: Hydraulic jumps in a flow down an inclined plate
We examine two-dimensional flows of a viscous liquid on an inclined plate. If the upstream depth h- of the liquid is larger than its downstream depth h+, a smooth hydraulic jump (bore) forms and
starts propagating down the slope. If the inclination angle of the plate is small, the bore can be described by the so-called lubrication theory. In this work we demonstrate that bores with h+/h-
<(√3-1)/2 either are unstable or do not exist as steady solutions of the governing equation (physically, these two possibilities are difficult to distinguish). The instability/evolution occurs near a
stagnation point and, generally, causes overturning - sometimes on the scale of the whole bore, sometimes on a shorter, local scale. The overturning occurs because the flow advects disturbances
towards the stagnation point and, thus, 'compresses' them, increasing the slope of the free surface. Interestingly, this effect is not captured by the lubrication theory, which formally yields a
well-behaved stable solution for all values of h+/h-.
• interfacial flows (free surface)
• lubrication theory
• wave breaking
Dive into the research topics of 'An example where lubrication theory comes short: Hydraulic jumps in a flow down an inclined plate'. Together they form a unique fingerprint. | {"url":"https://pure.ul.ie/en/publications/an-example-where-lubrication-theory-comes-short-hydraulic-jumps-i","timestamp":"2024-11-11T12:50:15Z","content_type":"text/html","content_length":"53973","record_id":"<urn:uuid:44fc3334-8169-4b03-ad1f-63228b9b0b6b>","cc-path":"CC-MAIN-2024-46/segments/1730477028230.68/warc/CC-MAIN-20241111123424-20241111153424-00276.warc.gz"} |
Temperature Conversion Calculator
Temperature Conversions: Understanding Thermal Measurements
Accurate temperature measurements are vital in various aspects of daily life and numerous professional fields such as meteorology, cooking, engineering, and healthcare. Whether you’re adjusting a
recipe, monitoring climate conditions, conducting scientific research, or managing patient care, understanding temperature conversions ensures precision and consistency. This comprehensive guide
explores the fundamentals of temperature conversions, examines common temperature scales and their conversion formulas, provides detailed example problems, and highlights practical applications to
enhance your understanding and use of temperature measurement tools.
Understanding Temperature Conversions
Temperature conversion involves translating a temperature value from one scale to another while maintaining its inherent thermal value. This process is essential in various fields to ensure
consistency, accuracy, and effective communication across different measurement systems. Unlike other unit conversions that typically rely on multiplication or division, temperature conversions often
require specific mathematical formulas due to the different starting points (offsets) of temperature scales.
At the core of temperature conversion are predefined formulas that define the relationship between different temperature scales. By applying these formulas, one can accurately convert measurements
from one temperature scale to another. For example, converting Celsius to Fahrenheit involves both multiplication and addition to account for the different zero points and scaling factors of the two
Common Temperature Scales and Their Conversion Formulas
Temperature conversions typically involve a variety of temperature scales used globally. Below, we explore the most commonly used temperature scales, accompanied by their conversion formulas.
Familiarizing yourself with these formulas will empower you to perform accurate conversions effortlessly.
1. Celsius (°C)
The Celsius scale is widely used around the world for most temperature measurements, including weather forecasts, cooking, and scientific applications. It is part of the metric system and is based on
the freezing and boiling points of water.
2. Fahrenheit (°F)
The Fahrenheit scale is primarily used in the United States and a few other countries for everyday temperature measurements such as weather forecasts, cooking, and indoor climate control.
3. Kelvin (K)
The Kelvin scale is the SI (International System of Units) base unit for temperature and is widely used in scientific research and engineering. It is an absolute temperature scale, starting at
absolute zero, the theoretical point where all thermal motion ceases.
4. Rankine (°R or °Ra)
The Rankine scale is primarily used in engineering fields in the United States, particularly in thermodynamics and engineering thermodynamics calculations.
Temperature Conversion Formulas
Understanding the formulas for converting between different temperature scales is essential for accurate temperature translations. Below are the primary conversion formulas used for temperature
From To Conversion Formula
Celsius (°C) Fahrenheit (°F) °F = (°C × 9/5) + 32
Fahrenheit (°F) Celsius (°C) °C = (°F – 32) × 5/9
Celsius (°C) Kelvin (K) K = °C + 273.15
Kelvin (K) Celsius (°C) °C = K – 273.15
Fahrenheit (°F) Kelvin (K) K = (°F – 32) × 5/9 + 273.15
Kelvin (K) Fahrenheit (°F) °F = (K – 273.15) × 9/5 + 32
Celsius (°C) Rankine (°R) °R = (°C + 273.15) × 9/5
Rankine (°R) Celsius (°C) °C = (°R – 491.67) × 5/9
Fahrenheit (°F) Rankine (°R) °R = °F + 459.67
Rankine (°R) Fahrenheit (°F) °F = °R – 459.67
Example Problem: Converting Celsius to Fahrenheit
Problem: Convert 25°C to Fahrenheit.
1. Apply the conversion formula: °F = (°C × 9/5) + 32.
2. Substitute the values: °F = (25 × 9/5) + 32 = 45 + 32 = 77°F.
3. Result: 25°C = 77°F.
Example Problem: Converting Fahrenheit to Celsius
Problem: Convert 98.6°F to Celsius.
1. Apply the conversion formula: °C = (°F – 32) × 5/9.
2. Substitute the values: °C = (98.6 – 32) × 5/9 = 66.6 × 5/9 ≈ 37°C.
3. Result: 98.6°F ≈ 37°C.
Example Problem: Converting Celsius to Kelvin
Problem: Convert 0°C to Kelvin.
1. Apply the conversion formula: K = °C + 273.15.
2. Substitute the values: K = 0 + 273.15 = 273.15 K.
3. Result: 0°C = 273.15 K.
Example Problem: Converting Kelvin to Fahrenheit
Problem: Convert 300 K to Fahrenheit.
1. Apply the conversion formula: °F = (K – 273.15) × 9/5 + 32.
2. Substitute the values: °F = (300 – 273.15) × 9/5 + 32 = 26.85 × 9/5 + 32 ≈ 48.33 + 32 = 80.33°F.
3. Result: 300 K ≈ 80.33°F.
Example Problem: Converting Rankine to Celsius
Problem: Convert 527.67°R to Celsius.
1. Apply the conversion formula: °C = (°R – 491.67) × 5/9.
2. Substitute the values: °C = (527.67 – 491.67) × 5/9 = 36 × 5/9 = 20°C.
3. Result: 527.67°R = 20°C.
Practical Applications of Temperature Conversions
Accurate temperature conversions are essential in numerous fields. Understanding how to convert between different temperature scales can enhance efficiency and precision in multiple contexts:
1. Cooking and Baking
Chefs and home cooks often convert oven temperatures between Celsius and Fahrenheit to follow recipes from different regions, ensuring dishes are cooked to perfection.
2. Meteorology and Weather Forecasting
Weather professionals convert temperatures to provide forecasts in units familiar to their audience, whether it’s Celsius, Fahrenheit, or Kelvin for scientific analysis.
3. Healthcare and Medicine
Medical professionals convert body temperatures between Celsius and Fahrenheit to maintain accurate patient records and communicate effectively with patients from different regions.
4. Engineering and Manufacturing
Engineers and manufacturers convert temperatures to meet industry standards, conduct experiments, and ensure the quality and safety of products under various thermal conditions.
5. Scientific Research
Researchers convert temperatures between different scales to maintain consistency in experiments, share findings internationally, and utilize standard measurement units like Kelvin.
6. Aviation and Aerospace
Aviation professionals convert temperatures to ensure the safety and efficiency of aircraft operations, including engine performance and environmental control systems.
7. International Business and Trade
Businesses involved in international trade convert temperatures to comply with regulations, manage logistics, and ensure the proper handling of temperature-sensitive goods.
Additional Example Problems
Problem 1: Converting Celsius to Fahrenheit
Question: What is 15°C in Fahrenheit?
1. Apply the conversion formula: °F = (°C × 9/5) + 32.
2. Substitute the values: °F = (15 × 9/5) + 32 = 27 + 32 = 59°F.
3. Result: 15°C = 59°F.
Problem 2: Converting Fahrenheit to Kelvin
Question: Convert 212°F to Kelvin.
1. Apply the conversion formula: K = (°F – 32) × 5/9 + 273.15.
2. Substitute the values: K = (212 – 32) × 5/9 + 273.15 = 180 × 5/9 + 273.15 = 100 + 273.15 = 373.15 K.
3. Result: 212°F = 373.15 K.
Problem 3: Converting Kelvin to Rankine
Question: What is 300 K in Rankine?
1. Apply the conversion formula: °R = K × 9/5.
2. Substitute the values: °R = 300 × 9/5 = 540°R.
3. Result: 300 K = 540°R.
Problem 4: Converting Rankine to Fahrenheit
Question: Convert 491.67°R to Fahrenheit.
1. Apply the conversion formula: °F = °R – 459.67.
2. Substitute the values: °F = 491.67 – 459.67 = 32°F.
3. Result: 491.67°R = 32°F.
Tips for Effective Temperature Conversions
• Understand the Conversion Formulas: Familiarize yourself with the primary temperature conversion formulas to enhance your ability to perform manual conversions when necessary.
• Use Reliable Sources: When in doubt, refer to reputable sources or standardized conversion tables to verify conversion formulas and factors.
• Double-Check Calculations: Temperature conversions often involve multiple steps. Double-check your calculations to ensure accuracy.
• Be Mindful of Decimal Places: Depending on the context, you may need to round the converted value to a specific number of decimal places for precision.
• Leverage Technology: Utilize temperature conversion tools and calculators for quick and accurate conversions, especially for complex or large-scale measurements.
• Remember Absolute Zero: Be aware that absolute zero (-273.15°C or 0 K) is the theoretical lowest possible temperature and cannot be surpassed.
Mastering temperature conversions is a valuable skill that enhances accuracy and efficiency across various aspects of life and work. Whether you’re a student, chef, engineer, healthcare professional,
or everyday user, understanding how to convert between different temperature scales ensures precision in your measurements and calculations. By familiarizing yourself with common temperature scales
and their conversion formulas, practicing with example problems, and applying practical tips, you can navigate the complexities of temperature measurement systems with confidence. Leveraging the
capabilities of temperature conversion tools and maintaining a solid grasp of conversion principles can significantly streamline your tasks, ensuring consistency and accuracy in your thermal | {"url":"https://turn2engineering.com/calculators/temperature-conversion-calculator","timestamp":"2024-11-07T00:21:39Z","content_type":"text/html","content_length":"209583","record_id":"<urn:uuid:f0642a6f-bd19-4143-990b-00661201ff21>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00380.warc.gz"} |
What Is a Risk-Free Interest Rate?
Last editedFeb 20222 min read
Even the safest investments have an element of risk. Because of this, investors will often demand a rate of return that reflects inflationary expectations. That’s where the risk-free interest rate
comes into play. Find out everything you need to know about the risk-free interest rate, including how to calculate it and what it means for businesses and investors. First off, what is a risk-free
interest rate?
Risk-free interest rate explained
The risk-free interest rate, also referred to as the risk-free rate of return, is a theoretical interest rate of an investment which carries zero risk. In actual terms, the risk-free interest rate is
assumed to be equal to the interest rate paid on a three-month government Treasury bill, which is considered to be one of the safest investments that it’s possible to make.
Technically, the risk-free interest rate is purely theoretical, as all investments have some type of risk attached to them. Having said that, although it is possible for the government to default on
its securities, the likelihood of this happening is enormously low. In fact, the level of risk is so low that it’s considered negligible. Hence, “risk-free”.
Real risk-free interest rate vs. nominal risk-free interest rate
When we’re discussing risk-free interest rates, we’re essentially talking about two different things: the real risk-free interest rate and the nominal risk-free interest rate. Although it may sound
complicated, the reasoning behind these concepts isn’t too difficult to grasp.
Essentially, the real risk-free interest rate refers to the rate of return required by investors on zero-risk financial instruments without inflation. Since this doesn’t exist, the real risk-free
interest rate is a theoretical concept. However, while it cannot be observed in any meaningful way, studies have indicated that the real risk-free interest rate is equal to the economy’s long-run
growth rate.
By contrast, the nominal risk-free interest rate is the observed return on a risk-free asset. You can work out the nominal risk-free interest rate using the real risk-free interest rate (however you
decide to calculate it) and the inflation rate. The most important thing you need to understand about these two concepts is that their relationship is determined by the inflation rate.
How to calculate the risk-free interest rate
There’s no consensus on how to calculate the risk-free interest rate, which means that there are a broad range of risk-free interest rate formulas that are purported to provide a direct measurement
of the risk-free rate. Because of this, some analysts decide to look at so-called “proxies” for the risk-free interest rate. Short-dated government bonds, inter-bank lending rates, or AAA-rated
corporate bonds from companies that are reportedly “too big to fail” are all good examples of potential proxies.
However, there are numerous issues with the proxy approach to calculating the risk-free interest rate. For example, government bonds can only really be risk-free if there’s no risk of default.
However, while it’s extremely unlikely for these types of bonds to default, it does happen on occasion, which means that government bonds may not be a suitable proxy. Furthermore, there’s always a
risk of the government “printing more money” to meet their obligations, leading to a loss of value.
Despite this, there are several risk-free interest rate formulas that can provide you with an effective way to calculate the risk-free interest rate. For example, you could simply subtract the
current inflation rate from the Treasury bill’s yield over the duration of your investment. This calculation can be expressed using the following risk-free interest rate formula:
Risk-Free Interest Rate = Real Risk-Free Interest Rate + Inflation Premium
What does the risk-free interest rate mean for businesses?
The risk-free interest rate has several important applications. First off, it plays an important role in a range of different financial calculations, including the Sharpe ratio and the Black-Scholes
formula. In addition, businesses will need to pay attention to the risk-free interest rate, as rising risk-free rates could lead to higher required return rates from investors, thereby driving up the
price of stock.
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R Programming: A Step-by-Step Guide for Absolute Beginners
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5R is a programming language and software environment for statistical analysis, graphics representation, and reporting. If you are trying to understand the R programming language as a beginner, this
short book will give you enough understanding of almost all the concepts of the R language. The author will guide you through examples, how to program in R and how to use R for effective data
Buy your copy Now Book Objectives
This book is about R programming. The following are the objectives of the author:
• To familiarize you with the basics of R programming language.
• To help you understand the various fields where R can be applied and its use cases in each field.
• To equip you with R programming skills, both beginner and advanced skills.
• To introduce you to R programming for data analysis.
• To introduce you to R programming for machine learning.
• To help you understand and appreciate the power of R in statistical computing, data analysis, and scientific research.
Who this Book is for?
• Anybody who is a complete beginner to R Programming.
• Anybody in need of advancing their R Programming skills.
• Professionals in computer programming.
• Professors, lecturers or tutors who are looking to find better ways to explain R programming to their students in the simplest and easiest way.
• Students and academicians, especially those focusing on R, Data Analysis, Machine Learning, computer science, and Databases development.
The author expects you to have a computer installed with an operating system such as Linux, Windows or Mac OS X.
What is inside the book?
• R BASICS
• R DATA TYPES
• R VARIABLES AND CONSTANTS
• R OPERATORS
• DECISION MAKING IN R
• R LOOPS
• R FUNCTIONS
• R CLASSES AND OBJECTS
• R FOR DATA SCIENCE
• R FOR MACHINE LEARNING
From the Back Cover.
R programming language is one of the most popular languages used by statisticians, data analysts, researchers to retrieve, clean, analyze, visualize and present data. This is a comprehensive book on
how to get started with R programming, why you should learn it and how you can learn it. Daniel Bell begins by introducing the readers to the foundations of the R programming language. The aim is to
help you understand, how the R interpreter works, the origin of the name R, how to set up the R programming environment, etc. The author has discussed the process of installing R on Windows, Linux
and Mac OS. Moreover, the author has explored the basics of R programming including writing comments, using the R console, creating R script files, etc. The various features provided by R have been
discussed in depth, including data types, variables, loops, decision making, functions, operators, classes, and objects, etc. The author has also discussed R for data science and R for machine
learning. The book has been organized into chapters, with each chapter having many sub-chapters. R code scripts have been provided, alongside thorough explanations of the code and images showing the
expected output upon the execution of every script. | {"url":"https://hartmannsoftware.com/books/COM-.-a7816a676a648","timestamp":"2024-11-12T10:13:29Z","content_type":"application/xhtml+xml","content_length":"14545","record_id":"<urn:uuid:c1be0d6a-8494-40c7-b4e5-76ac32e16d61>","cc-path":"CC-MAIN-2024-46/segments/1730477028249.89/warc/CC-MAIN-20241112081532-20241112111532-00378.warc.gz"} |
Printable Multiplication Table Of 12×12 | Multiplication Chart Printable
Printable Multiplication Table Of 12×12
Printable Multiplication Table Of 12×12
Printable Multiplication Table Of 12×12 – A Multiplication Chart is a valuable tool for children to learn exactly how to increase, separate, and find the smallest number. There are many usages for a
Multiplication Chart. These convenient tools assist children recognize the procedure behind multiplication by using tinted paths and also filling in the missing out on products. These charts are free
to print and download.
What is Multiplication Chart Printable?
A multiplication chart can be utilized to assist youngsters discover their multiplication facts. Multiplication charts come in lots of forms, from full web page times tables to solitary page ones.
While private tables work for providing chunks of information, a full page chart makes it less complicated to review facts that have currently been understood.
The multiplication chart will normally feature a leading row and also a left column. When you desire to find the item of 2 numbers, pick the first number from the left column as well as the second
number from the leading row.
Multiplication charts are valuable understanding tools for both youngsters and also grownups. Kids can use them in your home or in institution. Multiplication Chart 12×12 Printable are readily
available on the net as well as can be published out and also laminated flooring for sturdiness. They are a fantastic tool to make use of in math or homeschooling, and will give an aesthetic pointer
for kids as they discover their multiplication realities.
Why Do We Use a Multiplication Chart?
A multiplication chart is a layout that reveals exactly how to increase two numbers. You pick the first number in the left column, move it down the column, as well as after that pick the second
number from the top row.
Multiplication charts are useful for numerous factors, consisting of helping children discover how to separate and simplify portions. Multiplication charts can also be practical as desk resources
since they offer as a consistent pointer of the trainee’s progress.
Multiplication charts are additionally helpful for assisting students remember their times tables. They help them learn the numbers by minimizing the number of steps needed to complete each
operation. One method for remembering these tables is to concentrate on a solitary row or column at a time, and then relocate onto the following one. Ultimately, the entire chart will be committed to
memory. As with any skill, remembering multiplication tables takes time as well as technique.
Multiplication Chart 12×12 Printable
Printable Multiplication Chart 12 12 AlphabetWorksheetsFree
Multiplication Chart 12×12 Printable
If you’re looking for Multiplication Chart 12×12 Printable, you’ve come to the ideal place. Multiplication charts are offered in various layouts, consisting of complete size, half dimension, and a
selection of cute styles.
Multiplication charts as well as tables are indispensable tools for kids’s education. These charts are wonderful for usage in homeschool mathematics binders or as class posters.
A Multiplication Chart 12×12 Printable is a beneficial tool to enhance math realities and can aid a kid discover multiplication promptly. It’s additionally a terrific tool for skip checking and
learning the moments tables.
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Source code for ribs.emitters._iso_line_emitter
"""Provides the IsoLineEmitter."""
import numpy as np
from ribs._utils import check_batch_shape, check_shape, np_scalar
from ribs.emitters._emitter_base import EmitterBase
from ribs.emitters.operators import IsoLineOperator
[docs]class IsoLineEmitter(EmitterBase):
"""Emits solutions that are nudged towards other archive solutions.
If the archive is empty and ``self._initial_solutions`` is set, a call to
:meth:`ask` will return ``self._initial_solutions``. If
``self._initial_solutions`` is not set, we draw solutions from an isotropic
Gaussian distribution centered at ``self.x0`` with standard deviation
``self.iso_sigma``. Otherwise, to generate each new solution, the emitter
selects a pair of elites :math:`x_i` and :math:`x_j` and samples from
.. math::
x_i + \\sigma_{iso} \\mathcal{N}(0,\\mathcal{I}) +
\\sigma_{line}(x_j - x_i)\\mathcal{N}(0,1)
This emitter is based on the Iso+LineDD operator presented in `Vassiliades
2018 <https://arxiv.org/abs/1804.03906>`_.
archive (ribs.archives.ArchiveBase): An archive to use when creating and
inserting solutions. For instance, this can be
iso_sigma (float): Scale factor for the isotropic distribution used to
generate solutions.
line_sigma (float): Scale factor for the line distribution used when
generating solutions.
x0 (array-like): Center of the Gaussian distribution from which to
sample solutions when the archive is empty. Must be 1-dimensional.
This argument is ignored if ``initial_solutions`` is set.
initial_solutions (array-like): An (n, solution_dim) array of solutions
to be used when the archive is empty. If this argument is None, then
solutions will be sampled from a Gaussian distribution centered at
``x0`` with standard deviation ``iso_sigma``.
bounds (None or array-like): Bounds of the solution space. Solutions are
clipped to these bounds. Pass None to indicate there are no bounds.
Alternatively, pass an array-like to specify the bounds for each
dim. Each element in this array-like can be None to indicate no
bound, or a tuple of ``(lower_bound, upper_bound)``, where
``lower_bound`` or ``upper_bound`` may be None to indicate no bound.
batch_size (int): Number of solutions to return in :meth:`ask`.
seed (int): Value to seed the random number generator. Set to None to
avoid a fixed seed.
ValueError: There is an error in x0 or initial_solutions.
ValueError: There is an error in the bounds configuration.
def __init__(self,
self._rng = np.random.default_rng(seed)
self._batch_size = batch_size
self._iso_sigma = np_scalar(iso_sigma, dtype=archive.dtypes["solution"])
self._line_sigma = np_scalar(line_sigma, archive.dtypes["solution"])
self._x0 = None
self._initial_solutions = None
if x0 is None and initial_solutions is None:
raise ValueError("Either x0 or initial_solutions must be provided.")
if x0 is not None and initial_solutions is not None:
raise ValueError(
"x0 and initial_solutions cannot both be provided.")
if x0 is not None:
self._x0 = np.array(x0, dtype=archive.dtypes["solution"])
check_shape(self._x0, "x0", archive.solution_dim,
elif initial_solutions is not None:
self._initial_solutions = np.asarray(
initial_solutions, dtype=archive.dtypes["solution"])
check_batch_shape(self._initial_solutions, "initial_solutions",
archive.solution_dim, "archive.solution_dim")
self._operator = IsoLineOperator(line_sigma=self._line_sigma,
def x0(self):
"""numpy.ndarray: Center of the Gaussian distribution from which to
sample solutions when the archive is empty (if initial_solutions is not
return self._x0
def initial_solutions(self):
"""numpy.ndarray: The initial solutions which are returned when the
archive is empty (if x0 is not set)."""
return self._initial_solutions
def iso_sigma(self):
"""float: Scale factor for the isotropic distribution used to
generate solutions when the archive is not empty."""
return self._iso_sigma
def line_sigma(self):
"""float: Scale factor for the line distribution used when generating
return self._line_sigma
def batch_size(self):
"""int: Number of solutions to return in :meth:`ask`."""
return self._batch_size
[docs] def ask(self):
"""Generates ``batch_size`` solutions.
If the archive is empty and ``self._initial_solutions`` is set, we
return ``self._initial_solutions``. If ``self._initial_solutions`` is
not set, we draw solutions from an isotropic Gaussian distribution
centered at ``self.x0`` with standard deviation ``self.iso_sigma``.
Otherwise, each solution is drawn from a distribution centered at
a randomly chosen elite with standard deviation ``self.iso_sigma``.
If the archive is not empty, ``(batch_size, solution_dim)`` array
-- contains ``batch_size`` new solutions to evaluate. If the
archive is empty, we return ``self._initial_solutions``, which
might not have ``batch_size`` solutions.
if self.archive.empty and self._initial_solutions is not None:
return np.clip(self._initial_solutions, self.lower_bounds,
if self.archive.empty:
parents = np.repeat(self.x0[None],
repeats=2 * self._batch_size,
parents = self.archive.sample_elites(2 *
return self._operator.ask(
parents=parents.reshape(2, self._batch_size, -1)) | {"url":"https://docs.pyribs.org/en/latest/_modules/ribs/emitters/_iso_line_emitter.html","timestamp":"2024-11-05T00:43:01Z","content_type":"text/html","content_length":"32601","record_id":"<urn:uuid:9bc42209-b469-439f-8a09-7536c9cf4269>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.84/warc/CC-MAIN-20241104225856-20241105015856-00160.warc.gz"} |
Abelian surfaces with prescribed groups
Let A be an abelian surface over Fq, the field of q elements. The rational points on A/Fq form an abelian group A(Fq)≃Z/n1Z×Z/n1n2Z×Z/n1n2n3Z×Z/n1n2n3n4Z. We are interested in knowing which groups of
this shape actually arise as the group of points on some abelian surface over some finite field. For a fixed prime power q, a characterization of the abelian groups that occur was recently found by
Rybakov. One can use this characterization to obtain a set of congruences on certain combinations of coefficients of the corresponding Weil polynomials. We use Rybakov's criterion to show that groups
Z/n1Z×Z/n1n2Z×Z/n1n2n3Z×Z/n1n2n3n4Z do not occur if n1 is very large with respect to n2,n3,n4 (Theorem 1.1), and occur with density zero in a wider range of the variables (Theorem 1.2).
Repository Citation
David, C., D. Garton, Z. Scherr, A. Shankar, E. Smith, and L. Thompson. 2014. "Abelian surfaces with prescribed groups." Bulletin of the London Mathematical Society 46: 779-792.
London Mathematical Society
Publication Date
Publication Title
Bulletin of the London Mathematical Society | {"url":"https://digitalcommons.oberlin.edu/faculty_schol/1858/","timestamp":"2024-11-11T01:37:48Z","content_type":"text/html","content_length":"35202","record_id":"<urn:uuid:57bacb99-10d5-4517-90c7-4fad12db1ed1>","cc-path":"CC-MAIN-2024-46/segments/1730477028202.29/warc/CC-MAIN-20241110233206-20241111023206-00748.warc.gz"} |
Machine Learning for Investors: A Primer
If you are out to describe the truth, leave elegance to the tailor.
— Albert Einstein
Machine learning is everywhere now, from self-driving cars to Siri and Google Translate, to news recommendation systems and, of course, trading.
In the investing world, machine learning is at an inflection point. What was bleeding edge is rapidly going mainstream. It’s being incorporated into mainstream tools, news recommendation engines,
sentiment analysis, stock screeners. And the software frameworks are increasingly commoditized, so you don’t need to be a machine learning specialist to make your own models and predictions.
If you’re an old-school quant investor, you may have been trained in traditional statistics paradigms and want to see if machine learning can improve your models and predictions. If so, then this
primer is for you!
We’ll review the following items in this piece:
• How machine learning differs from statistical methods and why it’s a big deal
• The key concepts
• A couple of nuts-and-bolts examples to give a flavor of how it solves problems
• Code to jumpstart your own mad science experiments
• A roadmap to learn more about it.
Even if you’re not planning to build your own models, AI tools are proliferating, and investors who use them will want to know the concepts behind them. And machine learning is transforming society
with huge investing implications, so investors should know basically how it works.
Let’s dive in!
Why machine learning?
Machine learning can be described as follows:
• A highly empirical paradigm for inference and prediction
• Works pretty well with many noisy predictors
• Can generalize to a wide range of problems.
In school, when we studied modeling and forecasting, we were probably studying statistical methods. Those methods were created by geniuses like Pascal, Gauss, and Bernoulli. Why do we need something
new? What is different about machine learning?
Short answer: Machine learning adapts statistical methods to get better results in an environment with much more data and processing power.
In statistics, if you’re a Bayesian, you start from a prior distribution about what the thing you’re studying probably looks like. You identify a parametric function to model the distribution, you
sample data, and you estimate the parameters that yield the best model given the sample distribution and the prior.
If you’re not a Bayesian, you don’t think about a prior distribution, you just fit your model to the sample data. If you’re doing a linear regression, you specify a linear model and estimate its
parameters to minimize the sum of squared errors. If you believe any of your standard errors, you take an opinionated view that the underlying data fit a linear model plus a normally distributed
error. If your data violate the assumptions of ordinary least squares, those error estimates are misleading or meaningless.
I think we are all Bayesians, just to varying degrees. Even if we are frequentists, we have the power to choose the form of our regression model, but no power to escape the necessity of choice. We
have to select a linear or polynomial model, and select predictors to use in our model. Bayesians take matters a step further with a prior distribution. But if you don’t assume a prior distribution,
you are nevertheless taking an implied random distribution as a prior. If you assume all parameters are equally likely, your prior for the parameters is a uniform distribution. In the words of Rush,
“If you choose not to decide, you still have made a choice.”
But if OLS (ordinary least squares) is BLUE (the best linear unbiased estimator), why do we need machine learning?
When we do statistics, we most often do simple linear regression against one or a small number of variables. When you start adding many variables, interactions, nonlinearities, you get a
combinatorial explosion in the number of parameters you need to estimate. For instance, if you have 5 predictors and want to do a 3rd-degree polynomial regression, you have 56 terms, and in general
In real-world applications, when you go multivariate, even if you are well supplied with data, you rapidly consume degrees of freedom, overfit the data, and get results which don’t replicate well out
of sample. The curse of dimensionality multiplies outliers, and statistical significance dwindles.
The best linear unbiased estimator is just not very good when you throw a lot of noisy variables at it. And while it’s straightforward to apply linear regression to nonlinear interactions and
higher-order terms by generating them and adding them to the data set, that means adding a lot of noisy variables. Using statistics to model anything with a lot of nonlinear, noisy inputs is asking
for trouble. So economists look hard for ‘natural experiments’ that vary only one thing at a time, like a minimum wage hike in one section of a population.
When the discipline of statistics was created, Gauss and the rest didn’t have the computing power we have today. Mathematicians worked out proofs, closed form solutions, and computations that were
tractable with slide rules and table lookups. They were incredibly successful considering what they had to work with. But we have better tools today: almost unbounded computing resources and data.
And we might as well use them. So we get to ask more complicated questions. For instance, what is the nonlinear model that best approximates the data, where ‘best’ means it uses the number of degrees
of freedom that makes it optimally predictive out-of-sample?
More generally, how do you best answer any question to minimize out-of-sample error? Machine learning asks, what are the strongest statements I can make about some data with the maximum of cheap
tricks, in the finest sense of the word. Given a bunch of data, what is the best fitting nonlinear smoothing spline with x degrees of freedom or knots? And how many knots / degrees of freedom give
you the best bang for the buck, the best tradeoff of overfitting v. underfitting?
In machine learning, we do numerical optimizations, whereas in old-school statistics we solved a set of equations based on an opinionated view of what ‘clean’ data look like.
Those tricks, with powerful CPUs and GPUs, new algorithms, and careful cross-validation to prevent overfitting, are why machine learning feels like street–fighting statistics. If it works in a
well-designed test, use it, and don’t worry about proofs or elegance.
Pure statisticians might turn up their nose at a perceived lack of mathematical elegance or rigor in machine learning. Machine learning engineers might reply that statistical theory about an ideal
random variable may be a sweet science, but in the real world, it leads you to make unfounded assumptions that the underlying data don’t violate the assumptions of OLS, while fighting with one hand
tied behind your back.
In statistics, a lot of times you get not-very-good answers and know the reason: your data do not fit the assumptions of OLS. In machine learning, you often get better answers and you don’t really
know why they are so good.
There is more than one path to enlightenment. One man’s ‘data mining’ is another’s reasoned empiricism, and letting the data do the talking instead of leaning on theory and a prior about what data
should look like.
A 30,000 foot view of machine learning algorithms
In statistics, we have descriptive and inferential statistics. Machine learning deals with the same problems, uses them to attack higher-level problems like natural language, and claims for its
domain any problem where the solution isn’t programmed directly, but is mostly learned by the program.
Supervised learning – You have labeled data: a sample of ground truth with features and labels. You estimate a model that predicts the labels using the features. Alternative terminology: predictor
variables and target variables. You predict the values of the target using the predictors.
• Regression. The target variable is numeric. Example: you want to predict the crop yield based on remote sensing data. Algorithms: linear regression, polynomial regression, generalized linear
• Classification. The target variable is categorical. Example: you want to detect the crop type that was planted using remote sensing data. Or Silicon Valley’s “Not Hot Dog” application.^1
Algorithms: Naïve Bayes, logistic regression, discriminant analysis, decision trees, random forests, support vector machines, neural networks of many variations: feed-forward NNs, convolutional
NNs, recurrent NNs.
Unsupervised learning – You have a sample with unlabeled information. No single variable is the specific target of prediction. You want to learn interesting features of the data:
• Clustering. Which of these things are similar? Example: group consumers into relevant psychographics. Algorithms – k-means, hierarchical clustering.
• Anomaly detection. Which of these things are different? Example: credit card fraud detection. Algorithms: k-nearest-neighbor.
• Dimensionality reduction. How can you summarize the data in a high-dimensional data set using a lower-dimensional dataset which captures as much of the useful information as possible (possibly
for further modeling with supervised or unsupervised algorithms)? Example: image compression. Algorithms: principal component analysis (PCA), neural network autoencoders.^2
Reinforcement learning – You are presented with a game that responds sequentially or continuously to your inputs, and you learn to maximize an objective through trial and error.^3
All the complex tasks we assign to machine learning, from self-driving cars to machine translation, are solved by combining these building blocks into complex stacks.
The cost function.
Machine learning generally works by numerically minimizing something: a cost function or error.
Let’s try a classification problem. We want to train an algorithm to predict which dots are blue and which are orange, based on their position. Imagine we are trying to predict which stocks
outperformed the market, using 2 magical factors. Please follow along in the Google TensorFlow playground:
Hit the play button and watch it try to classify the dots using the simplest possible neural network: 1 layer of 1 unit (1×1).
Not very good, but we’ll work on it.
How does this work?
• Our objective is to train a function to predict if an observation is blue or orange based on its position in 2-dimensional space, i.e. using the x and y coordinates as predictors.
• For now, we are using a neural network with a single unit: a 1×1 neural network, 1 layer of 1 cell.
• The cell
□ Takes the x and y coordinates as inputs
□ Applies a linear function of the form ax + by + C to the inputs. This maps any inputs to a number between +∞ and -∞. We’ll call this number O.^4
□ Applies a nonlinear function to O called the sigmoid. The sigmoid function maps -∞ to a 100% probability of orange, +∞ is mapped to a 100% probability of blue, and 0 is mapped to 50/50.
□ We estimate the values of a, b, and C that give us the best classifier possible
• To further develop our intuition we can call O “log odds”. If the odds of blue are 3:1,
• We model log odds as a linear function of x and y
• The sigmoid function transforms log odds into probabilities between 0 and 1.
• Therefore the output of our simple network is the probability of blue.
How does the training of this neural network happen, in other words, how do we estimate a, b, and C?
• We initialize a and b to small random numbers, and C to 0 (good places to start for reasons beyond the scope of this introduction).
• We compute a loss function that describes how well this classifier worked. We construct it so if it did well, the error is small, if it did poorly, the error is large.
• We compute the gradient (partial derivative) of the loss function with respect to a, b, and C. In other words we determine if we increase a a little, does the loss improve or worsen, and by how
• By descending the gradient we find the values of a, b, and C that reduce the error to a minimum, in other words, we can’t improve in any direction.
OK, what we just did is logistic regression (applied example here). A single-cell neural network with sigmoid activation performs logistic regression and creates a linear decision boundary.^6
Now add a second cell. Once again, feed x and y to each unit. But now, instead of running the sigmoid on the output of a single cell, we run it on a linear combination of 2 unit outputs. We now have
more parameters to estimate…Try it!
We added a second boundary. If x and y are below one boundary, and above the other boundary, classify as blue, else orange.
Add a third cell.
By adding more units, we create a more and more complex boundary.
A single-hidden-layer neural network with sufficient units can approximate an arbitrarily complex decision boundary for classification.^7 In the case of regression, a neural network can approximate
any computable function to any desired level of precision. And, it works for any number of predictors, minimizing error and drawing decision boundaries in n-dimensional space.
After 3 boundaries, we have a pretty good classifier in this case. But how many do we need in a more difficult example? Look at the image below. How do we determine how many boundaries to add?
The next key concept is…
The bias-variance tradeoff.
As you add variables, interactions, relax linearity assumptions, add higher-order terms, and generally make your model more complex, your model should eventually fit the in-sample data pretty well.
And it should generalize to the out-of-sample data pretty well (as a general rule, no better but hopefully not much worse).
But also, as you add more and more complexity to your model, you start to fit the quirks in your training data too well, and your out-of-sample prediction gets worse. You start off being overly
constrained by the bias of your a priori model, and then as you add complexity, you become overly sensitive to random variance in the sample data you happen to have encountered.
The tradeoff looks like this, via Scott Fortmann-Roe:
Bias is error that comes from underfitting: an overly simplistic, insensitive or otherwise incorrectly specified model. Variance is error that comes from overfitting: from excessive sensitivity to
random variation in the sample itself. You have too-sparse data to get precise estimates in your too-numerous parameter estimates. If you happened to pick another random sample your parameter
estimates would change.
The data scientist is looking for a good model specification that efficiently uses as few parameters as possible to make the trough in the black line as deep as possible. The good model finds the
right bias-variance tradeoff between underfitting and overfitting. But how do we choose this in practice?
Cross-validation, hyperparameters, and test.
The next key concept is cross-validation. Data scientists use careful cross-validation to find the sweet spot in the bias-variance tradeoff, and avoid underfitting or overfitting.
So far we have used one hyperparameter: the number of units in our single layer. We call the number of units in a layer a hyperparameter, to distinguish it from the model parameters we are
estimating. Cross-validation is the process by which we tune the values of our hyperparameters to get the best results.
The simplest approach to cross-validation is to take a random 60% of your sample data and make it your training set. Take a random half of what’s left (20%) and make that your cross-validation set.
Take the 20% that’s left and make that your test set.
Create a set of hyperparameters to evaluate. In our simple case, vary the number of cells in the hidden layer. For each hyperparameter value, fit the best parameters using the training set, and
measure the error in the cross-validation set. Then pick the hyperparameter with the smallest error.
Maybe you have multiple hyperparameters. Perhaps you want to try a neural network of up to 3 10-unit layers. Say for each layer you will try 3, 10, 30, and 100 units. We have 1 hyperparameter for
the number of layers (3 values) and 3 hyperparameters for the number of units in each layer. This yields a grid of something like 4^3 (3 layers) + 4^2 (2 layers) + 4 combinations to test = 64+16+4=84
• Loop through all the combinations, training your parameters using the training set only
• Evaluate each model using the cross-validation set only
• Pick the best model
Finally, you may want to evaluate how the model will perform out of sample. To do this, evaluate the final model using the test set. Since you never used the test set to choose a model parameter,
this should give you a good estimate of real-world performance.
There are more sophisticated forms of cross-validation: You can split the training + cross-validation into 5 folds. Train 5 times on 4 of the folds, leaving a different fold out each time, measure
the error on the left-out fold each time, and then average them. Pick the hyperparameters yielding the smallest average error over all folds. This is called k-fold cross-validation with k=5. It uses
your data more efficiently and eliminates the possibility of a lucky or unlucky split between training and cross-validation sets. In the extreme, you can make k=n and leave a single observation out
each time… this is called leave-one-out cross-validation. In practice, unless you are starved for training data, k of around 5 is a reasonable balance between maximizing use of the training data, and
minimizing computation.
Cross-validating all possible combinations of hyperparameters is a brute-force, compute-intensive process, especially as the number of layers and hyperparameters grows. There are more sophisticated
methods like Bayesian optimization, which vary multiple hyperparameters simultaneously, and in theory allow you to find a more finely-tuned best set of hyperparameters with fewer attempts.
The key thing to remember about cross-validation is to make every decision about your model using the cross-validation set. Unlike statistics, where for instance we estimate standard errors based on
the assumptions of OLS, any result from training is assumed to not generalize until proved otherwise using cross-validation.
The firewall principle: Never make any decision to modify your model using the test set, and never use the training or cross-validation data sets to evaluate its out-of-sample performance.
The act of using the cross-validation set to choose the hyperparameter values contaminates the cross-validation set for testing purposes. You are most likely to pick a model with pretty good
hyperparameters, but you are also more likely to pick a model which also gets luckier in the cross-validation set. For this reason, training accuracy > cross-validation accuracy > test accuracy.^8
Hence the need for a test set to evaluate the real-world performance of the model. Anytime you use the test set to make a decision about how to use your model, you are tending to turn out-of-sample
data into in-sample data.
If you take away one thing from this post, it should be that machine learning allows you to add more noisy predictors, avoid overfitting, and get a fair estimate of out-of-sample performance in the
test set. But you need to do it carefully, in accordance with best practices.
That being said, all estimates are overfitted to the data you happen to have. If you train a hot-dog/not-hot-dog classifier, and in your training data, only hot dogs have ketchup, then your
classifier may learn that ketchup means hot dog. The higher the risk that your training data may not be totally representative, that the future may not be like the past, the more you should err on
the side of underfitting.
Regularization, ‘worse is better’, and other stupid tricks.
Data scientists improve the bias-variance tradeoff by using cost functions that are able to find good predictive signals while filtering noise.
This part always blows my mind a little. As we’ve seen, a good machine learning model is a model with a good bias-variance tradeoff: a low minimum in the cost function, resulting in a small error in
cross-validation. If you can find methods that make the model inherently resistant to chasing noise in the data, without disturbing the essence of prediction, you get a good bias-variance tradeoff.
A simple method is regularization. When an estimated parameter gets large, that may be because it captures a highly significant relationship. Another possibility is it got lucky with this dataset and
just happens to be strongly correlated with its random variation.
The regularization ‘stupid trick’ is to add a penalty for each parameter to the cost function, that typically scales linearly (L1 regularization) or quadratically (L2 regularization) with the
regularized parameter. This shrinks large parameters, whenever increasing the parameter doesn’t correspondingly improve the cost function.
Lo and behold, we almost always find that by making the model perform worse in-sample, it performs better out-of-sample. This is particularly true when you have noisy, multicollinear data.
Regularization reduces the impact of the noise and builds redundancy over the correlated variables.
The lasso regression is a popular application of regularization (applied example). You use a linear regularization and start with an unlimited ‘budget’ on the regularization cost, i.e. the sum of the
linear parameters multiplied by the regularization hyperparameter. Then you gradually shrink the budget. The regression will shrink the parameters that have high costs but don’t improve the
prediction much. Gradually it shrinks some parameters to zero, throwing out predictors and performing implicit variable selection. You keep shrinking the budget until you find the right bias-variance
There are some amazingly creative worse-is-better tricks in machine learning. Random forest is a ‘worse is better’ variation on decision trees.
A tree is an algorithm where you take a set of predictors, and you look for the best rule of the form ‘When return on equity is greater than x, book value is less than y, PE is less than z, and
momentum is higher than α… classify as X‘. When you use a random forest, instead of finding the best decision tree possible with all the predictors in you data set, you throw out a random half of
your predictors; find the best tree that uses only those predictors; repeat this process 10,000 times with a different random selection of predictors; and have the 10,000 simplified trees (the
‘random forest’) vote on the outcome. That turns out to be a better tree than the one using all the data. Crazy.
One reason it works is that it doesn’t become overly reliant on any one set of indicators, so it generalizes better out-of-sample.
Similarly with neural networks, you can use L1 or L2 regularization on all your parameters, but even more creatively, throw out half your units each training iteration, and call it dropout. Dropout
forces the other activations to pick up the slack and builds redundant reasoning. Then do cross-validation with the entire network. It’s like Bill Bradley training to dribble with glasses that
blocked his view so he couldn’t look down at the ball (and lead weights in his shoes).
Finally, you can use ensemble methods. First classify your data with a neural network, then with a random forest. Then use logistic regression to classify using the results of the previous two
It’s not always obvious at first glance why some of these tricks work, but in general, they make it harder for the algorithm to chase quirks in the data, and improve out-of-sample performance.
Finding clever ways to make your algorithm work a little harder for worse results in training can make it work better in the real world. Maybe you can see why one might call it street-fighting
The ROC curve helps us trade off false positives against false negatives, and find the most efficient classifier for a given problem.
Suppose I told you that I have a classifier that accurately classifies CAT scans as cancer/not cancer in 99% of cases. Is that good or bad? Suppose only 1% of scans are cancer. Then it’s not very
good: we could get the same accuracy by classifying all the scans as healthy. We need a measure of classifier quality that takes into account the base rate (1% incidence of cancer), the classifier’s
rate of false positives, and the rate of false negatives.
The ROC (Receiver Operating Characteristic) curve came from analyzing radar data in WW2. If your radar receiver detected a radar return, it could classify it as an incoming aircraft, or as noise. If
you made the receiver extremely sensitive, you would detect all the incoming aircraft (true positives), but you might also detect a lot of birds and random static (false positives). As you decrease
the sensitivity, you reduce the false positives, but you also increase the false negatives, simultaneously increasing the chance of missing a real aircraft.
Here’s an example from published research:
If 50% of the incoming data are labeled true and 50% false, and if you have a classifier with no real information rate, for instance, you classify observations randomly at the 50% base rate, then the
ROC curve will be a 45° line.
If you have a better classifier, it will be curvy and convex with respect to the best case of 100% true positives and 100% true negatives (top left). If you have a really great classifier, the ROC
curve will be a very sharp right angle. If it’s better at avoiding false positives or negatives, it may skew in one direction or the other.^9
The area under the curve (AUC) is a measure of overall classifier quality. A perfect classifier has a AUC close to 1, a poor classifier close to 0.5 in this example.
By selecting a threshold above which you classify as true, you can vary the sensitivity of your classifier. If the cost of a false negative is very high, you might set a low threshold and have high
sensitivity (% of positives correctly classified: true positives). If the cost of a false positive is very high, you set a high threshold and low sensitivity and high specificity (% of negatives
correctly classified: true negatives). You have to balance the possibility of being bombed against the possibility of launching all your expensive missiles at seagulls.
If the cost of a false positive always equals the cost of a false negative, you set a threshold to simply minimize the total number of errors, whether false positives or negatives. If the cost of a
false positive is not the same as the cost of a false negative, you can minimize the total cost by increasing the threshold until the cost of false positives avoided equals the cost of false
negatives added^10.
Deep learning
Modern GPUs and efficient algorithms allow us to combine these building blocks into complex stacks. Deep learning networks of more than 1000 layers are trained end-to-end to solve increasingly
complex problems with human-like intelligence.
Mind-blowing examples, such as facial recognition and natural language processing like Siri, seem indistinguishable from magic, but can be decomposed into a series of tractable steps.
Try this image labeling demo. It’s simple enough to run in your browser. The architecture is complex: a series of convolutional layers that essentially scan for specific features, interspersed with
rectified linear layers and max pooling layers that select the highest scoring ‘convolution’ or scan offset. The layers toward the left look for low-level features, and as you move toward the right,
you look for increasingly complex combinations of higher level features.
The hallmark of deep learning is that you train a very deep neural network with potentially hundreds of layers, each architected to solve a stage of the problem, and you train the entire network
Before neural networks, a state-of-the-art (SOTA) machine translation effort might consist a series of stages:
• Parse words and preprocess them (remove punctuation, common misspellings, maybe combine common n-grams or word combinations
• ‘Stem’ the words and classify by parts of speech, in other words ‘parsing’ -> ‘to parse’, verb, gerund
• Find a distribution of possible corresponding individual words and n-grams in the target language
• Try to diagram the sentence, build a graph of how the parts of speech relate to each other
• Given a distribution of possible sentences, and a language model that allows one to estimate the probability of different sentence structures and word co-occurrences, compute a probability
distribution and identify the most likely full sentence in the target language.
Each task might have a couple of Ph.D.s and the whole team might be dozens of engineers working for a year.
With a neural network, you start with a parallel corpus of several hundred books worth of e.g. English and French, you set up a network of, for instance, 16 layers. You need a cost function that
evaluates the quality of the translation. You repeatedly translate chunks, evaluate the gradient of the cost function across the entire network, and adjust your parameters. For a large network, this
might take weeks, even spread out over hundreds of GPUs in a cluster of computers.
End-to-end training of the entire network against the task-specific objective is the key to deep learning:
• Each layer optimizes activations directly against performance on the ultimate task.
• The optimization takes into account what other layers upstream and downstream are doing, and how they jointly impact the cost function and gradient.
• Complex interactions between layers emerge, optimized for the task at hand.
• Modern GPUs supporting massively parallel computation, and relatively efficient algorithms^11 allow computation of gradients to optimize large deep networks.
When you train a deep learning network end-to-end, the training process starts to feel like intelligent learning. Every new batch of data is analyzed for how to improve the whole process, and complex
new features and interactions between layers start to emerge.
The end result is that the network seems to come alive, layers find high-level features to look for in the noisy data, and the system teaches itself to translate almost as well as human translators,
and better than systems manually fine-tuned by teams of engineers.
Concluding remarks
These are the key concepts we’ve discussed:
• The cost function and gradient descent
• The bias-variance tradeoff
• Cross-validation and hyperparameter selection
• Regularization
• The ROC curve
If you understand them, you are off to a good start in your machine learning adventure.
The following heuristic should apply: If your data match the assumptions of OLS, machine learning should give you the same results as when you use linear regression. If your data don’t match the
assumptions of OLS, if you have a lot of noisy data with nonlinear relationships and complex interactions, you’ll probably get some improvement with machine learning tools. Sometimes a little,
sometimes a lot.
So, if your goal is the best possible predictions, and you have lots of noisy data that violate assumptions of OLS, it is worthwhile to understand the machine learning approach. In comparison with
traditional statistical methods, one can argue that machine learning is the least Bayesian approach, since it can dispense with not only any prior distribution, but also any notion of an ideal
variable or a priori functional form of the solution. It simply asks how to minimize some error or cost function defined in reference to a task.
Machine learning targets prediction; if it has an Achilles heel, it is attribution.^12
When you model a variable with multiple layers of complex non-linear interactions, it becomes impossible to explain why the model made the choice or estimate it did. For many applications, it doesn’t
matter. You don’t really care why your phone’s predictive text made the prediction it did (as long as it worked). But when parole boards apply machine learning to score applicants for early release,
data quirks can easily lead to arbitrary or inappropriately biased decisions.
When we do economics, we combine theoretical models and empirical evidence. If all you do is derive models from first principles, they usually don’t match the real world very well. If all you do is
study past relationships, you don’t have a framework to understand what happens when anything changes beyond past relationships. So typically you write down a model that uses a theory to encapsulate
the dynamics of the system, and then you estimate parameters using econometrics. This enables you to reason about what will happen if you change the underlying system through policy. If you don’t
have a theoretical framework, you don’t know why things happen. And you can’t predict anything that doesn’t closely resemble the past.
If you apply machine learning to credit decisions, it might decide that people of certain ethnicities or living in certain geographical locations, or who have Facebook friends with bad credit, are
poor credit risks. It might tend to exacerbate arbitrary existing patterns.
If your data represent existing social biases, the algorithm will tend to learn those biases. If your data is not representative, the algorithm will make poor predictions.
Investing needs heart as well as mind; theory as well as pattern recognition; wisdom, fortitude and nerve as well as analytical power. Long-term asset allocation isn’t necessarily a great candidate
for machine learning. There’s a lack of sufficient data, a need for attribution, and a future which may not be like the past.
But for much of what investors do, AI helpers will be the rule rather than the exception. It’s already happening in applications from stock screening for further analysis based on the stocks a
manager typically looks at, to technical analysis and developing algorithms for short-term trading against other machines.
In the not-too-distant future, AI will be assumed as a basic part of the investor toolkit, and managing without it will be unthinkable. We will see more amazing applications…possibly new theoretical
breakthroughs and investing paradigms…and probably a few disasters along the way as well.
Some fun examples, and a roadmap for study
The next steps are to pick a problem, write some code, maybe take an online course, dive into more complex algorithms like RNNs, CNNs, reinforcement learning.
Simple finance examples with code to get you started:
Amazing examples:
Books and courses
^1 Noteworthy and not atypical implementation details: A surprisingly complex trial-and-error process; a complex solution that looks deceptively simple.
^2 These three unsupervised learning tasks are somewhat closely related. If you can tell how things are similar in the important features, you can tell which ones are different, and throw away the
^3 One could view reinforcement as a type of iterated supervised learning where you sample more complex inputs and outputs, or where you attempt to learn a higher-level objective like playing a
game or driving a car through trial and error, using the same techniques as classification and regression. But it’s considered sufficiently different and important that many people put it in its own
^4 For simplicity, we’re not using a ‘standard’ notation, which exist using several conventions with scary Greek letters.
^5 We can’t apply gradient descent to accuracy directly, because it’s discrete. If you change your model an infinitesimal amount, it won’t change any individual prediction directly. The cost function
is a smooth, differentiable function that we can apply gradient descent to, that is a good proxy for what we are trying to minimize, in this case inaccurate classifications.
^6 Logistic regression is a generative method, in that for any x and y, it generates a numeric probability. A discriminant method just comes up with a boundary. Using logistic regression to come up
with a linear decision boundary is equivalent to linear discriminant analysis.
A note on activation functions. The Google playground gives a choice of activation functions including sigmoid, tanh, and ReLu. The sigmoid function has an intuitive mathematical foundation: if in
fact the log odds are some piecewise linear function of the coordinates, our neural network will converge on correct probabilities. But any of the nonlinearities can approximate any arbitrarily
complex boundary with sufficient units. If you try them, you might find that the other nonlinearities converge faster: tanh (squashes -∞ to +∞ down to -1 to +1) and ReLu (rectified linear unit, makes
everything above 0 linear and everything else 0. Which one works better might depend on the problem, one might just be faster because it’s less complex to differentiate and optimize.
^7 In practice, when a problem is amenable to being solved in steps, a multi-layer deep network is more common than single-layer. Layers end up performing different tasks, e.g. edge detection,
detection of complex objects with many edges, object classification.
^8 For the same reason, if there is a cross-validation that is not significantly worse than the best, but with more bias and less variance, you are usually better off selecting its hyperparameters.
^9 If one method is good at avoiding false positives and another at false positives, what does that tell you? Time to use an ensemble method combining both!
^10 If you reverse the y-axis of the ROC curve you can set a threshold where the slope equals the relative cost. The f score is an alternative approach.
^11 Backpropagation is the name of the algorithm that lets you efficiently compute all the gradients in a single pass, where earlier gradients depend on later gradients (or vice-versa depending on
how you look at it.
^12 People are working on this. One can, of course, make small changes in the inputs and see how they affect outputs. But the point of machine learning is to find complex nonlinear relationships and
interrelationships. So the whole model may act in ways that are hard to explain or to justify.
Druce Vertes, CFA is founder of
, a leading financial news aggregator, and previously worked as a consultant and IT executive for leading hedge funds.
Important Disclosures
For informational and educational purposes only and should not be construed as specific investment, accounting, legal, or tax advice. Certain information is deemed to be reliable, but its accuracy
and completeness cannot be guaranteed. Third party information may become outdated or otherwise superseded without notice. Neither the Securities and Exchange Commission (SEC) nor any other federal
or state agency has approved, determined the accuracy, or confirmed the adequacy of this article.
The views and opinions expressed herein are those of the author and do not necessarily reflect the views of Alpha Architect, its affiliates or its employees. Our full disclosures are available here.
Definitions of common statistics used in our analysis are available here (towards the bottom).
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element spaces, which do not necessarily satisfy the discrete inf-sup condition. We validate the framework's performance with numerical examples."
• artículo de publicación periódica.listelement.badge
(Integral-)ISS of switched and time-varying impulsive systems based on global state weak linearization
(2021) Mancilla-Aguilar, J. L.; Haimovich, Hernán
"It is shown that impulsive systems of nonlinear, time-varying and/or switched form that allow a stable global state weak linearization are jointly input-to-state stable (ISS) under small inputs
and integral ISS (iISS). The system is said to allow a global state weak linearization if its flow and jump equations can be written as a (time-varying, switched) linear part plus a (nonlinear)
pertubation satisfying a bound of affine form on the state. This bound reduces to a linear form under zero input but does not force the system to be linear under zero input. The given results
generalize and extend previously existing ones in many directions: (a) no (dwell-time or other) constraints are placed on the impulse-time sequence, (b) the system need not be linear under zero
input, (c) existence of a (common) Lyapunov function is not required, (d) the perturbation bound need not be linear on the input."
• artículo de publicación periódica.listelement.badge
Uniform input-to-state stability for switched and time-varying impulsive systems
(2020-12) Mancilla-Aguilar, J. L.; Haimovich, Hernán
"We provide a Lyapunov-function-based method for establishing different types of uniform input-to-state stability (ISS) for time-varying impulsive systems. The method generalizes to impulsive
systems with inputs the well established philosophy of assessing the stability of a system by reducing the problem to that of the stability of a scalar system given by the evolution of the
Lyapunov function on the system trajectories. This reduction is performed in such a way that the resulting scalar system has no inputs. Novel sufficient conditions for ISS are provided, which
generalize existing results for time-invariant and time-varying, switched and nonswitched, impulsive and nonimpulsive systems in several directions."
• artículo de publicación periódica.listelement.badge
Crude oil market and geopolitical events: an analysis based on information-theory-based quantifiers
(2016) Fernández Bariviera, Aurelio; Zunino, Luciano; Rosso, Osvaldo A.
"This paper analyzes the informational efficiency of oil market during the last three decades, and examines changes in informational efficiency with major geopolitical events, such as terrorist
attacks, financial crisis and other important events. The series under study is the daily prices of West Texas Intermediate (WTI) in USD/BBL, commonly used as a benchmark in oil pricing. The
analysis is performed using information-theory-derived quantifiers, namely permutation entropy and permutation statistical complexity. These metrics allow capturing the hidden structure in the
market dynamics, and allow discriminating different degrees of informational efficiency. We find that some geopolitical events impact on the underlying dynamical structure of the market."
• artículo de publicación periódica.listelement.badge
On a goodness-of-fit test for normality with unknown parameters and type-II censored data
(2010-07) Castro-Kuriss, Claudia; Kelmansky, Diana M.; Leiva, Víctor; Martínez, Elena J.
"We propose a new goodness-of-fit test for normal and lognormal distributions with unknown parameters and type-II censored data. This test is a generalization of Michael's test for censored
samples, which is based on the empirical distribution and a variance stabilizing transformation. We estimate the parameters of the model by using maximum likelihood and Gupta's methods. The
quantiles of the distribution of the test statistic under the null hypothesis are obtained through Monte Carlo simulations. The power of the proposed test is estimated and compared to that of the
Kolmogorov-Smirnov test also using simulations. The new test is more powerful than the Kolmogorov-Smirnov test in most of the studied cases. Acceptance regions for the PP, QQ and Michael's
stabilized probability plots are derived, making it possible to visualize which data contribute to the decision of rejecting the null hypothesis. Finally, an illustrative example is presented."
• artículo de publicación periódica.listelement.badge
Using linear difference equations to model nonlinear cryptographic sequences
(2010-03) Caballero-Gil, Pino; Fúster-Sabater, Amparo; Pazo-Robles, María Eugenia
"A new class of linear sequence generators based on cellular automata is here introduced in order to model several nonlinear keystream generators with practical applications in symmetric
cryptography. The output sequences are written as solutions of linear difference equations, and three basic properties (period, linear complexity and number of different output sequences) are
• artículo de publicación periódica.listelement.badge
A truncated version of the birnbaum-saunders distribution with an application in financial risk
(2010-01) Ahmed, Syed Ejaz; Castro-Kuriss, Claudia; Flores, Esteban; Leiva, Víctor; Sanhueza, Antonio
"In many Solvency and Basel loss data, there are thresholds or deductibles that affect the analysis capability. On the other hand, the Birnbaum-Saunders model has received great attention during
the last two decades and it can be used as a loss distribution. In this paper, we propose a solution to the problem of deductibles using a truncated version of the Birnbaum-Saunders distribution.
The probability density function, cumulative distribution function, and moments of this distribution are obtained. In addition, properties regularly used in insurance industry, such as
multiplication by a constant (inflation effect) and reciprocal transformation, are discussed. Furthermore, a study of the behavior of the risk rate and of risk measures is carried out. Moreover,
estimation aspects are also considered in this work. Finally, an application based on real loss data from a commercial bank is conducted."
• artículo de publicación periódica.listelement.badge
A state estimation strategy for a nonlinear switched system with unknown switching signals
(2021) Benítez, Oscar; García Galiñanes, Rafael
"A strategy is presented to estimate the state of a nonlinear autonomous switched system, with no knowledge of the switching signal, except its dwell time. To do so, algorithms to estimate the
switching times and the current mode of the system are developed. The estimation of the switching times is based on approximating the second ( generalised) derivative of the output of the system
via a convolution of this signal with a suitable function and on detecting the corresponding spikes. To estimate the modes, a scheme based on the use of a bank of observers (one for each mode)
and of a bank of subsystems (for each step of the estimation process a suitable subset of the subsystems of the switched system) is developed. The algorithms run regardless of the state observer
model, as long as its output error norm decays exponentially with a controlled decay rate."
• artículo de publicación periódica.listelement.badge
Nonrobustness of asymptotic stability of impulsive systems with inputs
(2020-12) Haimovich, Hernán; Mancilla-Aguilar, J. L.
"Suitable continuity and boundedness assumptions on the function f defining the dynamics of a time-varying nonimpulsive system with inputs are known to make the system inherit stability
properties from the zero-input system. Whether this type of robustness holds or not for impulsive systems was still an open question. By means of suitable (counter)examples, we show that such
stability robustness with respect to the inclusion of inputs cannot hold in general, not even for impulsive systems with time-invariant flow and jump maps. In particular, we show that zero-input
global uniform asymptotic stability (0-GUAS) does not imply converging input converging state (CICS), and that 0-GUAS and uniform bounded-energy input bounded state (UBEBS) do not imply integral
input-to-state stability (iISS). We also comment on available existing results that, however, show that suitable constraints on the allowed impulse–time sequences indeed make some of these
robustness properties possible."
• artículo de publicación periódica.listelement.badge
Strong ISS implies strong iISS for time-varying impulsive systems
(2020-12) Haimovich, Hernán; Mancilla-Aguilar, J. L.
"For time-invariant (nonimpulsive) systems, it is already well-known that the input-to-state stability (ISS) property is strictly stronger than integral input-to-state stability (iISS). Very
recently, we have shown that under suitable uniform boundedness and continuity assumptions on the function defining system dynamics, ISS implies iISS also for time-varying systems. In this paper,
we show that this implication remains true for impulsive systems, provided that asymptotic stability is understood in a sense stronger than usual for impulsive systems"
• artículo de publicación periódica.listelement.badge
Uniform stability of nonlinear time-varying impulsive systems with eventually uniformly bounded impulse frequency
(2020-11) Mancilla-Aguilar, J. L.; Haimovich, Hernán; Feketa, Petro
"We provide novel sufficient conditions for stability of nonlinear and time-varying impulsive systems. These conditions generalize, extend, and strengthen many existing results. Different types
of input-to-state stability (ISS), as well as zero-input global uniform asymptotic stability (0-GUAS), are covered by employing a two-measure framework and considering stability of both weak
(decay depends only on elapsed time) and strong (decay depends on elapsed time and the number of impulses) flavors. By contrast to many existing results, the stability state bounds imposed are
uniform with respect to initial time and also with respect to classes of impulse-time sequences where the impulse frequency is eventually uniformly bounded. We show that the considered classes of
impulse-time sequences are substantially broader than other previously considered classes, such as those having fixed or (reverse) average dwell times, or impulse frequency achieving uniform
convergence to a limit (superior or inferior). Moreover, our sufficient conditions are stronger, less conservative and more widely applicable than many existing results."
• artículo de publicación periódica.listelement.badge
Converging-input convergent-state and related properties of time-varying impulsive systems
(2020-07-03) Mancilla-Aguilar, J. L.; Haimovich, Hernán
"Very recently, it has been shown that the standard notion of stability for impulsive systems, whereby the state is ensured to approach the equilibrium only as continuous time elapses, is too
weak to allow for any meaningful type of robustness in a time-varying impulsive system setting. By strengthening the notion of stability so that convergence to the equilibrium occurs not only as
time elapses but also as the number of jumps increases, some facts that are well-established for time-invariant nonimpulsive systems can be recovered for impulsive systems. In this context, our
contribution is to provide novel results consisting in rather mild conditions under which stability under zero input implies stability under inputs that converge to zero in some appropriate
• artículo de publicación periódica.listelement.badge
Uniform asymptotic stability of switched nonlinear time-varying systems and detectability of reduced limiting control systems
(2019-07) Mancilla-Aguilar, J. L.; García Galiñanes, Rafael
"This paper is concerned with the study of both, local and global, uniform asymptotic stability for switched nonlinear time-varying (NLTV) systems through the detectability of output-maps. With
this aim, the notion of reduced limiting control systems for switched NLTV systems whose switchings verify time/state-dependent constraints, and the concept of weak zero-state detectability for
those reduced limiting systems are introduced. Necessary and sufficient conditions for the (global)uniform asymptotic stability of families of trajectories of the switched system are obtained in
terms of this detectability property. These sufficient conditions in conjunction with the existence of multiple weak Lyapunov functions yield a criterion for the (global) uniform asymptotic
stability of families of trajectories of the switched system. This criterion can be seen as an extension of the classical Krasovskii-LaSalle theorem. An interesting feature of the results is that
no dwell-Time assumptions are made. Moreover, they can be used for establishing the global uniform asymptotic stability of the switched NLTV system under arbitrary switchings. The effectiveness
of the proposed results is illustrated by means of various interesting examples, including the stability analysis of a semiquasi-Z-source inverter."
• artículo de publicación periódica.listelement.badge
ISS implies iISS even for switched and time-varying systems (if you are careful enough)
(2019-06) Haimovich, Hernán; Mancilla-Aguilar, J. L.
"For time-invariant systems, the property of input-to-state stability (ISS) is known to be strictly stronger than integral-ISS (iISS). Known proofs of the fact that ISS implies iISS employ
Lyapunov characterizations of both properties. For time-varying and switched systems, such Lyapunov characterizations may not exist, and hence establishing the exact relationship between ISS and
iISS remained an open problem, until now. In this paper, we solve this problem by providing a direct proof, i.e. without requiring Lyapunov characterizations, of the fact that ISS implies iISS,
in a very general time-varying and switched-system context. In addition, we show how to construct suitable iISS gains based on the comparison functions that characterize the ISS property, and on
bounds on the function f defining the system dynamics. When particularized to time-invariant systems, our assumptions are even weaker than existing ones. Another contribution is to show that for
time-varying systems, local Lipschitz continuity of f in all variables is not sufficient to guarantee that ISS implies iISS. We illustrate application of our results on an example that does not
admit an iISS-Lyapunov function."
• artículo de publicación periódica.listelement.badge
Robustness properties of an algorithm for the stabilisation of switched systems with unbounded perturbations
(2017-05) Mancilla-Aguilar, J. L.; García Galiñanes, Rafael
"In this paper, it is shown that an algorithm for the stabilisation of switched systems introduced by the authors is robust with respect to perturbations which are unbounded in the supremum norm,
but bounded in a power-like sense. The obtained stability results comprise, among others, both the exponential input-to-state stability and the exponential integral input-to-state stability
properties of the closed-loop system and give a better description of the behaviour of the closed-loop system. "
• artículo de publicación periódica.listelement.badge
On zero-input stability inheritance for time-varying systems with decaying-to-zero input power
(2017-06) Mancilla-Aguilar, J. L.; Haimovich, Hernán
"Stability results for time-varying systems with inputs are relatively scarce, as opposed to the abundant literature available for time-invariant systems. This paper extends to time-varying
systems existing results that ensure that if the input converges to zero in some specific sense, then the state trajectory will inherit stability properties from the corresponding zero-input
system. This extension is non-trivial, in the sense that the proof technique is completely novel, and allows to recover the existing results under weaker assumptions in a unifying way."
• artículo de publicación periódica.listelement.badge
Global stability results for switched systems based on weak Lyapunov functions
(2017-06) Mancilla-Aguilar, J. L.; Haimovich, Hernán; García Galiñanes, Rafael
"In this paper we study the stability of nonlinear and time-varying switched systems under restricted switching. We approach the problem by decomposing the system dynamics into a nominal-like
part and a perturbationlike one. Most stability results for perturbed systems are based on the use of strong Lyapunov functions, i.e. functions of time and state whose total time derivative along
the nominal system trajectories is bounded by a negative definite function of the state. However, switched systems under restricted switching may not admit strong Lyapunov functions, even when
asymptotic stability is uniform over the set of switching signals considered. The main contribution of the current paper consists in providing stability results that are based on the stability of
the nominal-like part of the system and require only a weak Lyapunov function. These results may have wider applicability than results based on strong Lyapunov functions. The results provided
follow two lines. First, we give very general global uniform asymptotic stability results under reasonable boundedness conditions on the functions that define the dynamics of the nominal-like and
the perturbation-like parts of the system. Second, we provide input-to-state stability (ISS) results for the case when the nominal-like part is switched linear-timevarying. We provide two types
of ISS results: standard ISS that involves the essential supremum norm of the input and a modified ISS that involves a power-type norm."
• artículo de publicación periódica.listelement.badge
A characterization of Integral ISS for switched and time-varying systems
(2018-02) Haimovich, Hernán; Mancilla-Aguilar, J. L.
"Most of the existing characterizations of the integral input-to-state stability (iISS) property are not valid for time-varying or switched systems in cases where converse Lyapunov theorems for
stability are not available. This paper provides a characterization that is valid for switched and time-varying systems, and shows that natural extensions of some of the existing
characterizations result in only sufficient but not necessary conditions. The results provided also pinpoint suitable iISS gains and relate these to supply functions and bounds on the function
defining the system dynamics." | {"url":"https://ri.itba.edu.ar/collections/d0a39f93-2461-47c2-aa81-3f0fc7f2ffba","timestamp":"2024-11-05T11:06:51Z","content_type":"text/html","content_length":"719188","record_id":"<urn:uuid:e08cdaa5-68fc-4d2c-b47d-a95d9dd2b9e7>","cc-path":"CC-MAIN-2024-46/segments/1730477027878.78/warc/CC-MAIN-20241105083140-20241105113140-00668.warc.gz"} |
Is a term or coefficient?
Each term in an algebraic expression is separated by a + sign or J sign. In , the terms are: 5x, 3y, and 8. When a term is made up of a constant multiplied by a variable or variables, that constant
is called a coefficient. In the term 5x, the coefficient is 5.
What does the term coefficient mean in math?
coefficients are the number when you multiply a number and a variable. For example 5a the coefficient in that term is 5 if you have 48e the coefficient is 48. So the coefficient is the number when
you multiply a number times a variable.
What are coefficient like terms?
The coefficients are the numbers that multiply the variables or letters. Thus in 5x + y – 7, 5 is a coefficient. Like terms are terms that contain the same variable raised to the same power. In 5x +
y – 7 the terms are 5x, y and -7 which all have different variables (or no variables) so there are no like terms.
What is the coefficient of x²?
A coefficient refers to a number or quantity placed with a variable. It is usually an integer that is multiplied by the variable next to it. Coefficient of x² is 1.
What does term mean in math?
A term is a single mathematical expression. It may be a single number (positive or negative), a single variable ( a letter ), several variables multiplied but never added or subtracted. Some terms
contain variables with a number in front of them. The number in front of a term is called a coefficient.
What is a term factor and coefficient?
An expression comprises the terms, factors, and coefficient. The terms are the numbers or the variables added together, factors are the numbers or the variables that are multiplied together and the
coefficient is the number multiplied to the variable. In an expression 3×2 + 5x + 2, there are 3 terms: 3×2, 5x and 2.
How do you find the coefficient of Class 9?
Coefficient of Polynomial: Each term of a polynomial has a coefficient. So, in p(x)=9×3 – 3×2 +8x – 2, the coefficient of x3 is 9, the coefficient of x2 is -3, the coefficient of x is 8 and –2 is the
coefficient of x0. Constant & Zero polynomial: 9 is also a polynomial. In fact, 4, –8, 32, etc.
What is a constant term example?
A constant term is a term that contains only a number. In other words, there is no variable in a constant term. Examples of constant terms are 4, 100, and -5.
What is the coefficient of x³?
In the term x3, the coefficient of x3 is 1. In 2x, the coefficient of 2x is 2, and 3 is a constant. Therefore, the coefficients are 1 and 2.
What is binomial coefficient give an example?
For example, (x+y)3=1⋅x3+3⋅x2y+3⋅xy2+1⋅y3, and the coefficients 1, 3, 3, 1 form row three of Pascal’s Triangle. For this reason the numbers (nk) are usually referred to as the binomial coefficients. | {"url":"https://quick-advices.com/is-a-term-or-coefficient/","timestamp":"2024-11-04T23:08:06Z","content_type":"text/html","content_length":"148290","record_id":"<urn:uuid:d5d2d66d-593b-456b-97ec-2233b5ff1c6f>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.84/warc/CC-MAIN-20241104225856-20241105015856-00157.warc.gz"} |