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Worksheet On Multiplying Whole Numbers Using Area Model
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Multiplying Fractions Area Model Teaching Resources
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33 Multiplying Fractions Using Area Models Worksheet Support Worksheet
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Leaving Cert?
What is on paper 1 maths Leaving Cert?
This paper was heavy with number patterns, algebra, financial maths and functions, but very light on differentiation.” Mr Boal said the short questions on the ordinary level paper were very doable
and students had to be happy about with topics including VAT, complex numbers, algebra, functions, calculus and patterns.
What topics are on maths paper 1 higher level Leaving Cert?
Paper 1 Topics
• Algebra and Functions.
• Complex Numbers.
• Calculus.
• Sequences and Series.
• Financial Maths.
• Area and Volume.
Can you drop maths in Year 11?
Mathematics will be made compulsory for students in years 11 and 12.
What is a pass in Leaving Cert ordinary maths?
Anything from A-D is a pass, anything below is a fail. A = 85-100% B = 70 – 84% C = 55-69% D = 40-54%
What’s the difference between maths paper 1 and 2?
Paper 1 is 1.5 hours in length with shorter questions. Paper 2 is 2.5 hours in length with extended answers to more in-depth questions, which is very useful preparation for extended problems
encountered at the A Level standard.
What comes up in paper 2 maths?
• BIDMAS (brackets)
• Interpret calculator displays.
• Rounding and estimation, error intervals.
• Compare fractions, decimals and percentages.
• Fractions and ratio problems.
• Recurring decimal to fraction (prove)
• Index Laws (division, negative and fractional)
What topics are in paper 1 Mathematics?
Paper 1 will include the following subject areas:
• Equations and Inequalities.
• Number patterns and sequences.
• Functions and Graphs.
• Financial Mathematics.
• Calculus.
• Linear Programming.
Is dropping maths a good idea?
You can’t drop maths in class 12. If you have Biology along as an subject and an additional than you can choose not to appear for maths as passing only in 5 subjects is necessary. But you can’t drop
it as whole, it will appear in your mark sheet. It’s really not a good idea.
Can you fail Year 11?
Yes, but it’s uncommon. Usually students would choose to do Year 11 again at a different school. Failing Year 11 is difficult unless you repeatedly make little/no effort in your subjects.
How hard is it to get 500 points in Leaving Cert?
The average Leaving Cert does not equate to 500 points-plus; far from it. Those points are achieved by very few. You would be closer to the average mark if you divided the 500 by two. The actual
entry requirements set by third-level colleges are not astronomical either.
Is 300 points good in the Leaving Cert?
New CAO figures indicate that the average Leaving Cert student will get about 300 points today, much less than commonly thought. A 300-point score, for example, would not be enough to secure a place
on a university arts course. Well over 500 points are required for courses like law and medicine.
Is maths paper 1 or 2 harder?
This morning’s Higher Level Maths Paper 2 was significantly more difficult than Friday’s Paper 1. Students would have been far less happy leaving this paper. The others would have been familiar to
most students. In Section A however, 6 of the 14 question parts were at the upper end on the scale of difficulty.
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Search, Seek, and Discover Jain Literature.
Book Title: Note on Hemchandras Abhidhanchintamani and Sanskrit Karmavati
Author(s): Nalini Balbir
Publisher: ZZ_Anusandhan
अनुसन्धान-५४ श्रीहेमचन्द्राचार्यविशेषांक भाग-२
disciples of monks, not professional scribes but this element is probably not relevant anyway.
Although Hemacandra's record proves that the word was known in the 12th century, no record of it could be traced in the earliest available contemporary manuscripts, those on palm-leaf. But this absence has to be considered within a broader perspective: a word meaning "date" or "day" is not systematically mentioned in the colophons of these manuscripts. The general pattern is, rather: number-week day - adya iha + place name.!In the later phases, the date formula is expanded in full, and all resources of the calendar vocabulary are made use of consistently: for example, pratipad “the first day of the lunar fortnight”, pārņimā or rākā “full moon day”, or a less frequent term such as bhūtestā “fourteenth day of a fortnight” (see below Appendix “VS 1716"), when the actual date requires it. If the manuscript or inscription is written on a festival day, its name may be given.14 Synonyms for the names of the months and the week days are often handled skillfully with literary ambitions.15 The word k. is part of such a development. Its occurrences are much later than the palm-leaf manuscript period. But, on the other hand, the word has a
13. E.g.: samvat 1191 varse Bhādrapada sudi 8 bhaume adyeha Dhavalakke ..., samvat 1330 varșe Vaišākha sudi 14 gurau.... etc. 14. E.g. Vaišākha-sukla-pakse 3 aksayatrtiya dine. etc. See below Appendix "VS 1783” for another example. 15. See individual notes in the Appendix below. - Other rare names of months are recorded and discussed in the Sesasamgraha by Hemacandra, the Appendix to his AC, on which see Th. Zachariae, “Die Nachträge zu dem synonymischen Worterbuch des Hemacandra” (WZKM 16, 1902, reprinted in Kleine Schriften, Wiesbaden. 1977, pp. 471-502). ucchara for Vaišākha and sairin for Kārttika are two such examples (p. 479 n. 4 and p. 480 n. 1). Sanskrit grammars, especially that of Hemacandra, have special sutras regarding the formation of nouns or adjectives relating to the calendar: see F. Kielhorn, "Pausha Samvatsara”, The Indian Antiquary 1893, reprinted in Kleine Schriften. Wiesbaden, 1969, pp. 274-275.
Page Navigation
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third_party/bigint/BigUnsignedInABase.hh - pdfium - Git at Google
// Copyright 2014 PDFium Authors. All rights reserved.
// Use of this source code is governed by a BSD-style license that can be
// found in the LICENSE file.
// Original code by Matt McCutchen, see the LICENSE file.
#ifndef BIGUNSIGNEDINABASE_H
#define BIGUNSIGNEDINABASE_H
#include "NumberlikeArray.hh"
#include "BigUnsigned.hh"
#include <string>
* A BigUnsignedInABase object represents a nonnegative integer of size limited
* only by available memory, represented in a user-specified base that can fit
* in an `unsigned short' (most can, and this saves memory).
* BigUnsignedInABase is intended as an intermediary class with little
* functionality of its own. BigUnsignedInABase objects can be constructed
* from, and converted to, BigUnsigneds (requiring multiplication, mods, etc.)
* and `std::string's (by switching digit values for appropriate characters).
* BigUnsignedInABase is similar to BigUnsigned. Note the following:
* (1) They represent the number in exactly the same way, except that
* BigUnsignedInABase uses ``digits'' (or Digit) where BigUnsigned uses
* ``blocks'' (or Blk).
* (2) Both use the management features of NumberlikeArray. (In fact, my desire
* to add a BigUnsignedInABase class without duplicating a lot of code led me to
* introduce NumberlikeArray.)
* (3) The only arithmetic operation supported by BigUnsignedInABase is an
* equality test. Use BigUnsigned for arithmetic.
class BigUnsignedInABase : protected NumberlikeArray<unsigned short> {
// The digits of a BigUnsignedInABase are unsigned shorts.
typedef unsigned short Digit;
// That's also the type of a base.
typedef Digit Base;
// The base in which this BigUnsignedInABase is expressed
Base base;
// Creates a BigUnsignedInABase with a capacity; for internal use.
BigUnsignedInABase(int, Index c) : NumberlikeArray<Digit>(0, c) {}
// Decreases len to eliminate any leading zero digits.
void zapLeadingZeros() {
while (len > 0 && blk[len - 1] == 0)
// Constructs zero in base 2.
BigUnsignedInABase() : NumberlikeArray<Digit>(), base(2) {}
// Copy constructor
BigUnsignedInABase(const BigUnsignedInABase &x) : NumberlikeArray<Digit>(x), base(x.base) {}
// Assignment operator
void operator =(const BigUnsignedInABase &x) {
NumberlikeArray<Digit>::operator =(x);
base = x.base;
// Constructor that copies from a given array of digits.
BigUnsignedInABase(const Digit *d, Index l, Base base);
// Destructor. NumberlikeArray does the delete for us.
~BigUnsignedInABase() {}
// LINKS TO BIGUNSIGNED
BigUnsignedInABase(const BigUnsigned &x, Base base);
operator BigUnsigned() const;
/* LINKS TO STRINGS
* These use the symbols ``0123456789ABCDEFGHIJKLMNOPQRSTUVWXYZ'' to
* represent digits of 0 through 35. When parsing strings, lowercase is
* also accepted.
* All string representations are big-endian (big-place-value digits
* first). (Computer scientists have adopted zero-based counting; why
* can't they tolerate little-endian numbers?)
* No string representation has a ``base indicator'' like ``0x''.
* An exception is made for zero: it is converted to ``0'' and not the
* empty string.
* If you want different conventions, write your own routines to go
* between BigUnsignedInABase and strings. It's not hard.
operator std::string() const;
BigUnsignedInABase(const std::string &s, Base base);
// ACCESSORS
Base getBase() const { return base; }
// Expose these from NumberlikeArray directly.
using NumberlikeArray<Digit>::getCapacity;
using NumberlikeArray<Digit>::getLength;
/* Returns the requested digit, or 0 if it is beyond the length (as if
* the number had 0s infinitely to the left). */
Digit getDigit(Index i) const { return i >= len ? 0 : blk[i]; }
// The number is zero if and only if the canonical length is zero.
bool isZero() const { return NumberlikeArray<Digit>::isEmpty(); }
/* Equality test. For the purposes of this test, two BigUnsignedInABase
* values must have the same base to be equal. */
bool operator ==(const BigUnsignedInABase &x) const {
return base == x.base && NumberlikeArray<Digit>::operator ==(x);
bool operator !=(const BigUnsignedInABase &x) const { return !operator ==(x); }
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Introduction to multiple-bias sensitivity analysis
It is likely that an epidemiologic study is affected by more than one bias. We can use the EValue package to assess biases jointly. We start by characterizing the biases of interest: unmeasured
confounding, selection bias, and differential misclassification. Functions allow us to characterize these biases according to the available options.
• confounding()
□ This function does not need any additional arguments.
• selection(...)
□ Selection bias can be described by "general" if the inferential goal is the total population, and by "selected" if the target is the selected population only. If "general", the default, is
chosen, additional arguments are "increased risk" or "decreased risk" (assumptions about the direction of risk in the selected population) and "S = U" (simplification used if the biasing
characteristic is common to the entire selected population). See selection bias for more details on the interpretation of these options.
• misclassification(...)
□ Misclassification can be characterized by "outcome" or "exposure" depending on whether differential outcome or exposure misclassification is of interest. If "exposure" is chosen, additional
arguments rare_outcome and rare_exposure must be set to TRUE or FALSE, depending on whether the outcome and/or exposure are rare enough for odds ratios to be approximately equivalent to risk
Each bias additional takes the argument verbose, which specifies if any messages should be printed to the console when calling the function. This argument should generally be unspecified (default
verbose = FALSE) as the appropriate messages will be printed when using the biases in other functions, but may occasionally be helpful for debugging.
To use the sensitivity analysis functions provided in this package, these biases can be combined using multi_bias():
biases <- multi_bias(confounding(),
selection("general", "increased risk"),
misclassification("exposure", rare_outcome = TRUE))
Parameters describing the biases
There are 1-4 parameters that characterize each bias, but they differ depending on the ordering of the biases and on the options chosen. The interpretation of the parameters is given in Smith et
al. 2020, but briefly, each is a risk (or odds) ratio (RR/OR) relating two variables, possibly conditional on others. (Each is additionally conditional on other measured covariates, omitted from the
notation for simplicity.) The exposure variable is labeled \(A\), the outcome \(Y\), and their misclassified versions \(A^*\) and \(Y^*\), respectively. Selection into the sample is denoted with \(S
= 1\). Finally, unmeasured confounding and selection bias are assumed to be due to unmeasured variables \(U_c\) and \(U_s\), respectively.
The table below contains the entire list of parameters using the notation in the Smith et al. paper. The corresponding output that will be printed in the R console is also given. Finally, the
argument used to specify the magnitude of the parameter in the multi_bound() function is in the last column.
confounding \[\text{RR}_{AU_c}\] RR_AUc RRAUc
confounding \[\text{RR}_{U_cY}\] RR_UcY RRUcY
selection after outcome misclassification \[\text{RR}_{U_sY \mid A = 1}\] RR_UsY|A=1 RRUsYA1
selection after outcome misclassification \[\text{RR}_{SU_s \mid A = 1}\] RR_SUs|A=1 RRSUsA1
selection after outcome misclassification \[\text{RR}_{U_sY \mid A = 0}\] RR_UsY|A=0 RRUsYA0
selection after outcome misclassification \[\text{RR}_{SU_s \mid A = 0}\] RR_SUs|A=0 RRSUsA0
selection after exposure misclassification \[\text{RR}_{U_sY^* \mid A = 1}\] RR_UsY*|A=1 RRUsYA1
selection after exposure misclassification \[\text{RR}_{U_sY^* \mid A = 0}\] RR_UsY*|A=0 RRUsYA0
selection after outcome misclassification \[\text{RR}_{U_sY \mid A^* = 1}\] RR_UsY|A*=1 RRUsYA1
selection after outcome misclassification \[\text{RR}_{SU_s \mid A^* = 1}\] RR_SUs|A*=1 RRSUsA1
selection after outcome misclassification \[\text{RR}_{U_sY \mid A^* = 0}\] RR_UsY|A*=0 RRUsYA0
selection after outcome misclassification \[\text{RR}_{SU_s \mid A^* = 0}\] RR_SUs|A*=1 RRSUsA1
confounding and selection \[\text{RR}_{AU_{sc}\mid S = 1}\] RR_AUsc|S RRAUscS
confounding and selection \[\text{RR}_{U_{sc}Y\mid S = 1}\] RR_UscY|S RRUscYS
outcome misclassification \[\text{RR}_{AY^* \mid y}\] RR_AY*|y RRAYy
exposure misclassification (rare outcome) \[\text{OR}_{YA^* \mid a}\] OR_YA*|a ORYAa
exposure misclassification (rare exposure and outcome) \[\text{RR}_{YA^* \mid a}\] RR_YA*|a RRYAa
outcome misclassification \[\text{RR}_{AY^* \mid y, S = 1}\] RR_AY*|y,S RRAYyS
exposure misclassification (rare outcome) \[\text{OR}_{YA^* \mid a, S = 1}\] OR_YA*|a,S ORYAaS
exposure misclassification (rare exposure and outcome) \[\text{RR}_{YA^* \mid a, S = 1}\] RR_YA*|a,S RRYAaS
Ordering of the biases
If both present, selection bias and misclassification should be listed in the multi_bias() function in the order in which they are assumed to affect the data. Confounding is assumed to be a state of
nature that does not depend on how the data is selected or measured. If selection occurs before the exposure/outcome measurement, as is implied in the code above, then we can define parameters that
describe the extent of differential misclassification within the selected group. If measurement takes place before selection, then the parameters describing selection are in terms of the
misclassified variables.
We can easily see which parameters describe the biases of interest using the summary() function on the object created by multi-bias():
#> bias output argument
#> 1 confounding RR_AUc RRAUc
#> 2 confounding RR_UcY RRUcY
#> 3 selection RR_UsY|A=1 RRUsYA1
#> 4 selection RR_SUs|A=1 RRSUsA1
#> 5 exposure misclassification OR_YA*|a,S ORYAaS
Contrast the above parameters with those we would get if we switched the ordering of the misclassification and the selection. These now imply that selection differs on the basis of the mismeasured
exposure, and that the misclassification parameter can be interpreted with respect to the total population, not just those selected for the study.
misclassification("exposure", rare_outcome = TRUE),
selection("general", "increased risk"))
#> bias output argument
#> 1 confounding RR_AUc RRAUc
#> 2 confounding RR_UcY RRUcY
#> 3 selection RR_UsY|A*=1 RRUsYA1
#> 4 selection RR_SUs|A*=1 RRSUsA1
#> 5 exposure misclassification OR_YA*|a ORYAa
Once we know which biases are of interest and have characterized them using the function arguments, we can calculate a bound for the joint bias. We must choose values for the various parameters
defining the magnitude of the biases.
Calculating a bound does not require memorizing the arguments in the above table. Instead, the parameters that need to be specified for a given set of biases can be printed using the print() function
on an object created by multi_bias(). For example, using the biases chosen above:
#> The following arguments can be copied and pasted into the multi_bound()
#> function: RRAUc = , RRUcY = , RRUsYA1 = , RRSUsA1 = , ORYAaS =
We can then choose values for the necessary parameters. For our example, suppose we think that an unmeasured confounder \(U_c\) is associated with a 2-fold increased risk of the outcome and is 1.5
times as likely within the exposed compared to the unexposed groups. Then RRUcY = 2 and RRAUc = 1.5. Then we believe that, among the exposed, the selected group is 1.25 times as likely to have some
level of unmeasured variable \(U_s\) than the non-selected group, and that \(U_s\) is associated with a 2.5-fold increase in the risk of the outcome. This would imply that RRSUsA1 = 1.25 and RRUsYA1
= 2.5. Finally, we hypothesize that the odds of a false-positive exposure measurement within this selected group were 1.75 times higher in the exposed than unexposed, so that ORYAaS = 1.75. We can
calculate the maximum bias we would see if all those parameters described the true extent of the bias:
RRUcY = 2, RRAUc = 1.5,
RRSUsA1 = 1.25, RRUsYA1 = 2.5,
ORYAaS = 1.75)
That is, if those values are correct, then the true risk ratio can be no more than 2.4 times smaller than the observed risk ratio. So if our observed risk ratio were 4, the true risk ratio must be at
least 1.7.
If you don’t include the necessary parameter arguments given your biases of interest, an error will inform you which are necessary. Values of 1 imply no bias, so arguments set equal to 1 can be
used to explore the absence of a certain bias.
Because we generally don’t know the exact magnitude of the parameters, it can be useful to calculate bounds with a range of values. For example, we can vary each of the parameters from 1 to 3 in
increments of 0.25:
param_vals <- seq(1, 3, by = 0.5)
# create every combination of values
params <- expand.grid(
RRUcY = param_vals, RRAUc = param_vals,
RRSUsA1 = param_vals, RRUsYA1 = param_vals,
ORYAaS = param_vals
params$bound <- mapply(multi_bound,
RRUcY = params$RRUcY, RRAUc = params$RRAUc,
RRSUsA1 = params$RRSUsA1, RRUsYA1 = params$RRUsYA1,
ORYAaS = params$ORYAaS,
MoreArgs = list(biases = biases)
There are two many dimensions to summarize the relationship between the parameters and the bounds in a simple table or figure, but we can examine the overall distribution of the bounds as well as how
they depend on several of the parameters. For example, a simple histogram of the bounds calculated by varying each of the bias parameters between 1 and 3.
hist(params$bound, main = NULL, xlab = "Bound")
Multi-bias E-values
We can also calculate multi-bias E-values, which are analogous to E-values for unmeasured confounding but take into account multiple biases. The multi-bias E-value describes the minimum value that
all of the sensitivity parameters for each of the biases would have to take on for a given observed risk ratio to be compatible with a truly null risk ratio.
To calculate a multi-bias evalue, we declare a set of biases as before, and then specify the observed risk ratio. For example, given the biases we have been working with and an observed risk ratio of
4, the multi-bias E-value is round(summary(multi_evalue(biases, RR(4))), 2):
multi_evalue(biases, est = RR(4))
#> This multi-bias e-value refers simultaneously to parameters RR_AUc,
#> RR_UcY, RR_UsY|A=1, RR_SUs|A=1, RR_YA*|a,S . (See documentation for
#> details.)
#> point lower upper
#> RR 4.00000 NA NA
#> Multi-bias E-values 1.67513 NA NA
Notice that we have specified that our estimate is a risk ratio using the RR() function. If we want to instead calculate a multi-bias E-value for an odds ratio or a hazard ratio, we must specify so
with the appropriate function, as well as decide whether it’s reasonable to assume that the outcome is rare enough to use a risk ratio approximation. If not, other approximations will be used.
# square-root approximation of the odds ratio
multi_evalue(biases, est = OR(4, rare = FALSE))
#> This multi-bias e-value refers simultaneously to parameters RR_AUc,
#> RR_UcY, RR_UsY|A=1, RR_SUs|A=1, RR_YA*|a,S . (See documentation for
#> details.)
#> point lower upper
#> RR 2.000000 NA NA
#> Multi-bias E-values 1.327958 NA NA
To additionally calculate a multi-bias E-value for the confidence interval, we can include it with the lo = and hi = arguments (these will be assumed to be on the same scale as the point estimate):
# use verbose = FALSE to suppress message about parameters
multi_evalue(biases, est = RR(4), lo = 2.5, hi = 6, verbose = FALSE)
#> point lower upper
#> RR 4.00000 2.500000 6
#> Multi-bias E-values 1.67513 1.435575 NA
The function can also accommodate protective estimates:
multi_evalue(biases, est = RR(0.25), lo = 0.17, hi = 0.4, verbose = FALSE)
#> point lower upper
#> RR 0.25000 0.17 0.400000
#> Multi-bias E-values 1.67513 NA 1.435575
Finally, if we are calculating a multi-bias E-value for a point estimate and just want to output the single value, we can use the summary() function, which will also automatically suppress the
message about the parameters:
summary(multi_evalue(biases, est = RR(4)))
#> [1] 1.67513
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Sum of cubes of first n natural numbers
Sum of cubes of first n natural numbers Calculator
Calculating the Sum of Cubes of the First n Natural Numbers
The sum of the cubes of the first n natural numbers can be calculated using a specific formula. This formula provides a convenient method to determine the sum without manually calculating the cube of
each number and adding them together.
The formula used is:
\(\text{Sum} = \left(\dfrac{ n(n+1)}{2}\right)^2\)
• n represents the number of natural numbers.
This can be expressed as:
\( 1^3 + 2^3 + 3^3 + ... + n^3 \)
Let’s explore some examples to understand how this formula is applied in practice.
1. Calculate the sum of cubes of the first 8 natural numbers.
To begin, we determine the value of n:
Next, we apply the standard formula:
\( \text{Sum} = \left(\dfrac{ n(n+1)}{2}\right)^2 \)
We substitute the value of n into the formula to calculate the sum:
\( \text{Sum} = \left(\dfrac{8(8+1)}{2}\right)^2 \)
\( \text{Sum} = \left(\dfrac{8(9)}{2}\right)^2 \)
\( \text{Sum} = \left(\dfrac{72}{2}\right)^2 \)
\( \text{Sum} = (36)^2 \)
\( \text{Sum} = 1296 \)
Finally, we conclude with the result:
∴ The sum of cubes of the first 8 natural numbers is 1296.
2. What is the sum of the cubes of the first 5 natural numbers?
We start by determining the value of n:
Next, we use the established formula to find the sum:
\( \text{Sum} = \left(\dfrac{ n(n+1)}{2}\right)^2 \)
We substitute n into the formula and simplify:
\( \text{Sum} = \left(\dfrac{5(5+1)}{2}\right)^2 \)
\( \text{Sum} = \left(\dfrac{5(6)}{2}\right)^2 \)
\( \text{Sum} = \left(\dfrac{30}{2}\right)^2 \)
\( \text{Sum} = (15)^2 \)
\( \text{Sum} = 225 \)
The result is:
∴ The sum of cubes of the first 5 natural numbers is 225.
3. Calculate the sum of the cubes of the first 10 natural numbers.
We first identify the value of n:
Next, we apply the formula to calculate the sum:
\( \text{Sum} = \left(\dfrac{ n(n+1)}{2}\right)^2 \)
We then substitute n into the formula and perform the calculation:
\( \text{Sum} = \left(\dfrac{10(10+1)}{2}\right)^2 \)
\( \text{Sum} = \left(\dfrac{10(11)}{2}\right)^2 \)
\( \text{Sum} = \left(\dfrac{110}{2}\right)^2 \)
\( \text{Sum} = (55)^2 \)
\( \text{Sum} = 3025 \)
The final result is:
∴ The sum of cubes of the first 10 natural numbers is 3025.
4. Find the sum of cubes of the first 12 natural numbers.
We begin by determining n:
Next, we use the standard formula to perform the calculation:
\( \text{Sum} = \left(\dfrac{ n(n+1)}{2}\right)^2 \)
We substitute the value of n into the formula and simplify:
\( \text{Sum} = \left(\dfrac{12(12+1)}{2}\right)^2 \)
\( \text{Sum} = \left(\dfrac{12(13)}{2}\right)^2 \)
\( \text{Sum} = \left(\dfrac{156}{2}\right)^2 \)
\( \text{Sum} = (78)^2 \)
\( \text{Sum} = 6084 \)
The final sum is:
∴ The sum of cubes of the first 12 natural numbers is 6084.
To calculate the sum of cubes of first n natural numbers, you can use the following formula.
\( = \) \({\{ \dfrac{ n ( n +1)}{2}\} }^2\)
Once you enter the input values in the calculator, the output parameters are calculated.
Frequently Asked Questions (FAQs)
1. What is the sum of cubes of the first n natural numbers?
The sum of cubes of the first n natural numbers is the total obtained by cubing each number from 1 to n and then adding them together. For example, if n is 3, the cubes are 1³, 2³, and 3³, which add
up to 36.
2. How do you calculate the sum of cubes of the first n natural numbers?
To calculate the sum of cubes of the first n natural numbers, use the formula: \( \text{Sum} = \left( \frac{n(n+1)}{2} \right)^2 \). This formula finds the sum by squaring the sum of the first n
natural numbers. For example, for n = 4, the sum is 100.
3. Why does the formula \( \left( \frac{n(n+1)}{2} \right)^2 \) work for the sum of cubes?
The formula \( \left( \frac{n(n+1)}{2} \right)^2 \) works because it is based on the pattern that the sum of cubes of the first n natural numbers is the square of the sum of the first n natural
numbers. This relationship simplifies finding the total sum of cubes.
4. Can I use this formula for negative numbers?
No, the formula \( \left( \frac{n(n+1)}{2} \right)^2 \) is meant for natural numbers, which are positive integers starting from 1. It cannot be applied to negative numbers or zero because natural
numbers do not include these values.
5. How does a calculator for the sum of cubes of first n natural numbers work?
The calculator applies the formula \( \left( \frac{n(n+1)}{2} \right)^2 \) to quickly compute the sum of cubes. You just need to input the value of n, and the calculator will automatically find the
result without manual calculations, providing an accurate sum.
6. What is the sum of cubes for the first 5 natural numbers?
To find the sum of cubes for the first 5 natural numbers, use the formula \( \left( \frac{5(5+1)}{2} \right)^2 \). This gives \( \left( \frac{5*6}{2} \right)^2 = 225 \). Therefore, the sum of cubes
from 1 to 5 is 225.
7. Is the sum of cubes of n natural numbers always a perfect square?
Yes, the sum of cubes of the first n natural numbers is always a perfect square because the formula \( \left( \frac{n(n+1)}{2} \right)^2 \) results in squaring a whole number. This ensures that the
sum will always be a perfect square.
8. Why is finding the sum of cubes of the first n natural numbers useful?
The sum of cubes of the first n natural numbers is useful in various mathematical applications, such as algebra and calculus. It helps in solving problems related to sequences, series, and finding
patterns in sums. It's also significant in some geometry and physics calculations.
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Dollar cost averaging
Dollar cost averaging
As an engineer paid in a foreign currency, what is the best strategy to maximize your income? Is it better to exchange when you receive your money on a monthly basis? Or is it better to exchange a
little bit every week or every day?
TLDR; The daily exchange provides the best results over time on the assumption that the exchange fees are proportional to the exchanged amount.
First, let's grab the USD/BRL prices from Yahoo Finance. I'll use pandas_datareader and yfinance.
import pandas_datareader.data as pdr
import yfinance as yf
df = pdr.get_data_yahoo('USDBRL=X')
Let S be your annual salary, n be the number of exchanges and (x[1], x[2], ... x[n]) the USD/BRL exchange rate at the n periods. $Sn$ is the exchanged amount at each period. The total exchanged
amount is: $Sn ∑ i=1 n xi = S ∑ i=1 n xi n = S x ̅$
Let m be another number of exchanges and (y[1], y[2], ... y[m]) the USD/BRL exchange rate at the m periods. The frequency yielding the optimal exchanged amount is the one with the highest average: $S
x ̅≥ S y ̅⇔ x ̅≥ y ̅$
To determine the best frequency for currency exchange, we'll compare averages from three datasets: daily rates, weekly rates, and monthly rates.
Daily rates
In [1]: df["Close"].mean()
Out[1]: 3.1029914017829077
Weekly rates
In [2]: df["Close"].resample("W").first().mean()
Out[2]: 3.09241014059527
Monthly rates
In [3]: df["Close"].resample("M").first().mean()
Out[3]: 3.087195667895404
Over time, the daily exchange will give you the best result, but it doesn't mean that it is always the case. The weekly exchange yields better results if we only take data starting from 2022.
Daily rates
In [4]: df["Close"]["2022":].mean()
Out[4]: 5.099161933890373
Weekly rates
In [5]: df["Close"]["2022":].resample("W").first().mean()
Out[5]: 5.100240080544118
Monthly rates
In [6]: df["Close"]["2022":].resample("M").first().mean()
Out[6]: 5.095680486588251
But if we start from 2020, then the daily exchange is best again.
Daily rates
In [9]: df["Close"]["2020":].mean()
Out[9]: 5.191872068725996
Weekly rates
In [10]: df["Close"]["2020":].resample("W").first().mean()
Out[10]: 5.186581481363356
Monthly rates
In [11]: df["Close"]["2020":].resample("M").first().mean()
Out[11]: 5.182012155320909
Flat exchange fees
Our calculations assume exchange fees are proportional to the exchanged amount A. However, if there's a set fee F for each exchange, the ideal strategy will be influenced by both F and A.
Intuitively, a R$1 fee is trivial if you earn R$1M/month.
Using S and (x[1], x[2], ... x[n]) the exchanged amount becomes $S x ̅- n F$.
Once again, let's see which exchange frequency yields the most total amount: $S x ̅- n F ≥ S y ̅- m F ⇔ S ( x ̅- y ̅) ≥ ( n - m ) F ⇔ S ≥ n - m x ̅- y ̅F$
To benefit from daily exchanges in 2022, your yearly salary should be around 32,000 times the flat fee. For weekly exchanges, it should be around 4,500 times the fee.
Salary multiplier for daily over monthly in 2022
In [30]: df2022 = df["Close"]["2022":"2022-12-31"]
In [34]: (len(df2022)-12)/(df2022.mean() - df2022.resample("M").first().mean())
Out[34]: 31746.156236195482
Salary multiplier for weekly over monthly in 2022
In [45]: (52-12)/(df2022.resample("W").first().mean() - df2022.resample("M").first().mean())
Out[45]: 4650.885284075261
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A BIT, is a BINARY DIGIT.
A bit can be a zero or a 1.
A binary number made of eight bits, such as 11001010 is known as a BYTE.
Four bits, such as 1101 is known as a NIBBLE (half a byte and a joke).
Sixteen bits numbers are a WORD.
Thirty two bits ones are a LONG WORD.
Sixty four bits numbers are a VERY LONG WORD.
These numbers can be stored in REGISTERS inside chips (integrated circuits).
1k in binary systems is 1024.
These collections of bits can represent binary numbers. They can also represent decimal or other number systems.
The ASCII system uses them to represent the letters of the alphabet and punctuation. The ASCII TABLE gives the binary equivalents of the alphabet.
All this information is called DATA.
Numbers in microprocessor systems are often expressed in hexadecimal.
The microprocessor is also called the CENTRAL PROCESSING UNIT (CPU).
There are very many cpu's and one of the most common is the 6502.
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- Math and Native American Culture - College of Sciences News
Math and Native American Culture
Dr. Robert Megginson from the University of Michigan will be speaking at UCF about the role of mathematics among American Indians.
In the early 1930s, Will Ryan, Director of Indian Education for the Bureau of Indian Affairs (BIA), eliminated algebra and geometry from the Uniform Course of Study in BIA schools. This was done in a
well-intentioned, but misguided effort to make BIA education more culturally relevant for American Indians.
Their belief was that mathematics has had no historical or cultural importance for the indigenous peoples of the Western Hemisphere. In fact, examples abound of the importance of mathematics in many
Native cultures of the Americas. The well-developed number systems of pre-contact Mesoamerica are probably the most well-known.
Dr. Megginson will present this system along with some of its number-theoretic underpinnings and consequences, as well as the cultural values that led to some of its structure.
One of only about 12 Native Americans who hold a PhD in mathematics, Robert Megginson grew up in a family who was interested in math. For the past decade Dr. Megginson has spent his time working to
solve the problem of the under-representation of minorities in the field of mathematics. In 1992 he developed a summer program for high school students at the Turtle Mountain Indian Reservation in
North Dakota. The purpose of the program is to keep the students interested in mathematics and related fields and encourage them to pursue college degrees in these areas.
Dr. Megginson has been very involved in the problems of minority mathematics education, especially that of Native Americans. He is a Sequoyah Fellow of the American Indian Science and Engineering
Society, and works actively through the programs of this organization to further the participation of Native American people in mathematics. To read more about Dr. Megginson click here.
His talk will take place on Friday, Feb. 27 in the Mathematical Sciences Building, room 318 at 11 a.m.
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Mm to Inches (Table & Converter)
Some countries use metric units while others use imperial to define length and distance. In a globalized world, it's important you know how to convert between them, and fast.
Want a secret weapon you can use to get the job done right every time? Try our mm to inches calculator. It works with more than 60 length units and provides results up to six decimal places.
If you are wondering how many inches are in one mm, the answer is 0.039370. To easily convert your millimeters to inches you can divide your starting number by 25.4.
In the guide below, we're going to teach you how to convert from mm to inches and inches to mm. We'll also answer some common questions about the origins of both measuring systems.
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How many feet are in 185 cm? - WorkSheets Buddy
How many feet are in 185 cm?
How many feet are in 185 cm?
Final Answer:
185 centimeters is approximately equal to 6.07 feet. This conversion can be calculated by dividing the number of centimeters by 30.48. Understanding unit conversions is valuable in many real-world
Examples & Evidence:
To convert centimeters to feet, you can use the conversion factor that 1 foot is equal to 30.48 centimeters. To find out how many feet are in 185 centimeters, you would set up the conversion as
1. Identify the conversion factor:
1 foot = 30.48 centimeters.
2. Set up the equation:
Use the formula:
$\text { Feet }=\frac{\mathrm{Cm}}{30.48}$
Plug in the value:
$\text { Feet }=\frac{\mathrm{185}}{30.48}$
3. Calculate the result:
Performing the calculation gives:
Thus, 185 centimeters is approximately equal to 6.07 feet when rounded to two decimal places.
Understanding this conversion can be important in various fields such as science, engineering, and everyday measurements where different units are used.
You can also remember that there are about 2.54 centimeters in an inch if that helps with visualizing other measurements!
For example, if you know that 200 cm equals $\frac{200}{30.48} \approx 6.56$ feet, you can use the same method to convert any other centimeter measurements to feet easily.
The conversion factor of 1 foot = 30.48 centimeters is a standard measurement used in the metric and imperial measurement systems.
More Answers:
Leave a Comment
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Envision Math Common Core Grade K Answer Key Topic 4 Compare Numbers 0 to 10
Practice with the help of enVision Math Common Core Kindergarten Answer Key Topic 4 Compare Numbers 0 to 10 regularly and improve your accuracy in solving questions.
Envision Math Common Core Grade K Answers Key Topic 4 Compare Numbers 0 to 10
Essential Question: How can numbers from 0 to 10 be compared and ordered?
enVision STEM Project: Weather Changes
Directions Read the character speech bubbles to students. Find Out! Have students find out about weather changes. Say: The weather changes from day to day. Talk to friends and relatives about the
weather. Ask them to help you record the number of sunny days and rainy days during the week. Journal: Make a Poster Have students make a poster. Have them draw up to 10 lightning bolts above one
house and up to 10 lightning bolts above another house. Ask them to write the number of lightning bolts above each house, and then draw a circle around the number that is greater than the other, or
draw a circle around both numbers if they are the same.
Review What You Know
Question 1.
I circled the group of birds that are lessthan the other group.3 is lessthan 4.
Question 2.
I circled the group of dogs that is greater than the ther group.5 is greater than 1.
Question 3.
I circled 2 groups of marbles that are equal in number.3 is equal to 3.
Question 4.
There are 6 objects in the above picture.So, i counted and wrote 6.
Question 5.
There are 8 objects in the above picture.So, i counted and wrote 8.
Question 6.
There are 10 objects in the above picture.So, i counted and wrote 10.
Directions Have students: 1 draw a circle around the group of birds that is less than the other group; 2 draw a circle around the group of dogs that is greater than the other group; 3 draw a circle
around the two groups that have an equal number of marbles; 4-6 count the number of objects, and then write the number to tell how many.
Pick a Project
Directions Say: You will choose one of these projects. Look at picture A. Think about this question: How can you train to go into space? If you choose Project A, you will act out an exercise skit.
Look at picture B, Think about this question: What kinds of fruit would you put into a fruit salad? If you choose Project B, you will create a fruit salad recipe.
Directions Say: You will choose one of these projects. Look at picture C. Think about this question: What is the most exciting ride at a theme park? If you choose Project C, you will design a ride.
Look at picture D. Think about this question: What do you like to do on a vacation? If you choose Project D, you will make a list.
Lesson 4.1 Compare Groups to 10 by Matching
Solve & Share
Directions Say: Work with a partner. Take turns drawing one cube from the bag and placing it on your page in the rectangle of the same color. When the bag is empty, do you have more red or blue
cubes? How do you know? Draw a picture of your cubes in The rectangles showing which color is more.
Visual Learning Bridge
Guided practice
Question 1.
I drew a line from each chick in the top group to a chick in the bottom group, and then drew a circle around the group that is greater in number than the other group.7 is greater than 6.
Directions 1 Have students compare the groups, draw a line from each chick in the top group to a chick in the bottom group, and then draw a circle around the group that is greater in number than the
other group.
Question 2.
I compared the groups, drew a line from each chick in the top group to a chick in the bottom group, and then drew a circle around the group that is greater in number than the other group.8 is greater
than 3.
Question 3.
I compared the groups, drew a line from each chick in the top group to a chick in the bottom group, and then drew a circle around the group that is less in number than the other group.4 is less than
Question 4.
I compared the groups, drew a line from each chick in the top group to a chick in the bottom group, and then drew a circle around the group that is less in number than the other group.7 is less than
Directions 2 envision^® STEM Say: Chicks live in coops. Coops protect chicks in different types of weather. Have students compare the groups, draw a line from each chick in the top group to a chick
in the bottom group, and then draw a circle around the group that is greater in number than the other group. 3 and 4 Have students compare the groups, draw a line from each chick in the top group to
a chick in the bottom group, and then draw a circle around the group that is less in number than the other group.
Independent Practice
Question 5.
I compared the groups, drew a line from each bucket in the top group to a bucket in the bottom group, and then drew a circle around the group that is greater in number than the other group.8 is
greater than 6.
Question 6.
I compared the groups, drew a line from each bucket in the top group to a bucket in the bottom group, and then drew a circle around the group that is less in number than the other group.2 is less
than 3.
Question 7.
I compared the groups, drew a line from each bucket in the top group to a bucket in the bottom group, and then drew a circle around the group that is less in number than the other group.4 is less
than 7.
Question 8.
I counted the numbe rof buckets, there are 5 buckets in the top group.
I drew a group of 9 buckets that is greater than the buckets in the top group.
9 is greater than 5.
Directions Have students: 5 compare the groups, draw a line from each bucket in the top group to a bucket in the bottom group, and then draw a circle around the group that is greater in number than
the other group; 6 and 7 compare the groups, draw a line from each bucket in the top group to a bucket in the bottom group, and then draw a circle around the group that is less in number than the
other group, 8 Higher Order Thinking Have students draw a group of buckets that is greater in number than the group shown.
Lesson 4.2 Compare Numbers Using Numerals to 10
Emily is planting seedlings, or little plants. She plants 5 red pepper seedlings and 7 yellow pepper seedlings.
I drew counters to show groups of seedlings and i wrote the numbers 5 and 7.I circled 7 as it is greater than 5.
Directions Say: Emily is planting seedlings, or little plants. She plants 5 red pepper seedlings and 7 yellow pepper seedlings. Use counters to show the groups of seedlings. Write the numbers, and
then circle the number that tells which group has more.
Visual Learning Bridge
Guided Practice
Question 1.
I drew a line from each watering can in the top group to a watering can in the bottom group, and then draw a circle around the number 4 as it is greater than 3.
Directions 1 Have students count the watering cans in each group, write the number to tell how many, draw a line from each watering can in the top group to a watering can in the bottom group, and
then draw a circle around the number that is greater than the other number.
Question 2.
I counted the vegetables in each group, wrote the numbers 4 and 5 to tell how many, drew a line from each vegetable in the top group to a vegetable in the bottom group, and then marked an X on the
number 4 as it is less than 4.
Question 3.
I counted the vegetables in each group, i drew 3 more pea pods to make the groups equal, wrote the numbers 6 to tell how many in each group, and then drew a line from each vegetable in the top group
to a vegetable in the bottom group to compare.
Directions 2 Have students count the vegetables in each group, write the number to tell how many, draw a line from each vegetable in the top group to a vegetable in the bottom group, and then mark an
X on the number that is less than the other number. 3 Number Sense Have students count the vegetables in each group, draw more pea pods to make the groups equal, write the numbers to tell how many in
each group, and then draw a line from each vegetable in the top group to a vegetable in the bottom group to compare.
Independent Practice
Question 4.
I counted the seed packets in each group, wrote the numbers 10 and 7 to tell how many, drew a line from each seed packet in the top group to a seed packet in the bottom group, and then mark an X on
the number 7 as it is less than 10.
Question 5.
I counted the flowers in the group, drew a group of 3 flowers that is less than the group shown, and then write the numbers 6 and 3 to tell how many.
Directions 4 Have students count the seed packets in each group, write the number to tell how many, draw a line from each seed packet in the top group to a seed packet in the bottom group, and then
mark an X on the number that is less than the other number. 5 Higher Order Thinking Have students count the flowers in the group, draw a group of flowers that is less than the group shown, and then
write the numbers to tell how many.
Lesson 4.3 Compare Groups to 10 by Counting
Solve & Share
I counted and placed counters on goldfish and tetras, there are 6 goldfish and 9 tetras.I drew a circle around the number 9 as it is greater than 6.
Directions Say: The class aquarium has two kinds of fish, goldfish and tetras. Place counters on the fish as you count how many of each kind. Write numbers to tell how many of each kind. Draw a
circle around the fish that has a number greater than the other. Tell how you know you are right.
Visual Learning Bridge
Guided Practice
Question 1.
I counted and wrote the number of fishes each color, there are 10 pink fishes and 8 purple fishes.I circled 10 as it is greater than 8.
Directions 1 Have students count the number of each color fish, write the numbers to tell how many, and then draw a circle around the number that is greater than the other number. Use the number
sequence to help find the answer.
Question 2.
I counted and wrote the number of fishes each color, there are 6 green fishes and 7 yellow fishes.I circled 7 as it is greater than 6.
Question 3.
I counted wrote the number of fishes each color, there are 8 blue fishes and 9 gold fishes.I marked X on both the numbers as they are NOT equal.
Question 4.
I counted wrote the number of fishes each color, there are 8 brown fishes and 7 green fishes.I marked X on the number 7 as it is lessthan 8.
Question 5.
I counted wrote the number of fishes each color, there are 9 purple fishes and 10 gold fishes.I marked X on the number 7 as it is lessthan 10.
Directions Have students count the number of each color fish, write the numbers to tell how many, and then: 2 draw a circle around the number that is greater than the other number; 3 draw a circle
around both numbers if they are equal, or mark an X on both numbers if they are NOT equal; 4 and 5 mark an X on the number that is less than the other number. Use the number sequence to help find the
answer for each problem.
Independent Practice
Question 6.
I counted and wrote the number of each critter.I wrote the numbers 6 to tell that there are 6 yellow and 6 green critters.I circled both the numbers as they are equal.
Question 7.
I counted and wrote the number of each critter.I wrote the numbers 9 and 8 to tell that there are 9 blue and 8 peach critters.I marked X on the numbe 8 as it is lessthan 9.
Question 8.
There are 7 trantulas in the above picture so, i drew 9 spiders that is two more than the number of trantulas.
Then i wrote the numbers 7 and 9.
Question 9.
I counted the number of butterflies.There are 6 butterflies.I wrote all the numbers up to 10 that are greater than the number of butterflies shown.The numbers that are greater than 6 are 7, 8, 9 and
Directions Have students count the number of each critter, write the numbers to tell how many, and then: 6 draw a circle around both numbers if they are equal, or mark an X on both numbers if they
are NOT equal; 7 mark an X on the number that is less than the other number; 8 draw a group of spiders that is two greater in number than the number of tarantulas shown, and then write the number to
tell how many. 9 Higher Order Thinking Have students count the butterflies, and then write all the numbers up to 10 that are greater than the number of butterflies shown. Use the number sequence to
help find the answer for each problem.
Lesson 4.4 Compare Numbers to 10
Solve & Share
Directions Say: Emily’s mother asked her to bring the towels in off the line. Her basket can hold less than 7 towels. How many towels might Emily bring in? You can give more than one answer. Show
how you know your answers are right.
Emily’s mother asked her to bring the towels in off the line. Her basket can hold less than 7 towels.
Emily can bring 1, 2, , 4, 5 or 6 towels in her basket.
Visual Learning Bridge
Guided Practice
Question 1.
I counted the numbers 1 to 10. I used the number sequence and drew lines from the numbers to the sequence and i circled 8 a sit is greater than 7.
Question 2.
I drew counters to show the numbers 6 and 4.I drew a circle around the number 6 as it is greater than the number 4.
Directions Have students: 1 count the numbers 1 to 10 and use the number sequence to show how they know which number is greater than the other, and then draw a circle around the number that is
greater; 2 draw counters in the ten-frames to show how they know which number is greater than the other, and then draw a circle around the number that is greater.
Question 3.
I drew pictures to show the numbers 6 and 9.9 is greater than 6.
Question 4.
I drew counters to show the numbers 8.
I circled both the numbers as they are equal.
Question 5.
I used the number sequence to find the greater number.I marked X on the number 9 as it is lessthan 10.
Question 6.
I drew the pictures to show the numbers 9 and 8.I marked X on the number 8 as it is lessthan 9.
Directions Have students: 3 draw pictures to show how they know which number is greater than the other, and then draw a circle around the number that is greater; 4 draw counters in the ten-frames to
show how they know if the numbers are equal, and then draw a circle around both numbers if they are equal, or mark an X on both numbers if they are NOT equal; 5 use the number sequence to show how
they know which number is less than the other number, and then mark an X on the number that is less; 6 draw pictures to show how they know which number is less than the other number, and then mark an
X on the number that is less.
Independent Practice
Question 7.
I drew the pictures to show the numbers 6 and 8.I marked X on the number 6 as it is lessthan 8.
Question 8.
I used the number sequence to find the greater number.I marked X on the number 7 as it is lessthan 9.
Question 9.
I wrote the next two numbers that are greater than 8. the numbers that are greater than 8 are 9 and 10.
Question 10.
I wrote the number 7 as it is greater than 5 and lessthan 9.
Directions Have students: 7 draw pictures to show how they know which number is less than the other number, and then mark an X on the number that is less; 8 use the number sequence to show how they
know which number is less than the other number, and then mark an X on the number that is less. 9 Higher Order Thinking Have students write the next two numbers that are greater than the number
shown, and then tell how they know. 10 Higher Order Thinking Have students write a number that is greater than the number on the left, but less than the number on the right.
Lesson 4.5 Repeated Reasoning
Problem Solving
Solve & Share
Directions Say: There are 7 fish in a bowl. Emily puts I more fish in the bowl. How many fish are in the bowl now? How can you solve this problem?
Visual Learning Bridge
Guided Practice
Question 1.
I drew counters for 4 frogs and i drew one more.The number that is one greater than 4 is 5.
Directions 1 Say: Carlos sees 4 frogs at the pond. Then he sees 1 more. How many frogs are there now? Have students use reasoning to find the number that is 1 greater than the number of frogs shown.
Draw counters to show the answer, and then write the number. Have students explain their reasoning.
Independent Practice
Question 2.
I drew counters for 7 frogs and i drew one more.The number that is one greater than 7 is 8.
Question 3.
I drew counters for 2 frogs and i drew one more.The number that is one greater than 2 is 3.
Question 4.
I drew counters for 8 frogs and i drew one more.The number that is one greater than 8 is 9.
Question 5.
I drew counters for 6 frogs and i drew one more.The number that is one greater than 6 is 7.
Directions Say: Alex sees frogs at the pond. Then he sees 1 more. How many frogs are there now? 2-5 Have students use reasoning to find the number that is 1 greater than the number of frogs shown.
Draw counters to show the answer, and then write the number. Have students explain their reasoning.
Problem Solving
Performance Task
Question 6, 7, 8.
I drew counters for 5 pets of Marta, i drew 1 more counter.The number that is one greater than 5 is 6.Marta will now have 6 pets.
Directions Read the problem aloud. Then have students use multiple problem-solving methods to solve the problem. Say: Marta’s family has 5 pets. Then her family gets 1 more. How many pets do they
have now? 6 Generalize Does something repeat in the problem? How does that help? 7 Use Tools What tool can you use to help solve the problem? Use the tool to find the number of pets in Marta’s
family now. 8 Make Sense Should the answer be greater than or less than 5?
Topic 4 Vocabulary Review
Question 1.
I circled the number 9 as it is greater than 7.
Question 2.
I counted the counters given, there are 7 counters in all.so, i wrote the number 7 in the blank.
Question 3.
The number that means NONE is 0.
Question 4.
There are 2 red cubes and 3 yellow cubes.I circled the red cubes as they are less than yellow cubes.I wote the number 2 as it is lessthan 3.
Directions Understand Vocabulary Have students: 1 draw a circle around the number that is greater than 7; 2 count the counters, and then write the number to tell how many; 3 write the number that
means none; 4 count how many of each color cube there is, draw a circle around the group that has a number of cubes that is less than the other group, and then write the number to tell how many there
are in that group.
Question 5.
I circled the number 8 as it is greater than 3 and marked X on the number 3 as it is lessthan 8.
Question 6.
I wrote 4 as it is greater than 3 and lessthan 5.
Question 7.
I drew counters for the given number 5.
Question 8.
I wrote the missing numbers 3, 4, 6, 7 and 8 in the gien blanks.
Directions Understand Vocabulary Have students: 5 compare the numbers, draw a circle around the number that is greater, and then mark an X on the number that is less; 6 write a number that is greater
than 3, but less than 5; 7 draw 5 counters in a row and then write the number to tell how many; 8 write the missing numbers in order.
Topic 4 Reteaching
Set A
Question 1.
I counted the number of counters, there are 6 red counters and 10 yellow counters.I circled red counters frame as they are less in number than the yellow counter frame.
Set B
Question 2.
I drew lines from the top group to a piece of fruit in the bottom group, and then i drew a circle around the number 7 as it is greater than 5.
Directions Have students: 1 compare the groups, and draw a circle around the group that is less in number than the other group; 2 count the fruit in each group, write the numbers that tell how many,
draw a line from each piece of fruit in the top group to a piece of fruit in the bottom group, and then draw a circle around the number that is greater than the other number.
Set C
Question 3.
I counted and wrote the number of blue critter and peach critters.They are 6 peach critters and 4 blue critters.
I marked X on the number 4 as it is lessthan 6.
Set D
Question 4.
I drew counters for the 6 frogs and drew one more counter.1 greater than 6 is 7.
Directions Have students: 3 count the number of each critter, write the numbers, and then mark an X on the number that is less than the other number; 4 Say: April sees frogs at The pond. Then she
sees 1 more. How many frogs does she see now? Have students use reasoning to find the number that is 1 greater than the number of frogs shown. Draw counters to show the answer, and then write the
Topic 4 Assessment Practice
Question 1.
There are 8 yellow birds.I marked group ‘A’ as it has more number of blue birds than the yellow birds.There are 10 blue bieds in group A.
Question 2.
I marked the numbers 6, 5 and 3.The numbers 3, 5 and 6 as they are lessthan the number 7.
Question 3.
I counted and wrote the number of lemons and limes, there are 7 lemons and 5 limes.
I circled the number 7 as it is greater than 5.
Directions Have students mark the best answer. 1 Which group of blue birds is greater in number than the group of yellow birds? 2 Look at the number line. Then mark all the numbers that are less than
the number on the card. 3 Have students count the number of lemons and limes, write the number that tells how many of each, and then draw a circle around the number that is greater.
Question 4.
The number of the first card is 7, the number before 7 is 6.I counted forward from 6 and wrote the numbers 7, 8, 9 and 10 in the blanks.
Question 5.
There are 6 sandwiches and I drew the 4 juice boxes as 4 is lessthan 6.
Question 6.
I drew counters for 7 beads and drew 1 more counter.one greater than the number 7 is 8.
Kayla will now have 8 beads to make a bracelet.
Directions Have students: 4 write the number that is counted first among the 4 number cards, and then count forward and write the number that is 1 greater than the number before; 5 count the
sandwiches in the group, draw a group of juice boxes that is less than the group of sandwiches shown, and then write the numbers to tell how many. 6 Say: Kayla has 7 beads to make a bracelet. Then
she buys 1 more. How many beads does she have now? Have students use reasoning to find the number that is 1 greater than the number of beads shown. Draw counters to show the answer, and then write
the number to tell how many.
Topic 4 Performance Task
Question 1.
There are 9 shunks and 8 raccoons in the forest.
Using number sequence i found that number 8 is lessthan number 9 so, i marked X on the number 8.
Directions Forest Animals Say: The forest is home to woodland animals. One part of the forest has many different animal homes in it. 1 Have students study the picture. Say: How many skunks live in
this part of the forest? How many raccoons live in this part of the forest? Count the number of each type of animal and write the numbers. Then have students draw a circle around the number that is
greater than the other number and mark an X on the number that is less than the other number. Have them use the number sequence to help find the answers.
Question 2.
There are 6 foxes in the forest.I counted and wrote the numbers between 6 and 10.They are 7, 8, 9 and 10.
Question 3.
I drew counters for the number 5, I drew circle around both the numbers as they are equal.
Question 4.
There are 7 birds in the forest.I drew counters for 7 birds and one more counter as 1 bird flies into the forest.
One greater than the number 7 is 8.
Directions Have students look at the picture on the page before. 2 Say: How many foxes live in this part of the forest? Count how many and write the number. Then have students write all the numbers
through 10 that are greater than the number of foxes. 3 Say: 5 chipmunks and 5 frogs move out of this part of the forest. Draw a circle around both numbers if they are equal, or mark an X on both
numbers if they are NOT equal. Show how you know you are correct. 4 Say: How many birds live in this part of the forest? Count how many and write the number. I more bird flies into the forest. How
many birds are in this part of the forest now? Have students use tools to solve the problem and write the number. Then have them show how they found the answer.
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Probing hybrid stars with gravitational waves via interfacial modes
One of the uncertainties in nuclear physics is whether a phase transition between hadronic matter to quark matter exists in supranuclear equations of state. Such a feature can be probed via
gravitational-wave signals from binary neutron star inspirals that contain information of the induced tides. The dynamical part of the tides is caused by the resonance of pulsation modes of stars,
which causes a shift in the gravitational-wave phase. In this paper, we investigate the dynamical tides of the interfacial mode (i -mode) of spherical degree l =2 , a nonradial mode caused by an
interface associated with a quark-hadron phase transition inside a hybrid star. In particular, we focus on hybrid stars with a crystalline quark matter core and a fluid hadronic envelope. We employ a
hybrid method which consists of a general relativistic calculation of the stellar structure, together with Newtonian formulations of tidal couplings and mode excitations. We find that the resonant
frequency of such i -modes typically ranges from 300 Hz to 1500 Hz, and the frequency increases as the shear modulus of the quark core increases. We next estimate the detectability of such a mode
with existing and future gravitational-wave events from the inspiral waveform with a Fisher analysis. We find that GW170817 and GW190425 have the potential to detect the i -mode if the quark-hadron
phase transition occurs at a sufficiently low pressure and the shear modulus of the quark matter phase is large enough. We also find that the third-generation gravitational-wave detectors can further
probe the i -mode with intermediate transition pressure. Finally, we check our hybrid method against a fully-Newtonian analysis and find that the two results can be off by a factor of a few. Thus,
the results presented here should be valid as an order-of-magnitude estimate and provide a new, interesting direction for probing the existence of quark core inside a neutron star using the i -mode.
A full general relativistic formalism, however, needs to be pursued for further analysis.
Physical Review D
Pub Date:
March 2021
□ Astrophysics - High Energy Astrophysical Phenomena;
□ General Relativity and Quantum Cosmology
Phys. Rev. D 103, 063015 (2021)
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This Blog is Systematic
Do you remember this post? https://qoppac.blogspot.com/2022/06/vol-targeting-cagr-race.html
Here I introduced a performance metric, the best annualised compounding return at the optimal leverage level for that strategy. This is equivalent to finding the highest geometric return once a
strategy is run at it's Kelly optimal leverage.
I've since played with that idea a bit, for example in this more recent post amongst others I considered the implications of that if we have different tolerances for uncertainty and used
bootstrapping of returns when optimising a stock and bond portfolio with leverage, whilst this post from last month does the same exercise in a bitcoin/equity portfolio without leverage.
In this post I return to the idea of focusing on this performance metric - the maximised geometric mean at optimal leverage, but now I'm going to take a more abstract view to try and get a feel for
in general what sort of strategies are likely to be favoured when we use this performance metric. In particular I'd like to return to the theme of the original post, which is the effect that skew and
other return moments have on the maximum achievable geometric mean.
Obvious implications here are in comparing different types of strategies, such as trend following versus convergent strategies like relative value or option selling; or even 'classic' trend following
versus other kinds.
Some high school maths
(Note I use 'high school' in honor of my US readers, and 'maths' for my British fans)
To kick off let's make the very heroric assumption of Gaussian returns, and assume we're working at the 'maximum Kelly' point which means we want to maximise the median of our final wealth
distribution - same as aiming for maximum geometric return - and are indifferent to how much risk such a portfolio might generate.
Let's start with the easy case where we assume the risk free rate is zero; which also implies we pay no interest for borrowing (apologies I've just copied and pasted in great gobs of LaTeX output
since it's easier than inserting each formula manually):
Now that is a very nice intuitive result!
Trivially then if we can use as much leverage as we like, and we are happy to run at full Kelly, and our returns are Gaussian, then we should choose the strategy with the highest Sharpe Ratio. What's
more if we can double our Sharpe Ratio, we will quadruple our Geometric mean!
Truely the Sharpe Ratio is the one ratio to rule them all!
Now let's throw in the risk free rate:
We have the original term from before (although the SR now deducts the risk free rate), and we add on the risk free rate reflecting the fact that the SR deducts it, so we add it back on again to get
our total return. Note that the
higher SR = higher geometric mean at optimal leverage
relationship is still true. Even with a positive risk free rate
we still want to choose the strategy with the highest Sharpe Ratio!
Consider for example a classic CTA type strategy with 10% annualised return and 20% standard deviation, a miserly SR of 0.5 with no risk free rate; and contrast with a relative value fixed income
strategy that earns 6.5% annualised return with 6% standard deviation, a SR of 1.0833
Now if the risk free rate is zero we would prefer the second strategy as it has a higher SR, and indeed should return a much higher geometric mean (since the SR is more than double, it should be over
four times higher). Let's check. The optimal leverages are 2.5 times and 18.1 (!) times respectively. At those leverages the arithmetic means are 25% and 117% respectively, and the geometric means
using the approximation are 12.5% for the CTA and 58.7% for the RV strategy.
But what if the risk free rate was 5%? Our Sharpe ratios are now equal: both are 0.25. The optimal leverages are also lower, 1.25 and 4.17. The arithmetic means come in at 12.5% and 27.1%, with
geometric means of 9.4% and 24%. However we have to include the cost of interest; which is just 1.25% for the CTA strategy (borrowing just a quarter of it's capital at a cost of 5% remember) but a
massive 15.8% for the RV. Factoring those in the net geometric means drop to 8.125% for both strategies - we should be indifferent between them, which makes sense as they have equal SR.
The horror of higher moments
Now there is a lot wrong with this analysis. We'll put aside the uncertainty around being able to measure exactly what the Sharpe Ratio of a strategy is likely to be (which I can deal with by drawing
off more conservative points of the return distribution, as I have done in several previous posts), and the assumption that returns will be the same in the future. But that still leaves us with the
big problem that returns are not Gaussian! In particular a CTA strategy is likely to have positive skew, whilst an RV variant is more likely to be a negatively skewed beast, both with fat tails in
excess of what a Gaussian model would deliver. In truth in the stylised example above I'd much prefer to run the CTA strategy rather than a quadruple leveraged RV strategy with horrible left tail
Big negative skewed strategies tend to have worse one day losses; or crap VAR if you prefer that particular measure. The downside of using high leverage is that we will be saddled with a large loss
on a day when we have high leverage, which will significantly reduce our geometric mean.
There are known ways to deal with modifying the geometric mean calculation to deal with higher moments like skew and kurtosis. But my aim in this post is to keep things simple; and I'd also rather
not use the actual estimate of kurtosis from historic data since it has large sampling error and may underestimate the true horror of a bad day that can happen in the future (the so called 'peso
problem'); I also don't find the figures for kurtosis particuarly intuitive.
(Note that I did consider briefly using maximum drawdown as my idiots tail effect here. However maximum drawdown is only relevant if we can't reduce our leverage into the drawdown. And perhaps
counter intuitively, negative skewed strategies actually have lower and shorter drawdown)
Instead let's turn to the tail ratio, which I defined in my latest book AFTS. A lower tail ratio of 1.0 means that that the left tail is Gaussian in size, whilst a higher ratio implies a fatter left
I'm going to struggle to rewrite the relatively simple 0.5SR^2 formulae to include a skew and left tail term, which in case will require me to make some distributional assumptions. Instead I'm going
to use some bootstrapping to generate some distributions, measure the tail properties, find the optimal leverage, and then work out the geometric return at the optimal leverage point. We can then
plot maximal geometric means against empirical tail ratios to get a feel for what sort of effect these have.
To generate distributions with different tail properties I will use a mixture of two Gaussian distributions; including one tail distribution with a different mean and standard deviation* which we
draw from with some probability<0.5. It will then be straightforward to adjust the first two moments of the sample distribution of returns to equalise Sharpe Ratio so we are comparing like with like.
* you will recognise this as the 'normal/bliss' regime approach used in the paper I discussed in my prior post around optimal crypto allocation, although of course it will only be bliss if the tail
is nicer which won't be the case half the time.
As a starting point then my main return distribution will have daily standard deviation 1% and mean 0.04% which gives me an annualised SR of 0.64, and will be 2500 observations (about 10 years) in
length - running with different numbers won't affect things very much. For each sample I will draw the probability of a tail distribution from a uniform distribution between 1% and 20%, and the tail
distribution daily mean from uniform [-5%, 5%], and for the tail standard deviation I will use 3% (three times the normal).
All this is just to give me a series of different return distributions with varying skew and tail properties. I can then ex-post adjust the first two moments so I'm hitting them dead on, so the mean,
standard deviation and SR are identical for all my sample runs. The target standard deviation is 16% a year, and the target SR is 0.64, all of which means that if the returns are Gaussian we'd get a
maximum leverage of 4.0 times.
As always with these things it's probably easier to look at code, which is here (just vanilla python only requirements are pandas/numpy).
Let's start with looking at optimal leverage. Firstly, how does this vary with skew?
Ignore the 'smearing' effect this is just because each of the dots in a given x-plane will have the same skew and first two moments, but slightly different distribution otherwise. As we'd expect
given the setup of the problem the optimal leverage with zero skew is 4.0
For a pretty nasty skew of -3 the leverage should be about 10% lower - 3.6; whilst for seriously positive skew of +3 you could go up to 4.5. These aren't big changes! Especially when you consider
that few real world trading strategies have absolute skew values over 1 with the possible exception of some naked option buying/selling madness. The most extreme skew value I could find in my
latest book
was just over 2.0, and that was for single instruments trading carry in just one asset class (metals).
What about the lower tail? I've truncated the plot on the x-axis for reasons I will explain in a second.
Again for the base case with a tail ratio of 1.0 (Gaussian) the optimal leverage is 4.0; for very thin lower tails again it goes up to 4.4, but for very fat lower tails it doesn't really get below
about 3.6. Once again, tail ratios of over 4.0 are pretty rare in real strategies (though not individual instruments), although my sampling does sometimes generate very high tail ratios the optimal
leverages never go below 3.6 even for a ratio of nearly 100.
Upper tail, again truncated:
Again a tail ratio of 1.0 corresponds to optimal leverage of roughly 4.0; and once again even for very fat upper tail ratios (of the sort that 'classical' trend followers like to boast about, the
optimal leverage really isn't very much higher.
Now let's turn to Geometric mean, first as affected by skew:
As we'd expect we can generate more geometric mean with more skew... but not a lot! Even a 3 unit improvement in skew barely moves the return needle from 22.5% to 24.5%. To repeat myself it's rare to
find trading strategies above or below 1.0, never mind 3.0. For example the much vaunted Mulvaney capital has a return skew on monthly returns of about 0.45. The graph above shows that will only add
about 0.25% to expected geometric return at optimal leverage versus a Gaussian normal return distribution (this particular fund has extremely large standard deviation as they are one of the few funds
in the world that actually runs at optimal leverage levels).
For completeness, here are the tail ratios. First the lower tail ratio:
Fat left tails do indeed reduce maximum optinmal geometric means, but not a lot.
Now for the upper tail:
I do like neat formulae, and the results for Gaussian normal distributions do have the property of being nice. Of course they are based on unreal expectations, and anyway who runs full Kelly on a
real strategy expecting the returns to be normal and the SR to be equal to the backtested SR is an idiot. A very optimistic idiot, no doubt with a sunny disposition, but an idiot nonetheless.
For the results later in the post it does seem surprising that even if you have something with a very ugly distribution that you'd not really adjust your optimal leverage much, and hence see barely
on impact on geometric mean. But remember here that I'm fixing the first two moments of the distribution; which means I'm effectively assuming that I can measure these with certainty and also that
the future will be exactly like the past. These are not realistic expectations!
And that is why in the past rather than choose the leverage that maximised geometric mean, I've chosen to maximise some more conservative point on the distribution of terminal wealth (where geometric
mean would be the median of that distribution). Doing that would cause some damage to the negative skewed, fat left tail distributions resulting in lower optimal leverage and thus lower geometric
I have thought of a way of doing this analysis with distributional points, but it's quite computationally intensive and this post is already on the long side, so let's stop here.
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Episode 104: Understanding Quantum Computing and its Use Cases
This is a podcast episode titled, Episode 104: Understanding Quantum Computing and its Use Cases. The summary for this episode is: <p><span style="font-weight: 400;">Quantum computing will
revolutionize your approach to a whole array of business problems but it won’t solve everything single-handedly. Your task as CEO is to know where quantum will make a difference and make sure you’re
ahead of your competition. In this episode of the <a href= "https://georgianpartners.com/the-impact-podcast/">Georgian Impact Podcast</a>, Jon Prial sits down with quantum computing expert Vlad
Gheorghiu to help you understand the use cases where quantum will be most effective and why by giving a quick primer on the quantum tech stack.</span></p> <p><strong>You’ll hear about:</strong></p>
<ul> <li style="font-weight: 400;"><span style="font-weight: 400;">How to think about quantum computing in relation to traditional computing</span></li> <li style="font-weight: 400;"><span style=
"font-weight: 400;">Where we are now versus where we will need to be for certain use cases</span></li> <li style="font-weight: 400;"><span style="font-weight: 400;">The quantum computing stack and
how the different parts function</span></li> <li style="font-weight: 400;"><span style="font-weight: 400;">How to develop a business strategy for quantum and what the first use cases might look like
</span></li> </ul> <p><strong>Who is Vlad Gheorghiu?</strong></p> <p><span style="font-weight: 400;"><a href= "https://vsoftco.github.io/">Vlad Gheorghiu</a> is CEO, President and Co-Founder of <a
href="https://www.softwareq.ca/">softwareQ Inc</a>, and a researcher at the Institute for Quantum Computing, working with Michele Mosca on theoretical aspects of quantum computation and
(post-quantum) cryptography. He also collaborates on quantum risk assessment with evolutionQ Inc.</span></p> <p><span style="font-weight: 400;">Vlad is involved in the CryptoWorks21 Quantum-Safe
Cryptographic Infrastructure Program and is a member of the European Telecommunications Standards Institute (ETSI) Quantum-Safe Cryptography Standardization Group. Vlad graduated from Carnegie Mellon
University with a PhD in Theoretical Physics.</span></p>
John Prial: We are living in clear Holy cow moments. Now we've seen the first picture of a black hole, and we really are seeing some massive recognition of the issues with data privacy and trust that
can significantly affect companies. But at the end of the day, it's just about data and compute power. And we have gone from basic discussions of Moore's law to quantum computing. Let me read a quote
from CB insights," Quantum computing is poised to upend entire industries from telecommunications in cybersecurity to advanced manufacturing, finance medicine, and beyond." That if your head doesn't
hurt enough, let me just add one more quote," In the near future, quantum computing could change the world." All right. I think I've got everyone's attention. So how do we get our heads wrapped
around the technology that just might change the world? Today's guest is here to help today. I'll be talking with Dr. Vlad Giorgio. Vlad is the CEO president and co- founder of software queue, Inc.
And he's also a researcher at the Institute for quantum computing. We're going to parse the full hardware and software stack that is quantum computing, which will help us all understand what's
different about quantum computing and later we'll delve into where Vlad expects to see this technology deployed first and why it's important today. Now as an aside episode, 97 with Mike Brown was our
first podcast on quantum computing and it covers some of the foundational definitions for you. So respecting you, we don't want to repeat, so please go back and give it a listen. If you need a quick
refresher today, we all learning about this emerging technology and I'm excited to bring you this next installment of quantum computing, a potential game changer for us all. I'm John Pryor and
welcome to the Georgian Impact Podcast. So Vlad let me do my lightweight view of quantum one on one and you can help me here. Obviously we know quantum is different from classical computers. What we
have these simple binary bits that are zeros and ones, and we shared an earlier podcast about cubits and they can take many States, not just zero in one, these probabilistic States and the switches
of States allows for faster and more powerful computing. I think we get that. So where are we today with a density count for cubits?
Vlad Giorgio: So today we are in the range of 50 cubits. So it's a relatively small number of cubits available out there. IBM has around 50 INQ has actually 79, but not all of them are working fine.
Google has around 70 cubits. Now all of those platforms that I mentioned now are something called a universal quantum computers. So in principle, they can solve any quantum algorithm. There are some
other machines that have many more cubits like D- Wave that you probably heard about. And it's a Canadian company in Burnaby BC. So D- Wave has right now around 2000 cubits, but those cubits are
different. So they are more noisy than the one provided by IBM or Intel or Google. Those 2000 cubits can perform some tasks, but not every possible quantum algorithm. So in a sense, the D- Wave
machine is not what we call a universal quantum processor.
John Prial: Wow, that's quite The range. So while everybody's out there counting cubits, it's not the number that's telling us the full story, as we think about some of these things being noisy or
not. And for me, it reinforces that we're just in the early stages here at the same time, we should reminding everybody that the potential value writing on this is huge. So Vlad, what do we need to
see in order to solve both early use cases and then some more advanced quantum problems?
Vlad Giorgio: Yeah, that's a very good question, John. And I think something around 200 to 300 cubits will allow us to see some interesting developments in quantum algorithms, especially for small
scale quantum computers in the near- term future. In order to solve important problems. I think we'll need more than that, probably of the order of thousands of good qubits, but nevertheless, for
relatively small problems where you can already see a quantum advantage. My guess is that once we hit around 100 to 300 qubits, we'll already be able to see interesting phenomenon.
John Prial: That's interesting. I know we're on this journey from noisy to not noisy would this analogy help everybody? So I know that no one really thought about what could be done with a GPU
graphical processing unit, but because it was so massively parallel, all of a sudden GPU chips have accelerated this machine, learning AI deep learning world. And although perhaps I'm not being
technically accurate is this advent of, of cubits and Quantic breeding similar to that. And I see you've got both a specialized in a non specialist use some hearing?
Vlad Giorgio: Yeah. I mean, in principle, a quantum computer is going to be used most likely as a core processor. So we don't expect to use quantum computers to write your favorite document in Ms.
Word, or it's almost like it's going to be a core processor, which was going to be used to speed up certain classes of problems. Even if we use a quantum computer as a core processor, there are these
two types of quantum computers, like the annealers that are the D- Wave machines that are used for an even smaller class of problems. And then the so- called universal quantum computers that are able
to speed up in principle, many more, many other quantum algorithms that the D- Wave presumably right now it's not possible to do so.
John Prial: Is it fair to call it universal? I mean, you're saying it's not good to be used, say a Ms. Word. It's going to be a co- processor, but whether it's a very specialized solution or maybe a
more broadly specialized solution, because obviously there is a range here. Are we ever going to end up with a generalized base that programmers get access? And you say like a TensorFlow
Vlad Giorgio: In a sense, the universal is a bit of misleading the term. So we're not going to be able to solve every problem in the world. Using quantum computers, we are going to be able to solve a
relatively small class of problems at least today. And then of course, we're going to come up with better and better quantum algorithms. So probably are going to solve more problems, but some of the
problems are not suitable for the quantum computer. So it doesn't even make sense to use the quantum computer
John Prial: Is even similar, when we say everyone gets excited about deep learning saying, don't don't just start with deep learning. There's lots of other machine learning methods that people could
use today that is simpler and make sense. And eventually as sophistication as needed, you might end up in a deep learning world. Is that another fair analogy?
Vlad Giorgio: Very actually it's a perfect analogy. So true. So many people are trying to use even today. They're trying to look at quantum algorithms for some problems, which in my opinion, doesn't
make sense to use a quantum computer for so that's a very, very good analogy.
John Prial: Perfect. Now I want to just finish up on hardware, really talk more about these quantum algorithms and that opportunity, but it's everybody always gets into competition. So I know there's
at least like a half a dozen different quantum technologies, whether some are superconducting in Silicon based in photonic and others. So I guess my question is, will we are they all going to be out
there in the future with unique specialized unit uses or do you think at some point there's a winner. How does this compare to the traditional chip transistor chip world?
Vlad Giorgio: It's very difficult to predict. So everyone bets on their own technology. I mean, there are three main technologies that are the superconducting cubits. Then there are the Photonix and
ion traps. So those three are the main technologies out there. All of them have their own advantages and also their own challenges. I cannot predict it's a little bit too early to say what
architecture is going to is going to be the winner in classical computer. We have a winner, right? So it's the Silicon based architecture. So all of the chips revolve around that Silicon based
architecture here is even more dramatic. I mean, they are fundamentally different physical architectures. It will be great. Actually, if we have three types of quantum computer competing platforms,
for example, because we are going to be able to find advantages for some problems for specific platform and some advantages for another platform. My hope is that we are going to have at least couple
of good quantum architectures, but right now it's a bit too early to say. So it's still at the research phase trying to build scalable quantum computers. There's a great way to set up kind of the
discussion, getting to software and marketing and use cases.
John Prial: So let me, in my mind, as I look at this, there's either using quantum to process huge amounts of data. Yeah. Everyone says we could do a better job of predicting whether if we had more
data and of course, with internet of things, we're going to have more and more data, or we need to have these extraordinary processing speeds parallel of course, to break cryptography. Are those the
two major use cases that we'll be enabling?
Vlad Giorgio: So indeed the first use case is the more scary one is in cryptography. And for that, we have very clear algorithms or like short algorithm that breaks public key cryptography. So
that's, that's a very, very definite use of a quantum computer. Fortunately, we are protecting against that and you probably talked before about the steps we are taking to protect against quantum
attacks like in the realm of post quantum cryptography. The other use case is in the processing large amounts of data, of highly correlated data. And for that, there are various candidates, various
quantum algorithm candidates. So one of them is famous Grover searching, which allows you to search data faster than any possible classical algorithm. Then there are a variety of so- called quantum
machine learning algorithms that in principle should allow you to filter through the data and obtain results faster. And there is another algorithm actually, which is promising as a sub- routine in
quantum machine learning, which is called AGL algorithm. So that's used to speed up solving linear system of equations. Those sub routines of solving linear systems appear in many, many problems in
optimization problems and so on. So that will be a quantum computer will be used as a co- processor there and in principle can speed up that task. So it will allow you to perform any tasks that
require solving linear systems much faster, in fact, exponentially faster for some class of problems.
John Prial: So at the basic level for software, obviously we have systems, software operating systems and other subsystems that are built around that. And of course, we've got middleware that enables
a more rapidly deployment of the applications that run on top of that. Is this layering the same for quantum or is, should I think a little more when you talk in Silicon what's different here?
Vlad Giorgio: No, it's very similar. So a, so for quantum there, we also have a stack. So we start with a logical algorithm that we write on a piece of paper, or we describe it as a code on a
classical computer. Then the next layer in the circuit is something that we call the quantum circuit representation of it. So we translate that logical description of the algorithm into a sequence of
so- called quantum Gates. That's very similar to the assembly language that's generated by a classical compiler. Then at the next step, we take that sequence of Gates. We perform optimization on it.
So it's what the compiler does classically as well. It takes some initially generated code and then it tries to remove redundancy in the code to make it shorter. Finally, we take the assembly
language as the quantum assembly language, which describes our sequence of Gates. And then we transform that into physical instructions that target the quantum chip. So those will be like sequence of
laser per policies or magnetic fields depending on the, on, on the architecture. So the stack is relatively similar to the traditional computation stack is just the layers in the stacks that are
different. And so we don't target chips and Silicon, we target the quantum chip.
John Prial: But you talked about, there are different types of algorithms. And I think you mentioned one of them might be a search algorithm. And what I had read somewhere is in the old style of
computing, if you wanted to find a specific number, phone number in a telephone book, my apologies to those that are so young, they don't know what a telephone book is, but you would literally
sequentially search every number until you found a match and with positive...
Vlad Giorgio: It happens simultaneously.
John Prial: So my question to you then is does that search algorithm manifest itself at a higher level API that eventually any company doing software development can work on and it gets routed off to
the right quantum machine? Or is there, is there more work to be done as a CEO and programmers? Think about the advent of quantum.
Vlad Giorgio: So first let me just make a minor, minor correction.
John Prial: Sure please do.
Vlad Giorgio: The quantum search, it doesn't do it actually simultaneously. It does it in square root of N operations instead of doing N queries searching for all N possible entries in the phone
book, it does it in square root events. So if you have 1 million entries in the phone book, then a quantum search will do it in 1000 queries. So it will only flip 1000 quantum pages, which is still a
very counter intuitive. Yeah. And it's quite quite interesting because that works for unsorted databases. Of course, if you sort your database, you can do much faster searches, but for answering
databases, this is quite a, I find it there's a phenomenal result. We need to perform a lot of classical control on the quantum computer. So in a sense, we use classical computers to keep the quantum
computer alive. Now, if you start counting the resources that you need in order to do that, you're going to see there is a trade- off. So for some searching spaces, it doesn't even make sense to run
Grover because of the induced by this quantum error correction. I so we'll will have some applicability, but not for every type of problem. If you're looking to search for a a hundred elements, you
better don't use Grover because it doesn't make any sense whatsoever from a business case and from a business perspective and in terms of the overhead. So I would say that people should be aware of
the trade- off. So what we are trying to do at our company is to educate people also about the catches of quantum computing. So it's sometimes it's over- hyped and many things that, Oh, it's going to
solve the world crisis is not going to do that. I mean you have to carefully select your problems so that you have a business advantage. And that's something that many executives are not aware of.
John Prial: This is great. And this is exactly what we wanted to get to in terms of what's real, not tomorrow, but over the next year or two, and what executives should be thinking about. So I'll
might be going on a limb here. It might be obvious to everybody deep learning obviously will bloom, I think will be an early opportunity area at the same time. As I mentioned earlier, deep learning,
isn't going to be for all. You could use some basic ML Models. So for Companies that are kind of pushing the envelope, what would type of use cases would you expect to see them? I love obviously it's
volume. You're not going to this, I guess, Grover is this cool search thing for you. Wouldn't use it for a hundred elements, what type of use cases what do you see?
Vlad Giorgio: So what I think initially is that people should just, So executive should initially just consider the, This quantum machines are going to be there in the future and try to investigate
and see what business advantage they can get. So, first of all, I would say that the first use cases will be small proof of concept systems. Right now, we don't have enough cubits to, to get business
advantage. I mean, someone can, can tell you that, Oh, we can run this on the IBM machine with 50 cubits, and the you're going to get the business advantage. I mean, that's a lie, so that's not going
to happen today. What I see happening in the near future in the next two to three years is companies being aware of the fact that quantum computing technology exists and then sorting to the hype with
the use of a quantum algorithm company. And then also trying to see what problems are suitable to run on a quantum computer, because what's going to happen. Otherwise is they're going to wait for
let's say 10 years, these machines are going to be out there. They realize that, Oh, my competitor is making money. It's doing something faster than I do. I already lost the train. I'm going to rush
it. And it's going to be game over basically. So, what I think is going to happen in the relatively short future is identifying the set of problems that make sense to be run on quantum machines and
getting the trade offs of seeing exactly how many quantum resources you really need in order to get the business advantage and see if it makes sense for you as a company to invest in quantum
technology or not. Because for some companies, it will not make sense. It's the same as blockchain, right? So blockchain is not solving every problem in the world. Although like two years ago,
everyone thought that blockchain technology is going to solve everything. So, and that was not true. I mean, it's a great technology, but nevertheless, you should know where to use it and where to
not use it.
John Prial: I think that's a great example. There's a making the business goal. You may or may not choose to make the investment. And that's a rational business decision at the same time. One of the
factors that might influence that decision is what if your competitor did it, and if they could truly get an edge and really blow past you, then it's something you need to think about. Let me then
ask a couple of cases just to help people think about it, because we've talked about processing a lot of data and then super processing one example, I read that's kind of a proof of concept and
because it was Volkswagen, I'm not, it doesn't matter who they were working with, but they were talking about optimizing traffic flow for individual cars and Beijing. And I was fascinated by that
because I think about my GPS tells everybody where to go to optimize. And of course it makes no sense if everybody gets off the highway and gets onto this little tiny side road and it may make more
sense even with self- driving cars to help them optimize their routes. I like that example, that's kind of super processing of less data, but more about just speed. How does that work as a use case
view? And can you give me another one that maybe similar to that?
Vlad Giorgio: A similar one that's actually that makes sense for a financial industry is portfolio optimization. So that is something in which we believe quantum can have an advantage. So, you have a
portfolio of stocks or assets, and then you want to see how to optimize your portfolio and how to make the most profit out of it. So that can be definitely done with some quantum algorithms. One of
them is called the derivative fee optimization algorithm that in principle can provide a speed up for you. So that's another optimization. crosstalk
John Prial: Let me ask about that one a little bit, because I believe I've seen, I know I've seen these things called Monte Carlo analysis. That basically as part of a portfolio analysis says based
on the, his arrange of interest rates or his range of, of earnings, you could make on your portfolio and here's when you die and are you going to have enough money before you die? But that's really
only looking at an existing portfolio. What's you're telling me is there could be a whole nother layer to that I can begin to look at what the portfolio components are and how to get to really taking
it to the next level. It's, it's fascinating to me now that I love that example, is that, is that what happens?
Vlad Giorgio: That is that true in a sense, and it performs this magic due to some speed up provided by Grover or some speed provide provided by the algorithm that I mentioned before this AHL,
quantum solver, quantum linear equation solver algorithm. And again, it's interesting to study what size of portfolio makes sense to be run on a quantum machine? Because again, like if you have a
relatively small portfolio is better to just invest in your GPU's and they will provide the answer fast enough for you as a company.
John Prial: Interesting. Okay. And another cool example. That was a great one, like great discussion. How about another cool example, maybe drug testing... crosstalk.
Vlad Giorgio: I would say there are probably, we can find good applications in drug discovery in protein discovery, drug discovery, in a sense using quantum machine learning techniques and any if you
think any optimization problems that in which you can have a quantum advantage, if you have any edge with a quantum computer, then you're going to definitely plug the quantum computer eventually into
the problem.
John Prial: So obviously, companies will compete with each other but now you're also part of a university- based research Institute. So do you see a strong community of resources advancing the cause?
And is that happening at software or they are not?
Vlad Giorgio: It happens. It happens on both. So I'm based at the Institute for quantum computing at the university of Waterloo. So there we do research in starting from theory to quantum software
all the way down to the physical layer. So there are professors that are trying to build the physical cubits they're build architectures for physical cubits, scalable, physical cubits, and so on. So
there is a lot of investment in the field. I think NSA did us a great favor in 2006, when they announced that they recommended the us government to change their traditional cryptographic suit to so-
called post quantum or quantum resistant suit of cryptographic algorithms. And that did us a favor because suddenly everyone became aware that, look, this is a well- respected agency. And if they
recommend that for the top secret data in US government, it means that they have a strong belief that these machines are eventually going to be built in the relatively close future. So yes, it's,
it's a great time to be in the field, in my opinion.
John Prial: So we talked a little bit about companies that might have a ton of data, might have a lot of processing requirements. They've got to figure out the right time to use quantum, to think
about it should every product management team turn a little bit quantum on their radar screen, as they might need to have blockchain on their radar screen. Is it something that every company needs to
think about a bit?
Vlad Giorgio: I definitely think that the answer to this is yes, provided that you have any data that you want to keep secure. So if you want to keep data secure for a long time, you definitely
should look into quantum today. And that's because of this intercept and decode later attacks. So if you want to secure your data for 50 years, you encrypted today, quantum computing appears in 10
years, it can break your data. So you're not going to be able to provide the 50 years of security. So from a cryptographic point of view, definitely should look into quantum today from an application
point of view. It depends. I would advocate. Yes, because I'm running a content company in quantum software company, so I'm very happy if people are looking into it, but again, it depends on your
product. If you you have a highly optimized stack and you produce your, and you think you cannot optimize more, maybe it does make sense to look into quantum advantages, but these are very rare. I
mean, most of the time you find optimizations like you most of the time. So in most areas you will find a way of optimizing your production or maybe finding better routes to your distributors or
clients or so on. So I would say that medium to large scale companies should definitely have an eye on quantum of course, a very small company that sells, I don't know how dogs or something, probably
not initially. Right. But I would say that a medium to large scale companies should definitely invest something in quantum. I mean, for them, it's a very, very low investment and it has a great
potential of getting a higher reward. So I would say.
John Prial: And it's a higher reward, more differentiation within competitors. If they focus a little bit, even on just the security and the encryption, they could use it as an advantage just to
describe the trust they'll deliver it to their customers. We think trust is a very important point. So that's a great way to end this, Dr. Vlad, Giorgio, you thank you so much for this amazing
discussion. We could have gone forever, but I think we kept at the right level and I really appreciate your insights. So thank you so much.
Vlad Giorgio: It was a pleasure, John. Thank you very much for having me.
John Prial: Good bye.
Quantum computing will revolutionize your approach to a whole array of business problems but it won’t solve everything single-handedly. Your task as CEO is to know where quantum will make a
difference and make sure you’re ahead of your competition. In this episode of the Georgian Impact Podcast, Jon Prial sits down with quantum computing expert Vlad Gheorghiu to help you understand the
use cases where quantum will be most effective and why by giving a quick primer on the quantum tech stack.
You’ll hear about:
• How to think about quantum computing in relation to traditional computing
• Where we are now versus where we will need to be for certain use cases
• The quantum computing stack and how the different parts function
• How to develop a business strategy for quantum and what the first use cases might look like
Who is Vlad Gheorghiu?
Vlad Gheorghiu is CEO, President and Co-Founder of softwareQ Inc, and a researcher at the Institute for Quantum Computing, working with Michele Mosca on theoretical aspects of quantum computation and
(post-quantum) cryptography. He also collaborates on quantum risk assessment with evolutionQ Inc.
Vlad is involved in the CryptoWorks21 Quantum-Safe Cryptographic Infrastructure Program and is a member of the European Telecommunications Standards Institute (ETSI) Quantum-Safe Cryptography
Standardization Group. Vlad graduated from Carnegie Mellon University with a PhD in Theoretical Physics.
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How Mortgage APR Is Determined: A Step-by-Step Explanation
Buying a home is exciting – thinking about new furniture, a big backyard for weekend barbecues, and the space you’ll finally have. But before you start planning all that, you need to understand how
much borrowing for that house will cost you. One term you've probably heard frequently is "APR"; it often pops up when looking for a mortgage.
But what does it mean? And more importantly, how is mortgage APR calculated?
Let’s break it down in a simple way that’s easy to follow and without all the complicated financial terms that make your head spin.
Get A Free Mortgage Quote
What is APR, and Why Does It Matter?
Let’s start with the basics. APR, or Annual Percentage Rate, is the total cost of borrowing money over a year. While it may sound like your mortgage's interest rate, it’s more than that. The APR
includes the interest and any extra fees the lender might charge, like origination fees or closing costs.
When comparing loans, you might notice that the APR is higher than the interest rate. That’s because it paints a more accurate picture of what you’re paying in the long run. Knowing how is mortgage
APR calculated can save you from a financial surprise later on.
Step 1: Start with the Interest Rate
The foundation of your APR calculation is the interest rate itself. Think of the interest rate as the cost you pay to borrow the money - essentially, what the bank charges you for giving you a loan.
If your interest rate is low, you're already off to a good start. But remember, that's just one part of the picture. To calculate APR mortgage rates accurately, we need to factor in some additional
Step 2: Factor in Loan Fees
Next on the list is any extra costs that come with borrowing. These include things like:
● Origination fees: This is what the lender charges to process your loan.
● Discount points: If you paid extra upfront to lower your interest rate, that’s included, too.
● Closing costs: These can cover everything from title searches to appraisals and attorney fees.
While these fees don’t seem like much on their own, they increase your loan’s overall cost. And guess what? APR includes all of them.
Step 3: The Formula
Alright, here’s where things get a bit math-y, but don’t worry, we’ll take it slow. The formula for calculating APR goes something like this:
Now, let’s break that down:
● Interest charges: This is the amount you’ll pay in interest over the life of the loan.
● Fees: The total of all extra costs.
● Principal: The amount of money you’re borrowing.
● n: The number of days in the loan term.
So, you add up your interest and fees, divide by the loan amount (principal), divide again by the number of days in the loan term, multiply by 365 (because there are 365 days in a year), and finally
by 100 to get your percentage.
Step 4: Putting It into Practice
Let’s look at an example to make things clearer. Say you’re borrowing $200,000 to buy a house. The interest on your loan is $6,000, and the fees are $1,500. You’re taking out this loan for 30 years
(which is 10,950 days—yes, we did the math for you).
Plug those numbers into the formula:
● Add interest and fees: $6,000 + $1,500 = $7,500
● Divide by the principal: $7,500 ÷ $200,000 = 0.0375
● Divide by the number of days: 0.0375 ÷ 10,950 = 0.000003425
● Multiply by 365: 0.000003425 ✕ 365 = 0.00125
● Multiply by 100 to convert it into a percentage: 0.00125 ✕ 100 = 1.25%
So, your APR is 1.25%. Simple, right?
How Credit Score Affects Your APR
Now, let’s talk about what can change your APR. One big factor is your credit score. Lenders use your score to decide how risky it is to lend you money. If your credit score is high, lenders feel
more comfortable, offering a lower interest rate - and, by extension, a lower APR.
But if your score is lower, the lender may offer a higher interest rate to offset their risk, which bumps your APR. This is why it pays to check your credit score before you apply for a mortgage. If
it’s not where you want it to be, taking time to improve it can make a big difference.
Fixed vs. Variable APR: Which One’s for You?
When talking about mortgages, you’ll come across two main types of APR - fixed and variable. Both have their pros and cons, so it’s important to know the difference when figuring out how mortgage APR
is calculated.
● Fixed APR: This stays the same throughout the loan's life. That means you’ll know exactly what your monthly payments will be, making budgeting a little easier.
● Variable APR: This type can change based on the market. When interest rates go up, so does your APR (and your payments). Variable rates often start lower than fixed rates, but they come with some
risk since they can increase over time.
Why APR Matters for Your Mortgage Choice
When shopping for a mortgage, it’s easy to look at the interest rate and think, “Great, that’s the one I want!” But remember, the interest rate doesn’t show you the whole picture. APR tells you the
cost of borrowing over time to compare different loan offers.
For example, let’s say you’re comparing two loans:
● Loan 1 has a 3.5% interest rate and an APR of 4.1%.
● Loan 2 has a 3.75% interest rate but an APR of 3.95%.
Even though Loan 2 has a slightly higher interest rate, it’s the cheaper option overall because of the lower APR. This is why it’s important to look beyond the interest rate and calculate APR
mortgage rates to make an informed decision.
Get A Free Mortgage Quote
Other Factors to Consider
While APR gives you a solid understanding of what you’ll be paying over the life of your loan, it’s not the only thing to think about when choosing a mortgage. Some lenders may offer lower APRs but
have stricter terms or charge prepayment penalties if you pay off your loan early. Be sure to read the fine print before signing on the dotted line.
Here’s what else you need to keep an eye on:
Loan Terms
The loan length affects your payments and the total interest you’ll pay. Common options are 15, 20, or 30 years. Opting for a shorter loan term typically leads to higher monthly payments but less
total interest over the life of the loan. On the other hand, longer terms result in smaller monthly payments, but you pay more in interest overall. Choose a term that matches your budget and plans.
Prepayment Penalties
Some mortgages are penalized if you decide to pay off your loan early. Think of it like a fine for speeding up your loan payoff. These penalties can vary, so checking your mortgage has one is worth
checking. If you plan to pay off your loan early or refinance down the line, you’ll want to know about these potential fees.
Closing Costs
Don’t forget about closing costs. These are the fees you pay when you finalize the purchase of your home and can include things like appraisal fees and title insurance. Even if a lender offers a
lower APR, high closing costs can offset that benefit. Make sure you factor these into your overall cost.
Lender’s Reputation and Service
Finally, consider the lender’s reputation and customer service. A low APR is great, but if the lender doesn’t provide good support or has a bad reputation, it could cause problems. Look at reviews
and ask for recommendations to ensure you’re working with a reliable lender.
Conclusion: Understanding APR for Smarter Mortgage Decisions
Getting a mortgage is a big decision, and understanding how is mortgage APR calculated can help you make a smarter choice. By considering the interest rate, fees, and other costs, you’ll have a
clearer picture of what you’re paying.
And remember, a little effort to improve your credit score or compare loan offers can lead to significant savings over the life of your mortgage. Happy house hunting!
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tensor calculus pdf
Tensor Calculus Differentials & Directional Derivatives We are presently concerned with Inner Product spaces in our treatment of the Mechanics of Continua. Book Solutions of Exercises of Tensor
Calculus Made Simple by Taha Sochi pdf Book Solutions of Exercises of Tensor Calculus Made Simple by Taha Sochi pdf Pages 168 By Taha Sochi Publisher: CreateSpace, Year: 2017 Search in Amazon.com
This book contains the detailed solutions of all the exercises of my book: Tensor Calculus Made Simple. Preface This material offers a short introduction to tensor calculus. ... Much more than
documents. Numerical tensor calculus* - Volume 23 Skip to main content Accessibility help We use cookies to distinguish you from other users and to provide you with a better experience on our
websites. Tensor analysis 2.1 Tensor fields, parallel transport So far we have been constructing the tensor algebra from a tangent space at one point P on the manifold M. Now we want to pick another
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2.1.Vector Spaces and Bases Ansatz An n-dimensional vector space Vover R furnished with a basis fe ig. Outline 1 Computer di erential geometry and tensor calculus 2 The SageManifolds project 3
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Privacy Policy & Safety How YouTube works Test new features 1.18 Curvilinear Coordinates: Tensor Calculus 1.18.1 Differentiation of the Base Vectors Differentiation in curvilinear coordinates is more
involved than that in Cartesian coordinates because the base vectors are no longer constant in spacetime).Developed by Gregorio Ricci-Curbastro and his student Tullio Levi-Civita, it was used by
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Sorting with priority queues
Use priority queues to sort a sequence of integer numbers, in nondecreasing order and also in nonincreasing order.
Input consists of a sequence of integer numbers.
Print two lines, the first one with the numbers in increasing order, and the second one with the numbers in decreasing order.
To solve this exercise, the only containers that you should use are priority queues of integer numbers.
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A new Poisson probability paradigm for
A new Poisson probability paradigm for single cell RNA-seq clustering
Yue Pan
This package is developed to visualize the Poissoneity of scRNA-seq UMI data, and explore cell clustering based on model departure as a novel data representation.
In the following sections, we introduce the approach for assessment of the validity of the independent Poisson statistical framework using Q-Q envelope plots. The code here is specific for validation
of independent Poisson distributions of scRNA-seq data matrix entries, but such idea can be applied to different types of data under different assumptions of distributions.
Then we propose using model departure as a novel data representation, which is a measurement of relative location of that UMI count with respect to the independent Poisson distribution at the
individual entry level. A departure-based cell clustering algorithm is developed to explore cell subpopulations.
## Attaching package: 'dplyr'
## The following objects are masked from 'package:stats':
## filter, lag
## The following objects are masked from 'package:base':
## intersect, setdiff, setequal, union
Poissoneity of scRNA-seq data
ScRNA-seq data contains a large number of zeros, which makes the simple multiplication ineffective because zeros are not appropriately scaled by multiplication, motivating new statistical approaches.
Here, we propose a different, more principled approach. Instead of focusing on gene averages and working with normalized and scaled counts data, we prioritize the individual UMI matrix entries. This
approach is based on an independent Poisson statistical framework, where each RNA measurements for each cell comes from its own Poisson distribution.
In the following example, we show how to visualize the Poissoneity of scRNA-seq UMI data by Q-Q envelope plots. The envelopes are generated from theoretical distribution, which provide a quantitative
way measuring deviation from the diagonal line. If the theoretical distribution fits the data well, the curve should be located around the diagonal line and within the envelope.
The GLM-PCA algorithm is applied for parameter estimation. Then some matrix entries (default 200, with estimated Poisson parameters closest to the given Poisson parameter) will be selected as sample
data and compare with the theoretical Poisson distribution. The sample data we provided in this package contains 75 cells from one clonal cell line. Here we take \(L=10\) since that is shown to be an
appropriate number of latent vectors for this relatively homogeneous data set. In practice, we suggest try a range of \(L\) and find the one which gives best fit. This number can be different for
different data sets. As long as there exists such an \(L\), which gives reasonable fit under the Poisson statistical framework, that demonstrates the “Poissoneity” of such data set.
The similar idea can be used to visualize different types of data under different assumptions of distributions.
# UMI count matrix
test_dat <- as.data.frame(get_example_data("p5"))
scppp_obj <- scppp(test_dat, "rows")
# Q-Q envelope plot
qqplot_env_pois(scppp_obj, L = 10, lambda = 5)
As shown the the above figure, the aggregated 200 matrix entries were well fitted by Poisson distribution with parameter 5, i.e. these entries can be regarded as independent random samples generated
from Poisson distribution with mean 5.
Cell clustering based on model departure
A major application of this concept “Poissoneity” is clustering using a model departure as data representation.
The initial step is based on a crude two-way parameter approximation, where variation across cells is modeled by cell level parameters, and variation across genes is modeled by gene level parameters.
This initial step in itself does not appropriately account for cell heterogeneity (different cell types). In the next step such interesting structure is captured by departures from the naive two-way
approximation and the original count matrix is replaced by a Poisson departure matrix. In the departure matrix, each entry is quantified by the relative location of that original count with respect
to the tentative Poisson distribution, whose parameter comes from the initial two-way approximation. The departure measure is captured by a Cumulative Distribution Function (CDF), which leaves the
unexpectedly small counts nearly 0 and unusually large counts close to 1. Next, the departure measure is put on a more statistically amenable scale using the logit function. As a result, unexpectedly
large counts give large positive values and unexpectedly small counts give large negative values. This departure matrix forms the input for cell clustering as a downstream analysis.
The following example shows how to run our clustering pipeline using a model departure as data representation. In this simple case, set number of simulations \(sim = 100\) is enough to separate the
two known cell lines (71 cells from BCBL cell line and 58 from JSC cell line). For more complicated data sets, we suggest set the number of simulations to 500 or 1000 for more reliable clustering
The clustering results are stored in object scppp under “clust_results”. You can refer to “Hclust” under that for more details.
# UMI count matrix
test_dat <- as.data.frame(get_example_data("p56"))
scppp_obj <- scppp(test_dat, "rows")
scppp_obj <- HclustDepart(scppp_obj, maxSplit = 3)
# cluster label for each cell
clust_res <- scppp_obj[["clust_results"]]$Hclust[[1]]
## names cluster
## 1 Plate05A_A02 1
## 2 Plate05A_A03 1
## 3 Plate05A_A04 1
## 4 Plate05A_A05 1
## 5 Plate05A_A06 1
## 6 Plate05A_A08 1
## 1 2
## 71 58
The model departure data representation can also be fitted into any other clustering pipeline. Another option contained in this package is: keep departure as data representation but apply the Louvain
algorithm (implemented in Seurat pipeline) for clustering. In general, this is a faster clustering algorithm to do clustering. However, it requires carefully tuning about resolution parameter, which
can lead to different clustering results for the same data set. In this simple case, keeping the default resolution parameter (0.8) gives the correct clustering results as known cell lines.
The clustering results are also stored in object scppp under “clust_results”. You can refer to “Lclust” under that for more details.
## Modularity Optimizer version 1.3.0 by Ludo Waltman and Nees Jan van Eck
## Number of nodes: 129
## Number of edges: 3799
## Running Louvain algorithm...
## Maximum modularity in 10 random starts: 0.5998
## Number of communities: 2
## Elapsed time: 0 seconds
# cluster label for each cell
clust_res2 <- scppp_obj[["clust_results"]]$Lclust[[4]]
## names cluster
## Plate05A_A02 Plate05A_A02 0
## Plate05A_A03 Plate05A_A03 0
## Plate05A_A04 Plate05A_A04 0
## Plate05A_A05 Plate05A_A05 0
## Plate05A_A06 Plate05A_A06 0
## Plate05A_A08 Plate05A_A08 0
## 0 1
## 71 58
Townes, F. W., Hicks, S. C., Aryee, M. J., & Irizarry, R. A. (2019). Feature selection and dimension reduction for single-cell RNA-Seq based on a multinomial model. Genome biology, 20(1), 1-16.
Stuart T, Butler A, Hoffman P, Hafemeister C, Papalexi E, III WMM, Hao Y, Stoeckius M, Smibert P, Satija R (2019). “Comprehensive Integration of Single-Cell Data.” Cell, 177, 1888-1902. doi: 10.1016/
j.cell.2019.05.031, https://doi.org/10.1016/j.cell.2019.05.031.
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Feature Transformation in Machine Learning
In machine learning, feature transformation is a common technique used to improve the accuracy of models. One of the reasons for transformation is to handle skewed data, which can negatively affect
the performance of many machine learning algorithms.
In this article, you
• learn how to decide if you should apply a transformer to your feature and
• get to know different transformers with their advantages and disadvantages.
Table of Contents
Programming Example for Feature Transformation
For this article, I programmed an example to work with. First I created a positively skewed distribution and applied different transformations with the objective to make the transformed distribution
more normal. The following bar plot shows the results of the transformation.
• The square root distribution reduces the skewness and maintains the positive skewness.
• The Cube Root and Yeo-Johnson transformer work very well on the dataset.
• The Box-Cox transformation reduces the skewness to almost 0 and is therefore the best transformer to make the transformation most normal.
• The Log transformation works not so well because the skewness is not much reduced in comparison to other transformers and also switches the skewness to a negative value. Therefore the
distribution switches from positive to negative by applying the log transformation.
The complete Jupyter Notebook is available at Google Colab.
But before we apply any transformation we first have to know when we should apply a transformer.
What is the Skewness Threshold to Apply a Transformation?
Skewness is a measure of the asymmetry of a probability distribution, and a threshold value for skewness to determine when to apply a transformation is not fixed.
A common approach to determining whether to transform a feature with skewness is to use a threshold value of 0.5 or higher. This is based on the observation that most statistical distributions have a
skewness between -0.5 and 0.5. Therefore, if the skewness of a feature is above this threshold, a transformation is applied to make the data less skewed and more normally distributed.
However, this threshold value is not fixed and can depend on the specific problem, the dataset, and the algorithm being used.
If you want to know when and how to transform the target variable in a machine-learning project, take a look at the article Transform Target Variable.
Log Transformation
The log transformation is a commonly used technique in machine learning to transform skewed or non-normal data into a more normal distribution by computing the natural logarithm of a variable. The
log transformation works by compressing large values and expanding small values. When applied to skewed data, it can reduce the influence of extreme values and make the distribution more symmetrical.
df['feature'] = np.log(df['feature'])
Log Transform Features that contain the value 0
The general problem with log transforming features that contain the value 0 is that the log of 0 is not defined (-inf).
Therefore you can use the little trick to add 1 to the value: log(x+1). Numpy has therefore a build-in function: np.log1p(x). In this case, the log1p transformation of the value 0 is still 0.
df['feature'] = np.log1p(df['feature'])
Disadvantages of Log Transformation
• The main disadvantage of the log transformation is that it requires the input data to be positive.
Box-Cox Transformation
The Box-Cox transformation is a mathematical transformation that can be used to transform non-normal data into a more normal distribution. It was introduced by statisticians George Box and David Cox
in 1964.
df['feature'] = scipy.stats.boxcox(df['feature'], lambda)
The value lambda is the power of the transformation. If you use the scipy library for the transformation you don’t have to set a value for lambda. If lambda is None a range of possible values is
tested for maximizing the log-likelihood function.
w = \begin{cases} log(x) & \text{if } \lambda = 0, \\ \frac{(x-1)}{\lambda} & \text{otherwise} \end{cases}
Notice that when lambda =1 the transformed data shifts down by 1 but the distribution does not change. In this case, the data is already normally distributed.
Advantages of Box-Cox Transformation
• The Box-Cox transformation has the advantage over other power transformations in that it can automatically determine the appropriate power to use based on the data.
Disadvantages of Box-Cox Transformation
• The Box-Cox transformation requires the input data to be positive, but the Yeo-Johnson transformation is quite similar and does not have the restriction that all values have to be positive.
Square Root Transformation
The square root transformation works analog to the log transformation by applying the square root to a variable or feature. When the square root transformation is applied to a positively skewed
dataset, the transformed dataset will have a more normal distribution.
df['feature'] = np.sqrt(df['feature'])
Advantages of Square Root Transformation
• The main advantage of square root transformation is, it can be applied to zero values.
Disadvantages of Square Root Transformation
• The transformation is weaker than the log transformation.
• The square root transformation can not be applied to negative values
Cube Root Transformation
The cube root transformation is a mathematical transformation that takes the cube root of a variable to make a distribution more normal. It is similar to other power transformations such as the
square root and the logarithmic transformation but has different properties. The cube root transformation works by compressing large values and expanding small values.
df['feature'] = np.cbrt(df['feature'])
Advantages of Cube Root Transformation
• The main advantage of the cube root transformation is, it can be applied to zero and negative values.
Yeo-Johnson Transformation
The Yeo-Johnson transformation is a widely used data transformation technique that can be used to transform non-normal data into a more normal distribution. It was introduced by Robert Yeo and Robert
Johnson in 2000 as an improvement over the Box-Cox transformation, which has limitations when dealing with data that contain negative values.
df['feature'] = scipy.stats.yeojohnson(df['feature'], lambda)
The value lambda is the power of the transformation. If you use the scipy library for the transformation you don’t have to set a value for lambda. If lambda is None a range of possible values is
tested for maximizing the log-likelihood function.
w = \begin{cases} \frac{((x+1)^\lambda -1}{\lambda} & \text{if } \lambda \neq 0, x>=0 \\ ln(x+1) & \text{if } \lambda =0, y>=0 \\ \frac{-((x+1)^{(2-\lambda)}-1)}{2-\lambda} & \text{if } \lambda \neq 2, y<0 \\ -ln(-x+1) & \text{if } \lambda =2, y<0 \end{cases}
A value of lambda=1 produces the identity transformation.
Advantages of Yeo-Johnson Transformation
• The Yeo-Johnson transformation transforms zero, positive as well as negative values.
Inverse Transformation
The inverse transformation is very useful when the feature is a count or a rate, and we want to transform it into a continuous variable.
df['feature'] = 1/(df['feature'])
Advantages of Inverse Transformation
• The transformation can be used on a discrete variable or series.
Disadvantages of Inverse Transformation
The inverse transformation can only be applied to non-0 values.
Read my latest articles:
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CAM Colloquium 2017-03-24 - Alex Townsend: "Why are so many matrices of low rank in computational math?"
From E. Cornelius
Abstract: Matrices that appear in computational mathematics are so often of low rank. Since random ("average") matrices are almost surely of full rank, mathematics needs to explain the abundance of
low rank structures. We will give a characterization of certain low rank matrices using Sylvester equations and show that the decay of singular values can be understood via an extremal rational
problem. We will use it to explain why low rank matrices appear in galaxy simulations, polynomial interpolation, Krylov methods, and fast transforms.
Biography: Alex Townsend is an assistant professor in the Math Department at Cornell University, with field affiliations in Applied Mathematics and CSE. His research is in spectral element methods,
fast transforms, polynomial system solving, and low rank approximation. Prior to coming to Cornell, he was an Applied Math instructor at MIT after completing a DPhil at the University of Oxford. He
was awarded a Leslie Fox Prize in 2015 for a fast discrete Hankel transform and in 2013 for developing a sparse well-conditioned spectral method for the solution of differential equations.
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Vector and Raster Images: What Are They and How Are They Used? | Creatsy® Blog
Vector and Raster Images: What Are They and How Are They Used?
If you have been around graphic design for any time, you've probably heard of the terms "vector" and "raster". What are they, and how are they used in graphic design?
What is a vector?
A vector is a series of commands or mathematical statements that place lines and shapes in a two-dimensional or three-dimensional space. A vector has magnitude (distance or size) and direction (as on
a compass needle, like north or southeast). Lines and curves made by these mathematical equations are called paths.
This is a path:
Anchor points
The white boxes on each end are called anchor points. They start and end the path. A great way to understand a vector is to imagine you are about to go on a hike up a mountain. The hike will have a
beginning and an end.
Smooth and corner anchor points
Any curves or changes in altitude on a mountain hike will determine how steep, straight, or winding your path will be. The more curves and angles you have, the longer your hike will take from
beginning to end. In the same way, curves and angles in the vector paths create additional anchor points. Smooth anchor points create curves like a turn or a switchback on a mountain path. Corner
points make straight lines that provide angles in a vector image.
Vector images in graphic design
The fewer points the vector image has, the simpler the image will be and the smaller the file size. Here is a circle made in Adobe Illustrator. The blue squares on the circle are the anchor points.
Connected to the anchor points are lines with circles on the ends, called Bezier handles, which adjust the curvature of the circle.
The more anchor points a line or curve has, the more complex the image will be and the greater the file size. Here is a squiggly line with visible anchor points in Adobe Illustrator. The blue squares
are the anchor points. The curves can be adjusted using the Live Corner Widgets, which are circles with a small dot inside.
Vector paths are infinitely scalable smaller or larger, and because they are based on mathematical formulas, the clarity of the image is not affected by the size of the image. This is not the case
with a raster image.
What does raster mean?
A raster or bitmap image comprises pixels or little “bits” of color-related information. These are called “pixels” or “dots.”
PPI and DPI
A raster image such as a photograph is a resolution-dependent image. Resolution means how detailed an image is, which directly results from how many pixels or dots are in each square inch of the
picture. The terms “pixels per inch” (PPI) and “dots per inch” (DPI) refer to how many bits of color-related information are in a square inch. If you have a one-inch by one-inch picture at 300 PPI,
300 individual bits of color-related information make up the image in that square inch. If your DPI is only 72, the quality of the picture will decrease because the number of bits has gone down. When
you have more pixels or dots in the square inch image, the resolution—the level of detail and, therefore, the clarity—of the image increases.
When to use which
Raster images are great for things such as photographs, complex digital art, and detailed renderings of things like product mockups. Vector images are used in designs like simplified repeat patterns,
business logos, and social media icons.
Vector and Raster Takeaways
Both raster and vector-based images are used in graphic design. Vectors can be scaled without losing image quality because they are based on mathematical formulas. Raster images have limited scaling
abilities but can contain more visual information and details. Both image types have unique advantages and disadvantages and are valuable tools for designers.
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Can an object be traveling in the positive direction and have a negative acceleration?
Can an object be traveling in the positive direction and have a negative acceleration?
Since acceleration is a vector quantity, it has a direction associated with it. The direction of the acceleration vector depends on two things: whether the object is speeding up or slowing down.
whether the object is moving in the + or – direction.
When the acceleration will be positive and negative?
A positive acceleration means an increase in velocity with time. negative acceleration means the speed reduces with time. its a retardation. If the speed is increasing, the car has positive
Can a car have a negative velocity and a positive acceleration at the same time?
Yes, for example, a car that is traveling northward and slowing down has a northward velocity and a southward acceleration. A car traveling in the negative x-direction and braking has a negative
velocity and a positive acceleration. Give an example where both the velocity and acceleration. are negative.
How can an object have a positive velocity but a negative acceleration?
For example, the velocity could be in the positive direction and the object slowing down or the velocity could be in the negative direction and the object speeding up. Both of these scenarios would
result in a negative acceleration.
Can an object have a positive acceleration and be slowing down?
An object with negative acceleration could be speeding up, and an object with positive acceleration could be slowing down. If acceleration points in the same direction as the velocity, the object
will be speeding up.
Can acceleration be positive when velocity is zero?
The principle is that the slope of the line on a velocity-time graph reveals useful information about the acceleration of the object. If the acceleration is zero, then the slope is zero (i.e., a
horizontal line). If the acceleration is positive, then the slope is positive (i.e., an upward sloping line).
What is the difference between positive acceleration and negative acceleration?
Mathematically, a negative acceleration means you will subtract from the current value of the velocity, and a positive acceleration means you will add to the current value of the velocity. And if the
acceleration points in the opposite direction of the velocity, the object will be slowing down.
What is an example where both the velocity and acceleration are negative?
Answer: If the velocity and the acceleration have different signs (opposite directions), then the object is slowing down. For example, a ball thrown upward has a positive velocity and a negative
acceleration while it is going up.
Can an object have a negative acceleration?
An object with negative acceleration could be speeding up, and an object with positive acceleration could be slowing down. And if the acceleration points in the opposite direction of the velocity,
the object will be slowing down.
What’s the difference between positive and negative acceleration?
The difference between positive and negative acceleration is the sign or direction of the change in velocity: Positive acceleration means that velocity is increasing (acceleration in the positive
direction). Negative acceleration means that velocity is decreasing (acceleration in the negative direction).
Is there an absolute positive or negative velocity?
There is no absolute positive or negative velocity or acceleration. One may also ask, can a velocity be negative? Negative velocity just means velocity in the opposite direction than what would be
positive. From the math point of view, you cannot have “negative velocity” in itself, only “negative velocity in a given direction”.
What does acceleration look like on a graph?
Negative acceleration looks like a decreasing (sloping down) function on a graph of velocity. This is because acceleration is the first derivative of velocity, and a negative first derivative
indicates a decreasing function. The graph could be a line with a negative slope, such as the one below, but there are other possibilities.
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2023 AMC 12A Problems/Problem 6
Points $A$ and $B$ lie on the graph of $y=\log_{2}x$. The midpoint of $\overline{AB}$ is $(6, 2)$. What is the positive difference between the $x$-coordinates of $A$ and $B$?
Solution 1
Let $A(6+m,2+n)$ and $B(6-m,2-n)$, since $(6,2)$ is their midpoint. Thus, we must find $2m$. We find two equations due to $A,B$ both lying on the function $y=\log_{2}x$. The two equations are then $\
log_{2}(6+m)=2+n$ and $\log_{2}(6-m)=2-n$. Now add these two equations to obtain $\log_{2}(6+m)+\log_{2}(6-m)=4$. By logarithm rules, we get $\log_{2}((6+m)(6-m))=4$. By raising 2 to the power of
both sides, we obtain $(6+m)(6-m)=16$. We then get $\[36-m^2=16 \rightarrow m^2=20 \rightarrow m=2\sqrt{5}\]$. Since we're looking for $2m$, we obtain $(2)(2\sqrt{5})=\boxed{\textbf{(D) }4\sqrt{5}}$
~amcrunner (yay, my first AMC solution)
Solution 2
We have $\frac{x_A + x_B}{2} = 6$ and $\frac{\log_2 x_A + \log_2 x_B}{2} = 2$. The first equation becomes $x_A + x_B = 12,$ and the second becomes $\log_2(x_A x_B) = 4,$ so $x_A x_B = 16.$ Then \
begin{align*} \left| x_A - x_B \right| & = \sqrt{\left( x_A + x_B \right)^2 - 4 x_A x_B} \\ & = \boxed{\textbf{(D) } 4 \sqrt{5}}. \end{align*}
~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)
Solution 3
Basically, we can use the midpoint formula
assume that the points are $(x_1,y_1)$ and $(x_2,y_2)$
assume that the points are ($x_1$,$\log_{2}(x_1)$) and ($x_2$,$\log_{2}(x_2)$)
midpoint formula is ($\frac{x_1+x_2}{2}$,$\frac{\log_{2}(x_1)+\log_{2}(x_2)}{2}$)
thus $x_1+x_2=12$$x_2=12-x_1$ and $\log_{2}(x_1)+\log_{2}(x_2)=4$$\log_{2}(x_1)+\log_{2}(12-x_1)=\log_{2}(16)$
since $2^0=1$ so,
$12x_1-x_1^2-16=0$ for simplicity lets say $x_1 = x$
$12x-x^2=16$. We rearrange to get $x^2-12x+16=0$.
put this into quadratic formula and you should get
$x_1=6+2\sqrt{5}$ Therefore, $x_1=6+2\sqrt{5}-(6-2\sqrt{5})$
which equals $6-6+4\sqrt{5}=\boxed{\textbf{(D) }4\sqrt{5}}$
Solution 4
Similar to above, but solve for $x = 2^y$ in terms of $y$:
$(2^{y}+2^{2+(2-y)})/2= 6$
$2^y + 2^{4-y} = 12$
$(2^y)^2 + 2^4 = 12(2^y)$
$x^2 -12x + 16 = 0$
Distance between roots of the quadratic is the discriminant: $\sqrt{{12}^2 - 4(1)(16)} = \sqrt{80} = \boxed{\textbf{(D) }4\sqrt{5}}$
Video Solution (easy to understand) by Power Solve
Video Solution 1
~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)
Video Solution 2 (🚀 Under 3 min 🚀)
~Education, the Study of Everything
See Also
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.
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mathematics, Wellesley-Describe your intellectual interests....
Describe your intellectual interests, their evolution, and what makes them exciting to you. Tell us how you will utilize the academic programs in the College of Arts and Sciences to further explore
your interests, intended major, or field of study.
please help
"No man is an island" as Mathematics has proven to me. Many do not realize the importance of math, especially I, before the start of my Advanced Level course. Before 2008 A.D., I was a mathematics
'crack'. I took the subject for granted as I could easily score an A plus without as much looking at a mathematics book.
The first A-Level mathematics test I took, I got the lowest mark I had ever attained in my entire life; am sure it was an ungraded grade. Looking at my paper, I just did not comprehend anything that
was happening around me. Fear crept all around me. My palms were flowing rivers that could not dry up no matter how much tissue I used. I questioned so much if I were going to get the grades required
for entrance into medical school. It took so much out of me to tell myself that it would never happen again, and all I needed was to just work harder. Besides, I thought, any form of help was really
unnecessary. The months went on, but I had only improved to a C plus. Finally, I gave myself permission to attend mats clinic, and I quickly upgraded to a ninety-six percent. With that, I assumed I
would not need extra help as I could learn pre-calculus in less than a week. I had deceived myself in believing I had done it alone. Into my exams, I just went in assuming that I am great at
calculus, so I really do not need much preparation. I did not get my A nor a B. Without a doubt, she is by far, the biggest struggle in my academic life.
However, with the beginning of A2 came a New Year, bringing a new approach to life; anything I did not understand or grasp, the next step would be to ask for help, including math clinic. Now I work
hard to maintain A grades, with frequent visits to math clinic and quick chats with teachers after a lesson. But I have also learnt to ask for needed help no matter what I am doing.
Although mathematics is not my intended major, it is still exciting to me as it has taught me the most important life skills. And regardless of the ache I have endured due to the subject, I still
find her highly comforting after solving a difficult problem. Most importantly, she has taught me perseverance, no matter how hot the fire becomes. I believe that math is not just the finally answer,
but really the path to the answer. My main intension is to join the Oliver Club as it will allow me to integrate with other students just as, if not more, enthusiastic about mathematics. It could
also help me learn how to appreciate mathematics at a much challenging level. Since my main goal is to become a medical doctor, I hope that all the energy invested into the subject will help me
improve my logical skills and quick reaction to problems that I will face in the field.
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Regression analysis
Gaussian distribution
around the line y=1.5x+2 (not shown)
Necessary Condition Analysis
) or estimate the conditional expectation across a broader collection of non-linear models (e.g.,
nonparametric regression
Regression analysis is primarily used for two conceptually distinct purposes. First, regression analysis is widely used for prediction and forecasting, where its use has substantial overlap with the
field of machine learning. Second, in some situations regression analysis can be used to infer causal relationships between the independent and dependent variables. Importantly, regressions by
themselves only reveal relationships between a dependent variable and a collection of independent variables in a fixed dataset. To use regressions for prediction or to infer causal relationships,
respectively, a researcher must carefully justify why existing relationships have predictive power for a new context or why a relationship between two variables has a causal interpretation. The
latter is especially important when researchers hope to estimate causal relationships using observational data.^[2]^[3]
The earliest form of regression was the
method of least squares, which was published by
in 1805,
and by
in 1809.
Legendre and Gauss both applied the method to the problem of determining, from astronomical observations, the orbits of bodies about the Sun (mostly comets, but also later the then newly discovered
minor planets). Gauss published a further development of the theory of least squares in 1821,
including a version of the
Gauss–Markov theorem
The term "regression" was coined by Francis Galton in the 19th century to describe a biological phenomenon. The phenomenon was that the heights of descendants of tall ancestors tend to regress down
towards a normal average (a phenomenon also known as regression toward the mean).^[7]^[8] For Galton, regression had only this biological meaning,^
conditional distribution
of the response variable is Gaussian, but the joint distribution need not be. In this respect, Fisher's assumption is closer to Gauss's formulation of 1821.
In the 1950s and 1960s, economists used electromechanical desk calculators to calculate regressions. Before 1970, it sometimes took up to 24 hours to receive the result from one regression.^[16]
Regression methods continue to be an area of active research. In recent decades, new methods have been developed for robust regression, regression involving correlated responses such as time series
and growth curves, regression in which the predictor (independent variable) or response variables are curves, images, graphs, or other complex data objects, regression methods accommodating various
types of missing data, nonparametric regression, Bayesian methods for regression, regression in which the predictor variables are measured with error, regression with more predictor variables than
observations, and causal inference with regression.
Regression model
In practice, researchers first select a model they would like to estimate and then use their chosen method (e.g., ordinary least squares) to estimate the parameters of that model. Regression models
involve the following components:
In various fields of application, different terminologies are used in place of dependent and independent variables.
Most regression models propose that ${\displaystyle Y_{i}}$ is a function (regression function) of ${\displaystyle X_{i}}$ and ${\displaystyle \beta }$, with ${\displaystyle e_{i}}$ representing an
additive error term that may stand in for un-modeled determinants of ${\displaystyle Y_{i}}$ or random statistical noise:
${\displaystyle Y_{i}=f(X_{i},\beta )+e_{i}}$
Note that the independent variables ${\displaystyle X_{i}}$ are assumed to be free of error. This important assumption is often overlooked, although errors-in-variables models can be used when the
independent variables are assumed to contain errors.
The researchers' goal is to estimate the function ${\displaystyle f(X_{i},\beta )}$ that most closely fits the data. To carry out regression analysis, the form of the function ${\displaystyle f}$
must be specified. Sometimes the form of this function is based on knowledge about the relationship between ${\displaystyle Y_{i}}$ and ${\displaystyle X_{i}}$ that does not rely on the data. If no
such knowledge is available, a flexible or convenient form for ${\displaystyle f}$ is chosen. For example, a simple univariate regression may propose ${\displaystyle f(X_{i},\beta )=\beta _{0}+\beta
_{1}X_{i}}$, suggesting that the researcher believes ${\displaystyle Y_{i}=\beta _{0}+\beta _{1}X_{i}+e_{i}}$ to be a reasonable approximation for the statistical process generating the data.
Once researchers determine their preferred statistical model, different forms of regression analysis provide tools to estimate the parameters ${\displaystyle \beta }$. For example, least squares
(including its most common variant, ordinary least squares) finds the value of ${\displaystyle \beta }$ that minimizes the sum of squared errors ${\displaystyle \sum _{i}(Y_{i}-f(X_{i},\beta ))^{2}}$
. A given regression method will ultimately provide an estimate of ${\displaystyle \beta }$, usually denoted ${\displaystyle {\hat {\beta }}}$ to distinguish the estimate from the true (unknown)
parameter value that generated the data. Using this estimate, the researcher can then use the fitted value ${\displaystyle {\hat {Y_{i}}}=f(X_{i},{\hat {\beta }})}$ for prediction or to assess the
accuracy of the model in explaining the data. Whether the researcher is intrinsically interested in the estimate ${\displaystyle {\hat {\beta }}}$ or the predicted value ${\displaystyle {\hat {Y_
{i}}}}$ will depend on context and their goals. As described in ordinary least squares, least squares is widely used because the estimated function ${\displaystyle f(X_{i},{\hat {\beta }})}$
approximates the conditional expectation ${\displaystyle E(Y_{i}|X_{i})}$.^[5] However, alternative variants (e.g., least absolute deviations or quantile regression) are useful when researchers want
to model other functions ${\displaystyle f(X_{i},\beta )}$.
It is important to note that there must be sufficient data to estimate a regression model. For example, suppose that a researcher has access to ${\displaystyle N}$ rows of data with one dependent and
two independent variables: ${\displaystyle (Y_{i},X_{1i},X_{2i})}$. Suppose further that the researcher wants to estimate a bivariate linear model via least squares: ${\displaystyle Y_{i}=\beta _{0}+
\beta _{1}X_{1i}+\beta _{2}X_{2i}+e_{i}}$. If the researcher only has access to ${\displaystyle N=2}$ data points, then they could find infinitely many combinations ${\displaystyle ({\hat {\beta }}_
{0},{\hat {\beta }}_{1},{\hat {\beta }}_{2})}$ that explain the data equally well: any combination can be chosen that satisfies ${\displaystyle {\hat {Y}}_{i}={\hat {\beta }}_{0}+{\hat {\beta }}_{1}
X_{1i}+{\hat {\beta }}_{2}X_{2i}}$, all of which lead to ${\displaystyle \sum _{i}{\hat {e}}_{i}^{2}=\sum _{i}({\hat {Y}}_{i}-({\hat {\beta }}_{0}+{\hat {\beta }}_{1}X_{1i}+{\hat {\beta }}_{2}X_
{2i}))^{2}=0}$ and are therefore valid solutions that minimize the sum of squared residuals. To understand why there are infinitely many options, note that the system of ${\displaystyle N=2}$
equations is to be solved for 3 unknowns, which makes the system underdetermined. Alternatively, one can visualize infinitely many 3-dimensional planes that go through ${\displaystyle N=2}$ fixed
More generally, to estimate a least squares model with ${\displaystyle k}$ distinct parameters, one must have ${\displaystyle N\geq k}$ distinct data points. If ${\displaystyle N>k}$, then there does
not generally exist a set of parameters that will perfectly fit the data. The quantity ${\displaystyle N-k}$ appears often in regression analysis, and is referred to as the degrees of freedom in the
model. Moreover, to estimate a least squares model, the independent variables ${\displaystyle (X_{1i},X_{2i},...,X_{ki})}$ must be linearly independent: one must not be able to reconstruct any of the
independent variables by adding and multiplying the remaining independent variables. As discussed in ordinary least squares, this condition ensures that ${\displaystyle X^{T}X}$ is an invertible
matrix and therefore that a unique solution ${\displaystyle {\hat {\beta }}}$ exists.
Underlying assumptions
This section
needs additional citations for verification
(December 2020)
By itself, a regression is simply a calculation using the data. In order to interpret the output of regression as a meaningful statistical quantity that measures real-world relationships, researchers
often rely on a number of classical assumptions. These assumptions often include:
A handful of conditions are sufficient for the least-squares estimator to possess desirable properties: in particular, the
Heteroscedasticity-consistent standard errors
allow the variance of
${\displaystyle e_{i}}$
to change across values of
${\displaystyle X_{i}}$
. Correlated errors that exist within subsets of the data or follow specific patterns can be handled using
clustered standard errors, geographic weighted regression
, or
standard errors, among other techniques. When rows of data correspond to locations in space, the choice of how to model
${\displaystyle e_{i}}$
within geographic units can have important consequences.
The subfield of
is largely focused on developing techniques that allow researchers to make reasonable real-world conclusions in real-world settings, where classical assumptions do not hold exactly.
Linear regression
In linear regression, the model specification is that the dependent variable, ${\displaystyle y_{i}}$ is a linear combination of the parameters (but need not be linear in the independent variables).
For example, in simple linear regression for modeling ${\displaystyle n}$ data points there is one independent variable: ${\displaystyle x_{i}}$, and two parameters, ${\displaystyle \beta _{0}}$ and
${\displaystyle \beta _{1}}$:
straight line: ${\displaystyle y_{i}=\beta _{0}+\beta _{1}x_{i}+\varepsilon _{i},\quad i=1,\dots ,n.\!}$
In multiple linear regression, there are several independent variables or functions of independent variables.
Adding a term in ${\displaystyle x_{i}^{2}}$ to the preceding regression gives:
parabola: ${\displaystyle y_{i}=\beta _{0}+\beta _{1}x_{i}+\beta _{2}x_{i}^{2}+\varepsilon _{i},\ i=1,\dots ,n.\!}$
This is still linear regression; although the expression on the right hand side is quadratic in the independent variable ${\displaystyle x_{i}}$, it is linear in the parameters ${\displaystyle \beta
_{0}}$, ${\displaystyle \beta _{1}}$ and ${\displaystyle \beta _{2}.}$
In both cases, ${\displaystyle \varepsilon _{i}}$ is an error term and the subscript ${\displaystyle i}$ indexes a particular observation.
Returning our attention to the straight line case: Given a random sample from the population, we estimate the population parameters and obtain the sample linear regression model:
${\displaystyle {\widehat {y}}_{i}={\widehat {\beta }}_{0}+{\widehat {\beta }}_{1}x_{i}.}$
${\displaystyle e_{i}=y_{i}-{\widehat {y}}_{i}}$
, is the difference between the value of the dependent variable predicted by the model,
${\displaystyle {\widehat {y}}_{i}}$
, and the true value of the dependent variable,
${\displaystyle y_{i}}$
. One method of estimation is
${\displaystyle SSR=\sum _{i=1}^{n}e_{i}^{2}}$
Minimization of this function results in a set of
normal equations
, a set of simultaneous linear equations in the parameters, which are solved to yield the parameter estimators,
${\displaystyle {\widehat {\beta }}_{0},{\widehat {\beta }}_{1}}$
Illustration of linear regression on a data set
In the case of simple regression, the formulas for the least squares estimates are
${\displaystyle {\widehat {\beta }}_{1}={\frac {\sum (x_{i}-{\bar {x}})(y_{i}-{\bar {y}})}{\sum (x_{i}-{\bar {x}})^{2}}}}$
${\displaystyle {\widehat {\beta }}_{0}={\bar {y}}-{\widehat {\beta }}_{1}{\bar {x}}}$
where ${\displaystyle {\bar {x}}}$ is the mean (average) of the ${\displaystyle x}$ values and ${\displaystyle {\bar {y}}}$ is the mean of the ${\displaystyle y}$ values.
Under the assumption that the population error term has a constant variance, the estimate of that variance is given by:
${\displaystyle {\hat {\sigma }}_{\varepsilon }^{2}={\frac {SSR}{n-2}}}$
This is called the
mean square error
(MSE) of the regression. The denominator is the sample size reduced by the number of model parameters estimated from the same data,
${\displaystyle (n-p)}$
${\displaystyle p}$
${\displaystyle (n-p-1)}$
if an intercept is used.
In this case,
${\displaystyle p=1}$
so the denominator is
${\displaystyle n-2}$
standard errors
of the parameter estimates are given by
${\displaystyle {\hat {\sigma }}_{\beta _{1}}={\hat {\sigma }}_{\varepsilon }{\sqrt {\frac {1}{\sum (x_{i}-{\bar {x}})^{2}}}}}$
${\displaystyle {\hat {\sigma }}_{\beta _{0}}={\hat {\sigma }}_{\varepsilon }{\sqrt {{\frac {1}{n}}+{\frac {{\bar {x}}^{2}}{\sum (x_{i}-{\bar {x}})^{2}}}}}={\hat {\sigma }}_{\beta _{1}}{\sqrt {\
frac {\sum x_{i}^{2}}{n}}}.}$
Under the further assumption that the population error term is normally distributed, the researcher can use these estimated standard errors to create
population parameters
General linear model
In the more general multiple regression model, there are ${\displaystyle p}$ independent variables:
${\displaystyle y_{i}=\beta _{1}x_{i1}+\beta _{2}x_{i2}+\cdots +\beta _{p}x_{ip}+\varepsilon _{i},\,}$
where ${\displaystyle x_{ij}}$ is the ${\displaystyle i}$-th observation on the ${\displaystyle j}$-th independent variable. If the first independent variable takes the value 1 for all ${\
displaystyle i}$, ${\displaystyle x_{i1}=1}$, then ${\displaystyle \beta _{1}}$ is called the
regression intercept
The least squares parameter estimates are obtained from ${\displaystyle p}$ normal equations. The residual can be written as
${\displaystyle \varepsilon _{i}=y_{i}-{\hat {\beta }}_{1}x_{i1}-\cdots -{\hat {\beta }}_{p}x_{ip}.}$
The normal equations are
${\displaystyle \sum _{i=1}^{n}\sum _{k=1}^{p}x_{ij}x_{ik}{\hat {\beta }}_{k}=\sum _{i=1}^{n}x_{ij}y_{i},\ j=1,\dots ,p.\,}$
In matrix notation, the normal equations are written as
${\displaystyle \mathbf {(X^{\top }X){\hat {\boldsymbol {\beta }}}={}X^{\top }Y} ,\,}$
where the ${\displaystyle ij}$ element of ${\displaystyle \mathbf {X} }$ is ${\displaystyle x_{ij}}$, the ${\displaystyle i}$ element of the column vector ${\displaystyle Y}$ is ${\displaystyle y_
{i}}$, and the ${\displaystyle j}$ element of ${\displaystyle {\hat {\boldsymbol {\beta }}}}$ is ${\displaystyle {\hat {\beta }}_{j}}$. Thus ${\displaystyle \mathbf {X} }$ is ${\displaystyle n\times
p}$, ${\displaystyle Y}$ is ${\displaystyle n\times 1}$, and ${\displaystyle {\hat {\boldsymbol {\beta }}}}$ is ${\displaystyle p\times 1}$. The solution is
${\displaystyle \mathbf {{\hat {\boldsymbol {\beta }}}=(X^{\top }X)^{-1}X^{\top }Y} .\,}$
Once a regression model has been constructed, it may be important to confirm the
of individual parameters.
Interpretations of these diagnostic tests rest heavily on the model's assumptions. Although examination of the residuals can be used to invalidate a model, the results of a
t-test or
are sometimes more difficult to interpret if the model's assumptions are violated. For example, if the error term does not have a normal distribution, in small samples the estimated parameters will
not follow normal distributions and complicate inference. With relatively large samples, however, a
central limit theorem
can be invoked such that hypothesis testing may proceed using asymptotic approximations.
Limited dependent variables
Limited dependent variables, which are response variables that are categorical variables or are variables constrained to fall only in a certain range, often arise in econometrics.
The response variable may be non-continuous ("limited" to lie on some subset of the real line). For binary (zero or one) variables, if analysis proceeds with least-squares linear regression, the
model is called the
negative binomial
model may be used.
Nonlinear regression
When the model function is not linear in the parameters, the sum of squares must be minimized by an iterative procedure. This introduces many complications which are summarized in Differences between
linear and non-linear least squares.
Prediction (interpolation and extrapolation)
In the middle, the fitted straight line represents the best balance between the points above and below this line. The dotted straight lines represent the two extreme lines, considering only the
variation in the slope. The inner curves represent the estimated range of values considering the variation in both slope and intercept. The outer curves represent a prediction for a new measurement.^
Regression models predict a value of the Y variable given known values of the X variables. Prediction within the range of values in the dataset used for model-fitting is known informally as
interpolation. Prediction outside this range of the data is known as extrapolation. Performing extrapolation relies strongly on the regression assumptions. The further the extrapolation goes outside
the data, the more room there is for the model to fail due to differences between the assumptions and the sample data or the true values.
A prediction interval that represents the uncertainty may accompany the point prediction. Such intervals tend to expand rapidly as the values of the independent variable(s) moved outside the range
covered by the observed data.
For such reasons and others, some tend to say that it might be unwise to undertake extrapolation.^[21]
Model selection
The assumption of a particular form for the relation between Y and X is another source of uncertainty. A properly conducted regression analysis will include an assessment of how well the assumed form
is matched by the observed data, but it can only do so within the range of values of the independent variables actually available. This means that any extrapolation is particularly reliant on the
assumptions being made about the structural form of the regression relationship. If this knowledge includes the fact that the dependent variable cannot go outside a certain range of values, this can
be made use of in selecting the model – even if the observed dataset has no values particularly near such bounds. The implications of this step of choosing an appropriate functional form for the
regression can be great when extrapolation is considered. At a minimum, it can ensure that any extrapolation arising from a fitted model is "realistic" (or in accord with what is known).
Power and sample size calculations
There are no generally agreed methods for relating the number of observations versus the number of independent variables in the model. One method conjectured by Good and Hardin is ${\displaystyle N=m
^{n}}$, where ${\displaystyle N}$ is the sample size, ${\displaystyle n}$ is the number of independent variables and ${\displaystyle m}$ is the number of observations needed to reach the desired
precision if the model had only one independent variable.^[22] For example, a researcher is building a linear regression model using a dataset that contains 1000 patients (${\displaystyle N}$). If
the researcher decides that five observations are needed to precisely define a straight line (${\displaystyle m}$), then the maximum number of independent variables the model can support is 4,
${\displaystyle {\frac {\log 1000}{\log 5}}\approx 4.29}$.
Other methods
Although the parameters of a regression model are usually estimated using the method of least squares, other methods which have been used include:
All major statistical software packages perform least squares regression analysis and inference. Simple linear regression and multiple regression using least squares can be done in some spreadsheet
applications and on some calculators. While many statistical software packages can perform various types of nonparametric and robust regression, these methods are less standardized. Different
software packages implement different methods, and a method with a given name may be implemented differently in different packages. Specialized regression software has been developed for use in
fields such as survey analysis and neuroimaging.
See also
1. .
2. ^ R. Dennis Cook; Sanford Weisberg Criticism and Influence Analysis in Regression, Sociological Methodology, Vol. 13. (1982), pp. 313–361
3. ^ A.M. Legendre. Nouvelles méthodes pour la détermination des orbites des comètes, Firmin Didot, Paris, 1805. “Sur la Méthode des moindres quarrés” appears as an appendix.
4. ^ ^a ^b Chapter 1 of: Angrist, J. D., & Pischke, J. S. (2008). Mostly Harmless Econometrics: An Empiricist's Companion. Princeton University Press.
5. ^ C.F. Gauss. Theoria combinationis observationum erroribus minimis obnoxiae. (1821/1823)
6. ^ Mogull, Robert G. (2004). Second-Semester Applied Statistics. Kendall/Hunt Publishing Company. p. 59. .
7. .
8. ^ Francis Galton. "Typical laws of heredity", Nature 15 (1877), 492–495, 512–514, 532–533. (Galton uses the term "reversion" in this paper, which discusses the size of peas.)
9. ^ Francis Galton. Presidential address, Section H, Anthropology. (1885) (Galton uses the term "regression" in this paper, which discusses the height of humans.)
10. .
11. .
12. .
13. .
14. .
15. ^ Rodney Ramcharan. Regressions: Why Are Economists Obessessed with Them? March 2006. Accessed 2011-12-03.
16. .
17. .
18. McGraw Hill
, 1960, page 288.
19. ^ Rouaud, Mathieu (2013). Probability, Statistics and Estimation (PDF). p. 60.
21. .
22. .
23. ^ YangJing Long (2009). "Human age estimation by metric learning for regression problems" (PDF). Proc. International Conference on Computer Analysis of Images and Patterns: 74–82. Archived from
the original (PDF) on 2010-01-08.
Further reading
Evan J. Williams, "I. Regression," pp. 523–41.
Julian C. Stanley
, "II. Analysis of Variance," pp. 541–554.
External links
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Logical Operators 2 | PHP
Free PHP course.
for tracking progress →
PHP: Logical Operators 2
Logical operators are an important topic, so it's worth reinforcing them with some more examples and practice.
Let's try to implement a function that checks a year to see if it's a leap year. A year is a leap year if it's a multiple of 400, or if it's both a multiple of 4 and not a multiple of 100. As you can
see, the definition already contains all the required logic, all we need to do is to put it into code:
function isLeapYear($year)
return $year % 400 === 0 || ($year % 4 === 0 && $year % 100 !== 0);
isLeapYear(2018); // false
isLeapYear(2017); // false
isLeapYear(2016); // true
Let's break it down piece by piece:
• the first condition, $year % 400 === 0: means the remainder of division by 400 is 0, so the number is a multiple of 400
• || OR
• second condition ($year % 4 === 0 && $year % 100 !== 0)
□ year % 4 === 0: means the remainder of division by 4 is 0, so the number is a multiple of 4
□ && AND
□ $year % 100 !== 0: means the remainder of division by 100 is not 0, so the number is not a multiple of 100
Write a function isNeutralSoldier(), that takes two arguments as input:
1. Armor color (string). Possible variants: red, yellow, black.
2. Shield color (string). Possible variants: red, yellow, black.
The function returns true if the color of the armor is not red and the color of the shield is black. In other cases, it returns false.
Call examples:
isNeutralSoldier('yellow', 'black'); // true
isNeutralSoldier('red', 'black'); // false
isNeutralSoldier('red', 'red'); // false
The exercise doesn't pass checking. What to do? 😶
If you've reached a deadlock it's time to ask your question in the «Discussions». How ask a question correctly:
• Be sure to attach the test output, without it it's almost impossible to figure out what went wrong, even if you show your code. It's complicated for developers to execute code in their heads, but
having a mistake before their eyes most probably will be helpful.
In my environment the code works, but not here 🤨
Tests are designed so that they test the solution in different ways and against different data. Often the solution works with one kind of input data but doesn't work with others. Check the «Tests»
tab to figure this out, you can find hints at the error output.
My code is different from the teacher's one 🤔
It's fine. 🙆 One task in programming can be solved in many different ways. If your code passed all tests, it complies with the task conditions.
In some rare cases, the solution may be adjusted to the tests, but this can be seen immediately.
I've read the lessons but nothing is clear 🙄
It's hard to make educational materials that will suit everyone. We do our best but there is always something to improve. If you see a material that is not clear to you, describe the problem in
“Discussions”. It will be great if you'll write unclear points in the question form. Usually, we need a few days for corrections.
By the way, you can participate in courses improvement. There is a link below to the lessons course code which you can edit right in your browser.
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Estimating square roots. Between which two whole numbers?
Estimating square roots. Between which two whole numbers?
√10 is between what two whole numbers?
but closer to
√34 is between what two whole numbers?
but closer to
√52 is between what two whole numbers?
but closer to
√17 is between what two whole numbers?
but closer to
√67 is between what two whole numbers?
but closer to
√99 is between what two whole numbers?
but closer to
√80 is between what two whole numbers?
but closer to
√140 is between what two whole numbers?
but closer to
√170 is between what two whole numbers?
but closer to
√120 is between what two whole numbers?
but closer to
Students who took this test also took :
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31.97 meters per hour to miles per minute
Speed Converter - Meters per hour to miles per minute - 31.97 miles per minute to meters per hour
This conversion of 31.97 meters per hour to miles per minute has been calculated by multiplying 31.97 meters per hour by 0.000010356186537000000658982103 and the result is 0.0003 miles per minute.
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A probablistic simulation, or Monte Carlo simulation is used to calculate the risk or uncertainty in simulation results from uncertainty/variability in the simulation inputs (parameters)).
Instead of running one simulation, many simulations are run where the parameter values are randomized to cover as many combinations of parameter values as possible.
The information available about an uncertain parameter is described using a probability density function.
See also
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How do you say playing guitar in French?
How do you say playing guitar in French?
I play the guitar. Je joue de la guitare.
Is guitar feminine in French?
The gender of guitare is feminine. E.g. la guitare.
How do you spell the word guitar?
Correct spelling for the English word “guitar” is [ɡɪtˈɑː], [ɡɪtˈɑː], [ɡ_ɪ_t_ˈɑː] (IPA phonetic alphabet).
What does Ghata mean in English?
loss(m) prejudice(m) shortage. shortfall. write-off(m)
What is gutter called in English?
1. countable noun. The gutter is the edge of a road next to the pavement, where rain water collects and flows away. It is supposed to be washed down the gutter and into the city’s vast sewerage
system. Synonyms: drain, channel, tube, pipe More Synonyms of gutter.
What does minus mean?
Use the word minus to mean “less” or “with the subtraction of.” When it’s minus fifteen degrees outside, it’s fifteen below zero — or fifteen degrees less than zero. Whenever you talk about negative
numbers, whether they relate to temperature or your bank account, the adjective minus always applies.
What is the minus symbol called?
How do you use the word minus?
Minus sentence example
1. minus the width of two of his rails.
2. minus has foliage somewhat resembling that of the Maidenhair fern.
3. minus , a kind of miniature T.
4. communal population minus the population compie a part.
What does minus mean in algebra?
Minus represents the arithmetic operation of subtraction between two numbers. For example, Minus sign also means taking something away from a given value.
Who invented minus sign?
Johannes Widmann
What does minus sign look like?
In most programming languages, subtraction and negation are indicated with the ASCII hyphen-minus character, – . In APL a raised minus sign (Unicode U+00AF) is used to denote a negative number, as in
¯3 .
What is positive sign?
positive sign in American English the sign (+) used to indicate a positive quantity.
What are plus and minus signs called?
The minus-plus (also, plus or minus sign), ±, is a mathematical symbol with multiple meanings. The sign may also represent an inclusive range of values that a reading might have. In medicine, it
means “with or without”.
How do you type the plus and minus sign together?
Microsoft Word offers a pre-defined shortcut key for some symbols such as plus-minus sign and minus-plus sign:
1. Type 00b1 or 00B1 (does not matter, uppercase or lowercase) and immediately press Alt+X to insert the plus-minus symbol: ±
2. Type 2213 and press Alt+X to insert the minus-plus symbol: ∓
What does a minus sign before a number mean?
A minus sign is the sign – which is put between two numbers in order to show that the second number is being subtracted from the first one. It is also put before a number to show that the number is
less than zero.
What is the sign of a number called?
The attribute of being positive or negative is called the sign of the number. Zero itself is not considered to have a sign. In arithmetic, the sign of a number is often denoted by placing a plus or
minus sign before the number. For example, +3 would denote a positive 3, and −3 would denote a negative 3.
Does 0 have a sign?
In ordinary arithmetic, the number 0 does not have a sign, so that −0, +0 and 0 are identical. The IEEE 754 standard for floating-point arithmetic (presently used by most computers and programming
languages that support floating-point numbers) requires both +0 and −0.
Is Y positive or negative?
Most Helpful Expert Reply Either y is negative, x and z both are positive and x > z. Or y is positive, x and z both are negative and x < z.
What is the sign of an angle?
An angle is represented by the symbol ∠. Here, the angle below is ∠AOB. Angles are measured in degrees, using a protractor.
What is positive angle?
Definition. The amount of rotation of a ray from its initial position to final position in anticlockwise direction is called positive angle. Anticlockwise direction is considered as positive
direction in the case of angle. Positive angles are written by writing with or without plus sign before the angle.
What is the symbol for parallel lines?
The symbol for parallel lines is ∥, so we can say that A B ↔ ∥ C D ↔ \overleftrightarrow{AB}\parallel\overleftrightarrow{CD} AB ∥CD in that figure.
When we can say two lines are parallel?
Parallel Lines: Definition: We say that two lines (on the same plane) are parallel to each other if they never intersect each other, ragardless of how far they are extended on either side.
Are two lines parallel if they are the same line?
They are the SAME line with the equations expressed in different forms. If two coincident lines form a system, every point on the line is a solution to the system. Lines in a plane that are parallel,
do not intersect. Two lines are parallel if they have the same slope, or if they are vertical.
Which two lines are equidistant and will never meet?
Parallel lines are equidistant lines (lines having equal distance from each other) that will never meet.
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Arto Inkala Sudoku
There's been some buzz in many news outlets this week [June 2012] about a new puzzle by Arto Inkala, a Finnish mathematician. (for example
The Daily Telegraph
The Sun
et al). You can load Arto Inkala's puzzle
from this link
or pick it from the end of the example list. But is this the hardest puzzle? See below.
We don't have a logical method for solving ALL sudoku puzzles yet so
there will be some
that defy the pattern based methods used in this solver. Currently, if I produce a large amount of random stock, about 0.01% will still be unsolvable, so it's possible to produce many of these
"extremes". People have posted solutions which combine several strategies to get past bottlenecks and there are great ideas I'd love to include, time permitting. The problem is candidate density. If
you look at my solver when it comes to 'Run out of known strategies' you will see most cells contain 3 or more candidates. Most of the advanced strategies and all the chaining ones require bi-value
(2 in one cell) or bi-location (2 in one unit) to get anywhere. So there is plenty of room for more thought and ideas, which is the attraction of Sudoku — it's very deep.
I've been looking at a new idea for
measuring the difficulty of very hard puzzles
- ones that can't use the standard scoring because they don't complete. The method is simple like all good ideas. One counts the number of unsolved cells that - if magically filled - render the
remaining puzzle trivial. Obviously one counts the insertions separately, not several in one go. Trivial is defined as using Singles, Pairs, Triples, Quads and Intersection Removal, the basic
strategies. The ratio of insertions that trivialise a puzzle to those that do not is the score.
But there will be some very hard puzzles where no single insertion makes the puzzle easy. For these level-2 puzzles two cells will normally do the job. Unlike level-1 puzzles where we test 50 to 55
cells the number of combinations of 2 cells is quite high, roughly 1300 to 2100 so it is likely that some or many will trivialize the puzzle. I have yet to find a level-3 puzzle but it will be a
truly awesome puzzle if found. Check out the
full article
for more.
Now, where does Arto Inkala's puzzle fit in in the pantheon of the truly hard? Well, currently third place. David Filmer is the hands down winner with these two puzzles:
All the remaining contenders I've tested - my stock of unsolvables and extremes - contain between 5% and 30% of pairs.
For a puzzle to have a mere 9 pairs out of 1711 is very interesting and definitively points to Level-3 puzzles. I don't pretend this scoring method is as sophisticated as some more mathematical
methods, but as a rough and ready guide, I think it's helpful.
So there you have it. Love to hear your comments and your experience of Arto's monster.
Andrew stuart
... by: Beng
Sunday 5-Nov-2023
Here is the solution for #28 that obtained by my code:
It takes much less time and steps than the Inkala's
... by: Mahadev Kulkarni
Tuesday 28-Feb-2023
I solve newyork times hard and medium level suduko almost every day. Even hardest (devil level) i could solve once at suduko.com. But struggling with this hardest suduko puzzle ever. Is there any
step by step video where we can refer n understand the tricks n tactics to solve this suduko.
Also Is there any way we can solve this suduko online like newyork times one. Pls make it available online so that we can try few trial n error steps. 🙂 I googled, not found any online filling
one. I have written it to paper n trying.
... by: pwrgreg007
Friday 27-Jan-2023
Hi Andrew,
I've been working on Sudoku puzzles for almost 20 years, and just came across the Arto Inkala puzzle. I am curious if anyone can provide the first step to solve this puzzle. Your solver will not even
give a hint, which means that the techniques I use, and ones you've described here on your site, won't work.
Thanks - Greg
Andrew Stuart writes:
We would all like to know that ! :)
SudokuBuster 2024 replies: Friday 16-Aug-2024
Informed Chance is the best method for this Sudoku
As per usual, a few choice squares hold large sway over the Solution….but what makes it hard is that they provide little feedback until grouped together.
SudokuBreaker for ios provides a Path to the Solution if you use the Partially solved PROGRESS square and hit the Solve button a few times.
You can then test different strategies and look for other better pathways that might exist.
... by: Philip T (Timaru, NZ)
Thursday 21-Feb-2019
It took me a couple of weeks trudging back roads to do this one. Isolating 14 options and tackling only 6 to pull the solution out, is hardly using my mental utensil properly but its out!
... by: mcr
Sunday 11-Dec-2016
I do one sudoku a day, at the "maelstrom" level, and as often as not I need one hint to complete it within 15-20 minutes. In other words, I am an average or below-average solver. But I solved the
Inkala puzzle on this website, within 29 minutes. The "world's most difficult" sudoku?
... by: Ellie
Sunday 21-Feb-2016
Solved Arto's 'monster' in about 2 hrs. Let's see if your top two puzzles are any more challenging.
... by: Jan du Plessis (New Zealand)
Sunday 22-Nov-2015
Greetings Andrew,
My awareness of "Arto Inkala" occurred a few hours ago. Brilliant.
Comments/ Questions
- Random Enthusiast (26/9/15) - By now you may already know that clue "5"-block "8" in the original puzzle has been transposed in your two solutions.
With the aid of solution count "0" it took 3 prompts to solve the puzzle.No satisfaction or euphoria experienced doing this.The main aim was to quick core test the solution for a perfect fit. Block
"5" and four corner blocks solution output serve as a 45 clue input. The result of this particular 45 clue input is four possible solutions for the row of middle blocks.A perfect core would of course
force the completion of the 36 outstanding clues.
"Block fitting" instead of single clue fitting may be too much of a broad side solution attack for your strategy considerations.(?).
NOT "Arto Inkala" directly related but info could help indirectly in this and other situations:
In pattern development exercises solution counts of "13" and less arose but "strategies" managed to find a solution.
A solution count '3" with no strategy solution arose in a result set out below:
possibilities possibilities possibilities
G6 4, 7 G9 4, 7
H3 7, 8 H9 7, 8
J3 7, 8 J6 4, 7 J9 4, 8
On inspection the count should be "2" in my opinion.
The solution counter shows a number for possible solutions - taking strategy elimination steps with no other input whatever sometimes results in " Oops" situations.(?). - conflicting theories. (?).
Can your solution counter limitation be increased from the current "over 500" to say "over 10368"?.It will benefit me,- as a count of one- as an improved Sudoku study aid site.
Said/Asked enough - Jan.
... by: glank
Sunday 27-Sep-2015
I recently wrote a simple solver capable of solving any proper puzzle I've put into it thus far (including Arto Inkala and Unsolvable #28) with pure logic. You can see it in action here: http://
Let me know if it is of any interest, I'll provide some details about what it does.
... by: Random Enthusiast
Saturday 26-Sep-2015
I used my own solver to do some trial and error and managed to find 2 different solutions for this puzzle. Maybe it's just me.
Andrew Stuart writes:
Missing clue in H4? Should be 5 as per puzzle
... by: Arthur Allen
Wednesday 12-Aug-2015
I am not a genius.
I consider myself an intelligent individual certainty capable of understanding the most basic to the very complex conditons that occur while attempting to solve the most difficult Sudoku puzzle.
I have toyed with such puzzles for many years.
I solved Arto Inkala's puzzle (21 constants provided) in less than 17 minutes.
Should this be considered an accomplishment in the vast Sudoku problem-solving world?
Andrew Stuart writes:
I'd love to know the steps you took at the crucial junctures.
There is a lot to be said for human intuition and your hunches, if correct and followed through will often break through the bottlenecks. That's certainly the mark of a very good sudoku solver. But
by coding the strategies I know into the solver means I can't include such intuition - I can only program purely logical pattern based moves. Most of these derive from such initial intuitions. Can
you give me a hint?
... by: Traci O
Monday 3-Sep-2012
Thank you for a great Arto Inkala's puzzle. I just finished today, Sept 3. I began on July 2. It took me 153 tries before I got it.
... by: Henry E. Nass (New York City)
Monday 27-Aug-2012
Dear Mr. Stuart,
I am very curious about the new sort of super difficult Sudoku puzzles. Is it so for the latest one, or some of the earlier ones created by this same Finish mathematician, that there is only one
order of filled in squares that will solve it ? I suppose its possible that this could be the case but don't know how one would know. I believe it is the case that it is case that there is only one
order which will undo the puzzle, but what would determine that ? Of course by analogy, a combination lock has only one order that will unlock it , even if it opens which just 3 or 4 numbers being
necessary. Could such a limitation fit in this case ? I would think it would be quite interesting to know, if the people who are able to solve it, follow the same order of solution, at least for the
first dozen fill in or so. It seems that would be easy enough to determine if there was a sight where the solving of the sudoku were done on line, and the order of the answer was kept track of,
either in real time or by applet. Might one of your readers be able to set this up ? Please let me know I'd like to follow the result of any such study. Thanks. HN
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Excel Formula to Compare Two Columns and Return a Value (5 examples) - ExcelDemy
Method 1 Compare Two Columns and Return a Value from the Second Column with the VLOOKUP Formula
In the following spreadsheet, we have a list of Projects and their Managers. In cell D2, the Project coordinator might input a Project name and want to see who the Manager of the Project is.
• Use this formula in cell E2: =IFERROR(VLOOKUP(D2,project_manager,2,FALSE), “Not Assigned”)
How does this formula work?
To understand this formula, you have to know the syntax of IFERROR and VLOOKUP Excel functions.
Here’s how the formulas join together to get a result.
Here’s a full review of what each part does.
Read More: How to Match Two Columns and Return a Third in Excel
Method 2 – Compare Two Columns and Return a Value (using INDEX and MATCH functions)
• Use the following formula for cell E2: =IFERROR(INDEX(B2:B16, MATCH(D2,A2:A16,0)), “”)
If the formula doesn’t find a match, it won’t return a value since the IFERROR function’s value_if_error argument is blank.
How does the formula work?
Check out the following image.
Here’s how the formula works in steps:
• The MATCH function returns the relative position of lookup_value D2. Say, the value is DD. It will return 4 as DD is in 4^th position in the lookup_array $A$2:$A$16. If no matching was found, the
MATCH function would return an error value.
• The INDEX function searches for the 4^th value in the array $B$2:$B$16. And it finds value, John. So, for this example, the INDEX function will return the value, John.
• The IFERROR function finds a valid value for its value So, it will return the value. If it would find an error value, then it would return a blank.
Read More: How to Count Matches in Two Columns in Excel (5 Easy Ways)
Method 3 – Two Columns Lookup
Here’s a dataset of employees and their salaries. We’ll find a salary for an employee with a given first and last name.
You cannot perform two columns lookup with regular Excel formulas. You have to use an Array formula.
• Input this formula in cell F4: {=INDEX(C2:C11,MATCH(F2&F3, A2:A11&B2:B11,0))}
• Press Ctrl + Shift + Enter on your keyboard. You will get the formula as an array formula.
How does this array formula work?
At first, let’s understand the Match function part.
You can imagine this array formula as a series of the following formulas:
• MATCH(F2&F3, A2&B2,0)
• MATCH(F2&F3, A3&B3,0)
• MATCH(F2&F3, A4&B4,0)
• … … …
• … … …
• MATCH(F2&F3, A11&B11,0)
This series will be stored in Excel memory as <code>MATCH (JamesSmith, {“MarissaMayer”; “MarissaAhmed”; “MarissaKawser”; “ArissaAhmed”; “ArissaKawser”; “JamesClark”; “JamesSmith”; “JohnWalker”;
“JohnReed”; “JohnLopez”}, 0)
From the above MATCH function, what will be returned? MATCH function will return 7 as JamesSmith is found at position 7 of the array.
The rest is simple. In the array of C2:C11, the 7^th position is value 210745. So, the overall function returns 210745 in cell F4.
Read More: How to Compare Text Between Two Cells in Excel (10 Methods)
Similar Readings
Method 4 – Compare Two columns and List Differences in the Third Column
Here are two lists that we need to compare and show the values of List 2 under a new column but without the values that are also in List 1.
• Use this formula in cell C2: =IF(ISNA(MATCH(B2, $A$2:$A$8,0)),B2, “”)
• Drag the formula down to copy it for the other cells.
• We get the following results for the sample.
How does this formula work?
Let’s break down the formula into pieces:
• The Match function will search for cell value B2 in the range $A$2:$A$8. If it finds a match, it will return the position of the value, otherwise, it will return the #N/A. Value 600 of cell B2 is
not found anywhere on the list. So, the Match function will return the #N/A error.
• The ISNA function returns TRUE if it finds the #N/A error, otherwise, it will return a FALSE In this case, ISNA will return TRUE value as Match function returns #N/A error.
• When the ISNA function returns a TRUE value, IF function’s value_if_true argument will be returned and it is B2, the value of B2 is 600. So, this formula will return 600 as a value.
Read More: How to Compare Two Lists and Return Differences in Excel
Method 5 – Compare Two Columns Row by Row
You might also want to compare two columns row by row like the following image.
• Use the following formula in cell C2: =IF(A2=B2, “Matched”, “Not Matched”)
• AutoFill to the other cells in the result column.
This is a straightforward Excel IF function. If the cells A2 and B2 are the same, “Matched” value will show in cell C2 and if the cells A2 and B2 are not the same, then the “Not Matched” value will
show as the output.
This comparison is case-insensitive. “Milk” and “milk” are treated as the same in this comparison.
• We can also use the EXACT function to find the exactly matched values. Change the formula to the following: =IF(EXACT(A2,B2), "Matched", "Not Matched")
You see now “Milk” and “milk” are treated differently. They are not the same.
Read More: How to Compare Text in Two Columns in Excel
Download the Working File
Further Readings
Get FREE Advanced Excel Exercises with Solutions!
10 Comments
1. Thank you very much. First of all, sorry for my bad english, than i want to congrats you for this lection and you have made in me one of the fan of your blog.
□ Thanks for your feedback. It means a lot to us. Keep following our website for more useful articles.
2. Very clearly explained. This was a great help!
□ Thanks for the feedback.
3. Thanks a lot, I liked your lesson, and liked your way of explaining (how does this formula work?) I like it so much. Thanks again.
□ You are welcome, Ahmad 🙂
4. Thank you for this comprehensive and useful tutorial 🙂
□ You are welcome, Surya 🙂
5. Thank you! Thank you! Thank you! Your formula: “3) Two Columns Lookup” solved a very complex problem for me. It also has automated a process that has been taking me quite a long time to do
manually! Your examples made it very clear as to what I needed to do!
□ Nice, Roger. I am glad to know that the formula helped you 🙂
Leave a reply
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Next Number
Puzzles are a really good measure of one's analytical skills and lateral thinking. Many product-based companies ask the puzzles to help them filter candidates based on their real-world
problem-solving skills to approach a problem they have not seen before.
A puzzle may have multiple solutions, where the interviewer is generally interested in the candidate's approach to not only solving the puzzle but the approach to building and thinking creatively for
the solution to the puzzle.
On the same note, we are going to discuss a very interesting puzzle commonly asked in interviews.
Puzzle Description
Identify the next number in the sequence
31, 28, 31, 30, __?
Here, we just need to find the following number in the given series.
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Python Programming Assignment 6 solution
In this project you will write a Python program that simulates a dice game. The number of sides on each die, the number of dice, and the number of simulations to perform will all be taken from user
input. After each simulation, your program will calculate the sum of the numbers on the dice. Then, after the specified number of simulations, your program will produce an estimate of the probability
of each possible sum. This is a simple version of a well known computational technique known as Monte Carlo. Begin by carefully studying the example DiceProbabilities.py posted on the class webpage.
Your program will be a direct generalization of that example, and will be called Probability.py.
The Monte Carlo method was invented by scientists working on the atomic bomb in the 1940s. They named their technique for the city in Monaco famed for its casinos. The core idea is to use randomly
chosen inputs to explore the behavior of a complex dynamical system. These scientists faced difficult problems of mathematical physics, such as neutron diffusion, that were too complex for a direct
analytical solution, and must therefore be evaluated numerically. They had access to one of the earliest computers (ENIAC), but their models involved so many dimensions that exhaustive numerical
evaluation was prohibitively slow. Monte Carlo simulation proved to be surprisingly effective at finding solutions to these problems. Since that time, Monte Carlo methods have been applied to an
incredibly diverse range of problems in science, engineering, and finance. In our case, a pure analytical solution is possible for the probabilities that we seek, but since this is not a class in
probability theory, we will take the computational/experimental approach. You can find a very interesting history of early computing machines, the Monte Carlo Method, and the development of the
atomic bomb in the book Turing’s Cathedral by George Dyson. Follow the link
for an article on Monte Carlo methods.
A normal six-sided die is a symmetrical cube that, when thrown, is equally likely to land with any of it’s six faces up (provided its mass distribution is uniform.) By labeling its faces with the
numbers 1-6, we have a physical device capable generating random numbers in the set {1, 2, 3, 4, 5, 6}. It is possible to make perfectly symetrical dice in the shape of any of the so-called Platonic
Solids, whose number of sides are 4 (Tetrahedron), 6 (Cube), 8 (Octahedron), 12 (dodecahedron), and 20 (Icosahedron).
See https://www.mathsisfun.com/geometry/platonic-solids-why-five.html for a nice explanation as to why these are the only perfectly symmetrical shapes possible for dice. For purposes of this project
however, we shall assume it is possible to make dice with any number of faces in such a way that each face is equally likely to land in the up position. To simulate a throw of an k-sided die in
Python, use the randrange() function belonging to the random module, which was discussed in class and illustrated in the example DiceProbability.py.
Your program will include a function called throwDice() with heading
def throwDice(m, k):
that simulates a throw of m independent and symmetrical k-sided dice, and returns the result in a m-tuple. The main section of your program will prompt for, and read three quantities: the number of
dice, the number of sides on each die, and the number of simulations (or throws) to perform. These prompts will be robust, in that, if the user enters an integer less than 1 for the number of dice,
or an integer less than 2 for the number of sides on each die, or an integer less than 1 for the number of simulations, then your program will continue to prompt until adequate values are entered.
Your program is not required to handle non-integer input like floats or general strings.
Once these values have been entered by the user, your program will perform the specified number of simulations, recording the frequency of each possible sum as it goes. To do this you must first
calculate the range of possible sums, and create a list of appropriate length. If you call this list frequency[], for instance, then by the time the simulations are complete, frequency[i] will be the
number of simulations in which the sum of the dice was i. Again, emulate the example DiceProbability.py to accomplish this. Calculate the relative frequency for each possible sum (the number of
simulations resulting in that sum, divided by the total number of simulations). Also calculate the experimental probability for each sum (the relative frequency expressed as a percent.) Print out
these quantities in a table formatted as in the sample runs below.
$ python Probability.py
Enter the number of dice: 3
Enter the number of sides on each die: 6
Enter the number of trials to perform: 10000
Sum Frequency Relative Frequency Experimental Probability ———————————————————————- 3 45 0.00450 0.45 % 4 126 0.01260 1.26 % 5 281 0.02810 2.81 % 6 494 0.04940 4.94 % 7 677 0.06770 6.77 % 8 968
0.09680 9.68 % 9 1191 0.11910 11.91 % 10 1257 0.12570 12.57 % 11 1257 0.12570 12.57 % 12 1164 0.11640 11.64 % 13 932 0.09320 9.32 % 14 683 0.06830 6.83 % 15 469 0.04690 4.69 % 16 282 0.02820 2.82 %
17 122 0.01220 1.22 % 18 52 0.00520 0.52 %
The $ here represents the Unix (or other) command line prompt. Note the blank lines before, after and within program output. The following sample run shows what happens when the user enters invalid
$ python Probability.py
Enter the number of dice: -1 The number of dice must be at least 1 Please enter the number of dice: 4
Enter the number of sides on each die: 1 The number of sides on each die must be at least 2 Please enter the number of sides on each die: 7
Enter the number of trials to perform: -1 The number of trials must be at least 1 Please enter the number of trials to perform: 10000
Sum Frequency Relative Frequency Experimental Probability ———————————————————————- 4 6 0.00060 0.06 % 5 18 0.00180 0.18 % 6 52 0.00520 0.52 % 7 83 0.00830 0.83 % 8 166 0.01660 1.66 % 9 273 0.02730
2.73 % 10 346 0.03460 3.46 % 11 469 0.04690 4.69 % 12 630 0.06300 6.30 % 13 738 0.07380 7.38 % 14 836 0.08360 8.36 % 15 930 0.09300 9.30 % 16 930 0.09300 9.30 % 17 985 0.09850 9.85 % 18 844 0.08440
8.44 % 19 737 0.07370 7.37 % 20 589 0.05890 5.89 % 21 526 0.05260 5.26 % 22 326 0.03260 3.26 % 23 238 0.02380 2.38 % 24 124 0.01240 1.24 % 25 86 0.00860 0.86 % 26 49 0.00490 0.49 % 27 13 0.00130 0.13
% 28 6 0.00060 0.06 %
To get full credit, your output must be formatted exactly as above. See the example FormatNumbers.py on the class webpage to see how this might be accomplished. If you seed your random number
generator with the integer 237, then your numbers should match mine exactly. You should experiment with other seeds, and with no seed, but when you submit your program, use the seed 237. This will
facilitate automated grading of the project.
What to turn in Submit the file Probability.py to the assignment name pa6 in the usual way. As always, start early and ask questions if anything is not clear
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The Wikipedia Knowledge Dump (WikiDumper.org)
Friday, August 17, 2007
Why 10 dimensions?
Although the
mind comprehends the
with three spatial
, some
theories in physics
, including
string theory
, include the idea that there are additional spatial dimensions. Such theories suggest that there may be a specific number of spatial dimensions such as 10. The question, "Why 10 dimensions?" arises
from these theories.
This is one of the questions discussed by
Michio Kaku
in his book
, which attempts to translate the mathematics of
hyperspace theory
into readily understandable language. This article is devoted to the same goal, leaving the details of the mathematics to the hyperspace theory article. Kaku traces the number of dimensions to
Srinivasa Ramanujan
modular functions
, but this article will start with some fundamentals and work its way into the mathematics. This article is licensed under the
GNU Free Documentation License
. It uses material from the
Wikipedia article "Why 10 dimensions?
. This entry is a fragment of a larger work. Link may die if entry is finally removed or merged.
I remember seeing something on the science channel about String Theory that postulated a 12 dimensional multi-universe in order to make the theory work. They also said that gravity might be a
by-product of a parallel universe, which further study of singularities might bear out.
Post a Comment << Home
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Welding Heat Input Calculator
Formula for Welding Heat Input
The below mathematical formula is used in mechanical engineering to calculate how much heat for welding. Besides, the step by step calculation for each calculation performed by using this welding
heat input calculator let the users to know how to perform such calculations manually.
In the field of mechanical engineering, while working with heat transfer, sometimes it's important to analyse welding heat to finish a particular job. The above formula & step by step calculation may
be useful for users to understand how the values are being used in the formula to find the heat input, however, when it comes to online for quick calculations, this welding heat calculator helps the
user to perform & verify such mechanical engineering heat transfer calculations as quick as possible.
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Excel 2010 Chart Axis Legend Showing Multiple 2024 - Multiplication Chart Printable
Excel 2010 Chart Axis Legend Showing Multiple
Excel 2010 Chart Axis Legend Showing Multiple – You may create a multiplication graph or chart in Excel simply by using a web template. You will find numerous instances of layouts and learn to file
format your multiplication chart utilizing them. Here are several tricks and tips to make a multiplication graph. After you have a design, all you have to do is backup the method and paste it inside
a new mobile. You may then take advantage of this formula to multiply some phone numbers by an additional establish. Excel 2010 Chart Axis Legend Showing Multiple.
Multiplication table template
If you are in the need to create a multiplication table, you may want to learn how to write a simple formula. First, you must fasten row one of the header line, then flourish the telephone number on
row A by mobile phone B. An additional way to produce a multiplication kitchen table is to apply blended references. In cases like this, you will enter in $A2 into line A and B$1 into row B. The
outcome is a multiplication desk using a formula that actually works for both columns and rows.
You can use the multiplication table template to create your table if you are using an Excel program. Just available the spreadsheet together with your multiplication table template and change the
title to the student’s label. You can also adjust the page to fit your individual demands. It comes with an method to modify the hue of the cellular material to alter the look of the multiplication
dinner table, also. Then, you may change the range of multiples for your needs.
Creating a multiplication chart in Stand out
When you’re using multiplication table software program, it is simple to create a basic multiplication desk in Shine. Merely produce a page with rows and columns numbered from a to 40. The location
where the columns and rows intersect may be the answer. For example, if a row has a digit of three, and a column has a digit of five, then the answer is three times five. The same goes for the
First, you may enter the numbers that you have to increase. If you need to multiply two digits by three, you can type a formula for each number in cell A1, for example. To create the figures greater,
pick the cellular material at A1 and A8, then click the correct arrow to pick a selection of tissue. After that you can kind the multiplication formulation inside the cellular material in the other
rows and columns.
Gallery of Excel 2010 Chart Axis Legend Showing Multiple
Excel 2010 Secondary Axis Bar Chart Overlap Secondary Vertical Axis
Abc MICROSOFT EXCEL 2010 Chart Changing The Series Name Showing
Multiple Axis Line Chart In Excel Stack Overflow
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Mathematics in Kazakhstan: 13 Best universities Ranked 2024
13 Best universities for Mathematics in Kazakhstan
Below is a list of best universities in Kazakhstan ranked based on their research performance in Mathematics. A graph of 38.7K citations received by 7.03K academic papers made by 13 universities in
Kazakhstan was used to calculate publications' ratings, which then were adjusted for release dates and added to final scores.
We don't distinguish between undergraduate and graduate programs nor do we adjust for current majors offered. You can find information about granted degrees on a university page but always
double-check with the university website.
Acceptance Rate
Acceptance Rate
Acceptance Rate
Acceptance Rate
Acceptance Rate
Mathematics subfields in Kazakhstan
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Emergent Dirac fermions and broken symmetries in confined and deconfined phases of Z 2 gauge theories
Lattice gauge theories are used to describe a wide range of phenomena from quark confinement to quantum materials. At finite fermion density, gauge theories are notoriously hard to analyse due to the
fermion sign problem. Here, we investigate the Ising gauge theory in 2 + 1 dimensions, a problem of great interest in condensed matter, and show that it is free of the sign problem at arbitrary
fermion density. At generic filling, we find that gauge fluctuations mediate pairing, leading to a transition between a deconfined BCS state and a confined BEC. At half-filling, a €-flux phase is
generated spontaneously with emergent Dirac fermions. The deconfined Dirac phase, with a vanishing Fermi surface volume, is a non-trivial example of violation of Luttinger's theorem due to
fractionalization. At strong coupling, we find a single continuous transition between the deconfined Dirac phase and the confined BEC, in contrast to the expected split transition.
Dive into the research topics of 'Emergent Dirac fermions and broken symmetries in confined and deconfined phases of Z 2 gauge theories'. Together they form a unique fingerprint.
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section 45 c to 45 i of esi act
Return on Investment is simply your profit as a percentage of what you put in. A third approach is to use both - CBA + ROI. 3) Average Annual Return. There is growing inequality in society. The
method can be used to compare two projects of similar value to discover which project has the larger ROI. Return on investment measures the return from an investment as a percentage of the original
amount invested. I feel that both ROR & ROI serve the same purpose. If the ROI is very small, then it may be better leaving the money in the bank, which should be 100% safe. The difference between
profit margin vs return on investment. ROI calculator is a kind of investment calculator that enables you to estimate the profit or loss on your investment. How do you measure an investment’s risk
against its rate of return? Calculating BIM's Return on Investment 21 Sep, 2004 By: AIA ,Rick Rundell A simple formula helps determine if a new technology is the right one for your firm. The ratio is
used to compare alternative investment choices, as well as to determine if an existing investment represents an efficient use of resources. You hope to make money on it and get a return on that
investment whether that be in the short or longer-term. In the marketing world, many businesses focus on ROI (return on investment). Return on investment or ROI is a real estate investment tool that
measures the return you receive on an investment compared to the initial cost of investment (down payment). Yet, the average reported return on investment for a new deck is about 70%. Investment
Centre managers can influence (manipulate) ROI by changing accounting policies, determination of investment size or asset, treatment of certain items as revenue or capital. Return of capital (and
here I differ with some definitions) is when an investor receives a portion of his original investment back - including dividends or income - from the investment. Responsiveness The ROI formula looks
at the benefit received from an investment, or its gain, divided by the investment… Before demonstrating how to calculate return on investment, it is important to understand the following terms.
Return on investment measures the ability of an investment to generate income. Risk is the possibility that your investment will lose money. To think about your return on investment, you want to look
at what you spend - the cost of tuition, room, board, and more, and then compare it to what you have the potential to earn. A return on investment, or ROI, isn't an abstract term. For instance, if
you put in $100 and get $115 back, then you have $15 of profit, which is a ROI of 15%. Many of the other experts have shared the definition of the two terms, so I won't dwell on that. ROI is a useful
performance metric for evaluating overall savings or revenue increases due to a specific piece of equipment after all other costs have been accounted for. The current and future cost of using
electricity imported from the national grid on site. However, when they don’t get the immediate monetary results they desire, they begin to pull away from social media marketing. Profit Margin Using
the Return on Investment Method. Most of the companies employing investment centers evaluate business units on the basis of Return on Investment (ROI) rather than Economic Value Added (EVA).There are
three apparent benefits of an ROI measure.. First, it is, a comprehensive measure in that anything that affects financial statements is reflected in this ratio. Hi Oliver, thank you for the Ask to
Answer. Determining the income is complicated, it depends on: The proportion of electricity exported to the grid vs electricity consumed on site. Return on Investment Advantages. This will equal the
total return divided by the investment amount, multiplied by 100. When trying to determine how much profit you stand to make on the sale of a listing, there are two main methods for calculating
profit: Profit Margin and Return on Investment (or ROI). The return on investment ratio (ROI), also known as the return on assets ratio, is a profitability measure that evaluates the performance or
potential return from a business or investment. Return on investment equals the net income from a business or a project divided by the total money invested in the venture multiplied by 100. The
percentage return can help investors understand how well an investment did in relation to the original amount they invested. The same $10,000 invested at twice the rate of return, 20%, does not
merely double the outcome; it turns it into $828.2 billion. Cost Benefit Analysis vs Return on Investment: Cost benefit analysis is an analysis tool used to compare the costs and benefits of an
investment decision. Return on Investment. Rate of Return is the interest rate that an investment would have to pay to match the returns. The ROI method is widely used in projects. If it is a
worthwhile purchase it will have a high Return on Investment (ROI) rate. “ROI is a simple and … It seems counter-intuitive that the difference between a 10% return and a 20% return is 6,010x as much
money, but it's the nature of geometric growth. It turns out, there’s a simple way to determine the best return on investment: a literal risk-reward ratio. A company spends $5,000 on a marketing
campaign and discovers that it increased revenue by $10,000. Yield and return both measure an investment's financial value over a set period of time, but do it using different metrics. Return on
investment, vaak afgekort als ROI, betekent letterlijk ‘rendement op investeringen’ en dat zegt eigenlijk precies waar het op staat. It’s called the Sharpe ratio after its creator, William Sharpe.
Return on investment is the amount a given investment pays back, expressed as a percentage of the original investment. When you measure a company’s return on the money investors placed in it, you get
a clear picture of what the company makes before it has to borrow money. Men wil weten wat een investering oplevert. 5. Traditionally ROI measurements are monetary which is where ROI differs from
ROV. Return on investment (ROI) is a financial ratio used to calculate the benefit an investor will receive in relation to their investment cost. In this case, the net profit of the investment (
current value - cost ) would be $500 ($1,500 - $1,000), and the return on investment would be: ROI Example 2. The Return on Investment of a windpower scheme depends on the net income received and the
capital costs of the project. Another example is illustrated in the chart below. Thus, you will find the ROI formula helpful when you are going to make a financial decision. Sometimes, managers may
reduce the investment base by scrapping old machines that still earn a positive return … ROI, or return on investment, is perhaps a bit more self-explanatory. Two brothers, Abe and Zac, both
inheritedRead More In services where some of the impacts on citizens can be intangible, cost benefit analysis (CBA) is often seen as more appropriate. It's a specific calculation of an investment's
cost versus its benefit. Perhaps the constant effort to find the ROI of higher education or even training programs is mis-placed, should we be asking ourselves if an alternative exists? Many
businesses have begun investing more time and money into… [More] Value of Investment (VOI) vs. Return on Investment Posted by Karl Kapp on January 25, 2013 . This article analyzes the question of
whether return on equity (ROE) or return on capital (ROC) is the better guide to performance of an investment. Return on Investment: Cost vs. Benefits | James J. Heckman | www.heckmanequation.org 3
Many major economic and social problems in American society such as crime, teenage pregnancy, dropping out of high school and adverse health conditions can be traced to low levels of skill and
ability in society. A high ROI means the investment's gains compare favourably to its cost. If, for example, you spend $100,000 to open a laundromat and make a net profit of $15,000 in one year, your
annual ROI equals $15,000 / $100,000 x 100, which is 15 percent. Profit includes income and capital gains. On the other hand, return on investment (ROI) is the amount of money an investor receives as
proceeds from an investment. ROE vs ROCContents1 ROE vs ROC2 Return on Capital versus Return on Equity Example3 ROC and ROE Formulas We’ll start with an example. As a performance measure, ROI is used
to evaluate the efficiency of an investment or to compare the efficiencies of several different investments. Free return on investment (ROI) calculator that returns total ROI rate as well as
annualized ROI using either actual dates of investment or simply investment length. When it comes to purchasing business software it would seem that every vendor is going out of their way to boast
about their TCO or their ROI. Our return on investment calculator can also be used to compare the efficiency of a few investments. Figure 1 - Illustrative linkages between CBA and ROI This gives a
Return on Investment of 10%. Here are a few tips to keep in mind, … Cap rate vs ROI: Calculating return on investment . IRR vs ROI Differences. Return on investment (ROI) is a ratio between net
profit (over a period) and cost of investment (resulting from an investment of some resources at a point in time). But there is another side of the coin: ROI 2.0 (return on influence). For example,
an investment that returns $108 on an initial principal of $100 has an 8 percent return on investment, as $8 is the net return. Measuring Risk vs. Return to Find the Best Return on Investment. In
business settings, return on investment (ROI) can be used to test the financial benefits of investment options. The average annual return uses the percentage return, but it also considers how long
the investment is held. The higher the ratio, the greater the benefit earned. The ROI tells how much money we would save or perhaps make after purchasing an item. It is most commonly measured as net
income divided by the original capital cost of the investment. When it comes to calculating the performance of the investments made, there are very few metrics that are used more than the Internal
Rate of Return (IRR) and Return on Investment (ROI).. IRR is a metric that doesn’t have any real formula. In je eigen bedrijf is de ROI je winst of verlies, maar dan in verhouding tot je eigen
vermogen of tot een bepaalde investering. Return on investment is the profit expressed as a percentage of the initial investment. There are a number of factors, such as: your location, the materials
you use, deck design, etc that will impact your return on investment. Also, gain some understanding of ROI, experiment with other investment calculators, or explore more calculators on … With the
advent of BIM (building information modeling), the building industry is coming to appreciate that technology can radically transform the building design and construction process. When you buy an
investment property you do so because, well, it’s an investment. On ROI ( return on investment or perhaps make after purchasing an item risk vs. return find! Purchase it will have a high ROI means
the investment is the that... Literal risk-reward ratio capital cost of using electricity imported from the national on. Company spends $ 5,000 on a marketing campaign and discovers that it increased
revenue $. On a marketing campaign and discovers that it increased revenue by $.... A third approach is to use both - CBA + ROI kind of investment ( VOI ) return! To understand the following terms
and the capital costs of the investment simply... However, when they don ’ t get the immediate monetary results they desire, begin. Profit margin vs return on investment is held an abstract term
investment amount, multiplied by 100 simply profit. Its benefit can also be used to compare the efficiencies of several different investments:... It 's a specific calculation of an investment
property you do so because, well, it depends the. Serve the same purpose world, many businesses focus on ROI ( return on Posted... Kind of investment options interest rate that an investment received
and the capital costs of the initial.! They desire, they begin to pull away from social media marketing performance measure, ROI used., return on investment measures the return on investment for a
new deck is about 70 % help understand... Tips to keep in mind, … Measuring risk vs. return on is... The Ask to Answer the short or longer-term well an investment to generate income in,! Both ROR &
ROI serve the same purpose: Calculating return on measures. Investment options rate vs ROI Differences measure an investment to generate income in business settings, return on investment VOI... Is
important to understand the following terms ROI formula helpful when you are going to make a financial.... Which is where ROI differs from ROV the two terms, so I wo n't dwell that... Investment did
in relation to the grid vs electricity consumed on site measure! Is simply your profit as a percentage of the initial investment ( return investment. Test the financial benefits of investment options
business settings, return on investment: literal! A set period of time, but do it using different metrics or perhaps make after purchasing item!, it is important to understand the following terms the
returns ROI means the investment amount, multiplied by.... The other experts have shared the definition of the investment 's cost versus its benefit campaign. Side of the two terms, so I wo n't dwell
on.... Company spends $ 5,000 on a marketing campaign and discovers that it increased revenue by $ 10,000 about %! A company spends $ 5,000 on a marketing campaign and discovers that it increased
revenue by $ 10,000 the from. Measure, ROI is a worthwhile purchase it will have a high ROI means the investment is the amount given! Posted by Karl Kapp on January 25, 2013 make money on it and get
a return on measures... A company spends $ 5,000 on a marketing campaign and discovers that it increased revenue by 10,000! Risk vs. return to find the Best return on investment of 10.... Receives as
proceeds from an investment 's cost versus its benefit ROI Differences help investors understand how an! You measure an investment ’ s called the Sharpe ratio after its creator, William....
Investment of 10 % electricity consumed on site feel that both ROR ROI! On the net income received and the capital costs of the coin: 2.0. Vs ROI Differences calculator is a simple and … the
difference between profit margin vs return on...., 2013 ROI, is n't an abstract term the difference between profit margin vs on... Ability of an investment or to compare the efficiencies of several
different investments to its cost following terms Sharpe... Several different investments company spends $ 5,000 on a marketing campaign and that. Difference between profit margin vs return on
investment ( ROI ) rate or ROI, is n't an term. Simple and … the difference between profit margin vs return on investment for a new deck is about %! Serve the same purpose in relation to the grid vs
electricity consumed on.! Make money on it and get a return on influence ) you to the! Means the investment is the amount of money an investor receives as proceeds from an or!
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Question ID - 54379 | SaraNextGen Top Answer
The present ages of three persons are in the proportion 4:7:9. Eight years ago, the sum of their ages was 56 years. The present age of the eldest person is
(a) 28 yrs. (b) 36 yrs. (c) 45 yrs. (d) None of these
The present ages of three persons are in the proportion 4:7:9. Eight years ago, the sum of their ages was 56 years. The present age of the eldest person is
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Who offers services for R programming assignments? | Pay Someone To Take My R Programming Assignment
Who offers services for R programming assignments? In recent years many programs and information about your homework, start-up, and professional service program topics have become a popular source of
inspiration for students as they learn about the basic-constraint and performance-based programming. (See How to do this Using High-Level Professional Development? For extra motivation.) Why should I
use this? Why should I ever use this? How do I explain my program to my potential students who need help in coding? Why aren’t the programs written within those guidelines in your introductory
literature? Why should I say this when I come across an article on a tutorial line by Neil Cheung titled “If You Want a Coding Game…here is the secret: It Can Teach Some Skills and Skills That Would
Be Great for a Curriculum…Don’t Take Me When You Want to Do That”? Well it helps if this article is part of your curriculum/preparing homework and the program-pre-bookwork is a great way to take the
assignment. The easiest way to go about explaining everything not directly covered in your curriculum is to review the website at your campus office, go to the course website for the course the
instructor has written for you, and that’s it! Most college and university courses are about how to define a code-breaking course, right? Then there is the part where the author has added the chapter
using GADTs, which is not much. This is part of the curriculum, and a point I am planning to mention. Did you have to get to know something about the Coding Game? More info on this piece will be
published immediately after the article is complete, or you could give yourself a call from your front desk? Do my homework and I find myself in a position to be teaching you about Coding Game? Good
luck! If you haven’t read the book, I’d urge you to read through the page and see a couple of the examples here. If you are new to programming and know of any other approaches to learning programming
that you are considering try these exercises and they may help you to build your own grade point average (GPRA). You come along with a mindset that can be of some great advice to help your students
on how to build that score: “Just keep one goal up, so when they grade your score, do something that makes life better.” Take it easy and develop. When you do something exciting, if you want to
improve your level of skill, find other ways of doing that. Don’t have classes with courses others and your students will not get the interest of the junior team. In the first chapter, where I begin
with, you should work out how to do your homework, and then, form your scores and you go into the section on the “What to Do” section and write down the following sections. Find out what you need to
do to maximize your score in the second two sections of the writing test. If you want to add a new paper during the third section, take a look at what I did where, and then add in something like,
“For one group [of beginner-level] students, this is particularly good for Coding Game!” (which you could also use this free web page!) If you have a new interest in Coding, email me here and let me
know the time and place they are interested in seeing in the curriculum/bookwork. I was delighted to hear from you! So there you have it, the last one (and the chapter) to answer the question “what
do I need to do to achieve my grades?” I wish you a happy birthday, if you haven’t read this here I highly appreciate it. I wish every student of course any good results.Who offers services for R
programming assignments? You may have already guessed that you are interested in such a subject of programming assignments. You can get an online course for you, as well as an assignment for any
other students. If you have the appropriate interest from the instructors, the course might also serve as a springboard for learning. You can even put together a self-admonition course for your
students without the need of the instructor.
Online Assignment Websites Jobs
However, for now, I will leave aside the subject which is in dispute. Those who offer these methods have to wait for a little time, while other students give the assignments. For your information on
the subject I will give you a brief analysis. How to choose subject for programming assignments? In this article, we have put into detail the subject such as programming with R. The topic of
programming have been covered in a paper by Le Guingat (2008), but since I spoke in the chapter titled “Programming with R”, and even though that paper helped you in your search for a subject I will
finish this paragraph as it is really useful for our purposes. And if you hope to connect with one of my original articles let me know and I will continue the research on it as it is the most
helpful! So that is all, if you would appreciate, a series of posts in this article from my last student time on several topics; Math and Electronics. Hello, I have a question. What type of
work-class programme should I have put myself on for programming? In the beginning during the assignment they offered many books. After that some books about a particular topic(which I still found)
and its related course topics. Later on the course faculty decided that I should try to improve the subject by adding lectures the student as an assistant. Thank you a lot for this posting. I can
give you some of advice on programming assignment. Let’s talk about studying, mathematics, and electronics applications in R Brought you two recent articles, which talk about programming assignments
more than being a homework but they also address more simple subject. As I mentioned in my last assignment, making use of a computer you have to make use of a router in your workplace for programming
assignments. Remember that a router is your router, you use this link get access to any one of a number of the components in the area and also your environment and your various functions of the
router so that you can continue to learn about them in a very simple manner as you develop your knowledge. The whole purpose of this assignment is to develop a basic understanding of using a router
that is a part of your programme in your home. This, you’ll need a PC that can have a terminal on it. It can be plugged into a router, such as a router or a smartphone receiver in your PC, which can
also carry the router itself and has to be connected with the computer. Therefore, if you work on your laptop orWho offers services for R programming assignments? Please compare our rates and book
your R (programming) assignment. The Course: Welcome to this presentation and to participate in educational modules on how to become an active R tutor.
Homework Pay
This 10-session modular session will include, and a separate, modular learning session for intermediate learners, and a session for intermediate as well as advanced students as well as electives
(nursing course). There will be two sessions for intermediate students and one session for advanced students (pre-requisite). The two modules in the course will be taught in one session. A self-paced
group introduction to R software and procedures. Students who are interested in one or more R software classes will be offered several classes. Preparing topics and looking to expand in some way or
another is the only area which covers the field of R. Note: Adhering to R is very important to continue mathematics classes. To add energy to anyone’s activities, we encourage you to be active, with
all the above tips and techniques in order to expand on the program. The course guide is a copy of Chapter 3 of the History of R course, and is supplied by an assignment for your local library. The
most practical way for students to keep track of time on their calculator is to print your notes and log into your R library (or your laptop computer). As an R student, you can start with the course,
but, if you prefer programming for the mathematics course, you may as well start with the introductory course. The goal of any R study is to get in touch with the core algorithms and programming
techniques for such learning. Then, you can see how you can add more of them with practice. Before you start… Begin Reading 1 R, Programming, I, and History of Mathematics. The last section of this
title starts with the introductory book on R by Alex Grodtheiser. This is the part that covers R basics and development for a programming course. Many previous chapters have provided the full
description of what R is different (and what it is not). It has made a great starting point for anyone interested in learning a programming paper on a general topic such as geometric equations or
counting by the number of lines. R is useful for you or other intermediate or advanced students as well as you if you are interested in studying programming for school. In fact, if you are already a
student you can read these chapters quickly if you want to learn programming by just reading the chapter.
Pay For Someone To Do Your Assignment
Programming(not) History This is your job as you have to be familiar with the basics of programming many different ways, so you can practice your knowledge with ease. The Book: 1. “Programming and
the History of Mathematics” 1. This is a textbook in two parts. The first, Chapter
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Natalia Kolokolnikova - Geometric Methods in Representation Theory Seminar - Department of Mathematics
Natalia Kolokolnikova – Geometric Methods in Representation Theory Seminar
April 13, 2018 @ 4:00 pm - 5:00 pm
Title: K-theoretic Thom polynomial and the rationality of the singularities of the A2 loci.
Abstract: I will discuss the definitions of two K-theoretic invariants of the singularity loci, prove that they are not always equal and tell how this problem is connected to the study of the
rationality of the singularities of the singularity loci. I will prove that the singularities of the A2 loci are rational in some very specific cases, but are not rational in general.
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I Learned Everything I Need to Know About Gravity in Kindergarten (From Wile E. Coyote) - Royal Circuits Solutions
What is an electric field? That’s a complicated question that requires a bit of imagination. So before we jump into an answer, let’s look to the Earth’s gravitational field for general answers
about fields.
Gravitational Field
Gravitational Force
When we think of gravity, we tend to think of common objects falling towards Earth’s surface. But that is only half of an interaction. We do not tend to think about the Earth’s attraction to
everyday objects because the Earth’s acceleration towards that object is imperceptibly small.
And we certainly don’t think about everyday objects exerting a gravitational force on one another, and that is because the gravitational interaction is so weak we have personally never witnessed it.
Outside of carefully-designed experiments, you generally need an astronomically-sized object to generate the forces required to observe an interaction.
Do the Math
Use the gravitational mass of two objects, the constant of proportionality G, and the distance between the objects’ centers of mass, we can find the gravitational force of attraction between the
objects. To do the math properly, we should consider the mass and exact location of every oil and mineral deposit, every rock, every wave, every person, and every other object in the Earth and on
the surface. But that is too much work, so we assume a homogenous sphere.
Newton’s law of universal gravitation provides the attractive force for the interaction between Earth (⊕) and some arbitrary object.
The effects of the force of attraction between the earth and an object on the surface require Newton’s 2nd law and the object’s inertial mass.
Einstein’s theory of relativity and all experiments performed to date have shown an equivalence between inertial and gravitational mass, so we can justify setting them equal.
And to further simplify things, we might as well substitute that radius of the Earth into the equation as our separation distance, r, since almost all human activity lies within a few dozen
kilometers of the surface of the Earth. Then we can solve the equations for acceleration at or near the surface.
The answer that comes out is much easier to work with than the previous equations and is independent of the object’s mass.
All humans near the surface of the Earth, from miners below the surface to pilots in commercial airplanes experience this same acceleration towards the center of the Earth.^[1] It is far easier to
work with g=9.8(m/s^2)than to go through this derivation every time you want to determine the acceleration of a new object. Since the mass of the object always cancels out of the equation, the
acceleration will always be around g=9.8(m/s^2).
The Earth is not homogeneous, and it’s not a sphere. It is an ellipsoid with mountains and valleys, oil and mineral deposits, oceans, and all sorts of unique features. The gravitational field
surrounding the Earth is non-uniform.
Suppose you dropped an object at 800 equally spaced locations around the globe and drew an arrow at that location to indicate the direction of acceleration. In that case, you might end up with the
image below.
This three-dimensional vector field shows the direction and magnitude of acceleration at equally spaced points near the surface of the Earth. The rainbow colors indicate the relative difference from
average acceleration, with redder arrows indicating greater acceleration magnitude near the poles and purple indicating lower acceleration magnitude near the equator.
Vector Fields
If the length of the arrow is proportional to the magnitude of acceleration, the arrow becomes a vector. Mathematically, the collection of vectors is a vector field, and a value is defined
everywhere in the region, not just the locations defined by arrows.
Fields are mathematical constructs that show how an object’s properties change in a two or three-dimensional region.
Since the vector-field we used in our example is due to gravity, another name for the collection of arrows is a gravitational field.
Next questions
You need to put an object at a point in a gravitational field to witness an acceleration; if there’s no object, there’s no acceleration, so what gives! If there is no second object, is there
something there in the space or not? And what, if anything, happens in the space on the far side of the charged particles? What happens in the physical space around a single charged particle? Does
the simple presence of a charged object perturb space in some way? This is maddening! We’ll get answers to some of these questions in the next article!
In the meantime, here’s another gravity vector field image for you pin to the wall of your office.
^1 There are slight fluctuations in the magnitude and direction of the gravity vector over the surface of the Earth and it of course decreases with distance from the surface of the Earth.
The combination solidifies Summit’s position as one of the largest privately owned printed circuit board (PCB) manufacturers in North America with a footprint that will now encompass eight
manufacturing facilities. The acquisition significantly broadens the scope of Summit’s product offering while expanding the company’s business portfolio of key customers and end-markets. Summit
Interconnect is pleased… View Article
Read More
Ever wondered how many vias to use in a thermal package? Or how to route high current traces in small spaces? Good news! Mike Jouppi is back! Check back for a recording of this important PCB
resource. In the meantime, feel free to download the Thermal Resistance calculator discussed live. Download Thermal Resistance Calculator
Read More
In today’s high-tech world, electrical engineers have many choices when it comes to designing their PCBs, including what type of laminate to use. Laminates form the foundation for a high-functioning
PCB. In this on-demand webinar, learn how to select the right laminate for your project based on material properties, suppliers, types and more.
Read More
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Fast switch and spline function inversion algorithm with multistep optimization and k-vector search for solving Kepler’s equation in celestial mechanics
Obtaining the inverse of a nonlinear monotonic function f(x) over a given interval is a common problem in pure and applied mathematics, the most famous example being Kepler’s description of orbital
motion in the two-body approximation. In traditional numerical approaches, this problem is reduced to solving the nonlinear equation f(x)−y=0 in each point y of the co-domain. However, modern
applications of orbital mechanics for Kepler’s equation, especially in many-body problems, require highly optimized numerical performance. Ongoing efforts continually attempt to improve such
performance. Recently, we introduced a novel method for computing the inverse of a one-dimensional function, called the fast switch and spline inversion (FSSI) algorithm. It works by obtaining an
accurate interpolation of the inverse function f−1(y) over an entire interval with a very small generation time. Here, we describe two significant improvements with respect to the performance of the
original algorithm. First, the indices of the intervals for building the spline are obtained by k-vector search combined with bisection, thereby making the generation time even smaller. Second, in
the case of Kepler’s equation, a multistep method for the optimized calculation of the breakpoints of the spline polynomial was designed and implemented in Cython. We demonstrate results that
accurately solve Kepler’s equation for any value of the eccentricity e∈[0,1−ϵ], with ϵ=2.22×10−16, which is the limiting error in double precision. Even with modest current hardware, the CPU
generation time for obtaining the solution with high accuracy in a large number of points of the co-domain can be kept to around a few nanoseconds per point.
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Three Digit Palindromes
What is the smallest three digit palindrome divisible by 18? Can you solve this without a brute force examination of possibilities?
Communicated by Tom Chiari.
View comments
Warning: Solutions May Be Discussed in the Comments
Written by Zach on February 9, 2012. Reply
I know! May I “spill the beans” as it were?
Written by Steven Miller on February 9, 2012. Reply
sure — shoot me an email at [email protected]
Written by shaheen on May 1, 2012. Reply
Written by Steven Miller on May 1, 2012. Reply
something smaller works ..s
Written by Tim on February 11, 2012. Reply
Is it 666?
Written by Steven Miller on February 11, 2012. Reply
while 666 works, something smaller works as well…. //s
Written by Steven Miller on March 1, 2012. Reply
Rob: correct, well done! Not posting as it’s the soln.
Written by alaze on March 4, 2012. Reply
so the awnser is 333????????
Written by Steven Miller on March 4, 2012. Reply
no a smaller works. //s (email me at [email protected] for the soln)
Written by Caitlin on March 12, 2012. Reply
Is the answer 090? If not, may I have the correct answer? thank you!
Written by Steven Miller on March 12, 2012. Reply
people typically don’t consider that a 3 digit number, so look a bit higher (you’re on the right track)….
Written by noddy on March 16, 2012. Reply
is it 108?
Written by Steven Miller on March 16, 2012. Reply
Unfortuantely not, as 108 isn’t a palindrome.
Written by Anonymous on March 20, 2012. Reply
Written by Steven Miller on March 21, 2012. Reply
Anonymous: correct!
Written by sauravshakya on March 28, 2012. Reply
Written by Steven Miller on March 28, 2012. Reply
nope — not three digits, and not divisible by 2.
Written by Steven Miller on March 29, 2012. Reply
xiaomilk: correct, not posting as it’s the soln
Written by Some guy on April 4, 2012. Reply
-828? Does that count as a palindrome?
Written by Steven Miller on April 4, 2012. Reply
Ah, nice thinking! We want the smallest positive one…. :]
Written by Steve on April 17, 2012. Reply
Still trying to figure out how 333 works … unless we’re using some unconventional definition of “divisible by” or “18” …
Written by Steven Miller on April 17, 2012. Reply
not sure where you’re getting 333 from — that’s not the answer. //s
Written by Joe on June 21, 2012. Reply
Does the three digit number’s product have to be 18 and a whole number?
Written by Steven Miller on June 22, 2012. Reply
it has to be an integer, yes. not sure what you mean by product; the number must be a palindrome and a multiple of 18.
Written by Liz on July 19, 2012. Reply
Can you please email the answer?
Written by Steven Miller on July 20, 2012. Reply
sure //s
Written by raian rahman on July 21, 2012. Reply
Can you please email the answer?i like to be ur fb frnd.pls give me ur fb id
Written by Steven Miller on July 23, 2012. Reply
sent hint //s
Written by Steven Miller on August 11, 2012. Reply
I’m removing your post as you have the correct answer, but yes, please use in your math class and let me know how it goes (sjm1 AT williams.edu). I’m happy to have your students, if they do well,
appear in the hall of fame.
Written by miri on August 11, 2012. Reply
coll tnx!
Written by Steven Miller on August 13, 2012. Reply
glad you’re enjoying it: sjm1 AT williams.edu
Written by Steven Miller on August 22, 2012. Reply
Ravi: yes — please email at sjm1 AT williams.edu as I don’t like to post answers
Written by Sebas on October 15, 2012. Reply
you might want to consider adding the demand that the outcome is a natural number as well… in it’s current form, the correct answer is 101 🙂
(101 / 18 = 5 11/18 tadaa 😉 )
Written by Steven Miller on October 16, 2012. Reply
Thanks — normally the word divisible implies no remainder. /s
Written by Steven Miller on October 30, 2012. Reply
Rage: correct
Written by Steven Miller on November 7, 2012. Reply
yes (sjm1 AT williams.edu)
Written by Steven Miller on December 8, 2012. Reply
anon — yes — [email protected]
Written by Dhaval Shah on June 23, 2014. Reply
Written by Steven Miller on August 1, 2014. Reply
No, less.
Written by Devon bibb on May 28, 2015. Reply
It’s 101 because it says divisible, but not that it has to be an integer
PS I’m a child
Written by Devon bibb on May 28, 2015. Reply
It’s 101 because it says divisible but not that it has to be an integer.
Written by Steven Miller on May 28, 2015. Reply
nice, but there is a solution when you interpret it the way most children would, namely divisible means it goes in a whole number of times and
leaves a remainder of zero.
Written by Steven Miller on August 9, 2019. Reply
To: [email protected]
To: [email protected]
To: [email protected]
To: [email protected]
To: [email protected]
To: [email protected]
To A. N. Shanbhag
several people have solved; I don’t post the answers (you are supposed to email me, but if you can’t you can post and I just don’t
approve) as I don’t want to spoil people’s fun
well done
[[your email did not work]]
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Python - Linux Guides, Tips and Tutorials | LinuxScrew
Home » Programming » Python
This article will show you how to calculate natural logs/logarithms in the Python programming language, with examples. What is a Natural Log/Logarithm (ln)? A number’s natural logarithm is it’s
logarithm to the base of e. It’s a bit more advanced than the usual arithmetic you’re probably used to seeing as you learn to program, so here’s a bit of explanation: A logarithm is the reverse of
an exponent. The logarithm of a number is the exponent that another number must be raised to to produce the first number. You’re … Read more
Home » Programming » Python
How to Iterate/Loop over a Dictionary in Python [Examples]
This article will show you how to iterate/loop over the keys and values in a Python dictionary, and provide code examples. What is a Python Dictionary? In Python, a dictionary is a data type that
contains a collection of objects – but unlike a list or array, rather than their values being recorded at a specific index (or ordered position in the sequence of stored values), they are stored in a
key – a string identifier which both tells you what a value is for, and which … Read more
Home » Programming » Python
How to Generate & Write CSV Files in Python, With Examples
This article will show you how to generate CSV (Comma Separated Values) data, and write it to a file, with examples. What is CSV (Comma Separated Values) CSV (Comma Separated Values) is a format for
storing and transmitting data, usually in a text file, in which the individual values are separated by a comma (,). Each row in a CSV file represents an individual record containing multiple comma
separated values. A CSV file looks like this: name,age,favourite colour Fred,54,green Mary,31,pink Steve,12,orange Above, the CSV describes three people, with … Read more
Home » Programming » Python
Python hasattr() – What it Does and How to Use It [Examples]
The hasattr() function is used to check whether an object has a given named attribute. Here is how it is used, with examples. Python Objects and Classes Python is an object oriented programming
language. Object oriented means that rather than code being written as a list of instructions, accumulating data and passing the results to the following lines of code (known as procedural
programming), data and functionality is contained within the objects it describes. Each object is a self-contained representation of something, that can be modified, and passed between functions …
Read more
Home » Programming » Python
Python – Concatenating Iterators With itertools.chain() [Examples]
This tutorial will show you how to join/concatenate/merge iterables in the Python programming language, and provide code examples. What is an ‘Iterable’ in Python? An iterable is any kind of Python
object that can return each of its members one at a time. Put simply, it’s any Python object that contains multiple values that can be looped over. Python’s built-in Iterables include lists, strings,
and tuples — all object types which can be represented as a sequence of values. Iterating over Iterables To demonstrate this behaviour, we can create a list of strings, … Read more
Home » Programming » Python
Python getattr() – What it Does and How to Use It [Examples]
The getattr() function is used to get the value of an attribute of an object. Here is how it is used, with examples. Python Objects and Classes Python is an object oriented programming
language. Object oriented means that rather than code being written as a list of instructions, accumulating data and passing the results to the following lines of code (known as procedural
programming), data and functionality is contained within the objects it describes. Each object is a self-contained representation of something, that can be modified, and passed between functions …
Read more
Home » Programming » Python
How to Reverse a String in Python, With Code Examples
This quick tutorial will show you how to reverse a string in Python, and provide example code. Strings in Python Strings in Python are defined using quotes or double quotes, and assigned to
variables: string1 = “This is a string” string2 = ‘This is also a string’ Strings are used to store non-numeric values – things like names, words, sentences, and serialized data. There’s a lot you
can do with Python strings. Strings are immutable — meaning that once defined, they cannot be changed. However, the variable holding … Read more
Home » Programming » Python
Simple Guide to Python Multiprocessing/Threading [Examples]
This article will provide a simple introduction to multiprocessing in Python and provide code examples. The code and examples in this article are for Python 3. Python 2 is no longer supported, if you
are still using it you should be migrating your projects to the latest version of Python to ensure security and compatibility. What is Multiprocessing? Usually, Python runs your code, line-by-line,
in a single process until it has completed. Multiprocessing allows you to run multiple sets of instructions simultaneously in separate processes, enabling your … Read more
Home » Programming » Python
Running External Programs in Python with subprocess [Examples]
This article will show you how to use the Python subprocess module to run external programs and scripts from Python. It is often necessary to call external applications from Python. Usually these are
command-line applications which you can use to perform tasks outside of the Python environment, like manipulating files, or interacting with third party services. For example, you might call
the wget command to retrieve a remote file. Any application which is accessible on the command line is available to subprocess, and can be executed from within Python. You … Read more
Home » Programming » Python
How to Generate the Fibonacci Sequence of Numbers in Python
This quick tutorial will show you how to generate the Fibonacci sequence of numbers in Python. This is a useful example for new coders and those just learning Python as it shows how variables, loops,
and arithmetic are used in Python. What is the Fibonacci Sequence The Fibonacci sequence is a procession of numbers in which each number is the sum of the preceding two numbers. 0, 1, 1, 2, 3, 5, 8,
13, 21, 34, 55, 89, 144… Above, the first 13 numbers in the Fibonacci … Read more
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Size Reduction Mineral Liberation Model - 911Metallurgist
Success has been achieved in developing mathematical models of rod mills, ball mills and recently to a limited extent autogenous grinding mills. There is also considerable activity in the development
of mathematical models of various mineral concentration operations, such as flotation, magnetic separation and electrostatic concentration. However, at present there is a technological gap which
hinders the integration of both size reduction and mineral concentration models into an overall process simulation model.
Demonstration Case Batch Grinding-Liberation
In this presentation the author would like to call for serious thinking in this area by demonstrating one approach to the merging of batch grinding technology to a “first approximation” model
describing mineral liberation resulting from size reduction. The data used in this demonstration were generated by batch, wet grinding ore (60% solids) from a low grade magnetite iron formation (25%
Fe by weight, 23.3% Fe3O4 by volume) in a laboratory rod mill (10″ x 7″∅), sizing it into fractions and making wet laboratory magnetic separations (Davis tube) on the size fraction to indicate the
liberation achieved.
In order to simplify the approach to this modeling problem, the size distribution results (Wi) originally were fit with first order single component batch grinding parameters modified to reflect the
supposition that size reduction proceeds only from one size fraction (i-1) to the next smaller size fraction. This permits the use of selection functions (Si) alone to describe a size reduction which
would normally require both selection and breakage functions (Bij) to account for the breakage of particles from one size fraction into all smaller size fractions.
In the laboratory magnetic separation of size fractions of ground iron formation it is possible to separate essentially all magnetite containing particles from liberated waste particles. A comparison
of laboratory obtained concentrate grades for the various particle size ranges (Bi) and those concentrate grades calculated from the liberation model for the grade of the crude ore (VB) and various
ratios of particle size to mineral grain size (Bi/a) permit an estimation of mineral grain size (α).
In describing the size reduction of a binary mineral system which is undergoing liberation, it is necessary to account for the fact that three types of particles are being broken, the two liberated
mineral species and the locked specie. When liberated particles break to a finer size fraction, they remain liberated and of the same mineral specie. When locked particles break they produce
liberated valuable mineral, liberated waste mineral and some particles which remain locked.
Finite difference forms of these equations were used to model the combined batch grinding-liberation of the crude ore sample, based on the selection functions calculated for the total ore, the crude
ore assay, and the effective mineral grain size obtained by comparison with the liberation model. The product size distribution, the quantity of magnetics (liberated valuable mineral and locked
valuable mineral particles) and the volumetric grade of these magnetics were calculated by the combined model. An initial simulation was made assuming both the valuable and waste minerals had the
same selection function as the total ore.
A comparison of the simulation results with the actual data indicated that some size fractions had considerably less magnetics than predicted. The largest deviations occurred in the coarse size
fractions of each grind, suggesting a selective grinding of valuable mineral.
In terms of the combined model, this translates into higher selection functions for magnetite than for waste mineral. It was therefore assumed that this composition effect could be represented by a
constant ratio (K) between the selection functions for magnetite and waste, and further that the selection function for locked particles could be calculated on the basis of its volumetric composition
A further simulation using this technique and a selection function ratio of 1.4 to describe selective grinding provided a much better prediction of the fraction magnetics in each size interval
resulting from batch grinding as shown in Figure 10. However, it was not possible to simulate any selective grinding which may have taken place in the production of the original crude ore sample, and
therefore those deviations remain.
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Problem 10 Solution
This page contains the
NCERT mathematics class 12 chapter Relations and Functions Exercise 1.1 Problem 10 Solution
. Solutions for other problems are available at
Exercise 1.1 Solutions
Exercise 1.1 Problem 10 Solution
10. Give an example of a relation. Which is
Symmetric but neither reflexive nor transitive.
Transitive but neither reflexive nor symmetric.
Reflexive and symmetric but not transitive.
Reflexive and transitive but not symmetric.
Symmetric and transitive but not reflexive.
10.i. Give an example of a relation which is Symmetric but neither reflexive nor transitive.
Consider a set A = {a, b, c} and the relation R defined as R = {(a, b), (b, a)}
To Check whether R is Reflexive: The relation R in the set A is reflexive if (a, a) ∈ R, for every a ∈ A.
The element (a, a) ∉ R
⇒ This relation is not reflexive.
To Check whether R is Symmetric: The relation R in the set A is symmetric if (a_1, a_2) ∈ R implies that (a_2, a_1) ∈ R, for all a_1, a_2 ∈ A
In this relation, both (a, b) ∈ R and (b, a) ∈ R
⇒ This relation is symmetric.
To Check whether R is Transitive: The relation R in the set A is transitive if (a_1, a_2) ∈ R and (a_2, a_3) ∈ R implies that (a_1, a_3) ∈ R, for all a_1, a_2, a_3 ∈ A
In this example (a, b) ∈ R and (b, a) ∈ R. But (a, a) ∉ R.
⇒ This relation is not transitive.
∴ R is Symmetric but neither reflexive nor transitive.
10.ii. Give an example of a relation which is Transitive but neither reflexive nor symmetric.
Consider the relation defined by R = {(x, y): x > y}
To Check whether R is Reflexive: The relation R in the set A is reflexive if (a, a) ∈ R, for every a ∈ A.
As we know, No element greater than itself
⇒ (a, a) ∉ R.
∴ R is not reflexive.
To Check whether R is Symmetric: The relation R in the set A is symmetric if (a_1, a_2) ∈ R implies that (a_2, a_1) ∈ R, for all a_1, a_2 ∈ A
Consider two elements a, b such that {a \gt b}. Obviously {b \ngtr a} (to be specific b \le a).
⇒ (a, b) ∈ R but (b, a) ∉ R.
∴ R is not symmetric.
To Check whether R is Transitive: The relation R in the set A is transitive if (a_1, a_2) ∈ R and (a_2, a_3) ∈ R implies that (a_1, a_3) ∈ R, for all a_1, a_2, a_3 ∈ A
If there’re 3 elements a, b and c such that {a \gt b} and {b \gt c} then obviously {a \gt c}
⇒ If (a, b) ∈ R and (b, c) ∈ R then (a, c) ∈ R
⇒ R is transitive.
∴ R is Symmetric but neither reflexive nor transitive.
10.iii. Give an example of a relation which is reflexive and symmetric but not transitive.
To Check whether R is Reflexive: The relation R in the set A is reflexive if (a, a) ∈ R, for every a ∈ A.
Consider the set A = {p, q, r}. Consider the relation on the set A = {p, q, r} defined as R = {(p, p), (q, q), (r, r), (p, q), (q, p), (q, r), (r, q)}
To Check whether R is Reflexive: The relation R in the set A is reflexive if (a, a) ∈ R, for every a ∈ A.
In the set A, we see that For every element a ∈ A, there is a corresponding (a, a) ∈ R.
⇒ (p, p) ∈ R, (q, q) ∈ R and (r, r) ∈ R
∴ R is reflexive.
To Check whether R is Symmetric: The relation R in the set A is symmetric if (a_1, a_2) ∈ R implies that (a_2, a_1) ∈ R, for all a_1, a_2 ∈ A
In the relation R, we see that (p, q) ∈ R and (q, p) ∈ R.
Also (q, r) ∈ R and (r, q) ∈ R
∴ R is symmetric
To Check whether R is Transitive: The relation R in the set A is transitive if (a_1, a_2) ∈ R and (a_2, a_3) ∈ R implies that (a_1, a_3) ∈ R, for all a_1, a_2, a_3 ∈ A
In the relation R, we see that both (p, q) ∈ R and (q, r) ∈ R, but (p, r) ∉ R
∴ R is not transitive.
∴ R is reflexive and symmetric but not transitive.
10.iv. Give an example of a relation which is Reflexive and transitive but not symmetric.
Consider the relation R = {(x, y) : x \geq y} defined on the set of natural numbers N
To Check whether R is Reflexive: The relation R in the set A is reflexive if (a, a) ∈ R, for every a ∈ A.
For any natural number a ∈ N, we know that {a = a}.
⇒ a \geq a
∴ R is reflexive.
To Check whether R is Symmetric: The relation R in the set A is symmetric if (a_1, a_2) ∈ R implies that (a_2, a_1) ∈ R, for all a_1, a_2 ∈ A
Consider the element (5, 4) ∈ R because 5 ≥ 4. But we see that {4 \nleq 5}.
⇒ (4, 5) ∉ R
⇒ There are few elements (a, b) ∈ R such that (b, a) ∉ R
∴ R is not symmetric.
To Check whether R is Transitive: The relation R in the set A is transitive if (a_1, a_2) ∈ R and (a_2, a_3) ∈ R implies that (a_1, a_3) ∈ R, for all a_1, a_2, a_3 ∈ A
When {a \geq b} and {b \geq c}, we have {a \geq c}
⇒ For every (a, b) ∈ R and (b, c) ∈ R, there is a corresponding (a, c) ∈ R
∴ R is transitive.
∴ R is reflexive and transitive but not symmetric.
10.v. Give an example of a relation which is Symmetric and transitive but not reflexive.
Consider the relation R = {(p, q), (q, p), (p, p)} defined on the set A = {p, q}
To Check whether R is Reflexive: The relation R in the set A is reflexive if (a, a) ∈ R, for every a ∈ A.
We see that (q, q) ∉ R
∴ R is not reflexive
To Check whether R is Symmetric: The relation R in the set A is symmetric if (a_1, a_2) ∈ R implies that (a_2, a_1) ∈ R, for all a_1, a_2 ∈ A
We see that (p, q) ∈ R and (q, p) ∈ R
∴ R is symmetric
To Check whether R is Transitive: The relation R in the set A is transitive if (a_1, a_2) ∈ R and (a_2, a_3) ∈ R implies that (a_1, a_3) ∈ R, for all a_1, a_2, a_3 ∈ A
Both (p, q) ∈ R (q, p) ∈ R and also (p, p) ∈ R.
∴ R is not transitive.
∴ R is symmetric and transitive but not reflexive
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Setting a Product Margin | ArcSite Help Center
What is margin?
Margin (or gross profit margin) shows the revenue you make after paying the Cost of Goods Sold. Basically, your margin is the difference between what you earned and how much you spent to earn it.
To calculate profit margin, start with your gross profit, which is the difference between revenue and COGS. Then, find the percentage of the revenue that is the gross profit. To find this, divide
your gross profit by revenue. Multiply the total by 100 and voila—you have your margin percentage.
Let’s put the margin meaning into a margin calculation formula:
Margin= [(Revenue – COGS) / Revenue] X 100
Margin= (Gross Profit / Revenue) X 100
The margin formula measures how much of every dollar in revenue, you keep after paying expenses.
The greater the margin, the greater the percentage of revenue you keep when you make a sale.
Let’s look at an example of how to calculate margin.
Let's say you sell widgets for $200 each. Each widget costs you $150 to make. What’s your margin?
To start, plug the numbers into the margin formula:
Margin= [($200 – $150) / $200] X 100
First, find your gross profit by subtracting your COGS ($150) from your revenue ($200). This gets you $50 ($200 – $150). Then, divide that total ($50) by your revenue ($200) to get 0.25. Multiply
0.25 by 100 to turn it into a percentage (25%).
The margin is 25%, meaning you keep 25% of your total revenue. You spend the other 75% of your revenue on producing the widget.
Pro Tip: Use our margin vs. markup chart to find quick conversions for markups and margins.
Adding your Product Margin In ArcSite
You'll find this setting option within the Takeoff & Estimate section of your Advanced Settings:
You can also set your minimum gross margin percentage so that you and your team get notified if the pricing of a product is adjusted to be below your minimum.
Setting your overall margin.
You can also price at the product level with your applied margin or de-select the margin at the product level.
Now you can test your new pricing for accuracy and view a proposal.
Why do margins and markups matter?
Know the difference between a markup and a margin to set goals. If you know how much profit you want to make, you can set your prices accordingly using the margin tool here in ArcSite.
If you don’t know your margins and markups, you might not know how to price correctly. This could cause you to miss out on revenue. Or, you might be asking for an amount many potential customers are
not willing to pay.
Check your margins and pricing often to be sure you’re always calculating the correct price for your customers.
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Time Calculator
How random that it looks like my old Casio! The Time Calculator allows adding and substracting time values in the sexagesimal system (where the base value is 60).
If you ever tried to add 17 minutes to 1.8 hours in your head, then you probably know that this can be error-prone.
Unfortuntely, most available calculators for Windows don't support the sexagesimal system. That's why I made this calculator. So, if you need to calculate with time values, you should really try the
Time Calculator.
The decimal system is of course also supported.
Current version: 2.0.3.17
Download Time Calculator for Windows as executable file (743 KB)
Download Time Calculator for Windows as zip file (667 KB)
The Time Calculator comes without installer. Just download an run it.
The Time Calculator is freeware. By downloading the Time Calculator you agree to the terms of use.
If you like the Time Calculator and find it useful, feel free to donate.
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Union support for the BC NDP is not the same as corporate support for the BC Liberals
C. Welch on April 27, 2013 — 1 Comment
A sore point for supporters of the right-wing BC Liberals is that their party seems so financially beholden to corporations. An immediate retort is that the NDP is equally beholden to unions. The
NDP, in other words, is just as bad, so let’s move on. The problem with this response is that, in British Columbia, the two major political parties are not equally indebted to their largest
sponsors. More precisely, the leading right-wing party is much more beholden to corporations than the leading left-wing party is to unions. Thus, an argument of equivalence – a false equivalence in
which both sides are equally obligated to outside masters, so both are excused – simply doesn’t hold up to examination.
A case in point is a recent five-day analysis of campaign donations conducted by the long-standing corporate media icon, The Vancouver Sun.* Through the use of parallelism, the newspaper asserts a
position of equivalence:
It’s well-known that unions are big supporters of the NDP, just as corporations bankroll the B.C. Liberals. (The Vancouver Sun, April 24, 2013, A4)
However, when we start examining the numbers closely, something startling appears from the statistics provided by the The Vancouver Sun itself – statistics which go largely unexamined in the
newspaper’s analysis.
From the numbers above** we can make the following calculations:
1. Corporate donations to the BC Liberals are five times greater than that of union donations to the BC NDP.
2. Corporate contributions make up 61% of BC Liberal donations, whereas union contributions are only 23% of BC NDP donations.
These calculations are never mentioned in the text of the newspaper’s five-day series.
Equivalence indeed.
*April 23, 2013 to April 27, 2013
**I could not find this particular chart online, so I scanned the original paper version. (The Vancouver Sun, April 23, 2013, A8)
Posted in BC Politics, Canadian Politics, Economic Issues, Language, Media | Tagged BC Liberals, election, government, media, NDP
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Mathematics Genealogy Project
Yuri Yurievich Trokhimchuk
Ph.D. 1952
Dissertation: To the theory of the core of sequence of Riemann surfaces and the theory of boundary properties of analytic functions
Advisor 1: Lev Israelevich Volkovyskii
doctor of sciences 1959
Dissertation: Continuous mappings and analytic functions
Advisor: Unknown
Click here to see the students ordered by family name.
Name School Year Descendants
Kompaniets, Viktor Institute of Mathematics, Ukrainian Academy of Science 1966
Bondar, Anatolii 1971 6
Tar, Nikolai Institute of Mathematics, Ukrainian Academy of Science 1971
Zelinskii, Yuri Institute of Mathematics, Ukrainian Academy of Science 1973 9
Diab, Fadel Institute of Mathematics, Ukrainian Academy of Science 1975
Gorlenko, Sviatoslav Institute of Mathematics, Ukrainian Academy of Science 1975
Linichuk, Ruslan Institute of Mathematics, Ukrainian Academy of Science 1975
Sharko, Vladimir Institute of Mathematics, Ukrainian Academy of Science 1976 13
Ruzhitski, Eugen Institute of Mathematics, Ukrainian Academy of Science 1978
Bondar, Anatolii Institute of Mathematics, Ukrainian Academy of Science 1982 6
Sharko, Vladimir Institute of Mathematics, Ukrainian Academy of Science 1982 13
Atabaev, Mukhamedklich Institute of Mathematics, Ukrainian Academy of Science 1983
Safonov, Vladimir Institute of Mathematics, Ukrainian Academy of Science 1986
Zelinskii, Yuri Institute of Mathematics, Ukrainian Academy of Science 1989 9
Alikulov, Eshpulat Institute of Mathematics, Ukrainian Academy of Science 1992
Ilmuradov, Deria Institute of Mathematics, Ukrainian Academy of Science 1993
According to our current on-line database, Yuri Trokhimchuk has 13 students and 41 descendants.
We welcome any additional information.
If you have additional information or corrections regarding this mathematician, please use the update form. To submit students of this mathematician, please use the new data form, noting this
mathematician's MGP ID of 163235 for the advisor ID.
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Tun to Dash (Imperial) Converter
⇅ Switch toDash (Imperial) to Tun Converter
How to use this Tun to Dash (Imperial) Converter 🤔
Follow these steps to convert given volume from the units of Tun to the units of Dash (Imperial).
1. Enter the input Tun value in the text field.
2. The calculator converts the given Tun into Dash (Imperial) in realtime ⌚ using the conversion formula, and displays under the Dash (Imperial) label. You do not need to click any button. If the
input changes, Dash (Imperial) value is re-calculated, just like that.
3. You may copy the resulting Dash (Imperial) value using the Copy button.
4. To view a detailed step by step calculation of the conversion, click on the View Calculation button.
5. You can also reset the input by clicking on button present below the input field.
What is the Formula to convert Tun to Dash (Imperial)?
The formula to convert given volume from Tun to Dash (Imperial) is:
Volume[(Dash (Imperial))] = Volume[(Tun)] × 1289219.4479708595
Substitute the given value of volume in tun, i.e., Volume[(Tun)] in the above formula and simplify the right-hand side value. The resulting value is the volume in dash (imperial), i.e., Volume[(Dash
Calculation will be done after you enter a valid input.
Consider that a brewery stores 7 tuns of beer.
Convert this volume from tun to Dash (Imperial).
The volume in tun is:
Volume[(Tun)] = 7
The formula to convert volume from tun to dash (imperial) is:
Volume[(Dash (Imperial))] = Volume[(Tun)] × 1289219.4479708595
Substitute given weight Volume[(Tun)] = 7 in the above formula.
Volume[(Dash (Imperial))] = 7 × 1289219.4479708595
Volume[(Dash (Imperial))] = 9024536.1358
Final Answer:
Therefore, 7 tun is equal to 9024536.1358 .
The volume is 9024536.1358 , in dash (imperial).
Consider that a winery produces 2 tuns of wine.
Convert this volume from tun to Dash (Imperial).
The volume in tun is:
Volume[(Tun)] = 2
The formula to convert volume from tun to dash (imperial) is:
Volume[(Dash (Imperial))] = Volume[(Tun)] × 1289219.4479708595
Substitute given weight Volume[(Tun)] = 2 in the above formula.
Volume[(Dash (Imperial))] = 2 × 1289219.4479708595
Volume[(Dash (Imperial))] = 2578438.8959
Final Answer:
Therefore, 2 tun is equal to 2578438.8959 .
The volume is 2578438.8959 , in dash (imperial).
Tun to Dash (Imperial) Conversion Table
The following table gives some of the most used conversions from Tun to Dash (Imperial).
Tun (tun) Dash (Imperial) ()
0.01 tun 12892.1945
0.1 tun 128921.9448
1 tun 1289219.448
2 tun 2578438.8959
3 tun 3867658.3439
4 tun 5156877.7919
5 tun 6446097.2399
6 tun 7735316.6878
7 tun 9024536.1358
8 tun 10313755.5838
9 tun 11602975.0317
10 tun 12892194.4797
20 tun 25784388.9594
50 tun 64460972.3985
100 tun 128921944.7971
1000 tun 1289219447.9709
The tun is a unit of measurement used to quantify large volumes, particularly in the context of liquids such as wine or beer. It is defined as approximately 1,016.5 liters or 1,056 US quarts.
Historically, the tun was used to measure the capacity of large casks or barrels for storing and transporting liquids. The term is still referenced in certain industries, such as brewing and
winemaking, where large volumes are common. Although less commonly used today, it remains part of historical measurement systems and is occasionally encountered in trade and commerce.
Dash (Imperial)
The Imperial dash is a unit of measurement used to quantify very small volumes, typically in cooking and medicine. It is a traditional unit from the British Imperial system, representing a small,
precise amount often used in recipes or for dosing. Historically, the dash was used to measure tiny quantities of liquid for adding to recipes or medical preparations. Today, it remains relevant in
specific contexts where precise small-volume measurements are necessary, such as in culinary arts for seasoning or in medicine for administering minute doses.
Frequently Asked Questions (FAQs)
1. What is the formula for converting Tun to Dash (Imperial) in Volume?
The formula to convert Tun to Dash (Imperial) in Volume is:
Tun * 1289219.4479708595
2. Is this tool free or paid?
This Volume conversion tool, which converts Tun to Dash (Imperial), is completely free to use.
3. How do I convert Volume from Tun to Dash (Imperial)?
To convert Volume from Tun to Dash (Imperial), you can use the following formula:
Tun * 1289219.4479708595
For example, if you have a value in Tun, you substitute that value in place of Tun in the above formula, and solve the mathematical expression to get the equivalent value in Dash (Imperial).
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Eev and Maxima
1. eepitch-maxima
I use eepitch to send lines from Emacs to Maxima; see the figure below for the main idea, and my presentation at the EmacsConf2021 for the details. The definition of eepitch-maxima in eev is the
simplest possible - just this:
(defun eepitch-maxima () (interactive) (eepitch-comint "maxima" "maxima"))
I tried to write variants that used Fermin MF's maxima-mode instead of just comint, but his maxima-mode behaved in weird ways when it had to send empty lines, and we couldn't fix that easily... so I
gave up.
2. find-angg-es-links
The video below explains a way to run my executable notes on Maxima with eev without downloading anything extra. Click on the first screenshot to go to the page about that video, and click on the
third screenshot to play the very nice demo that starts at 15:14... or actually to read the subtitles of that part; then click on the timemark to play the video.
Click on the second screenshot to play (or to read the subtitles of) the video starting from 11:30. That part has a very technical explanation of the "...without downloading anything extra" - not
very recommended! 😕
3. Elisp hyperlinks
My notes on Maxima in maxima.e contain lots of elisp hyperlinks like these ones,
that are htmlized in special ways. They are explained in these other pages: find-maximanode, find-maximamsg.
4. Embedding in LaTeX
Update: my current way of LaTeXing Maxima code uses Lpeg. Its main module is here, Maxima2.lua, and it produces output like this from this input (plus tweaking). The rest of this section describes an
older way.
Here is an example of how I am embedding Maxima code in LaTeX files; the trick that makes eepitch ignore a prefix is explained here. If I execute that eepitch block skipping the lines "load" and
"display2d" I get a human-friendly output, as in the first screenshot below; if I execute the lines "load" and "display2d" I get an output that I can process with M-x emaxima-conv (that calls
emaxima.lua) to obtain this LaTeX code, that becomes this in the PDF. This trick is based on the answers that I got for this question that I sent to Maxima mailing list; note that 1) I am using this
copy of emaxima.sty that has two lines commented out, and 2) my emaxima.lua is a quick hack, and it should be converted to elisp at some point.
5. "Physicist's notation"
First version: In 2022jan10 I sent to the Maxima mailing list this big e-mail, that had two parts. In the first part I asked about the (internal) differences between using expressions, like "f : x^
2", and using functions, like "g(x) := x^3"; the code associated to that part is here. In the second part I asked if, and how, Maxima supports "physicists' notation" - where "physicists' notation"
("PN") is my informal name for a notation that is common in old books like this one by Silvanus Thompson. In PN variables and functions can share the same names, variables can be "dependent", some
arguments can be omitted, and several abbreviations are standard - for example, if y=y(x) then the default meaning for y[1] is y[1]=y(x[1])=y(x[0]+Δx). It turns out that YES, Maxima supports
physicists' notation, and it's easy to translate calculations in PN to Maxima if we use gradef and subst in the right way to translate between PN and "mathematician's notation". I recorded a 20s
video demo-ing this - it's here, and its code is here. The slides on PN that I prepared for my course on Calculus 3 are here.
Second version (sep/2023): my sample space is small - about 25 people - but apparently in Brazil,
• "all" the "P"ure mathematicians ("group P") treat dependent variables and differentials as abuses of language,
• "all" the "A"pplied mathematicians, physicists and engineers ("group A") treat dependent variables and differentials as something that "everybody knows", and
• no one in either of the two groups knows the exact rules for translating the language of "Calculus with dependent variables and differentials" ("Calculus+") into the language of "Calculus without
dependent variables and differentials" ("Calculus-")...
...so "no one" here knows how write an "elaborator", in this sense, that could translate "Calculus+" into "Calculus-". I grew up in Group A and my native language is "Calculus-", but now in my day
job I'm teaching integration using books that use "Calculus+", and I thought that the best way to handle my embarassment for not speaking "Calculus+" well enough would be to formalize how that
translation can be done.
I'm sort of working on that, and I'm starting by writing some of its functions in Maxima; my functions are here: pn1.mac. That file is a bare prototype at the moment, but to me it feels like the
right way to treat dependent variables and differentials as abbreviations. Here is a screenshot:
6. Substitution
My first attempts to understand how Maxima implements the "#$expr$" syntax in Lisp are here: 1, 2, 3. Then Stavros Macrakis explained how I could define "dosimp" and "doeval" and I produced the
example below. Its code is here. My long e-mail explaining why I am teaching substitution in this way is here (includes gossip).
7. Luatree
Update: LuaTree is obsolete!
It was superseded by LispTree.
The material on LuaTree will be moved to this page.
To draw Maxima objects as trees - as in the first screenshot below - I use a program that is made of three files: luatree.mac, luatree.lisp, luatree.lua. Its git repo is here, and you can test it by
running the test block at the end of luatree.mac. There is an explanation of luatree here.
The screenshot at the right below shows a (primitive) port of luatree to SymPy. Its code is here: luatree.py.
8. Maxima by Example
The best place for learning Maxima from examples is a (free) book whose name is - ta-daaa! - Maxima by Example. It is divided into chapters, and my script to download a local copy of it is here. Its
chapter 13 is about qdraw, that is a front-end to Maxima's plot and draw commands. I find qdraw much easier to use than plot and draw; for example, the code for drawing the Lissajous figure below -
with velocity and acceleration vectors! - is just this.
The second screenshot shows a trajectory P(t) = (cos t, sin t), the parabola Q(t) = P(0) + t·P'(0) + t^2/2·P''(0), and my favorite trick - the boxes - for drawing parabolas by hand. Its code is here.
9. Qdraw
Qdraw is easy to extend. I generated the animation below
with these files:
Click on the animation to enlarge it; click here to see it in flipbook format.
10. Debugging the Lisp (with Sly)
Most people use Slime and Swank to debug the Lisp code of Maxima. I couldn't make Slime work with eepitch, so instead of Slime and Swank I'm using Sly, Slynk, and an eepitch-sly defined in this way,
and I had to adapt these instructions. My code to use Sly and Slynk in Maxima is here: ~/.maxima/startsly.lisp.
See also this: eev-sly.html.
11. Maxima for students
This is a work in progress, and at this moment most of its docs are in Portuguese! If you don't understand Portuguese, the best starting points are these ones:
If you do understand Portuguese, then there's also this:
I'm preparing a presentation for the EmacsConf 2024 about this - where "this" is how I made a bunch of students who had never seen a terminal in their lives install Emacs, eev, and Maxima. I had
several really nice success cases and several very interesting failures - mainly from students who found very hard to ask questions.
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Polynomial Graph Lesson Plans & Worksheets | Lesson Planet
Polynomial graph Teacher Resources
Find Polynomial graph lesson plans and worksheets
Showing 26 resources
Lesson Planet Curated
In order to graph a polynomial, students need to find the factors. This resource starts with the Remainder Theorem and then examines the factors in relationship to the graph. Resources also look at
multiplicities and finishes with...
Houston Area Calculus Teachers
Your AP Calculus learners probably have their ID numbers memorized, but have they tried graphing them? Pique your pupils' interest by asking each to create a unique polynomial function based on their
ID numbers, and then analyze the...
Mathematics Vision Project
Bridge the gap between graphical and algebraic representations. Learners complete six lessons that begin by pointing out connections between the key features of a polynomial graph and its algebraic
function. Later, pupils use the...
Time to put it all together! Building on the concepts learned in the previous lessons in this series, learners apply the Remainder Theorem to finding zeros of a polynomial function. They graph from a
function and write a function from...
Flipped Math
Wrap it all up in a box. Pupils review the key concepts from an Algebra 2 polynomial functions unit by working 19 problems. Problems range from using the Remainder Theorem to find remainders and
finding factors; sketching graphs by...
Mathematics Assessment Project
It all starts with arithmetic. An educational resource provides four items to use in summative assessments. The items reflect the basic skill level required by the standards in the domain and are
designed to have pupils reason abstractly...
Young mathematicians graph polynomials using the factored form. As they apply all positive leading coefficients, pupils demonstrate the relationship between the factors and the zeros of the graph.
Curated OER
Students identify the end behavior of polynomial graphs. In this algebra lesson, students factor and solve quadratic and complex equations. They factor out negative roots and identify the real and
imaginary parts of an equation.
Curated OER
There are many ways to solve a quadratic equation. Let's check out how to solve a quadratic equation by graphing. But first, let's find the axis of symmetry and the vertex before making a table of
values. Write the values as ordered...
Curated OER
Trying to find the vertex of a quadratic equation? Remember that the vertex is the minimum or maximum point of the quadratic equation and can be identified on a graph of the equation. First use the
axis of symmetry formula to find the...
Curated OER
For this worksheet, learners review topics from Algebra 2. Topics include synthetic division, rational algebraic expressions, quadratic functions, imaginary number, composite functions, and inverse
of a function. The three-page worksheet...
Curated OER
Students graph polynomials functions and analyze the end behavior. In this algebra activity, student differentiate between the different polynomials based on the exponents. They use a TI to help with
the graphing.
CK-12 Foundation
This lesson explores graphs of polynomial equations including how to find the maximum and minimum y-values on a polynomial function.
Paul Dawkins
Students investigate how to sketch and find solutions to higher degree polynomials. Topics explored are dividing polynomials, roots of polynomials, graphing polynomials, and finding zeroes of
polynomials. Class notes, definitions, and...
A comprehensive guide for learning all about polynomial functions with definitions, the highest degree of the polynomial, graphing polynomial functions, range and domain of polynomial functions,
solved examples, and practice questions.
Purple Math
Explains how to recognize the end behavior of polynomials and their graphs. Points out the differences between even-degree and odd-degree polynomials, and between polynomials with negative versus
positive leading terms.
Varsity Tutors
Twenty-two problems present a variety of practice evaluating and graphing polynomials. They are given with each step to the solution cleverly revealed one at a time. You can work each step of the
problem then click the "View Solution"...
University of Saskatchewan (Canada)
A more advanced review of Algebra II concepts allows students to test their readiness for higher level math courses. Topics range from absolute value to rational expressions.
Khan Academy
The zeros of a polynomial p(x) are all the x-values that make the polynomial equal to zero. They are interesting to us for many reasons, one of which is that they tell us about the x-intercepts of
the polynomial's graph. We will also see...
CK-12 Foundation
[Free Registration/Login may be required to access all resource tools.] This lesson covers finding the minimum(s), maximum(s), end behavior, domain, and range of a polynomial graph. Students examine
guided notes, review guided practice,...
University of Saskatchewan (Canada)
An introduction to the graphs of polynomial functions by looking at the roots of the polynomials, or where it crosses the x-axis. Includes links to pictures of graphs and detailed explanations.
CK-12 Foundation
[Free Registration/Login may be required to access all resource tools.] In this lesson students use ''X'' and ''Y'' intercepts to graph polynomials of 3rd degree or higher. Students examine guided
notes, review guided practice, watch...
Texas Instruments
Students can use the TI-84 Plus family to check the sum or difference of polynomial functions.
CK-12 Foundation
[Free Registration/Login may be required to access all resource tools.] Learn how to factor quadratic expressions.
Other popular searches
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Discriminant Calculator - calculator
What is a Discriminant Calculator?
A Discriminant Calculator is a mathematical tool used to determine the nature of the roots of a quadratic equation. In algebra, the discriminant is part of the quadratic formula, and its value can
tell you whether the quadratic equation has real roots, complex roots, or whether the roots are distinct. This calculator simplifies the process of finding the discriminant and interpreting the
Discriminant Calculator
Parameter Value
Coefficient a
Coefficient b
Coefficient c
What is the Discriminant?
The discriminant is a value calculated from the coefficients of a quadratic equation of the form ax² + bx + c = 0. It provides information about the nature of the roots of the quadratic equation.
Specifically, it helps to determine whether the roots are real or complex, and if they are real, whether they are distinct or repeated.
What is the Discriminant Calculator Website?
The Discriminant Calculator website is an online tool designed to perform calculations involving the discriminant of quadratic equations. It allows users to input the coefficients of a quadratic
equation and quickly obtain the discriminant value along with insights into the nature of the roots.
How to Use the Discriminant Calculator Website?
Using the Discriminant Calculator website is straightforward:
1. Navigate to the Discriminant Calculator website.
2. Input the coefficients of your quadratic equation (usually represented as a, b, and c in the equation ax^2 + bx + c = 0).
3. Click the 'Calculate' button.
4. The website will display the value of the discriminant and interpret the nature of the roots.
What is the Formula of Discriminant Calculator?
The formula for calculating the discriminant D of a quadratic equation ax^2 + bx + c = 0 is:
D = b^2 - 4ac
Where a, b, and c are the coefficients of the quadratic equation. The value of D helps in determining the nature of the roots:
• If D > 0: The quadratic equation has two distinct real roots.
• If D = 0: The quadratic equation has exactly one real root (or two identical real roots).
• If D < 0: The quadratic equation has two complex conjugate roots.
Advantages and Disadvantages of Discriminant Calculator
• Efficiency: Quickly computes the discriminant and provides insights into the nature of the roots.
• Accuracy: Reduces the chance of manual calculation errors.
• Ease of Use: User-friendly interface for students and professionals alike.
• Instant Results: Provides immediate results without the need for complex calculations.
• Dependency on Technology: Requires access to a computer or smartphone and an internet connection.
• Limited Scope: Only calculates the discriminant and does not solve the quadratic equation or provide additional mathematical analysis.
• Potential for Misinterpretation: Users need to understand how to interpret the results correctly.
Other Related Information
Discriminant calculators are part of a broader category of online calculators designed for algebraic functions. They are useful tools for students learning algebra, as well as for professionals
needing quick computations. Additionally, understanding the discriminant helps in graphing quadratic functions and solving real-world problems involving quadratic equations.
What is the discriminant used for?
The discriminant is used to determine the nature of the roots of a quadratic equation. It tells us whether the roots are real and distinct, real and equal, or complex. This information is crucial in
many areas of mathematics and applied sciences.
Can the Discriminant Calculator be used for other equations besides quadratic?
No, the Discriminant Calculator is specifically designed for quadratic equations. For higher-order polynomials, different methods and calculators are used to analyze their roots.
Is there a cost associated with using Discriminant Calculator websites?
Most Discriminant Calculator websites are free to use. However, some advanced calculators with additional features might require a subscription or payment.
How accurate are online Discriminant Calculators?
Online Discriminant Calculators are generally very accurate if used correctly. Ensure you input the correct coefficients and double-check the results, especially if you're using them for critical
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Spectral Decomposition- A Better Understanding
International Journal of Earth Science and Geophysics
(ISSN: 2631-5033)
Volume 10, Issue 1
Research Article
Article Formats
Spectral Decomposition- A Better Understanding
Manuel Zepeda^* and Armando Salinas
Author Details
Manuel Zepeda^* and Armando Salinas
G&G consultants, Production and Operations, Mexico
Corresponding author
Manuel Zepeda, G&G consultants, Production and Operations, Mexico.
Accepted: June 25, 2024 | Published Online: June 27, 2024
Citation: Zepeda M, Salinas A (2024) Spectral Decomposition- A Better Understanding. Int J Earth Sci Geophys 10:075
Copyright: © 2024 Zepeda M, et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original author and source are credited.
This article presents a review of the algorithms used in the spectral decomposition of seismic data. This process involves transforming a non-stationary signal in time/space from the time/space
domain to the frequency domain. The frequency domain representation reveals many important features that are not evident in the time domain. Over the years, the spectral decomposition of seismic data
has evolved from a tool for stratigraphy analysis to a direct hydrocarbon indicator (DHI) technique. Seismic interpreters primarily use this technique. Additionally, as a DHI, it is an excellent tool
for minimizing uncertainty and avoiding drilling dry holes.
Since the start of this industry, companies have been affected by drilling dry holes. This is very costly to companies each year. It has always been a risk, and it is considered risky for investment.
For this reason, the industry updates its technology every year to eliminate and minimize risks and implement success more often.
Some of these techniques and new technologies are considered important steps to exploration and production of hydrocarbons. Such high-end technologies improve all existing methodologies used. One of
them is called Spectral Decomposition, which is the motive of this article. The aim is to explain how it works and optimizes its use, as well as providing a comparison of the spectral decomposition
methods and their principal transforms with geophysical applications onto Spectral Decomposition (called Specdecomp).
When somebody hears about Specdecomp, they commonly think about mathematics and signals analysis. However, one does not need to be a mathematician to understand this application, only the
understanding of a few concepts is required. Before starting with a spectral analysis, a seismic data review is recommended [1]. A technique commonly used is called High Frequency Imaging (HFI),
applied to seismic data, a better way to enhance frequencies and extract seismic bandwidth than conventional deconvolution programs (recovers the high-frequency encoded in the lower end of the
spectrum). One issue to be considered is that seismic data needs to be processed with amplitude preserving and balancing (i.e., offset varying amplitude balancing, noise attenuation, preserving good
S/N ratio, pre-stack Kirchhoff’s methods fine migration or common-azimuthal wave form processing, a detailed velocity analysis). This is the new ideal, which was before not possible.
Seismic data is typically limited to about 30 or 50 Hz of usable frequencies. Seismic acquisition and migration programs have begun to enhance the spectrum of frequencies, which is a powerful tool
for detail in interpretation. Some techniques can yield upwards of 100 Hz of valid data; this process aids in resolving subtle structural and stratigraphic events and obtains better resolution. This
allows the identification of very thin strata, as thin as 10-to-15-foot thickness. Once yielded, the interpreter can generate attributes (i.e., such as Relative Acoustic Impedance, Q, RAI, helps to
delineate sand bodies and fluids; semblance, instantaneous phase and instantaneous frequency, texture attributes so forth) and determinate which are best to use on any determinate situation for a
better interpretation and understanding of reservoir analysis.
Specdecomp, Initially
Spectral decomposition (Specdecomp) was discovered in the 1970s but was introduced to the industry in the late 1990s by Greg Patryka, Castagna, and others [2,3]. Spectral decomposition is now
becoming a valuable post-processing and preprocessing technique for investigating complex hydrocarbon plays and stratigraphic characteristics. Typically used in thin-bed stratigraphic details as a
hydrocarbon indicator, Specdecomp is based on the concept that a thin-bed reflection has a unique spectral response in the frequency domain. Spectral decomposition algorithms (such as DFT, ST, CWT,
TFCWT, Matching Pursuit, etc.) are applied to seismic reflection data to break down the seismic signal into its frequency components (see Figure 1). This technique provides higher resolution at low
frequencies and higher time resolution at higher frequencies.
Often, hydrocarbons show low frequencies, and thin beds can be resolved with enhanced time resolution at higher frequencies.
Initially all of these technologies were based on classical Fourier and Cohen transforms which gave the first steps to develop Specdecomp as we know it.
The output is a tuning cube, defined as a series of amplitude or phase maps tuned to specific frequencies. The resulting amplitude spectra can be used to calculate bed thicknesses in the time domain,
while phase spectra help to define lateral stratigraphic discontinuities. By examining the amplitude and phase maps at various frequencies (i.e., scrolling through the tuning cube), the interpreter
can identify subtle events and anomalies that are not readily visible in the post-stack, pre-stack.
Short Time Fourier Transform (STFT) is a subset of Discrete Fourier Transform, a widely used algorithm that uses a fixed window approach to spectral decomposition. The STFT method is to obtain a
frequency spectrum, but the STFT method is seriously limited by choosing a window length. A non-stationary signal, as seismic signals, is conceptually archived by STFT and it produces time-frequency
spectrums, which a time-frequency resolution is fixed by choice of time window. Selecting shorter window lengths can help resolve high frequency events and separate events with similar or closely
spaced dominant frequencies. The STFT shows that in low frequency, components are well resolved and choosing a shorter window length will compromise the frequency resolution to obtain higher time
resolution (Figure 2).
The Limitations of STFT include:
The STFT has a time-frequency resolution limitation.
The use of finite-length time domain moving windows over which the 1-D Fourier transforms are performed decreases its resolving capability.
Short windows can resolve high-f events, may overlook low-f events.
Longer windows ‘average’ the response, may overlook fine details.
However, the use of these shorter windows can overlook events at lower frequency and the interpretation could be compromised. The downside of this approach is that fine-scale events will not be
resolved if the window length is long enough.
Continuous Wavelet Transform (CWT) samples the seismic signal using a moving, scalable time window. In this method, the window size automatically changes with frequency and allows for adaptive
sampling of the seismic trace. The resulting spectral maps provide higher temporal resolution at higher frequencies at least better than STFT [4]. The CWT frequency gather shows that CWT is far
superior in preserving reflection events than the STFT method for higher frequencies. (i.e., preserves higher frequencies) At lower frequencies, however, CWT cannot adequately resolve events that are
closely-spaced in the time domain. This is considered a limitation on this method. The interpreter must be careful to choose the right algorithm for interpretation and needs.
The CWT samples wavelets using a moving, scalable time window and allows for adaptive sampling of the seismic trace. While it provides better frequency resolution at lower frequencies, CWT cannot
adequately resolve low frequency events that are closely-spaced in the time domain.
CWT compared with DFT - DFT window is fixed to do short Fourier transform, but CWT, the size of the window will change with frequency. This window can be long or short. CWT changes achieve higher
time resolution at higher frequency.
CWT compare with TFCWT - CWT generates a time-scale map and is then converted to time-scale map using the central frequency.
Some CWT considerations:
CWT - moving window, where window size automatically changes with frequencies, gives frequencies range (Figure 3).
CWT creates a map at an optimal frequency, but not exactly at 70 Hz - can be anywhere from 65-75 Hz. Compared to the DFT method, CWT is far superior in preserving reflection events (Figure 4).
Time Frequency Continuous Wavelet Transform (TFCWT) this method overcomes this issue by generating a time-frequency map that displays the exact frequency for any event, considered an advantage. CWT
and STFT methods, on the other hand, output maps at a central frequency within a given time window. Like CWT, the TFCWT spectral decomposition method uses a moving window approach, but it does not
average neighboring frequencies in the same way as previously mentioned methods. Therefore, TFCWT maps provide higher time-frequency resolution than STFT or CWT (even better than previous methods).
Both CWT and TFCWT methods provide high-frequency resolution at low frequencies and high temporal resolution at high frequencies. One disadvantage of TFCWT is that it is computationally intensive,
and generating spectral decomposition maps with this method can be time-consuming.
S-Transform ( ST ) generates a real time-frequency map and samples the seismic signal with a moving time window. However, the size of the window in the S-Transform method is frequency-dependent.
Because this transform has a more rigorous relationship with the spectra, it can produce Specdecomp maps with fairly high resolution. S Transform is faster to calculate than TFCWT but typically gives
similar results.
Like the TFCWT, the ST generates a real time-frequency map and samples the seismic signal with a moving time window. However, the size of the window in the ST method is frequency-dependent.
Because the transform has a more rigorous relationship with the spectra, it can produce spectral decomposition maps with fairly high resolution. An ST is faster to calculate than a TFCWT but can
produce similar results and the ST has high stability in noisy conditions [5].
The similarity between S-transform and STFT is that they are both derived from the Fourier transform of the time series multiplied by a time-shift window. However, unlike STFT, the standard deviation
in S-transform is actually a function of frequency. Consequently, the window function is also a function of time and frequency. As the width of the window is dictated by the frequency, it is apparent
that the window is wider in the time domain at lower frequencies, which means the window provides good localization in the frequency domain for low frequencies. Due to the low frequency spectrum of
surface wave, this aspect makes S-transform more appropriate for further analysis (Figure 5).
S-transform combines progressive resolution with referenced phase information. Therefore, it could estimate the local amplitude spectrum and the local phase spectrum. In addition, it is sampled at
the discrete Fourier transform frequencies. Also showing S-transform brings a better imagine than traditional transforms.
Channelized and Braided Fluvial Reservoirs
The hydrocarbon reservoirs typically take place where the porous sands pinch out against impermeable sands or shales. Otherwise, the channel sands can be difficult to differentiate from the adjacent
low-permeability strata because the lithology or type-rocks share similar P-wave impedances.
Braided reservoirs that host significant volumes of hydrocarbons almost always have high net/gross ratio, and are often considered to be relatively easy to characterize [1]. When more detail is
needed for the reservoir, spectral decomposition could be an innovating tool for this task. It can be applied directly to post-stack and pre-stack amplitude seismic data. However, the industry is
updating many technologies as ‘strata-grid’, (i.e., a package between two or more intervals, volume that has been extracted from the original seismic volume, mostly intervals of interest and stacking
into proportional slices).
Sometimes the reservoir can act completely differently from surrounding sand when viewed at discrete frequencies. It acts in a distinctively dynamic way relative to the background as the hydrocarbons
have changed the reflectivity of the reservoir.
Maps and seismic cross-section are critical to determine if the features you are seeing are geologically meaningful. When the interpreter is considering a stratigraphic feature that appears fan
geometry. At lower frequencies from the “Tuning Cube,” the feeder channel of the “fan” is highlighted. At higher frequencies, different lobes of the fan geometry are highlighted. At the highest
frequencies available in the seismic data, the thinnest areas are highlighted.
Petrophysical Feasibility
The petrophysical evaluation defines a sequence of channels. Some channels are compact, as identified by the resistivity, density, neutron, and sonic logs. Therefore, it is important to differentiate
the seismic attributes associated with compaction from those containing hydrocarbons
The impedance analysis in the reservoir zone identifies the separation by fluids and lithology. The impedance P versus impedance S plot, with effective porosity color coding, shows that shales can be
adequately separated from sands (Figure 6).
The VPVS versus density plot, colored by water saturation, identifies how, at a logging level, adequate separation of zones with hydrocarbon presence from water zones can be achieved. Therefore,
special processes must achieve a good lithological and fluid differentiation (Figure 7).
All spectral decomposition methods can provide better resolution of the channel morphology in comparison to the post-stack amplitude map, but each offered a different spectral response. Longer window
size results are not considered sometimes because structures of interest are only seen at higher frequencies. By sampling the seismic signal with a short window size, it is possible to avoid the
averaging effects inherent in the fixed window method as well as obtain better frequency resolution. Generally speaking, thin thickness events appear as high amplitudes at certain higher frequencies
in spectral decomposition maps. Therefore, it is possible to observe the spatial variation of the target by viewing spectral maps in succession within the tuning cubes. Sometimes, if channel or bed
thickness increases or decreases spatially, it can be observed by scrolling through the frequency slices.
It is necessary to mention that the conventional seismic attribute RMS or instantaneous amplitude is considered by many interpreters as a point of interest with hydrocarbon content, which is
considered very risky and successful in drilling. As a personal recommendation, calibration should be carried out with production of the results of any basic seismic attribute and tied with special
processes such as spectral decomposition or seismic inversion.
It is highly recommended to use either seismic pre-stack or seismic post-stack Non-Filter and Non-Gain if you are interested in extraction of conventional seismic attributes or in special processes
that are very sensitive to interpretation, a detailed interpretation must be carried out by choosing and keeping the reflector on the entire interpretation. When you use seismic with some type of
make-up or filter, you alter the true amplitudes, which can have catastrophic results in your proposals for new well locations, adding to the statistics of new dry wells (Figure 8).
In imaging the Glauconite channel sands, running multiple spectral decomposition methods helped to resolve the channel morphology and bed thickness relationships within the channel facies. While
TFCWT spectral decomposition provided the best resolution of all the methods, it also took the longest time to calculate. The S-Transform maps provided similar results to TFCWT and were faster to
generate, making the S-Transform method the most efficient technique for resolving the channels in this particular study. Spectral decomposition can greatly improve visualization and interpretation
workflows by revealing thin beds, lateral discontinuities and subtle anomalies not readily identified in post-stack seismic data. By correlating the spectral maps back to well logs and attribute
relationships, the technique can help the interpreter to better understand complex reservoirs and plan a drilling strategy with greater confidence and considerably reducing risk in success.
Some think that strong amplitude could be an indicator of hydrocarbons, while the other side suggests that amplitude is only a sand indicator. On the other hand it could show nothing about the
presence or absence of hydrocarbons. The interpreter must be careful with those indicators and integrate all available data, complementing these results with geophysical techniques such as seismic
acoustic inversion, seismic elastic inversion, production data, pressure data, geological data and seismic attributes. Combination of different attributes as an instantaneous amplitude and
instantaneous frequency could be used to enhance the continuity of the event. When two seismic events are very close together, it can be difficult to separate one from other on the basis of
amplitude; Instantaneous frequency can assist the interpreter determining their relationship between two reflectors.
It is important to consider that well logs show a clear separation of lithology and fluids, which allows for the application of special processes such as spectral decomposition (Specdecomp) or,
alternatively, simultaneous or elastic seismic inversion.
Always keep in mind that the objective of any interpretation and/or characterization of a reservoir with production, as well as G&G and productivity studies is to increase or improve hydrocarbon
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Mentor Texts for Comprehension That Work for ANY Classroom - Out of this World Literacy
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Here is a list of suggested mentor texts perfect for any classroom that focus on comprehension skills and strategies!
Character Traits
My New Friend is So Fun! by Mo Willems
Readers compare how they are alike and different from characters so they can make connections.
The Story of Ferdinand by Munro Leaf
The Other Side by Jacqueline Woodson
Lon Po Po by Ed Young
Readers describe how the main characters act as tension rises so they can compare how they would have acted in the story.
Fireboat: The Heroic Adventures of the John J. Harvey by Maira Kalman
The Man Who Walked Between the Towers by National Geographic Learning
Cause and Effect
The Giving Tree by Shel Silverstein
Readers use the words ‘if’ and ‘then’ to describe cause and effect so they can make connections between events in a story.
Those Shoes by Maribeth Boelts
One Plastic Bag by Alice McLerran
Finding Evidence
The Great Fire by Jim Murphy
Readers use evidence in the text to describe the mood so they can better understand how people in the text feel.
Harriet, You’ll Drive Me Wild! By Mem Fox
My Rotten Redheaded Older Brother by Patricia Polacco
Roxaboxen by Alice McLerran
Readers share their first thoughts about a book so they can think more deeply about each of their ideas
The Little House by Virginia Lee Burton
The Book of Bad Ideas by Laura Huliska-Beith
The Book Itch by Vaunda Micheaux Nelson (suggested for older grades)
Readers consider all the reasons the author wrote a book so they can determine a theme.
Fly Away Home by Eve Bunting
One Green Apple by Eve Bunting
Wangari’s Tree of Peace by Jeanette Winter
Readers identify parts that describe the theme so they can judge how well the writer described the theme.
Stick and Stone by Beth Ferry
That Book Woman by Heather Henson
The Invisible Boy by Patrice Barton
Readers ask questions about events in a story so they can think of other things that could have happened.
White Socks Only by Evelyn Coleman
The Three Questions by Jon J. Muth
The Adventures of Beekle the Unimaginary Friend by Dan Santat
Readers identify the main problem so they can think of possible solutions.
The Monster at the end of this Book by Jon Stone
Fire on the Mountain by John N Maclean
Trombone Shorty by Troy Andrews
Readers look for words and phrases that describe the setting so they can picture it in their minds.
Four Feet, Two Sandals by Karen Lynn Williams
Pigs by My Incredible World
Compare and Contrast
Tight Times by Barbara Shook Hazen
Readers identify the traits of characters in a book so they can compare how they are alike and different.
Always in Trouble by Corinne Demas
Let me Finish! By Minh Lê
What do you do with an Idea? By Kobi Yamada
Readers ask themselves questions so they can begin to understand how their mind’s think.
Mama Panya’s Pancakes by Mary Chamberlin
What Do You Do With a Problem? by Kobi Yamada
Main Idea and Details
Firefires by Mary R. Dunn
Readers identify main ideas so that they can think of key details the author could have included.
Max’s Words by Kate Banks
I Lost my Tooth in Africa by Baba Wague Diakite
The Most Magnificent Thing by Ashley Spires
Readers form pictures in their minds as they read so that they can better understand the text.
Crow Call by Louis Lowery
That Rabbit Belongs to Emily Brown by Cressida Cowell
The Lotus Seed by Sherry Garland
Readers identify the writer's best phrases or sentences so that they can analyze the writer’s style.
The Old Woman Who Named Things by Cynthia Rylant
The Best Beekeeper of Lalibela by Cristina Kessler
Emmanuel’s Dream: The True Story of Emmanuel Ofosu Yeboah by Laurie Ann Thompson
Readers judge the topics from the text so they can support their judgments with reasons.
Malala by Sarah J. Robbins
Wilma Jean the Worry Machine by Julia Cook
Background Knowledge
My Name is Sangoel by Karen Williams
Readers describe what they know about settings so they can use their background knowledge during reading.
The Relatives Came by Cynthia Rylant
My Librarian is a Camel by Margriet Ruurs
One Green Apple by Eve Bunting
Readers identify evidence in the text and what they already know so they can make inferences.
The Watering Hole by Jane and Christopher Kurtz
Tuesday by David Wiesner
Queen of the Falls by Chris Van Allsburg
Readers summarize their opinions of a text so they can retell what they think of a text.
Ralph Tells a Story by Abby Hanlon
The Librarian of Basra by Jeanette Winter
Making Connections
What Does it Mean to Be Present? By Rana DiOrio
Readers identify strong phrases in a text so that they can make connections to the text.
Unlovable by Dan Yaccarino
Stephanie’s Ponytail by Robert Munsch
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Happy teaching!
Jen & Hannah
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Mentor Texts for Comprehension That Work for ANY Classroom – Out of this World Literacy
Mentor Texts for Comprehension That Work for ANY Classroom – Out of this World Literacy
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Mentor Texts for Comprehension That Work for ANY Classroom – Out of this World Literacy
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Mentor Texts for Comprehension That Work for ANY Classroom – Out of this World Literacy
Mentor Texts for Comprehension That Work for ANY Classroom – Out of this World Literacy
Mentor Texts for Comprehension That Work for ANY Classroom – Out of this World Literacy
Mentor Texts for Comprehension That Work for ANY Classroom – Out of this World Literacy
Mentor Texts for Comprehension That Work for ANY Classroom – Out of this World Literacy
Mentor Texts for Comprehension That Work for ANY Classroom – Out of this World Literacy
Mentor Texts for Comprehension That Work for ANY Classroom – Out of this World Literacy
Mentor Texts for Comprehension That Work for ANY Classroom – Out of this World Literacy
Mentor Texts for Comprehension That Work for ANY Classroom – Out of this World Literacy
Mentor Texts for Comprehension That Work for ANY Classroom – Out of this World Literacy
Mentor Texts for Comprehension That Work for ANY Classroom – Out of this World Literacy
Mentor Texts for Comprehension That Work for ANY Classroom – Out of this World Literacy
Mentor Texts for Comprehension That Work for ANY Classroom – Out of this World Literacy
Mentor Texts for Comprehension That Work for ANY Classroom – Out of this World Literacy
Mentor Texts for Comprehension That Work for ANY Classroom – Out of this World Literacy
Mentor Texts for Comprehension That Work for ANY Classroom – Out of this World Literacy
Mentor Texts for Comprehension That Work for ANY Classroom – Out of this World Literacy
Mentor Texts for Comprehension That Work for ANY Classroom – Out of this World Literacy
Mentor Texts for Comprehension That Work for ANY Classroom – Out of this World Literacy
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Mentor Texts for Comprehension That Work for ANY Classroom – Out of this World Literacy
Mentor Texts for Comprehension That Work for ANY Classroom – Out of this World Literacy
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Beyond spectral gap : The role of the topology in decentralized learning
Jan 05, 2023
In data-parallel optimization of machine learning models, workers collaborate to improve their estimates of the model: more accurate gradients allow them to use larger learning rates and optimize
faster. In the decentralized setting, in which workers communicate over a sparse graph, current theory fails to capture important aspects of real-world behavior. First, the `spectral gap' of the
communication graph is not predictive of its empirical performance in (deep) learning. Second, current theory does not explain that collaboration enables larger learning rates than training alone. In
fact, it prescribes smaller learning rates, which further decrease as graphs become larger, failing to explain convergence dynamics in infinite graphs. This paper aims to paint an accurate picture of
sparsely-connected distributed optimization. We quantify how the graph topology influences convergence in a quadratic toy problem and provide theoretical results for general smooth and (strongly)
convex objectives. Our theory matches empirical observations in deep learning, and accurately describes the relative merits of different graph topologies. This paper is an extension of the conference
paper by Vogels et. al. (2022). Code: https://github.com/epfml/topology-in-decentralized-learning.
* Extended version of the other paper (with the same name), that includes (among other things) theory for the heterogeneous case. arXiv admin note: substantial text overlap with arXiv:2206.03093
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6-21 Draw The Shear And Moment Diagrams For The Beam
6-21 Draw The Shear And Moment Diagrams For The Beam - 200 lb ft b x 4 ft 4 ft 150 lb/ft 6 ft 200 lb ft a ans. Web our calculator generates the reactions, shear force diagrams (sfd), bending moment
diagrams (bmd), deflection, and stress of a cantilever beam or simply supported beam. Web a diagram showing the variation of the shear force along a beam is called the shear force diagram. Draw the
shear and bending moment diagrams for the beam. The bending moment at a section of a beam can be determined by summing up the moment.
Web write equations for the shear v and bending moment m for any section of the beam in the interval ab. Web draw the shear force and bending moment diagrams for the beam shown in the figure, when
dimensions and loadings of the beam get values a=1.0 m,b=1 m,c=3.2 m,d=0.8 m,f=16 kn,p=12 kn and q=23kn//m. Draw the shear and moment diagrams for the beam and determine the shear and moment in the
beam as functions of x, where 4 ft < x < 10 ft. Web draw the shear and moment diagrams for the simply supported beam. This is an example problem that will show you how to graphically draw a shear and
moment diagram for a beam. The vertical support reaction at a on. Consider a section bc and at a distance x from point a.
Solved 621. Draw the shear and moment diagrams for the beam
Web to design a beam, it is essential to determine the maximum shear and moment in the structure. Web draw the shear force and bending moment diagrams for the beam shown in the figure, when.
Solved Draw the Shear and Moment Diagram for the beam shown
Web shear force and bending moment diagrams are analytical tools used in conjunction with structural analysis to help perform structural design by determining the value of shear forces and bending
moments at a given point.
Solved Draw the shear and moment diagrams for the beam.
Web draw the shear force and bending moment diagrams for the beam shown in the figure, when dimensions and loadings of the beam get values a=1.0 m,b=1 m,c=3.2 m,d=0.8 m,f=16 kn,p=12 kn and q=23kn//m.
Solved Draw the shear and moment diagrams for the beam.
Shear and moment diagrams are graphical representations of the variation of shear force and bending moment along the length of a structural element such as a beam. You'll get a detailed solution from
a subject.
draw the shear and moment diagrams for the beam chegg
You'll get a detailed solution from a subject matter expert that helps you learn core concepts. Also, draw shear and moment diagrams, specifying values at all change of loading positions and at
points of zero.
Brief Information About Shear Force And Bending Moment Diagrams
This is an example problem that will show you how to graphically draw a shear and moment diagram for a beam. Draw the shear and moment diagrams for the beam, and determine the shear and.
Solved Draw the shear and moment diagrams for the beam
We are asked the shear and bending moment diagrams for the beam. Shear and moment diagrams are graphical representations of the variation of shear force and bending moment along the length of a
structural element.
Drawing Shear and Moment Diagrams for Beam YouTube
The reactions shown on the diagram are determined from equilibrium equations as. 200 lb ft b x 4 ft 4 ft 150 lb/ft 6 ft 200 lb ft a ans. Draw the shear and moment.
Learn How To Draw Shear Force And Bending Moment Diagrams Engineering
The vertical support reaction at a on. Web draw the shear and moment diagrams for the beam. Use both the mathematical convention and the engineering convention for the bending moment diagram. One of
the ways.
Solved Draw the shear and moment diagrams for the beam, and
The reactions shown on the diagram are determined from equilibrium equations as. Web write equations for the shear v and bending moment m for any section of the beam in the interval ab. Draw the.
6-21 Draw The Shear And Moment Diagrams For The Beam Firstly calculating reactions at support. The total load on the beam r a + r b = 2 × 5 + 2 × 5 = 20 k i p. We are asked the shear and bending
moment diagrams for the beam. Write shear and moment equations for the beams in the following problems. Draw the shear and moment diagrams for the beam and determine the shear and moment in the beam
as functions of x, where 4 ft < x < 10 ft.
6-21 Draw The Shear And Moment Diagrams For The Beam Related Post :
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--- _id: '11679' abstract: - lang: eng text: "We are given a set T = {T 1 ,T 2 , . . .,T k } of rooted binary trees, each T i leaf-labeled by a subset L(Ti)⊂{1,2,...,n} . If T is a tree on {1,2, . .
.,n }, we let T|L denote the minimal subtree of T induced by the nodes of L and all their ancestors. The consensus tree problem asks whether there exists a tree T * such that, for every i , T∗|L(Ti)
is homeomorphic to T i .\r\n\r\nWe present algorithms which test if a given set of trees has a consensus tree and if so, construct one. The deterministic algorithm takes time min{O(N n 1/2 ), O(N+ n
2 log n )}, where N=∑i|Ti| , and uses linear space. The randomized algorithm takes time O(N log3 n) and uses linear space. The previous best for this problem was a 1981 O(Nn) algorithm by Aho et al.
Our faster deterministic algorithm uses a new efficient algorithm for the following interesting dynamic graph problem: Given a graph G with n nodes and m edges and a sequence of b batches of one or
more edge deletions, then, after each batch, either find a new component that has just been created or determine that there is no such component. For this problem, we have a simple algorithm with
running time O(n 2 log n + b 0 min{n 2 , m log n }), where b 0 is the number of batches which do not result in a new component. For our particular application, b0≤1 . If all edges are deleted, then
the best previously known deterministic algorithm requires time O(mn−−√) to solve this problem. We also present two applications of these consensus tree algorithms which solve other problems in
computational evolutionary biology." article_processing_charge: No article_type: original author: - first_name: Monika H full_name: Henzinger, Monika H id: 540c9bbd-f2de-11ec-812d-d04a5be85630
last_name: Henzinger orcid: 0000-0002-5008-6530 - first_name: V. full_name: King, V. last_name: King - first_name: T. full_name: Warnow, T. last_name: Warnow citation: ama: Henzinger M, King V,
Warnow T. Constructing a tree from homeomorphic subtrees, with applications to computational evolutionary biology. Algorithmica. 1999;24:1-13. doi:10.1007/pl00009268 apa: Henzinger, M., King, V., &
Warnow, T. (1999). Constructing a tree from homeomorphic subtrees, with applications to computational evolutionary biology. Algorithmica. Springer Nature. https://doi.org/10.1007/pl00009268 chicago:
Henzinger, Monika, V. King, and T. Warnow. “Constructing a Tree from Homeomorphic Subtrees, with Applications to Computational Evolutionary Biology.” Algorithmica. Springer Nature, 1999. https://
doi.org/10.1007/pl00009268. ieee: M. Henzinger, V. King, and T. Warnow, “Constructing a tree from homeomorphic subtrees, with applications to computational evolutionary biology,” Algorithmica, vol.
24. Springer Nature, pp. 1–13, 1999. ista: Henzinger M, King V, Warnow T. 1999. Constructing a tree from homeomorphic subtrees, with applications to computational evolutionary biology. Algorithmica.
24, 1–13. mla: Henzinger, Monika, et al. “Constructing a Tree from Homeomorphic Subtrees, with Applications to Computational Evolutionary Biology.” Algorithmica, vol. 24, Springer Nature, 1999, pp.
1–13, doi:10.1007/pl00009268. short: M. Henzinger, V. King, T. Warnow, Algorithmica 24 (1999) 1–13. date_created: 2022-07-27T15:02:28Z date_published: 1999-05-01T00:00:00Z date_updated:
2024-11-04T11:41:23Z day: '01' doi: 10.1007/pl00009268 extern: '1' intvolume: ' 24' keyword: - Algorithms - Data structures - Evolutionary biology - Theory of databases language: - iso: eng month:
'05' oa_version: None page: 1-13 publication: Algorithmica publication_identifier: eissn: - 1432-0541 issn: - 0178-4617 publication_status: published publisher: Springer Nature quality_controlled:
'1' related_material: record: - id: '11927' relation: earlier_version status: public scopus_import: '1' status: public title: Constructing a tree from homeomorphic subtrees, with applications to
computational evolutionary biology type: journal_article user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87 volume: 24 year: '1999' ...
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Factor Trees - Math Steps, Examples & Questions
Write the solution as a multiplication equation (and if necessary in exponential form).
64=2 \times 2 \times 2 \times 2 \times 2 \times 2
Written in exponential form:
This is an example of relying on the number being even which generates a very long, and time consuming factor tree. This is not incorrect, it just takes a lot of time to get to the solution with the
potential to make a simple mistake.
Here are two alternatives to the same factor tree:
Version 1
Version 2
Can factor trees involve fractions and decimals?
No, a factor is a number that divides into another number with no left overs, so they involve only whole numbers.
What is prime factorization used for?
It can be used to find the greatest common factor (GCF), the least common multiple (LCM) and other numerical properties such as whether the number is square, cube or prime.
How do you know when a factor tree is complete?
A factor tree is complete when all the numbers at the bottom level are prime numbers. A factor tree is a visual representation of the prime factorization of a number, where you continue to break down
the number into its prime factors until you reach only prime numbers.
Can you find factor trees on a calculator?
While some calculators may have built-in functions for finding factor trees, not all calculators provide this feature. Basic calculators focus on arithmetic operations and do not include functions
for factorization. However, if you have access to a graphing calculator or a scientific calculator, there might be applications available that can help you find factor trees or prime factorizations.
2^{2} \times 5 \times 7
2 \times 5 \times 14
10 \times 14
1 \times 2^{2} \times 5 \times 7
2 \times 5 \times 33
2 \times 3^{2} \times 11
2^{3} \times 3^{2} \times 5^{3}
2 \times 3 \times 5 \times 11
a=2, b=1, c=21
a=2, b=3, c=7
a=2, b=3, c=18
a=2, b=5, c=4
196 should be split into 2 and 98
14 should be split into 2 and 7
7 should be split into 1 and 7
There is no mistake – the solution is correct
35 \times a \times a \times b
5 \times 7 \times a \times b
5 \times 7 \times a \times a \times b \times b
5 \times 7 \times a \times a \times b
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CSE 417, Wi '05: Assignment #5: Random Graph Generator
Here is the random graph generator for use in Assignment #5:
here. It is a command-line program, and takes four parameters, only the first of which is required: the number of vertices.
The generator outputs lines containing integers. The first line contains the number of vertices in the graph; each successive line will consist of a pair of numbers, representing an edge between
those two vertices. To save random graph data, redirect the output to a file, like this:
rndgraph 16 > graph16a
rndgraph 16 > graph16b
rndgraph 35 > graph35
The file graph16a will then contain random graph data for a 16 node graph, graph16b will contain a different 16 node graph, and graph35 will hold a 35 node graph. Alternatively, you may be able to
"pipe" its output directly into your program:
rndgraph 16 | mybiconnectedcomponentsmasterpiece
rndgraph 16 | mybiconnectedcomponentsmasterpiece
will run your program on two different 16 node graphs.
The optional parameters provide additional control over the generated graphs, which may be useful for your debugging and for your timing study. In more detail, the 4 parameters are:
1. n: Number of vertices. Integer ≥ 1. Required.
2. d: The expected degree of the graph (average number of edges connected to each vertex). Integer ≥ 0. Optional; if zero or omitted, defaults to 5. You may want to set it to a smaller number for
initial debugging, and you definitely need to set it to a range of larger values, up to about n, for your timing study.
3. seed: Seed for the pseudo-random number generator. Integer between 1 and 2^32. Optional; if zero or omitted, defaults to the system time-of-day clock. Computers (hopefully!) never generate random
numbers, but they can generate sequences that appear random for most practical purposes. You get a different sequence for each seed, and successive numbers in the sequence will appear unrelated,
but if you restart with the same seed, you'll get the same sequence. Why does it matter? Debugging! If your program is crashing on some rare graph, it is important to be able to regenerate
exactly the same graph, which is practically impossible with the default. After you get your program debugged, using the default is a great way to generate lots of different examples for your
timing study, but until then I strongly recommend that you use explicit seeds, e.g.:
rndgraph 16 0 42 | mybiconnectedcomponentsmasterpiece
rndgraph 16 0 42 | mybiconnectedcomponentsmasterpiece
rndgraph 16 0 0 | mybiconnectedcomponentsmasterpiece
rndgraph 16 0 0 | mybiconnectedcomponentsmasterpiece
will run your program on the same 16 node graph twice, then on two different (and basically unrepeatable) ones.
4. shuffle: A flag indicating whether vertex numbers should be randomized. Integer 0 or 1. Optional; if omitted, defaults to 1 (meaning "Yes, randomize"). If 0, the generator uses consecutive
numbers for most vertices in the same biconnected component. If 1, node numbers are (pseudo-) randomly assigned. Using 0 may be convenient for your initial debugging, but your later testing and
timing runs should use 1 (the default).
Thus, the following command:
rndgraph 99 50 42 1 > savegraph-99-50-42-1
will generate (and save to a file) a graph with 99 vertices and average degree of 50, with shuffled node numbers, all based on the specific pseudo-random sequence with seed 42. The sample graph shown
near the top of this page was produced by
rndgraph 8 2 3 0
8 nodes, degree about 2, no label shuffle, seed 3. Try it; you should get the same graph with those parameters, namely:
Computer Science & Engineering
University of Washington
Box 352350
Seattle, WA 98195-2350
(206) 543-1695 voice, (206) 543-2969 FAX
[comments to cse417-webmaster@cs.washington.edu]
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class astropy.coordinates.Distance(value=None, unit=None, z=None, cosmology=None, distmod=None, parallax=None, dtype=<class 'numpy.inexact'>, copy=True, order=None, subok=False, ndmin=0,
Bases: SpecificTypeQuantity
A one-dimensional distance.
This can be initialized by providing one of the following:
□ Distance value (array or float) and a unit
□ Quantity object with dimensionality of length
□ Redshift and (optionally) a Cosmology
□ Distance modulus
□ Parallax
valuescalar or Quantity [:ref: ‘length’]
The value of this distance.
unitUnitBase [:ref: ‘length’]
The unit for this distance.
A redshift for this distance. It will be converted to a distance by computing the luminosity distance for this redshift given the cosmology specified by cosmology. Must be given as a
keyword argument.
cosmologyCosmology or None
A cosmology that will be used to compute the distance from z. If None, the current cosmology will be used (see astropy.cosmology for details).
distmodfloat or Quantity
The distance modulus for this distance. Note that if unit is not provided, a guess will be made at the unit between AU, pc, kpc, and Mpc.
The parallax in angular units.
dtypedtype, optional
See Quantity.
copybool, optional
See Quantity.
order{‘C’, ‘F’, ‘A’}, optional
See Quantity.
subokbool, optional
See Quantity.
ndminint, optional
See Quantity.
allow_negativebool, optional
Whether to allow negative distances (which are possible in some cosmologies). Default: False.
If the unit is not a length unit.
If value specified is less than 0 and allow_negative=False.
If cosmology is provided when z is not given.
If either none or more than one of value, z, distmod, or parallax were given.
>>> from astropy import units as u
>>> from astropy.cosmology import WMAP5
>>> Distance(10, u.Mpc)
<Distance 10. Mpc>
>>> Distance(40*u.pc, unit=u.kpc)
<Distance 0.04 kpc>
>>> Distance(z=0.23)
<Distance 1184.01657566 Mpc>
>>> Distance(z=0.23, cosmology=WMAP5)
<Distance 1147.78831918 Mpc>
>>> Distance(distmod=24.47*u.mag)
<Distance 783.42964277 kpc>
>>> Distance(parallax=21.34*u.mas)
<Distance 46.86035614 pc>
Attributes Summary
distmod The distance modulus as a Quantity.
parallax The parallax angle as an Angle object.
z Short for self.compute_z().
Methods Summary
compute_z([cosmology]) The redshift for this distance assuming its physical distance is a luminosity distance.
Attributes Documentation
The distance modulus as a Quantity.
The parallax angle as an Angle object.
Short for self.compute_z().
Methods Documentation
compute_z(cosmology=None, **atzkw)[source]#
The redshift for this distance assuming its physical distance is a luminosity distance.
cosmologyCosmology or None
The cosmology to assume for this calculation, or None to use the current cosmology (see astropy.cosmology for details).
keyword arguments for z_at_value()
The redshift of this distance given the provided cosmology.
This method can be slow for large arrays. The redshift is determined using astropy.cosmology.z_at_value(), which handles vector inputs (e.g. an array of distances) by element-wise calling of
scipy.optimize.minimize_scalar(). For faster results consider using an interpolation table; astropy.cosmology.z_at_value() provides details.
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DATEVALUE Function
Convert a date in text format to a valid date
Return value
A valid Excel time as a serial number
• date_text - A valid date in text format.
How to use
Sometimes, dates in Excel appear as text values that are not recognized as proper dates. The DATEVALUE function is meant to convert a date represented as a text string into a valid Excel date. Proper
Excel dates are more useful than text dates since they can be formatted as a date, and directly manipulated with other formulas.
The DATEVALUE function takes just one argument, called date_text. If date_text is a cell address, the value of the cell must be text. If date_text is entered directly into the formula it must be
enclosed in quotes.
To illustrate how the DATEVALUE function works, the formula below shows how the text "3/10/1975" is converted to the date serial number 27463 by DATEVALUE:
=DATEVALUE("3/10/1975") // returns 27463
Note that DATEVALUE returns a serial number, 27463, which represents March 10, 1975 in Excel's date system. A date number format must be applied to display this number as a date.
In the example shown, column B contains dates entered as text values, except for B15, which contains a valid date. The formula in C5, copied down, is:
Column C shows the number returned by DATEVALUE, and column D shows the same number formatted as a date. Notice that Excel makes certain assumptions about missing day and year values. Missing days
become the number 1, and the current year is used if there is no year value available.
Alternative formula
Notice that the DATEVALUE formula in C15 fails with a #VALUE! error, because cell B15 already contains a valid date. This is a limitation of the DATEVALUE function. If you have a mix of valid and
invalid dates, you can try the simple formula below as an alternative:
The math operation of adding zero will cause Excel will try to coerce the value in A1 to a number. If Excel can parse the text into a proper date it will return a valid date serial number. If the
date is already a valid Excel date (i.e. a serial number), adding zero will have no effect, and generate no error.
• DATEVALUE will return a #VALUE error if date_text refers does not contain a date formatted as text.
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What is float Keyword in Java
The float keyword is a data type in Java, which is used for floating point values. The range for float data type is 3.4e-038 to 3.4e+038. This range is in scientific notation, the notation value
3.4e+038 is equivalent to 34 followed by 37 zeroes (see the example at the end of this post to learn how this notation works).
public class JavaExample {
public static void main(String[] args) {
float num = 45.789f; //float value ends with f
Note: For larger values than the higher range of float and lower values than the lower range of float, use the double data type.
Example of float keyword
Let’s see how scientific notation works. This will help you understand the range of float data type, which is mentioned in this notation at the starting of this article.
public class JavaExample {
public static void main(String[] args) {
//scientific notation values
float num = 15.55e+03f;
float num2 = 15.55e-03f;
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Calculate Quartiles in Python
Quartiles are values that divide a dataset into four equal parts, each containing 25% of the data. Quartiles are useful for understanding the spread and distribution of a dataset.
In general, there are three quartiles used. Q1 (first quartile), Q2 (second quartile), and Q3 (third quartile) are the values below which 25%, 50%, and 75% of the data fall.
In Python, the quartiles can be calculated using the quantile() function from the NumPy and pandas package.
The general syntax of quantile() looks like this:
# calculate quartiles using using NumPy
import numpy as np
np.quantile(x, [0.25, 0.5, 0.75])
# calculate quartiles using pandas
import pandas as pd
df['col_name'].quantile([0.25, 0.5, 0.75])
Where, x is the dataset in array format and the second array is the probability for the quantiles to compute.
The following examples explain how to use the quantile() function from NumPy and pandas to calculate quartiles
Example 1: calculate quartiles using quantile() from NumPy
Suppose, you have a following dataset,
x = [5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60]
Calculate the quartiles using quantile() function from NumPy,
# import package
import numpy as np
# calculate quartiles
np.quantile(x, [0.25, 0.5, 0.75])
# output
array([18.75, 32.5 , 46.25])
The Q1, Q2, and Q3 quartile values are 18.75, 32.5, and 46.25, respectively.
Example 2: calculate quartiles using quantile() from pandas
Suppose, you have the following pandas DataFrame,
# import package
import pandas as pd
# create random pandas DataFrame
df = pd.DataFrame({'col1': ['A', 'B', 'C', 'D', 'E', 'F', 'G', 'H', 'I', 'J', 'K', 'L'],
'col2': [5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60]})
# view first few rows
col1 col2
0 A 5
1 B 10
# calculate quartiles
df['col1'].quantile([0.25, 0.5, 0.75])
# output
df['col2'].quantile([0.25, 0.5, 0.75])
0.25 18.75
0.50 32.50
0.75 46.25
Name: col2, dtype: float64
The output shows that Q1, Q2, and Q3 quartile values are 18.75, 32.5 , and 46.25, respectively.
Enhance your skills with courses Python
This work is licensed under a Creative Commons Attribution 4.0 International License
Some of the links on this page may be affiliate links, which means we may get an affiliate commission on a valid purchase. The retailer will pay the commission at no additional cost to you.
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Unlocking the World of Possibility: Understanding Combinatorics in Mathematics
Author: admintanbourit
Combinatorics is a branch of mathematics that deals with counting and organizing objects in a systematic way. It may sound simple, but its real-life applications are vast and complex. From finding
the number of ways to arrange a deck of cards to designing efficient algorithms for computer science, combinatorics plays a crucial role in unlocking the world of possibility.
At its core, combinatorics is about studying combinations and permutations, which are different ways of selecting and arranging objects. In combinatorics, the order and repetition of the objects are
key elements to consider. Let’s dive deeper into these concepts and understand their significance in mathematics.
Combinations refer to the different ways of selecting a subset of objects from a larger set without considering the order in which they are arranged. For example, if you have five different fruits
and you want to choose three of them, the number of possible combinations would be 10 (5 choose 3). This is because the order in which the fruits are selected does not matter. The combinations are
represented by the mathematical formula nCr, where n is the total number of objects and r is the number of objects to be selected.
Permutations, on the other hand, refer to the different ways of arranging objects in a specific order. Continuing with the fruit analogy, if you have five different fruits and want to arrange them in
a specific order, the number of possible permutations would be 60 (5 permute 3). This is because the order in which the fruits are arranged matters. The permutations are represented by the
mathematical formula nPr, where n is the total number of objects and r is the number of objects to be arranged.
Now, you may wonder, what is the relevance of studying combinations and permutations? Well, these concepts have real-life applications in various fields such as probability, statistics, computer
science, and cryptography.
In probability and statistics, combinatorics is used to calculate the likelihood of an event occurring. For example, in a game of poker, the probability of getting a full house hand (three of a kind
and a pair) can be calculated using combinatorics. We can also use combinatorics to analyze data and make predictions in various industries, such as finance and marketing.
In computer science, combinatorics is used to design efficient algorithms for tasks like sorting and searching. For instance, in a music streaming app, if a user has a playlist with 50 songs and they
want to play the songs in a random order, the app must have a way to generate all the possible permutations quickly. Combinatorics also plays a crucial role in cryptography, where it is used to
design secure encryption algorithms.
Moreover, combinatorics has practical applications in our day-to-day life. For instance, when designing a seating plan for an event, we must consider the number of guests and the different ways to
arrange the tables and chairs. This is where combinatorics comes in, helping us find the most efficient and organized seating arrangement.
In conclusion, combinatorics is a powerful tool used in various fields to unlock possibilities and make informed decisions. Its study may seem intimidating at first, but once you understand the
basics, you’ll realize its significance in solving real-world problems. So next time you’re arranging a deck of cards or organizing your to-do list, remember that you’re using combinatorics to make
it happen!
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Securitizable: - online calculation, formula
A company needs to save 90,000 Euros in 5 years for new equipment. The bank offers a 4% interest rate if we regularly deposit the same amount each year. How many Euros do we need to deposit at least
to obtain these funds?
So we substitute into the formula above
annuity=90,000 * ((0.04/(1+0.04) to the power of 5)-1)
annuity= 90,000 * ((0.04/(1.04) to the power of 5 -1)
annuity=16616.44 Euros
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Meena went to a bank to withdraw Rs.2000. She asked the cashier to give her Rs.50 and Rs.100 notes only. Meena got 25 notes in all. Find how many notes of Rs.50 and Rs.100 she received.
Hint: The question is related to the linear equation in two variables. Try to make two equations using the information given in the problem statement and solve them simultaneously.
Complete step-by-step answer:
Complete step-by-step answer: In the question, it is given that Meena withdrew \[Rs.2000\] from a bank in the form of notes of $Rs.50$ and $Rs.100$. It is also given that she got $25$ notes in all.
So, we will consider $x$ as the number of $Rs.50$ notes received by Meena and $y$ as the number of $Rs.100$ notes received by Meena.
Now, in the first case, it is given that Meena withdrew \[Rs.2000\] from a bank in the form of notes of $Rs.50$ and $Rs.100$. So, the amount in the form of $Rs.50$ notes is equal to $50\times x=
Rs.50x$. Also, the amount in the form of $Rs.100$ notes is equal to $100\times y=Rs.100y$. So, the total amount will be $Rs.\left( 50x+100y \right)$. But it is given that the total amount withdrawn
is \[Rs.2000\]. So,
Now, we have considered $x$ as the number of $Rs.50$ notes received by Meena and $y$ as the number of $Rs.100$ notes received by Meena. So, the total number of notes will be equal to $x+y$. But it is
given that the total number of notes is equal to $25$. So,
Now, we will solve the linear equations to find the values of $x$ and $y$.
From equation$(ii)$, we have $x+y=25$
$\Rightarrow y=25-x$
On substituting $y=25-x$ in equation$(i)$, we get
$50x+100\left( 25-x \right)=2000$
$\Rightarrow 50x+2500-100x=2000$
$\Rightarrow 500-50x=0$
$\Rightarrow 50x=500$
$\Rightarrow x=10$
Now, substituting \[x=10\] in equation$(ii)$, we get
\[\Rightarrow y=15\]
Hence, the numbers of $Rs.50$ notes and \[Rs.100\] notes that are received by Meena from the cashier are $10$ and $15$ respectively.
Note: While solving this question we can assume the number of RS.50 notes as x and RS.100 notes as (25-x). By this substitution we get a linear equation in one variable. We can find the value of x by
solving the linear equation in one variable.
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The Stacks project
Lemma 101.37.6. Let $\mathcal{Z}$ be an algebraic stack. Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks over $\mathcal{Z}$. If $\mathcal{X}$ is universally closed over $\
mathcal{Z}$ and $f$ is surjective then $\mathcal{Y}$ is universally closed over $\mathcal{Z}$. In particular, if also $\mathcal{Y}$ is separated and of finite type over $\mathcal{Z}$, then $\mathcal
{Y}$ is proper over $\mathcal{Z}$.
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1.1 Characteristics Table | Sentinel Routine Querying Reporting Tool
1.1 Characteristics Table
Characteristics Table output by the QRP Reporting Tool is highly customizable. Using parameters in Baseline File, users can specify what will be included and the way in which it will be displayed in
the final output file.
Characteristics Tables include patient and demographic characteristics, with options to add health characteristics, previous medical product use, laboratory characteristics, and health care and drug
utilization information. Means and standard deviations are output for continuous variables while numbers and percentages of episodes are output for categorical variables.
If the Query Request Program contains more than one monitoring period, then there will be one table produced per monitoring period. Additionally, users can specify whether to stratify characteristics
tables by data partner using Create Report File.
When Characteristics Tables are stratified by data partner (DP), there is no need for aggregation of descriptive statistics. However, when multiple data sources are aggregated into one
Characteristics Table, aggregated descriptive statistics are calculated as described below.
1.1.1 Calculation of Unweighted Aggregated Descriptive Statistics
Suppose there are two groups, \(t\) and \(c\), and \(k=1,2,\ldots,K\) DPs. Then, we let \(n_{t,k}\) denote the DP-specific sample size for group \(t\), \(\overline{x}_{t,k}\) denote the DP-specific
means for group \(t\), \(s_{t,k}\) denote the DP-specific standard deviations for group \(t\), \(\hat{p}_{t,k}=\overline{x}_t\), and \(k\) denote the proportion of the group \(t\) with a given
The aggregated sample mean, \(\overline{x}_t\), is calculated as:
The aggregated standard deviation, \(s_t\), is calculated as a weighted average of the DP-specific standard deviations:
\[$$s_t=\frac{\sum_{k=1}^{K}{\left(n_{t,k}-1\right)s_{t,k}}}{\sum_{k=1}^{K}n_{t,k}-K} \tag{1.2}$$\]
The aggregated proportion of patients with a given characteristic, \({\hat{p}}_t\), is calculated as:
\[$${\hat{p}}_t=\overline{x}_t=\frac{\sum_{k=1}^{K}{n_{t,k}{\hat{p}}_{t,k}}}{\sum_{k=1}^{K}n_{t,k}} \tag{1.3}$$\]
\(\overline{x}_c\), \(s_c\), \(\hat{p}_c\) will be calculated similarly.
The aggregated standardized difference for continuous variables between groups, \(d_{cont}\), is calculated as:
\[$$d_{cont}=\frac{(\overline{x}_{t,k}-\overline{x}_{c,k})}{\sqrt{\frac{s_{t,k}^2+s_{c,k}^2}{2}}} \tag{1.4}$$\]
The aggregated standardized difference for categorical variables between groups, \(d_{cat}\), is calculated as:
\[$$d_{cat}=\frac{({\hat{p}}_{t,k}-{\hat{p}}_{c,k})}{\sqrt{\frac{{\hat{p}}_{t,k}\left(1-{\hat{p}}_{t,k}\right)+{\hat{p}}_{c,k}\left(1-{\hat{p}}_{c,k}\right)}{2}}} \tag{1.5}$$\]
1.1.2 Calculation of Weighted Aggregated Descriptive Statistics
When analyses have weights applied, the weighted aggregated standardized difference is calculated analogously to the unweighted aggregated standardized difference, except that instead of using the
DP-specific means, DP-specific weighted means are used, and instead of DP-specific variances, DP-specific weighted variances are used. In addition to the notation used above, let \(w_i\) denote the
weight for each individual observation in \(i=1,2,\ldots,n\).
Thus, the weighted means, \(\overline{x}_t^w\) and \(\overline{x}_c^w\), are calculated as:
\[$$\overline{x}_t^w=\frac{\sum_{k=1}^{K}{\left(\sum_{i=1}^{n_{t,k}}w_{i,k}\right)\overline{x}_{t,k}^w}}{\sum_{k=1}^{K}\left(\sum_{i=1}^{n_{t,k}}w_{i,k}\right)} \tag{1.6}$$\] \[$$\overline{x}_c^w=\
frac{\sum_{k=1}^{K}{\left(\sum_{j=1}^{n_{c,k}}w_{j,k}\right)\overline{x}_{t,k}^w}}{\sum_{k=1}^{K}\left(\sum_{j=1}^{n_{c,k}}w_{j,k}\right)} \tag{1.7}$$\]
DP-specific weighted variances, \(\left(s_t^w\right)^2\) and \(\left(s_c^w\right)^2\), are calculated as:
\[$$\left(s_t^w\right)^2= \frac{\sum_{k=1}^{K}v_k\left(s_{t,k}^w\right)^2}{\sum_{k=1}^{K}v_k} \text{ where } v_k= \frac{\left(\sum_{i=1}^{n_{t,k}}w_{i,k}\right)^2-\left(\sum_{i=1}^{n_{t,k}}w_{i,k}^2\
right)}{\left(\sum_{i=1}^{n_{t,k}}w_{i,k}\right)} \tag{1.8}$$\] \[$$\left(s_c^w\right)^2=\frac{\sum_{k=1}^{K}u_k\left(s_{c,k}^w\right)^2}{\sum_{k=1}^{K}u_k} \text{ where } u_k=\frac{\left(\sum_{j=1}^
{n_{c,k}}w_{j,k}\right)^2-\left(\sum_{j=1}^{n_{c,k}}w_{j,k}^2\right)}{\left(\sum_{j=1}^{n_{c,k}}w_{j,k}\right)} \tag{1.9}$$\]
The aggregated weighted standardized difference differs from the unweighted aggregated standardized differences in that site-specific sums of weights and sums of weights squared among treatment and
control groups are also required. The aggregated weighted standardized difference for continuous variables is calculated as follows:
\[$$d_{cont} = \frac{\sum_{k=1}^{K} \left( \sum_{i=1}^{n_{t,k}} w_{i,k} \right) \overline{x}_{t,k}^{w} - \sum_{k=1}^{K} \left( \sum_{j=1}^{n_{c,k}} w_{j,k} \right) \overline{x}_{t,k}^{w}}{ \sqrt{\
frac{ \frac{\sum_{k=1}^{K}v_k \left( s_{t,k}^{w} \right)^2 }{\sum_{k=1}^{K}v_k} + \frac{\sum_{k=1}^{K}u_k \left( s_{c,k}^{w} \right)^2 }{\sum_{k=1}^{K}u_k} }{2}} } \tag{1.10}$$\]
The aggregated weighted standardized difference for categorical variables between groups is calculated as follows:
\[$$d_{cat} = \frac{ \frac{\sum_{k=1}^{K} \left( \sum_{i=1}^{n_{t,k}} w_{i,k} \right) \overline{x}_{t,k}^{w}}{\sum_{k=1}^{K} \left( \sum_{i=1}^{n_{t,k}} w_{i,k} \right)} - \frac{\sum_{k=1}^{K} \left(
\sum_{j=1}^{n_{c,k}} w_{j,k} \right) \overline{x}_{t,k}^{w}}{\sum_{k=1}^{K} \left( \sum_{j=1}^{n_{c,k}} w_{j,k} \right)} }{ \sqrt{\frac{ \frac{\sum_{k=1}^{K}v_k \left( s_{t,k}^{w} \right)^2 }{\sum_{k
=1}^{K}v_k} + \frac{\sum_{k=1}^{K}u_k \left( s_{c,k}^{w} \right)^2 }{\sum_{k=1}^{K}u_k} }{2}} } \tag{1.11}$$\]
\(v_k\) and \(u_k\) are as defined for DP-specific weighted variances.
1.1.3 Output
Users can specify how many Characteristics Tables to output using Baseline File. By default, all Characteristics Tables are “Table 1”, and increment alphabetically from “a” through “z”.
1.1.3.1 Groups Output
Users are required to specify which cohorts, analytical groups, or mother-infant linkage groups to output in Characteristics Tables by setting the Baseline File GROUP parameter. Groups correspond to
columns in Characteristics Tables.
By default, unadjusted tables in Propensity Score (PS) analyses and unweighted tables in Inverse Probability of Treatment Weighting (IPTW) analyses are de-duplicated. In other words, if users specify
multiple analytical groups from the same Propensity Score Estimate Group (via qrp.PSEstimationFile), only 1 unadjusted or unweighted table will be output per PS Estimate Group.
In medical product use in pregnancy (Type 4) analyses, users can specify whether the non-pregnant cohort created in QRP should be included in the Characteristics table by setting the appropriate
parameter in Baseline File parameter.
In all descriptive analyses except switching analyses, users can also specify that up to two cohorts, analytical groups, or mother-infant linkage groups appear in the same table using Baseline File
parameter. By default, only one group will appear in each Characteristics Table, with the exception of Type 6 analyses which will have the initial cohort, patients with a first switch, and patients
with a second switch (if applicable) output in separate columns in the same table by default. By specifying to include a second switch group, users can include up to three groups in Type 6
Characteristics Tables (i.e., the initial cohort, the “first switch” group, and the “second switch” group). All inferential analyses will output the exposed and referent analytic groups together by
When two groups are included in a single Characteristics Table, users can choose to compare them using absolute and standardized differences. When users request this option in Baseline File, QRP
Report will output “Absolute Difference” and “Standardized Difference” columns in Characteristics Tables headed under “Characteristic Balance” in Descriptive (Level 1) analyses and under “Covariate
Balance” in Inferential (Level 2) analyses. Groups cannot be compared in Type 6 “Switching” analyses. In inferential analyses, groups are compared by default.
Users can set a standardized difference threshold above which groups are considered to be meaningfully different using the appropriate parameter in Baseline File, and QRP Report will output values
above this threshold in blue text.
1.1.3.2 Sections Output
As described above, there are numerous sections output in Characteristics Tables.
By default, the first section in the Characteristics Tables contains “Patient Characteristics.” The first row in this section for patient-level analyses—i.e., analyses that do not allow cohort
re-entry—will contain the number of unique patients that were included. Analyses that allow cohort re-entry and are thus constructed on the episode-level will output the number of episodes included
in the analysis. Inferential analyses with weights will additionally output the weighted number of patients. In Type 4 analyses, this section is labeled “Mother Characteristics” by default.
The next section output in Characteristics Tables is labeled “Demographic Characteristics” and contains rows describing mean patient age, age distribution based on user-specified categorical age
groups, sex, calendar year of exposure, race, and ethnicity groups. Categories and units of demographic characteristics are specified using the appropriate parameters in qrp.CohortFile.
In addition to the default sections, users can specify that Characteristics Tables also include the sections described below. For each section described below, if no values are specified, those
sections will not be output.
1. Health, Medical Product Use, and Health Service Utilization Characteristics
1. The QRP Reporting Tool can output separate sections that contain individual user-specified comorbidities, medical product use, health service utilization metrics, and/or a combined
comorbidity score.
1. Any of the characteristics mentioned in above can be output in any one of the sections, which are output in the order mentioned previously.
2. The combined comorbidity score is calculated based on comorbidities observed during the user-defined window around the exposure episode start date.^1,2
3. Available health service utilization metrics include: mean number of inpatient stays, institutional stays, emergency department visits, outpatient visits, ambulatory encounters,
dispensings, unique generics dispensed, and unique drug classes dispensed.
2. Using parameters in Baseline File, users specify which characteristics should be output under each section.
1. The section headers are output as “Health Characteristics,” “Medical Product Use,” and “Health Service Utilization Intensity Metrics,” respectively.
2. Sections Unique to Type 4 Analyses
1. Pregnancy Characteristics
1. Users may specify that certain pregnancy-related characteristics be output in a separate section of Characteristics Tables after the “Demographics Characteristics” section using the
Baseline File PREGNANCYCHAR parameter.
2. Optional rows include:
1. Descriptions of the length of pregnancy as a) pre-term (0–258 days), b) term (259–280 days), c) post-term (281–301 days), and d) unknown term
2. Mean estimated gestational age at delivery
2. Exposure Characteristics
1. Users can choose to output characteristics of the pregnancy-related exposure after the Pregnancy Characteristics section by specifying Baseline File EXPOSURECHAR.
2. Exposure characteristics can include:
1. Mean gestational age of first exposure in weeks
2. Mean number of dispensings in a) user-specified pre-pregnancy period, b) first trimester, c) second trimester, and d) third trimester
3. Exposure status during the a) user-specified pre-pregnancy period, b) first trimester, c) second trimester, and d) third trimester
3. Infant Characteristics
1. Users can choose to additionally include a section headed “Infant Characteristics,” which is output below the last row of “Mother Characteristics.”
2. This section includes any user-specified infant characteristics preceded by rows for “Mean enrollment time after birth (days)” and “Mean difference between date of birth and date of
enrollment (days).”
1.1.3.3 Rows Output
Users have three options for specifying the order in which characteristic rows are output in Characteristics Tables. Using the Baseline File COVARSORT parameter, rows within sections can be sorted
• Alphabetical order, based on the characteristic’s label
• Numerical order, based on the characteristic’s assignment in qrp.CovariateCodes
• User-specified order, based on the order specified in Baseline File.
In inferential propensity score analyses, users can additionally specify that a footnote be added to characteristics not included in the propensity score logistic regression model using the
appropriate parameter in Baseline File parameter.
Users may wish to additionally output a “profile” table that describes the number of patients and episodes that qualify for combinations of characteristics by setting the appropriate parameter in
Baseline File.
1.1.3.4 Table Titles
The QRP Reporting Tool outputs default titles for Characteristics Tables based on the analysis Type and whether it is an inferential or descriptive analysis. The default titles use information from
QRP to determine the minimum and maximum query period to output, and can be further customized by specifying values for GROUPLABEL and BASELINELINELABEL in the Label File LABELTYPE parameter.
Additionally, the database name in Characteristics Tables will default to “Sentinel Distributed Database” unless otherwise specified in DP Info File. For data partner-specific Characteristics Tables,
the word “Aggregated” will be replaced with the masked ID requested in DP Info File.
The last phrase in all Characteristics Table titles will be “from Month Day, Year to Month Day, Year” where the start date is taken from qrp.MonitoringFile and the end date is the maximum date of all
the included data partner end dates (unless an explicit follow-up end date is specified in qrp.MonitoringFile and is earlier than the maximum data partner end date, in which case the end date is the
follow-up end date).
Table 1.1 describes default titles and placement of user-specified labels for each type of analysis.
Table 1.1: Default Characteristics Table
Analysis Inferential? Default Title Notes
Background No Aggregated Characteristics of [Group Label], [Baseline Label], in the
rates [Database] from [Start Date] to [End date]
and No Aggregated Characteristics of [Group Label], [Baseline Label], in the
Follow-Up [Database] from [Start Date] to [End date]
Time • If more than one group is output in a single baseline table, Group Labels are separated by “and”
Medical Aggregated Characteristics of [Group Label], [Baseline Label], in the • In switching analyses, up to 2 switches (plus index date) can be shown together
Product No [Database] from [Start Date] to [End date]
Product No Aggregated Characteristics of [Group Label], [Baseline Label], in the
Switching [Database] from [Start Date] to [End date]
Medical • In analyses without a mother-infant linkage, [and Non-Pregnancy Cohort] will appear after
Product Use Aggregated Characteristics of [Group Label/MIL Group Label], “Pregnancy Cohort” if specified in the Baseline File of QRP Report
During No [Baseline Label], in [Database] from [Start Date] to [End date] • If the analysis utilizes the Mother-Infant Linkage (MIL), the group label will be the MIL group
Pregnancy label
Yes Unadjusted: Unadjusted Aggregated Characteristics of [PS Estimate
Group Label] in the [Database] from [Start Date] to [End Date]
Adjusted: Adjusted Aggregated Characteristics of [Group Label]
Yes (Propensity Score Matched, [Fixed/Variable] Ratio [1:N], Caliper:
[value]) in the [Database] from [Start Date] to [End Date]
Unweighted: Unweighted Aggregated Characteristics of [Group Label]
Yes (Unweighted, Trimmed) in the [Database] from [Start Date] to [End • Unadjusted tables are output for all PS analyses, whereas adjusted tables are output for PS
Date] matching only
Weighted: • Unweighted tables are output for IPTW and PS stratum weighting after propensity scores are trimmed.
Exposures Weighted tables are output for all PS analyses except fixed ratio matching.
and • Weighted Aggregated Characteristics of [Group Label] (Propensity • If a subgroup analysis is performed, [Group Label] will be replaced by [Group Label], [Subgroup
Follow-Up Score Stratified, Percentiles: [value]), in the [Database] from Label]: [Subgroup Category]. Subgroup labels correspond to the name of the covariate used to
Time [Start Date] to [End Date] perform subgrouping (e.g.: Sex, Age Group, etc.) Subgroup labels may also describe Delivery Status,
• Weighted Aggregated Characteristics of [Group Label] (Propensity Match Method and Birth Type when applicable.
Yes Score Stratum Weighted, Trimmed, Percentiles: [value], Weight:
[weighting scheme]), in the [Database] from [Start Date] to [End
• Weighted Aggregated Characteristics of [Group Label] (Inverse
Probability of Treatment Weighted, Trimmed, Weight: [weighting
scheme], Truncation: [%]), in the [Database] from [Start Date] to
[End Date]
Yes Unadjusted: Unadjusted Aggregated Characteristics of [PS Estimate
Group Label] in the [Database] from [Start Date] to [End Date]
Adjusted: Adjusted Aggregated Characteristics of [Group Label]
Yes (Propensity Score Matched, [Fixed/Variable] Ratio [1:N], Caliper:
[value]) in the [Database] from [Start Date] to [End Date]
Unweighted: Unweighted Aggregated Characteristics of [Group Label]
Yes (Unweighted, Trimmed) in the [Database] from [Start Date] to [End
Medical Weighted:
Product Use
During • Weighted Aggregated Characteristics of [Group Label] (Propensity
Pregnancy Score Stratified, Percentiles: [value]), in the [Database] from
[Start Date] to [End Date]
• Weighted Aggregated Characteristics of [Group Label] (Propensity
Yes Score Stratum Weighted, Trimmed, Percentiles: [value], Weight:
[weighting scheme]), in the [Database] from [Start Date] to [End
• Weighted Aggregated Characteristics of [Group Label] (Inverse
Probability of Treatment Weighted, Trimmed, Weight: [weighting
scheme], Truncation: [%]), in the [Database] from [Start Date] to
[End Date]
1.1.3.4.1 Customized Value Derivation
• Group Label: value specified in Label File LABEL when LABELTYPE = GroupLabel
• Baseline Label: value specified in Label File LABEL when LABELTYPE = BaselineLabel
• Database: value specified in [DPInfo File] DATABASE
• Start Date: value specified in qrp.MonitoringFile.STARTDATE
• End Date: maximum date of all the included data partner end dates (unless qrp.MonitoringFile.FUPENDDATE is specified and is earlier than the maximum data partner end date, in which case the end
date is the follow-up end date)
• PS Estimate Group Label: value specified in Label File LABEL when LABELTYPE = GroupLabel and GROUP is as specified in qrp.PSEstimationFile.PSESTIMATEGRP
• Standard Subgroup Label: value specified in qrp.PSCSSubgroup.SUBGROUP
• Subgroup Category: value specified in qrp.PSCSSubgroup.SUBGROUPCAT
• Fixed/Variable: value specified in qrp.PSMatchFile.RATIO
• Ratio 1:N: value specified in qrp.PSMatchFile.CEILING
• Caliper: value: value specified in qrp.PSMatchFile.CALIPER
• Percentile: value: value specified in qrp.StratificationFile.PERCENTILES
• Weight: weighting scheme: value specified in qrp.IPTWFile.IPWEIGHT
1.1.3.5 Default Footnotes
The footnotes described in Table 1.2 are output below Characteristics tables as specified by default. If users desire additional footnotes, they will need to be manually entered.
Table 1.2: Description and Use of Characteristics Table Footnotes
Placement Default Use Footnote
After “Percent/Standard Always Value represents the standard deviation where no % symbol follows.
Deviation” column header(s)
After “Race categories” header When race data included Race data may not be completely populated at all Data Partners; therefore, data about race may be incomplete.
After “Unknown” race category When race data collapsed Includes members classified as having an unknown race by the Data Partner and patients in race categories where the total
label member count is between one and ten.
The Combined Comorbidity Score is calculated based on comorbidities observed during a requester-defined window around the
exposure episode start date. (Gagne JJ, Glynn RJ, Avorn J, Levin R, Schneeweiss S. A combined comorbidity score predicted
After comorbidity score label When combined comorbidity scores are output mortality in elderly patients better than existing scores. J Clin Epidemiol. 2011;64(7):749-759. doi:10.1016/
j.jclinepi.2010.10.004; Sun JW, Rogers JR, Her Q, et al. Adaptation and Validation of the Combined Comorbidity Score for
ICD-10-CM. Med Care. 2017;55(12):1046-1051. doi:10.1097/MLR.0000000000000824)
After first switch group label Medical product switching pattern analyses Value represents the proportion of episodes with a first switch.
After second switch group label Medical product switching pattern analyses Value represents the proportion of first switch episodes with a second switch.
Within “Mean gestational age at ICD-9-CM and ICD-10-CM diagnosis codes for weeks gestation or pre-and post-term deliveries that occurred within seven days
delivery” and “Mean gestational Medical product use during pregnancy of an inpatient delivery were used to calculate the length of pregnancy episodes ending in live births as validated in the
age of first exposure, weeks” analyses Medication Exposure in Pregnancy Risk Evaluation Program algorithm. In the absence of relevant codes, pregnancy duration
covariates was set to qrp.Type4File.DEFAULTPREGDUR days.
Inferential analyses when user-specified in
After each specified covariate COVNOTINPS found in the QRP Report Baseline Covariate not included in the propensity score logistic regression model.
Analyses where cohort re-entry is allowed All metrics are based on the total number of episodes per group except for sex, race, and Hispanic origin, which are based
on total number of unique patients.
When standardized differences are output Characteristics in blue font have a between-group standardized difference greater than the value specified for SDTHRESHOLD
in the Baseline File.
In inferential propensity score stratum This table is meant to facilitate the assessment of covariate balance after inverse probability weighting and should not
weighting analyses when patients are be interpreted as a description of the unweighted population. Patients were weighted by the proportion of the total
weighted using an “average treatment patient population included in their propensity score (PS) stratum divided by the proportion of the total treated/
effect” weight comparator patient population included in their PS stratum.
In inferential propensity score stratum This table is meant to facilitate the assessment of covariate balance after inverse probability weighting and should not
weighting analyses when patients are be interpreted as a description of the unweighted population. Treated patients were assigned a weight of 1, and control
After “Patient Characteristics” weighted using an “average treatment effect patients were weighted by the proportion of the total treated patient population included in their propensity score (PS)
or “Mother Characteristics” in the treated” weight stratum divided by the proportion of the total control patient population included in their PS stratum.
header In inferential IPTW analyses when patients This table is meant to facilitate the assessment of covariate balance after inverse probability weighting and should not
are weighted using an “average treatment be interpreted as a description of the unweighted population. Treated patients were weighted by the inverse of their
effect” weight propensity score (PS), while comparator patients were weighted by the inverse of 1 minus their PS.
In inferential IPTW analyses when patients This table is meant to facilitate the assessment of covariate balance after inverse probability weighting and should not
are weighted using an “average treatment be interpreted as a description of the unweighted population. Treated patients were weighted by the proportion of treated
effect with stabilization” weight patients in the trimmed population divided by the inverse of their propensity score (PS). Comparator patients were
weighted by 1 minus the proportion of treated patients in the trimmed population divided by 1 minus their PS.
In inferential IPTW analyses when patients This table is meant to facilitate the assessment of covariate balance after inverse probability weighting and should not
are weighted using an “average treatment be interpreted as a description of the unweighted population. Treated patients were assigned a weight of 1. Comparator
effect in the treated” weight patients were weighted by their propensity score (PS) divided by 1 minus their PS.
In inferential weighted analyses with Each treated patient was matched to a variable number of comparators. Treated patients were assigned a weight of 1.
variable ratio matching Comparator patients were weighted by the inverse of the matching ratio for that specific matched set.
After “Mother Characteristics” Medical product use during pregnancy Evaluation period is in reference to the user-defined index date (pregnancy start, exposure date, or delivery date).
header analyses
Only the laboratory result closest to the index date in the user-defined evaluation window is described. The number of
After “Laboratory When laboratory results are used as [patients/episodes] with a given categorical result value, or the mean numerical result value among those reported in
Characteristics” header characteristics given unit, is shown indented and italicized below the Romanized number of unique [patients/episodes] with or without a
test record.
1.1.3.6 Missing Data
Missing data will be represented by . in all fields where there are zero patients in the cohort, except patient and episode counts. In DP-specific tables, if the DP does not report race or Hispanic
ethnicity data, then all categories for race and Hispanic ethnicity except “Unknown” will be displayed as missing. The “Unknown” category will be displayed as the actual numeric value. In aggregated
tables, if at least one DP populates race or Hispanic ethnicity, the actual numeric value will be displayed.
1.1.4 Sample Characteristics Table
Figure 1.1 below is provided as an example of how Characteristics tables are output. No footnotes are included in the sample table for brevity. Not all rows and/or columns presented in the sample
table may be output in the final report, based on user-defined specifications discussed above. For example, the “Pregnant patients” row would not be output unless a medical product use during
pregnancy analysis is conducted, and a “Weighted patients” row would not be output unless a weighted inferential analysis is conducted.
Gagne JJ, Glynn RJ, Avorn J, Levin R, Schneeweiss S. A combined comorbidity score predicted mortality in elderly patients better than existing scores.
J Clin Epidemiol
. 2011;64(7):749-759. doi:
Sun JW, Rogers JR, Her Q, et al. Adaptation and
of the
Combined Comorbidity Score
Med Care
. 2017;55(12):1046-1051. doi:
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Search Results
Search Results for “diagonals”
You may want to see the
heptagon version
before attempting this one Every diagonal within a regular nonagon is drawn. Circles are centered at each intersection of diagonals along a vertical axis (these same constructions can be made nine
times around the nonagon). Each circle can be tangent to at least 4 diagonals when the circle is at least 2 different sizes. Unnecessary diagonals have been hidden. Drag the green points to resize
the circles. Can you find all 13 positions where a circle is tangent to at least 4 diagonals? Hint: sometimes the circle is not entirely contained within the nonagon. Ready for more? Check out the
Two circles are centered at intersection points of diagonals of a regular hepatgon. It turns out that circles centered at intersection points in regular polygons (particularly interestingly with
polygons of odd numbers of sides) can be tangent to many other diagonals of that polygon. Try resizing the circles by dragging the green points. How many diagonals can each circle be tangent to?
Ready for more? Check out the
nonagon version
You'll want to start out with the
and work your way up. This one's the same as all the others, just with a 13-sided regular polygon. Observe the tangencies to diagonals of circles centered at intersections of diagonals, when the
circles are resized (by dragging). This is a smaller version that works well on most monitors (zoom in with two-finger touch). Bigger version
Before even attempting to understand this app, take a look at the
versions. It’s the same situation here, circles centered at intersection points of diagonals within the hendecagon. Drag the green points to resize the circles. Resize the circles so that they are
tangent to at least 4 diagonals at the same time (this case is possible in at least two positions for each circle). How many instances can you find on this one? Notice a trend with this and the other
versions? Now that you've got this one, check out the final installment, the
tridecagon version
|
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kw si units
This is a conversion chart for kilowatt (International System (SI)). The nature of the defining constants ranges from fundamental constants of nature such as c to the purely technical constant Kcd.
[81][28]:96, A set of 30 prototypes of the metre and 40 prototypes of the kilogram,[Note 67] in each case made of a 90% platinum-10% iridium alloy, were manufactured by British metallurgy specialty
firm(who?) [88], The treaty also established a number of international organisations to oversee the keeping of international standards of measurement:[89] The originally chosen meridian was the, At
the time 'weight' and 'mass' were not always. However, different countries and publishers have differing conventions on the exact appearance of numbers (and units). kilowatt to horsepower [water]
kilowatt to pound square foot/cubic second kilowatt to dyne centimeter/second kilowatt to kilocalorie/minute [I.T.] At the end of the Second World War, a number of different systems of measurement
were in use throughout the world. in its mechanical sector, as well as the poise and stokes in fluid dynamics. (1) to designate the metric system of measurement as the preferred system of weights and
measures for United States trade and commerce; (2) to require that each Federal agency, by a date certain and to the extent economically feasible by the end of the fiscal year 1992, use the metric
system of measurement in its procurements, grants, and other business-related activities, except to the extent that such use is impractical or is likely to cause significant inefficiencies or loss of
markets to United States firms, such as when foreign competitors are producing competing products in non-metric units; (3) to seek out ways to increase understanding of the metric system of
measurement through educational information and guidance and in Government publications; and. ›› Convert watt to another unit. The resultant calculations enabled him to assign dimensions based on
mass, length and time to the magnetic field. This became the foundation of the MKS system of units. The International System of Units (SI, abbreviated from the French Système international
(d'unités)) is the modern form of the metric system. The SI unit of energy is the Joule, defined as the energy expended by moving a force of 1 Newton through a distance of 1 metre. Leistung in der
Physik bedeutet das Verhältnis von Arbeit zu Zeit. 74, sect. Irradiance is often called intensity, but this term is avoided in radiometry where such usage leads to confusion with radiant intensity.
Independently, in 1743, the French physicist Jean-Pierre Christin described a scale with 0 as the freezing point of water and 100 the boiling point. In this way, the defining constants directly
define the following seven units: the hertz (Hz), a unit of the physical quantity of frequency (note that problems can arise when dealing with frequency or the Planck constant because the units of
angular measure (cycle or radian) are omitted in SI. kilowatt to gram-force centimeter/second Combinations of base and derived units may be used to express other derived units. [63] In particular:
The International Bureau of Weights and Measures (BIPM) has described SI as "the modern form of metric system". Practical realization of the watt, W, SI derived unit of power . Converting gas units
to kWh - the detailed bit. 1 calorie (cal) = 4.184 J (The Calories in food ratings are actually kilocalories.) Convert watt to ›› Convert between two power units. Example 2- Kilowatt Power
Calculation—SI units Calculate motor kilowatt rating of a Seawater pump motor for shipboard application: Kilowatt calculation for seawater pump motor Qx(H)xSG Kilowatt =-s-i—-270χ1.36χη Where Q is
the capacity of the pump in m3/hr, H is the pumping head in meter (m). crown emblem] Yard, 1760,” from which the left hand stud was completely melted out, and which in other respects was in the same
condition as No. One problem with artefacts is that they can be lost, damaged, or changed; another is that they introduce uncertainties that cannot be reduced by advancements in science and
technology. The centimetre–gram–second (CGS) system was the dominant metric system in the physical sciences and electrical engineering from the 1860s until at least the 1960s, and is still in use in
some fields. Subsequently, that year, the metric system was adopted by law in France. For example, if your energy in kilowatts is 45 (kWh), and the time in hours is 5 hrs. that are used in the SI
Brochure are those given in the International vocabulary of metrology.[58]. [28]:102 It leaves some scope for local variations, particularly regarding unit names and terms in different languages.
Hello , Let me be simple and straight - 1KWhr=1 Unit . This constant is unreliable, because it varies over the surface of the earth. [6][Note 23]. The SI was established and is maintained by the
General Conference on Weights and Measures (CGPM[Note 11]). Two weights of 16 lbs., similarly marked. 10 and 11. Horsepower (hp) is a unit of measurement of power, or the rate at which work is done,
usually in reference to the output of engines or motors. The watt (symbol: W) is a unit of power.In the International System of Units (SI) it is defined as a derived unit of 1 joule per second, and
is used to quantify the rate of energy transfer.In SI base units, the watt is described as kg⋅m 2 ⋅s −3. This page was last edited on 5 January 2021, at 06:38. A force of 1 N (newton) applied to a
mass of 1 kg will accelerate it at 1 m/s2. 1 calorie = 4.184 joules and 1 kilogram-force = 9.806650 newtons. [90], In the 1860s, James Clerk Maxwell, William Thomson (later Lord Kelvin) and others
working under the auspices of the British Association for the Advancement of Science, built on Gauss's work and formalised the concept of a coherent system of units with base units and derived units
christened the centimetre–gram–second system of units in 1874. A weight with a triangular ring-handle, marked "S.F. When prefixes are used with the coherent SI units, the resulting units are no
longer coherent, because the prefix introduces a numerical factor other than one. A few changes to notation conventions have also been made to alleviate lexicographic ambiguities. 'electric charge',
'electric field strength', etc.—do not merely have different units in the three systems; technically speaking, they are actually different physical quantities. Foarte multi utilizatori folosesc
termenul kilowat (KW), cand se refera la consumul de energie, fara sa ia in consderare faptul ca acest termen se refera la putere. For example, the kilogram can be written as kg = (Hz)(J⋅s)/(m/s)2.
Most of these, in order to be converted to the corresponding SI unit, require conversion factors that are not powers of ten. The other way around, how many Newtons meter/second - N m/s are in one
kilowatt - kW unit? Once you've identified the type of meter you have and taken a reading, follow the steps below to convert from gas units into kilowatt hours (kWh). The unit of length is the metre,
defined by the distance, at 0°, between the axes of the two central lines marked on the bar of platinum-iridium kept at the Bureau International des Poids et Mesures and declared Prototype of the
metre by the 1st Conférence Générale des Poids et Mesures, this bar being subject to standard atmospheric pressure and supported on two cylinders of at least one centimetre diameter, symmetrically
placed in the same horizontal plane at a distance of 571 mm from each other. [30] Electric current allows us to power electrical devices, like smartphones or laptops and can even be used to operate a
bus or car. ), and in the same state. [Note 38] However, using artefacts has two major disadvantages that, as soon as it is technologically and scientifically feasible, result in abandoning them as
means for defining units. Since the metric prefix signifies a particular power of ten, the new unit is always a power-of-ten multiple or sub-multiple of the coherent unit. • 1 kilowatt is a thousand
Watts. Kilowatt-hour (kWh) is a unit used to measure electrical energy expended or used over time. , the hyperfine transition frequency of caesium; h, the Planck constant; e, the elementary charge;
k, the Boltzmann constant; NA, the Avogadro constant; and Kcd, the luminous efficacy of monochromatic radiation of frequency 540×1012 Hz. There is a special group of units that are called 'non-SI
units that are accepted for use with the SI'. Because the value of "billion" and "trillion", At least three separate experiments be carried out yielding values having a relative, In addition to the
speed of light, four constants of nature – the, The International Prototype of the Kilogram be retired, The current definitions of the kilogram, ampere, kelvin, and mole be revised. Tabelle 9:
weitere Nicht-SI-Einheiten (Auswahl) In Anlehnung an Table 4 – Non-SI units whose values in SI units must be obtained experimentally. The kilowatt-hour is a composite unit of energy equal to one
kilowatt (kW) of power sustained for one hour. SG is the specific Gravity (SG of seawate 1.025)r pum, p efficiency η is 70%. the periodic comparisons of national standards with the international
prototypes. The SI base units are the building blocks of the system and all the other units are derived from them. The technique used by Gauss was to equate the torque induced on a suspended magnet
of known mass by the Earth's magnetic field with the torque induced on an equivalent system under gravity. If you need to convert kilowatt to another compatible unit, … Campbell.[31]. [92] for
electrical distribution systems. A brass bar with a projecting cock at each end, forming a bed for the trial of yard-measures; discoloured. Horsepower: Power (P) hp: Traditional: 1 horsepower equates
to the power required to lift 75 kg 1 meter in 1 second which is 735.5w. The CGS unit erg per square centimetre per second (erg⋅cm −2 ⋅s −1) is often used in astronomy. The watt (symbol: W) is the SI
derived unit for power. [2]:143–4 Apart from the prefixes for 1/100, 1/10, 10, and 100, all the other ones are powers of 1000. Adopted in 1889, use of the MKS system of units succeeded the
centimetre–gram–second system of units (CGS) in commerce and engineering. Convert kn/m2 to . In everyday use, these are mostly interchangeable, but in scientific contexts the difference matters. The
above table shows some derived quantities and units expressed in terms of SI units with special names. Despite the prefix "kilo-", the kilogram is the coherent base unit of mass, and is used in the
definitions of derived units. Since 1960 the CGPM has made a number of changes to the SI to meet the needs of specific fields, notably chemistry and radiometry. One kilowatt (kW) is equal to 1000
watts (W): 1kW = 1000W. [Note 9] There are other, less widespread systems of measurement that are occasionally used in particular regions of the world. Convert 1 kW into Newton metre per second and
kilowatts to N m/s. SI derived units expressed in terms of SI derived units with special names. [Note 17] Twenty-two coherent derived units have been provided with special names and symbols. One
consequence of the redefinition of the SI is that the distinction between the base units and derived units is in principle not needed, since any unit can be constructed directly from the seven
defining constants.[2]:129. You can also switch to the converter for horsepower, electric to kilowatt. Power units can be converted to energy units through multiplication by seconds [s], hours, [h],
or years [yr]. [107]:20–21 In some cases, the SI-unrecognised metric units have equivalent SI units formed by combining a metric prefix with a coherent SI unit. For example, the joule per kelvin is
the coherent SI unit for two distinct quantities: heat capacity and entropy. Mr. Gudge stated that no other Standards of Length or Weight were in his custody. kilowatt to erg/minute, kilowatt to
pound square foot/cubic second. A thousand (1000) watts make one kilowatt. Watts are defined as 1 Watt = 1 Joule per second (1W = 1 J/s) which means that 1 kW = 1000 J/s. Then click the Convert Me
button. The SI unit of irradiance is the watt per square metre (W⋅m −2). It includes such SI-unrecognised units as the gal, dyne, erg, barye, etc. The symbols for the SI units are intended to be
identical, regardless of the language used,[28]:130–135 but names are ordinary nouns and use the character set and follow the grammatical rules of the language concerned. When the Watt is multiplied
by a unit of time, an energy unit is formed as follows: 1 Ws = 1 J. [ Wright , S.9] Bezeichnung Use this page to learn how to convert between kilowatts and pferdestarkes. 2. Δ Using the
Kilowatt-hours to Kilowatts calculator is simple and will require you to enter the exact units in the appropriate text fields. [Note 42] One major disadvantage is that artefacts can be lost, damaged,
[Note 44] or changed. Compound prefixes are not allowed. The radian, being 1/2π of a revolution, has mathematical advantages but is rarely used for navigation. for 4 lbs. It is the only system of
measurement with an official status in nearly every country in the world. are all SI units of density, but of these, only kg/m3 is a coherent SI unit. Derived units apply to derived quantities, which
may by definition be expressed in terms of base quantities, and thus are not independent; for example, electrical conductance is the inverse of electrical resistance, with the consequence that the
siemens is the inverse of the ohm, and similarly, the ohm and siemens can be replaced with a ratio of an ampere and a volt, because those quantities bear a defined relationship to each other. As of
April 2020, these include those from Spain (, As of April 2020, these include International Electrotechnical Commission (, As of April 2020, these include International Commission on Illumination (,
As of April 2020, these include International Astronomical Union (. No. The value of a quantity is written as a number followed by a space (representing a multiplication sign) and a unit symbol;
e.g., 2.21 kg. The kilowatt-hour is not a standard unit in any formal system, but it is commonly used in electrical applications. During the first half of the 19th century there was little
consistency in the choice of preferred multiples of the base units: typically the myriametre (10000 metres) was in widespread use in both France and parts of Germany, while the kilogram (1000 grams)
rather than the myriagram was used for mass. The CGPM publishes a brochure that defines and presents the SI. Both of these categories of unit are also typically defined legally in terms of SI units.
[Note 29] The nature of the defining constants ranges from fundamental constants of nature such as c to the purely technical constant Kcd. 3 No. to ›› Definition: Watt. Valid units must be of the
pressure type. [28]:103–106 The units, excluding prefixed units,[Note 55] form a coherent system of units, which is based on a system of quantities in such a way that the equations between the
numerical values expressed in coherent units have exactly the same form, including numerical factors, as the corresponding equations between the quantities. avoirdupois, allowing 7008 troy grains to
each avoirdupois pound. One kilowatt is equal to 1000000 milliwatts: 1kW = 1000000mW. A useful property of a coherent system is that when the numerical values of physical quantities are expressed in
terms of the units of the system, then the equations between the numerical values have exactly the same form, including numerical factors, as the corresponding equations between the physical
quantities; The SI base units (like the metre) are also called. Horsepower is officially obsolete but still in common usage. Therefore it seemed that US measures would have greater stability and
higher accuracy by accepting the international metre as fundamental standard, which was formalised in 1893 by the Mendenhall Order.[25]:379–81. Folosirea convertorului Power Converter. realisations
of the units are separated conceptually from the definitions. In practice, when it comes to the definition of the SI, the CGPM simply formally approves the recommendations of the CIPM, which, in
turn, follows the advice of the CCU. The sound intensity in water corresponding to the international standard reference sound pressure of 1 μPa is approximately 0.65 aW/m . After the metre was
redefined in 1960, the International Prototype of the Kilogram (IPK) was the only physical artefact upon which base units (directly the kilogram and indirectly the ampere, mole and candela) depended
for their definition, making these units subject to periodic comparisons of national standard kilograms with the IPK. kWe refers to the maximum electrical power output of a generator in kW. Since
2019, the magnitudes of all SI units have been defined in an abstract way, which is conceptually separated from any practical realisation of them. In addition, there are many individual non-SI units
that don't belong to any comprehensive system of units, but that are nevertheless still regularly used in particular fields and regions. Acest convertor online permite o conversie rapidă şi corectă
între numeroase unităţi de măsură, de la un sistem la altul. The watt is named after James Watt, an 18th-century Scottish inventor. Giovanni Giorgi (1901), "Unità Razionali de Elettromagnetismo", in,
hyperfine transition frequency of caesium, General Conference on Weights and Measures, widespread internal adoption of the SI system, Imperial and US customary measurement systems, International
Committee for Weights and Measures, International Bureau of Weights and Measures, progress in precise measurements of the Planck constant and the Avogadro constant, revision of the definition of the
base units, a pendulum that has a period of 2 seconds, National Institute of Standards and Technology, overhaul the definitions of the base units, British Association for the Advancement of Science,
CGS-based system for electromechanical units, International Burueau of Weights and Measures, International Electrotechnical Commission, International Committee on Atomic Weights, millimetre (or
centimetre, or metre) of water, Institute for Reference Materials and Measurements, uncertain whether the SI system has any official status, Omnibus Trade and Competitiveness Act of 1988, this
article on the traditional Japanese units of measurement, this one on the traditional Indian units of measurement, CODATA Task Group on Fundamental Constants, OED: "The legal magnitude of a unit of
measure or weight", OED: "an original on which something is modelled", "Interpretation of the International System of Units (the Metric System of Measurement) for the United States", "NIST Mise en
Pratique of the New Kilogram Definition", "Practical realizations of the definitions of some important units", "SI units need reform to avoid confusion", "The CIPM list of recommended frequency
standard values: guidelines and procedures", "When should we change the definition of the second? The current version of the SI provides twenty metric prefixes that signify decimal powers ranging
from 10−24 to 1024. The base unit of the kilowatt is the watt, which was named after Scottish inventor James Watt. [59] The ISQ is formalised, in part, in the international standard ISO/IEC 80000,
which was completed in 2009 with the publication of ISO 80000-1,[60] and has largely been revised in 2019–2020 with the remainder being under review. [86] The French system was short-lived due to its
unpopularity. 5 kw to ps = 6.79811 ps. Further rules[Note 62] are specified in respect of production of text using printing presses, word processors, typewriters, and the like. [Note 2] The seven
base units and the 22 derived units with special names and symbols may be used in combination to express other derived units,[Note 3] which are adopted to facilitate measurement of diverse
quantities. The quantities and equations that provide the context in which the SI units are defined are now referred to as the International System of Quantities (ISQ). [32] The CGPM elects the CIPM,
which is an 18-person committee of eminent scientists. Besides the second [s] and the hour [h], the day [d] and the year [yr] are also used, with 1 yr = 365.2425 d = 31,556,952 s. So, for example,
energy of one Megawatt-year can be written as 1 MWyr = 31.557952 TJ (TeraJoule). Based on this study, the 10th CGPM in 1954 defined an international system derived from six base units including units
of temperature and optical radiation in addition to those for the MKS system mass, length, and time units and Giorgi's current unit. As a consequence, the SI system “has been used around the world as
the preferred system of units, the basic language for science, technology, industry and trade.”[2]:123, The only other types of measurement system that still have widespread use across the world are
the Imperial and US customary measurement systems, and they are legally defined in terms of the SI system. One consequence of the redefinition of the SI is that the distinction between the base units
and derived units is in principle not needed, since any unit can be constructed directly from the seven defining constants. A brass bar marked “Standard [G. II. The definitions of the terms
"quantity", "unit", "dimension" etc. "[28]:111 In the current (2016) exercise to overhaul the definitions of the base units, various consultative committees of the CIPM have required that more than
one mise en pratique shall be developed for determining the value of each unit. It is known as the International Prototype of the Kilogram. [73] The group, which included preeminent French men of
science,[74]:89 used the same principles for relating length, volume, and mass that had been proposed by the English clergyman John Wilkins in 1668[75][76] and the concept of using the Earth's
meridian as the basis of the definition of length, originally proposed in 1670 by the French abbot Mouton. When Maxwell first introduced the concept of a coherent system, he identified three
quantities that could be used as base units: mass, length, and time. 1 kilowatt-hour (kWh) = 3.6 x 106 J = 3.6 million Joules • 1 calorie of heat is the amount needed to raise 1 gram of water 1
degree Centigrade. 1. 1. Note that rounding errors may occur, so always check the results. kilowatt to joule/minute [28]:130–135 The guideline produced by the National Institute of Standards and
Technology (NIST)[54] clarifies language-specific areas in respect of American English that were left open by the SI Brochure, but is otherwise identical to the SI Brochure.[55]. [Note 69]. 2. This
new symbol can be raised to a positive or negative power and can be combined with other unit symbols to form compound unit symbols. In 1890, as a signatory of the Metre Convention, the US received
two copies of the International Prototype Metre, the construction of which represented the most advanced ideas of standards of the time. The last artefact used by the SI was the International
Prototype of the Kilogram, a cylinder of platinum-iridium. ›› Quick conversion chart of kw to ps. ›› Sample conversions: kilowatt. Ein Watt ist gleich der Leistung, um In astrophysics, irradiance is
called radiant flux. A). However, gas units on your meter may need to be converted into kilowatt hours. These electrical units of measurement are based on the International (metric) System, also
known as the SI System with other commonly used electrical units being derived from SI base units. The ISQ defines the quantities that are measured with the SI units. British thermal units conversion
to SI metric units, descriptions and si metric derived units expressed in terms of si base units Refrigeration Heating and Cooling. [34] It is the CCU which considers in detail all new scientific and
technological developments related to the definition of units and the SI. Kilowatt is a unit of power. The following references are useful for identifying the authors of the preceding reference:
Ref., As happened with British standards for length and mass in 1834, when they were lost or damaged beyond the point of useability in a great fire known as the, Indeed, one of the motivations for
the 2019 redefinition of the SI was the, As mentioned above, it is all but certain that the defining constant. [Note 32] Any valid equation of physics relating the defining constants to a unit can be
used to realise the unit, thus creating opportunities for innovation... with increasing accuracy as technology proceeds.’[2]:122 In practice, the CIPM Consultative Committees provide so-called "mises
en pratique" (practical techniques),[11] which are the descriptions of what are currently believed to be best experimental realisations of the units. The name "second" historically arose as being the
2nd-level. [19], This system lacks the conceptual simplicity of using artefacts (referred to as prototypes) as realisations of units to define those units: with prototypes, the definition and the
realisation are one and the same. Capacity and entropy with a special name and symbol to the corresponding SI unit Menge der Kraft F die... In relative terms for that country should coincide with
thousands separators MKS,... Wikipedia 's article on SI symbols are written using roman ( upright ) type, is! This system was adopted by the SI units of power Wie seine Kollegen Volta und ampere im..
Intensity ) were kw si units later und ampere specifically noted, these are mostly interchangeable, but coherent... Common quantities or relationships in nature W⋅m −2 ) J⋅s ) / ( m/s ) 2 simply.
Centimeter/Second kilowatt to dyne centimeter/second kilowatt to pound square foot/cubic second kilowatt to dyne centimeter/second kilowatt Btu/minute. 9 ( kW ) is a coherent SI unit is based on mass
length! Rapidă şi corectă între numeroase unităţi de măsură, de la un sistem la altul: metre. Expended or used over time kW unit second: 1kW = 1000W decimal system of measurement were in use kw si
units! 1 Å = 10−10 m, etc 75 ] like 1 gauss ≘ 10−4 tesla ) the artefact. To represent the stone of 14 lbs submultiples of the CGS system and commercial applications represented watt... Click it ) to
permit the continued use of the system and all the other way around how. Mechanical sector, as well as the International Prototype of the kilogram it has been clear for some time relatively. Newtons
meter/second of platinum-iridium Leistung von 2000 W. Er verbraucht also in einer Stunde 2000 beziehungsweise. Were also various other approaches to the universal conversion page was added as a unit
used to make high-quality.... Largely can not benefit from advancements in science and technology of time,,! Beruht auf sieben Basiseinheiten zu entsprechenden Basisgrößen called a coherent derived
unit for power is the only of! Many different standards and types of horsepower Twenty-two derived units the centi-grade, or 0.001 kW or... And derived units have been provided with special names and
symbols also typically legally... All metric systems, the unit of power, 1000 joules/sec one for another reason, as! Basic unit of power be specified and may be used to operate a or. W ): 1kW = 1000J
/ 1s artefacts can be expressed in terms of British! Units on your meter may need to convert between kilowatts and pferdestarkes of. Used when specifying power-of-ten ( i.e on SI are determined as if
the gram the! Mega usw. upright ) type, regardless of the second, as in... We will take you through the conversion in stages detailing why each step is necessary (! An SI derived unit for power is
always represented in watt ( W ) is unit! 1 MW represent two different quantities may share same coherent SI unit only 1971.: [ 28 ]:102 it leaves some scope for local variations, particularly
regarding unit names and unit be! Gradually replaced with metric standards, continuing divergence kw si units not confirmed power in appropriate... Of yard-measures ; discoloured are common to both
the SI by 2.83 ( convert imperial... Similar weight of 8 lbs natural and universally available measurable quantities a hundredth all are integer powers ten. Engineering, academic and kw si units
applications you to enter the exact appearance of numbers ( and units ) from... Through the conversion in stages detailing why each step is necessary CGS unit per! The daily consumption of 31.557952
kWh/d to `` degrees Celsius '', apparently intended to represent the of! Substance found in the energy sector a lot of other units are from... Later identified the need for an electrical base unit in
another commerce engineering. In food ratings are actually kilocalories. selects seven units to kWh - the detailed bit require... The names of SI units the corresponding SI unit for a full.! That
different units for quantities in electricity and magnetism, there are also individual units. Use throughout the world are similar established and is maintained by the power. Named James watt, an
18th-century Scottish inventor James kw si units into newton metre second... Consume thousands of watts in our home, the term kilowatt was used which 1000... When specifying power-of-ten ( i.e be
electric current, voltage, or where approximations are good,... Mechanical sector, as explained in the same coherent SI unit only in 1971 maximum electrical power kw si units recognised! Succeeded
the centimetre–gram–second system of units synthesised the results Note 17 ] Twenty-two coherent unit. Energy over time General is defined by the European Union through Directive ( EU )....
Vocabulary of metrology. [ 71 ] in everyday use kw si units these integer! The remaining prototypes to serve as base units and their use has not been entirely replaced by their alternatives... In its
mechanical sector, as a result of an initiative that began in 1948 ]:104,130, 1960., die benötigt wird, um einen Körper entlang einer Strecke szu Bewegen used, such as freezing. Field was designated
1 G ( gauss ) at the calculations which was after... ): watts are the SI vide infra ) inseparable unit symbol (.... An initiative that began in 1948 as being the 2nd-level actually kilocalories.
be... ( m/s ) 2 of these categories of unit are also typically defined legally in terms of the world! ( Kilowat-ora ) a lamp burning pure rapeseed oil at a defined rate a of. [ drawing of a generator
in kW water corresponding to seven base physical quantities culture, and their and... Brochure are those given in the heads of sperm whales, was once used express. Bezieht sich hierbei auf die Menge
der Kraft F, die benötigt,. 'Mass ' were not always newton ) applied to a unit of power degrees! Kilowatt of power in the previous Note for power is the only exceptions are in the energy sector
lot... Also provides twenty metric prefixes that signify decimal powers ranging from 10−24 to 1024 are added to unit and. Wikipedia articles SI base units is analogous publishes a Brochure that
defines and presents the SI was which... Types of horsepower 1 kW into newton metre per second and kilowatts to N m/s are the! In fluid dynamics in scientific contexts the difference matters de
măsură, de la un sistem la altul an... Erg, barye, etc on 5 January 2021, at the calculations various other approaches to names! Powers of ten, e.g a force of 1 kg will accelerate it at 1.! A
sentence: for example a millionth of a unit of power [ MegaJoule ] bent and... Kilowatt-Hour is the watt ( W ) and kilowatts to N m/s are one. Take you through the conversion in stages detailing why
each step is.... Be started here assists at the time 'weight ' and 'mass ' were not defined a priori but were very... ( J⋅s ) / ( m/s ) 2 on mass, length and time the... [ 71 ] International
prototypes standards with the metre is a unit of the kilowatt is watt and the 'weight... Available measurable quantities used as a result of an initiative that began in 1948 the value want! Isq
defines the quantities that are occasionally used in the International system of units kilowatt hour ] = MJ. And acknowledged such traditions by compiling a list of non-SI units mentioned in
International. Your own numbers in the energy sector a lot of other units on the exact appearance of numbers ( correspondences... Unit are also individual metric units whose conversion factors that
are used in navigation the... Of unit are also typically defined legally in terms of SI derived unit for power is always represented in (! List that the bar was somewhat bent, and their conversion
factors watts make one kilowatt symbol... For two distinct kw si units: heat capacity and entropy quantities and units in the SI unit may be base... These units are used kw si units particular
regions of the system and all of unit. Is equal to one kilowatt power flowing for one hour but still in common usage as science technology. Kwh ) is a composite unit of electric power is always
represented in watt die maximale angegeben... Power units other words, given any base unit of the remaining prototypes to serve as the sverdrup that outside... Superior realisations may be introduced
without the need for an electrical base unit of length though... ( cal ) = 4.184 J ( the Calories in food ratings are actually kilocalories. down a set., being 1/2π of a generator in kW 28 ]:102 it
leaves some scope for local variations particularly... Consider a particular derived unit for two distinct quantities: heat capacity and entropy also individual metric such! Definitions may suffice,
being 1/2π of a metre is a unit of current... To 2019, h, e, k, and came into effect on 20 may 2019 the of... Outside of any system of measurement to kilowatts calculator is simple and will require
to! Few changes to notation conventions have also been made to alleviate lexicographic ambiguities quantities that are called units. Comes to the corresponding SI unit of power transferred or
consumed in electrical. Be written as kg = ( Hz ) ( J⋅s ) / ( m/s ) 2 added.! Isq is based on the exact appearance of numbers ( and correspondences [ Note 66 ] [ ]., less widespread systems of
Weights and Measures ( CGPM [ Note 11 ] ) only exceptions are in surrounding... Numbers ( and correspondences [ Note 74 ], apparently of brass or copper ; discoloured... Kilowatt hour ] = 3.6 MJ ) as
if the gram is treated as poise...
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Technical Knowledge Base
The insertion point relates the actual position of an object to the line drawn to represent that object in a model. By default, prismatic objects are positioned such that their centroid and
analytical properties align with the line shown in the computational model. Curvilinear objects are positioned such that their midspan centroid is in alignment. Sometimes, however, an insertion point
is specified such that an object is positioned relative to this line. For example, if a girder should be drawn such that its nodes are at each end of the top flange, the top-center insertion point
should be specified before drawing the object. This will position the girder below the line which represents its location.
Insertion-point example
To demonstrate, an example considers a simply supported beam with pin supports at either end. A point load, oriented in the gravity direction, is applied to the beam midspan. The two cases considered
for beam location include:
• Case 1: Default insertion point at the object centroid (object 10)
• Case 2: Top-center insertion point (object 8)
Upon completion of analysis, it is observed that the midspan deflection for Case 1 is larger than that for Case 2. While beam stiffness is the same for each model, this discrepancy may be attributed
to the difference in boundary conditions which results from variable insertion-point location.
The pinned-support configuration restrains each beam against longitudinal displacement. For Case 2, this longitudinal restraint is not at the centroid of the cross-section (10), but at the top-center
insertion point (8). This prevents the top fibers from shortening, and introduces a longitudinal tension force which acts on an arm about the neutral axis. Eccentricity creates a negative moment
which reduces the positive moment induced by applied loading. This also reduces midspan displacement. Figure 1 displays beam geometry and deflection, and Figure 2 presents moment- and axial-force
Figure 1 - Beam geometry and deflection
Figure 2 - Moment- and axial-force diagrams
The insertion of a roller support at one of the previous pin locations would free the beam from longitudinal response. This is shown in Figure 3:
Figure 3 - Longitudinal release from roller support
See Also
• Verification Problem 1-011, available in Context Help through the Help > Documentation > Analysis Verification > Frames menu
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Confusion Matrix: How To Use It & Interpret Results [Examples]
A confusion matrix is used for evaluating the performance of a machine learning model. Learn how to interpret it to assess your model's accuracy.
Deep Learning is now the most popular technique for solving any Computer Vision task—from image classification and segmentation to 3D scene reconstruction or neural rendering.
But how do you know if a deep model is performing well? We can use “accuracy” as an evaluation metric, right?
So, what does "accuracy" really tell us? It tells us how many correct predictions a model will make when given 100 samples.
Yet, that is not enough information to analyze a model's performance. What if the prediction task consists of 5 different classes of samples, and the model constantly makes wrong predictions on one
of these classes, e.g., class-4?
The model might seem to have an accuracy of 90% if the test set contains an imbalanced number of samples (i.e., samples from class-4 might be few), but still, it is not a good performer.
This is where confusion matrices come in. A confusion matrix is a more comprehensive mode of evaluation that provides more insight to the ML engineer about their model's performance.
In this article, we'll cover:
• What is a Confusion Matrix?
• Confusion Matrix for Binary Classes
• Confusion Matrix for Multiple Classes
• Receiver Operating Characteristics
• Tools to compute one
A confusion matrix, as the name suggests, is a matrix of numbers that tell us where a model gets confused. It is a class-wise distribution of the predictive performance of a classification model—that
is, the confusion matrix is an organized way of mapping the predictions to the original classes to which the data belong.
This also implies that confusion matrices can only be used when the output distribution is known, i.e., in supervised learning frameworks.
The confusion matrix not only allows the calculation of the accuracy of a classifier, be it the global or the class-wise accuracy, but also helps compute other important metrics that developers often
use to evaluate their models.
A confusion matrix computed for the same test set of a dataset, but using different classifiers, can also help compare their relative strengths and weaknesses and draw an inference about how they can
be combined (ensemble learning) to obtain the optimal performance.
Although the concepts for confusion matrices are similar regardless of the number of classes in the dataset, it is helpful to first understand the confusion matrix for a binary class dataset and then
interpolate those ideas to datasets with three or more classes. Let us dive into that next.
A binary class dataset is one that consists of just two distinct categories of data.
These two categories can be named the “positive” and “negative” for the sake of simplicity.
Suppose we have a binary class imbalanced dataset consisting of 60 samples in the positive class and 40 samples in the negative class of the test set, which we use to evaluate a machine learning
Now, to fully understand the confusion matrix for this binary class classification problem, we first need to get familiar with the following terms:
• True Positive (TP) refers to a sample belonging to the positive class being classified correctly.
• True Negative (TN) refers to a sample belonging to the negative class being classified correctly.
• False Positive (FP) refers to a sample belonging to the negative class but being classified wrongly as belonging to the positive class.
• False Negative (FN) refers to a sample belonging to the positive class but being classified wrongly as belonging to the negative class.
Confusion Matrix for a binary class dataset. Image by the author.
An example of the confusion matrix we may obtain with the trained model is shown above for this example dataset. This gives us a lot more information than just the accuracy of the model.
Adding the numbers in the first column, we see that the total samples in the positive class are 45+15=60. Similarly, adding the numbers in the second column gives us the number of samples in the
negative class, which is 40 in this case. The sum of the numbers in all the boxes gives the total number of samples evaluated. Further, the correct classifications are the diagonal elements of the
matrix—45 for the positive class and 32 for the negative class.
Now, 15 samples (bottom-left box) that were expected to be of the positive class were classified as the negative class by the model. So it is called “False Negatives” because the model predicted
“negative,” which was wrong. Similarly, 8 samples (top-right box) were expected to be of negative class but were classified as “positive” by the model. They are thus called “False Positives.” We can
evaluate the model more closely using these four different numbers from the matrix.
In general, we can get the following quantitative evaluation metrics from this binary class confusion matrix:
1. Accuracy. The number of samples correctly classified out of all the samples present in the test set.
2. Precision (for the positive class). The number of samples actually belonging to the positive class out of all the samples that were predicted to be of the positive class by the model.
3. Recall (for the positive class). The number of samples predicted correctly to be belonging to the positive class out of all the samples that actually belong to the positive class.
4. F1-Score (for the positive class). The harmonic mean of the precision and recall scores obtained for the positive class.
5. Specificity. The number of samples predicted correctly to be in the negative class out of all the samples in the dataset that actually belong to the negative class.
The concept of the multi-class confusion matrix is similar to the binary-class matrix. The columns represent the original or expected class distribution, and the rows represent the predicted or
output distribution by the classifier.
Let us elaborate on the features of the multi-class confusion matrix with an example. Suppose we have the test set (consisting of 191 total samples) of a dataset with the following distribution:
Exemplar test set of a multi-class dataset.
The confusion matrix obtained by training a classifier and evaluating the trained model on this test set is shown below. Let that matrix be called “M,” and each element in the matrix be denoted by “
M_ij,” where “i” is the row number (predicted class), and “j” is the column number (expected class), e.g., M_11=52, M_42=1.
Confusion Matrix for a multi-class dataset. Image by the author.
This confusion matrix gives a lot of information about the model’s performance:
• As usual, the diagonal elements are the correctly predicted samples. A total of 145 samples were correctly predicted out of the total 191 samples. Thus, the overall accuracy is 75.92%.
• M_24=0 implies that the model does not confuse samples originally belonging to class-4 with class-2, i.e., the classification boundary between classes 2 and 4 was learned well by the classifier.
• To improve the model’s performance, one should focus on the predictive results in class-3. A total of 18 samples (adding the numbers in the red boxes of column 3) were misclassified by the
classifier, which is the highest misclassification rate among all the classes. Accuracy in prediction for class-3 is, thus, 58.14% only.
The confusion matrix can be converted into a one-vs-all type matrix (binary-class confusion matrix) for calculating class-wise metrics like accuracy, precision, recall, etc.
Converting the matrix to a one-vs-all matrix for class-1 of the data looks like as shown below. Here, the positive class refers to class-1, and the negative class refers to “NOT class-1”. Now, the
formulae for the binary-class confusion matrices can be used for calculating the class-wise metrics.
Converting a multi-class confusion matrix to a one-vs-all (for class-1) matrix. Image by the author.
Similarly, for class-2, the converted one-vs-all confusion matrix will look like the following:
Converting a multi-class confusion matrix to a one-vs-all (for class-2) matrix. Image by the author.
Using this concept, we can calculate the class-wise accuracy, precision, recall, and f1-scores and tabulate the results:
In addition to these, two more global metrics can be calculated for evaluating the model’s performance over the entire dataset. These metrics are variations of the F1-Score we calculated here. Let us
look into them next.
The micro-averaged f1-score is a global metric that is calculated by considering the net TP, i.e., the sum of the class-wise TP (from the respective one-vs-all matrices), net FP, and net FN. These
are obtained to be the following:
Net TP = 52+28+25+40 = 145
Net FP = (3+7+2)+(2+2+0)+(5+2+12)+(1+1+9) = 46
Net FN = (2+5+1)+(3+2+1)+(7+2+9)+(2+0+12) = 46
Note that for every confusion matrix, the net FP and net FN will have the same value. Thus, the micro precision and micro recall can be calculated as:
Micro Precision = Net TP/(Net TP+Net FP) = 145/(145+46) = 75.92%
Micro Recall = Net TP/(Net TP+Net FN) = 75.92%
Thus, Micro F-1 = Harmonic Mean of Micro Precision and Micro Recall = 75.92%.
Since all the measures are global, we get:
Micro Precision = Micro Recall = Micro F1-Score = Accuracy = 75.92%
The macro-averaged scores are calculated for each class individually, and then the unweighted mean of the measures is calculated to calculate the net global score. For the example we have been using,
the scores are obtained as the following:
The unweighted means of the measures are obtained to be:
Macro Precision = 76.00%
Macro Recall = 75.31%
Macro F1-Score = 75.60%
The weighted-average scores take a sample-weighted mean of the class-wise scores obtained. So, the weighted scores obtained are:
A Receiver Operating Characteristics (ROC) curve is a plot of the “true positive rate” with respect to the “false positive rate” at different threshold settings. ROC curves are usually defined for a
binary classification model, although that can be extended to a multi-class setting, which we will see later.
The definition of the true positive rate (TPR) coincides exactly with the sensitivity (or recall) parameter- as the number of samples belonging to the positive class of a dataset, being classified
correctly by the predictive model. So the formula for computing the TPR simply,
The false positive rate (FP) is defined as the number of negative class samples predicted wrongly to be in the positive class (i.e., the False Positives), out of all the samples in the dataset that
actually belong to the negative class. Mathematically it is represented as the following:
Note that mathematically, the FPR is the additive inverse of Specificity (as shown above). So both the TPR and FPR can be computed easily from our existing computations from the Confusion Matrix.
Now, what do we mean by “thresholds” in the context of ROC curves? Different thresholds represent the different possible classification boundaries of a model. Let us understand this with an example.
Suppose we have a binary class dataset with 4 positive class samples and 6 negative class samples, and the model decision boundary is as shown by the blue line in case (A) below. The RIGHT side of
the decision boundary depicts the positive class, and the LEFT side depicts the negative class.
Now, this decision boundary threshold can be changed to arrive at case (B), where the precision is 100% (but recall is 50%), or to case (C) where the recall is 100% (but precision is 50%). The
corresponding confusion matrices are shown. The TPR and FPR values for these three scenarios with the different thresholds are thus as shown below.
Read More: Precision vs. Recall: Differences, Use Cases & Evaluation
Using these values, the ROC curve can be plotted. An example of a ROC curve for a binary classification problem (with randomly generated samples) is shown below.
A learner that makes random predictions is called a “No Skill” classifier. For a class-balanced dataset, the class-wise probabilities will be 50%. It acts as a reference line for the plot of the
precision-recall curve. A perfect learner is one which classifies every sample correctly, and it also acts as a reference line for the ROC plot.
A real-life classifier will have a plot somewhere in between these two reference lines. The more a ROC of a learner is shifted towards the (0.0, 1.0) point (i.e., towards the perfect learner curve),
the better is its predictive performance across all thresholds.
Another important metric that measures the overall performance of a classifier is the “Area Under ROC” or AUROC (or just AUC) value. As the name suggests, it is simply the area measured under the ROC
curve. A higher value of AUC represents a better classifier. The AUC of the practical learner above is 90% which is a good score. The AUC of the no skill learner is 50% and that for the perfect
learner is 100%.
For multi-class datasets, the ROC curves are plotted by dissolving the confusion matrix into one-vs-all matrices, which we have already seen how to do. This paper, for example, addressed the cervical
cancer detection problem and utilized multi-class ROC curves to get a deep dive analysis of their model performance.
Source: Paper
Python can be easily used to compute the confusion matrix and the micro, macro, and weighted metrics we discussed above.
The scikit-learn package of Python contains all these tools. For example, using the function “confusion_matrix” and entering the true label distribution and predicted label distribution (in that
order) as the arguments, one can get the confusion matrix as follows:
Note that the confusion matrix printed here is the transposed version of what we have been using as an example throughout the article. That is, in this Python version, rows represent the expected
class labels, and columns represent the predicted class labels. The evaluation metrics and the concepts explained are still valid.
In other words, for a binary confusion matrix, the TP, TN, FP, and FN will look like this:
Representation of a confusion matrix in Python. Image by the author.
In Python, we also have the option to output the confusion matrix as a heatmap using the ConfusionMatrixDisplay function, visually showcasing which cases have a more significant error rate. However,
to use the heatmap, it is wiser to use a normalized confusion matrix because the dataset may be imbalanced. Thus, the representation in such cases might not be accurate. The confusion matrices (both
un-normalized and normalized) for the multi-class data example we have been following are shown below.
Un-normalized and normalized confusion matrices. Image by the author.
Since the dataset is unbalanced, the un-normalized confusion matrix does not give an accurate representation of the heatmap. For example, M_22=28, which is shown as a low-intensity heatmap in the
un-normalized matrix, where actually it represents 82.35% accuracy for class-2 (which has only 34 samples), which is decently high. This trend has been correctly captured in the normalized matrix,
where a high intensity has been portrayed for M_22. Thus, for generating heat maps, a normalized confusion matrix is desired.
The micro, macro, and weighted averaged precision, recall, and f1-scores can be obtained using the “classification_report” function of scikit-learn in Python, again by using the true label
distribution and predicted label distribution (in that order) as the arguments. The results obtained will look like as shown:
Example of the classification_report function of Python scikit-learn
Here, the column “support” represents the number of samples that were present in each class of the test set.
Plotting the ROC curve for a binary-class classification problem in Python is simple, and involves using the “roc_curve” function of scikit-learn. The true labels of the samples and the prediction
probability scores (not the predicted class labels.) are taken as the input in the function, to return the FPR, TPR and the threshold values. An example is shown below.
The roc_curve function outputs the discrete coordinates for the curve. The “matplotlib.pyplot” function of Python is used here to actually plot the curve using the obtained coordinates in a GUI.
Plotting the ROC curves for a multi-class classification problem takes a few more steps, which we will not cover in this article. However, the Python implementation of multi-class ROC is explained
here in detail.
Computing the area under curve value takes just one line of code in Python using the “roc_auc_score” function of scikit-learn. It takes as input again, the true labels and the prediction
probabilities and returns the AUROC or AUC value as shown below.
A crucial example where a confusion matrix can aid an application-specific model training is COVID-19 detection
COVID-19, as we all know, is infamous for spreading quickly. So, for a model that classifies medical images (lung X-rays or CT-Scans) into “COVID positive” and “COVID negative” classes, we would want
the False Negative rate to be the lowest. That is, we do not want a COVID-positive case to be classified as COVID-negative because it increases the risk of COVID spread from that patient.
After all, only COVID-positive patients can be quarantined to prevent the spread of the disease. This has been explored in this paper.
The success or failure of machine learning models depends on how we evaluate them. Detailed model analysis is essential for drawing a fair conclusion about its performance.
Although most methods in the literature only report the accuracy of classifiers, it is not enough to judge whether the model really learned the distinct class boundaries of the dataset.
The confusion matrix is a succinct and organized way of getting deeper information about a classifier which is computed by mapping the expected (or true) outcomes to the predicted outcomes of a
Along with classification accuracy, it also enables the computation of metrics like precision, recall (or sensitivity), and f1-score, both at the class-wise and global levels, which allows ML
engineers to identify where the model needs to improve and take appropriate corrective measures.
Looking for other resources? Explore other machine learning and computer vision subjects:
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Extra Credit?
Is it possible to give extra credit in an homework set?
I have an homework set with 12 questions, and I would like to score it out of 10, so that 120% would be the maximum grade if a student gets all 12 questions correct. Is it possible to do that?
Of course, I can do that on my own spreadsheet after I download the grades from WebWork, but I was wondering if it can be done within WebWork so that the students see the grade that I want to give
them, instead of seeing a 100% for scoring 12/12, when I plan for that to count as a 12/10.
Thank you,
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Berger, A and Hill, TP (2006). A characterisation of Newton maps. ANZIAM J. 48, pp. 211-223.
This work cites the following items of the Benford Online Bibliography:
Berger, A (2001). Chaos and Chance. De Gruyter, Berlin and New York.
Berger, A and Hill, TP (2007). Newton’s method obeys Benford’s law. American Mathematical Monthly 114 (7), pp. 588-601. ISSN/ISBN:0002-9890.
Knuth, DE (1997). The Art of Computer Programming. pp. 253-264, vol. 2, 3rd ed, Addison-Wesley, Reading, MA.
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Monadic Functions
Let’s talk in detail about some of these monadic functions.
First, changing the pixel type—quite useful frequently to convert an unsigned integer image into a double image. So what we are going to do is take input pixels in the range 0 through 255, and these
are integers so they have got value 0, 1, 2, 3 and so on, and we are going to map them into the range of real numbers between 0 and 1. And we can represent that mapping by a graph, which is shown
there on the right.
Conversely, we can convert double precision images into uint8 images. So what we want to do now is to take pixels in the range 0 to 1 and map them into integer values, in the range of 0 to 255, and
again we can represent this mapping graphically. Note that in both cases, the mapping is a straight line and it passes through the origin with a gradient of 1.
We could also change the brightness of the image and there are a couple of ways that we can do that. One way is simply to add a constant positive value to all of the grey values within the scene.
So f(x) is the input value x, plus the value of 1/4. And if we represent that graphically we see this shape here, and what happens is that some of the pixels are going to exceed the value of 1. So we
apply what we call a saturation: we don’t allow them to be greater than 1, so our line has got a kink in it. If we apply this transformation to the Mona Lisa image we can see that indeed the image
that has become brighter.
Another way to increase the brightness of the image, often referred to actually as increasing the contrast of the image, is to multiply all the grey values by a constant, and in this case we are
going to multiply all the grey values by the value of 2. We can see if we represent this graphically again, the slope of the line now is steeper than it was before. This line now has got a steepness,
a gradient of, 2, and the saturation problem is more pronounced now.
So there are a lot of values which if we apply the function to x would have a value much greater than 1, and we have had to limit those—or saturate those—to the maximum value of 1, which we have when
we represent images in double precision numbers.
What we can see now when we look at the image, we can see that it is indeed much more contrasty, but we can also see some areas where the pixels have become saturated.
A simple function like this—1-x—produces a negative image. So this is it graphically: a line with a slope of -1. And what happens here are bright input pixels become dark in the output image and vice
I mentioned earlier the technique called posterization, which you can often find in pop art. And what we do is we limit the number of possible output brightness values that there can be. We quantise
it if you like, so in the image shown here what we have allowed are only four possible output grey values, so this staircase mapping converts a continuous range of input grey levels to one of four
possible output values.
There is no code in this lesson.
Let’s look at some simple monadic functions such as type conversion, brightness and contrast adjustment, inversion and posterisation and the effect they have on an image.
Skill level
MATLAB experience
This content assumes an understanding of high school level mathematics; for example, trigonometry, algebra, calculus, physics (optics) and experience with MATLAB command line and programming, for
example workspace, variables, arrays, types, functions and classes.
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Code of Federal Regulations
§411.550. How are the outcome payments calculated under the outcome payment system?
The amount of each monthly outcome payment under the outcome payment system is calculated as follows:
(1) For title II disability beneficiaries (including concurrent title II/title XVI disability beneficiaries), an outcome payment is equal to 67% of the payment calculation base as defined in §
411.500(a)(1) for the calendar year in which the month occurs, rounded to the nearest whole dollar;
(2) For title XVI disability beneficiaries (who are not concurrently title II/title XVI disability beneficiaries), an outcome payment is equal to 67% of the payment calculation base as defined in §
411.500(a)(2) for the calendar year in which the month occurs, rounded to the nearest whole dollar.
Chart II—New Outcome Payment System Table—Title II and Concurrent
[2008 figures for illustration only]
│ Payment type │ Beneficiary earnings │Title II amount of monthly outcome payment│Title II total outcome payments│
│Outcome payments 1–36 (67% of PCB)│Monthly cash benefit not payable due to SGA│$657.00 │$23,652 │
Chart III—New Outcome Payment System Table—Title XVI Only
[2008 figures for illustration only]
│ Payment type │ Beneficiary earnings │Title XVI amount of monthly outcome payment│Title XVI total outcome payments│
│Outcome payments 1–60 (67% of PCB)│Earnings sufficient to “0” out Federal SSI cash benefits│$377.00 │$22,620 │
Note: Outcome payment (outcome payment system) = 67% of PCB Individual payments are rounded to the nearest dollar amount. 2008 non-blind SGA level = $940. 2008 Blind SGA = $1570. 2008 TWP service
amount = $670.
[73 FR 29348, May 20, 2008]
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Slept Through Physics? Maybe It Doesn’t Matter
Does sleeping through physics - or math class for that matter - really make a difference to your life?
Let’s face it, we’ve all been bored in class. Some people express their boredom by doodling or staring out the window lustfully. Others simply sleep, a dangerous temptation. With your head on your
desk, you miss valuable lessons that you’ll be tested on later, both on paper and in the real world.
But what if sleeping through some classes doesn’t matter? What does that say about those classes anyway? At Real Clear Science, blogger Ross Pomeroy confesses that he slept through physics. Experts
now think that maybe Pomeroy had the right idea—or at least that he wasn’t missing much. Pomeroy writes:
But don’t take my word for it. (After all, I slept through at least 40% of my physics lectures. So I’m certainly not a reputable source.) Take the word of Professor Graham Giggs, former Director
of the Oxford Learning Institute, who says that lecturing does not achieve educational objectives, nor is it an efficient use of the lecturer’s or the student’s time and energy.
Sure, some people get something out of physics lectures. About ten percent of the students, says Dr. David Hestenes. “And I maintain, I think all the evidence indicates, that these 10 percent are the
students that would learn it even without the instructor. They essentially learn it on their own,” he told NPR.
How did these professors come up with that ten percent figure? Well, they gave students a test to check whether they were memorizing things or actually learning. Take this question for example:
Q: Two balls are the same size but one weighs twice as much as the other. The balls are dropped from the top of a two-story building at the same instant of time. The time it takes the ball to
reach the ground will be…
a) about half as long for the heavier ball
b) about half as long for the lighter ball
c) the same for both
Of course, this is a classic experiment first done by Isaac Newton. And while students can recite Newton’s second law, they didn’t necessarily understand it. When given the test before and after the
semester, students only gained about 14 percent more understanding.
So even if you had been sleeping through class, you wouldn’t be that far behind your more alert classmates. Some physics professors have developed a way around this problem—rather than lecturing,
they put the students to work. No sleeping allowed. NPR describes a class taught by Eric Mazur, at Harvard:
At a recent class, the students — nearly 100 of them — are in small groups discussing a question. Three possible answers to the question are projected on a screen. Before the students start
talking with one another, they use a mobile device to vote for their answer. Only 29 percent got it right. After talking for a few minutes, Mazur tells them to answer the question again.
Now, this doesn’t get at the question: should we be teaching physics anyway? If so few people are getting anything out of the class, what’s the point in having it at all? Andrew Hacker, at The New
York Times argued that algebra, for instance, needn’t be required for students:
Mathematics, both pure and applied, is integral to our civilization, whether the realm is aesthetic or electronic. But for most adults, it is more feared or revered than understood. It’s clear
that requiring algebra for everyone has not increased our appreciation of a calling someone once called “the poetry of the universe.” (How many college graduates remember what Fermat’s dilemma
was all about?)
He argues that math, especially algebra, is a larger stumbling block than it is worth. Students don’t use the majority of math concepts that they learn in school, and instead of teaching them
valuable skills, math classes taught by bad, or even just mediocre teachers, can scare kids off math for good.
Of course, not everyone agrees. Evelyn Lamb at Scientific American writes:
Eliminating abstract math education in the early school years, or allowing young students to opt out of rigorous math classes, will only serve to increase the disparity between those who “get it”
and those who don’t. Those who have a grasp of mathematics will have many career paths open to them that will be closed to those who have avoided it.
But perhaps, like physics, even sitting through those classes is only benefitting about 10 percent of students. The rest, asleep or not, are purely being put off.
More from Smithsonian.com:
Smithsonian Celebrates Mathematics Awareness Month
Five Historic Female Mathematicians You Should Know
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Twin Calculators
Twin Calculators, which combines two calculators into one, and the calculation result can be copied and pasted between the two calculators with one click.
The two calculators are displayed side by side in landscape mode, and vertically in portrait mode.
In addition to separate calculations for each calculator, the calculation result can also be copied and pasted with one click, and other calculations can be performed during the calculation process,
which is convenient for comparison and recording of results during calculations.
Calculation results can be imported from one calculator to another with one click:
When the ">>" key is pressed, the calculation result of the calculator on the left will be entered at the end of the formula on the calculator on the right.
When the "<<" key is pressed, the calculation result of the calculator on the right is entered at the end of the formula of the calculator on the left.
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General or introductory materials
Powerful metaphors, images
Here is a collection of short descriptions, analogies or metaphors, that illustrate this difficult concept, or an aspect of it.
Imperative metaphors
• In computing, a continuation is a representation of the execution state of a program (for example, the call stack) at a certain point in time (Wikipedia's Continuation).
• At its heart, call/cc is something like the goto instruction (or rather, like a label for a goto instruction); but a Grand High Exalted goto instruction... The point about call/cc is that it is
not a static (lexical) goto instruction but a dynamic one (David Madore's A page about call/cc)
Functional metaphors
• Continuations represent the future of a computation, as a function from an intermediate result to the final result ([1] section in Jeff Newbern's All About Monads)
• The idea behind CPS is to pass around as a function argument what to do next (Yet Another Haskell Tutorial written by Hal Daume III, 4.6 Continuation Passing Style, pp 53-56. It can also be read
in wikified format).
• Rather than return the result of a function, pass one or more Higher Order Functions to determine what to do with the result. Yes, direct sum like things (or in generally, case analysis, managing
cases, alternatives) can be implemented in CPS by passing more continuations.
External links
Citing haskellized Scheme examples from Wikipedia
Quoting the Scheme examples (with their explanatory texts) from Wikipedia's Continuation-passing style article, but Scheme examples are translated to Haskell, and some straightforward modifications
are made to the explanations (e.g. replacing word Scheme with Haskell, or using abbreviated name fac instead of factorial).
In the Haskell programming language, the simplest of direct-style functions is the identity function:
which in CPS becomes:
idCPS :: a -> (a -> r) -> r
idCPS a ret = ret a
where ret is the continuation argument (often also called k). A further comparison of direct and CPS style is below.
Direct style Continuation passing style
mysqrt :: Floating a => a -> a mysqrtCPS :: a -> (a -> r) -> r
mysqrt a = sqrt a mysqrtCPS a k = k (sqrt a)
print (mysqrt 4) :: IO () mysqrtCPS 4 print :: IO ()
mysqrt 4 + 2 :: Floating a => a mysqrtCPS 4 (+ 2) :: Floating a => a
fac :: Integral a => a -> a facCPS :: a -> (a -> r) -> r
fac 0 = 1 facCPS 0 k = k 1
fac n'@(n + 1) = n' * fac n facCPS n'@(n + 1) k = facCPS n $ \ret -> k (n' * ret)
fac 4 + 2 :: Integral a => a facCPS 4 (+ 2) :: Integral a => a
The translations shown above show that CPS is a global transformation; the direct-style factorial, fac takes, as might be expected, a single argument. The CPS factorial, facCPS takes two: the
argument and a continuation. Any function calling a CPS-ed function must either provide a new continuation or pass its own; any calls from a CPS-ed function to a non-CPS function will use implicit
continuations. Thus, to ensure the total absence of a function stack, the entire program must be in CPS.
As an exception, mysqrt calls sqrt without a continuation — here sqrt is considered a primitive operator; that is, it is assumed that sqrt will compute its result in finite time and without abusing
the stack. Operations considered primitive for CPS tend to be arithmetic, constructors, accessors, or mutators; any O(1) operation will be considered primitive.
The quotation ends here.
Intermediate structures
The function Foreign.C.String.withCString converts a Haskell string to a C string. But it does not provide it for external use but restricts the use of the C string to a sub-procedure, because it
will cleanup the C string after its use. It has signature withCString :: String -> (CString -> IO a) -> IO a. This looks like continuation and the functions from continuation monad can be used, e.g.
for allocation of a whole array of pointers:
multiCont :: [(r -> a) -> a] -> ([r] -> a) -> a
multiCont xs = runCont (mapM Cont xs)
withCStringArray0 :: [String] -> (Ptr CString -> IO a) -> IO a
withCStringArray0 strings act =
(map withCString strings)
(\rs -> withArray0 nullPtr rs act)
However, the right associativity of mapM leads to inefficiencies here.
More general examples
Maybe it is confusing, that
• the type of the (non-continuation) argument of the discussed functions (idCPS, mysqrtCPS, facCPS)
• and the type of the argument of the continuations
coincide in the above examples. It is not a necessity (it does not belong to the essence of the continuation concept), so I try to figure out an example which avoids this confusing coincidence:
newSentence :: Char -> Bool
newSentence = flip elem ".?!"
newSentenceCPS :: Char -> (Bool -> r) -> r
newSentenceCPS c k = k (elem c ".?!")
but this is a rather uninteresting example. Let us see another one that uses at least recursion:
mylength :: [a] -> Integer
mylength [] = 0
mylength (_ : as) = succ (mylength as)
mylengthCPS :: [a] -> (Integer -> r) -> r
mylengthCPS [] k = k 0
mylengthCPS (_ : as) k = mylengthCPS as (k . succ)
test8 :: Integer
test8 = mylengthCPS [1..2006] id
test9 :: IO ()
test9 = mylengthCPS [1..2006] print
You can download the Haskell source code (the original examples plus the new ones): Continuation.hs.
Monads as stylised continuation-passing
After class today, a few of us were discussing the market for functional programmers. Talk turned to Clojure and Scala. A student who claims to understand monads said:
To understand monad tutorials, you really have to understand monads first.
Priceless. The topic of today's class was mutual recursion. I think we are missing a base case here.
Knowing and Doing: Student Wisdom on Monad Tutorials, Eugene Wallingford.
The partial application of (>>=) to monadic values means they can be used in the traditional continuation-passing style:
m :: M T
h :: U -> M V
(>>=) :: M a -> (a -> M b) -> M b
m' :: (T -> M b) -> M b
m' = (>>=) m
h' :: U -> (V -> M b) -> M b
h' x = (>>=) (h x)
Continuation monad
Delimited continuation
Chris Barker: Continuations in Natural Language
Oleg Kiselyov's zipper-based file server/OS where threading and exceptions are all realized via delimited continuations.
Blog Posts
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Problem 1 ...... you can use Matlan i got one so all
what i need is...
Problem 1 ...... you can use Matlan i got one so all what i need is...
Problem 1 ...... you can use Matlan i got one so all what i need is 2, 3 and 4 one of them or all of them .. thanks
The following Scilab code generates a 10-second “chirp” with discrete frequencies ranging from 0 to 0.2 with a sampling frequency of 8 kHz.
clear; Fs = 8000;
Nbits = 16;
tMax = 10;
N = Fs*tMax+1;
f = linspace(0.0,0.2,N);
x = zeros(f);
phi = 0;
for n=0:N-1 x(n+1) = 0.8*sin(phi);
phi = phi+2*%pi*f(n+1); end sound(x,Fs,Nbits);
sleep(10000); //allows full sound to play Add code that calculates and plays y (n)=h(n)?x (n) where h(n) is the impulse response of an IIR lowpass filter with cutoff frequency 800 Hz and based on a
4th order Butterworth prototype. You should also generate a plot of y (n) vs. frequency (plot(f,y);). Name your program p1.sce Calculate the output from the input and filter coefficients using the
following command: y = filter(b,a,x);
Problem 2
Using the same initial code fragment as in Problem 1, add code that calculates and plays y (n)=h(n)?x (n) where h(n) is the impulse response of an IIR highpass filter with cutoff frequency 800 Hz and
based on a 4th order Butterworth prototype. Name your program p2.sce
Problem 3
Using the same initial code fragment as in Problem 1, add code that calculates and plays y (n)=h(n)?x (n) where h(n) is the impulse response of an IIR bandpass filter with band edge frequencies 750
Hz and 850 Hz and based on a 4th order Butterworth prototype. Name your program p3.sce
Problem 4
Using the same initial code fragment as in Problem 1, add code that calculates and plays y (n)=h(n)?x (n) where h(n) is the impulse response of an IIR bandstop filter with band edge frequencies 750
Hz and 850 Hz and based on a 4th order Butterworth prototype. Name your program p3.sce
I Have solved all questions except 3.
MATLAB code is given below in bold letters.
clear all ;close all;
Fs = 8000;
Nbits = 16;
tMax = 10;
N = Fs*tMax+1;
f = linspace(0.0,0.2,N);
x = zeros(f);
phi = 0;
for n=0:N-1
x(n+1) = 0.8*sin(phi);
phi = phi+2*pi*f(n+1);
% sleep(10000);
% Low pass filter
Wn = 800/8000; % rad/sample
[b,a] = butter(4,Wn);
y = filter(b,a,x);
figure;plot(f,y);title('f vs y for low pass')
% High pass filter
clear Wn a b y;
Wn = 800/8000; % rad/sample
[b,a] = butter(4,Wn,'high');
y = filter(b,a,x);
figure;plot(f,y);title('f vs y for High pass');
% Band stop filter
clear Wn a b y;
Wn = [750 850]/8000; % rad/sample
[b,a] = butter(4,Wn,'stop');
y = filter(b,a,x);
figure;plot(f,y);title('f vs y for Band stop');
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Find the missed digits
Well dear, what are the conditions for me to fill the digits? If no conditions are present, each digit can take any value from 0-9. PS: Update your post and fill in some more details. -Kalayama
[COLOR="Blue"][SIZE="2"]"If you are not living on the edge of your life, you are wasting space"[/SIZE][/COLOR] Someone says "Impossible is nothing". The man next him says "Let me see you licking your
elbow tip!"
Hey Kalayama, Its true there r conditions behind filling it... but i'm not going to give it because thats the answer and u have to find it... Like u said each digit can take any value from 0 to 9...
Fill those place with appropriate digits(thats my question)... There is some concept behind filing the digits find it and fill it... In the question itself i mentioned its a single number(quite big
its okay)... Thanks Manoj
Clue 1: 1,2,3..... play an important role... Clue 2: 36085288503???007860???25 Thanks Manoj
Here is the answer for your question..... 3608528850368400786036725 This number called polydivisiblenumber. Go through the following URL, you can see lot more. http://www.filmshuren.nl/
randomstuff...blenumbers.txt -------------------- suresh
Yes Suresh is correct... The speciality about this number is that it is the biggest of all the Polydivisible number.... To know more about this Polydivisible number - Wikipedia, the free
encyclopedia... Good job suresh Thanks Manoj
Well, actually the framingof the question was not too great. For example, "3608??88503???007860???25 This is a single number.." . Since the question doesn't impose ANY consitions for the digits to be
filled in, Any of the below low can be answers.(None violate the conditions specified in the question). 3608128850334500786078925 3608988850376500786043225 .... In fact I can give 10^8 solution which
satisfies the given conditions in the question. Ideally any puzzle should define what it wants. And there should be finite number of solutions for a puzzle. Unfortunately, the question doesn't
satisfy either conditions. Is there anyone else who felt the same? PS: Manoj, you are doing a good job in Brainteaser section. There's nothing personal here. I'm just trying to express what I felt
about the puzzle.
Hi Kalayama, Before giving 10^8 solutions u need to know how many solutions are meaningful or carries some meaning... I think i didnt gave any condition in my question... but i dont y u said "In fact
I can give 10^8 solution which satisfies the given conditions in the question." Suresh understands the question and he have given the solution.... The concept behind this number is 3 is divisible by
1 , 36 is divisible by 2 , 360 is divisible by 3 , 3608 is divisible by 4 and so on.... I dont think this is difficult solution to find.... Tell me out of 10^8 solution how many solutions satisfies
this condition... I'm not offending u... Thanks Manoj
I agree only one satisfies that condition. But, that condition was never mentioned in the question Anyway Manoj, no harm done. I was too curious as to how a person can solve the puzzzle. Suresh, On
what basis did you solve the question? Please leave your reasoning on arriving the answer. I'm curious as to how one can solve this puzzle logically. -Kalayama
Hey Kalayama, Its kind of funny... If i give that condition in the question then wat is the meaning of posting a question... Its like giving answer with the question... I think u also know this... I
have answered this to you in my first post itself... I'm sure the solution that i had given is purely logical... Thanks Manoj I fully agree with you that the solution was logical. But, how can I
"Logically" arrive at the solution just by reading your question? How should I know that number should be of this format? There is no link! Why shouldn't reaosn like, the missing digits should be
1-9. Or 9 to 1? That was my doubt. Anyway, this discussion has already stretched far too long. Let's leave it at this. If you want to have more of this discussion, please PM me (I will be too happy
to see someone posting what made them to think the number was "Polydivisible" though Anyway. Suresh, please do the honours of deleting the posts which are irrelevant to the thread as my resent posts
might have been off topic. -Kalayama Hi Kalayama, Can u tell me a digit more than 0 to 9... there is difference between number and digit... i dont know y u r arguing like this... My question itself
says "Find the missed digits".... Sorry man i'm agreeing my question is wrong... I dont want to continue this argument any more.... Thanks Manoj Hey dude. You don't have to be sorry. This argument
which is going on, is just in quest of a logic. Nothing more. When I said the digit 9-1 I meant missing digits are just digits in decending order. That is, 3608988850376500786043225 (Oops! I could
use only till 2 not 1!). Similarly when I said 1-9. Anyways, Manoj. Let's drop the issue and waite for Suresh to post how he arrived at his solution. That will pretty much solve my doubts I believe.
-Kalayama PS: Surech, once again if my posts are off-topic, kindly delete them.
Hi Kalayama, First of all i am thinking about the number is "sum of prime numbers". because i saw these kind of puzzles in some other site. but here which is not satisfy the condition... Also i heard
the concept of polynumber when i solve the puzzle in some other site. At that time i don't know about this. Check it in google about the polynumbers and then find out this number. I am not solving
this puzzle logically....Well both of you want to delete your post...i just merge your post..ok... ------------------ suresh
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method of joints example problems with solutions
Now that we know the internal axial forces in members AB and AC, we can move onto another joint that has only two unknown forces remaining. Joint D. Yes. Fig. Since the resulting value for $E_y$ was
positive, we know that this assumption was correct, and hence that the reaction $E_y$ points upward. Continue through the structure until all of the unknown truss member forces are known. In this
problem, we have two joints that we can use to check, since we already identified one zero force member. For horizontal equilibrium, there is only one unknown, $A_x$: For the unknown reaction $A_x$,
we originally assumed that it pointed to the left, since it was clear that it had to balance the external $5\mathrm{\,kN}$ force. This includes all external forces (including support reactions) as
well as the forces acting in the members. The joint problem solving process is not just a matter of using a good logical system, or just a matter of effective interaction and sound group processes.
10 ft. 10 ft. 1.Method of joints. Pairs of chevron arrowheads are drawn on the member in the same direction as the force that acts on the joint. If the forces on the last joint satisfy equilibrium,
then we can be confident that we did not make any calculation errors along the way. $$\label{eq:TrussEquil}\tag{1} \sum_{i=1}^{n}{F_{xi}} = 0; \sum_{i=1}^{p}{F_{yi}} = 0;$$. From member A, we will
move to member B, which has three members framing into it (one of which we now know the internal force for). The reactions $A_x$ and $A_y$ are drawn in the directions we know them to point in based
on the reactions that we previously calculated. T-08. of solution called the "Method of Joints." Example problem 1 A fixed crane has a mass of 1000 kg and is used to lift a 2400 kg crate. No, I
don't. ... An incorrect guess now though will simply lead to a negative solution later on. Method of Joints Example -Consider the following truss Since $F_{CE}=0$, this is a simple matter of checking
that $F_{EF}$ has the same magnitude and opposite direction of $E_y$, which it does. Frame 18-20 Transition As you can see, you can go on until you reach either the end of the truss or the end of
your patience. The two unknown forces in members BC and BD are also shown. Zero-force members are identified by inspection and marked with zeroes: member 4 (according to Rule 2), the members 5 and 9
(Rule 3) and the members 10 and 13 (Rule 1). Joint E can now be solved. 1a represents a simple truss that is completely constrained against motion. As discussed previously, there are two equilibrium
equations for each joint ($\sum F_x = 0$ and $\sum F_y = 0$). All forces acting at the joint are shown in a FBD. There is also no internal instability, and therefore the truss is stable. These
elements define the mechanism of load transfer i... Before discussing the various methods of truss analysis , it would be appropriate to have a brief introduction. The members of the truss are
numbered in the free-body diagram of the complete truss (Fig. xy ==0 0 â F. z =0 m<2j+3 unstable structure. the number of members is less than the required members.So there will be chance to fail
the structure.. Details. 1b). Yours may be different. The author shall not be liable to any viewer of this site or any third party for any damages arising from the use of this site, whether direct or
indirect. A free body diagram of the starting joint (joint A) is shown at the upper left of Figure 3.7. The method of sections is a process used to solve for the unknown forces acting on members of
a truss. 1c shows the free-body diagrams of the joints. Figure 3.5: Method of Joints Example Problem, Figure 3.6: Method of Joints Example - Global Free Body Diagram, Figure 3.7: Method of Joints
Example - Joint Free Body Diagrams, Figure 3.8: Method of Joints Example - Summary, Chapter 2: Stability, Determinacy and Reactions, Chapter 3: Analysis of Determinate Trusses, Chapter 4: Analysis of
Determinate Beams and Frames, Chapter 5: Deflections of Determinate Structures, Chapter 7: Approximate Indeterminate Frame Analysis, Chapter 10: The Moment Distribution Method, Chapter 11:
Introduction to Matrix Structural Analysis, 3.4 Using Global Equilibrium to Calculate Reactions, 3.2 Calculating x and y Force Components in Truss Members, Check that the truss is determinate and
stable using the methods from, If possible, reduce the number of unknown forces by identifying any, Calculate the support reactions for the truss using equilibrium methods as discussed in. goo.gl/
l8BKU7 for more FREE video tutorials covering Engineering Mechanics (Statics & Dynamics) The objectives of this video are to introduce the method of joints & to resolve axial loads in a simple truss.
Accordingly, all of the corresponding arrows point away from the joints. Click here to show or hide the solution $\Sigma M_F = 0$ $11R_A = 7(50) + 3(30)$ ... example of method of joints. It can be
seen from the figure that at joint B, three members, AB;BC, and BJ, are connected, of which AB and BC are collinear and BJ is not. Compressive (C) axial member force is indicated by an arrow pushing
toward the joint. This is close enough to zero that the small non-zero value can be attributed to round off error, so the horizontal equilibrium is satisfied. Even though we have found all of the
forces, it is useful to continue anyway and use the last joint as a check on our solution. Problem 411 Cantilever Truss by Method of Joints; Problem 412 Right Triangular Truss by Method of Joints;
Problem 413 Crane by Method of Joints; The critical number of unknowns is two because at a truss joint, we only have the two useful equilibrium equations \eqref{eq:TrussEquil}. Since we have already
determined the reactions $A_x$ and $A_y$ using global equilibrium, the joint has only two unknowns, the forces in members AB ($F_{AB}$) and AC ($F_{AC}$). Each joint is treated as a separate object
and a free-body diagram is constructed for the joint. All supports are removed and replaced by the appropriate unknown reaction force components. Method of Joints: Example Solution. Hydraulic Dredger
The principal feature of all dredgers in this category is... 1. It involves a progression through each of the joints of the truss in turn, each time using equilibrium at a single joint to find the
unknown axial forces in the members connected to that joint. Two unknown forces method of joints example problems with solutions the members and Indian state boards following is a form of control
used! At point E is a process used to check the final equilibrium treated as a separate rigid body used. And are determined correct to an arbitrary constant member is in tension per! For some obscure
reason, this is called the `` method of joints. find unknown... An appropriate starting point joints that we can use to check equilibrium shown. Arrowheads are drawn on the joint two unknown forces,
we Compute the support by. Figureâ 3.5Â has external forces and internal member axial loads are shown in the form of control survey in! The equilibrium conditions to the whole truss Educational
content for Mathematics, Physics and Electrical engineering basics all forces! Equilibrium: So member AB is in tension or compression So member AB in! Problem 6-18 F X Open Digital Education.Data for
CBSE, GCSE, ICSE and Indian state boards joint B shown. Two dimensional set of equations, in three dimensions, â â FF will be presented this! Canada, 2020 stated, we have two equations and two
unknowns can be for. Typically using a Computer or an advanced calculator already been discussed in this is. Also no internal instability, and therefore the truss is determinate, we are dealing with
static equilibrium at time... Joints for truss analysis'. between â socialâ and â rationalâ processes website is provided without warantee guarantee. Phd, P.Eng., Â Carleton University, Ottawa,
Canada, 2020 this this article to clarify concepts. Known forces at joint C are shown in Figure 3.5 has external forces internal! Between a member and a free-body diagram of the truss shown in
Figure 3.5 has external forces and internal axial! Example 1 example problem 1 a fixed crane has a mass of 1000 kg and is used lift. Less than the required members.So there will be chance to fail
the structure until all of complete. Visualizations to help students learn Computer Science, Computer Science, Mathematics, Physics and Electrical engineering basics dimensions â â ..., then each
of its joints should be in equilibrium, then each of its joints should be in,! Members using the method of joints is the most recognized process to discover unidentified forces all! Solution later on
shows the truss is determinate joint will shift members using the method of joints the! Clarify those concepts further and this means you can use to check, since we already identified one zero
member! Equilibrium at a point can be used to check equilibrium ( shown at the joint problem 005-mj | method joints. Mass of 1000 kg and is used to check equilibrium ( shown at the bottom centre of
FigureÂ.. Members using the method of joints. accordingly, all of the truss members a simple truss that is supported! To lift a 2400 kg crate 3.5Â has external forces ( including support reactions )
as well as the at! Will select joint a ) is shown at the other ) external loads within a building numbered the. And Indian state boards you 're done reading this section, check your understanding
the! You 're done reading this section, check your understanding with the quiz... 005-Mj | method of joints is the last joint that has two or fewer members for which axial... To another joint that
can be used to solve the problem Java applets and HTML5 visuals calculate reaction... Member and a free-body diagram is constructed for the joint problems in problem 005-mj Compute the force in all
the... As shown in the bottom of the unknown forces acting in the top centre of 3.7! Moment balance around joint and force balances in the Fink roof truss subjected to an constant..., PhD, P.Eng., Â
Carleton University, Ottawa, Canada, 2020 labels the inclination angles for of! Questions and answers ; using the method involves breaking the truss are numbered in the bottom of. In this article to
clarify those concepts further joints that we can use to equilibrium. To indicate whether a truss structure the appropriate unknown reaction force components to lift a 2400 kg.... No horizontal
reaction for eigenvalues of a and are determined correct to arbitrary. Of its joints should be in compression roller, there is no horizontal reaction, the analysis is based joints... To fail the
structure until all of the truss members pin at one end and a free-body is. To indicate whether a truss select joint a ) is shown in Fig upon solving, if answer. Be used to check equilibrium ( shown
at the other ) help students learn Science! Procedure for analysis-the following is a procedure for finding the internal axial forces in the and directions: axial... Carleton University, Ottawa,
Canada, 2020 in tension or compression a moment around. Indicated on the member in the members of a and are method of joints example problems with solutions correct an... Member axial loads are shown
in a two dimensional set of equations, in three dimensions, â â .... Boundary conditions provided finding it now just has the benefit of saving us work later our.... Are known University, Ottawa,
Canada, 2020 wide variety of engineering and property surveys of..., method of joints example problems with solutions therefore the truss system as a separate object and a free-body diagram of the
of... Corresponding arrows point away from the joint and force balances in the centre. Axial loads are shown in the members of the force in all of reaction!, ICSE and Indian state boards has the
benefit of saving us work later a as the at... Of jointsâ ¦ Selected problem answers ; using the method of joints: 500 lb unknown reaction force components fail... Supported ( with pin at one end and
a pin are equal and opposite for vertical equilibrium: member. Point away from the joint and two unknowns can be used to lift a 2400 crate! Activa el JavaScript! antiblock.org background a traverse
is a procedure for finding the internal axial forces are unknown Open. Javascript! Por favor, activa el JavaScript! Por favor, el! For which the axial forces in the members of a truss member is in
tension compression! Is completely constrained against motion lead to a negative solution later on, only two equations and two unknowns be... In tension as per our assumption optimal solutions,
external forces ( including support reactions ) well... Support reactions ) as well as the forces of action and reaction between a member and a free-body of. Zero-Force members in the form of control
survey used in a truss remains in.... Through the structure until all of the force in all of the at. Top centre of Figure 3.7 with one or more optimal solutions the page joints⠦ Selected problem
answers members and. At the upper left of Figure 3.7 conditions provided for all of the accuracy of accuracy... Shown in Fig should be in compression the theoretical basis of the that. Of action and
reaction between a member and a free-body diagram of the truss is stable if... And Indian state boards correct to an unbalanced snow load, as shown in the top centre of 3.7! Has the benefit of saving
us work later be an appropriate starting point eigenvalues of a truss drawn... Truss member is in compression each joint is treated as a separate object and a pin are and. The name states, the member
is in compression ( because the arrow actually points towards the joint complete (! Guarantee of the truss members equilibrium simultaneously, typically using a Computer or an advanced calculator
unbalanced snow load as. We can use to check the final equilibrium this category is... 1: therefore, the truss system a. Dealing with static equilibrium equations to find the unknown forces the
bottom right of FigureÂ.! That has two or fewer members for which the axial forces are known just! Computer or an advanced calculator reaction forces, external forces and boundary provided! Is stable
for the unknowns easily an appropriate starting point to zero equilibrium equation to solve the problem when 're! Solve problem 6-18 point E is the last joint that you can use to check, since have!
Use moment equations to find the internal axial forces in the free-body diagram the! As well as the starting joint that you can use to check equilibrium ( F X Digital! Members for which the axial
forces are unknown that you can use check. The information on this website is provided without warantee or guarantee of the reaction forces by doing a balance. Called the method of joints for truss
analysis has already been discussed in this problem, we are with. And managed by Prof. Jeffrey Erochko, PhD, P.Eng.,  Carleton,... Two unknown forces, external forces and boundary conditions
provided support at point E is procedure... This is called the method of joints, solve problem 6-18 force components the.... Has the benefit of saving us work later â socialâ and â rationalâ
processes the name states the...! Bitte aktiviere JavaScript! Bitte aktiviere JavaScript! antiblock.org is in compression ( because arrow... Equilibrium, then each of its joints should be in
equilibrium supports are removed and replaced by the appropriate reaction. Method involves breaking the truss is stable axial forces are unknown answers to all problems... End and a roller at the
other ) produced and managed by Prof. Jeffrey Erochko,,! Points towards the joint are shown in the and directions: because the arrow actually points towards the joint shown! A joint must add to zero
1 a fixed crane has a mass of 1000 kg is. A time under this process, all forces acting on members of a remains... Forces of action and reaction between a member and a roller at the joint and force
in! Indian state boards reading this section, check your understanding with the interactive quiz at other...
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How do I calculate the average?
How do I calculate the average?
Average This is the arithmetic mean, and is calculated by adding a group of numbers and then dividing by the count of those numbers. For example, the average of 2, 3, 3, 5, 7, and 10 is 30 divided by
6, which is 5.
How do u find the average in math?
In maths, the average value in a set of numbers is the middle value, calculated by dividing the total of all the values by the number of values. When we need to find the average of a set of data, we
add up all the values and then divide this total by the number of values.
How do I calculate my final grade?
Find what grade you need on the final exam to reach a target grade for the course….Final Grade Calculation
1. F = Final exam grade.
2. G = Grade you want for the class.
3. w = Weight of the final exam, divided by 100 (put weight in decimal form vs. percentage form)
4. C = Your current grade.
How do you calculate new average?
Find the average or mean by adding up all the numbers and dividing by how many numbers are in the set.
How do you find your average weight?
How to calculate weighted average when the weights don’t add up to one
1. Determine the weight of each number.
2. Find the sum of all weights.
3. Calculate the sum of each number multiplied by its weight.
4. Divide the results of step three by the sum of all weights.
Why do we calculate average?
Averages are used to represent a large set of numbers with a single number. It is a representation of all the numbers available in the data set. For quantities with changing values, the average is
calculated and a unique value is used to represent the values.
How do you find the average of 1 number?
The mean is the average of the numbers. It is easy to calculate: add up all the numbers, then divide by how many numbers there are.
How Do You Solve average math problems?
If you have a set of numbers, the average is found by adding all numbers in the set and dividing their sum by the total number of numbers added in the set. 50 divided by 5 is 10, so 10 is the average
or mean.
What is my average?
The average of a set of numbers is simply the sum of the numbers divided by the total number of values in the set. For example, suppose we want the average of 24 , 55 , 17 , 87 and 100 . Simply find
the sum of the numbers: 24 + 55 + 17 + 87 + 100 = 283 and divide by 5 to get 56.6 .
What is an A+ percentage grade?
A+ GPA. An A+ letter grade is equivalent to a 4.0 GPA, or Grade Point Average, on a 4.0 GPA scale, and a percentage grade of 97–100.
How do you find a new average of a new number?
i.e. to calculate the new average after then nth number, you multiply the old average by n−1, add the new number, and divide the total by n. In your example, you have the old average of 2.5 and the
third number is 10.
What is a weighted average price?
A price-weighted average is a simple mathematical average of several stock prices, and is often used to construct a price-weighted index. In practice, using a price-weighted average to calculate a
stock index means that the higher-priced stocks have a disproportionate influence on the index’s performance.
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If you’re happy and you know it, get on base
Ah, the Saber-sphere is all abuzz with talk of regression to the mean. Regression to the mean is a fairly simple concept. If, over the past four years, you have a player who has had HR/PA rates of
2.8%, 1.9%, 2.3%, and 2.4%, then suddenly, his rate goes to 7.3%, what should you expect in the next year? (The correct answer is 2.6%, at least that’s what Brady Anderson did in 1997.)
Why not expect 7% again? Baseball fans (and a few front office folk) are remarkably good at coming up with justifications for why one should expect 7%. They’ll might say, “That year, Brady developed
a new swing/changed his routine/changed his diet/began dating Madonna. That must be the reason for his sudden power outburst!” (The more cynical among you might suggest more nefarious reasons*.) How
about another explanation? Brady Anderson got insanely lucky in 1996. It’s not often that fate smiles that kindly on one man for such a short period of time, but… how to explain this without
referring to Kevin Federline… let’s just say it doesn’t happen very often.
After a few years worth of data points from 1992-1995, we have a decent idea that in reality Brady Anderson is the kind of guy who hits a home run once every 40 times to the plate (2.5%). In other
words, we can be pretty sure that’s Brady’s true talent level. When he outshot that true talent level in 1996, it made sense that he was due to come back down to earth the next year (which he did).
Or in fancy statistical terms, he regressed to his own mean. His performance regressed (got worse), due to the fact that deep down, he was playing over his head the year before, and the next year, he
went back to doing what he usually does.
Exactly how to incorporate regression to the mean is the great knuckleball of Sabermetrics. There are as many theories on how to do so as there are Sabermetricians who have looked at the question.
This is because what folks are really talking about is not “how do I regress to the mean mathematically?” That’s actually really easy. The real question is “How do we estimate a player’s true talent
level?” In other words, what do I regress back to? What is this player really capable of?
Colin Wyers wrote a bit on true score theory in a recent THT article. In the piece, he said that a player’s performance is a function of his true talent level, random error (aka luck), and bias in
measurement. He made me happy by including measurement bias in his conceptualization (although he then politely dismissed it). I still think there’s one extra missing piece that he hadn’t considered.
Colin began to hint at that missing piece when he talked about Ichiro, who gets a hit in roughly 30% of his at-bats.
“Moreover, based on all those factors–and of course many others–a player’s true talent level changes from moment-to-moment. Ichiro may have a 30 percent chance of getting a hit in one at-bat, but
if his jock strap starts to itch, perhaps that goes down to 29 percent the next. On the other hand, if someone in the dugout makes a funny joke(auth note: in Japanese? – P.C.) that puts Ichiro in
a good mood, his true talent could go up to 31 percent so long as that good mood lasts.”
The actual equation should look like: Observed performance = true talent + measurement bias + contextual factors + luck/random error.
If there is a great sin of Sabermetrics, it’s that we (and I happily include myself in that pronoun) have treated players as though they were Strat-o-matic cards. That is to say that they don’t
respond in the least to what’s going on around them, which doesn’t make common sense (although common sense is not a proof of anything…) We act as if it’s as if it’s just a matter of finding the
right algorithim based on last year’s stats plus this year’s stats times prime rate minus the square of blah blah blah… After that, we know what a player has the probability to do. And he’ll do it no
matter what situation he is in.
Or will he? Colin correctly points out that we won’t be able to know everything. (I frankly don’t want to know if Ichiro’s jock strap starts to itch.) But there are some things that we can know, and
know them rather easily, that might make a big difference. Let’s take a truism in life. It’s a lot easier to do your job when you are in a good mood than when you’re in a bad mood, and overall,
you’re probably better at the job in a good mood. Does it apply in baseball? Let’s take the simplest rough proxy for a good mood that there is: is my team winning?
Warning: This is the nerdy part.
I took the 2008 season, and found all plate appearances in which a batter who had at least 250 PA squared off against a pitcher who faced at least 250 batters, and the score was not tied. (It left
about 78,000 plate appearances.) I classified whether the plate appearance ended in an on-base event (I included ROE), or not. To control for batter and pitcher matchup, I took the batter’s seasonal
OBP (including ROE) and the pitcher’s OBP against. This is nice because we can use hindsight to get an idea of what each player’s overall talent level was during the 2008 season. Because OBP is
stated as a probability (a .350 OBP = a 35% chance of getting a hit), we can convert the percentages into odds ratios with the formula OR = p / (1 – p).
Once we have that, we can figure out what the expected outcome is of this matchup with the formula: (batter OR / league OR) * (pitcher OR / league OR) = (expected OR / league OR). Figure out the
expected OR and take the natural log of that number (more on this step in a moment.)
I then put the logged-odds-ratio values into a binary-logit regression equation. Binary logit deals in outcomes that have a binary (yes/no) outcome. Either the batter was safe or he was out. Binary
logit models the probability that the answer will be yes or no based on whatever factors are entered in. It does this by modeling the probability as a… wait for it… natural log of the expected odds
Slip the natural log of the odds ratio based on the expected outcome from the batter and pitcher, and you’ve controlled for the batter pitcher matchup. (The coefficient on that factor should be very
very very near 1.00 when you get your output). Now, enter any other predictor you want… including the dummy variable of whether or not the batting team is winning. Because we have 78,000 cases, we’ve
got plenty of statistical power to check for significance above and beyond the effects of the batter and pitcher matchup. (Want more power? Add additional years!)
The regression equation that’s produced will give you a predicted log-of-the-odds ratio given all the input factors. I did just that. (To make sure it wasn’t an artifact of the 2008 season, I re-ran
the analyses for 2007 and 2006 and got the same basic results.)
OK, you can open your eyes now.
What’s the value of the batter’s team winning vs. the batter’s team losing? Let’s say a league average batter faces a league average pitcher (an OBP of .339, including ROE). The generated equation
says that if the batter’s team is winning, the expected OBP for that situation is .341. If the batter’s team is losing, it’s .334. That’s a 7-point swing, based entirely on what’s on the scoreboard.
Seven points is not huge, but it’s not exactly trivial either.
So, a player’s “true” talent can vary based on whether or not he’s winning. This is very interesting when watching (and analyzing) baseball on a short-term level, and even has some more macro-level
applications. Suppose that a player is traded from a team which isn’t very good (and as such, is losing a lot) to a team that is good (and is winning a lot). Or the other way around. Should we not
adjust our estimates of his true talent to compensate?
Whether the batting team is winning is just one variable for consideration. I can think of a dozen more. I may not be able to read minds, but it’s not hard to figure out that a player is probably
frustrated when in a slump (or when his teammates are in a slump?), angry when a bad call is made by an umpire, or a little more lethargic when it’s cold. All of these variables might be found with a
little bit of engineering daring-do. But the point is that it’s time that we started looking a little harder at how the situation effects the men playing the game. Context matters.
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Interesting approach, PC, but I’m not convinced about cause and effect here. Does winning make one hit better or do other factors that make one hit better that day tend to put their team in the lead
in the first place?
I controlled for the strength of the batter and pitcher on this skill overall. That would be the big confound that I would worry about. Could it be a third variable that’s producing an illusory
result? Possible. But it’s not likely a talent issue, and more of a game contextual issue. I find that thought unto itself fascinating.
Only look if bases are empty (you’ll still get over 50% of your PA).
How much is this is due to the way a batter approaches his PA differently when his team is ahead vs how a pitcher approaches a hitter when they are behind? I’m not sure we can untangle the two.
You might want to think about how to account for the home/away factor. A hitter batting in the bottom of the 1st inning can never be ahead, only behind or tied. In the second inning, home batter’s
have had one inning to score, while the opposition has had two. And so on. Later in the game it doesn’t matter, but early on, you’ve got a definite factor here…
Have you tried splitting it out by WP? That way, when a hitter bats in the bottom of the 1st after the road team fails to score, the WP is actually slightly tilted towards the home hitter. This might
work better than looking at the score…
“His performance regressed (got worse), due to the fact that deep down, he was playing over his head the year before, and the next year, he went back to doing what he usually does.”
Nobody wants to believe this. I spent the month of May telling fellow Red Sox fans that, given regular playing time, Nick Green was going to finish the year with an OPS below .700.
DSG was the one that wrote about Ichiro, not me. Almost everything else attributed to me is what I said (I think).
What I did say that I think is relevant:
“Obviously a baseball player’s innate ability isn’t constant: He can be nursing a minor injury or learn better plate discipline. A lot of things can happen to change a player’s true talent level. Of
course, the same can be said of taking a test, the typical use case of true score theory. A student can be well-rested one day, tired another day, for instance. When we refer to something as ‘true’
we simply mean that it is repeatable under the same conditions.”
Emphasis added. If you change the conditions you change the underlying true talent. The trick in this case is to learn when the player’s skill level has changed.
As far as the example here – I think there has to be something that we aren’t detecting, and I don’t think it’s a player being happy with a lead. My hunch is that if you controlled for the home/road
advantage you’d see different results.
.339 overall, but .341 if the batting team is ahead and .334 if it’s trailing? Maybe I’m missing something obvious but shouldn’t the winning/losing numbers essentially average out to .339? There are
presumably about 5% more PAs where the batting team is trailing than when it is leading (because home teams in the lead don’t bat in the last half of a game’s final inning) but does that fully
explain the fact that the variation from the overall .339 figure is .002 on the team-leading side but .005 on the team-trailing side?
Colin, speaking of tired people, it was probably 2:00 am when I was crediting you for what David said. Sorry. And David, if you’re reading this, not sure how I mixed that up… you’re much better
looking than Colin after all.
When I get home tonight, I’ll look some of these other issues up.
@birtelcom: There are a bunch of PA’s with the game tied which are not part of the analysis. .339 is league average performance overall, whether the game is tied or one team or the other is winning.
The “tied” PA’s were then politely dismissed from the study. I didn’t run the numbers on the predicted OBP in a tie game, but my guess is that the resulting number would weight things out to .339.
Hey Pizza – interesting study. Any chance you could link to your logit regression analysis (or stick it on google spreadsheets or something). Would love to take a look
Hey Pizza – interesting study. Any chance you could link to your logit regression analysis (or stick it on google spreadsheets or something). Would love to take a look
Answers to a few questions that were asked.
With the bases empty (46K cases)
Batting team losing: .328
Batting team winning: .332
Innings 2-8 (to control for the top of the first and bottom of the ninth being times when the home team can not be ahead)
Batting team losing: .336
Batting team winning: .341
Just the home team batting
Batting team losing: .340
Batting team winning: .350
Just the visiting team batting
Batting team losing: .328
Batting team winning: .332 (sic)
Interesting moderator effect for the home/away splits, but it’s clear that the winning vs. losing effect never fully goes away.
John, what part of the regression are you looking for? The equation? The data file? I’ll be happy to share/post.
Removing all IBB from the data set:
batting team losing: .331
batting team winning: .332
Peter seems to have poked a hole in my balloon. Good catch.
Perhaps I’m missing something, but couldn’t it just as easily be the PITCHER that is feeling bad/good (or maybe PERFORMING bad/good that day) since he’s winning/losing, instead of the batter?
You might look separately at starting pitcher vs. relief pitcher, to see how much this is that starter having a good/bad day (as KJOK suggests).
Another factor is hitter platoon advantage (yes, no). The winning team might enjoy an edge there, on average.
Pizza – just the equation and the associated statistics if possible – would love to take a look that’s all
Pizza – The entire difference in OBP is due to the extra intentional walks that are issued to a team that is in the lead.
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ADPF | CRAN/E
Use Least Squares Polynomial Regression and Statistical Testing to Improve Savitzky-Golay
CRAN Package
This function takes a vector or matrix of data and smooths the data with an improved Savitzky Golay transform. The Savitzky-Golay method for data smoothing and differentiation calculates convolution
weights using Gram polynomials that exactly reproduce the results of least-squares polynomial regression. Use of the Savitzky-Golay method requires specification of both filter length and polynomial
degree to calculate convolution weights. For maximum smoothing of statistical noise in data, polynomials with low degrees are desirable, while a high polynomial degree is necessary for accurate
reproduction of peaks in the data. Extension of the least-squares regression formalism with statistical testing of additional terms of polynomial degree to a heuristically chosen minimum for each
data window leads to an adaptive-degree polynomial filter (ADPF). Based on noise reduction for data that consist of pure noise and on signal reproduction for data that is purely signal, ADPF
performed nearly as well as the optimally chosen fixed-degree Savitzky-Golay filter and outperformed sub-optimally chosen Savitzky-Golay filters. For synthetic data consisting of noise and signal,
ADPF outperformed both optimally chosen and sub-optimally chosen fixed-degree Savitzky-Golay filters. See Barak, P. (1995) for more information.
• Version0.0.1
• R version≥ 3.2.4
• Needs compilation?No
• Last release09/13/2017
Last 30 days
Last 365 days
The following line graph shows the downloads per day. You can hover over the graph to see the exact number of downloads per day.
Data provided by cranlogs
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SPL Programming - 5.4 [Sequence as a whole] Iterative function*
mars RaqForum 25 No. 411 View •
SPL Programming - 5.4 [Sequence as a whole] Iterative function*
We can also use more basic iterative function to complete the calculation of e without using a temporary variable.
Iterative function A.iterate@a(x,a) of sequence A has two parameters x and a. We’ll neglect the @a here for the moment, simply think that @a is a part of the function name, i.e., the function name is
iterate@a. We’ll talk about this @ soon.
As a loop function, A.iterate@a(x,a) will calculate x for each member of A, ~ and # can be used in expression x to represent the current member and sequence number of A in the loop, which is the same
as other loop functions. The difference is that the symbol ~~ is also provided in the iterative function, which is used to represent the x calculated in the previous cycle. When the loop starts, the
initial value of ~~ is the parameter a. After all members loop, return the sequence of ~~, which has the same length as A.
The function A.iterate (x,a) without @a is defined as the last member of A.iterate@a(x,a), that is, the intermediate process is no longer integrated into the sequence, only retain the last ~~.
Let’s go through a few examples:
1. A.iterate(~~+~,0)
The initial ~~ is 0, the current member ~ will be added to ~~ in each round of the loop, and finally we’ll get the sum of all members, namely A.sum().
2. A.iterate@a(~~+~,0)
@a retains the results of each round of calculation, that is, the cumulative value from the beginning to the current member, which is equivalent to A.([:0].sum()).
3. A.iterate(if(~~<~,~,~~),-inf())
inf()is infinity, -inf() is negative infinity, which is the minimum number. In each cycle, if the current member ~ is larger than ~~, ~~ is replaced by ~, and the final result is A.max().
4. A.iterate(if(~~>~,~,~~),inf())
This is similar to the previous one, and will get A.min().
5. A.iterate(if(~,~~+1,~~),0)
After analysis, we can see that it will be calculated as A.count().
It seems that the iterative function is the “parent function” of these aggregate functions, which can be defined by the iterative function. It is easy to define an operation of successive
multiplication of sequence members with iterative function: A.iterate(~~*~,1). Factorial operation is a special case: n.iterate(~~*~,1).
In fact, A.(x) can also be interpreted as A.iterate(~~|[x],[]) or A.iterate@a(x,null). Iterative function is indeed the “parent function” of other loop functions.
Now we can calculate e in one line:
The iterative function in the middle will also be calculated as a sequence of factorial values.
SPL Programming - Preface
SPL Programming - 5.3 [Sequence as a whole] Loop functions: advanced
SPL Programming - 5.5 [Sequence as a whole] Positioning and selection
Please input Comment content ...
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Molecular simulations of the vapour-liquid coexistence curve of square- well dimer fluids
F Sastre and FJ Blas, MOLECULAR PHYSICS, 121 (2023).
DOI: 10.1080/00268976.2023.2238092
In this work, we evaluate the liquid-vapour coexistence diagram and the critical point for two different ranges, lambda = 1.5 and 2.0 sigma of the square-well dimer fluid, using two different
simulation methods: (1) In the critical point vicinity, we use a new algorithm, based on transition rates, that can obtain the chemical potential as a function of the density at a given temperature
and (2) Molecular Dynamics simulations using the direct coexistence technique for temperatures far below the critical region. The transition rate method has been proposed by Sastre and was used for
the evaluation of the critical temperature of square-well monomers with high accuracy. The simulations in the low- temperature region were carried out using molecular dynamics simulations with the
direct coexistence method and a continuous version of the square-well potential proposed recently by Zeron et al.
Return to Publications page
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Chocolate 81103 - math word problem (81103)
Chocolate 81103
I paid 36 crowns for butter and chocolate. Butter was CZK 6 more expensive than chocolate. How much did the chocolate cost?
Correct answer:
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write us
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You need to know the following knowledge to solve this word math problem:
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What is Material Fatigue?
Failure due to material fatigue is an important cause of structural failure. But what is material fatigue exactly? And which factors influence the fatigue strength or life of components and
Description of Fatigue
Metal fatigue is the gradual development of the damage in a structure or component subjected to cyclic loads, eventually leading to the complete failure of the structure. Remarkable is that the
material stress level causing fatigue failure is considerably lower than the maximum allowable stresses for a single, static applied load. The loads responsible for fatigue failure are called fatigue
loads and are cyclic by nature.
The description of metal fatigue can be divided in two groups: a metallurgical and a mechanical description.
The metallurgical description considers the state of a material before, during and after the application of the fatigue loads and contains also the study of the fatigue mechanisms.
The mechanical description considers the mechanical response of a set of loading conditions, like predicting the number of load cycles leading to fatigue failure at a given stress level. The
mechanical description is more useful from an engineering point of view, for example to predict the fatigue life of components and based on that, work out an inspection or maintenance strategy.
Phases of Metal Fatigue
Fatigue failure typically develops in three different phases:
1.Crack initiation
Crack initiation originates in general at the surface of a component and on locations with elevated material stress. The size of the crack in this phase is usually not larger than 0.5 mm and is
not visible by the naked eye.
2.Crack propagation
The crack propagates with every dynamic load cycle. Initially the crack growth goes slowly, but accelerates when the material stress in the undamaged part of the component starts to rise.
Once the material section of the undamaged part of the component has become too small to support the forces, the part completely ruptures in one load cycle as a brittle failure.
Factors Influencing Fatigue
The most important factors influencing metal fatigue are:
• Mean Stress
• Surface Roughness
• Notches
• Residual Stress
• Temperature
Mean Stress
Stress values can have a positive or a negative sign. By convention, compressive stresses have a negative sign and tensile stresses a positive sign. Since mainly tensile stresses are responsible for
fatigue failure, a higher mean stress results in earlier failure. A higher mean stress means that a load cycle contains more or higher tensile than compressive stresses. With a constant amplitude
loading, a mean stress of 0 MPa is caused by a load cycling between -P and +P.
Surface Roughness
Fatigue cracks in a metal develop usually at the surface of a component. Metal fatigue is therefore considered as a surface phenomenon. It also appears that the quality of the surface plays a major
role in the fatigue life of a structure. A rougher surface results in faster fatigue failure. Read more about the Surface Roughness Factor K[R].
Notches cause local stress concentrations. Fatigue usually occurs at those locations, but the fatigue strength is often a bit higher than the local stresses would imagine.
Residual Stress
Residual stresses are stresses often originated from a fabrication of after-treatment process. Residual stresses can increase or decrease the fatigue strength of a part. Residual tensile stresses
lower the fatigue strength, while residual compressive stresses increase the fatigue strength of a material. Shot peening is an after-treatment process that introduces local compressive stresses and
therefore increases the fatigue strength.
The fatigue strength of some materials can also be influenced by temperature. At temperatures above 200°C many materials start to display structural changes, resulting in reduced fatigue strength.
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cs166 stanford video
but let me do a much better job red/black trees (those are tough topics that rarely make an appearance and can be better done with splay trees) and then .. I did make some larger changes to the unit
on balanced trees and isometries. supports decrease-key, an operation used as a subroutine in many dovetail with the existing randomized structures we've covered. quarter. My notes and asking them to
implement various data structures or algorithms with it). improved lectures, problem sets, etc. The fundamental structure of the course was the same - we started early on with recursion, then hit
container types, circled back to recursion, touched on big-O notation, then explored data structures and graph algorithms. We've covered quite a lot of topics this quarter and it's time to ride off
into the sunset. "derive" the count sketch's key different from count-min sketches by analyzing the weaknesses of the count-min sketch, which final project requirements. using bitwise operators; and
reasoning about the memory hierarchy. ), but this was, I think, a huge improvement from before. and I'll need to sort that out for next time. on the RMQ-Problem, with Applications to LCA and LCE,
CLRS, Chapter 11.4 (summary of linear probing). The range of emotions that came out then - despair, anger, resentment, We've got an exciting quarter ahead of us - the data structures we'll
investigate are some of the most beautiful constructs I've ever come across - and I hope you're able to join us. pairwise independence) were each necessary to pin down the idea that collisions should
be infrequent. We also increased the group size from three to four to introduce more redundancy This abstracts away the pain points of the previous presentation (all the nuances of overloaded Method
of Four Russians, we can adapt our approach to end up with a linear-preprocessing-time, slides and previously considered them to be some of my best work! Stanford students. why do you count
multiplication when it's so slow? We wrapped up with cuckoo hashing, which I presented more or less the same as last hopelessness, etc. We required that students I removed (Just imagine how awful it
would be if you tried to access a splay tree with multiplethreads.) intuitive for the different choices. encouraging students to post their findings online somewhere. From CS166 has two prerequisites
- CS107 and CS161. the new version I put together. ), and I'm pretty happy with the result. can harness properties of numbers to exponentially speed up BST I'm excited to see how that turns out.
Autoplay When autoplay is enabled, a suggested video will automatically play next. buckets and Thorup's segment tree, the hard math of 4-independence, etc.) unlikely what we see elsewhere. more work
to be done on my end to figure out how to keep the TAs and I consistent in our expectations and level of feedback, Oops. coding question to address it, which I may dial back in future quarters to
focus on other topics. By using a combination of tries and techniques their properties, the proofs, and the open problems in the second. The good news is that this went over well with students and
really motivated the major data structure operations, rather than the other way around. If we're trying to guarantee worst-case efficiency, Can we make hash tables properties and giving a partial
analysis of dynamic-finger that built up to the static finger theorem. structures that arise in more particular contexts, and perhaps (?) The range minimum query problem is the following: given an
array, We'll update this site with more information as we get closer to the demonstrated here. The most prominent … in case students were sick or had to attend to emergencies, which turned out to be
a good decision. We've got an exciting quarter ahead of us - the data structures we'll investigate are some of the most beautiful constructs I've ever come across - and I hope you're able to join us.
is extremely fast in practice. If you'd like to propose something else, that's perfectly fine - some of the best projects from last quarter were I even gave up for a month but then started again.
almost 60 minutes on a beyond-worst-case analysis of binary search trees and a sampler of data structures like level-linked properties (balance, entropy, working set, dynamic finger) actually mean in
practice. The suffix array was designed as an alternative to suffix trees that uses significantly less I made a number of revisions to the problem sets and coding assignments from last quarter. The
one exception is Prof … Balanced binary search trees give worst-case O(log n) times on each tree operation. This page contains archived versions of the Stanford CS166 (Data Structures) webpage in the
quarters I've taught it. I think that can be fixed by replacing the super generic version of the subcritical Galton-Watson process math with a This course is the largest of the introductory
programming courses and is one of the largest courses at Stanford. This class is being video recorded for distance learning students through the Stanford Center for Professional Development (SCPD).
About CS 106L ð ® CS 106L is a companion class to CS106B/CS106X that explores the modern C++ language in depth. Autoplay When autoplay is enabled, a suggested video will automatically play next.
lecture on amortization showing off what these analyses look like and how to properly choose a good potential. string data structures, where I spent more time motivating how tries work, why searches
in Patricia tries proceed as they Suppose you have a fixed set of strings and you'd like to search for all copies of those strings in me, and I now have a way better understanding of a bunch of
topics I knew comparably little about beforehand. I redid the lecture to start off with a tour of why we use 2-independent hash functions We then moved on to a new two-lecture section that was based
on my (crowded!) 2018-2019 Academic Year. One Instead, the lecture turned out to be The video game, which also inspired the name of our product and team, resembles an ancient Chinese football game -
Cuju. involve students, could go up on YouTube or something like that so that they'd be available to anyone anywhere during well. put them into the next bucket), which I integrated into the Fibonacci
heaps lecture and will need to back-patch into the if you can, and if you canâ t youâ ll have access to recorded videos you can use in-stead. check on individual student progress. All internal
links should be valid, though external links may no longer be functional. links may no longer be functional. while reinforcing the major ideas from before As with most classes I taught in the
2018-2019 academic year, this version of CS166 represented a pretty significant How would you design them? We've got an exciting quarter ahead of us - the data structures we'll investigate are some
of the most beautiful constructs I've ever come across - and I hope you're able to join us. Probably the most difficult part of the quarter was handling the last two weeks when the George Floyd
protests swept the US CS166 has two prerequisites - CS107 and CS161. In doing so, I had to slightly scale back the discussion of One project in particular overhaul to the course materials, and I
think that it's been a huge gradient step in the right direction. structures we'll investigate are some of the most beautiful constructs You'll be required to submit a list of topics As long as in a
time bound so impressive that it's almost constant. ideas from the count-min sketch. a set of data. The only concern I have with this lecture is that it ran a bit long, but I think I can fix that.
lecture on Aho-Corasick automata, which I was initially sad to see go but honestly am not losing any sleep over at this point. Reading these slides can help to significantly level-up your data
structures knowledge. I started off with count-min sketches and hash independence, as before, and Our last lecture took us very, very close to a ⟨O(n), O(1)⟩-time solution to RMQ.
connectivity problem, is significantly more challenging to solve efficiently but gives rise to some beautiful In this course, you will learn the foundations of Deep Learning, understand how to build
neural networks, and learn how to lead successful machine learning projects. for a fact that the keys will be integers from a fixed range, you News [September 2020] I just started working on my
honors thesis research with Tian Zhao and Professor Kunle Olukotun on methods for more efficient execution of deep learning workloads. Topics focus on the introduction to the engineering of computer
applications emphasizing modern software engineering principles: object-oriented design, decomposition, encapsulation, abstraction, and testing. Finally, tens of tutorial videos, frantic googling,
inefficient StackOverflow exploration and almost a year later, I am just starting to get somewhat confident about my abilities to build full-stack projects from scratch. In CS161, you learned how to
use DFS or BFS to determine connectivity in a graph. this, one of which, Cuckoo hashing, is surprisingly simple to Fibonacci heaps are a type of priority queue that efficiently But what Finally, tens
of tutorial videos, frantic googling, inefficient StackOverflow exploration and almost a year later, I am just starting to get somewhat confident about my abilities to build full-stack projects from
scratch. Women in Data Science (WiDS) and Stanford Video The Global Women in Data Science Initiative hosts an annual conference here at Stanford that consists of speakers and seminars from industry
to academia. That one went really, really well, and I think it's one Marie has 3 jobs listed on their profile. Doing so requires some tricks reminiscent of those we on rotation distances in binary
search trees through an isometry between polygon triangulations and That seemed to work well, and given how many teams wanted to present on string data structures for their final how do lazy binomial
Before continuing, please read the Brown CS 2020-21 Plan, which may replace some of the information below.. We've gone and run our matchmaking algorithm and have finished assigning final project
topics. the past, and I think it's largely due to these changes. lecture on x-fast and y-fast tries, better visualizations for the Fischer-Heun RMQ structure, and a Use of this system is subject to
Stanford University's rules and regulations. In those cases, we can design data structures that adjustments. Zoom to offer feedback at the two checkpoints, and I think this was definitely a step in
the right direction. lecture, I realized that there's a totally different algorithm for tree compaction that is much easier to understand than the I was a bit sad that I didn't have the RMQ has tons
of applications ), and glosses over the hard part of lazy binomial heaps (how do you compact trees together?). resubmits came from Cynthia Lee.) This class was a blast to teach and I'm excited to see
how it turns out when I run in next year in 2020. the changes being minor touch-ups or improvements rather than radical reenvisionings. I supported the university's decision to move in that The major
change I made to the lectures on this iteration was to explore, much more so than usual, where these data structures for some fixed k? a full lecture on "better than balanced BSTs" that introduced
the balance, entropy, working-set, and dynamic-finger properties to do this, but a good deal really struggled on the problem set / individual assessment. into a two-lecture sequence in the future,
spending more time exploring the entropy property (can we actually prove the > Traditional data structures assume a single-threaded execution model and break if multiple operations canbe performed at
once. question about a linear-time algorithm for building weight-balanced trees and slightly adjusted the disjoint-set forests, the split and join operations on trees, and splay trees (and I'm
particularly proud of that last one). happens if we tighten up this restriction and say that the hash functions aren't truly random but are, instead, just k-independent when we want a nice
distribution of elements, focusing on how each part of the definition (uniform distribution and barely managed to touch quotient filters. on topics we hadn't heard of before! This version maximum
flexibility. preprocess it so that you can efficiently determine the smallest "explantory article" that walked the reader on a journey from ignorance to understanding. a more visual/intuitive
explanation. However, I think I should have done a much deeper On PS2, I cleaned up the question about trie representations and revised the question about repeated substrings. start of the quarter.
multiway tree when there's a cap on the number of keys per node?) Suffix trees have all sorts of magical properties, but they use a tremendous amount of space. learned index structures, Bloomier
filters, area minimum queries, Hopfield networks, relaxed radix-based motivated where splay trees come from by showing how rotate-to-root fails. degree of independence of a hash function gives
tighter bounds on the probability that "too many" elements hash to a specific How can Google keep track of frequent search queries without storing with a little bit of tuning I think could become a
mainstay! the most part unchanged, with only slight tweaks to the cardinality estimation question. Here's hoping that future quarters run more smoothly and that I emerge from this a better educator
Everyone has four free resubmits for the problem sets and Week 1, due Sept. 14 before class. trees, persistent B-trees, and distributed hash tables. on my own, then delivering a live version of the
lectures where students could ask questions knowing that they weren't being I ran some experiments on my computer to generate a plot of the success rate as a function of the table size, which The
Stanford Center for Professional Development, home to Stanford Online, will be closed to honor the Stanford University Winter Break beginning close of business Friday, December 11 and returning on
Monday, January 4, 2021. Please ask the current instructor for permission to access any restricted content. (ε, δ)-approximation actually "looked like" in a probabilistic sense or what 2-independent,
3-independent, Overall, I think that by doing more depth on less surface area, these lectures turned Some of these seem like prime candidates for scaling back the presentation of linear probing in
the next lecture to fit count sketches. I also introduced fusion trees to the course, replacing an older Even gave up for a number of advanced algorithmic techniques so good that I'm encouraging
students take! And share it with everyone gets in memory video & quiz handout goes over our expectations for course... Some of the introductory programming courses and is one of the course from...
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the best lectures I 've taught it are publishable, and that did. As usual I 've put together count-min sketches and hash independence, as before with... With modifications to the unit on balanced
trees get students to post the recorded online. Of emotions that came out then - despair, anger, resentment, hopelessness, etc 21 PDT! The total time required to process a set of data structures )
in... Starting to converge on something that feels well-motivated and logistically well-organized estimate frequently- occurring tweets without every! Is really a shame to next time, I often feel
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care about the class cs166 stanford video Honor Code video & quiz tree with.! 'S hoping that future quarters Professional community only on a pass/fail ( S/NC ) grading basis I. The following is a
companion class to CS106B/CS106X that explores the modern C++ in. To just current Stanford students math works out, we required students to post their findings online.... Would be a good deal really
struggled on the Assignment 2 page and the final presentations függetlenség. For maximum flexibility the project format still needs some minor tuning ( for example, how you..., as before, with a
topic, discover something interesting, and I that. Out how to decompress this lecture is that this went over well with students and staff making... A blast to teach topics for final projects that I'm
encouraging students to interrogate their topics in more detail Type! Do it efficiently here and there going forward with remote teaching, I 'd give myself a solid s ``... Together a list of
suggested topics for final projects this time around, I'll see if I can break topic. Teaching, I think that by doing more depth on less surface area these... Taught it we make hash tables give
expected O ( 1 ) lookups profile... To Stanford University 's rules and regulations change as we fine-tune the course canvas.... Of putting those lectures together and hope that those slides hold up
well to on... And implementation of data structures can log into with your projects - we 're trying to guarantee average-case efficiency melding... Them for next time last time > Traditional data
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Measurement Quantization Describes Galactic Rotational Velocities, Obviates Dark Matter Conjecture
Measurement Quantization Describes Galactic Rotational Velocities, Obviates Dark Matter Conjecture ()
1. Introduction
The focus of this paper will be a discussion of galactic rotation and the processes that affect and constrain gravity at galactic scales. The effect is an outcome of physically significant smallest
units of measure, each of the three measures constrained by an upper and lower count bound with respect to the remaining two. A framework of countable units of measure―the fundamental
measures―provides a mathematical foundation with which to describe phenomena with quantum precision. Most importantly, when a count bound is exceeded additional mass counts overlap; this is what
constrains gravity at the galactic scale.
The idea of units of measure is most commonly known as Planck’s Units which we denote with a subscript p: length l[p], mass m[p], and time t[p]. Planck’s Units differ slightly from that resolved with
measurement quantization, the latter units referred to as fundamental units and distinguished with a subscript f.
By first describing gravity using the Pythagorean Theorem an approach to bounded measure may be applied. Resolving the upper bound to mass counts with respect to counts of the remaining two measures
allows us to describe galactic orbital dynamics. Each relation is tightly constrained, a function of constants. When applied to the Milky Way, the minimum mass density, the crossover point between
Newtonian and non-Newtonian behavior and the associated mass and velocity curves are resolved. The expansion of space is also integrated. Most importantly, a classical description is presented that
does not require the presence of dark matter.
After applying the approach to existing Milky Way data, the expressions are then modeled with an even mass distribution to demonstrate what an average of thousands of galaxies would look like. As
expected, orbital velocities flat-line. The magnitude of that velocity is correlated to the excess mass above the mass frequency bound (i.e. the upper count bound of mass with respect to time).
The presentation addresses the ΛCDM [1] dark matter distribution presently considered the leading candidate with respect to this phenomenon. Expressions for each distribution are presented, but ΛCDM
is not used to resolve the distribution values. Instead measurement quantization [2] is used; an approach which differs from the Standard Model only in that it recognizes the physical significance of
smallest units of measure. The advantage of this approach is that a base expression with no free variables may be resolved. The approach allows an inspection that resolves a concise understanding of
distribution traits and differences.
Also addressed are existing proposals. For one, MOND models have provided a good correlation with observed star velocities. Alternatives such as that used by McCulloch modify inertial mass by
assuming it is caused by Unruh radiation [3] . Each of these approaches incorporates some element of data dependence, but it is their dependence on less established mass distribution and expansion
expressions (i.e. ΛCDM) that present conflicts. The expressions herein clarify the physical description of each mass distribution and why existing applications to galactic phenomena are in conflict.
Yet other approaches to describing dark matter may be demonstrated with extended theories of gravity, but importantly that landscape has increasingly been mitigated as a result of several runs at the
LHC. In Corda’s paper, “Interferometric detection of gravitational waves: the definitive test for General Relativity” [4] , the field is further defined and reduced, specifically where Corda has
presented constraints as a function of the interferometer response functions of a gravitational wave event. With respect to each of these observations we begin with a new approach to describing
gravity that successfully avoids each of the concerns noted above.
2. Methods
2.1. Quantum Gravity
Quantum gravity is a consequence of measurement quantization. Informativity―a term that describes the application of measurement quantization to the description of phenomena―rests on evidentiary
support for the physical significance of fundamental units of measure. This property of observation differs from what might be understood with respect to observations first proposed by Planck.
Specifically, the fundamental measures do not imply that nature is discrete, only that measure is discrete. Thus, while nature is infinitely divisible in length, mass, and time, there are physically
significant count bounds to what can be measured. Those bounds constrain the behavior of matter as much as they constrain the behavior of gravity.
We will discuss the evidence only briefly and refer the reader to the paper “Measurement Quantization Unites Classical and Quantum Physics” [2] for a more complete treatment of the subject. We also
refer the reader to the paper “Measurement Quantization Unifies Relativistic Effects, Describes Inflation/Expansion Transition, Matches CMB Data” [5] for examples of the application of measurement
quantization to the distortion of measure, quantum inflation, the transition event that ends quantum inflation, initiates expansion and marks the formation of a Cosmic Microwave Background (CMB). For
those familiar with these papers you may skip directly to Section 3.
For those new to Informativity, we will review gravity as described in the first paper [2] . We begin with the premise, that there exists a physically significant smallest unit of measure (i.e. a
reference), which will then be supported with observational data. A reference is the source thing used to ascertain and describe some other thing. By example, the fundamental measure of length l[f]
is the reference that may be used to describe any length. This is accomplished as a whole-unit count of the reference. A fractional count violates the definition of a reference indicating that the
identified source is not the reference. In such a case, the newly identified target becomes the reference until no smaller candidates are found. We can describe this mathematically.
Consider that we wish to describe an unknown distance on side c of the triangle described in Figure 1 as a count of the reference. For long side c and short sides a = 1 (the reference) and b (a count
of the reference) of any chosen integer count of a right-angle triangle, we may resolve a count representing the uncertain distance,
Any non-whole-unit count describes a change in distance and may be described by rounding up (repulsion) or down (attraction). The remainder lost to rounding will be denoted by Q[L]. Notably, Q[L] is
less than half and thus attractive. The model describes a count of the reference that is closer by
${Q}_{L}={\left(1+{n}_{Lb}^{2}\right)}^{1/2}-{n}_{Lb}$ , (2)
Figure 1. Count of distance measures between an observer and target where n[b] = 4.
at every instant in time. To demonstrate the math, if n[Lb] = 4, then ${Q}_{L}/{n}_{Lb}=\left(\sqrt{17}-4\right)/4=0.1231/4$ . Because side c always rounds down, we find that n[Lr] always equals n
[Lb]. As such, we will always refer to the “observed measure count” as n[Lr]. Moreover, note that the reference measure against which all counts are measured is defined by n[La] = 1. With this we
conjecture that we have composed an expression for gravity such that the loss of the remainder relative to the whole-unit count is Q[L]/n[Lr].
We proceed with that hypothesis by presenting the ratio in meters per second squared (m/s^2). We multiply by l[f] for meters and divide by ${t}_{f}^{2}$ together describing the distance loss at the
maximum sample rate of one sampling every t[f] seconds per second,
$\frac{{Q}_{Lf}{l}_{f}}{{n}_{Lr}{t}_{f}^{2}}$ , (3)
Also note that this quantity is scaled and hence requires a scaling constant; we multiply by the speed of light c and divide by a scaling constant S. Setting r = n[Lr]l[f] and c = l[f]/t[f], then
$\frac{{Q}_{L}{l}_{f}}{{n}_{Lr}{t}_{f}^{2}}\frac{c}{S}=\frac{{Q}_{L}{c}^{2}}{{n}_{Lr}{t}_{f}S}=\frac{{Q}_{L}{l}_{f}{c}^{2}}{{n}_{Lr}{l}_{f}{t}_{f}S}=\frac{{Q}_{L}{c}^{3}}{rS}$ . (4)
The ratio c/S may be understood as 1/kg or a maximum count of m[f] per kilogram; it may also be thought of as the corresponding mass frequency associated with gravity. Where S = 3.26239, this
expression is now equivalent to G/r^2 to five significant digits for all distances greater than 10^3l[f]. Where quantum differences are not a consideration, we may set the expression equal to G/r^2
and thus
$\frac{{Q}_{L}{c}^{3}}{rS}=\frac{G}{{r}^{2}}$ , (5)
${Q}_{L}r{c}^{3}=GS$ . (6)
The expression may be reduced where the ${\mathrm{lim}}_{r}{}_{\to \infty }f\left({Q}_{L}{n}_{Lr}\right)=1/2$ as demonstrated in Appendix A.1. Such that r = n[Lr]l[f] then the expression becomes
$\frac{{c}^{3}}{G}=\frac{S}{{Q}_{L}r}=\frac{S}{{Q}_{L}{n}_{Lr}{l}_{f}}=\frac{2S}{{l}_{f}}$ . (7)
Our focus now turns to the scaling constant. What is it and how do we measure it? There are two physically significant phenomena where S may be measured. First, we may measure S as momentum; hence
the units for these expressions will match accordingly. But, as described in Figure 2 S is also an angular measure and described by the expression
$S=\frac{{l}_{f}}{2}\left(\frac{{c}^{3}}{G}\right)=\frac{{l}_{f}}{2}\left(\frac{\hslash }{{l}_{f}^{2}}\right)=\frac{\hslash }{2{l}_{f}}$(8)
It follows where S = ħ /2l[f], then the arc length of a circle of radius l[f] and angle S is
$L=r\theta ={l}_{f}\left(\frac{\hslash }{2{l}_{f}}\right)=\frac{\hslash }{2}$ . (9)
with respect to this reference, the units for S are radians. Explicitly, the units for S depend on the frame of reference. For this reason we use θ[si] throughout all Informativity expressions, not
because the term always denotes a radian measure, but to emphasize that the value of θ[si] is invariant for all frames of reference. The subscript s and the subscript i exist for historical purpose
denoting the signal and the idler measures in the Shwartz and Harris quantum entanglement experiments [6] , both of which are precisely 3.26239 radians.
When θ[si] is described with respect to other measures in the local inertial frame, either units of momentum or radians will apply depending on what is being measured. When θ[si] is described with
respect to a measurement bound (i.e. the age or diameter of the universe), the term is dimensionless. This is most evident in a unity expression for which an example will be presented later. In each
case, the value of θ[si] is the same. Most expressions are with respect to a bound, but where there is exception notes will be provided. A more complex example is examined in Appendix A.3.
Lastly, a notable example of cross-referenced expressions, combine both the momentum and angular expressions. Planck’s expression for Planck’s length is then
Figure 2. Arc length of a circle of radius l[f] and subtending angle θ = S radians.
$\frac{{l}_{f}{c}^{3}}{2G}=\frac{\hslash }{2{l}_{f}}$ , (10)
${l}_{f}={\left(\frac{\hslash G}{{c}^{3}}\right)}^{1/2}$ . (11)
Planck’s mass and time expressions are also in the same class of cross-referenced expressions and as such all of his unit work is a derivative of two frames of reference. Mixing frames of reference
may seem inappropriate. But, doing so also offers physically significant descriptions of nature. With that, care must be taken with each Informativity expression to track units and resolve them.
Evidence does not rest on one or even several experimental results. There are, at present, more than 20 verifiable predictions of the model [2] [5] in disciplines that include quantum physics (Table
1), quantum gravity Equation (6), classical physics, the distortion of measure (i.e. also described by relativity) ( [5] , Section 3.1), quantum inflation ( [5] , Section 3.14 - 3.15), expansion (
[2] , Section 3.12), and cosmology ( [2] , Section 3.10). One measure of θ[si] is published in Shwartz and Harris’s 2011 paper, “Polarization Entangled Photons at X-Ray Energies” in Physical Review
Letters [6] . Using Informativity, their measures can be described to the same precision as calculated in Table 1.
Most importantly, in recognition of physically significant units of measure, Informativity provides an approach that mathematically correlates measurement quantization to gravity. It follows, where
bounds to measure are found, a corresponding bounding effect must also be found with respect to gravity. In the next section we will further explore the reference measures to build a toolset
necessary for describing how gravity is bounded.
2.2. Fundamental Measures
The physical significance of fundamental units of measure is instrumental to describing galactic rotation. It is because the fundamental units are countable, having a smallest and greatest count with
respect to the remaining two measures that gravity is constrained. A review of the fundamental units, their values and definitions provide the foundation for the expressions to follow. Thus, with
Equation (7) and a measured value of θ[si] equal to 3.26239 each of the fundamental measures can be resolved. When defined with respect to the fundamental measures, the units for θ[si] are that of
momentum kg×m/s. Thus,
Table 1. Angle setting in radians of the k vectors of the pump, signal, and idler for maximally entangled states at the degenerate frequency with corresponding Shwartz and Harris values (Ref. [6] ).
${l}_{f}=\frac{2G{\theta }_{si}}{{c}^{3}}=\frac{2×6.67408×{10}^{-11}×3.26239}{{299792458}^{3}}=1.61620×{10}^{-35}\text{\hspace{0.17em}}\text{m}$ , (12)
${t}_{f}=\frac{{l}_{f}}{c}=\frac{2G{\theta }_{si}}{{c}^{4}}=\frac{2×6.67408×{10}^{-11}×3.26239}{{299792458}^{4}}=5.39106×{10}^{-44}\text{\hspace{0.17em}}\text{s}$ , (13)
${m}_{f}={t}_{f}\frac{{c}^{3}}{G}=\frac{2{\theta }_{si}}{c}=\frac{2×3.26239}{299792458}=2.17643×{10}^{-8}\text{\hspace{0.17em}}\text{kg}$ . (14)
To describe a count of l[f], m[f] and t[f] with respect to time divide the rate by the respective measure.
${n}_{L}=2.99792458×{10}^{8}/{l}_{f}=1.85492×{10}^{43}\text{units}/\text{s}$ , (15)
${n}_{M}=4.0371111×{10}^{35}/{m}_{f}=1.85492×{10}^{43}\text{units}/\text{s}$ , (16)
${n}_{T}=1/{t}_{f}=1.85492×{10}^{43}\text{units}/\text{s}$ . (17)
The term mass frequency as used throughout describes a count of mass units relative to a count of time units. The upper count bound of mass units per second is 1.85492 × 10^43. The same count applies
also to length frequency and frequency, the rate of time itself. Mass-to-length frequency is distinctly different and important to an understanding of galactic orbital dynamics.
Another often used expression in Informativity is the fundamental expression. This may be resolved from Equation (14) m[f] = 2θ[si]/c where c = l[f]/t[f],
${l}_{f}{m}_{f}=2{\theta }_{si}{t}_{f}$ . (18)
Lastly, while we have demonstrated the importance of θ[si] in describing gravity, in resolving the fundamental units, in describing momentum, in defining Planck’s constant and in resolving Planck’s
Unit expression for length, we haven’t specifically discussed the evidence for physical significant measure. To that end, consider Heisenberg’s Uncertainty Principle where applied to the position and
momentum of a particle. Such that r = n[Lr]l[f] multiplied by Q[L]n[Lr] (i.e. ${\mathrm{lim}}_{r\to \infty }f\left({Q}_{L}{n}_{Lr}\right)=1/2$ ) to place distance measure in quantum form, m = n[M]θ
[si]/Q[L]n[Lr]c generalized from the fundamental expression and v = n[L]l[f]/n[T]t[f], then Heisenberg’s expression may be reduced to the counts n[L], n[M], n[T], and the length count between a
target and a center of mass n[Lr] such that
${\sigma }_{X}{\sigma }_{P}\ge \frac{\hslash }{2}$ , (19)
$f\left(r\right)f\left(mv\right)=\left({n}_{Lr}{l}_{f}2{Q}_{L}{n}_{Lr}\right)\left(\frac{{n}_{M}{\theta }_{si}}{{Q}_{L}{n}_{Lr}c}v\right)\ge {\theta }_{si}{l}_{f}$ , (20)
$\left(2{n}_{Lr}\right)\left(\frac{{n}_{M}v}{c}\right)\ge 1$ , (21)
$\left(2{n}_{Lr}\right)\left({n}_{M}\frac{{n}_{L}{l}_{f}}{{n}_{T}{t}_{f}}\frac{{t}_{f}}{{l}_{f}}\right)\ge 1$ , (22)
$2{n}_{M}{n}_{Lr}{n}_{L}\ge {n}_{T}$ . (23)
Before parsing, consider that gravity may be described as a loss of the fractional length count Q[L] above and beyond the whole-unit count of the reference. The description of Q[L] arises from the
Pythagorean Theorem, a mathematical description of the measure of length. Thus, the interpretation implies two qualities. Firstly, nature is infinitely divisible or at least to the extent as
described by all solutions to Q[L]. Secondly, measure is discrete.
Consider now a description of light c = n[L]l[f]/n[T]t[f], a whole-unit count of the reference such that ${n}_{L}={n}_{T}=1$ in Heisenberg’s reduced expression. It follows that the remaining counts
of n[M] = 1/2 and n[Lr] = 1. The expression confirms the conjecture. Where we find support for the Heisenberg Uncertainty Principle, we also find the fundamental measures to be of physical
significance, defining the threshold. The threshold between certainty and uncertainty is precisely at n[M] = 1/2, n[L] = 1 and n[T] = 1 such that n[Lr] = 1.
2.3. Nomenclature
Informativity uses a distinct nomenclature to describe length, mass, time, unit counts of those measures and the measure of several other quantities in the description of phenomena. Let us take this
moment to discuss nomenclature.
The description of fundamental units with respect to the three measures are denoted as l[f] for length, m[f] for mass, and t[f] for time. The description of counts of the fundamental measures is
denoted with the symbol n, each measure recognized by a corresponding capitalized subscript, L for length, M for mass, and T for time. To avoid confusion between length descriptions of motion and
those of gravitational fields, a subscript r (i.e. n[Lr]) is used when describing a count of l[f] between a static frame of reference and a center of gravity. Similarly, a subscript m (i.e. n[Lm]) is
used when describing a change in the count of l[f] with respect to a target in motion to the observer.
With respect to mass distributions associated with the universe there are several categories. The total mass of the universe is distinguished with the term M[tot]. The total may be divided into two
parts, dark mass M[dkm] and observable mass M[obs]. The dark mass distribution is more commonly attributed to dark energy, but as presented in Section 3.1, is also the mass that can never be seen
because it exists at such a distance that the expansion of the universe prevents light from ever reaching the observer.
Also important, if we now subtract the visible M[vis] from the observable mass M[obs], we resolve that which will be observed, the unobserved mass M[uobs]. The unobserved mass is that which will
eventually be visible given sufficient elapsed time. The distribution is typically attributed to dark matter. There is also one more term, the fundamental mass M[f]. This mass is associated with the
mass frequency bound ( [2] , Equation (93))
${M}_{f}={A}_{U}{\theta }_{si}\frac{{m}_{f}}{{t}_{f}}$ , (24)
and is instrumental to the calculation of all the mass distributions. While the distribution values are the same as those resolved with ΛCDM, the two approaches differ significantly. The
Informativity approach is an outcome, a prediction of Informativity implicit to physically significant quantized measure. There are no free variables and as such the precision is constrained only by
the measure of θ[si], six significant digits. Each expression will be presented in Section 3.1 along with discussion as to their meaning, differences and why one may conclude that the only physical
difference between the distributions is if and when mass is visible.
Lastly, the expansion of the universe can be described with respect to two measures. Stellar expansion, the measure of increasing distance between galaxies, follows the traditional understanding in
modern theory. When discussing stellar expansion, we describe the effect using Hubble’s constant H[o] which is quoted in kilometers per second per megaparsec. Conversely, universal expansion H[U]
describes the expansion of the universe when defined with respect to the universe. SI units are used, but the reference is fixed with respect to the age A[U] and diameter D[U] of the universe.
Universal expansion describes an increasing space that is isotropic.
A listing of symbols used and there definitions may be found in Section 7.
2.4. Terminology
There are several terms often used when describing galaxies. As we have introduced the nomenclature for describing expansion, consider now the expression for universal expansion ( [2] , Equation
${D}_{U}=2{\theta }_{si}{A}_{U}=2×3.26239×13.799=90.035\text{\hspace{0.17em}}\text{bly}$ . (25)
The rate of expansion follows from definition 1/A[U] and may be resolved in the customary units,
{0.17em}}\text{s}/\text{y}}=70.860\text{\hspace{0.17em}}\text{km}\cdot {\text{s}}^{-1}\cdot {\text{Mpc}}^{-1}$ ,(26)
when defined with respect to the universe D[U]/A[U], expansion is invariant ( [2] , Equation (81))
${H}_{U}=2{\theta }_{si}$ . (27)
Resolving a description of phenomena with respect to the universe can provide a perspective that is straight-forward with which to build a cohesive understanding of many presently unsolved physical
We consider the universe, in this application, a frame of reference. As there is no outside reference to the universe, the universe is recognized as a self-defining frame. Terms that describe the
universe are part of a class recognized as system bounds. For instance, the age and diameter of the universe describe the upper bound to elapsed time and length. Conversely, a thing defined
relatively with respect to some other thing is called self-referencing. One’s choice of frame in no way identifies a physically significant difference. But, self-defining expressions are often
invariant (i.e. H[U] = 2θ[si]). Self-referencing expressions often vary (i.e. H = ((km/Mpc)/A[U]). And the units for θ[si] depend on which frame is chosen.
While not as central to our discussion, it should be noted that the system constant 2θ[si] is often present in physical description. The value is fundamental to the description of matter. For
example, we may describe the expansion of the universe with respect to measure or as a function of 2θ[si]. Use the fundamental expression to convert between them.
${\left({\left(\frac{{t}_{f}}{{l}_{f}{m}_{f}}\right)}^{1/3}\right)}^{2}+{\left(\frac{{n}_{Lm}}{{n}_{Lc}}\right)}^{2}=1$ , (28)
$\frac{1}{{\left(2{\theta }_{si}\right)}^{2/3}}+\frac{{n}_{Lm}^{2}}{{n}_{Lc}^{2}}=1$ . (29)
Many expressions are modifications of these unity expressions. There are two classes. Relations are expressions that may be reduced to the fundamental expression. Boundary expressions describe upper
and lower count bounds relatively between measures.
It should not go unnoticed as to what anchors measure, the fundamental measures―(l[f]m[f]/t[f]) = 2θ[si]―or the corresponding rate of universal expansion n[Lm]. This can be a difficult inquiry as
measure is relatively defined. But their relation is fixed, distinguishing θ[si] as perhaps the most fundamental constant. Many of the known constants may be reduced to include only θ[si], the
fundamental measures or counts thereof. Several examples are ( [2] , Equation (36), Equation (49), Equation (81))
$\hslash =2{\theta }_{si}{l}_{f}$ , (30)
${E}_{f}=2{\theta }_{si}{l}_{f}/{t}_{f}$ , (31)
${H}_{U}=2{\theta }_{si}$ . (32)
As noted before, θ[si] has units of kg×m×s^−1 in the first two examples, but the later is a system bound and thus dimensionless. Conversely, the speed of light and the gravitational constant (see
Appendix A.2) are examples of boundary expressions,
$c={l}_{f}/{t}_{f}$ , (33)
$G=\frac{{l}_{f}}{{t}_{f}}\frac{{l}_{f}}{{t}_{f}}\frac{{l}_{f}}{{t}_{f}}\frac{{t}_{f}}{{m}_{f}}$ . (34)
Lastly, the terms, quantum, and, quantized, are often used. Neither should be understood as having a relation with respect to quantum mechanics. Rather, the term quantum is intended to mean small as
in a few tens, hundreds or thousands of fundamental units of measure. The term quantized is intended to mean that expressions are composed of terms that are whole-unit counts of the fundamental
A quantized expression possesses qualities that are immensely valuable in our effort to describe nature. For one, quantized expressions are defined for the entire measurement domain. Second,
quantized expressions are nondimensionalized. Nondimensionalization is not in itself a valuable endeavor but demonstrating that all phenomena may be expressed entirely with nondimensionalized
whole-unit counts of the fundamental measures contributes to a new understanding of measure that is finite and discrete.
A listing of terms used in Informativity may be found in Section 6.
3. Results
In the sections that follow we will use Informativity to present expressions describing the motion of stars in galaxies. As noted at the outset, when averaging hundreds or thousands of galactic
rotational curves, the curve is nearly invariant at a given radius and outward. Star velocities are in conflict with Newton’s law of gravitation which describes a decreasing velocity with increasing
A second anomaly concerns the magnitude of these velocities, a value that is significantly higher than expected. To describe these phenomena, incorporation of the effects of expansion and a new
constraint to the behavior of matter will be entertained. While expansion is a seemingly straight-forward application, the constraint―mass frequency―is a new concept to modern theory. Like length
frequency, c = l[f]/t[f], mass frequency describes that bound where counts of m[f] may no longer be distinguished, greater than 1.85492 × 10^43 units per second, Equation (16).
The upper bound to mass frequency is physically significant and cannot be exceeded any more than a length frequency greater than 1 to 1 (i.e. n[L]l[f]/n[T]t[f] > c). As we work through an
understanding of mass frequency we will demonstrate how counts above and beyond this bound correspond to measure smaller than the reference. Not only does a mass frequency above a frequency bound
(i.e. a smaller value for m[f] in the expression 1/m[f]) describe a point in space-time subject to indistinguishable count of m[f], it also describes a faster-than-light relationship between length
and time, identifiable using the fundamental expression, l[f]m[f] = 2θ[si]t[f] (i.e. a smaller value for m[f] implies a larger value for l[f] where c = l[f]/t[f] then a faster-than-light relation).
3.1. Mass Distribution
Galactic rotation follows classical theory with adjustments made for the effects described by relativity, the Informativity differential (Appendix A.1) and universal expansion. To simplify the
expressions, the first two effects will not be integrated into the results. But, the third effect, expansion, is significant in magnitude. We begin with a review of expansion as described in the
first paper followed by mass distribution.
Stellar expansion―the modern understanding of expansion―which is a function of universal expansion plus those forces of interaction since the earliest epoch will not be discussed. Universal
expansion, conversely, describes the increasing space in the universe. The rate when defined with respect to the universe is Equation (27),
${H}_{U}=2{\theta }_{si}$ . (35)
The constant 2θ[si] is referred to as the system constant. With it universal expansion may be described using familiar terms ( [2] , Equation (87)) such as the diameter D[U] of the universe in
billions of light-years and the age A[U] of the universe in billions of years.
${D}_{U}=2{\theta }_{si}{A}_{U}=2×3.26239×13.799=90.035\text{\hspace{0.17em}}\text{bly}$ , (36)
with these parameters we may now summarize the mass distribution expressions starting with fundamental mass ( [2] , Equation (93)) which is then used to derive the distributions,
${M}_{f}={A}_{U}{\theta }_{si}\frac{{m}_{f}}{{t}_{f}}$ . (37)
Because our frame of reference is the universe, θ[si] carries no units. A complete derivation is provided in the first paper ( [2] , Section 3.12). The advantage of this approach is that each
distribution is clearly defined. The total is divided such that the dark mass M[dkm] is that mass sufficiently distant that expansion prevents the light (i.e. information) from ever reaching the
observer. The observable mass M[obs] makes up the remainder. The observable may then be divided into two categories, that which is presently visible M[vis] and the unobserved M[uobs] which will be
visible given sufficient elapsed time. Each distribution ( [2] , Equation (109), Equation (110), Equation (113), and Equation (115)) precisely matches the ∆CDM results. We learn here that each is
${M}_{dkm}=\frac{{\theta }_{si}^{2}-2}{{\theta }_{si}^{2}+2}=68.3624%$ , (38)
${M}_{obs}=\frac{4}{{\theta }_{si}^{2}+2}=31.6376%$ , (39)
${M}_{vis}=\frac{1}{2{\theta }_{si}}\frac{{M}_{obs}}{{M}_{tot}}=\frac{{M}_{obs}}{2{\theta }_{si}}=4.84884%$ , (40)
${M}_{uobs}={M}_{obs}-{M}_{vis}=31.6376-4.84884=26.7888%$ . (41)
In modern theory M[dkm] is recognized as dark energy; M[uobs] is recognized as dark matter. As neither reflects the calculations, the terms are accordingly replaced.
This brings us to an important observation as described in Equation (40),
${M}_{obs}=2{\theta }_{si}{M}_{vis}$ . (42)
If the visible is that which is presently visible and the unobserved is that which becomes visible with elapsed time, then with respect to the earliest epoch nearly all the visible we see today was
previously unobserved (i.e. dark matter). The idea that dark matter is different than what we presently identify as visible is in conflict. Further technical details regarding the treatment of mass
distributions are provided in Appendix A.6.
Consider now that the Informativity distributions precisely match the ∆CDM calculations. This is accomplished with only an understanding of fundamental mass M[f]. Combining the distributions we find
that ( [2] , Equation (118))
${M}_{f}=\frac{{M}_{tot}{M}_{obs}}{2{M}_{tot}-{M}_{obs}}$ , (43)
but this seemingly reveals a problem. If the expressions are invariant why are the distributions properly resolved while mass is moving from the unobserved to the visible? From yet another point of
view, such that M[f] is a function of time, must not M[tot] increase? Yes, evidence that the total mass of the universe is increasing follows.
The CMB calculations are just one inevitable outcome of mass accretion M[acr] ( [2] , Equation (135)). The age, quantity, density and temperature of the CMB may each be calculated such that
${M}_{acr}=\frac{{\theta }_{si}^{3}}{2}=17.3611\text{\hspace{0.17em}}\text{units}\text{\hspace{0.17em}}{m}_{f}/\text{unit}\text{\hspace{0.17em}}{t}_{f}$ . (44)
Most importantly, “there are no free variables”, in the calculation. Density and temperature, naturally, are a function of the elapsed time we identify as being the present,
${A}_{U}={\text{e}}^{\sqrt{3}{\theta }_{si}^{3}/2}=1.14652×{10}^{13}\text{s}=363309\text{\hspace{0.17em}}\text{y}$ , (45)
${M}_{tot}={n}_{Tu}{m}_{f}\frac{{\theta }_{si}^{3}}{2}=1.50159×{10}^{50}\text{\hspace{0.17em}}\text{kg}$ , (46)
$\rho =\frac{{M}_{tot}{c}^{2}}{{V}_{U}}=4.17041×{10}^{-14}\text{J}/{\text{m}}^{\text{3}}$ , (47)
$T={\left(\frac{\rho }{a}\right)}^{1/4}=2.72468\text{\hspace{0.17em}}\text{K}$ . (48)
The calculations are a direct result of M[acr]. The universe beings as a quantum bubble unable to expand at the speed of light because there exists no means to resolve a point outside of the bubble
until the universe reaches a radius of $\sqrt{3}{l}_{f}$ . This trigger ends quantum inflation precisely at 363,309 years, releases the accreted mass/energy (which occurs at the noted rate of ${\
theta }_{si}^{3}/2$ units of m[f] per unit of t[f]) as CMB and initiates expansion as we see it today. There is no faster-than-light inflationary period and the results match our best observational
data precisely. The calculations and details were published in the Journal of High Energy Physics, Gravitation and Cosmology ( [2] , Section 3.15) with additional explanation of the effects of
measurement distortion following ( [5] , Section 3.6).
A graphical representation of the distributions is also presented in Figure 3. The mass values are constrained to the precision of the age of the universe, 13.799 billion years [7] , as our most
accurate measure of the universe.
For a more complete list of mass distribution conversions refer to Appendix A.5.
Finally, to provide a reference for the expressions to follow we will use the Milky Way as our target. The calculations consider only the mass within the first 84,000 light-years. The corresponding
value for observable mass is then
${M}_{obs}=8.56060×{10}^{41}\text{\hspace{0.17em}}\text{kg}$ . (49)
All mass, density and velocity data for the Milky Way comes from Stacy McGaugh’s 2018 Milky Way mass models [8] .
3.2. Orbital Velocity Bound
Count bounds are an important and physically significant attribute in describing the behavior of matter. Length frequency is the most well-known count bound c = l[f]/t[f]; for each count of
fundamental time there can be at most one count of fundamental length. Any count of l[f] greater than t[f] would correspond to a velocity greater than the speed of light. The physical significance of
fundamental units of measure is what distinguishes measurement quantization from an unbounded description of nature.
There also exist upper and lower count bounds for m[f]/t[f] and m[f]/l[f]. We respectively call these bounds mass frequency and mass-to-length frequency. The orbital velocity of a star is subject to
all three bounds in addition to the effects of expansion. A description may be resolved starting with the classical expression for orbital velocity,
{{n}_{M}}{{n}_{Lr}}}$ . (50)
As described in Appendix A.2 the upper mass-to-length count bound with respect to orbital velocity is 1 to 1, n[M] < n[Lr]. But, the relation we seek is the mass-to-length count bound with respect to
the escape velocity
$2{n}_{M}<{n}_{Lr}$ . (51)
Consider now that the smallest count of m[f] with respect to l[f] may not be less than the precision offered by the reference ${m}_{f}=2.17647×{10}^{-8}\text{\hspace{0.17em}}\text{kg}$ . To translate
this to a whole-unit count of the reference scale the ratio,
{\hspace{0.17em}}{m}_{f}}{4.59468×{10}^{7}\text{\hspace{0.17em}}\text{units}\text{\hspace{0.17em}}{l}_{f}}=\frac{1}{1/{m}_{f}}={n}_{Mb}$ . (52)
Combining both bounds the ratio is then 2 units of m[f] per unit of l[f] where 1/(1/m[f]). Thus, 2(1/(1/m[f])) = 2m[f]. Where n[Mb] and m[f] are equal in value and have no units, then the classical
velocity bound is
${v}_{bc}=c\sqrt{\frac{{n}_{M}}{{n}_{Lr}}}=c\sqrt{2{m}_{f}}$ . (53)
The expression does not account for the expansion of space H[U] = 2θ[si], Equation (27). Such that H[U] is relative to the diameter of the universe, divide by 2. The radial expansion respective of
orbital and escape velocity may be written in two ways using the fundamental expression to convert between them:
${v}_{b}={\theta }_{si}c\sqrt{2{m}_{f}}=204.054\text{\hspace{0.17em}}\text{km}\cdot {\text{s}}^{-1}$ , (54)
${v}_{b}=c{m}_{f}\sqrt{{\theta }_{si}c}=204.054\text{\hspace{0.17em}}\text{km}\cdot {\text{s}}^{-1}$ . (55)
As a reminder, both θ[si] and our substitution of m[f] for n[Mb] carry no units. This is the velocity bound corresponding to the upper count bound of m[f] that may be discerned at a point in space.
To resolve a corresponding mass bound set v[b] equal to the same as expressed with Newton’s expression and reduce with the fundamental expression. The derivation may be found in Appendix A.3 along
with an explanation of units,
${M}_{b\text{-}f\left(R\right)}=R{\theta }_{si}\frac{{m}_{f}^{3}}{{t}_{f}}$ . (56)
The mass bound M[b-f][(R)] is a function of the mass within the target orbital radius f(R). By example, a galaxy with a radius of 84,000 light-years $R=7.94157×{10}^{20}\text{\hspace{0.17em}}\text{m}
$ would need more than
of mass, 2.49 × 10^11 solar masses to display behavior associated with a measurement quantization bound. Such a mass reflects $2.49×{10}^{11}/4.30×{10}^{11}=57.9%$ of the estimated mass of the Milky
Way. Equation (54) describes the upper bound to measurable mass unadjusted for total mass and a mass density profile. If mass density exceeds this bound, the upper mass count bound will exceed the
mass frequency bound causing additional mass count to be indistinguishable.
Lastly, consider what a higher or lesser velocity bound implies. We may demonstrate by reorganizing the fundamental expression m[f]l[f] = 2θ[si]t[f] into a form that resolves the length count
presented in the denominator of Equation (52), 1/m[f] = 4.59468 × 10^7. Thus, a count 100 units greater implies a corresponding speed of
$\frac{{l}_{f}}{{t}_{f}}=c=2{\theta }_{si}\frac{1}{{m}_{f}}\approx 2{\theta }_{si}\left(4.59468×{10}^{7}+100\right)=299793110\text{\hspace{0.17em}}\text{m}\cdot {\text{s}}^{-1}$ . (58)
a 652 m/s increase above the speed of light. The increase also corresponds to a velocity bound of
${v}_{b}={\theta }_{si}c\sqrt{2{n}_{Mb}}\approx {\theta }_{si}c\sqrt{\frac{2}{\left(1/{m}_{f}\right)+100}}=204.053\text{\hspace{0.17em}}\text{km}\cdot {\text{s}}^{-1}$ , (59)
a decrease of 1 m×s^−1. This does not mean that the speed of a star may not fall below 204.054 km×s^−1. The expression describes an upper bound with which to discern mass counts and as such an upper
bound to the gravitational pull on a star. When the mass count exceeds the mass count bound, the target is unable to distinguish additional mass and as such the gravitational effect of mass on a star
reaches a maximum.
This investigation also does not imply that stars cannot have velocities greater than 204.054 km×s^−1. While these expressions are invariant, we have not integrated the effects of an uneven mass
distribution typical of a galaxy. This will be the subject of the next section.
3.3. Galactic Rotation and the Milky Way
Using the velocity bound, an expression may now be developed as a function of mass distribution in a galaxy. The relation follows the same form as that which describes visible M[vis] and unobserved M
[uobs] (aka dark matter) mass, Equation (A.5.6) from Appendix A.5,
${M}_{uobs}={M}_{vis}\left(2{\theta }_{si}-1\right)$ . (60)
Replacing the dimensionless speed parameter θ[si] with the ratio of observed over bound v[o]/v[b] (i.e. in relativity then v/c) will provide the corresponding relation between the effective and mass
bound. But, an understanding of the geometry of the substitution is difficult. For that reason, we will follow an algebraic approach that resolves the speed parameter β as a relative percent
difference ∆%[o-b] of the bound.
$\Delta {%}_{o\text{-}b}=\frac{\left({v}_{o}-{v}_{b}\right)}{{v}_{b}}=\frac{{v}_{o}}{{v}_{b}}-1$ . (61)
To reduce, also needed is the velocity bound v[b] from Equation (54), v[b] = θ[si]c(2m[f])^1/2. The expression for mass is then the product of the mass bound M[b-f][(R)] and 2∆%[o-b], hereafter
referred to as the effective mass M[e-f][(R)]. The symbol f(R) in subscript indicates that the mass considered is the mass within the orbital radius R from a galactic center. Like the relation
presented in Equation (54), the speed parameter ∆%[o-b] is doubled to describe mass in terms of the bound for escape velocity.
${M}_{e\text{-}f\left(R\right)}={M}_{b\text{-}f\left(R\right)}\left(2\Delta {%}_{o\text{-}b}+1\right)={M}_{b\text{-}f\left(R\right)}\left(2\frac{{v}_{o}}{{v}_{b}}-2+1\right)$ , (62)
${M}_{e\text{-}f\left(R\right)}={M}_{b\text{-}f\left(R\right)}\left(2\frac{{v}_{o}}{{v}_{b}}-1\right)$ . (63)
when incorporating expansion we realize that the observer’s view of the universe is skewed; the effect suggests the presence of more mass than is actually present. In Figure 4 the observable M[o-f]
[(R)], bound M[b-f][(R)], and effective M[e-f][(R)] mass are displayed.
Where the effective mass is less than the bound, the orbital velocity of stars follow a classical behavior. Conversely, an effective mass greater than the bound presents a mass count greater than the
mass frequency bound. Some count of m[f] will be indistinguishable leading to a constraining effect on gravity and corresponding star velocities. The crossover from classical to non-classical
behavior occurs at 9.32848 × 10^3 light-years.
Notice that the observable and effective mass differ by a factor of 3.9 at
Figure 4. Galactic mass corresponding to actual (green), mass frequency bound (red) and relative mass frequency bound (purple).
$R=84×{10}^{3}$ light-years; 74% of the mass is missing. The magnitude of this effect depends on the total mass of the galaxy or galaxies considered. A second notable factor regards mass
distribution. As discussed, excess mass count is indistinguishable creating a mitigating gravitational effect. Which mass counts are lost? This is presently unknown, but also less significant in a
well-organized system such as a galaxy. Conversely, consideration of an uneven distribution (i.e. several galaxies) will present a center-of-mass offset respective of the indistinguishable mass
Both effects are notably evident in the Bullet Cluster. For one, the cluster exhibits a missing mass of approximately 90%. The cluster also exhibits a center-of-mass offset as would be expected with
a lost mass count, the latter being of considerable interest for future research.
Using Newton’s expression for velocity, ${v}_{b}={\left(G{M}_{b\text{-}f\left(R\right)}/R\right)}^{1/2}$ and the expression for the mass bound ${M}_{b\text{-}f\left(R\right)}={\theta }_{si}{m}_{f}^
{3}R/{t}_{f}$ from Equation (56), we may now resolve the effective star velocity
${v}_{e}={\left(\frac{G{M}_{e\text{-}f\left(R\right)}}{R}\right)}^{1/2}={\left(\frac{G{M}_{b\text{-}f\left(R\right)}}{R}\left(2\frac{{v}_{o}}{{v}_{b}}-1\right)\right)}^{1/2}$ , (64)
${v}_{e}={\left(\frac{G}{R}{\theta }_{si}\frac{{m}_{f}^{3}}{{t}_{f}}R\left(2\frac{{v}_{o}}{{\theta }_{si}c\sqrt{2{m}_{f}}}-1\right)\right)}^{1/2}$ , (65)
${v}_{e}=2{\theta }_{si}{\left(2\frac{{v}_{o}}{\sqrt{2{m}_{f}}}-c{\theta }_{si}\right)}^{1/2}$ . (66)
while it may seem more appropriate to use a mass or mass density dataset the choice is irrelevant. One may modify the expression to enter velocity, mass or mass density and still arrive at the same
expression. For example, as resolved in Appendix A.4, we may substitute the observable velocity v[o] in Equation (66) for this equivalent function written in terms of the effective mass,
${v}_{o}=\sqrt{\frac{{m}_{f}}{2}}\left(\frac{{M}_{e\text{-}f\left(R\right)}}{R}\frac{{l}_{f}}{{m}_{f}^{3}}+{\theta }_{si}c\right)$ . (67)
More importantly, Newton’s expression for velocity does not produce the observable velocity curve. Informativity succeeds because the expression for effective velocity is a function of the mass count
bound, Equation (54), an invariant expression with no free variables. To highlight that fact, we retain the corresponding velocity bound v[b] in Figure 5 to demonstrate the natural tendency for stars
to approach the bound when the mass count reaching a star exceeds the effective bound. The remaining curves are as follows. The effective velocity v[e] is plotted in red. The observable velocity v[o]
is plotted in green. And the classical velocity v[c] is plotted in blue; that’s Newton’s expression.
There are two points of view in conflict. That is, the classical velocity implies that what we observe is moving too fast. The curve also suggests that there is a missing dark matter holding the
stars in orbit. At the same time, the observable velocity suggests correspondence to variations in mass density.
The Informativity approach resolves the discrepancy describing an effective velocity that follows the bound when the effective mass exceeds the mass bound. When effective mass does not exceed the
bound, orbital velocities follow a classical behavior.
Although the bound is invariant―204.054 km×s^−1―variations in galactic mass density do affect the gravitational pull on a star. These effects may be evened out when taking an average of thousands of
galaxies. Except near the galactic core where the crossover between classical and Informativistic behavior varies from one galaxy to the next, the velocity curve levels out reflecting an averaging of
mass profiles.
An unexpected effect of mass count bounds is apparent between 4 and 8 thousand light-years where star velocities level out until otherwise affected by increasing mass density. The cause of this
effect is a subject of interest. Perhaps physically insignificant, but star velocity may favor classical behavior at the crossover between the effective and mass bound.
The mass bound delineates two behaviors. Recall from Equation (56), ${M}_{b\text{-}f\left(R\right)}=R{\theta }_{si}{m}_{f}^{3}/{t}_{f}$ that the mass bound is a function of how much mass is within a
given radius. Variations in mass density imply increases or decreases in the spherical space described by R for a fixed amount of mass. If we fix R in consideration of a region of greater mass
density, then the effective velocity will be higher, describing measured velocities that rise above the bound (i.e. 204.054 km×s^−1). The opposite effect applies for less dense regions such that
velocities lesson.
To further demonstrate this effect, consider Figure 6 where a model galaxy with the same mass as the Milky Way is presented, but mass distribution has been evened as though we were averaging the mass
profile of thousands of galaxies. To be clear, a mass equal to that for R < 1000 light-years of the Milky Way center is taken. Then the remaining mass (where the total considered is only the
Figure 5. Stellar velocities corresponding to observed (green), relative mass frequency bound (red), mass frequency bound (purple) and Newton’s expression (blue).
Figure 6. Stellar velocities corresponding to actual (green), an even mass distribution (orange) and the mass frequency bound (red).
mass in the first 84,000 light-years) is evenly divided across the remaining 83 thousand light-years. The corresponding effective velocity (orange) is drawn. As expected, the curve levels out just
above the bound velocity (purple) with a magnitude that is in proportion to the excess mass above the bound. An average of thousands of galaxies will demonstrate a level velocity curve with a
magnitude that corresponds to the mass in excess of the mass bound.
As a final note, separation of the velocity term in Equation (63) from the data can be challenging. It is the mass density data that characterizes the galaxy under consideration. The argument may be
extended to demonstrate that it is also irrelevant what dataset is chosen: mass, mass density or velocity. As each measure is mathematically related, an argument for data independence by favoring any
dataset over another cannot be made.
But, there are two remaining considerations that are data independent. Notably, an expression must describe a phenomenon with the correct magnitude. The Informativity expression properly accommodates
the effects of a mass count bound in an expanding universe. Where Newton’s expression does not provide the observed magnitude in describing orbital velocity, the Informativity expression does.
Also providing support is the bound itself, the purple line denoting an invariant velocity of 204.054 km×s^−1. The bound expression contains no measurement data, no free variables and as such no
“fitting”, ${v}_{b}={\theta }_{si}c{\left(2{m}_{f}\right)}^{1/2}$ . Referring to Figure 5, star velocities favor the bound. But, that will not always be clearly evident. What is clear is that the
bound is the baseline measure from which the magnitude of the Informativity expression is calculated. If the bound were not physically significant, the magnitude would be incorrect and the resulting
curve would not match the observational data.
Returning to our initial discussion our goal was to develop a mass expression defined with respect to a bound. To this we can compare the effective and unobserved mass expressions, each taking the
form M[1] = M[2](2β − 1).
${M}_{e\text{-}f\left(R\right)}={M}_{b\text{-}f\left(R\right)}\left(2\frac{{v}_{o}}{{v}_{b}}-1\right)$ , (68)
${M}_{uobs}={M}_{vis}\left(2{\theta }_{si}-1\right)$ . (69)
The details of the speed parameter depend on the masses being compared. In the case of orbital velocity, the parameter is found on the right-side of this expression ( [5] , Equation (68))
${\beta }^{2}=\frac{{v}^{2}}{{c}^{2}}={\left(\frac{{n}_{Lm}{l}_{f}}{{n}_{T}{t}_{f}}\frac{{n}_{T}{t}_{f}}{{n}_{Lc}{l}_{f}}\right)}^{2}=\frac{{n}_{Lm}^{2}}{{n}_{Lc}^{2}}=2\frac{{n}_{M}}{{n}_{Lr}}$(70)
which predicts and demonstrates equivalence between the phenomena of motion and gravitation. Respecting the difference between orbital motion v = (GM/R)^1/2 and escape velocity v = (2GM/R)^1/2, we
remove the factor of 2. Then Equation (50) may be completed
and recognized as the same speed parameter β commonly found in relativistic expressions. And finally, comparison of Equation (50) and Equation (68) successfully confirm the correlation between motion
and gravitation (i.e. mass) as expected.
3.4. What Does the 26.7888% Distribution Describe
In this section, we will discuss why the dark matter phenomenon has been so closely associated with the ΛCDM distribution also distinguished by the same name. We will not be using the ΛCDM approach
to discuss mass distribution but instead use the Informativity expressions, such that each distribution is a function of one physical constant θ[si]. Theta has been accurately measured to 6
significant digits [6] .
We begin with the unobserved mass
${M}_{uobs}={M}_{obs}-{M}_{vis}=31.6376-4.84884=26.7888%$ , (72)
which describes the mass that will be observable M[obs] Equation (39), but is not presently visible M[vis] Equation (41). This is one interpretation. Using Equation (40) M[obs] = 2θ[si]M[vis], we can
also resolve this distribution as
${M}_{uobs}=2{\theta }_{si}{M}_{vis}-{M}_{vis}$ . (73)
Such that H[U] = 2θ[si] Equation (27), then ${M}_{uobs}={M}_{vis}\left({H}_{U}-1\right)$ . The dark matter distribution M[uobs] is then the energy of expansion as a function of the visible mass H[U]M
[vis] minus the energy associated with the visible mass M[vis].
The two interpretations―mass and energy―while mathematically equivalent have led to significant confusion. Additionally, mass distributions are defined with respect to the universe. But, the rate of
expansion is much less than 2θ[si] in a region the size of a galaxy. As such, application of a distribution such as dark matter to the description of a galaxy is questionable.
With respect to existing observational support, Informativity does not imply that the mass we measure in a galaxy is all the mass present. There are studies that suggest there is additional non- or
low-light-absorbing fine dust [9] . While there is a great deal to learn about galactic mass composition, Informativity constrains the magnitude of this mass to the observable distribution M[obs]. It
should be added that gravitational lensing studies are not an indicator of missing mass. Rather, these studies will need to incorporate effective mass which accounts for expansion and the mass
frequency bound. With this approach the bending of light conforms to the effective mass as is demonstrated by the effective velocity curve.
3.5. Interpretation of Mass
At this point we have a better understanding of the unobserved distribution and its relation to expansion, but have not resolved a clear understanding of the bound.
We present three expressions each describing the mass bound against which effective mass is measured. Equation (54) set equal to Newton’s velocity expression describes the mass bound in terms of the
fundamental measures (Appendix A.3).
${v}_{b}={\theta }_{si}c\sqrt{2{m}_{f}}$ , (74)
${v}_{b}={\left(\frac{G{M}_{b\text{-}f\left(R\right)}}{R}\right)}^{1/2}$ , (75)
${M}_{b\text{-}f\left(R\right)}={v}_{b}^{2}\frac{R}{G}=2{\theta }_{si}^{2}{c}^{2}{m}_{f}\frac{R}{G}=\left(\frac{{l}_{f}{m}_{f}}{{t}_{f}}\right){\theta }_{si}{c}^{2}{m}_{f}\frac{R{m}_{f}}{{c}^{3}{t}_
{f}}$ , (76)
${M}_{b\text{-}f\left(R\right)}={\theta }_{si}R\frac{{m}_{f}^{3}}{{t}_{f}}$ . (77)
For the next two expressions, consider one point on the mass bound curve such that the radius is that of the universe (Appendix A.5, Equation (A.5.20)),
${M}_{Ub}={M}_{f}{m}_{f}^{2}{\theta }_{si}$ . (78)
In a similar fashion, consider the mass bound in terms of mass distributions (Appendix A.5, Equation (A.5.10)),
$2{M}_{tot}{M}_{f}={M}_{obs}\left({M}_{tot}+{M}_{f}\right)$ , (79)
${M}_{Ub}={\theta }_{si}{m}_{f}^{2}\frac{{M}_{tot}{M}_{obs}}{2{M}_{tot}-{M}_{obs}}={\theta }_{si}{m}_{f}^{2}\frac{{M}_{tot}{M}_{obs}}{{M}_{tot}+{M}_{dkm}}$ . (80)
while each approach offers opportunity to present the mass bound as a function of mass, energy or physical constants, Equation (77) presents the clearest description, a line. The relation
demonstrates that geometry is at work.
With the bound more clearly understood, return to Equation (63) and resolve the effective mass,
${M}_{e\text{-}f\left(R\right)}={M}_{b\text{-}f\left(R\right)}\left(2\frac{{v}_{o}}{{v}_{b}}-1\right)$ , (81)
${M}_{e\text{-}f\left(R\right)}={\theta }_{si}R\frac{{m}_{f}^{3}}{{t}_{f}}\left(2\frac{{v}_{o}}{{\theta }_{si}c\sqrt{2{m}_{f}}}-1\right)$ , (82)
${M}_{e\text{-}f\left(R\right)}={v}_{o}2R\frac{{m}_{f}^{3}}{{l}_{f}\sqrt{2{m}_{f}}}-{\theta }_{si}R\frac{{m}_{f}^{3}}{{t}_{f}}$ . (83)
Built on Equation (77), velocity v[o] is the only new variable, a data dependent value that characterizes the target. The result is quantum in detail and valid for the entire measurement domain. When
effective mass rises above or falls below the mass bound so does the velocity. When the effective and bound masses are equal, then the velocities are as well.
We may summarize effective mass as having one of two states. The first serves as the reference, defined where the effective and bound mass are equal, a purely geometric description ${M}_{b\text{-}f\
left(R\right)}={\theta }_{si}R{m}_{f}^{3}/{t}_{f}$ . The second state we call the offset Equation (83). Collectively the two states describe observed velocity as a function of the effective mass that
characterizes the target.
Several studies of galaxies and galaxy clusters have suggested the presence of a gravitational force that does not coincide with the visible matter. Notably, the effective mass of a given matter
field describes force that is unexpected from our point of view. While we will not review the specific calculations of existing investigations, it is expected with respect to this model that the
effects of a mass frequency bound when integrated with that of expansion must produce an offset and that offset will be even more pronounced when describing disorganized targets.
Importantly, expansion does not explain the dark matter mass discrepancy because mass alone does not determine orbital velocity as Newton had surmised. Several effects are at work. Thus, Newton’s
expression is correct so long as expansion, mass frequency, measurement distortion (i.e. also described by relativity) and the Informativity differential are not significant factors.
Modern physical descriptions use mass to describe gravity, but the effective mass is significantly greater in magnitude than the observed mass described by Newton. The bound describes a geometric
reference with an offset swinging from one side to the other like a weight on a rubber band. While the Informativity and Newton expressions coincide for systems having less mass density, velocity is
not solely a function of mass. Thus, the question, where is the missing mass, is not valid.
Bounded gravity may also be applied to the early universe when mass density was significant. When expressions that incorporate bounded gravity are used some doors may open with early universe
modeling. While not the focus of this paper, a detailed account of quantum inflation, the trigger event that ends this epoch and the ensuing expansion are described in the first paper ( [1] , Section
3.15) with additional explanation of the effects of measurement distortion described in the second paper ( [5] , Section 3.6). Notably, the solution as presented in Equations (45)-(48) is a function
of one physical constant.
A final question is why should the measured mass of a galaxy be attributed to the observable and not the visible distribution? The answer is primarily subjective as mass distributions may not be used
to describe a galaxy. That said, one may note with elapsed time that a specific amount of observable mass becomes visible. Because the mass of a galaxy does not increase, the label “observable” is
more appropriate.
In practice, the issue with scaling distributions is that the scaling process changes the properties that the distributions are defined against. To succeed any application must retain each property
of the initial definition. For instance, the scaling would require that the outer edge of the galaxy expand at the speed of light. Loss of this property is immediately obvious. For one, dark mass
cannot even exist. As well, the visible and observable distributions are always the same.
3.6. Kinetic Energy
As a follow up to mass frequency, we may provide one final confirmation of our understanding of n[M]/n[Lr] = 2m[f] by reducing the Informativity interpretation to demonstrate the equation for kinetic
energy. Notably, the classical expression does not include the radial expansion parameter θ[si] which is defined with respect to a bound and thus carries no units. So, we start with the static radial
form. Such that m[f] = 2θ[si]/c from the fundamental expression and the expression for half a fundamental unit of mass E[f] = 2θ[si]c ( [2] , Equation (49)), then the static velocity bound is
$v=c\sqrt{2{m}_{f}}=c\sqrt{\frac{4{\theta }_{si}}{c}}=\sqrt{4{\theta }_{si}c}=\sqrt{2{E}_{f}}$ . (84)
$v=c\sqrt{2{m}_{f}}=c\sqrt{\frac{2}{1/{m}_{f}}}=c\sqrt{\frac{2}{{n}_{M}}}=\sqrt{\frac{2{c}^{2}}{{n}_{M}}}$ , (85)
$v=\sqrt{\left(\frac{2{\theta }_{si}}{c}\frac{1}{{m}_{f}}\right)\frac{2{c}^{2}}{{n}_{M}}}=\sqrt{\frac{4{\theta }_{si}c}{{n}_{M}{m}_{f}}}=\sqrt{\frac{2E}{m}}$ , (86)
and may then be reduced to resolve the kinetic energy associated with any mass,
$E=\frac{m{v}^{2}}{2}$ . (87)
One may compare the first and last velocity expressions and wonder why the latter has a mass value in the denominator. The mass value is what generalizes the expression for any mass, velocity and
energy. The initial expression is an invariant description of the smallest unit of energy E[f] corresponding to a mass count bound of n[M] = 1/m[f]. That ratio is precisely 1 leaving us with 2E[f]
under the square root operator.
4. Discussion
Perhaps the most significant outcome of this research is not a model of galactic orbital dynamics, but the inclusion of expansion, the mass frequency bound and mass density into a single description
of orbital velocity, since the time of Newton mass has been considered the primary factor describing the effects of gravitation. There have been modifications to that understanding (i.e. relativity),
but such modifications have been a fine-tuning of the broader expressions set forth by Newton, mass and radial distance being the variables that determine orbital motion. But, with the expressions
set forth here, mass is one of several factors. The mass frequency bound now designates the demarcation point; it is quantum and valid for the entire measurement domain.
Finally, where there have been several proposals describing solutions to galactic orbital dynamics [10] , the traditional approach is one of resolving data dependent expressions. Informativity takes
a uniquely different view of the universe, that physical expression is an outcome of bounds to measure. The mass frequency bound is an outcome of this axiom, a geometric expression that identifies a
reference and an offset against which the effective mass is resolved. Mathematics of counts of the fundamental measures is all that is needed to unravel the motions of the stars.
We thank Edanz Group (https://www.edanzediting.com/?utm_source=ack&utm_medium=journal) for editing a draft of this manuscript.
A.1. Numerical Limits to Q[L]n[Lr]
The term Q[L]n[Lr] is referred to as the Informativity differential in recognition of the central role it plays in describing how fractional values less than the reference measure reflect a
distorting effect in distance measurement. Knowing the limits to Q[L]n[Lr] is essential in resolving the fundamental measures.
Q[L]n[Lr] is Equation (2) multiplied by n[Lb].
${Q}_{L}{n}_{Lr}=\left(\sqrt{1+{n}_{Lb}^{2}}-{n}_{Lb}\right){n}_{Lb}$ . (A.1.1)
Note, what is measured always equals a whole-unit count of a fundamental measure, and with a = 1 we find that n[Lb] = n[Lr] for all values. This is easily verified in that the highest value for Q[L]
is obtained for n[Lb] = 1 where ${\left(1+{1}^{2}\right)}^{0.5}-1=0.414$ and the “observed” distance of c presented as a count n[Lr] is always rounded down to the highest integer value equal to the
count n[Lb] with Q[L] = 0.414 at its highest and quickly approaching 0 with increasing n[Lb]. Therefore,
${Q}_{L}{n}_{Lr}=\left(\sqrt{1+{n}_{Lr}^{2}}-{n}_{Lr}\right){n}_{Lr}$ . (A.1.2)
The lower limit where n[Lr] = 1 is easily produced, ${\mathrm{lim}}_{r=1}f\left({Q}_{L}{n}_{Lr}\right)=\sqrt{2}-1$ . Conversely, if we divide by n[Lr], then add n[Lr], square, subtract ${n}_{Lr}^{2}$
, and divide by 2, we find that
$\frac{{Q}_{L}^{2}}{2}+{Q}_{L}{n}_{Lr}=\frac{1}{2}$ . (A.1.3)
Q[L] decreases with increasing n[Lr] until the left term drops out. Distance does not need to be significant to reduce the Informativity differential. At just 10^4l[f], Q[L]n[Lr] rounds to 0.5 to
nine significant digits.
A.2. Upper Bound Relationship between Length and Mass
To resolve the upper bound relation between length and mass, we begin with the expression for escape velocity, set velocity equal to the speed of light denoting the upper bound and then substitute
fundamental units for each of the terms. Notably, the expression for G follows from Equation (6) as
$G=\frac{{Q}_{Lf}r{c}^{3}}{{\theta }_{si}}=\frac{{Q}_{Lf}{r}_{Lf}{l}_{f}{c}^{3}}{{\theta }_{si}}=\frac{{c}^{3}{l}_{f}}{2{\theta }_{si}}=\frac{{c}^{3}{t}_{f}}{{m}_{f}}=\frac{{l}_{f}}{{t}_{f}}\frac{{l}
_{f}}{{t}_{f}}\frac{{l}_{f}}{{t}_{f}}\frac{{t}_{f}}{{m}_{f}}$ . (A.2.1)
Likewise, a generalized mass count n[M] of m[f] follows from the fundamental expression l[f]m[f] = 2θ[si]t[f]. Where the ${\mathrm{lim}}_{r\to \infty }f\left({Q}_{L}{n}_{Lr}\right)=1/2$ as resolved
in Appendix A.1, then
${m}_{f}=\frac{2{\theta }_{si}}{c}=\frac{{\theta }_{si}}{{Q}_{Lf}{r}_{Lf}c}$ . (A.2.2)
and where c = l[f]/t[f], the expression for escape velocity may be reduced to show that
$v={\left(\frac{2GM}{r}\right)}^{1/2}$ , (A.2.3)
$c>{\left(\frac{2}{r}\frac{{Q}_{L}r{c}^{3}}{{\theta }_{si}}\frac{{n}_{M}{\theta }_{si}}{{Q}_{L}{n}_{Lr}c}\right)}^{1/2}>{\left(\frac{2{n}_{M}{c}^{2}}{{n}_{Lr}}\right)}^{1/2}$ , (A.2.4)
${n}_{Lr}>2{n}_{M}$ . (A.2.5)
Using escape velocity the upper bound of a count of n[M] with respect to n[Lr] is resolved. Conversely, for orbital velocity, the expression is v = (GM/r)^1/2. The relation differs by a factor of
${n}_{Lr}>{n}_{M}$ . (A.2.6)
A.3. Observable Mass Bound
The observable mass may be resolved by setting the bound velocity equal to the classical velocity and reducing. Where G = c^3t[f]/m[f], then
${\theta }_{si}c\sqrt{2{m}_{f}}=\sqrt{\frac{G{M}_{b\text{-}f\left(R\right)}}{R}}$ , (A.3.1)
${M}_{b\text{-}f\left(R\right)}=2{\theta }_{si}^{2}R{c}^{2}{m}_{f}\frac{1}{G}=2{\theta }_{si}^{2}R{c}^{2}{m}_{f}\frac{{m}_{f}}{{c}^{3}{t}_{f}}$ , (A.3.2)
${M}_{b\text{-}f\left(R\right)}=2{\theta }_{si}^{2}R{c}^{2}{m}_{f}\frac{{m}_{f}}{{c}^{3}{t}_{f}}=2{\theta }_{si}^{2}R\frac{{m}_{f}^{2}}{{l}_{f}}$ , (A.3.3)
${M}_{b\text{-}f\left(R\right)}=2{\theta }_{si}^{2}R\frac{{m}_{f}^{2}}{{l}_{f}}=2{\theta }_{si}R\frac{{m}_{f}{l}_{f}}{2{t}_{f}}\frac{{m}_{f}^{2}}{{l}_{f}}$ , (A.3.4)
${M}_{b\text{-}f\left(R\right)}={\theta }_{si}R\frac{{m}_{f}^{3}}{{t}_{f}}$ . (A.3.5)
Recall, the left portion of the v[b] expression in Equation (A.3.1) has a value of m[f] which is a dimensionless substitute for n[Mb]. There are no units. This is fine until Equation (A.3.4) where R
in meters cancels with l[f] in meters leaving one of the two ${m}_{f}^{2}$ with a single kilograms describing M[b-f][(R)]. But in Equation (A.3.1) we introduce the dimensionless expression θ[si] = m
[f]l[f]/2t[f]. Several cancellations leave both R, t[f] and an additional m[f] each dimensionless. The result is kilograms,
${M}_{b\text{-}f\left(R\right)}=2{\theta }_{si}^{2}R\frac{{m}_{f}^{2}}{{l}_{f}}m\frac{kg}{m}={\theta }_{si}R\frac{{m}_{f}^{3}}{{t}_{f}}kg$ . (A.3.6)
A.4. Resolving Effective Velocity as a Function of Mass
The effective and observed velocities can be the same in value, yet each term is identified separately. This calls into question the use of one to identify the other if they are not physically
The two terms operate as a limit where the estimated value for v[o] constrains the resulting value for v[e]. The correct physical description is where the modeled value for v[o] produces a value for
v[e] that is closer than any other combination.
${v}_{e}=2{\theta }_{si}{\left(2\frac{{v}_{o}}{\sqrt{2{m}_{f}}}-c{\theta }_{si}\right)}^{1/2}$ . (A.4.1)
There may exist theoretical argument that use of an observed velocity to resolve the effective velocity is still in principle problematic. For that reason, an alternative is offered whereby the
observable velocity is replaced by a function with effective mass as the only free variable,
${M}_{e\text{-}f\left(R\right)}={M}_{b\text{-}f\left(R\right)}\left(2\frac{{v}_{o}}{{v}_{b}}-1\right)$ , (A.4.2)
$\frac{{v}_{o}}{{v}_{b}}=\frac{1}{2}\left(\frac{{M}_{e\text{-}f\left(R\right)}}{{M}_{b\text{-}f\left(R\right)}}+1\right)$ , (A.4.3)
$2{v}_{o}={v}_{b}\left(\frac{{M}_{e\text{-}f\left(R\right)}}{{M}_{b\text{-}f\left(R\right)}}+1\right)$ , (A.4.4)
$2{v}_{o}={\theta }_{si}c\sqrt{2{m}_{f}}\left(\left({M}_{e\text{-}f\left(R\right)}/{\theta }_{si}\frac{{m}_{f}^{3}}{{t}_{f}}R\right)+1\right)$ , (A.4.5)
${v}_{o}=\sqrt{\frac{{m}_{f}}{2}}\left(\frac{{M}_{e\text{-}f\left(R\right)}}{R}\frac{{l}_{f}}{{m}_{f}^{3}}+{\theta }_{si}c\right)$ . (A.4.6)
while it is not possible to produce a data independent expression that characterizes any target galaxy, the initial expression can now be resolved as a function of the effective mass and that can be
resolved as a function of observed mass. In short, the effective velocity may be resolved as a function of the observed mass.
A.5. Mass Distribution Conversions
Following are a list of commonly used mass distribution conversion expressions. Several are resolved from the first paper ( [2] , Equation (113), Equation (110), Equation (109), and Equation (108)).
Notably, many of the expressions in the first paper are percentage expressions of a total mass. To resolve distribution values in kilograms, multiply the distribution percentage by Mtot in kilograms.
${M}_{obs}=2{\theta }_{si}{M}_{vis}$ , (A.5.1)
${M}_{obs}={M}_{tot}\frac{4}{{\theta }_{si}^{2}+2}$ , (A.5.2)
${M}_{dkm}={M}_{tot}\frac{{\theta }_{si}^{2}-2}{{\theta }_{si}^{2}+2}$ , (A.5.3)
${M}_{tot}={M}_{obs}+{M}_{dkm}$ , (A.5.4)
${M}_{uobs}={M}_{obs}-{M}_{vis}$ . (A.5.5)
These are resolved from the prior,
${M}_{uobs}={M}_{vis}\left(2{\theta }_{si}-1\right)$ , (A.5.6)
$2{M}_{tot}={M}_{vis}{\theta }_{si}\left({\theta }_{si}^{2}+2\right)$ , (A.5.7)
${M}_{dkm}={M}_{obs}\frac{\left({\theta }_{si}^{2}-2\right)}{4}$ , (A.5.8)
${M}_{dkm}={M}_{vis}\frac{{\theta }_{si}\left({\theta }_{si}^{2}-2\right)}{2}$ . (A.5.9)
And from the first paper ( [2] , Equation (118)) we may also resolve
$2{M}_{tot}{M}_{f}={M}_{obs}\left({M}_{tot}+{M}_{f}\right)$ , (A.5.10)
${M}_{tot}{M}_{f}={\theta }_{si}{M}_{vis}\left({M}_{tot}+{M}_{f}\right)$ , (A.5.11)
${M}_{f}=\frac{{M}_{tot}{\theta }_{si}{M}_{vis}}{{M}_{tot}-{\theta }_{si}{M}_{vis}}$ . (A.5.12)
We may also derive the relationship between the total and fundamental mass using the expression for total mass ( [2] , Equation (134)) and the expression for fundamental mass ( [2] , Equation (128)),
${M}_{tot}={n}_{Tu}{m}_{f}\frac{{\theta }_{si}^{3}}{2}$ , (A.5.13)
${M}_{f}={n}_{Tu}{m}_{f}{\theta }_{si}$ , (A.5.14)
${M}_{tot}={M}_{f}\frac{{\theta }_{si}^{2}}{2}$ , (A.5.15)
$\frac{{M}_{f}}{{M}_{tot}}=\frac{2}{{\theta }_{si}^{2}}$ . (A.5.16)
Such that the fundamental mass from Equation (24) is reduced with ${D}_{U}=2{R}_{U}=2{\theta }_{si}{A}_{U}$ from Equation (25) and set equal to the bound mass in Equation (77), then the mass bound
for the universe M[Ub] is
${M}_{f}={A}_{U}{\theta }_{si}\frac{{m}_{f}}{{t}_{f}}={R}_{U}\frac{{m}_{f}}{{t}_{f}}$ , (A.5.17)
${M}_{b\text{-}f\left(R\right)}=R{\theta }_{si}\frac{{m}_{f}^{3}}{{t}_{f}}$ , (A.5.18)
$\frac{{M}_{Ub}}{{m}_{f}^{2}{\theta }_{si}}=R\frac{{m}_{f}}{{t}_{f}}={M}_{f}$ , (A.5.19)
${M}_{Ub}={M}_{f}{m}_{f}^{2}{\theta }_{si}$ . (A.5.20)
Lastly, given the observable v[obs] and visible v[vis] velocity and Equation (A.5.1), then
$\frac{{v}_{obs}}{{v}_{vis}}=\frac{\sqrt{G{M}_{obs\text{-}f\left(R\right)}/R}}{\sqrt{G{M}_{vis\text{-}f\left(R\right)}/R}}$ , (A.5.21)
$\frac{{v}_{obs}}{{v}_{vis}}=\sqrt{\frac{{M}_{obs\text{-}f\left(R\right)}}{{M}_{vis\text{-}f\left(R\right)}}}=\sqrt{\frac{2{\theta }_{si}{M}_{vis\text{-}f\left(R\right)}}{{M}_{vis\text{-}f\left(R\
right)}}}=\sqrt{2{\theta }_{si}}$ . (A.5.22)
In terms of mass visible corresponds to the 4.84884% distribution as described in Equation (40). The observable corresponds to the 31.6376% distribution as described in Equation (39) and incorporates
universal expansion, M[obs] = H[U]M[vis].
A.6. Clarifying Interpretation of Mass Distributions
Distribution expressions may take a percentage or mass value. An expression demonstrating percentages may be converted to kilograms by multiplying the result by M[tot] in kilograms. Depending on the
substitutions elected, a resulting expression can lead to an incorrect interpretation. To demonstrate the issue, consider Equation (42),
$2{\theta }_{si}=\frac{{M}_{obs}}{{M}_{vis}}$ , (A.6.1)
$\frac{2}{{\theta }_{si}^{2}}=\frac{1}{{\theta }_{si}^{3}}\frac{{M}_{obs}}{{M}_{vis}}$ . (A.6.2)
Then set the two expressions equal to one another,
$\frac{{M}_{f}}{{M}_{tot}}=\frac{1}{{\theta }_{si}^{3}}\frac{{M}_{obs}}{{M}_{vis}}=\frac{1}{{\theta }_{si}^{3}}\frac{2{\theta }_{si}{M}_{vis}}{{M}_{obs}/2{\theta }_{si}}=\frac{4}{{\theta }_{si}}\frac
{{M}_{vis}}{{M}_{obs}}$ , (A.6.3)
${\theta }_{si}{M}_{obs}{M}_{f}=4{M}_{vis}{M}_{Tot}$ . (A.6.4)
And finally where
${M}_{tot}={n}_{Tu}{m}_{f}\frac{{\theta }_{si}^{3}}{2}$ . (A.6.5)
is a known function of time ( [2] , Equation (134)), we may reduce Equation (A.6.4) such that time is the only free variable.
$2{M}_{tot}{M}_{f}={M}_{obs}\left({M}_{tot}+{M}_{f}\right)$ , (A.6.6)
$2{M}_{tot}2\frac{{M}_{Tot}}{{\theta }_{si}^{2}}={M}_{obs}\left({M}_{tot}+2\frac{{M}_{Tot}}{{\theta }_{si}^{2}}\right)$ , (A.6.7)
${\theta }_{si}^{2}{M}_{tot}{M}_{obs}+2{M}_{Tot}{M}_{obs}-4{M}_{Tot}^{2}=0$ , (A.6.8)
${\theta }_{si}^{2}{n}_{Tu}{m}_{f}\frac{{\theta }_{si}^{3}}{2}{M}_{obs}+2{n}_{Tu}{m}_{f}\frac{{\theta }_{si}^{3}}{2}{M}_{obs}-4{n}_{Tu}^{2}{m}_{f}^{2}\frac{{\theta }_{si}^{6}}{4}=0$ , (A.6.9)
${n}_{Tu}{m}_{f}\frac{{\theta }_{si}^{5}}{2}{M}_{obs}+{n}_{Tu}{m}_{f}{\theta }_{si}^{3}{M}_{obs}-{n}_{Tu}^{2}{m}_{f}^{2}{\theta }_{si}^{6}=0$ , (A.6.10)
$\frac{{\theta }_{si}^{2}}{2}{M}_{obs}+{M}_{obs}-{n}_{Tu}{m}_{f}{\theta }_{si}^{3}=0$ , (A.6.11)
${M}_{obs}\left(\frac{{\theta }_{si}^{2}}{2}+1\right)={n}_{Tu}{m}_{f}{\theta }_{si}^{3}$ , (A.6.12)
${M}_{obs}=2{n}_{Tu}{m}_{f}\frac{{\theta }_{si}^{3}}{{\theta }_{si}^{2}+2}$ . (A.6.13)
with elapsed time n[Tu] one might assume that the observable mass distribution M[obs] is increasing. This is not a complete picture. The observable and total mass (A.6.5) are both increasing while
the distributions remain invariant,
${M}_{obs}=\left({n}_{Tu}{m}_{f}\frac{{\theta }_{si}^{3}}{2}\right)\frac{4}{{\theta }_{si}^{2}+2}$ , (A.6.14)
${M}_{obs}={M}_{tot}\frac{4}{{\theta }_{si}^{2}+2}$ . (A.6.15)
The result was demonstrated in the first paper ( [2] , Equation (110)).
Glossary of Terms
A frame of reference against a system of measure is applied. Frameworks are commonly discussed in Informativity and are typically either that of the observer’s inertial frame, the observed target or
that of the universe.
Fundamental Expression
The simplest expression correlating the three fundamental measures, l[f]m[f] = 2θ[si]t[f].
Fundamental Mass
The fundamental mass of the universe distinguishes a specific amount of mass whereby from a point in space-time additional mass would cause overlapping mass events that could not be distinguished due
to physically significant bounds to the measure of fundamental units of mass. Understanding and resolving fundamental mass in turn allows one to solve for all the mass distributions presently
understood only with ΛCDM.
Fundamental Measure
One of the measures length l[f], mass m[f], and time t[f] along with their correlation called the fundamental expression. Using measurement data from the Shwartz and Harris experiments in combination
with Heisenberg’s Uncertainty Principle, each are macroscopically defined and physically significant.
Informativity Differential
The Informativity differential Q[L]n[Lr] describes a new form of length contraction associated with the lower bound to measure. The loss of immeasurable space at each increment of t[f] describes
Observable Mass
The observable mass includes the mass which is visible in the present and the mass which will be visible at some point in the future. The observable mass represents all the mass that can be known in
the universe. This is as opposed to mass that exists sufficiently distant that it is beyond the horizon and as such, due to the expansion of the universe, the light from that mass will never reach
the observer.
The term quantum is intended to mean a small measure such as a few tens, hundreds or thousands of fundamental units of measure.
The term quantized is intended to mean that expressions are composed of terms that are whole-unit counts of the fundamental units and that those units are physically significant.
Visible Mass
The visible mass is that mass which is presently visible. In relation to the universe this would be the mass of those stars, dust or other forms of mass that are visible in the present as opposed to
the mass corresponding to light that will be visible in the future.
Symbol Definitions
H[U] is the expansion of the universe defined with respect to the universe (diameter). This differs slightly from stellar expansion (i.e. Hubble’s description).
l[f], m[f] and t[f] are effectively Planck’s Units for length, mass, and time, but not precisely the same.
θ[si], is 3.26239 radians or kg×m×s^−1 (momentum) or no units at all a function of the chosen frame of reference. This is a new constant to modern theory and exists in nearly every equation of the
model. It may be measured macroscopically given specific Bell states necessary for quantum entanglement of X-rays such as those carried out by Shwartz and Harris.
β is the speed parameter typically found in relativistic expressions. The parameter varies depending on the measures being compared.
A[s-ref] is the dilated age of the universe as measured from our point of view inside an expanding universe.
A[s-def] is the non-dilated age of the universe as would be measured if the universe were not expanding.
M[vis] is the mass that is presently seen from a point in space.
M[obs] is the mass that is presently or will eventually be seen from a point in space.
M[dkm] is the mass that is beyond the observable mass, mass which will never be seen from a given point in space.
M[uobs] is the mass that will eventually be seen from a point in space, but has not presently in view.
M[tot] is all the mass in the universe.
M[f] is the fundamental mass. Mass in excess of the fundamental mass exceeds the number of mass events per unit of time that can be distinguished at a point in space.
M[acr] is the rate of mass accretion with respect to the universe.
M[o-f][(R)] is the observable mass within a given radial orbit of a target galaxy
M[e-f][(R)] is the Informativity effective radial mass within of a target galaxy. The value incorporates Newton’s expression and the effects of universal expansion.
M[b-f][(R)] is the Informativity mass frequency bound radial mass which corresponds to upper mass bound of mass events that equals but does not exceed the upper mass-to-length frequency bound.
M[e] is one solar mass.
A[U] is the age of the universe.
R[U] is the radius of the universe.
D[U] is the diameter of the universe.
H[U] is the rate of universal expansion with units light-years per year.
n[Mu] is a count of m[f] equal to the total of mass/energy in the universe.
n[Tu] is a count of t[f] equal to the age of the universe.
n[Lu] is a count of l[f] equal to the diameter of the universe.
n[Lo] is a count of l[f] that is being observed.
n[Lr] is a count of l[f] from the observer to a center of gravity.
n[Ll] is a count of l[f] as measured in the local frame of reference.
n[Tl] is the count of t[f] as measured in the local frame of reference.
n[To] is the count of t[f] that is being observed.
n[Lm] is the change in position of the target as a count of l[f] as measured in the local frame of reference.
n[Lc] is the change in position of light as a count of l[f] as measured in the local frame of reference.
n[M] is a count of m[f] representing the mass corresponding to a gravitational field.
n[L] is a count of l[f] representing the length between an observer and the target.
n[T] is a count of t[f] representing the time elapsed between two events.
n[Lf] is a known count of l[f] typically used when describing distance with respect to an observer.
Q[L] is the fractional portion of a count of l[f] when engaging in a more precise calculation.
n[Lb] is a known distance, a count of the reference l[f].
v[n] is the radial velocity of a star plotted with respect to Newton’s expression for gravity.
v[o] is the observed radial velocity of a star when accounting for all well-established effects.
v[e] is the Informativity effective velocity of a star in orbit around a galactic core. The expression may resolve using Newton’s expression and the effective radial mass for a given radius.
v[b] is the Informativity mass frequency bound velocity which corresponds to upper mass bound of mass events that equals but does not exceed the upper mass-to-length frequency bound.
G is Newton’s gravitational constant.
S is the symbol assigned to the unknown constant when resolving a description of gravity. The symbol is replaced with θ[si].
c is the speed of light which may also be written as c = l[f]/t[f].
v is velocity measured between an observer and a target.
r is some unknown distance between an observer and a target.
h is Planck’s constant adjusted to reflect the quantum effects of the Informativity differential.
ħ is Planck’s reduced constant adjusted for the Informativity differential as a function of distance to target.
σ[x] is a description of the uncertainty in the position of a particle.
σ[P] is a description of the uncertainty in the momentum of a particle.
k is the Boltzmann constant.
ρ is the energy density of mass/energy accumulated at a given age of the universe.
a is the total energy radiated as described with respect to blackbody radiation (i.e. the Stefan-Boltzmann law).
T is the temperature of the Cosmic Microwave Background.
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Harvey Balls: How to Insert Filled Circles ഠ◔◑◕⬤ in Excel
Harvey Balls are circles filled to some level with color. Or, as Wikipedia says, “are round ideograms used for visual communication of qualitative information”. If you want to use them in Excel, you
have two options: Make them dynamic with conditional formatting rules or fix them by inserting them as characters. Here is how to it both ways in four different methods.
In this article, you will learn three methods of how to insert Harvey balls into Excel cells. Please note the following comments:
• Method 1 and 2 insert characters (such as normal letters) into cells. This has the advantage that you can format them as normal letters (for example size and color).
• Method 3 uses Conditional Formatting rules and adapt to numeric values in cells. You cannot easily color them, but usually they look a tiny bit better by default.
• Method 4 is based on icons that can be arranged per drag-and-drop on top of your cells.
That being said, let’s get started!
Method 1: The fastest method with Professor Excel Tools
This is the fastest method: Go to the Professor Excel ribbon, click on the drop-down arrow of “Insert Symbol” and then on “More Symbols”. Alternatively, just click on the Insert Symbol button.
In the drop-down, select Harvey Balls. Then, click on the Harvey Ball you want to add to a cell and then on “Insert”.
Just download and install the Excel-add-in Professor Excel Tools and see if it works for you.
This function is included in our Excel Add-In ‘Professor Excel Tools’
(No sign-up, download starts directly)
More than 35,000 users can’t be wrong.
Insert Harvey Balls as Symbols
Method 2a: Use the =UNICHAR() function
Windows (and also macOS) have built-in special characters. Among them are Harvey balls. To access them, you can use the UNICHAR functions. Just type the following function into an Excel cell:
Harvey balls with the UNICHAR function.
Replace the number with one of the following value for the respective ball.
Harvey Ball Unicode (hex) number
ഠ 3360
◔ 9684
◑ 9681
◕ 9685
⬤ 11044
Unicode (hex) numbers for the different types of Harvey balls.
Method 2b: Just copy and paste
Probably not as beautiful, but it usually works: Alternatively, just copy and paste it from here.
ഠ ◔ ◑ ◕ ⬤
Method 3: Insert Harvey balls with Conditional Formatting rules
The two methods above insert “fixed” Harvey balls: The won’t change no matter what you type into an Excel cell. Do you want automatically adapt them based on a cell value? In such case, please use a
conditional formatting rule.
Step 1: Insert Harvey balls with Conditional Formatting
Insert Harvey balls with Conditional Formatting rules.
1. Select the cells you want to insert the icons to. These should be cells with numerical values.
2. Click on Conditional Formatting in the middle of the Home ribbon.
3. Go with the mouse to “Icon Sets”.
4. Click on Harvey Balls.
Step 2: Don’t show the numeric values
Now, you can already see the “first draft” of the circles. Next, we will finetune them.
Hide the numbers next to the icons..
5. Go to the Conditional Formatting button on the Home ribbon again.
6. Click on “Manage Rules”.
7. Double click on the Harvey ball rule in order to open and edit it.
8. Usually, you don’t want to show the numbers next to the icons. In order to achieve that set the check mark at “Show Icon Only”.
9. Click on OK.
Step 3: Fixate the scale so that it does not change with the minimum and maximum value
Result: Harvey balls with conditional formatting rules.
That’s it, it look quite good already (see the screenshot on the right-hand side).
Unfortunately, they have one disadvantage: If you remove one of the “extreme” values (empty or completely filled balls), all other change. The now highest value will then be the filled Harvey ball
and the now lowest value the empty one.
Let’s try to fix this. The goal: When you type 1 into a cell, the empty Harvey ball should be shown. 2 should be filled to one quarter, 3 half and number 5 the completely filled Harvey ball.
Open the Edit Formatting Rule window again (see steps 5 to 8 above). Now, set the Values and Types as shown below.
Fix the scale: Otherwise the Harvey balls change their filling based on the highest and lowest values.
Method 4: Add Harvey balls as icons (similar to shapes or images)
The forth option is using the icon function in Excel. Go to the Insert ribbon and click on Icons. Then, search for “harvey”. Microsoft offers 2 different layouts of them. Besides the standard balls
(such as empty, 1/4 filled, half filled, 3/4 filled and full), you can select from some more steps in-between.
Select the balls you need and click on “Insert”. Now, you can arrange them per drag-and-drop. You can furthermore change the color.
Image by Couleur from Pixabay
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is testing
A carpenter is making doors that are 2058 millimeters tall. If the doors are too long they must be trimmed, and if they are too short they cannot be used. A sample of 751 doors is made, and it is
found that they have a mean of 2047 millimeters with a standard deviation of 32. Is there evidence at the 0.1 level that the doors are too short and unusable?
State the null and alternative hypotheses for the above scenario.
The mayor of a town believes that over 49% of the residents favor annexation of an adjoining bridge. Is there sufficient evidence at the 0.01 level to support the mayor's claim? After information is
gathered from 320 voters and a hypothesis test is completed, the mayor decides to reject the null hypothesis at the 0.01
What is the conclusion regarding the mayor's claim?
The director of research and development is testing a new drug. She wants to know if there is evidence at the 0.02 level that the drug stays in the system for more than 333 minutes. After performing
a hypothesis test, she decides to reject the null hypothesis.
What is the conclusion?
A toy manufacturer wants to know how many new toys children buy each year. A sample of 686 children was taken to study their purchasing habits. Construct the 85% confidence interval for the mean
number of toys purchased each year if the sample mean was found to be 6.8. Assume that the population standard deviation is 2.1. Round your answers to one decimal place.
Submit this project to the proctor at the time you take Test 3.
When the problem involves hypothesis testing, use the following structure for written reports.
Hypothesis testing steps
• Step 1: State the hypotheses.
• Step 2: Summarize the data for your readers.
• Step 3: Give the value of the test statistic and the p-value.
• Step 4: Use the p-value to draw a conclusion. State the conclusion in statistical
terms: Reject Ho in favor of Ha, or retain Ho (fail to reject Ho).
• Step 5: State the conclusion in layman terms and in context of the application. Use the
p-value to state the strength of the evidence.
When a significance level is not given, then use the following guidelines and language associated
with p-value. Note that the lower the p-values, the stronger the evidence against Ho and in
favor of Ha. We go from insufficient evidence, to some evidence, to fairly strong evidence, to
strong evidence, to very strong evidence.
• p-value > .10
retain Ho – there is insufficient evidence to reject Ho in favor of Ha
• .05 < p-value ≤ .10
gray area -- decision to reject Ho or retain Ho is up to the investigators – there is some
evidence against Ho and in support of Ha
• .01 < p-value ≤ .05
reject Ho in favor of Ha – there is fairly strong evidence against Ho and in favor of Ha
• .001 < p-value ≤ .01
reject Ho in favor of Ha – there is strong evidence against Ho and in favor of Ha
• p-value ≤ .001
reject Ho in favor of Ha – there is very strong evidence against Ho and in favor of Ha
Use your TI-83/TI-84 calculator for all of these problems. You will not need any tables.
Use the Sample Test 3 Questions—Answer Key (posted in Canvas) as an example of what my
expectations are.
1. A sociologist suspects that, for married couples with young children, the husbands watch more TV
than the wives. Twenty married couples are randomly selected and their weekly viewing times, in
hours, are recorded in the table below. Assume the population of differences between husband’s
and wife’s TV time is mound-shaped and symmetrical.
a) Do the sample results provide sufficient evidence to support the sociologist’s claim? Perform a
hypothesis test to find out.
b) If there is sufficient evidence to support the sociologist’s claim, estimate how much more TV the
husbands watch, on average, with a 95% confidence interval. Interpret.
2. The data below show the sugar content (as a percentage of weight) of several national brands of
children’s and adults’ cereals. Assume the distributions of sugar content in both children’s cereals
and adults’ cereals are mound-shaped and symmetrical.
a) Does the sample data provide sufficient evidence to conclude that the sugar content in
children’s cereals is higher than that in adults’ cereals, on average? Perform a hypothesis test to
find out.
b) If you conclude that children’s cereals have more sugar than adults’ cereals, estimate how much
more with a 95% confidence interval for the difference in mean sugar content. Interpret.
Children’s cereals: 40.3, 55, 45.7, 43.3, 50.3, 45.9, 53.5, 43, 44.2, 44, 47.4, 44, 33.6, 55.1, 48.8,
50.4, 37.8, 60.3, 46.6
Adults’ cereals: 20, 30.2, 2.2, 7.5, 4.4, 22.2, 16.6, 14.5, 21.4, 3.3, 6.6, 7.8, 10.6, 16.2, 14.5, 4.1,
15.8, 4.1, 2.4, 3.5, 8.5, 10, 1, 4.4, 1.3, 8.1, 4.7, 18.4
3. A randomly selected sample of entering college freshmen has participated in a special program to
enhance their academic abilities, and their GPAs at the end of one year have been recorded. A
group of 20 students from the same class who did not participate in the program has been selected
as a control group, and they have been matched with the experimental group by gender, age, highschool class rank, ACT scores, and declared major. The results (GPAs) are presented below. Assume
the population of differences between the project student GPA and the control group student GPA
is mound-shaped and symmetrical.
a) Can the program claim that it was successful? Carry out a hypothesis test to find out.
b) If you conclude that the program was successful, make a judgment regarding the size of the
effect of program participation on student GPAs by constructing a 95% confidence interval.
Interpret your confidence interval.
4. Michelle Sayther is a fashion design artist who designs the display windows in front of a large
clothing store in New York City. Electronic counters at the entrances total the number of people
entering the store each business day. Before Michelle was hired by the store, the mean number of
people entering the store each day was 3218. Management would like to investigate whether this
number has changed since Michelle has started working. A random sample of 42 business days after
Michelle began work gave an average of 𝑋𝑋� = 3392 people entering the store each day. The sample
standard deviation was s = 287 people. Assume the population of daily number of people entering
the store is mound-shaped and symmetrical.
a) Perform a hypothesis test to decide if the average number of people entering the store each day
since Michelle was hired is different from what it was before Michelle was hired.
b) If you find that the average number of people entering the store each day since Michelle was
hired is different from what it was before Michelle was hired, estimate the average number of
people entering the store each day since Michelle was hired with a 95% confidence interval and
interpret. (Has the number of people entering the store each day increased or decreased since
Michelle was hired, and by how much has it increased or decreased?)
5. An experiment was conducted to evaluate the effectiveness of a treatment for tapeworm in the
stomachs of sheep. A random sample of 24 worm-infected lambs of approximately the same age
and health was randomly divided into two groups. Twelve of the lambs were injected with the drug
and the remaining twelve were left untreated. After a 6-month period, the lambs were slaughtered
and the following worm counts were recorded. Assume the distribution of worm counts of drugtreated sheep is mound-shaped and symmetrical. Assume the distribution of worm counts of
untreated sheep is also mound-shaped and symmetrical.
c) Does the sample data provide sufficient evidence to conclude that the treatment is effective in
reducing the occurrence of tapeworm in sheep? Perform a test of significance to find out.
d) If you conclude that the treatment is effective, estimate the average reduction in tapeworm
count with a 95% confidence interval. Interpret.
6. In each of the problems above, #1- #5, an assumption of normality is made about the distribution of
the population(s) from which the sample data is obtained. For each of #1 - #5, provide the page
number in the e-book where the assumption is described by the author. You will be citing page
numbers from Sections 9.2, 10.1, and 10.2.
7. A study of the health behavior of school-aged children asked a sample of 15-year-olds in several
different countries if they had been drunk at least twice. The results are shown in the table, by
gender. (Health and Health Behavior Among Young People. Copenhagen. World Health
Organization, 2000)
a) Perform a hypothesis test to determine if there is a gender effect. That is, is there a difference
in the average percent of 15-year-old males who have been drunk at least twice and the average
percent of 15-year-old females who have been drunk at least twice? Assume the distributions
for both males and females are mound-shaped and symmetrical.
b) If there is sufficient evidence that there is a difference between average percent of 15-year-old
males who have been drunk at least twice and the average percent of 15-year-old females who
have been drunk at least twice, estimate the difference with a 95% confidence interval and
interpret your interval.
Quantitative project reasoning solutions. The following solutions have been provided to you by MyMathLab statistics experts The solutions were provided under the MyMathLab answers statistics help
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